| NId _ -> "nid"
| NIntro (_,n) -> "#" ^ n
| NRewrite (_,dir,n,where) -> "nrewrite" ^ assert false
- | NReduce _ | NGeneralize _ | NLetIn _ | NAssert _ | NAuto _ -> assert false
+ | NAuto (_,(l,flgs)) ->
+ "nauto" ^
+ (if l <> [] then (" by " ^
+ (String.concat "," (List.map CicNotationPp.pp_term l))) else "") ^
+ String.concat " " (List.map (fun a,b -> a ^ "=" ^ b) flgs)
+ | NReduce _ | NGeneralize _ | NLetIn _ | NAssert _ -> assert false
;;
let rec pp_tactic ~map_unicode_to_tex ~term_pp ~lazy_term_pp =
| None -> "")
| Print (_,s) -> "print " ^ s
| Set (_, name, value) -> Printf.sprintf "set \"%s\" \"%s\"" name value
- | NObj (_,o) -> "not supported"
+ | NObj (_,o) ->
+
+
| NUnivConstraint (_) -> "not supported"
- | NQed (_) -> "not supported"
+ | NQed (_) -> "nqed"
| Pump (_) -> "not supported"
let pp_punctuation_tactical =
parser.cmi: ast.cmx
+tptp2grafite.cmi:
+ast.cmo:
+ast.cmx:
lexer.cmo: parser.cmi
lexer.cmx: parser.cmx
parser.cmo: ast.cmx parser.cmi
EXTRA_OBJECTS_TO_INSTALL =
EXTRA_OBJECTS_TO_CLEAN =
-TPTPDIR=/home/$(USER)/TPTP-v3.1.1/
+TPTPDIR= /home/$(USER)/work-area/TPTP-v3.2.0/
all: tptp2grafite
clean: clean_tests
> ../../matita/tests/TPTP/$$X.ma || echo Failed: $$X; \
done
+ngenerate-%:
+ for X in `cat $*`; do\
+ ./tptp2grafite -ng -tptppath $(TPTPDIR) $$X.p \
+ > ../../matita/contribs/ng_TPTP/$$X.ma || echo Failed: $$X; \
+ done
+
parse-%:
for X in `cat $*`; do\
echo "Parsing $$X"; \
(* OPTIONS *)
let tptppath = ref "./";;
+let ng = ref false;;
let spec = [
+ ("-ng",Arg.Set ng,"Matita ng syntax");
("-tptppath",
Arg.String (fun x -> tptppath := x),
"Where to find the Axioms/ and Problems/ directory")
exit 1
end;
print_endline
- (Tptp2grafite.tptp2grafite ~filename:!inputfile ~tptppath:!tptppath ());
+ (Tptp2grafite.tptp2grafite ~filename:!inputfile ~tptppath:!tptppath ~ng:!ng ());
exit 0
| A.Atom a -> check_if_atom_is_negative a
;;
-let convert_ast statements context = function
+let ng_generate_tactics fv ueq_case context arities =
+ [ GA.Executable(floc,GA.NTactic(floc,
+ (GA.NIntro (floc,"Univ")),GA.Dot(floc))) ]
+ @
+ (HExtlib.list_mapi
+ (fun (name,_) _->
+ GA.Executable(floc,GA.NTactic(floc,
+ (GA.NIntro (floc,name)),GA.Dot(floc))))
+ arities)
+ @
+ (HExtlib.list_mapi
+ (fun _ i->
+ GA.Executable(floc,GA.NTactic(floc,
+ (GA.NIntro (floc,"H"^string_of_int i)),GA.Dot(floc))))
+ context)
+ @
+(if fv <> [] then
+ (List.flatten
+ (List.map
+ (fun _ ->
+ [GA.Executable(floc,GA.NTactic(floc,
+ (GA.NApply (floc,mk_ident "ex_intro")),GA.Branch floc));
+ GA.Executable(floc,GA.NTactic(floc, GA.NId floc ,
+ (GA.Pos (floc,[2]))))])
+ fv))
+ else [])@
+ [GA.Executable(floc,GA.NTactic(floc, (
+ if (*ueq_case*) true then
+ GA.NAuto (floc,(
+ HExtlib.list_mapi
+ (fun _ i ->
+ mk_ident ("H" ^ string_of_int i))
+ context
+ ,[]))
+ else
+ GA.NAuto (floc,([],[
+ "depth",string_of_int 5;
+ "width",string_of_int 5;
+ "size",string_of_int 20;
+ "timeout",string_of_int 10;
+ ]))
+ ),
+ GA.Semicolon(floc)));
+(*
+ GA.Executable(floc,GA.NTactic(floc, Some (GA.Try(floc,
+ GA.Assumption floc)), GA.Dot(floc)))
+*)
+ ]@
+(if fv <> [] then
+ (List.flatten
+ (List.map
+ (fun _ ->
+ [GA.Executable(floc,GA.NTactic(floc, GA.NId floc, GA.Shift floc));
+ GA.Executable(floc,GA.NNonPunctuationTactical(floc, GA.Skip floc,
+ (GA.Merge floc)))])
+ fv))
+ else [])@
+ [GA.Executable(floc,GA.Command(floc, GA.NQed(floc)))]
+;;
+
+let generate_tactics fv ueq_case =
+ [GA.Executable(floc,GA.Tactic(floc, Some
+ (GA.Intros (floc,(None,[]))),GA.Dot(floc)))] @
+(if fv <> [] then
+ (List.flatten
+ (List.map
+ (fun _ ->
+ [GA.Executable(floc,GA.Tactic(floc, Some
+ (GA.Exists floc),GA.Branch floc));
+ GA.Executable(floc,GA.Tactic(floc, None,
+ (GA.Pos (floc,[2]))))])
+ fv))
+ else [])@
+ [GA.Executable(floc,GA.Tactic(floc, Some (
+ if true (*ueq_case*) then
+ GA.AutoBatch (floc,([],["paramodulation","";
+ "timeout",string_of_int !paramod_timeout]))
+ else
+ GA.AutoBatch (floc,([],[
+ "depth",string_of_int 5;
+ "width",string_of_int 5;
+ "size",string_of_int 20;
+ "timeout",string_of_int 10;
+ ]))
+ ),
+ GA.Semicolon(floc)));
+ GA.Executable(floc,GA.Tactic(floc, Some (GA.Try(floc,
+ GA.Assumption floc)), GA.Dot(floc)))
+ ]@
+(if fv <> [] then
+ (List.flatten
+ (List.map
+ (fun _ ->
+ [GA.Executable(floc,GA.Tactic(floc, None, GA.Shift floc));
+ GA.Executable(floc,GA.NonPunctuationTactical(floc, GA.Skip floc,
+ (GA.Merge floc)))])
+ fv))
+ else [])@
+ [GA.Executable(floc,GA.Command(floc, GA.Print(floc,"proofterm")));
+ GA.Executable(floc,GA.Command(floc, GA.Qed(floc)))]
+;;
+
+let convert_ast ng statements context = function
| A.Comment s ->
let s = String.sub s 1 (String.length s - 1) in
let s =
| A.Negated_conjecture ->
let ueq_case = is_formulae_1eq_negated f in
let fv = collect_fv_1stord_from_formulae f in
-(*
- if fv <> [] then
- prerr_endline ("FREE VARIABLES: " ^ String.concat "," fv);
-*)
+ let old_f = f in
let f =
PT.Binder
(`Forall,
- (mk_ident universe,Some (PT.Sort `Set)),
+ (mk_ident universe,Some (PT.Sort (`Type (CicUniv.fresh ())))),
convert_formula fv false context f)
in
let o = PT.Theorem (`Theorem,name,f,None) in
- statements @ [
- GA.Executable(floc,GA.Command(floc,GA.Obj(floc,o)));
- GA.Executable(floc,GA.Tactic(floc, Some
- (GA.Intros (floc,(None,[]))),GA.Dot(floc)))] @
- (if fv <> [] then
- (List.flatten
- (List.map
- (fun _ ->
- [GA.Executable(floc,GA.Tactic(floc, Some
- (GA.Exists floc),GA.Branch floc));
- GA.Executable(floc,GA.Tactic(floc, None,
- (GA.Pos (floc,[2]))))])
- fv))
- else [])@
- [GA.Executable(floc,GA.Tactic(floc, Some (
- if ueq_case then
- GA.AutoBatch (floc,([],["paramodulation","";
- "timeout",string_of_int !paramod_timeout]))
- else
- GA.AutoBatch (floc,([],[
- "depth",string_of_int 5;
- "width",string_of_int 5;
- "size",string_of_int 20;
- "timeout",string_of_int 10;
- ]))
- ),
- GA.Semicolon(floc)));
- GA.Executable(floc,GA.Tactic(floc, Some (GA.Try(floc,
- GA.Assumption floc)), GA.Dot(floc)))
- ]@
- (if fv <> [] then
- (List.flatten
- (List.map
- (fun _ ->
- [GA.Executable(floc,GA.Tactic(floc, None, GA.Shift floc));
- GA.Executable(floc,GA.NonPunctuationTactical(floc, GA.Skip floc,
- (GA.Merge floc)))])
- fv))
- else [])@
- [GA.Executable(floc,GA.Command(floc, GA.Print(floc,"proofterm")));
- GA.Executable(floc,GA.Command(floc, GA.Qed(floc)))],
+ (statements @
+ [ GA.Executable(floc,GA.Command(floc,
+ (*if ng then GA.NObj (floc,o) else*) GA.Obj(floc,o))); ] @
+ if ng then
+ ng_generate_tactics fv ueq_case context
+ (let atom = atom_of_formula old_f in collect_arities atom context)
+ else generate_tactics fv ueq_case),
context
| A.Definition
| A.Lemma
;;
(* MAIN *)
-let tptp2grafite ?(timeout=600) ?(def_depth=10) ?raw_preamble ~tptppath ~filename () =
+let tptp2grafite ?(timeout=600) ?(def_depth=10) ?raw_preamble ~tptppath ~filename ~ng () =
paramod_timeout := timeout;
depth := def_depth;
let rec aux = function
let grafite_ast_statements,_ =
List.fold_left
(fun (st, ctx) f ->
- let newst, ctx = convert_ast st ctx f in
+ let newst, ctx = convert_ast ng st ctx f in
newst, ctx)
([],[]) statements
in
let markup = CicNotationPres.render ~lookup_uri pres_term in
let s = BoxPp.render_to_string List.hd width markup ~map_unicode_to_tex:false in
Pcre.substitute
- ~pat:"\\\\forall [Ha-z][a-z0-9_]*" ~subst:(fun x -> "\n" ^ x) s
+ ~rex:(Pcre.regexp ~flags:[`UTF8] "∀[Ha-z][a-z0-9_]*") ~subst:(fun x -> "\n" ^ x)
+ s
in
CicNotationPp.set_pp_term term_pp;
let lazy_term_pp = fun x -> assert false in
let obj_pp = CicNotationPp.pp_obj CicNotationPp.pp_term in
- GrafiteAstPp.pp_statement
- ~map_unicode_to_tex:false ~term_pp ~lazy_term_pp ~obj_pp t
+ Pcre.replace ~pat:"theorem" ~templ:"ntheorem"
+ (GrafiteAstPp.pp_statement
+ ~map_unicode_to_tex:false ~term_pp ~lazy_term_pp ~obj_pp t)
in
let buri = Pcre.replace ~pat:"\\.p$" ("cic:/matita/TPTP/" ^ filename) in
let extra_statements_start = [
- GA.Executable(floc,GA.Command(floc,
- GA.Set(floc,"baseuri",buri)))]
+ (*GA.Executable(floc,GA.Command(floc,
+ GA.Set(floc,"baseuri",buri)))*)]
in
let preamble =
match raw_preamble with
?timeout:int ->
?def_depth:int ->
?raw_preamble:(string -> string) ->
- tptppath:string -> filename:string -> unit ->
+ tptppath:string -> filename:string -> ng:bool -> unit ->
string
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: ALG005-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : ALG005-1 : TPTP v3.2.0. Released v2.2.0. *)
+
+(* Domain : General Algebra *)
+
+(* Problem : Associativity of intersection in terms of set difference. *)
+
+(* Version : [MP96] (equality) axioms : Especial. *)
+
+(* English : Starting with Kalman's basis for families of sets closed under *)
+
+(* set difference, we define intersection and show it to be *)
+
+(* associative. *)
+
+(* Refs : [McC98] McCune (1998), Email to G. Sutcliffe *)
+
+(* : [MP96] McCune & Padmanabhan (1996), Automated Deduction in Eq *)
+
+(* Source : [McC98] *)
+
+(* Names : SD-2-a [MP96] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.07 v3.1.0, 0.22 v2.7.0, 0.00 v2.2.1 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 1 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 9 ( 1 singleton) *)
+
+(* Maximal term depth : 3 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Kalman's axioms for set difference: *)
+
+(* ----Definition of intersection: *)
+
+(* ----Denial of associativity: *)
+ntheorem prove_associativity_of_multiply:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀difference:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (multiply X Y) (difference X (difference X Y)).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (difference (difference X Y) Z) (difference (difference X Z) (difference Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.eq Univ (difference X (difference X Y)) (difference Y (difference Y X)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (difference X (difference Y X)) X.eq Univ (multiply (multiply a b) c) (multiply a (multiply b c))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#difference.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: ALG006-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : ALG006-1 : TPTP v3.2.0. Released v2.2.0. *)
+
+(* Domain : General Algebra *)
+
+(* Problem : Simplification of Kalman's set difference basis (part 1) *)
+
+(* Version : [MP96] (equality) axioms : Especial. *)
+
+(* English : This is part 1 of a proof that one of the axioms in Kalman's *)
+
+(* basis for set difference can be simplified. *)
+
+(* Refs : [McC98] McCune (1998), Email to G. Sutcliffe *)
+
+(* : [MP96] McCune & Padmanabhan (1996), Automated Deduction in Eq *)
+
+(* Source : [McC98] *)
+
+(* Names : SD-3-a [MP96] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.2.1 *)
+
+(* Syntax : Number of clauses : 4 ( 0 non-Horn; 4 unit; 1 RR) *)
+
+(* Number of atoms : 4 ( 4 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 7 ( 1 singleton) *)
+
+(* Maximal term depth : 3 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Kalman's axioms for set difference: *)
+
+(* ----Denial of simplified third axiom: *)
+ntheorem prove_set_difference_3_simplified:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀difference:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (difference (difference X Y) Z) (difference (difference X Z) (difference Y Z)).
+∀H1:∀X:Univ.∀Y:Univ.eq Univ (difference X (difference X Y)) (difference Y (difference Y X)).
+∀H2:∀X:Univ.∀Y:Univ.eq Univ (difference X (difference Y X)) X.eq Univ (difference (difference a c) b) (difference (difference a b) c)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#difference.
+#H0.
+#H1.
+#H2.
+nauto by H0,H1,H2;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: ALG007-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : ALG007-1 : TPTP v3.2.0. Released v2.2.0. *)
+
+(* Domain : General Algebra *)
+
+(* Problem : Simplification of Kalman's set difference basis (part 2) *)
+
+(* Version : [MP96] (equality) axioms : Especial. *)
+
+(* English : This is part 2 of a proof that one of the axioms in Kalman's *)
+
+(* basis for set difference can be simplified. *)
+
+(* Refs : [McC98] McCune (1998), Email to G. Sutcliffe *)
+
+(* : [MP96] McCune & Padmanabhan (1996), Automated Deduction in Eq *)
+
+(* Source : [McC98] *)
+
+(* Names : SD-3-b [MP96] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v3.1.0, 0.11 v2.7.0, 0.00 v2.2.1 *)
+
+(* Syntax : Number of clauses : 4 ( 0 non-Horn; 4 unit; 1 RR) *)
+
+(* Number of atoms : 4 ( 4 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 7 ( 1 singleton) *)
+
+(* Maximal term depth : 3 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Kalman's axioms for set difference: *)
+
+(* ----Simplified third axiom: *)
+
+(* ----Denial of original third axiom: *)
+ntheorem prove_set_difference_3:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀difference:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (difference (difference X Y) Z) (difference (difference X Z) Y).
+∀H1:∀X:Univ.∀Y:Univ.eq Univ (difference X (difference X Y)) (difference Y (difference Y X)).
+∀H2:∀X:Univ.∀Y:Univ.eq Univ (difference X (difference Y X)) X.eq Univ (difference (difference a b) c) (difference (difference a c) (difference b c))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#difference.
+#H0.
+#H1.
+#H2.
+nauto by H0,H1,H2;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: BOO001-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO001-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Boolean Algebra (Ternary) *)
+
+(* Problem : In B3 algebra, inverse is an involution *)
+
+(* Version : [OTTER] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : *)
+
+(* Source : [OTTER] *)
+
+(* Names : tba_gg.in [OTTER] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.2.1, 0.11 v2.2.0, 0.14 v2.1.0, 0.25 v2.0.0 *)
+
+(* Syntax : Number of clauses : 6 ( 0 non-Horn; 6 unit; 1 RR) *)
+
+(* Number of atoms : 6 ( 6 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-3 arity) *)
+
+(* Number of variables : 13 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include ternary Boolean algebra axioms *)
+
+(* Inclusion of: Axioms/BOO001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Algebra (Ternary Boolean) *)
+
+(* Axioms : Ternary Boolean algebra (equality) axioms *)
+
+(* Version : [OTTER] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* : [Win82] Winker (1982), Generation and Verification of Finite M *)
+
+(* Source : [OTTER] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 0 RR) *)
+
+(* Number of literals : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 1-3 arity) *)
+
+(* Number of variables : 13 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : These axioms appear in [Win82], in which ternary_multiply_1 is *)
+
+(* shown to be independant. *)
+
+(* : These axioms are also used in [Wos88], p.222. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_inverse_is_self_cancelling:
+ ∀Univ:Type.∀V:Univ.∀W:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (multiply X Y (inverse Y)) X.
+∀H1:∀X:Univ.∀Y:Univ.eq Univ (multiply (inverse Y) Y X) X.
+∀H2:∀X:Univ.∀Y:Univ.eq Univ (multiply X X Y) X.
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (multiply Y X X) X.
+∀H4:∀V:Univ.∀W:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply V W X) Y (multiply V W Z)) (multiply V W (multiply X Y Z)).eq Univ (inverse (inverse a)) a
+.
+#Univ.
+#V.
+#W.
+#X.
+#Y.
+#Z.
+#a.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+nauto by H0,H1,H2,H3,H4;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: BOO002-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO002-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Boolean Algebra (Ternary) *)
+
+(* Problem : In B3 algebra, X * X^-1 * Y = Y *)
+
+(* Version : [OTTER] (equality) axioms : Reduced > Incomplete. *)
+
+(* English : *)
+
+(* Refs : [LO85] Lusk & Overbeek (1985), Reasoning about Equality *)
+
+(* : [Ove90] Overbeek (1990), ATP competition announced at CADE-10 *)
+
+(* : [Ove93] Overbeek (1993), The CADE-11 Competitions: A Personal *)
+
+(* : [LM93] Lusk & McCune (1993), Uniform Strategies: The CADE-11 *)
+
+(* : [Zha93] Zhang (1993), Automated Proofs of Equality Problems in *)
+
+(* Source : [Ove90] *)
+
+(* Names : Problem 5 [LO85] *)
+
+(* : CADE-11 Competition Eq-3 [Ove90] *)
+
+(* : THEOREM EQ-3 [LM93] *)
+
+(* : PROBLEM 3 [Zha93] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.07 v3.1.0, 0.00 v2.7.0, 0.09 v2.6.0, 0.00 v2.2.1, 0.33 v2.2.0, 0.43 v2.1.0, 0.38 v2.0.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 1 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 2 constant; 0-3 arity) *)
+
+(* Number of variables : 11 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Don't include ternary Boolean algebra axioms, as one is omitted *)
+
+(* include('axioms/BOO001-0.ax'). *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----This axiom is omitted *)
+
+(* input_clause(right_inverse,axiom, *)
+
+(* [++equal(multiply(X,Y,inverse(Y)),X)]). *)
+ntheorem prove_equation:
+ ∀Univ:Type.∀V:Univ.∀W:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (multiply (inverse Y) Y X) X.
+∀H1:∀X:Univ.∀Y:Univ.eq Univ (multiply X X Y) X.
+∀H2:∀X:Univ.∀Y:Univ.eq Univ (multiply Y X X) X.
+∀H3:∀V:Univ.∀W:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply V W X) Y (multiply V W Z)) (multiply V W (multiply X Y Z)).eq Univ (multiply a (inverse a) b) b
+.
+#Univ.
+#V.
+#W.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: BOO002-2.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO002-2 : TPTP v3.2.0. Released v1.1.0. *)
+
+(* Domain : Boolean Algebra (Ternary) *)
+
+(* Problem : In B3 algebra, X * X^-1 * Y = Y *)
+
+(* Version : [OTTER] (equality) axioms : Reduced & Augmented > Incomplete. *)
+
+(* English : *)
+
+(* Refs : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [Wos88] *)
+
+(* Names : Test Problem 13 [Wos88] *)
+
+(* : Lemma for Axiom Independence [Wos88] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.7.0, 0.09 v2.6.0, 0.00 v2.2.1, 0.33 v2.2.0, 0.43 v2.1.0, 0.38 v2.0.0 *)
+
+(* Syntax : Number of clauses : 6 ( 0 non-Horn; 6 unit; 1 RR) *)
+
+(* Number of atoms : 6 ( 6 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 2 constant; 0-3 arity) *)
+
+(* Number of variables : 13 ( 3 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : This version contains an extra lemma *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Don't include ternary Boolean algebra axioms, as one is omitted *)
+
+(* include('axioms/BOO001-0.ax'). *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----This axiom is omitted *)
+
+(* input_clause(right_inverse,axiom, *)
+
+(* [++equal(multiply(X,Y,inverse(Y)),X)]). *)
+ntheorem prove_equation:
+ ∀Univ:Type.∀V:Univ.∀W:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (multiply X Y X) X.
+∀H1:∀X:Univ.∀Y:Univ.eq Univ (multiply (inverse Y) Y X) X.
+∀H2:∀X:Univ.∀Y:Univ.eq Univ (multiply X X Y) X.
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (multiply Y X X) X.
+∀H4:∀V:Univ.∀W:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply V W X) Y (multiply V W Z)) (multiply V W (multiply X Y Z)).eq Univ (multiply a (inverse a) b) b
+.
+#Univ.
+#V.
+#W.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+nauto by H0,H1,H2,H3,H4;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: BOO003-2.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO003-2 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Problem : Multiplication is idempotent (X * X = X) *)
+
+(* Version : [ANL] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : *)
+
+(* Source : [ANL] *)
+
+(* Names : prob2_part1.ver2.in [ANL] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.1.0, 0.38 v2.0.0 *)
+
+(* Syntax : Number of clauses : 15 ( 0 non-Horn; 15 unit; 1 RR) *)
+
+(* Number of atoms : 15 ( 15 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 24 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include boolean algebra axioms for equality formulation *)
+
+(* Inclusion of: Axioms/BOO003-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO003-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Axioms : Boolean algebra (equality) axioms *)
+
+(* Version : [ANL] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 14 ( 0 non-Horn; 14 unit; 0 RR) *)
+
+(* Number of literals : 14 ( 14 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 24 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_a_times_a_is_a:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀additive_identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiplicative_identity:Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.eq Univ (add additive_identity X) X.
+∀H1:∀X:Univ.eq Univ (add X additive_identity) X.
+∀H2:∀X:Univ.eq Univ (multiply multiplicative_identity X) X.
+∀H3:∀X:Univ.eq Univ (multiply X multiplicative_identity) X.
+∀H4:∀X:Univ.eq Univ (multiply (inverse X) X) additive_identity.
+∀H5:∀X:Univ.eq Univ (multiply X (inverse X)) additive_identity.
+∀H6:∀X:Univ.eq Univ (add (inverse X) X) multiplicative_identity.
+∀H7:∀X:Univ.eq Univ (add X (inverse X)) multiplicative_identity.
+∀H8:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y Z)) (add (multiply X Y) (multiply X Z)).
+∀H9:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X Y) Z) (add (multiply X Z) (multiply Y Z)).
+∀H10:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add X (multiply Y Z)) (multiply (add X Y) (add X Z)).
+∀H11:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add (multiply X Y) Z) (multiply (add X Z) (add Y Z)).
+∀H12:∀X:Univ.∀Y:Univ.eq Univ (multiply X Y) (multiply Y X).
+∀H13:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).eq Univ (multiply a a) a
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#add.
+#additive_identity.
+#inverse.
+#multiplicative_identity.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: BOO003-4.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO003-4 : TPTP v3.2.0. Released v1.1.0. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Problem : Multiplication is idempotent (X * X = X) *)
+
+(* Version : [Ver94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Ver94] Veroff (1994), Problem Set *)
+
+(* Source : [Ver94] *)
+
+(* Names : TA [Ver94] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.1.0, 0.13 v2.0.0 *)
+
+(* Syntax : Number of clauses : 9 ( 0 non-Horn; 9 unit; 1 RR) *)
+
+(* Number of atoms : 9 ( 9 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 14 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include boolean algebra axioms for equality formulation *)
+
+(* Inclusion of: Axioms/BOO004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Axioms : Boolean algebra (equality) axioms *)
+
+(* Version : [Ver94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Ver94] Veroff (1994), Problem Set *)
+
+(* Source : [Ver94] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 14 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_a_times_a_is_a:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀additive_identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiplicative_identity:Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.eq Univ (multiply X (inverse X)) additive_identity.
+∀H1:∀X:Univ.eq Univ (add X (inverse X)) multiplicative_identity.
+∀H2:∀X:Univ.eq Univ (multiply X multiplicative_identity) X.
+∀H3:∀X:Univ.eq Univ (add X additive_identity) X.
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y Z)) (add (multiply X Y) (multiply X Z)).
+∀H5:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add X (multiply Y Z)) (multiply (add X Y) (add X Z)).
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (multiply X Y) (multiply Y X).
+∀H7:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).eq Univ (multiply a a) a
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#add.
+#additive_identity.
+#inverse.
+#multiplicative_identity.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: BOO004-2.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO004-2 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Problem : Addition is idempotent (X + X = X) *)
+
+(* Version : [ANL] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : *)
+
+(* Source : [ANL] *)
+
+(* Names : prob2_part2.ver2.in [ANL] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.1.0, 0.13 v2.0.0 *)
+
+(* Syntax : Number of clauses : 15 ( 0 non-Horn; 15 unit; 1 RR) *)
+
+(* Number of atoms : 15 ( 15 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 24 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include boolean algebra axioms for equality formulation *)
+
+(* Inclusion of: Axioms/BOO003-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO003-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Axioms : Boolean algebra (equality) axioms *)
+
+(* Version : [ANL] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 14 ( 0 non-Horn; 14 unit; 0 RR) *)
+
+(* Number of literals : 14 ( 14 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 24 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_a_plus_a_is_a:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀additive_identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiplicative_identity:Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.eq Univ (add additive_identity X) X.
+∀H1:∀X:Univ.eq Univ (add X additive_identity) X.
+∀H2:∀X:Univ.eq Univ (multiply multiplicative_identity X) X.
+∀H3:∀X:Univ.eq Univ (multiply X multiplicative_identity) X.
+∀H4:∀X:Univ.eq Univ (multiply (inverse X) X) additive_identity.
+∀H5:∀X:Univ.eq Univ (multiply X (inverse X)) additive_identity.
+∀H6:∀X:Univ.eq Univ (add (inverse X) X) multiplicative_identity.
+∀H7:∀X:Univ.eq Univ (add X (inverse X)) multiplicative_identity.
+∀H8:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y Z)) (add (multiply X Y) (multiply X Z)).
+∀H9:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X Y) Z) (add (multiply X Z) (multiply Y Z)).
+∀H10:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add X (multiply Y Z)) (multiply (add X Y) (add X Z)).
+∀H11:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add (multiply X Y) Z) (multiply (add X Z) (add Y Z)).
+∀H12:∀X:Univ.∀Y:Univ.eq Univ (multiply X Y) (multiply Y X).
+∀H13:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).eq Univ (add a a) a
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#add.
+#additive_identity.
+#inverse.
+#multiplicative_identity.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: BOO004-4.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO004-4 : TPTP v3.2.0. Released v1.1.0. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Problem : Addition is idempotent (X + X = X) *)
+
+(* Version : [Ver94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Ver94] Veroff (1994), Problem Set *)
+
+(* Source : [Ver94] *)
+
+(* Names : TA [Ver94] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 9 ( 0 non-Horn; 9 unit; 1 RR) *)
+
+(* Number of atoms : 9 ( 9 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 14 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include boolean algebra axioms for equality formulation *)
+
+(* Inclusion of: Axioms/BOO004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Axioms : Boolean algebra (equality) axioms *)
+
+(* Version : [Ver94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Ver94] Veroff (1994), Problem Set *)
+
+(* Source : [Ver94] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 14 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_a_plus_a_is_a:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀additive_identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiplicative_identity:Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.eq Univ (multiply X (inverse X)) additive_identity.
+∀H1:∀X:Univ.eq Univ (add X (inverse X)) multiplicative_identity.
+∀H2:∀X:Univ.eq Univ (multiply X multiplicative_identity) X.
+∀H3:∀X:Univ.eq Univ (add X additive_identity) X.
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y Z)) (add (multiply X Y) (multiply X Z)).
+∀H5:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add X (multiply Y Z)) (multiply (add X Y) (add X Z)).
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (multiply X Y) (multiply Y X).
+∀H7:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).eq Univ (add a a) a
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#add.
+#additive_identity.
+#inverse.
+#multiplicative_identity.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: BOO005-2.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO005-2 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Problem : Addition is bounded (X + 1 = 1) *)
+
+(* Version : [ANL] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : *)
+
+(* Source : [ANL] *)
+
+(* Names : prob3_part1.ver2.in [ANL] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.1.0, 0.14 v2.0.0 *)
+
+(* Syntax : Number of clauses : 15 ( 0 non-Horn; 15 unit; 1 RR) *)
+
+(* Number of atoms : 15 ( 15 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 24 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* Bugfixes : v1.2.1 - Clause prove_a_plus_1_is_a fixed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include boolean algebra axioms for equality formulation *)
+
+(* Inclusion of: Axioms/BOO003-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO003-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Axioms : Boolean algebra (equality) axioms *)
+
+(* Version : [ANL] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 14 ( 0 non-Horn; 14 unit; 0 RR) *)
+
+(* Number of literals : 14 ( 14 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 24 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_a_plus_1_is_a:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀additive_identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiplicative_identity:Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.eq Univ (add additive_identity X) X.
+∀H1:∀X:Univ.eq Univ (add X additive_identity) X.
+∀H2:∀X:Univ.eq Univ (multiply multiplicative_identity X) X.
+∀H3:∀X:Univ.eq Univ (multiply X multiplicative_identity) X.
+∀H4:∀X:Univ.eq Univ (multiply (inverse X) X) additive_identity.
+∀H5:∀X:Univ.eq Univ (multiply X (inverse X)) additive_identity.
+∀H6:∀X:Univ.eq Univ (add (inverse X) X) multiplicative_identity.
+∀H7:∀X:Univ.eq Univ (add X (inverse X)) multiplicative_identity.
+∀H8:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y Z)) (add (multiply X Y) (multiply X Z)).
+∀H9:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X Y) Z) (add (multiply X Z) (multiply Y Z)).
+∀H10:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add X (multiply Y Z)) (multiply (add X Y) (add X Z)).
+∀H11:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add (multiply X Y) Z) (multiply (add X Z) (add Y Z)).
+∀H12:∀X:Univ.∀Y:Univ.eq Univ (multiply X Y) (multiply Y X).
+∀H13:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).eq Univ (add a multiplicative_identity) multiplicative_identity
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#add.
+#additive_identity.
+#inverse.
+#multiplicative_identity.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: BOO005-4.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO005-4 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Problem : Addition is bounded (X + 1 = 1) *)
+
+(* Version : [Ver94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Ver94] Veroff (1994), Problem Set *)
+
+(* Source : [Ver94] *)
+
+(* Names : TB [Ver94] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 9 ( 0 non-Horn; 9 unit; 1 RR) *)
+
+(* Number of atoms : 9 ( 9 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 14 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* Bugfixes : v1.2.1 - Clause prove_a_plus_1_is_a fixed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include boolean algebra axioms for equality formulation *)
+
+(* Inclusion of: Axioms/BOO004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Axioms : Boolean algebra (equality) axioms *)
+
+(* Version : [Ver94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Ver94] Veroff (1994), Problem Set *)
+
+(* Source : [Ver94] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 14 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_a_plus_1_is_a:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀additive_identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiplicative_identity:Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.eq Univ (multiply X (inverse X)) additive_identity.
+∀H1:∀X:Univ.eq Univ (add X (inverse X)) multiplicative_identity.
+∀H2:∀X:Univ.eq Univ (multiply X multiplicative_identity) X.
+∀H3:∀X:Univ.eq Univ (add X additive_identity) X.
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y Z)) (add (multiply X Y) (multiply X Z)).
+∀H5:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add X (multiply Y Z)) (multiply (add X Y) (add X Z)).
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (multiply X Y) (multiply Y X).
+∀H7:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).eq Univ (add a multiplicative_identity) multiplicative_identity
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#add.
+#additive_identity.
+#inverse.
+#multiplicative_identity.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: BOO006-2.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO006-2 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Problem : Multiplication is bounded (X * 0 = 0) *)
+
+(* Version : [ANL] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : *)
+
+(* Source : [ANL] *)
+
+(* Names : prob3_part2.ver2.in [ANL] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.1.0, 0.38 v2.0.0 *)
+
+(* Syntax : Number of clauses : 15 ( 0 non-Horn; 15 unit; 1 RR) *)
+
+(* Number of atoms : 15 ( 15 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 24 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include boolean algebra axioms for equality formulation *)
+
+(* Inclusion of: Axioms/BOO003-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO003-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Axioms : Boolean algebra (equality) axioms *)
+
+(* Version : [ANL] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 14 ( 0 non-Horn; 14 unit; 0 RR) *)
+
+(* Number of literals : 14 ( 14 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 24 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_right_identity:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀additive_identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiplicative_identity:Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.eq Univ (add additive_identity X) X.
+∀H1:∀X:Univ.eq Univ (add X additive_identity) X.
+∀H2:∀X:Univ.eq Univ (multiply multiplicative_identity X) X.
+∀H3:∀X:Univ.eq Univ (multiply X multiplicative_identity) X.
+∀H4:∀X:Univ.eq Univ (multiply (inverse X) X) additive_identity.
+∀H5:∀X:Univ.eq Univ (multiply X (inverse X)) additive_identity.
+∀H6:∀X:Univ.eq Univ (add (inverse X) X) multiplicative_identity.
+∀H7:∀X:Univ.eq Univ (add X (inverse X)) multiplicative_identity.
+∀H8:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y Z)) (add (multiply X Y) (multiply X Z)).
+∀H9:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X Y) Z) (add (multiply X Z) (multiply Y Z)).
+∀H10:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add X (multiply Y Z)) (multiply (add X Y) (add X Z)).
+∀H11:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add (multiply X Y) Z) (multiply (add X Z) (add Y Z)).
+∀H12:∀X:Univ.∀Y:Univ.eq Univ (multiply X Y) (multiply Y X).
+∀H13:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).eq Univ (multiply a additive_identity) additive_identity
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#add.
+#additive_identity.
+#inverse.
+#multiplicative_identity.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: BOO006-4.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO006-4 : TPTP v3.2.0. Released v1.1.0. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Problem : Multiplication is bounded (X * 0 = 0) *)
+
+(* Version : [Ver94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Ver94] Veroff (1994), Problem Set *)
+
+(* Source : [Ver94] *)
+
+(* Names : TB [Ver94] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.1.0, 0.13 v2.0.0 *)
+
+(* Syntax : Number of clauses : 9 ( 0 non-Horn; 9 unit; 1 RR) *)
+
+(* Number of atoms : 9 ( 9 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 14 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include boolean algebra axioms for equality formulation *)
+
+(* Inclusion of: Axioms/BOO004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Axioms : Boolean algebra (equality) axioms *)
+
+(* Version : [Ver94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Ver94] Veroff (1994), Problem Set *)
+
+(* Source : [Ver94] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 14 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_right_identity:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀additive_identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiplicative_identity:Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.eq Univ (multiply X (inverse X)) additive_identity.
+∀H1:∀X:Univ.eq Univ (add X (inverse X)) multiplicative_identity.
+∀H2:∀X:Univ.eq Univ (multiply X multiplicative_identity) X.
+∀H3:∀X:Univ.eq Univ (add X additive_identity) X.
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y Z)) (add (multiply X Y) (multiply X Z)).
+∀H5:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add X (multiply Y Z)) (multiply (add X Y) (add X Z)).
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (multiply X Y) (multiply Y X).
+∀H7:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).eq Univ (multiply a additive_identity) additive_identity
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#add.
+#additive_identity.
+#inverse.
+#multiplicative_identity.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: BOO007-2.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO007-2 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Problem : Product is associative ( (X * Y) * Z = X * (Y * Z) ) *)
+
+(* Version : [ANL] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Ver92] Veroff (1992), Email to G. Sutcliffe *)
+
+(* Source : [Ver92] *)
+
+(* Names : associativity [Ver92] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.21 v3.2.0, 0.14 v3.1.0, 0.00 v2.7.0, 0.18 v2.6.0, 0.17 v2.5.0, 0.00 v2.2.1, 0.44 v2.2.0, 0.57 v2.1.0, 0.75 v2.0.0 *)
+
+(* Syntax : Number of clauses : 15 ( 0 non-Horn; 15 unit; 1 RR) *)
+
+(* Number of atoms : 15 ( 15 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 8 ( 5 constant; 0-2 arity) *)
+
+(* Number of variables : 24 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include boolean algebra axioms for equality formulation *)
+
+(* Inclusion of: Axioms/BOO003-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO003-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Axioms : Boolean algebra (equality) axioms *)
+
+(* Version : [ANL] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 14 ( 0 non-Horn; 14 unit; 0 RR) *)
+
+(* Number of literals : 14 ( 14 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 24 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_associativity:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀additive_identity:Univ.
+∀b:Univ.
+∀c:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiplicative_identity:Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.eq Univ (add additive_identity X) X.
+∀H1:∀X:Univ.eq Univ (add X additive_identity) X.
+∀H2:∀X:Univ.eq Univ (multiply multiplicative_identity X) X.
+∀H3:∀X:Univ.eq Univ (multiply X multiplicative_identity) X.
+∀H4:∀X:Univ.eq Univ (multiply (inverse X) X) additive_identity.
+∀H5:∀X:Univ.eq Univ (multiply X (inverse X)) additive_identity.
+∀H6:∀X:Univ.eq Univ (add (inverse X) X) multiplicative_identity.
+∀H7:∀X:Univ.eq Univ (add X (inverse X)) multiplicative_identity.
+∀H8:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y Z)) (add (multiply X Y) (multiply X Z)).
+∀H9:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X Y) Z) (add (multiply X Z) (multiply Y Z)).
+∀H10:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add X (multiply Y Z)) (multiply (add X Y) (add X Z)).
+∀H11:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add (multiply X Y) Z) (multiply (add X Z) (add Y Z)).
+∀H12:∀X:Univ.∀Y:Univ.eq Univ (multiply X Y) (multiply Y X).
+∀H13:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).eq Univ (multiply a (multiply b c)) (multiply (multiply a b) c)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#add.
+#additive_identity.
+#b.
+#c.
+#inverse.
+#multiplicative_identity.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: BOO007-4.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO007-4 : TPTP v3.2.0. Released v1.1.0. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Problem : Product is associative ( (X * Y) * Z = X * (Y * Z) ) *)
+
+(* Version : [Ver94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Ver94] Veroff (1994), Problem Set *)
+
+(* Source : [Ver94] *)
+
+(* Names : TD [Ver94] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.14 v3.2.0, 0.07 v3.1.0, 0.11 v2.7.0, 0.09 v2.6.0, 0.00 v2.2.1, 0.33 v2.2.0, 0.43 v2.1.0, 0.62 v2.0.0 *)
+
+(* Syntax : Number of clauses : 9 ( 0 non-Horn; 9 unit; 1 RR) *)
+
+(* Number of atoms : 9 ( 9 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 8 ( 5 constant; 0-2 arity) *)
+
+(* Number of variables : 14 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include boolean algebra axioms for equality formulation *)
+
+(* Inclusion of: Axioms/BOO004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Axioms : Boolean algebra (equality) axioms *)
+
+(* Version : [Ver94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Ver94] Veroff (1994), Problem Set *)
+
+(* Source : [Ver94] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 14 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_associativity:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀additive_identity:Univ.
+∀b:Univ.
+∀c:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiplicative_identity:Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.eq Univ (multiply X (inverse X)) additive_identity.
+∀H1:∀X:Univ.eq Univ (add X (inverse X)) multiplicative_identity.
+∀H2:∀X:Univ.eq Univ (multiply X multiplicative_identity) X.
+∀H3:∀X:Univ.eq Univ (add X additive_identity) X.
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y Z)) (add (multiply X Y) (multiply X Z)).
+∀H5:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add X (multiply Y Z)) (multiply (add X Y) (add X Z)).
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (multiply X Y) (multiply Y X).
+∀H7:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).eq Univ (multiply a (multiply b c)) (multiply (multiply a b) c)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#add.
+#additive_identity.
+#b.
+#c.
+#inverse.
+#multiplicative_identity.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: BOO008-2.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO008-2 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Problem : Sum is associative ( (X + Y) + Z = X + (Y + Z) ) *)
+
+(* Version : [ANL] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : *)
+
+(* Source : [ANL] *)
+
+(* Names : prob1.ver2.in [ANL] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.14 v3.2.0, 0.07 v3.1.0, 0.00 v2.7.0, 0.09 v2.6.0, 0.17 v2.5.0, 0.00 v2.2.1, 0.44 v2.2.0, 0.57 v2.1.0, 0.75 v2.0.0 *)
+
+(* Syntax : Number of clauses : 15 ( 0 non-Horn; 15 unit; 1 RR) *)
+
+(* Number of atoms : 15 ( 15 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 8 ( 5 constant; 0-2 arity) *)
+
+(* Number of variables : 24 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include boolean algebra axioms for equality formulation *)
+
+(* Inclusion of: Axioms/BOO003-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO003-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Axioms : Boolean algebra (equality) axioms *)
+
+(* Version : [ANL] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 14 ( 0 non-Horn; 14 unit; 0 RR) *)
+
+(* Number of literals : 14 ( 14 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 24 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The ANL version of this problem includes the idempotence lemmas *)
+
+(* input_clause(idempotence_of_add,axiom, *)
+
+(* [++equal(add(X,X),X)]). *)
+
+(* input_clause(idempotence_of_multiply,axiom, *)
+
+(* [++equal(multiply(X,X),X)]). *)
+ntheorem prove_associativity:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀additive_identity:Univ.
+∀b:Univ.
+∀c:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiplicative_identity:Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.eq Univ (add additive_identity X) X.
+∀H1:∀X:Univ.eq Univ (add X additive_identity) X.
+∀H2:∀X:Univ.eq Univ (multiply multiplicative_identity X) X.
+∀H3:∀X:Univ.eq Univ (multiply X multiplicative_identity) X.
+∀H4:∀X:Univ.eq Univ (multiply (inverse X) X) additive_identity.
+∀H5:∀X:Univ.eq Univ (multiply X (inverse X)) additive_identity.
+∀H6:∀X:Univ.eq Univ (add (inverse X) X) multiplicative_identity.
+∀H7:∀X:Univ.eq Univ (add X (inverse X)) multiplicative_identity.
+∀H8:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y Z)) (add (multiply X Y) (multiply X Z)).
+∀H9:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X Y) Z) (add (multiply X Z) (multiply Y Z)).
+∀H10:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add X (multiply Y Z)) (multiply (add X Y) (add X Z)).
+∀H11:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add (multiply X Y) Z) (multiply (add X Z) (add Y Z)).
+∀H12:∀X:Univ.∀Y:Univ.eq Univ (multiply X Y) (multiply Y X).
+∀H13:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).eq Univ (add a (add b c)) (add (add a b) c)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#add.
+#additive_identity.
+#b.
+#c.
+#inverse.
+#multiplicative_identity.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: BOO008-4.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO008-4 : TPTP v3.2.0. Released v1.1.0. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Problem : Sum is associative ( (X + Y) + Z = X + (Y + Z) ) *)
+
+(* Version : [Ver94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Ver94] Veroff (1994), Problem Set *)
+
+(* Source : [Ver94] *)
+
+(* Names : TD [Ver94] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.07 v3.2.0, 0.00 v2.2.1, 0.33 v2.2.0, 0.43 v2.1.0, 0.75 v2.0.0 *)
+
+(* Syntax : Number of clauses : 9 ( 0 non-Horn; 9 unit; 1 RR) *)
+
+(* Number of atoms : 9 ( 9 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 8 ( 5 constant; 0-2 arity) *)
+
+(* Number of variables : 14 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include boolean algebra axioms for equality formulation *)
+
+(* Inclusion of: Axioms/BOO004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Axioms : Boolean algebra (equality) axioms *)
+
+(* Version : [Ver94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Ver94] Veroff (1994), Problem Set *)
+
+(* Source : [Ver94] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 14 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_associativity:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀additive_identity:Univ.
+∀b:Univ.
+∀c:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiplicative_identity:Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.eq Univ (multiply X (inverse X)) additive_identity.
+∀H1:∀X:Univ.eq Univ (add X (inverse X)) multiplicative_identity.
+∀H2:∀X:Univ.eq Univ (multiply X multiplicative_identity) X.
+∀H3:∀X:Univ.eq Univ (add X additive_identity) X.
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y Z)) (add (multiply X Y) (multiply X Z)).
+∀H5:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add X (multiply Y Z)) (multiply (add X Y) (add X Z)).
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (multiply X Y) (multiply Y X).
+∀H7:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).eq Univ (add a (add b c)) (add (add a b) c)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#add.
+#additive_identity.
+#b.
+#c.
+#inverse.
+#multiplicative_identity.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: BOO009-2.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO009-2 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Problem : Multiplication absorption (X * (X + Y) = X) *)
+
+(* Version : [ANL] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : *)
+
+(* Source : [ANL] *)
+
+(* Names : prob4_part1.ver2.in [ANL] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.2.1, 0.11 v2.2.0, 0.14 v2.1.0, 0.50 v2.0.0 *)
+
+(* Syntax : Number of clauses : 15 ( 0 non-Horn; 15 unit; 1 RR) *)
+
+(* Number of atoms : 15 ( 15 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 24 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include boolean algebra axioms for equality formulation *)
+
+(* Inclusion of: Axioms/BOO003-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO003-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Axioms : Boolean algebra (equality) axioms *)
+
+(* Version : [ANL] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 14 ( 0 non-Horn; 14 unit; 0 RR) *)
+
+(* Number of literals : 14 ( 14 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 24 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_operation:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀additive_identity:Univ.
+∀b:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiplicative_identity:Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.eq Univ (add additive_identity X) X.
+∀H1:∀X:Univ.eq Univ (add X additive_identity) X.
+∀H2:∀X:Univ.eq Univ (multiply multiplicative_identity X) X.
+∀H3:∀X:Univ.eq Univ (multiply X multiplicative_identity) X.
+∀H4:∀X:Univ.eq Univ (multiply (inverse X) X) additive_identity.
+∀H5:∀X:Univ.eq Univ (multiply X (inverse X)) additive_identity.
+∀H6:∀X:Univ.eq Univ (add (inverse X) X) multiplicative_identity.
+∀H7:∀X:Univ.eq Univ (add X (inverse X)) multiplicative_identity.
+∀H8:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y Z)) (add (multiply X Y) (multiply X Z)).
+∀H9:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X Y) Z) (add (multiply X Z) (multiply Y Z)).
+∀H10:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add X (multiply Y Z)) (multiply (add X Y) (add X Z)).
+∀H11:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add (multiply X Y) Z) (multiply (add X Z) (add Y Z)).
+∀H12:∀X:Univ.∀Y:Univ.eq Univ (multiply X Y) (multiply Y X).
+∀H13:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).eq Univ (multiply a (add a b)) a
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#add.
+#additive_identity.
+#b.
+#inverse.
+#multiplicative_identity.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: BOO009-4.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO009-4 : TPTP v3.2.0. Released v1.1.0. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Problem : Multiplication absorption (X * (X + Y) = X) *)
+
+(* Version : [Ver94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Ver94] Veroff (1994), Problem Set *)
+
+(* Source : [Ver94] *)
+
+(* Names : TC [Ver94] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.2.1, 0.22 v2.2.0, 0.29 v2.1.0, 0.38 v2.0.0 *)
+
+(* Syntax : Number of clauses : 9 ( 0 non-Horn; 9 unit; 1 RR) *)
+
+(* Number of atoms : 9 ( 9 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 14 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include boolean algebra axioms for equality formulation *)
+
+(* Inclusion of: Axioms/BOO004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Axioms : Boolean algebra (equality) axioms *)
+
+(* Version : [Ver94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Ver94] Veroff (1994), Problem Set *)
+
+(* Source : [Ver94] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 14 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_operation:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀additive_identity:Univ.
+∀b:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiplicative_identity:Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.eq Univ (multiply X (inverse X)) additive_identity.
+∀H1:∀X:Univ.eq Univ (add X (inverse X)) multiplicative_identity.
+∀H2:∀X:Univ.eq Univ (multiply X multiplicative_identity) X.
+∀H3:∀X:Univ.eq Univ (add X additive_identity) X.
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y Z)) (add (multiply X Y) (multiply X Z)).
+∀H5:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add X (multiply Y Z)) (multiply (add X Y) (add X Z)).
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (multiply X Y) (multiply Y X).
+∀H7:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).eq Univ (multiply a (add a b)) a
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#add.
+#additive_identity.
+#b.
+#inverse.
+#multiplicative_identity.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: BOO010-2.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO010-2 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Problem : Addition absorbtion (X + (X * Y) = X) *)
+
+(* Version : [ANL] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : *)
+
+(* Source : [ANL] *)
+
+(* Names : prob4_part2.ver2.in [ANL] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.2.1, 0.11 v2.2.0, 0.14 v2.1.0, 0.38 v2.0.0 *)
+
+(* Syntax : Number of clauses : 15 ( 0 non-Horn; 15 unit; 1 RR) *)
+
+(* Number of atoms : 15 ( 15 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 24 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include boolean algebra axioms for equality formulation *)
+
+(* Inclusion of: Axioms/BOO003-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO003-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Axioms : Boolean algebra (equality) axioms *)
+
+(* Version : [ANL] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 14 ( 0 non-Horn; 14 unit; 0 RR) *)
+
+(* Number of literals : 14 ( 14 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 24 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_a_plus_ab_is_a:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀additive_identity:Univ.
+∀b:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiplicative_identity:Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.eq Univ (add additive_identity X) X.
+∀H1:∀X:Univ.eq Univ (add X additive_identity) X.
+∀H2:∀X:Univ.eq Univ (multiply multiplicative_identity X) X.
+∀H3:∀X:Univ.eq Univ (multiply X multiplicative_identity) X.
+∀H4:∀X:Univ.eq Univ (multiply (inverse X) X) additive_identity.
+∀H5:∀X:Univ.eq Univ (multiply X (inverse X)) additive_identity.
+∀H6:∀X:Univ.eq Univ (add (inverse X) X) multiplicative_identity.
+∀H7:∀X:Univ.eq Univ (add X (inverse X)) multiplicative_identity.
+∀H8:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y Z)) (add (multiply X Y) (multiply X Z)).
+∀H9:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X Y) Z) (add (multiply X Z) (multiply Y Z)).
+∀H10:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add X (multiply Y Z)) (multiply (add X Y) (add X Z)).
+∀H11:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add (multiply X Y) Z) (multiply (add X Z) (add Y Z)).
+∀H12:∀X:Univ.∀Y:Univ.eq Univ (multiply X Y) (multiply Y X).
+∀H13:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).eq Univ (add a (multiply a b)) a
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#add.
+#additive_identity.
+#b.
+#inverse.
+#multiplicative_identity.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: BOO010-4.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO010-4 : TPTP v3.2.0. Released v1.1.0. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Problem : Addition absorbtion (X + (X * Y) = X) *)
+
+(* Version : [Ver94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Ver94] Veroff (1994), Problem Set *)
+
+(* Source : [Ver94] *)
+
+(* Names : TC [Ver94] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.7.0, 0.09 v2.6.0, 0.00 v2.2.1, 0.22 v2.2.0, 0.29 v2.1.0, 0.38 v2.0.0 *)
+
+(* Syntax : Number of clauses : 9 ( 0 non-Horn; 9 unit; 1 RR) *)
+
+(* Number of atoms : 9 ( 9 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 14 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include boolean algebra axioms for equality formulation *)
+
+(* Inclusion of: Axioms/BOO004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Axioms : Boolean algebra (equality) axioms *)
+
+(* Version : [Ver94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Ver94] Veroff (1994), Problem Set *)
+
+(* Source : [Ver94] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 14 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_a_plus_ab_is_a:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀additive_identity:Univ.
+∀b:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiplicative_identity:Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.eq Univ (multiply X (inverse X)) additive_identity.
+∀H1:∀X:Univ.eq Univ (add X (inverse X)) multiplicative_identity.
+∀H2:∀X:Univ.eq Univ (multiply X multiplicative_identity) X.
+∀H3:∀X:Univ.eq Univ (add X additive_identity) X.
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y Z)) (add (multiply X Y) (multiply X Z)).
+∀H5:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add X (multiply Y Z)) (multiply (add X Y) (add X Z)).
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (multiply X Y) (multiply Y X).
+∀H7:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).eq Univ (add a (multiply a b)) a
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#add.
+#additive_identity.
+#b.
+#inverse.
+#multiplicative_identity.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: BOO011-2.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO011-2 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Problem : Inverse of additive identity = Multiplicative identity *)
+
+(* Version : [ANL] (equality) axioms. *)
+
+(* English : The inverse of the additive identity is the multiplicative *)
+
+(* identity. *)
+
+(* Refs : *)
+
+(* Source : [ANL] *)
+
+(* Names : prob7.ver2.in [ANL] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 15 ( 0 non-Horn; 15 unit; 1 RR) *)
+
+(* Number of atoms : 15 ( 15 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 24 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* Bugfixes : v1.2.1 - Clause prove_inverse_of_1_is_0 fixed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include boolean algebra axioms for equality formulation *)
+
+(* Inclusion of: Axioms/BOO003-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO003-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Axioms : Boolean algebra (equality) axioms *)
+
+(* Version : [ANL] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 14 ( 0 non-Horn; 14 unit; 0 RR) *)
+
+(* Number of literals : 14 ( 14 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 24 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_inverse_of_1_is_0:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀additive_identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiplicative_identity:Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.eq Univ (add additive_identity X) X.
+∀H1:∀X:Univ.eq Univ (add X additive_identity) X.
+∀H2:∀X:Univ.eq Univ (multiply multiplicative_identity X) X.
+∀H3:∀X:Univ.eq Univ (multiply X multiplicative_identity) X.
+∀H4:∀X:Univ.eq Univ (multiply (inverse X) X) additive_identity.
+∀H5:∀X:Univ.eq Univ (multiply X (inverse X)) additive_identity.
+∀H6:∀X:Univ.eq Univ (add (inverse X) X) multiplicative_identity.
+∀H7:∀X:Univ.eq Univ (add X (inverse X)) multiplicative_identity.
+∀H8:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y Z)) (add (multiply X Y) (multiply X Z)).
+∀H9:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X Y) Z) (add (multiply X Z) (multiply Y Z)).
+∀H10:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add X (multiply Y Z)) (multiply (add X Y) (add X Z)).
+∀H11:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add (multiply X Y) Z) (multiply (add X Z) (add Y Z)).
+∀H12:∀X:Univ.∀Y:Univ.eq Univ (multiply X Y) (multiply Y X).
+∀H13:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).eq Univ (inverse additive_identity) multiplicative_identity
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#add.
+#additive_identity.
+#inverse.
+#multiplicative_identity.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: BOO011-4.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO011-4 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Problem : Inverse of additive identity = Multiplicative identity *)
+
+(* Version : [Ver94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Ver94] Veroff (1994), Problem Set *)
+
+(* Source : [Ver94] *)
+
+(* Names : TG [Ver94] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 9 ( 0 non-Horn; 9 unit; 1 RR) *)
+
+(* Number of atoms : 9 ( 9 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 14 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* Bugfixes : v1.2.1 - Clause prove_inverse_of_1_is_0 fixed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include boolean algebra axioms for equality formulation *)
+
+(* Inclusion of: Axioms/BOO004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Axioms : Boolean algebra (equality) axioms *)
+
+(* Version : [Ver94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Ver94] Veroff (1994), Problem Set *)
+
+(* Source : [Ver94] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 14 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_inverse_of_1_is_0:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀additive_identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiplicative_identity:Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.eq Univ (multiply X (inverse X)) additive_identity.
+∀H1:∀X:Univ.eq Univ (add X (inverse X)) multiplicative_identity.
+∀H2:∀X:Univ.eq Univ (multiply X multiplicative_identity) X.
+∀H3:∀X:Univ.eq Univ (add X additive_identity) X.
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y Z)) (add (multiply X Y) (multiply X Z)).
+∀H5:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add X (multiply Y Z)) (multiply (add X Y) (add X Z)).
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (multiply X Y) (multiply Y X).
+∀H7:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).eq Univ (inverse additive_identity) multiplicative_identity
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#add.
+#additive_identity.
+#inverse.
+#multiplicative_identity.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: BOO012-2.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO012-2 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Problem : Inverse is an involution *)
+
+(* Version : [ANL] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : *)
+
+(* Source : [ANL] *)
+
+(* Names : prob8.ver2.in [ANL] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.2.1, 0.11 v2.2.0, 0.14 v2.1.0, 0.25 v2.0.0 *)
+
+(* Syntax : Number of clauses : 15 ( 0 non-Horn; 15 unit; 1 RR) *)
+
+(* Number of atoms : 15 ( 15 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 24 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include boolean algebra axioms for equality formulation *)
+
+(* Inclusion of: Axioms/BOO003-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO003-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Axioms : Boolean algebra (equality) axioms *)
+
+(* Version : [ANL] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 14 ( 0 non-Horn; 14 unit; 0 RR) *)
+
+(* Number of literals : 14 ( 14 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 24 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_inverse_is_an_involution:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀additive_identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiplicative_identity:Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀x:Univ.
+∀H0:∀X:Univ.eq Univ (add additive_identity X) X.
+∀H1:∀X:Univ.eq Univ (add X additive_identity) X.
+∀H2:∀X:Univ.eq Univ (multiply multiplicative_identity X) X.
+∀H3:∀X:Univ.eq Univ (multiply X multiplicative_identity) X.
+∀H4:∀X:Univ.eq Univ (multiply (inverse X) X) additive_identity.
+∀H5:∀X:Univ.eq Univ (multiply X (inverse X)) additive_identity.
+∀H6:∀X:Univ.eq Univ (add (inverse X) X) multiplicative_identity.
+∀H7:∀X:Univ.eq Univ (add X (inverse X)) multiplicative_identity.
+∀H8:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y Z)) (add (multiply X Y) (multiply X Z)).
+∀H9:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X Y) Z) (add (multiply X Z) (multiply Y Z)).
+∀H10:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add X (multiply Y Z)) (multiply (add X Y) (add X Z)).
+∀H11:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add (multiply X Y) Z) (multiply (add X Z) (add Y Z)).
+∀H12:∀X:Univ.∀Y:Univ.eq Univ (multiply X Y) (multiply Y X).
+∀H13:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).eq Univ (inverse (inverse x)) x
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#add.
+#additive_identity.
+#inverse.
+#multiplicative_identity.
+#multiply.
+#x.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: BOO012-4.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO012-4 : TPTP v3.2.0. Released v1.1.0. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Problem : Inverse is an involution *)
+
+(* Version : [Ver94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Ver94] Veroff (1994), Problem Set *)
+
+(* Source : [Ver94] *)
+
+(* Names : TF [Ver94] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.2.1, 0.22 v2.2.0, 0.29 v2.1.0, 0.25 v2.0.0 *)
+
+(* Syntax : Number of clauses : 9 ( 0 non-Horn; 9 unit; 1 RR) *)
+
+(* Number of atoms : 9 ( 9 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 14 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include boolean algebra axioms for equality formulation *)
+
+(* Inclusion of: Axioms/BOO004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Axioms : Boolean algebra (equality) axioms *)
+
+(* Version : [Ver94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Ver94] Veroff (1994), Problem Set *)
+
+(* Source : [Ver94] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 14 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_inverse_is_an_involution:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀additive_identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiplicative_identity:Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀x:Univ.
+∀H0:∀X:Univ.eq Univ (multiply X (inverse X)) additive_identity.
+∀H1:∀X:Univ.eq Univ (add X (inverse X)) multiplicative_identity.
+∀H2:∀X:Univ.eq Univ (multiply X multiplicative_identity) X.
+∀H3:∀X:Univ.eq Univ (add X additive_identity) X.
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y Z)) (add (multiply X Y) (multiply X Z)).
+∀H5:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add X (multiply Y Z)) (multiply (add X Y) (add X Z)).
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (multiply X Y) (multiply Y X).
+∀H7:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).eq Univ (inverse (inverse x)) x
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#add.
+#additive_identity.
+#inverse.
+#multiplicative_identity.
+#multiply.
+#x.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: BOO013-2.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO013-2 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Problem : The inverse of X is unique *)
+
+(* Version : [ANL] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : *)
+
+(* Source : [ANL] *)
+
+(* Names : prob9.ver2.in [ANL] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.2.1, 0.11 v2.2.0, 0.14 v2.1.0, 0.14 v2.0.0 *)
+
+(* Syntax : Number of clauses : 19 ( 0 non-Horn; 19 unit; 5 RR) *)
+
+(* Number of atoms : 19 ( 19 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 8 ( 5 constant; 0-2 arity) *)
+
+(* Number of variables : 24 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* Bugfixes : v1.2.1 - Clauses b_and_multiplicative_identity and *)
+
+(* c_and_multiplicative_identity fixed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include boolean algebra axioms for equality formulation *)
+
+(* Inclusion of: Axioms/BOO003-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO003-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Axioms : Boolean algebra (equality) axioms *)
+
+(* Version : [ANL] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 14 ( 0 non-Horn; 14 unit; 0 RR) *)
+
+(* Number of literals : 14 ( 14 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 24 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_b_is_a:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀additive_identity:Univ.
+∀b:Univ.
+∀c:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiplicative_identity:Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:eq Univ (multiply a c) additive_identity.
+∀H1:eq Univ (multiply a b) additive_identity.
+∀H2:eq Univ (add a c) multiplicative_identity.
+∀H3:eq Univ (add a b) multiplicative_identity.
+∀H4:∀X:Univ.eq Univ (add additive_identity X) X.
+∀H5:∀X:Univ.eq Univ (add X additive_identity) X.
+∀H6:∀X:Univ.eq Univ (multiply multiplicative_identity X) X.
+∀H7:∀X:Univ.eq Univ (multiply X multiplicative_identity) X.
+∀H8:∀X:Univ.eq Univ (multiply (inverse X) X) additive_identity.
+∀H9:∀X:Univ.eq Univ (multiply X (inverse X)) additive_identity.
+∀H10:∀X:Univ.eq Univ (add (inverse X) X) multiplicative_identity.
+∀H11:∀X:Univ.eq Univ (add X (inverse X)) multiplicative_identity.
+∀H12:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y Z)) (add (multiply X Y) (multiply X Z)).
+∀H13:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X Y) Z) (add (multiply X Z) (multiply Y Z)).
+∀H14:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add X (multiply Y Z)) (multiply (add X Y) (add X Z)).
+∀H15:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add (multiply X Y) Z) (multiply (add X Z) (add Y Z)).
+∀H16:∀X:Univ.∀Y:Univ.eq Univ (multiply X Y) (multiply Y X).
+∀H17:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).eq Univ b c
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#add.
+#additive_identity.
+#b.
+#c.
+#inverse.
+#multiplicative_identity.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+#H16.
+#H17.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15,H16,H17;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: BOO013-4.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO013-4 : TPTP v3.2.0. Released v1.1.0. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Problem : The inverse of X is unique *)
+
+(* Version : [Ver94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Ver94] Veroff (1994), Problem Set *)
+
+(* Source : [Ver94] *)
+
+(* Names : TE [Ver94] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.2.1, 0.22 v2.2.0, 0.29 v2.1.0, 0.25 v2.0.0 *)
+
+(* Syntax : Number of clauses : 11 ( 0 non-Horn; 11 unit; 3 RR) *)
+
+(* Number of atoms : 11 ( 11 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 14 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include boolean algebra axioms for equality formulation *)
+
+(* Inclusion of: Axioms/BOO004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Axioms : Boolean algebra (equality) axioms *)
+
+(* Version : [Ver94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Ver94] Veroff (1994), Problem Set *)
+
+(* Source : [Ver94] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 14 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_a_inverse_is_b:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀additive_identity:Univ.
+∀b:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiplicative_identity:Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:eq Univ (multiply a b) additive_identity.
+∀H1:eq Univ (add a b) multiplicative_identity.
+∀H2:∀X:Univ.eq Univ (multiply X (inverse X)) additive_identity.
+∀H3:∀X:Univ.eq Univ (add X (inverse X)) multiplicative_identity.
+∀H4:∀X:Univ.eq Univ (multiply X multiplicative_identity) X.
+∀H5:∀X:Univ.eq Univ (add X additive_identity) X.
+∀H6:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y Z)) (add (multiply X Y) (multiply X Z)).
+∀H7:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add X (multiply Y Z)) (multiply (add X Y) (add X Z)).
+∀H8:∀X:Univ.∀Y:Univ.eq Univ (multiply X Y) (multiply Y X).
+∀H9:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).eq Univ b (inverse a)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#add.
+#additive_identity.
+#b.
+#inverse.
+#multiplicative_identity.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: BOO014-2.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO014-2 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Problem : DeMorgan for inverse and product (X+Y)^-1 = (X^-1) * (Y^-1) *)
+
+(* Version : [ANL] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : *)
+
+(* Source : [ANL] *)
+
+(* Names : prob10.ver2.in [ANL] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.2.1, 0.22 v2.2.0, 0.29 v2.1.0, 0.75 v2.0.0 *)
+
+(* Syntax : Number of clauses : 17 ( 0 non-Horn; 17 unit; 3 RR) *)
+
+(* Number of atoms : 17 ( 17 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 9 ( 6 constant; 0-2 arity) *)
+
+(* Number of variables : 24 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include boolean algebra axioms for equality formulation *)
+
+(* Inclusion of: Axioms/BOO003-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO003-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Axioms : Boolean algebra (equality) axioms *)
+
+(* Version : [ANL] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 14 ( 0 non-Horn; 14 unit; 0 RR) *)
+
+(* Number of literals : 14 ( 14 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 24 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_c_inverse_is_d:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀additive_identity:Univ.
+∀b:Univ.
+∀c:Univ.
+∀d:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiplicative_identity:Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:eq Univ (multiply (inverse a) (inverse b)) d.
+∀H1:eq Univ (add a b) c.
+∀H2:∀X:Univ.eq Univ (add additive_identity X) X.
+∀H3:∀X:Univ.eq Univ (add X additive_identity) X.
+∀H4:∀X:Univ.eq Univ (multiply multiplicative_identity X) X.
+∀H5:∀X:Univ.eq Univ (multiply X multiplicative_identity) X.
+∀H6:∀X:Univ.eq Univ (multiply (inverse X) X) additive_identity.
+∀H7:∀X:Univ.eq Univ (multiply X (inverse X)) additive_identity.
+∀H8:∀X:Univ.eq Univ (add (inverse X) X) multiplicative_identity.
+∀H9:∀X:Univ.eq Univ (add X (inverse X)) multiplicative_identity.
+∀H10:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y Z)) (add (multiply X Y) (multiply X Z)).
+∀H11:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X Y) Z) (add (multiply X Z) (multiply Y Z)).
+∀H12:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add X (multiply Y Z)) (multiply (add X Y) (add X Z)).
+∀H13:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add (multiply X Y) Z) (multiply (add X Z) (add Y Z)).
+∀H14:∀X:Univ.∀Y:Univ.eq Univ (multiply X Y) (multiply Y X).
+∀H15:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).eq Univ (inverse c) d
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#add.
+#additive_identity.
+#b.
+#c.
+#d.
+#inverse.
+#multiplicative_identity.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: BOO014-4.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO014-4 : TPTP v3.2.0. Released v1.1.0. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Problem : DeMorgan for inverse and product (X+Y)^-1 = (X^-1) * (Y^-1) *)
+
+(* Version : [Ver94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Ver94] Veroff (1994), Problem Set *)
+
+(* Source : [Ver94] *)
+
+(* Names : TH [Ver94] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.2.1, 0.33 v2.2.0, 0.43 v2.1.0, 0.62 v2.0.0 *)
+
+(* Syntax : Number of clauses : 9 ( 0 non-Horn; 9 unit; 1 RR) *)
+
+(* Number of atoms : 9 ( 9 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 14 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include boolean algebra axioms for equality formulation *)
+
+(* Inclusion of: Axioms/BOO004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Axioms : Boolean algebra (equality) axioms *)
+
+(* Version : [Ver94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Ver94] Veroff (1994), Problem Set *)
+
+(* Source : [Ver94] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 14 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_c_inverse_is_d:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀additive_identity:Univ.
+∀b:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiplicative_identity:Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.eq Univ (multiply X (inverse X)) additive_identity.
+∀H1:∀X:Univ.eq Univ (add X (inverse X)) multiplicative_identity.
+∀H2:∀X:Univ.eq Univ (multiply X multiplicative_identity) X.
+∀H3:∀X:Univ.eq Univ (add X additive_identity) X.
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y Z)) (add (multiply X Y) (multiply X Z)).
+∀H5:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add X (multiply Y Z)) (multiply (add X Y) (add X Z)).
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (multiply X Y) (multiply Y X).
+∀H7:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).eq Univ (inverse (add a b)) (multiply (inverse a) (inverse b))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#add.
+#additive_identity.
+#b.
+#inverse.
+#multiplicative_identity.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: BOO015-2.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO015-2 : TPTP v3.2.0. Bugfixed v1.0.1. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Problem : DeMorgan for inverse and sum (X^-1 + Y^-1) = (X * Y)^-1 *)
+
+(* Version : [ANL] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : *)
+
+(* Source : [ANL] *)
+
+(* Names : prob10.ver2.in [ANL] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.2.1, 0.22 v2.2.0, 0.29 v2.1.0, 0.62 v2.0.0 *)
+
+(* Syntax : Number of clauses : 17 ( 0 non-Horn; 17 unit; 3 RR) *)
+
+(* Number of atoms : 17 ( 17 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 9 ( 6 constant; 0-2 arity) *)
+
+(* Number of variables : 24 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* Bugfixes : v1.0.1 - Clause a_inverse_plus_b_inverse_is_d fixed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include boolean algebra axioms for equality formulation *)
+
+(* Inclusion of: Axioms/BOO003-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO003-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Axioms : Boolean algebra (equality) axioms *)
+
+(* Version : [ANL] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 14 ( 0 non-Horn; 14 unit; 0 RR) *)
+
+(* Number of literals : 14 ( 14 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 24 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_c_inverse_is_d:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀additive_identity:Univ.
+∀b:Univ.
+∀c:Univ.
+∀d:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiplicative_identity:Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:eq Univ (add (inverse a) (inverse b)) d.
+∀H1:eq Univ (multiply a b) c.
+∀H2:∀X:Univ.eq Univ (add additive_identity X) X.
+∀H3:∀X:Univ.eq Univ (add X additive_identity) X.
+∀H4:∀X:Univ.eq Univ (multiply multiplicative_identity X) X.
+∀H5:∀X:Univ.eq Univ (multiply X multiplicative_identity) X.
+∀H6:∀X:Univ.eq Univ (multiply (inverse X) X) additive_identity.
+∀H7:∀X:Univ.eq Univ (multiply X (inverse X)) additive_identity.
+∀H8:∀X:Univ.eq Univ (add (inverse X) X) multiplicative_identity.
+∀H9:∀X:Univ.eq Univ (add X (inverse X)) multiplicative_identity.
+∀H10:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y Z)) (add (multiply X Y) (multiply X Z)).
+∀H11:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X Y) Z) (add (multiply X Z) (multiply Y Z)).
+∀H12:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add X (multiply Y Z)) (multiply (add X Y) (add X Z)).
+∀H13:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add (multiply X Y) Z) (multiply (add X Z) (add Y Z)).
+∀H14:∀X:Univ.∀Y:Univ.eq Univ (multiply X Y) (multiply Y X).
+∀H15:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).eq Univ (inverse c) d
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#add.
+#additive_identity.
+#b.
+#c.
+#d.
+#inverse.
+#multiplicative_identity.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: BOO015-4.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO015-4 : TPTP v3.2.0. Released v1.1.0. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Problem : DeMorgan for inverse and sum (X^-1 + Y^-1) = (X * Y)^-1 *)
+
+(* Version : [Veroff, 1993] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Ver94] Veroff (1994), Problem Set *)
+
+(* Source : [Ver94] *)
+
+(* Names : TH [Ver94] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.2.1, 0.33 v2.2.0, 0.43 v2.1.0, 0.62 v2.0.0 *)
+
+(* Syntax : Number of clauses : 9 ( 0 non-Horn; 9 unit; 1 RR) *)
+
+(* Number of atoms : 9 ( 9 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 14 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include boolean algebra axioms for equality formulation *)
+
+(* Inclusion of: Axioms/BOO004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Axioms : Boolean algebra (equality) axioms *)
+
+(* Version : [Ver94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Ver94] Veroff (1994), Problem Set *)
+
+(* Source : [Ver94] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 14 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_c_inverse_is_d:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀additive_identity:Univ.
+∀b:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiplicative_identity:Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.eq Univ (multiply X (inverse X)) additive_identity.
+∀H1:∀X:Univ.eq Univ (add X (inverse X)) multiplicative_identity.
+∀H2:∀X:Univ.eq Univ (multiply X multiplicative_identity) X.
+∀H3:∀X:Univ.eq Univ (add X additive_identity) X.
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y Z)) (add (multiply X Y) (multiply X Z)).
+∀H5:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add X (multiply Y Z)) (multiply (add X Y) (add X Z)).
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (multiply X Y) (multiply Y X).
+∀H7:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).eq Univ (inverse (multiply a b)) (add (inverse a) (inverse b))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#add.
+#additive_identity.
+#b.
+#inverse.
+#multiplicative_identity.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: BOO016-2.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO016-2 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Problem : Relating product and sum (X * Y = Z -> X + Z = X) *)
+
+(* Version : [ANL] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.2.1, 0.11 v2.2.0, 0.14 v2.1.0, 0.38 v2.0.0 *)
+
+(* Syntax : Number of clauses : 16 ( 0 non-Horn; 16 unit; 2 RR) *)
+
+(* Number of atoms : 16 ( 16 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 8 ( 5 constant; 0-2 arity) *)
+
+(* Number of variables : 24 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include boolean algebra axioms for equality formulation *)
+
+(* Inclusion of: Axioms/BOO003-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO003-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Axioms : Boolean algebra (equality) axioms *)
+
+(* Version : [ANL] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 14 ( 0 non-Horn; 14 unit; 0 RR) *)
+
+(* Number of literals : 14 ( 14 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 24 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_sum:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀additive_identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiplicative_identity:Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀x:Univ.
+∀y:Univ.
+∀z:Univ.
+∀H0:eq Univ (multiply x y) z.
+∀H1:∀X:Univ.eq Univ (add additive_identity X) X.
+∀H2:∀X:Univ.eq Univ (add X additive_identity) X.
+∀H3:∀X:Univ.eq Univ (multiply multiplicative_identity X) X.
+∀H4:∀X:Univ.eq Univ (multiply X multiplicative_identity) X.
+∀H5:∀X:Univ.eq Univ (multiply (inverse X) X) additive_identity.
+∀H6:∀X:Univ.eq Univ (multiply X (inverse X)) additive_identity.
+∀H7:∀X:Univ.eq Univ (add (inverse X) X) multiplicative_identity.
+∀H8:∀X:Univ.eq Univ (add X (inverse X)) multiplicative_identity.
+∀H9:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y Z)) (add (multiply X Y) (multiply X Z)).
+∀H10:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X Y) Z) (add (multiply X Z) (multiply Y Z)).
+∀H11:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add X (multiply Y Z)) (multiply (add X Y) (add X Z)).
+∀H12:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add (multiply X Y) Z) (multiply (add X Z) (add Y Z)).
+∀H13:∀X:Univ.∀Y:Univ.eq Univ (multiply X Y) (multiply Y X).
+∀H14:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).eq Univ (add x z) x
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#add.
+#additive_identity.
+#inverse.
+#multiplicative_identity.
+#multiply.
+#x.
+#y.
+#z.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: BOO017-2.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO017-2 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Problem : Relating sum and product (X + Y = Z -> X * Z = X) *)
+
+(* Version : [ANL] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.2.1, 0.11 v2.2.0, 0.14 v2.1.0, 0.50 v2.0.0 *)
+
+(* Syntax : Number of clauses : 16 ( 0 non-Horn; 16 unit; 2 RR) *)
+
+(* Number of atoms : 16 ( 16 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 8 ( 5 constant; 0-2 arity) *)
+
+(* Number of variables : 24 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include boolean algebra axioms for equality formulation *)
+
+(* Inclusion of: Axioms/BOO003-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO003-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Axioms : Boolean algebra (equality) axioms *)
+
+(* Version : [ANL] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 14 ( 0 non-Horn; 14 unit; 0 RR) *)
+
+(* Number of literals : 14 ( 14 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 24 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_sum:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀additive_identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiplicative_identity:Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀x:Univ.
+∀y:Univ.
+∀z:Univ.
+∀H0:eq Univ (add x y) z.
+∀H1:∀X:Univ.eq Univ (add additive_identity X) X.
+∀H2:∀X:Univ.eq Univ (add X additive_identity) X.
+∀H3:∀X:Univ.eq Univ (multiply multiplicative_identity X) X.
+∀H4:∀X:Univ.eq Univ (multiply X multiplicative_identity) X.
+∀H5:∀X:Univ.eq Univ (multiply (inverse X) X) additive_identity.
+∀H6:∀X:Univ.eq Univ (multiply X (inverse X)) additive_identity.
+∀H7:∀X:Univ.eq Univ (add (inverse X) X) multiplicative_identity.
+∀H8:∀X:Univ.eq Univ (add X (inverse X)) multiplicative_identity.
+∀H9:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y Z)) (add (multiply X Y) (multiply X Z)).
+∀H10:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X Y) Z) (add (multiply X Z) (multiply Y Z)).
+∀H11:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add X (multiply Y Z)) (multiply (add X Y) (add X Z)).
+∀H12:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add (multiply X Y) Z) (multiply (add X Z) (add Y Z)).
+∀H13:∀X:Univ.∀Y:Univ.eq Univ (multiply X Y) (multiply Y X).
+∀H14:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).eq Univ (multiply x z) x
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#add.
+#additive_identity.
+#inverse.
+#multiplicative_identity.
+#multiply.
+#x.
+#y.
+#z.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: BOO018-4.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO018-4 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Problem : Inverse of multiplicative identity = Additive identity *)
+
+(* Version : [Ver94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Ver94] Veroff (1994), Problem Set *)
+
+(* Source : [Ver94] *)
+
+(* Names : TG [Ver94] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 9 ( 0 non-Horn; 9 unit; 1 RR) *)
+
+(* Number of atoms : 9 ( 9 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 14 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* Bugfixes : v1.2.1 - Clause prove_inverse_of_1_is_0 fixed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include boolean algebra axioms for equality formulation *)
+
+(* Inclusion of: Axioms/BOO004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Axioms : Boolean algebra (equality) axioms *)
+
+(* Version : [Ver94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Ver94] Veroff (1994), Problem Set *)
+
+(* Source : [Ver94] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 14 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_inverse_of_1_is_0:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀additive_identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiplicative_identity:Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.eq Univ (multiply X (inverse X)) additive_identity.
+∀H1:∀X:Univ.eq Univ (add X (inverse X)) multiplicative_identity.
+∀H2:∀X:Univ.eq Univ (multiply X multiplicative_identity) X.
+∀H3:∀X:Univ.eq Univ (add X additive_identity) X.
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y Z)) (add (multiply X Y) (multiply X Z)).
+∀H5:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add X (multiply Y Z)) (multiply (add X Y) (add X Z)).
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (multiply X Y) (multiply Y X).
+∀H7:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).eq Univ (inverse multiplicative_identity) additive_identity
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#add.
+#additive_identity.
+#inverse.
+#multiplicative_identity.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: BOO019-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO019-1 : TPTP v3.2.0. Released v1.2.0. *)
+
+(* Domain : Boolean Algebra (Ternary) *)
+
+(* Problem : Prove the independance of Ternary Boolean algebra axiom *)
+
+(* Version : Especial. *)
+
+(* English : *)
+
+(* Refs : [Win82] Winker (1982), Generation and Verification of Finite M *)
+
+(* : [BCP94] Bourely et al. (1994), A Method for Building Models Au *)
+
+(* : [Pel98] Peltier (1998), A New Method for Automated Finite Mode *)
+
+(* Source : [BCP94] *)
+
+(* Names : A1 [Win82] *)
+
+(* : Example 4 [BCP94] *)
+
+(* : 4.2.1 [Pel98] *)
+
+(* Status : Satisfiable *)
+
+(* Rating : 0.33 v3.2.0, 0.67 v3.1.0, 0.33 v2.4.0, 0.67 v2.2.1, 0.75 v2.2.0, 0.67 v2.1.0, 1.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 1 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 2 constant; 0-3 arity) *)
+
+(* Number of variables : 11 ( 1 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Thought to be satisfiable. *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_ternary_multiply_1_independant:
+ ∀Univ:Type.∀V:Univ.∀W:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.∀_:Univ.Univ.
+∀x:Univ.
+∀y:Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (multiply X Y (inverse Y)) X.
+∀H1:∀X:Univ.∀Y:Univ.eq Univ (multiply (inverse Y) Y X) X.
+∀H2:∀X:Univ.∀Y:Univ.eq Univ (multiply X X Y) X.
+∀H3:∀V:Univ.∀W:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply V W X) Y (multiply V W Z)) (multiply V W (multiply X Y Z)).eq Univ (multiply y x x) x
+.
+#Univ.
+#V.
+#W.
+#X.
+#Y.
+#Z.
+#inverse.
+#multiply.
+#x.
+#y.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: BOO021-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO021-1 : TPTP v3.2.0. Released v2.2.0. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Problem : A Basis for Boolean Algebra *)
+
+(* Version : [MP96] (equality) axioms. *)
+
+(* English : This ntheorem starts with a (self-dual independent) basis_ *)
+
+(* for Boolean algebra and derives commutativity of product. *)
+
+(* Refs : [McC98] McCune (1998), Email to G. Sutcliffe *)
+
+(* : [MP96] McCune & Padmanabhan (1996), Automated Deduction in Eq *)
+
+(* Source : [McC98] *)
+
+(* Names : DUAL-BA-1 [MP96] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.07 v3.1.0, 0.11 v2.7.0, 0.00 v2.2.1 *)
+
+(* Syntax : Number of clauses : 7 ( 0 non-Horn; 7 unit; 1 RR) *)
+
+(* Number of atoms : 7 ( 7 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 12 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : The other part of this problem is to prove associativity. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Boolean Algebra: *)
+
+(* ----Denial of conclusion: *)
+ntheorem prove_commutativity_of_multiply:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀n0:Univ.
+∀n1:Univ.
+∀H0:∀X:Univ.eq Univ (multiply X (inverse X)) n0.
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add X (multiply Y Z)) (multiply (add Y X) (add Z X)).
+∀H2:∀X:Univ.∀Y:Univ.eq Univ (add (multiply X Y) Y) Y.
+∀H3:∀X:Univ.eq Univ (add X (inverse X)) n1.
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y Z)) (add (multiply Y X) (multiply Z X)).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (multiply (add X Y) Y) Y.eq Univ (multiply b a) (multiply a b)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#add.
+#b.
+#inverse.
+#multiply.
+#n0.
+#n1.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+nauto by H0,H1,H2,H3,H4,H5;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: BOO022-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO022-1 : TPTP v3.2.0. Released v2.2.0. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Problem : A Basis for Boolean Algebra *)
+
+(* Version : [MP96] (equality) axioms. *)
+
+(* English : This ntheorem starts with a (self-dual independent) 6-basis *)
+
+(* for Boolean algebra and derives associativity of product. *)
+
+(* Refs : [McC98] McCune (1998), Email to G. Sutcliffe *)
+
+(* : [MP96] McCune & Padmanabhan (1996), Automated Deduction in Eq *)
+
+(* Source : [McC98] *)
+
+(* Names : DUAL-BA-1 [MP96] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.14 v3.2.0, 0.07 v3.1.0, 0.22 v2.7.0, 0.00 v2.2.1 *)
+
+(* Syntax : Number of clauses : 7 ( 0 non-Horn; 7 unit; 1 RR) *)
+
+(* Number of atoms : 7 ( 7 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 8 ( 5 constant; 0-2 arity) *)
+
+(* Number of variables : 12 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : The other part of this problem is to prove commutativity. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Boolean Algebra: *)
+
+(* ----Denial of conclusion: *)
+ntheorem prove_associativity_of_multiply:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀c:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀n0:Univ.
+∀n1:Univ.
+∀H0:∀X:Univ.eq Univ (multiply X (inverse X)) n0.
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add X (multiply Y Z)) (multiply (add Y X) (add Z X)).
+∀H2:∀X:Univ.∀Y:Univ.eq Univ (add (multiply X Y) Y) Y.
+∀H3:∀X:Univ.eq Univ (add X (inverse X)) n1.
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y Z)) (add (multiply Y X) (multiply Z X)).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (multiply (add X Y) Y) Y.eq Univ (multiply (multiply a b) c) (multiply a (multiply b c))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#add.
+#b.
+#c.
+#inverse.
+#multiply.
+#n0.
+#n1.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+nauto by H0,H1,H2,H3,H4,H5;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: BOO023-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO023-1 : TPTP v3.2.0. Released v2.2.0. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Problem : Half of Padmanabhan's 6-basis with Pixley, part 1. *)
+
+(* Version : [MP96] (equality) axioms : Especial. *)
+
+(* English : Part 1 (of 3) of the proof that half of Padmanaban's self-dual *)
+
+(* independent 6-basis for Boolean Algebra, together with a Pixley *)
+
+(* polynomial, is a basis for Boolean algebra. *)
+
+(* Refs : [McC98] McCune (1998), Email to G. Sutcliffe *)
+
+(* : [MP96] McCune & Padmanabhan (1996), Automated Deduction in Eq *)
+
+(* Source : [McC98] *)
+
+(* Names : DUAL-BA-2-a [MP96] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.50 v3.1.0, 0.33 v2.7.0, 0.36 v2.6.0, 0.17 v2.5.0, 0.00 v2.2.1 *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 1 RR) *)
+
+(* Number of atoms : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 8 ( 4 constant; 0-3 arity) *)
+
+(* Number of variables : 15 ( 2 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Half of Padmanabhan's self-dual independent 6-basis for Boolean Algebra: *)
+
+(* ----pixley(X,Y,Z) is a Pixley polynomial: *)
+
+(* ----Denial of conclusion: *)
+ntheorem prove_add_multiply_property:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀c:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀n1:Univ.
+∀pixley:∀_:Univ.∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (pixley X Y X) X.
+∀H1:∀X:Univ.∀Y:Univ.eq Univ (pixley X Y Y) X.
+∀H2:∀X:Univ.∀Y:Univ.eq Univ (pixley X X Y) Y.
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (pixley X Y Z) (add (multiply X (inverse Y)) (add (multiply X Z) (multiply (inverse Y) Z))).
+∀H4:∀X:Univ.eq Univ (add X (inverse X)) n1.
+∀H5:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y Z)) (add (multiply Y X) (multiply Z X)).
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (multiply (add X Y) Y) Y.eq Univ (add a (multiply b c)) (multiply (add a b) (add a c))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#add.
+#b.
+#c.
+#inverse.
+#multiply.
+#n1.
+#pixley.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+nauto by H0,H1,H2,H3,H4,H5,H6;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: BOO024-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO024-1 : TPTP v3.2.0. Released v2.2.0. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Problem : Half of Padmanabhan's 6-basis with Pixley, part 2. *)
+
+(* Version : [MP96] (equality) axioms : Especial. *)
+
+(* English : Part 2 (of 3) of the proof that half of Padmanaban's self-dual *)
+
+(* independent 6-basis for Boolean Algebra, together with a Pixley *)
+
+(* polynomial, is a basis for Boolean algebra. *)
+
+(* Refs : [McC98] McCune (1998), Email to G. Sutcliffe *)
+
+(* : [MP96] McCune & Padmanabhan (1996), Automated Deduction in Eq *)
+
+(* Source : [McC98] *)
+
+(* Names : DUAL-BA-2-b [MP96] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.2.1 *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 1 RR) *)
+
+(* Number of atoms : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 3 constant; 0-3 arity) *)
+
+(* Number of variables : 15 ( 2 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Half of Padmanabhan's self-dual independent 6-basis for Boolean Algebra: *)
+
+(* ----pixley(X,Y,Z) is a Pixley polynomial: *)
+
+(* ----Denial of conclusion: *)
+ntheorem prove_add_multiply:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀n1:Univ.
+∀pixley:∀_:Univ.∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (pixley X Y X) X.
+∀H1:∀X:Univ.∀Y:Univ.eq Univ (pixley X Y Y) X.
+∀H2:∀X:Univ.∀Y:Univ.eq Univ (pixley X X Y) Y.
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (pixley X Y Z) (add (multiply X (inverse Y)) (add (multiply X Z) (multiply (inverse Y) Z))).
+∀H4:∀X:Univ.eq Univ (add X (inverse X)) n1.
+∀H5:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y Z)) (add (multiply Y X) (multiply Z X)).
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (multiply (add X Y) Y) Y.eq Univ (add (multiply a b) b) b
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#add.
+#b.
+#inverse.
+#multiply.
+#n1.
+#pixley.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+nauto by H0,H1,H2,H3,H4,H5,H6;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: BOO025-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO025-1 : TPTP v3.2.0. Released v2.2.0. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Problem : Half of Padmanabhan's 6-basis with Pixley, part 3. *)
+
+(* Version : [MP96] (equality) axioms : Especial. *)
+
+(* English : Part 3 (of 3) of the proof that half of Padmanaban's self-dual *)
+
+(* independent 6-basis for Boolean Algebra, together with a Pixley *)
+
+(* polynomial, is a basis for Boolean algebra. *)
+
+(* Refs : [McC98] McCune (1998), Email to G. Sutcliffe *)
+
+(* : [MP96] McCune & Padmanabhan (1996), Automated Deduction in Eq *)
+
+(* Source : [McC98] *)
+
+(* Names : DUAL-BA-2-c [MP96] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.14 v3.2.0, 0.07 v3.1.0, 0.11 v2.7.0, 0.00 v2.2.1 *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 1 RR) *)
+
+(* Number of atoms : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 3 constant; 0-3 arity) *)
+
+(* Number of variables : 15 ( 2 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Half of Padmanabhan's self-dual independent 6-basis for Boolean Algebra: *)
+
+(* ----pixley(X,Y,Z) is a Pixley polynomial: *)
+
+(* ----Denial of conclusion: *)
+ntheorem prove_equal_identity:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀n1:Univ.
+∀pixley:∀_:Univ.∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (pixley X Y X) X.
+∀H1:∀X:Univ.∀Y:Univ.eq Univ (pixley X Y Y) X.
+∀H2:∀X:Univ.∀Y:Univ.eq Univ (pixley X X Y) Y.
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (pixley X Y Z) (add (multiply X (inverse Y)) (add (multiply X Z) (multiply (inverse Y) Z))).
+∀H4:∀X:Univ.eq Univ (add X (inverse X)) n1.
+∀H5:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y Z)) (add (multiply Y X) (multiply Z X)).
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (multiply (add X Y) Y) Y.eq Univ (multiply b (inverse b)) (multiply a (inverse a))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#add.
+#b.
+#inverse.
+#multiply.
+#n1.
+#pixley.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+nauto by H0,H1,H2,H3,H4,H5,H6;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: BOO026-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO026-1 : TPTP v3.2.0. Released v2.2.0. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Problem : Absorption from self-dual independent 2-basis *)
+
+(* Version : [MP96] (eqiality) axioms : Especial. *)
+
+(* English : This is part of a proof that there exists an independent self-dual *)
+
+(* 2-basis for Boolean Algebra. You may note that the basis *)
+
+(* below has more than 2 equations; but don't worry, it can be *)
+
+(* reduced to 2 (large) equations by Pixley reduction. *)
+
+(* Refs : [Wos98] Wos (1998), Automating the Search for Elegant Proofs *)
+
+(* : [McC98] McCune (1998), Email to G. Sutcliffe *)
+
+(* : [MP96] McCune & Padmanabhan (1996), Automated Deduction in Eq *)
+
+(* Source : [McC98] *)
+
+(* Names : DUAL-BA-3 [MP96] *)
+
+(* : DUAL-BA-3 [Wos98] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.07 v3.1.0, 0.11 v2.7.0, 0.00 v2.2.1 *)
+
+(* Syntax : Number of clauses : 11 ( 0 non-Horn; 11 unit; 1 RR) *)
+
+(* Number of atoms : 11 ( 11 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 20 ( 0 singleton) *)
+
+(* Maximal term depth : 5 ( 3 average) *)
+
+(* Comments : The original proof has 816 steps. Wos later found a 99-step *)
+
+(* proof. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Two Boolean algebra properties and their duals: *)
+
+(* ----Expanded Pixley properties and their duals: *)
+
+(* ----Denial of the conclusion: *)
+ntheorem prove_multiply_add:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀n0:Univ.
+∀n1:Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (multiply (add X (inverse Y)) (multiply (add X X) (add (inverse Y) X))) X.
+∀H1:∀X:Univ.∀Y:Univ.eq Univ (multiply (add X (inverse Y)) (multiply (add X Y) (add (inverse Y) Y))) X.
+∀H2:∀X:Univ.∀Y:Univ.eq Univ (multiply (add X (inverse X)) (multiply (add X Y) (add (inverse X) Y))) Y.
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (add (multiply X (inverse Y)) (add (multiply X X) (multiply (inverse Y) X))) X.
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (add (multiply X (inverse Y)) (add (multiply X Y) (multiply (inverse Y) Y))) X.
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (add (multiply X (inverse X)) (add (multiply X Y) (multiply (inverse X) Y))) Y.
+∀H6:∀X:Univ.eq Univ (multiply X (inverse X)) n0.
+∀H7:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add X (multiply Y Z)) (multiply (add Y X) (add Z X)).
+∀H8:∀X:Univ.eq Univ (add X (inverse X)) n1.
+∀H9:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y Z)) (add (multiply Y X) (multiply Z X)).eq Univ (multiply (add a b) b) b
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#add.
+#b.
+#inverse.
+#multiply.
+#n0.
+#n1.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: BOO027-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO027-1 : TPTP v3.2.0. Released v2.2.0. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Problem : Independence of self-dual 2-basis. *)
+
+(* Version : [MP96] (eqiality) axioms : Especial. *)
+
+(* English : Show that half of the self-dual 2-basis in DUAL-BA-3 is not *)
+
+(* a basis for Boolean Algebra. *)
+
+(* Refs : [McC98] McCune (1998), Email to G. Sutcliffe *)
+
+(* : [MP96] McCune & Padmanabhan (1996), Automated Deduction in Eq *)
+
+(* Source : [McC98] *)
+
+(* Names : DUAL-BA-4 [MP96] *)
+
+(* Status : Satisfiable *)
+
+(* Rating : 0.00 v3.2.0, 0.33 v3.1.0, 0.00 v2.2.1 *)
+
+(* Syntax : Number of clauses : 6 ( 0 non-Horn; 6 unit; 1 RR) *)
+
+(* Number of atoms : 6 ( 6 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 10 ( 0 singleton) *)
+
+(* Maximal term depth : 5 ( 3 average) *)
+
+(* Comments : There is a 2-element model. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Two properties of Boolean algebra: *)
+
+(* ----Pixley properties: *)
+
+(* ----Denial of a property of Boolean Algebra: *)
+ntheorem prove_idempotence:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀one:Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (add (multiply X (inverse Y)) (add (multiply X X) (multiply (inverse Y) X))) X.
+∀H1:∀X:Univ.∀Y:Univ.eq Univ (add (multiply X (inverse Y)) (add (multiply X Y) (multiply (inverse Y) Y))) X.
+∀H2:∀X:Univ.∀Y:Univ.eq Univ (add (multiply X (inverse X)) (add (multiply X Y) (multiply (inverse X) Y))) Y.
+∀H3:∀X:Univ.eq Univ (add X (inverse X)) one.
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y Z)) (add (multiply Y X) (multiply Z X)).eq Univ (add a a) a
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#add.
+#inverse.
+#multiply.
+#one.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+nauto by H0,H1,H2,H3,H4;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: BOO028-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO028-1 : TPTP v3.2.0. Released v2.2.0. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Problem : Self-dual 2-basis from majority reduction, part 1. *)
+
+(* Version : [MP96] (equality) axioms : Especial. *)
+
+(* English : This is part of a proof that there exists an independent *)
+
+(* self-dual-2-basis for Boolean algebra by majority reduction. *)
+
+(* Refs : [McC98] McCune (1998), Email to G. Sutcliffe *)
+
+(* : [MP96] McCune & Padmanabhan (1996), Automated Deduction in Eq *)
+
+(* Source : [McC98] *)
+
+(* Names : DUAL-BA-5-a [MP96] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.14 v3.2.0, 0.07 v3.1.0, 0.00 v2.7.0, 0.09 v2.6.0, 0.00 v2.2.1 *)
+
+(* Syntax : Number of clauses : 11 ( 0 non-Horn; 11 unit; 1 RR) *)
+
+(* Number of atoms : 11 ( 11 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 26 ( 8 singleton) *)
+
+(* Maximal term depth : 4 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Properties L1, L3, and B1 of Boolean Algebra: *)
+
+(* ----The corresponding dual properties L2, L4, and B2. *)
+
+(* ----Associativity and Commutativity of both operations: *)
+
+(* ----Denial of conclusion: *)
+ntheorem prove_multiply_add_property:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀c:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add (add X Y) Z) (add X (add Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.eq Univ (multiply X Y) (multiply Y X).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (add (multiply X Y) (multiply X (inverse Y))) X.
+∀H5:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply (add X Y) (add Y Z)) Y) Y.
+∀H6:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y (add X Z))) X.
+∀H7:∀X:Univ.∀Y:Univ.eq Univ (multiply (add X Y) (add X (inverse Y))) X.
+∀H8:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add (add (multiply X Y) (multiply Y Z)) Y) Y.
+∀H9:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add X (multiply Y (multiply X Z))) X.eq Univ (multiply a (add b c)) (add (multiply b a) (multiply c a))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#add.
+#b.
+#c.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: BOO029-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO029-1 : TPTP v3.2.0. Released v2.2.0. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Problem : Self-dual 2-basis from majority reduction, part 3. *)
+
+(* Version : [MP96] (equality) axioms : Especial. *)
+
+(* English : This is part of a proof that there exists an independent *)
+
+(* self-dual-2-basis for Boolean algebra by majority reduction. *)
+
+(* Refs : [McC98] McCune (1998), Email to G. Sutcliffe *)
+
+(* : [MP96] McCune & Padmanabhan (1996), Automated Deduction in Eq *)
+
+(* Source : [McC98] *)
+
+(* Names : DUAL-BA-5-c [MP96] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.2.1 *)
+
+(* Syntax : Number of clauses : 11 ( 0 non-Horn; 11 unit; 1 RR) *)
+
+(* Number of atoms : 11 ( 11 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 26 ( 8 singleton) *)
+
+(* Maximal term depth : 4 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Properties L1, L3, and B1 of Boolean Algebra: *)
+
+(* ----The corresponding dual properties L2, L4, and B2. *)
+
+(* ----Associativity and Commutativity of both operations: *)
+
+(* ----Denial of conclusion: *)
+ntheorem prove_equal_inverse:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add (add X Y) Z) (add X (add Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.eq Univ (multiply X Y) (multiply Y X).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (add (multiply X Y) (multiply X (inverse Y))) X.
+∀H5:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply (add X Y) (add Y Z)) Y) Y.
+∀H6:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y (add X Z))) X.
+∀H7:∀X:Univ.∀Y:Univ.eq Univ (multiply (add X Y) (add X (inverse Y))) X.
+∀H8:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add (add (multiply X Y) (multiply Y Z)) Y) Y.
+∀H9:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add X (multiply Y (multiply X Z))) X.eq Univ (add b (inverse b)) (add a (inverse a))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#add.
+#b.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: BOO030-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO030-1 : TPTP v3.2.0. Released v2.2.0. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Problem : Independence of a BA 2-basis by majority reduction. *)
+
+(* Version : [MP96] (equality) axioms : Especial. *)
+
+(* English : This shows that the self-dual 2-basis for Boolean algebra *)
+
+(* (majority reduction) of problem DUAL-BA-5 is independent, *)
+
+(* in particular, that half of the 2-basis is not a basis. *)
+
+(* Refs : [McC98] McCune (1998), Email to G. Sutcliffe *)
+
+(* : [MP96] McCune & Padmanabhan (1996), Automated Deduction in Eq *)
+
+(* Source : [McC98] *)
+
+(* Names : DUAL-BA-6 [MP96] *)
+
+(* Status : Satisfiable *)
+
+(* Rating : 0.33 v3.2.0, 0.67 v3.1.0, 0.33 v2.4.0, 0.67 v2.3.0, 1.00 v2.2.1 *)
+
+(* Syntax : Number of clauses : 7 ( 0 non-Horn; 7 unit; 1 RR) *)
+
+(* Number of atoms : 7 ( 7 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 14 ( 5 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : There is a 2-element model. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Properties L1, L3, and B1 of Boolean Algebra: *)
+
+(* ----Majority reduction properties: *)
+
+(* ----Denial of a property of Boolean Algebra. *)
+ntheorem prove_inverse_involution:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (multiply (add (multiply X Y) Y) (add X Y)) Y.
+∀H1:∀X:Univ.∀Y:Univ.eq Univ (multiply (add (multiply X X) Y) (add X X)) X.
+∀H2:∀X:Univ.∀Y:Univ.eq Univ (multiply (add (multiply X Y) X) (add X Y)) X.
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (multiply (add X Y) (add X (inverse Y))) X.
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add (add (multiply X Y) (multiply Y Z)) Y) Y.
+∀H5:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add X (multiply Y (multiply X Z))) X.eq Univ (inverse (inverse a)) a
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#add.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+nauto by H0,H1,H2,H3,H4,H5;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: BOO031-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO031-1 : TPTP v3.2.0. Released v2.2.0. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Problem : Dual BA 3-basis, proof of distributivity. *)
+
+(* Version : [MP96] (equality) axioms : Especial. *)
+
+(* English : This is part of a proof of the existence of a self-dual *)
+
+(* 3-basis for Boolean algebra by majority reduction. *)
+
+(* Refs : [McC98] McCune (1998), Email to G. Sutcliffe *)
+
+(* : [MP96] McCune & Padmanabhan (1996), Automated Deduction in Eq *)
+
+(* Source : [McC98] *)
+
+(* Names : DUAL-BA-8-a [MP96] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.29 v3.2.0, 0.21 v3.1.0, 0.22 v2.7.0, 0.09 v2.6.0, 0.17 v2.5.0, 0.00 v2.4.0, 0.00 v2.2.1 *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 1 RR) *)
+
+(* Number of atoms : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 8 ( 5 constant; 0-2 arity) *)
+
+(* Number of variables : 27 ( 8 singleton) *)
+
+(* Maximal term depth : 4 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Self-dual distributivity: *)
+
+(* ----3 properties of Boolean algebra and the corresponding duals. *)
+
+(* ----Existence of 0 and 1. *)
+
+(* ----Associativity of the 2 operations. *)
+
+(* ----Denial of conclusion: *)
+ntheorem prove_multiply_add_property:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀c:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀n0:Univ.
+∀n1:Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add (add X Y) Z) (add X (add Y Z)).
+∀H2:∀X:Univ.eq Univ (multiply X (inverse X)) n0.
+∀H3:∀X:Univ.eq Univ (add X (inverse X)) n1.
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (add (multiply X (inverse X)) Y) Y.
+∀H5:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply (add X Y) (add Y Z)) Y) Y.
+∀H6:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y (add X Z))) X.
+∀H7:∀X:Univ.∀Y:Univ.eq Univ (multiply (add X (inverse X)) Y) Y.
+∀H8:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add (add (multiply X Y) (multiply Y Z)) Y) Y.
+∀H9:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add X (multiply Y (multiply X Z))) X.
+∀H10:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add (multiply X Y) (add (multiply Y Z) (multiply Z X))) (multiply (add X Y) (multiply (add Y Z) (add Z X))).eq Univ (multiply a (add b c)) (add (multiply b a) (multiply c a))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#add.
+#b.
+#c.
+#inverse.
+#multiply.
+#n0.
+#n1.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: BOO032-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO032-1 : TPTP v3.2.0. Released v2.2.0. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Problem : Independence of a system of Boolean algebra *)
+
+(* Version : [MP96] (equality) axioms : Especial. *)
+
+(* English : This is part of a proof that a self-dual 3-basis for Boolean *)
+
+(* algebra is independent. *)
+
+(* Refs : [McC98] McCune (1998), Email to G. Sutcliffe *)
+
+(* : [MP96] McCune & Padmanabhan (1996), Automated Deduction in Eq *)
+
+(* Source : [McC98] *)
+
+(* Names : DUAL-BA-9 [MP96] *)
+
+(* Status : Satisfiable *)
+
+(* Rating : 0.33 v3.2.0, 0.67 v3.1.0, 0.33 v2.5.0, 0.67 v2.4.0, 0.67 v2.3.0, 1.00 v2.2.1 *)
+
+(* Syntax : Number of clauses : 13 ( 0 non-Horn; 13 unit; 1 RR) *)
+
+(* Number of atoms : 13 ( 13 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 28 ( 10 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : The smallest model has 5 elements. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----3 properties of Boolean algebra and the corresponding duals. *)
+
+(* ----Majority polynomials: *)
+
+(* ----Duals of majority polynomials: *)
+
+(* ----A propery of Boolean Algebra fails to hold. *)
+ntheorem prove_inverse_involution:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (multiply (add (multiply X Y) Y) (add X Y)) Y.
+∀H1:∀X:Univ.∀Y:Univ.eq Univ (multiply (add (multiply X X) Y) (add X X)) X.
+∀H2:∀X:Univ.∀Y:Univ.eq Univ (multiply (add (multiply X Y) X) (add X Y)) X.
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (add (multiply (add X Y) Y) (multiply X Y)) Y.
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (add (multiply (add X X) Y) (multiply X X)) X.
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (add (multiply (add X Y) X) (multiply X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (add (multiply X (inverse X)) Y) Y.
+∀H7:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply (add X Y) (add Y Z)) Y) Y.
+∀H8:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y (add X Z))) X.
+∀H9:∀X:Univ.∀Y:Univ.eq Univ (multiply (add X (inverse X)) Y) Y.
+∀H10:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add (add (multiply X Y) (multiply Y Z)) Y) Y.
+∀H11:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add X (multiply Y (multiply X Z))) X.eq Univ (inverse (inverse a)) a
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#add.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: BOO033-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO033-1 : TPTP v3.2.0. Released v2.2.0. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Problem : Independence of a system of Boolean algebra. *)
+
+(* Version : [MP96] (equality) axioms : Especial. *)
+
+(* English : This is part of a proof that a self-dual 3-basis for *)
+
+(* Boolean algebra is independent. *)
+
+(* Refs : [McC98] McCune (1998), Email to G. Sutcliffe *)
+
+(* : [MP96] McCune & Padmanabhan (1996), Automated Deduction in Eq *)
+
+(* Source : [McC98] *)
+
+(* Names : DUAL-BA-10 [MP96] *)
+
+(* Status : Satisfiable *)
+
+(* Rating : 0.33 v3.2.0, 0.67 v3.1.0, 0.33 v2.4.0, 0.67 v2.3.0, 1.00 v2.2.1 *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 1 RR) *)
+
+(* Number of atoms : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 17 ( 5 singleton) *)
+
+(* Maximal term depth : 4 ( 3 average) *)
+
+(* Comments : There is a model of size 2. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Self-dual distributivity: *)
+
+(* ----3 properties of Boolean algebra and the corresponding duals. *)
+
+(* ----Majority polynomials: *)
+
+(* ----A simple propery of Boolean Algebra fails to hold. *)
+ntheorem prove_inverse_involution:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (multiply (add (multiply X Y) Y) (add X Y)) Y.
+∀H1:∀X:Univ.∀Y:Univ.eq Univ (multiply (add (multiply X X) Y) (add X X)) X.
+∀H2:∀X:Univ.∀Y:Univ.eq Univ (multiply (add (multiply X Y) X) (add X Y)) X.
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (multiply (add X (inverse X)) Y) Y.
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add (add (multiply X Y) (multiply Y Z)) Y) Y.
+∀H5:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add X (multiply Y (multiply X Z))) X.
+∀H6:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add (multiply X Y) (add (multiply Y Z) (multiply Z X))) (multiply (add X Y) (multiply (add Y Z) (add Z X))).eq Univ (inverse (inverse a)) a
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#add.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+nauto by H0,H1,H2,H3,H4,H5,H6;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: BOO034-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO034-1 : TPTP v3.2.0. Released v2.2.0. *)
+
+(* Domain : Boolean Algebra (Ternary) *)
+
+(* Problem : Ternary Boolean Algebra Single axiom is sound. *)
+
+(* Version : [MP96] (equality) axioms. *)
+
+(* English : We show that that an equation (which turns out to be a single *)
+
+(* axiom for TBA) can be derived from the axioms of TBA. *)
+
+(* Refs : [McC98] McCune (1998), Email to G. Sutcliffe *)
+
+(* : [MP96] McCune & Padmanabhan (1996), Automated Deduction in Eq *)
+
+(* Source : [McC98] *)
+
+(* Names : TBA-1-a [MP96] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.29 v3.2.0, 0.21 v3.1.0, 0.11 v2.7.0, 0.27 v2.6.0, 0.33 v2.5.0, 0.00 v2.2.1 *)
+
+(* Syntax : Number of clauses : 6 ( 0 non-Horn; 6 unit; 1 RR) *)
+
+(* Number of atoms : 6 ( 6 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 9 ( 7 constant; 0-3 arity) *)
+
+(* Number of variables : 13 ( 2 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include ternary Boolean algebra axioms *)
+
+(* Inclusion of: Axioms/BOO001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Algebra (Ternary Boolean) *)
+
+(* Axioms : Ternary Boolean algebra (equality) axioms *)
+
+(* Version : [OTTER] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* : [Win82] Winker (1982), Generation and Verification of Finite M *)
+
+(* Source : [OTTER] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 0 RR) *)
+
+(* Number of literals : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 1-3 arity) *)
+
+(* Number of variables : 13 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : These axioms appear in [Win82], in which ternary_multiply_1 is *)
+
+(* shown to be independant. *)
+
+(* : These axioms are also used in [Wos88], p.222. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Denial of single axiom: *)
+ntheorem prove_single_axiom:
+ ∀Univ:Type.∀V:Univ.∀W:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀d:Univ.
+∀e:Univ.
+∀f:Univ.
+∀g:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (multiply X Y (inverse Y)) X.
+∀H1:∀X:Univ.∀Y:Univ.eq Univ (multiply (inverse Y) Y X) X.
+∀H2:∀X:Univ.∀Y:Univ.eq Univ (multiply X X Y) X.
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (multiply Y X X) X.
+∀H4:∀V:Univ.∀W:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply V W X) Y (multiply V W Z)) (multiply V W (multiply X Y Z)).eq Univ (multiply (multiply a (inverse a) b) (inverse (multiply (multiply c d e) f (multiply c d g))) (multiply d (multiply g f e) c)) b
+.
+#Univ.
+#V.
+#W.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#d.
+#e.
+#f.
+#g.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+nauto by H0,H1,H2,H3,H4;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: BOO036-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO036-1 : TPTP v3.2.0. Released v2.5.0. *)
+
+(* Domain : Algebra (Ternary Boolean) *)
+
+(* Problem : Ternary Boolean algebra (equality) axioms *)
+
+(* Version : [OTTER] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* : [Win82] Winker (1982), Generation and Verification of Finite M *)
+
+(* Source : [OTTER] *)
+
+(* Names : *)
+
+(* Status : Satisfiable *)
+
+(* Rating : 0.00 v3.2.0, 0.33 v3.1.0, 0.00 v2.5.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 0 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 1-3 arity) *)
+
+(* Number of variables : 13 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Ternary Boolean algebra (equality) axioms *)
+
+(* Inclusion of: Axioms/BOO001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Algebra (Ternary Boolean) *)
+
+(* Axioms : Ternary Boolean algebra (equality) axioms *)
+
+(* Version : [OTTER] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* : [Win82] Winker (1982), Generation and Verification of Finite M *)
+
+(* Source : [OTTER] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 0 RR) *)
+
+(* Number of literals : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 1-3 arity) *)
+
+(* Number of variables : 13 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : These axioms appear in [Win82], in which ternary_multiply_1 is *)
+
+(* shown to be independant. *)
+
+(* : These axioms are also used in [Wos88], p.222. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: BOO037-2.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO037-2 : TPTP v3.2.0. Released v2.5.0. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Problem : Boolean algebra (equality) axioms *)
+
+(* Version : [ANL] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : Satisfiable *)
+
+(* Rating : 0.00 v3.2.0, 0.33 v3.1.0, 0.00 v2.5.0 *)
+
+(* Syntax : Number of clauses : 14 ( 0 non-Horn; 14 unit; 0 RR) *)
+
+(* Number of atoms : 14 ( 14 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 24 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Boolean algebra (equality) axioms *)
+
+(* Inclusion of: Axioms/BOO003-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO003-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Axioms : Boolean algebra (equality) axioms *)
+
+(* Version : [ANL] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 14 ( 0 non-Horn; 14 unit; 0 RR) *)
+
+(* Number of literals : 14 ( 14 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 24 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: BOO037-3.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO037-3 : TPTP v3.2.0. Released v2.5.0. *)
+
+(* Domain : Algebra (Boolean) *)
+
+(* Problem : Boolean algebra (equality) axioms *)
+
+(* Version : [Ver94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Ver94] Veroff (1994), Problem Set *)
+
+(* Source : [Ver94] *)
+
+(* Names : *)
+
+(* Status : Satisfiable *)
+
+(* Rating : 0.00 v3.2.0, 0.33 v3.1.0, 0.00 v2.5.0 *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of atoms : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 14 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Boolean algebra (equality) axioms *)
+
+(* Inclusion of: Axioms/BOO004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Axioms : Boolean algebra (equality) axioms *)
+
+(* Version : [Ver94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Ver94] Veroff (1994), Problem Set *)
+
+(* Source : [Ver94] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 14 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: BOO067-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO067-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Boolean Algebra (Ternary) *)
+
+(* Problem : Ternary Boolean Algebra Single axiom is complete, part 1 *)
+
+(* Version : [MP96] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC98] McCune (1998), Email to G. Sutcliffe *)
+
+(* : [MP96] McCune & Padmanabhan (1996), Automated Deduction in Eq *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.21 v3.1.0, 0.33 v2.7.0, 0.27 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 5 constant; 0-3 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 5 ( 3 average) *)
+
+(* Comments : A UEQ part of BOO035-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_tba_axioms_1:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.∀E:Univ.∀F:Univ.∀G:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀d:Univ.
+∀e:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.∀E:Univ.∀F:Univ.∀G:Univ.eq Univ (multiply (multiply A (inverse A) B) (inverse (multiply (multiply C D E) F (multiply C D G))) (multiply D (multiply G F E) C)) B.eq Univ (multiply (multiply d e a) b (multiply d e c)) (multiply d e (multiply a b c))
+.
+#Univ.
+#A.
+#B.
+#C.
+#D.
+#E.
+#F.
+#G.
+#a.
+#b.
+#c.
+#d.
+#e.
+#inverse.
+#multiply.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: BOO068-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO068-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Boolean Algebra (Ternary) *)
+
+(* Problem : Ternary Boolean Algebra Single axiom is complete, part 2 *)
+
+(* Version : [MP96] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC98] McCune (1998), Email to G. Sutcliffe *)
+
+(* : [MP96] McCune & Padmanabhan (1996), Automated Deduction in Eq *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v3.1.0, 0.11 v2.7.0, 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 2 constant; 0-3 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : A UEQ part of BOO035-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_tba_axioms_2:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.∀E:Univ.∀F:Univ.∀G:Univ.
+∀a:Univ.
+∀b:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.∀E:Univ.∀F:Univ.∀G:Univ.eq Univ (multiply (multiply A (inverse A) B) (inverse (multiply (multiply C D E) F (multiply C D G))) (multiply D (multiply G F E) C)) B.eq Univ (multiply b a a) a
+.
+#Univ.
+#A.
+#B.
+#C.
+#D.
+#E.
+#F.
+#G.
+#a.
+#b.
+#inverse.
+#multiply.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: BOO069-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO069-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Boolean Algebra (Ternary) *)
+
+(* Problem : Ternary Boolean Algebra Single axiom is complete, part 3 *)
+
+(* Version : [MP96] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC98] McCune (1998), Email to G. Sutcliffe *)
+
+(* : [MP96] McCune & Padmanabhan (1996), Automated Deduction in Eq *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v3.1.0, 0.11 v2.7.0, 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 2 constant; 0-3 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : A UEQ part of BOO035-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_tba_axioms_3:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.∀E:Univ.∀F:Univ.∀G:Univ.
+∀a:Univ.
+∀b:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.∀E:Univ.∀F:Univ.∀G:Univ.eq Univ (multiply (multiply A (inverse A) B) (inverse (multiply (multiply C D E) F (multiply C D G))) (multiply D (multiply G F E) C)) B.eq Univ (multiply a b (inverse b)) a
+.
+#Univ.
+#A.
+#B.
+#C.
+#D.
+#E.
+#F.
+#G.
+#a.
+#b.
+#inverse.
+#multiply.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: BOO070-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO070-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Boolean Algebra (Ternary) *)
+
+(* Problem : Ternary Boolean Algebra Single axiom is complete, part 4 *)
+
+(* Version : [MP96] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC98] McCune (1998), Email to G. Sutcliffe *)
+
+(* : [MP96] McCune & Padmanabhan (1996), Automated Deduction in Eq *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v3.1.0, 0.11 v2.7.0, 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 2 constant; 0-3 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : A UEQ part of BOO035-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_tba_axioms_4:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.∀E:Univ.∀F:Univ.∀G:Univ.
+∀a:Univ.
+∀b:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.∀E:Univ.∀F:Univ.∀G:Univ.eq Univ (multiply (multiply A (inverse A) B) (inverse (multiply (multiply C D E) F (multiply C D G))) (multiply D (multiply G F E) C)) B.eq Univ (multiply a a b) a
+.
+#Univ.
+#A.
+#B.
+#C.
+#D.
+#E.
+#F.
+#G.
+#a.
+#b.
+#inverse.
+#multiply.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: BOO071-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO071-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Boolean Algebra (Ternary) *)
+
+(* Problem : Ternary Boolean Algebra Single axiom is complete, part 5 *)
+
+(* Version : [MP96] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC98] McCune (1998), Email to G. Sutcliffe *)
+
+(* : [MP96] McCune & Padmanabhan (1996), Automated Deduction in Eq *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v3.1.0, 0.11 v2.7.0, 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 2 constant; 0-3 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : A UEQ part of BOO035-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_tba_axioms_5:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.∀E:Univ.∀F:Univ.∀G:Univ.
+∀a:Univ.
+∀b:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.∀E:Univ.∀F:Univ.∀G:Univ.eq Univ (multiply (multiply A (inverse A) B) (inverse (multiply (multiply C D E) F (multiply C D G))) (multiply D (multiply G F E) C)) B.eq Univ (multiply (inverse b) b a) a
+.
+#Univ.
+#A.
+#B.
+#C.
+#D.
+#E.
+#F.
+#G.
+#a.
+#b.
+#inverse.
+#multiply.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: BOO072-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO072-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Problem : DN-1 is a single axiom for Boolean algebra, part 1 *)
+
+(* Version : [EF+02] axioms. *)
+
+(* English : *)
+
+(* Refs : [EF+02] Ernst et al. (2002), More First-order Test Problems in *)
+
+(* : [MV+02] McCune et al. (2002), Short Single Axioms for Boolean *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.07 v3.1.0, 0.00 v2.7.0, 0.09 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 4 ( 2 singleton) *)
+
+(* Maximal term depth : 9 ( 4 average) *)
+
+(* Comments : A UEQ part of BOO038-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem huntinton_1:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.
+∀a:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀inverse:∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.eq Univ (inverse (add (inverse (add (inverse (add A B)) C)) (inverse (add A (inverse (add (inverse C) (inverse (add C D)))))))) C.eq Univ (add b a) (add a b)
+.
+#Univ.
+#A.
+#B.
+#C.
+#D.
+#a.
+#add.
+#b.
+#inverse.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: BOO073-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO073-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Problem : DN-1 is a single axiom for Boolean algebra, part 2 *)
+
+(* Version : [EF+02] axioms. *)
+
+(* English : *)
+
+(* Refs : [EF+02] Ernst et al. (2002), More First-order Test Problems in *)
+
+(* : [MV+02] McCune et al. (2002), Short Single Axioms for Boolean *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.36 v3.1.0, 0.33 v2.7.0, 0.36 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 4 ( 2 singleton) *)
+
+(* Maximal term depth : 9 ( 4 average) *)
+
+(* Comments : A UEQ part of BOO038-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem huntinton_2:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.
+∀a:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀c:Univ.
+∀inverse:∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.eq Univ (inverse (add (inverse (add (inverse (add A B)) C)) (inverse (add A (inverse (add (inverse C) (inverse (add C D)))))))) C.eq Univ (add (add a b) c) (add a (add b c))
+.
+#Univ.
+#A.
+#B.
+#C.
+#D.
+#a.
+#add.
+#b.
+#c.
+#inverse.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: BOO074-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO074-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Problem : DN-1 is a single axiom for Boolean algebra, part 3 *)
+
+(* Version : [EF+02] axioms. *)
+
+(* English : *)
+
+(* Refs : [EF+02] Ernst et al. (2002), More First-order Test Problems in *)
+
+(* : [MV+02] McCune et al. (2002), Short Single Axioms for Boolean *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.07 v3.1.0, 0.00 v2.7.0, 0.18 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 4 ( 2 singleton) *)
+
+(* Maximal term depth : 9 ( 4 average) *)
+
+(* Comments : A UEQ part of BOO038-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem huntinton_3:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.
+∀a:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀inverse:∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.eq Univ (inverse (add (inverse (add (inverse (add A B)) C)) (inverse (add A (inverse (add (inverse C) (inverse (add C D)))))))) C.eq Univ (add (inverse (add (inverse a) b)) (inverse (add (inverse a) (inverse b)))) a
+.
+#Univ.
+#A.
+#B.
+#C.
+#D.
+#a.
+#add.
+#b.
+#inverse.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: BOO075-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO075-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Problem : Sh-1 is a single axiom for Boolean algebra, part 1 *)
+
+(* Version : [EF+02] axioms. *)
+
+(* English : *)
+
+(* Refs : [EF+02] Ernst et al. (2002), More First-order Test Problems in *)
+
+(* : [MV+02] McCune et al. (2002), Short Single Axioms for Boolean *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 3 ( 1 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : A UEQ part of BOO039-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_meredith_2_basis_1:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a:Univ.
+∀b:Univ.
+∀nand:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (nand (nand A (nand (nand B A) A)) (nand B (nand C A))) B.eq Univ (nand (nand a a) (nand b a)) a
+.
+#Univ.
+#A.
+#B.
+#C.
+#a.
+#b.
+#nand.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: BOO076-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO076-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Problem : Sh-1 is a single axiom for Boolean algebra, part 2 *)
+
+(* Version : [EF+02] axioms. *)
+
+(* English : *)
+
+(* Refs : [EF+02] Ernst et al. (2002), More First-order Test Problems in *)
+
+(* : [MV+02] McCune et al. (2002), Short Single Axioms for Boolean *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.71 v3.1.0, 0.78 v2.7.0, 0.91 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 3 ( 1 singleton) *)
+
+(* Maximal term depth : 5 ( 4 average) *)
+
+(* Comments : A UEQ part of BOO039-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_meredith_2_basis_2:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀nand:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (nand (nand A (nand (nand B A) A)) (nand B (nand C A))) B.eq Univ (nand a (nand b (nand a c))) (nand (nand (nand c b) b) a)
+.
+#Univ.
+#A.
+#B.
+#C.
+#a.
+#b.
+#c.
+#nand.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: BOO077-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO077-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Problem : Axiom C1 for Boolean algebra in the Sheffer stroke, part 1 *)
+
+(* Version : [EF+02] axioms. *)
+
+(* English : *)
+
+(* Refs : [EF+02] Ernst et al. (2002), More First-order Test Problems in *)
+
+(* : [MV+02] McCune et al. (2002), Short Single Axioms for Boolean *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unknown *)
+
+(* Rating : 1.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 3 ( 1 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : A UEQ part of BOO040-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_meredith_2_basis_1:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a:Univ.
+∀b:Univ.
+∀nand:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (nand (nand A (nand (nand B A) A)) (nand B (nand A C))) B.eq Univ (nand (nand a a) (nand b a)) a
+.
+#Univ.
+#A.
+#B.
+#C.
+#a.
+#b.
+#nand.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: BOO078-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO078-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Problem : Axiom C1 for Boolean algebra in the Sheffer stroke, part 2 *)
+
+(* Version : [EF+02] axioms. *)
+
+(* English : *)
+
+(* Refs : [EF+02] Ernst et al. (2002), More First-order Test Problems in *)
+
+(* : [MV+02] McCune et al. (2002), Short Single Axioms for Boolean *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unknown *)
+
+(* Rating : 1.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 3 ( 1 singleton) *)
+
+(* Maximal term depth : 5 ( 4 average) *)
+
+(* Comments : A UEQ part of BOO040-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_meredith_2_basis_2:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀nand:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (nand (nand A (nand (nand B A) A)) (nand B (nand A C))) B.eq Univ (nand a (nand b (nand a c))) (nand (nand (nand c b) b) a)
+.
+#Univ.
+#A.
+#B.
+#C.
+#a.
+#b.
+#c.
+#nand.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: BOO079-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO079-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Problem : Axiom C2 for Boolean algebra in the Sheffer stroke, part 1 *)
+
+(* Version : [EF+02] axioms. *)
+
+(* English : *)
+
+(* Refs : [EF+02] Ernst et al. (2002), More First-order Test Problems in *)
+
+(* : [MV+02] McCune et al. (2002), Short Single Axioms for Boolean *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unknown *)
+
+(* Rating : 1.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 3 ( 1 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : A UEQ part of BOO041-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_meredith_2_basis_1:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a:Univ.
+∀b:Univ.
+∀nand:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (nand (nand A (nand A (nand B A))) (nand B (nand C A))) B.eq Univ (nand (nand a a) (nand b a)) a
+.
+#Univ.
+#A.
+#B.
+#C.
+#a.
+#b.
+#nand.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: BOO080-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO080-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Problem : Axiom C2 for Boolean algebra in the Sheffer stroke, part 2 *)
+
+(* Version : [EF+02] axioms. *)
+
+(* English : *)
+
+(* Refs : [EF+02] Ernst et al. (2002), More First-order Test Problems in *)
+
+(* : [MV+02] McCune et al. (2002), Short Single Axioms for Boolean *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unknown *)
+
+(* Rating : 1.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 3 ( 1 singleton) *)
+
+(* Maximal term depth : 5 ( 4 average) *)
+
+(* Comments : A UEQ part of BOO041-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_meredith_2_basis_2:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀nand:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (nand (nand A (nand A (nand B A))) (nand B (nand C A))) B.eq Univ (nand a (nand b (nand a c))) (nand (nand (nand c b) b) a)
+.
+#Univ.
+#A.
+#B.
+#C.
+#a.
+#b.
+#c.
+#nand.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: BOO081-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO081-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Problem : Axiom C3 for Boolean algebra in the Sheffer stroke, part 1 *)
+
+(* Version : [EF+02] axioms. *)
+
+(* English : *)
+
+(* Refs : [EF+02] Ernst et al. (2002), More First-order Test Problems in *)
+
+(* : [MV+02] McCune et al. (2002), Short Single Axioms for Boolean *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unknown *)
+
+(* Rating : 1.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 3 ( 1 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : A UEQ part of BOO042-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_meredith_2_basis_1:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a:Univ.
+∀b:Univ.
+∀nand:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (nand (nand A (nand A (nand A B))) (nand B (nand C A))) B.eq Univ (nand (nand a a) (nand b a)) a
+.
+#Univ.
+#A.
+#B.
+#C.
+#a.
+#b.
+#nand.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: BOO082-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO082-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Problem : Axiom C3 for Boolean algebra in the Sheffer stroke, part 2 *)
+
+(* Version : [EF+02] axioms. *)
+
+(* English : *)
+
+(* Refs : [EF+02] Ernst et al. (2002), More First-order Test Problems in *)
+
+(* : [MV+02] McCune et al. (2002), Short Single Axioms for Boolean *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unknown *)
+
+(* Rating : 1.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 3 ( 1 singleton) *)
+
+(* Maximal term depth : 5 ( 4 average) *)
+
+(* Comments : A UEQ part of BOO042-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_meredith_2_basis_2:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀nand:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (nand (nand A (nand A (nand A B))) (nand B (nand C A))) B.eq Univ (nand a (nand b (nand a c))) (nand (nand (nand c b) b) a)
+.
+#Univ.
+#A.
+#B.
+#C.
+#a.
+#b.
+#c.
+#nand.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: BOO083-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO083-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Problem : Axiom C4 for Boolean algebra in the Sheffer stroke, part 1 *)
+
+(* Version : [EF+02] axioms. *)
+
+(* English : *)
+
+(* Refs : [EF+02] Ernst et al. (2002), More First-order Test Problems in *)
+
+(* : [MV+02] McCune et al. (2002), Short Single Axioms for Boolean *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unknown *)
+
+(* Rating : 1.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 3 ( 1 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : A UEQ part of BOO043-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_meredith_2_basis_1:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a:Univ.
+∀b:Univ.
+∀nand:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (nand (nand A (nand A (nand A B))) (nand B (nand A C))) B.eq Univ (nand (nand a a) (nand b a)) a
+.
+#Univ.
+#A.
+#B.
+#C.
+#a.
+#b.
+#nand.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: BOO084-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO084-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Problem : Axiom C4 for Boolean algebra in the Sheffer stroke, part 2 *)
+
+(* Version : [EF+02] axioms. *)
+
+(* English : *)
+
+(* Refs : [EF+02] Ernst et al. (2002), More First-order Test Problems in *)
+
+(* : [MV+02] McCune et al. (2002), Short Single Axioms for Boolean *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unknown *)
+
+(* Rating : 1.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 3 ( 1 singleton) *)
+
+(* Maximal term depth : 5 ( 4 average) *)
+
+(* Comments : A UEQ part of BOO043-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_meredith_2_basis_2:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀nand:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (nand (nand A (nand A (nand A B))) (nand B (nand A C))) B.eq Univ (nand a (nand b (nand a c))) (nand (nand (nand c b) b) a)
+.
+#Univ.
+#A.
+#B.
+#C.
+#a.
+#b.
+#c.
+#nand.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: BOO085-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO085-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Problem : Axiom C5 for Boolean algebra in the Sheffer stroke, part 1 *)
+
+(* Version : [EF+02] axioms. *)
+
+(* English : *)
+
+(* Refs : [EF+02] Ernst et al. (2002), More First-order Test Problems in *)
+
+(* : [MV+02] McCune et al. (2002), Short Single Axioms for Boolean *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unknown *)
+
+(* Rating : 1.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 3 ( 0 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : A UEQ part of BOO044-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_meredith_2_basis_1:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a:Univ.
+∀b:Univ.
+∀nand:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (nand (nand A (nand A (nand B C))) (nand B (nand C A))) B.eq Univ (nand (nand a a) (nand b a)) a
+.
+#Univ.
+#A.
+#B.
+#C.
+#a.
+#b.
+#nand.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: BOO086-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO086-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Problem : Axiom C5 for Boolean algebra in the Sheffer stroke, part 2 *)
+
+(* Version : [EF+02] axioms. *)
+
+(* English : *)
+
+(* Refs : [EF+02] Ernst et al. (2002), More First-order Test Problems in *)
+
+(* : [MV+02] McCune et al. (2002), Short Single Axioms for Boolean *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unknown *)
+
+(* Rating : 1.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 3 ( 0 singleton) *)
+
+(* Maximal term depth : 5 ( 4 average) *)
+
+(* Comments : A UEQ part of BOO044-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_meredith_2_basis_2:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀nand:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (nand (nand A (nand A (nand B C))) (nand B (nand C A))) B.eq Univ (nand a (nand b (nand a c))) (nand (nand (nand c b) b) a)
+.
+#Univ.
+#A.
+#B.
+#C.
+#a.
+#b.
+#c.
+#nand.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: BOO087-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO087-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Problem : Axiom C6 for Boolean algebra in the Sheffer stroke, part 1 *)
+
+(* Version : [EF+02] axioms. *)
+
+(* English : *)
+
+(* Refs : [EF+02] Ernst et al. (2002), More First-order Test Problems in *)
+
+(* : [MV+02] McCune et al. (2002), Short Single Axioms for Boolean *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unknown *)
+
+(* Rating : 1.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 3 ( 0 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : A UEQ part of BOO045-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_meredith_2_basis_1:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a:Univ.
+∀b:Univ.
+∀nand:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (nand (nand A (nand A (nand B C))) (nand C (nand A B))) C.eq Univ (nand (nand a a) (nand b a)) a
+.
+#Univ.
+#A.
+#B.
+#C.
+#a.
+#b.
+#nand.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: BOO088-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO088-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Problem : Axiom C6 for Boolean algebra in the Sheffer stroke, part 2 *)
+
+(* Version : [EF+02] axioms. *)
+
+(* English : *)
+
+(* Refs : [EF+02] Ernst et al. (2002), More First-order Test Problems in *)
+
+(* : [MV+02] McCune et al. (2002), Short Single Axioms for Boolean *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unknown *)
+
+(* Rating : 1.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 3 ( 0 singleton) *)
+
+(* Maximal term depth : 5 ( 4 average) *)
+
+(* Comments : A UEQ part of BOO045-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_meredith_2_basis_2:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀nand:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (nand (nand A (nand A (nand B C))) (nand C (nand A B))) C.eq Univ (nand a (nand b (nand a c))) (nand (nand (nand c b) b) a)
+.
+#Univ.
+#A.
+#B.
+#C.
+#a.
+#b.
+#c.
+#nand.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: BOO089-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO089-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Problem : Axiom C7 for Boolean algebra in the Sheffer stroke, part 1 *)
+
+(* Version : [EF+02] axioms. *)
+
+(* English : *)
+
+(* Refs : [EF+02] Ernst et al. (2002), More First-order Test Problems in *)
+
+(* : [MV+02] McCune et al. (2002), Short Single Axioms for Boolean *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unknown *)
+
+(* Rating : 1.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 3 ( 1 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : A UEQ part of BOO046-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_meredith_2_basis_1:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a:Univ.
+∀b:Univ.
+∀nand:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (nand (nand A (nand A (nand B B))) (nand B (nand C A))) B.eq Univ (nand (nand a a) (nand b a)) a
+.
+#Univ.
+#A.
+#B.
+#C.
+#a.
+#b.
+#nand.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: BOO090-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO090-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Problem : Axiom C7 for Boolean algebra in the Sheffer stroke, part 2 *)
+
+(* Version : [EF+02] axioms. *)
+
+(* English : *)
+
+(* Refs : [EF+02] Ernst et al. (2002), More First-order Test Problems in *)
+
+(* : [MV+02] McCune et al. (2002), Short Single Axioms for Boolean *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unknown *)
+
+(* Rating : 1.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 3 ( 1 singleton) *)
+
+(* Maximal term depth : 5 ( 4 average) *)
+
+(* Comments : A UEQ part of BOO046-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_meredith_2_basis_2:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀nand:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (nand (nand A (nand A (nand B B))) (nand B (nand C A))) B.eq Univ (nand a (nand b (nand a c))) (nand (nand (nand c b) b) a)
+.
+#Univ.
+#A.
+#B.
+#C.
+#a.
+#b.
+#c.
+#nand.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: BOO091-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO091-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Problem : Axiom C8 for Boolean algebra in the Sheffer stroke, part 1 *)
+
+(* Version : [EF+02] axioms. *)
+
+(* English : *)
+
+(* Refs : [EF+02] Ernst et al. (2002), More First-order Test Problems in *)
+
+(* : [MV+02] McCune et al. (2002), Short Single Axioms for Boolean *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unknown *)
+
+(* Rating : 1.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 3 ( 1 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : A UEQ part of BOO047-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_meredith_2_basis_1:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a:Univ.
+∀b:Univ.
+∀nand:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (nand (nand (nand A (nand A B)) A) (nand B (nand C A))) B.eq Univ (nand (nand a a) (nand b a)) a
+.
+#Univ.
+#A.
+#B.
+#C.
+#a.
+#b.
+#nand.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: BOO092-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO092-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Problem : Axiom C8 for Boolean algebra in the Sheffer stroke, part 2 *)
+
+(* Version : [EF+02] axioms. *)
+
+(* English : *)
+
+(* Refs : [EF+02] Ernst et al. (2002), More First-order Test Problems in *)
+
+(* : [MV+02] McCune et al. (2002), Short Single Axioms for Boolean *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unknown *)
+
+(* Rating : 1.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 3 ( 1 singleton) *)
+
+(* Maximal term depth : 5 ( 4 average) *)
+
+(* Comments : A UEQ part of BOO047-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_meredith_2_basis_2:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀nand:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (nand (nand (nand A (nand A B)) A) (nand B (nand C A))) B.eq Univ (nand a (nand b (nand a c))) (nand (nand (nand c b) b) a)
+.
+#Univ.
+#A.
+#B.
+#C.
+#a.
+#b.
+#c.
+#nand.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: BOO093-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO093-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Problem : Axiom C9 for Boolean algebra in the Sheffer stroke, part 1 *)
+
+(* Version : [EF+02] axioms. *)
+
+(* English : *)
+
+(* Refs : [EF+02] Ernst et al. (2002), More First-order Test Problems in *)
+
+(* : [MV+02] McCune et al. (2002), Short Single Axioms for Boolean *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unknown *)
+
+(* Rating : 1.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 3 ( 1 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : A UEQ part of BOO048-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_meredith_2_basis_1:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a:Univ.
+∀b:Univ.
+∀nand:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (nand (nand (nand A (nand B B)) A) (nand B (nand C A))) B.eq Univ (nand (nand a a) (nand b a)) a
+.
+#Univ.
+#A.
+#B.
+#C.
+#a.
+#b.
+#nand.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: BOO094-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO094-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Problem : Axiom C9 for Boolean algebra in the Sheffer stroke, part 2 *)
+
+(* Version : [EF+02] axioms. *)
+
+(* English : *)
+
+(* Refs : [EF+02] Ernst et al. (2002), More First-order Test Problems in *)
+
+(* : [MV+02] McCune et al. (2002), Short Single Axioms for Boolean *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unknown *)
+
+(* Rating : 1.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 3 ( 1 singleton) *)
+
+(* Maximal term depth : 5 ( 4 average) *)
+
+(* Comments : A UEQ part of BOO048-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_meredith_2_basis_2:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀nand:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (nand (nand (nand A (nand B B)) A) (nand B (nand C A))) B.eq Univ (nand a (nand b (nand a c))) (nand (nand (nand c b) b) a)
+.
+#Univ.
+#A.
+#B.
+#C.
+#a.
+#b.
+#c.
+#nand.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: BOO095-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO095-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Problem : Axiom C10 for Boolean algebra in the Sheffer stroke, part 1 *)
+
+(* Version : [EF+02] axioms. *)
+
+(* English : *)
+
+(* Refs : [EF+02] Ernst et al. (2002), More First-order Test Problems in *)
+
+(* : [MV+02] McCune et al. (2002), Short Single Axioms for Boolean *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unknown *)
+
+(* Rating : 1.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 3 ( 0 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : A UEQ part of BOO049-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_meredith_2_basis_1:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a:Univ.
+∀b:Univ.
+∀nand:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (nand (nand (nand A (nand B C)) A) (nand B (nand A C))) B.eq Univ (nand (nand a a) (nand b a)) a
+.
+#Univ.
+#A.
+#B.
+#C.
+#a.
+#b.
+#nand.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: BOO096-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO096-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Problem : Axiom C10 for Boolean algebra in the Sheffer stroke, part 2 *)
+
+(* Version : [EF+02] axioms. *)
+
+(* English : *)
+
+(* Refs : [EF+02] Ernst et al. (2002), More First-order Test Problems in *)
+
+(* : [MV+02] McCune et al. (2002), Short Single Axioms for Boolean *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unknown *)
+
+(* Rating : 1.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 3 ( 0 singleton) *)
+
+(* Maximal term depth : 5 ( 4 average) *)
+
+(* Comments : A UEQ part of BOO049-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_meredith_2_basis_2:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀nand:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (nand (nand (nand A (nand B C)) A) (nand B (nand A C))) B.eq Univ (nand a (nand b (nand a c))) (nand (nand (nand c b) b) a)
+.
+#Univ.
+#A.
+#B.
+#C.
+#a.
+#b.
+#c.
+#nand.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: BOO097-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO097-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Problem : Axiom C11 for Boolean algebra in the Sheffer stroke, part 1 *)
+
+(* Version : [EF+02] axioms. *)
+
+(* English : *)
+
+(* Refs : [EF+02] Ernst et al. (2002), More First-order Test Problems in *)
+
+(* : [MV+02] McCune et al. (2002), Short Single Axioms for Boolean *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unknown *)
+
+(* Rating : 1.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 3 ( 0 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : A UEQ part of BOO050-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_meredith_2_basis_1:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a:Univ.
+∀b:Univ.
+∀nand:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (nand (nand (nand A (nand B C)) A) (nand C (nand A B))) C.eq Univ (nand (nand a a) (nand b a)) a
+.
+#Univ.
+#A.
+#B.
+#C.
+#a.
+#b.
+#nand.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: BOO098-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO098-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Problem : Axiom C11 for Boolean algebra in the Sheffer stroke, part 2 *)
+
+(* Version : [EF+02] axioms. *)
+
+(* English : *)
+
+(* Refs : [EF+02] Ernst et al. (2002), More First-order Test Problems in *)
+
+(* : [MV+02] McCune et al. (2002), Short Single Axioms for Boolean *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unknown *)
+
+(* Rating : 1.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 3 ( 0 singleton) *)
+
+(* Maximal term depth : 5 ( 4 average) *)
+
+(* Comments : A UEQ part of BOO050-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_meredith_2_basis_2:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀nand:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (nand (nand (nand A (nand B C)) A) (nand C (nand A B))) C.eq Univ (nand a (nand b (nand a c))) (nand (nand (nand c b) b) a)
+.
+#Univ.
+#A.
+#B.
+#C.
+#a.
+#b.
+#c.
+#nand.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: BOO099-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO099-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Problem : Axiom C12 for Boolean algebra in the Sheffer stroke, part 1 *)
+
+(* Version : [EF+02] axioms. *)
+
+(* English : *)
+
+(* Refs : [EF+02] Ernst et al. (2002), More First-order Test Problems in *)
+
+(* : [MV+02] McCune et al. (2002), Short Single Axioms for Boolean *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unknown *)
+
+(* Rating : 1.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 3 ( 1 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : A UEQ part of BOO051-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_meredith_2_basis_1:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a:Univ.
+∀b:Univ.
+∀nand:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (nand (nand (nand A (nand A B)) A) (nand B (nand A C))) B.eq Univ (nand (nand a a) (nand b a)) a
+.
+#Univ.
+#A.
+#B.
+#C.
+#a.
+#b.
+#nand.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: BOO100-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO100-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Problem : Axiom C12 for Boolean algebra in the Sheffer stroke, part 2 *)
+
+(* Version : [EF+02] axioms. *)
+
+(* English : *)
+
+(* Refs : [EF+02] Ernst et al. (2002), More First-order Test Problems in *)
+
+(* : [MV+02] McCune et al. (2002), Short Single Axioms for Boolean *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unknown *)
+
+(* Rating : 1.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 3 ( 1 singleton) *)
+
+(* Maximal term depth : 5 ( 4 average) *)
+
+(* Comments : A UEQ part of BOO051-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_meredith_2_basis_2:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀nand:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (nand (nand (nand A (nand A B)) A) (nand B (nand A C))) B.eq Univ (nand a (nand b (nand a c))) (nand (nand (nand c b) b) a)
+.
+#Univ.
+#A.
+#B.
+#C.
+#a.
+#b.
+#c.
+#nand.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: BOO101-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO101-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Problem : Axiom C13 for Boolean algebra in the Sheffer stroke, part 1 *)
+
+(* Version : [EF+02] axioms. *)
+
+(* English : *)
+
+(* Refs : [EF+02] Ernst et al. (2002), More First-order Test Problems in *)
+
+(* : [MV+02] McCune et al. (2002), Short Single Axioms for Boolean *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unknown *)
+
+(* Rating : 1.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 3 ( 1 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : A UEQ part of BOO052-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_meredith_2_basis_1:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a:Univ.
+∀b:Univ.
+∀nand:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (nand (nand (nand (nand A B) A) A) (nand B (nand C A))) B.eq Univ (nand (nand a a) (nand b a)) a
+.
+#Univ.
+#A.
+#B.
+#C.
+#a.
+#b.
+#nand.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: BOO102-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO102-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Problem : Axiom C13 for Boolean algebra in the Sheffer stroke, part 2 *)
+
+(* Version : [EF+02] axioms. *)
+
+(* English : *)
+
+(* Refs : [EF+02] Ernst et al. (2002), More First-order Test Problems in *)
+
+(* : [MV+02] McCune et al. (2002), Short Single Axioms for Boolean *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unknown *)
+
+(* Rating : 1.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 3 ( 1 singleton) *)
+
+(* Maximal term depth : 5 ( 4 average) *)
+
+(* Comments : A UEQ part of BOO052-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_meredith_2_basis_2:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀nand:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (nand (nand (nand (nand A B) A) A) (nand B (nand C A))) B.eq Univ (nand a (nand b (nand a c))) (nand (nand (nand c b) b) a)
+.
+#Univ.
+#A.
+#B.
+#C.
+#a.
+#b.
+#c.
+#nand.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: BOO103-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO103-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Problem : Axiom C14 for Boolean algebra in the Sheffer stroke, part 1 *)
+
+(* Version : [EF+02] axioms. *)
+
+(* English : *)
+
+(* Refs : [EF+02] Ernst et al. (2002), More First-order Test Problems in *)
+
+(* : [MV+02] McCune et al. (2002), Short Single Axioms for Boolean *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unknown *)
+
+(* Rating : 1.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 3 ( 1 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : A UEQ part of BOO053-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_meredith_2_basis_1:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a:Univ.
+∀b:Univ.
+∀nand:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (nand (nand (nand (nand A B) A) A) (nand B (nand A C))) B.eq Univ (nand (nand a a) (nand b a)) a
+.
+#Univ.
+#A.
+#B.
+#C.
+#a.
+#b.
+#nand.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: BOO104-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO104-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Problem : Axiom C14 for Boolean algebra in the Sheffer stroke, part 2 *)
+
+(* Version : [EF+02] axioms. *)
+
+(* English : *)
+
+(* Refs : [EF+02] Ernst et al. (2002), More First-order Test Problems in *)
+
+(* : [MV+02] McCune et al. (2002), Short Single Axioms for Boolean *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unknown *)
+
+(* Rating : 1.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 3 ( 1 singleton) *)
+
+(* Maximal term depth : 5 ( 4 average) *)
+
+(* Comments : A UEQ part of BOO053-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_meredith_2_basis_2:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀nand:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (nand (nand (nand (nand A B) A) A) (nand B (nand A C))) B.eq Univ (nand a (nand b (nand a c))) (nand (nand (nand c b) b) a)
+.
+#Univ.
+#A.
+#B.
+#C.
+#a.
+#b.
+#c.
+#nand.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: BOO105-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO105-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Problem : Axiom C15 for Boolean algebra in the Sheffer stroke, part 1 *)
+
+(* Version : [EF+02] axioms. *)
+
+(* English : *)
+
+(* Refs : [EF+02] Ernst et al. (2002), More First-order Test Problems in *)
+
+(* : [MV+02] McCune et al. (2002), Short Single Axioms for Boolean *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unknown *)
+
+(* Rating : 1.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 3 ( 0 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : A UEQ part of BOO054-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_meredith_2_basis_1:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a:Univ.
+∀b:Univ.
+∀nand:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (nand (nand (nand (nand A B) C) C) (nand B (nand A C))) B.eq Univ (nand (nand a a) (nand b a)) a
+.
+#Univ.
+#A.
+#B.
+#C.
+#a.
+#b.
+#nand.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: BOO106-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO106-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Problem : Axiom C15 for Boolean algebra in the Sheffer stroke, part 2 *)
+
+(* Version : [EF+02] axioms. *)
+
+(* English : *)
+
+(* Refs : [EF+02] Ernst et al. (2002), More First-order Test Problems in *)
+
+(* : [MV+02] McCune et al. (2002), Short Single Axioms for Boolean *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unknown *)
+
+(* Rating : 1.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 3 ( 0 singleton) *)
+
+(* Maximal term depth : 5 ( 4 average) *)
+
+(* Comments : A UEQ part of BOO054-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_meredith_2_basis_2:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀nand:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (nand (nand (nand (nand A B) C) C) (nand B (nand A C))) B.eq Univ (nand a (nand b (nand a c))) (nand (nand (nand c b) b) a)
+.
+#Univ.
+#A.
+#B.
+#C.
+#a.
+#b.
+#c.
+#nand.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: BOO107-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO107-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Problem : Axiom C16 for Boolean algebra in the Sheffer stroke, part 1 *)
+
+(* Version : [EF+02] axioms. *)
+
+(* English : *)
+
+(* Refs : [EF+02] Ernst et al. (2002), More First-order Test Problems in *)
+
+(* : [MV+02] McCune et al. (2002), Short Single Axioms for Boolean *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unknown *)
+
+(* Rating : 1.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 3 ( 0 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : A UEQ part of BOO055-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_meredith_2_basis_1:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a:Univ.
+∀b:Univ.
+∀nand:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (nand (nand (nand (nand A B) C) C) (nand B (nand C A))) B.eq Univ (nand (nand a a) (nand b a)) a
+.
+#Univ.
+#A.
+#B.
+#C.
+#a.
+#b.
+#nand.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: BOO108-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO108-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Boolean Algebra *)
+
+(* Problem : Axiom C16 for Boolean algebra in the Sheffer stroke, part 2 *)
+
+(* Version : [EF+02] axioms. *)
+
+(* English : *)
+
+(* Refs : [EF+02] Ernst et al. (2002), More First-order Test Problems in *)
+
+(* : [MV+02] McCune et al. (2002), Short Single Axioms for Boolean *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unknown *)
+
+(* Rating : 1.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 3 ( 0 singleton) *)
+
+(* Maximal term depth : 5 ( 4 average) *)
+
+(* Comments : A UEQ part of BOO055-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_meredith_2_basis_2:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀nand:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (nand (nand (nand (nand A B) C) C) (nand B (nand C A))) B.eq Univ (nand a (nand b (nand a c))) (nand (nand (nand c b) b) a)
+.
+#Univ.
+#A.
+#B.
+#C.
+#a.
+#b.
+#c.
+#nand.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL001-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL001-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Weak fixed point for S and K *)
+
+(* Version : [WM88] (equality) axioms. *)
+
+(* English : The weak fixed point property holds for the set P consisting *)
+
+(* of the combinators S and K alone, where ((Sx)y)z = (xz)(yz) *)
+
+(* and (Kx)y = x. *)
+
+(* Refs : [Smu85] Smullyan (1978), To Mock a Mocking Bird and Other Logi *)
+
+(* : [WM88] Wos & McCune (1988), Challenge Problems Focusing on Eq *)
+
+(* Source : [WM88] *)
+
+(* Names : C1 [WM88] *)
+
+(* : Problem 1 [WM88] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.07 v3.1.0, 0.11 v2.7.0, 0.00 v2.1.0, 0.13 v2.0.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 6 ( 1 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_fixed_point:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀apply:∀_:Univ.∀_:Univ.Univ.
+∀combinator:Univ.
+∀k:Univ.
+∀s:Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (apply (apply k X) Y) X.
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply s X) Y) Z) (apply (apply X Z) (apply Y Z)).∃Y:Univ.eq Univ Y (apply combinator Y)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#apply.
+#combinator.
+#k.
+#s.
+#H0.
+#H1.
+napply ex_intro[
+nid2:
+nauto by H0,H1;
+nid|
+skip]
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL001-2.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL001-2 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Weak fixed point for S and K *)
+
+(* Version : [WM88] (equality) axioms : Augmented. *)
+
+(* English : The weak fixed point property holds for the set P consisting *)
+
+(* of the combinators S and K alone, where ((Sx)y)z = (xz)(yz) *)
+
+(* and (Kx)y = x. *)
+
+(* Refs : [WM88] Wos & McCune (1988), Challenge Problems Focusing on Eq *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.07 v3.1.0, 0.11 v2.7.0, 0.00 v2.1.0, 0.13 v2.0.0 *)
+
+(* Syntax : Number of clauses : 6 ( 0 non-Horn; 6 unit; 1 RR) *)
+
+(* Number of atoms : 6 ( 6 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 6 constant; 0-2 arity) *)
+
+(* Number of variables : 11 ( 1 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : This allows the use of B and I in the proof, as done in the *)
+
+(* "Proof of Theorem C1" in [WM88]. *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_fixed_point:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀apply:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀combinator:Univ.
+∀i:Univ.
+∀k:Univ.
+∀s:Univ.
+∀x:Univ.
+∀H0:∀X:Univ.eq Univ (apply (apply (apply s (apply b X)) i) (apply (apply s (apply b X)) i)) (apply x (apply (apply (apply s (apply b X)) i) (apply (apply s (apply b X)) i))).
+∀H1:∀X:Univ.eq Univ (apply i X) X.
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply b X) Y) Z) (apply X (apply Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (apply (apply k X) Y) X.
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply s X) Y) Z) (apply (apply X Z) (apply Y Z)).∃Y:Univ.eq Univ Y (apply combinator Y)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#apply.
+#b.
+#combinator.
+#i.
+#k.
+#s.
+#x.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+napply ex_intro[
+nid2:
+nauto by H0,H1,H2,H3,H4;
+nid|
+skip]
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL002-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL002-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Weak fixed point for S, B, C, and I *)
+
+(* Version : [WM88] (equality) axioms. *)
+
+(* English : The weak fixed point property holds for the set P consisting *)
+
+(* of the combinators S, B, C, and I, where ((Sx)y)z = (xz)(yz), *)
+
+(* ((Bx)y)z = x(yz), ((Cx)y)z = (xz)y, and Ix = x. *)
+
+(* Refs : [WM88] Wos & McCune (1988), Challenge Problems Focusing on Eq *)
+
+(* Source : [WM88] *)
+
+(* Names : C1.1 [WM88] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.1.0, 0.13 v2.0.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 1 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 5 constant; 0-2 arity) *)
+
+(* Number of variables : 11 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_fixed_point:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀apply:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀c:Univ.
+∀fixed_pt:Univ.
+∀i:Univ.
+∀s:Univ.
+∀H0:∀X:Univ.eq Univ (apply i X) X.
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply c X) Y) Z) (apply (apply X Z) Y).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply b X) Y) Z) (apply X (apply Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply s X) Y) Z) (apply (apply X Z) (apply Y Z)).∃Y:Univ.eq Univ Y (apply fixed_pt Y)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#apply.
+#b.
+#c.
+#fixed_pt.
+#i.
+#s.
+#H0.
+#H1.
+#H2.
+#H3.
+napply ex_intro[
+nid2:
+nauto by H0,H1,H2,H3;
+nid|
+skip]
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL002-4.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL002-4 : TPTP v3.2.0. Bugfixed v3.1.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Weak fixed point for S, B, C, and I *)
+
+(* Version : [WM88] (equality) axioms. *)
+
+(* Theorem formulation : The fixed point is provided and checked. *)
+
+(* English : The weak fixed point property holds for the set P consisting *)
+
+(* of the combinators S, B, C, and I, where ((Sx)y)z = (xz)(yz), *)
+
+(* ((Bx)y)z = x(yz), ((Cx)y)z = (xz)y, and Ix = x. *)
+
+(* Refs : [WM88] Wos & McCune (1988), Challenge Problems Focusing on Eq *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.07 v3.1.0 *)
+
+(* Syntax : Number of clauses : 6 ( 0 non-Horn; 6 unit; 1 RR) *)
+
+(* Number of atoms : 6 ( 6 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 5 constant; 0-2 arity) *)
+
+(* Number of variables : 11 ( 0 singleton) *)
+
+(* Maximal term depth : 5 ( 3 average) *)
+
+(* Comments : This is the one found in proofs 1 and 2 of C1.1 in [WM88]. *)
+
+(* Bugfixes : Fixed clauses weak_fixed_point and prove_weak_fixed_point. *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_weak_fixed_point:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀apply:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀c:Univ.
+∀fixed_pt:Univ.
+∀i:Univ.
+∀s:Univ.
+∀weak_fixed_point:∀_:Univ.Univ.
+∀H0:∀X:Univ.eq Univ (weak_fixed_point X) (apply (apply (apply s (apply b X)) i) (apply (apply s (apply b X)) i)).
+∀H1:∀X:Univ.eq Univ (apply i X) X.
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply c X) Y) Z) (apply (apply X Z) Y).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply b X) Y) Z) (apply X (apply Y Z)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply s X) Y) Z) (apply (apply X Z) (apply Y Z)).eq Univ (weak_fixed_point fixed_pt) (apply fixed_pt (weak_fixed_point fixed_pt))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#apply.
+#b.
+#c.
+#fixed_pt.
+#i.
+#s.
+#weak_fixed_point.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+nauto by H0,H1,H2,H3,H4;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL002-5.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL002-5 : TPTP v3.2.0. Bugfixed v3.1.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Weak fixed point for S, B, C, and I *)
+
+(* Version : [WM88] (equality) axioms. *)
+
+(* Theorem formulation : The fixed point is provided and checked. *)
+
+(* English : The weak fixed point property holds for the set P consisting *)
+
+(* of the combinators S, B, C, and I, where ((Sx)y)z = (xz)(yz), *)
+
+(* ((Bx)y)z = x(yz), ((Cx)y)z = (xz)y, and Ix = x. *)
+
+(* Refs : [WM88] Wos & McCune (1988), Challenge Problems Focusing on Eq *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.43 v3.1.0 *)
+
+(* Syntax : Number of clauses : 6 ( 0 non-Horn; 6 unit; 1 RR) *)
+
+(* Number of atoms : 6 ( 6 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 5 constant; 0-2 arity) *)
+
+(* Number of variables : 11 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : This is the one found in proof 3 of C1.1 in [WM88]. *)
+
+(* Bugfixes : Fixed clauses weak_fixed_point and prove_weak_fixed_point. *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_weak_fixed_point:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀apply:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀c:Univ.
+∀fixed_pt:Univ.
+∀i:Univ.
+∀s:Univ.
+∀weak_fixed_point:∀_:Univ.Univ.
+∀H0:∀X:Univ.eq Univ (weak_fixed_point X) (apply (apply (apply s (apply c (apply b X))) (apply s (apply c (apply b X)))) (apply s (apply c (apply b X)))).
+∀H1:∀X:Univ.eq Univ (apply i X) X.
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply c X) Y) Z) (apply (apply X Z) Y).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply b X) Y) Z) (apply X (apply Y Z)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply s X) Y) Z) (apply (apply X Z) (apply Y Z)).eq Univ (weak_fixed_point fixed_pt) (apply fixed_pt (weak_fixed_point fixed_pt))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#apply.
+#b.
+#c.
+#fixed_pt.
+#i.
+#s.
+#weak_fixed_point.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+nauto by H0,H1,H2,H3,H4;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL003-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL003-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Strong fixed point for B and W *)
+
+(* Version : [WM88] (equality) axioms. *)
+
+(* English : The strong fixed point property holds for the set *)
+
+(* P consisting of the combinators B and W alone, where ((Bx)y)z *)
+
+(* = x(yz) and (Wx)y = (xy)y. *)
+
+(* Refs : [Smu85] Smullyan (1978), To Mock a Mocking Bird and Other Logi *)
+
+(* : [MW87] McCune & Wos (1987), A Case Study in Automated Theorem *)
+
+(* : [WM88] Wos & McCune (1988), Challenge Problems Focusing on Eq *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* : [Ove90] Overbeek (1990), ATP competition announced at CADE-10 *)
+
+(* : [LW92] Lusk & Wos (1992), Benchmark Problems in Which Equalit *)
+
+(* : [Wos93] Wos (1993), The Kernel Strategy and Its Use for the St *)
+
+(* : [Ove93] Overbeek (1993), The CADE-11 Competitions: A Personal *)
+
+(* : [LM93] Lusk & McCune (1993), Uniform Strategies: The CADE-11 *)
+
+(* : [Zha93] Zhang (1993), Automated Proofs of Equality Problems in *)
+
+(* Source : [WM88] *)
+
+(* Names : C2 [WM88] *)
+
+(* : Problem 2 [WM88] *)
+
+(* : Test Problem 17 [Wos88] *)
+
+(* : Sages and Combinatory Logic [Wos88] *)
+
+(* : CADE-11 Competition Eq-8 [Ove90] *)
+
+(* : CL2 [LW92] *)
+
+(* : THEOREM EQ-8 [LM93] *)
+
+(* : Question 3 [Wos93] *)
+
+(* : Question 5 [Wos93] *)
+
+(* : PROBLEM 8 [Zha93] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.71 v3.2.0, 0.79 v3.1.0, 0.78 v2.7.0, 0.73 v2.6.0, 0.67 v2.5.0, 0.25 v2.4.0, 0.33 v2.3.0, 0.67 v2.2.1, 1.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 6 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_strong_fixed_point:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀apply:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀f:∀_:Univ.Univ.
+∀w:Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (apply (apply w X) Y) (apply (apply X Y) Y).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply b X) Y) Z) (apply X (apply Y Z)).∃Y:Univ.eq Univ (apply Y (f Y)) (apply (f Y) (apply Y (f Y)))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#apply.
+#b.
+#f.
+#w.
+#H0.
+#H1.
+napply ex_intro[
+nid2:
+nauto by H0,H1;
+nid|
+skip]
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL003-2.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL003-2 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Strong fixed point for B and W *)
+
+(* Version : [WM88] (equality) axioms : Augmented > Especial. *)
+
+(* Theorem formulation : The fixed point is provided and checked. *)
+
+(* English : The strong fixed point property holds for the set *)
+
+(* P consisting of the combinators B and W alone, where ((Bx)y)z *)
+
+(* = x(yz) and (Wx)y = (xy)y. *)
+
+(* Refs : [Smu85] Smullyan (1978), To Mock a Mocking Bird and Other Logi *)
+
+(* : [MW87] McCune & Wos (1987), A Case Study in Automated Theorem *)
+
+(* : [WM88] Wos & McCune (1988), Challenge Problems Focusing on Eq *)
+
+(* : [Wos93] Wos (1993), The Kernel Strategy and Its Use for the St *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.29 v3.1.0, 0.33 v2.7.0, 0.17 v2.6.0, 0.29 v2.5.0, 0.20 v2.4.0, 0.33 v2.2.1, 0.67 v2.2.0, 0.71 v2.1.0, 1.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 4 ( 0 non-Horn; 3 unit; 2 RR) *)
+
+(* Number of atoms : 5 ( 3 equality) *)
+
+(* Maximal clause size : 2 ( 1 average) *)
+
+(* Number of predicates : 2 ( 0 propositional; 1-2 arity) *)
+
+(* Number of functors : 4 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 6 ( 0 singleton) *)
+
+(* Maximal term depth : 5 ( 3 average) *)
+
+(* Comments : This the J sage of [McCune & Wos, 1987], found by Statman. *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_strong_fixed_point:
+ ∀Univ:Type.∀Strong_fixed_point:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀apply:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀fixed_point:∀_:Univ.Prop.
+∀fixed_pt:Univ.
+∀w:Univ.
+∀H0:∀Strong_fixed_point:Univ.∀_:eq Univ (apply Strong_fixed_point fixed_pt) (apply fixed_pt (apply Strong_fixed_point fixed_pt)).fixed_point Strong_fixed_point.
+∀H1:∀X:Univ.∀Y:Univ.eq Univ (apply (apply w X) Y) (apply (apply X Y) Y).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply b X) Y) Z) (apply X (apply Y Z)).fixed_point (apply (apply b (apply w w)) (apply (apply b w) (apply (apply b b) b)))
+.
+#Univ.
+#Strong_fixed_point.
+#X.
+#Y.
+#Z.
+#apply.
+#b.
+#fixed_point.
+#fixed_pt.
+#w.
+#H0.
+#H1.
+#H2.
+nauto by H0,H1,H2;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL004-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL004-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Find combinator equivalent to U from S and K *)
+
+(* Version : [WM88] (equality) axioms. *)
+
+(* English : Construct from S and K alone a combinator that behaves as the *)
+
+(* combinator U does, where ((Sx)y)z = (xz)(yz), (Kx)y *)
+
+(* = x, (Ux)y = y((xx)y). *)
+
+(* Refs : [WM88] Wos & McCune (1988), Challenge Problems Focusing on Eq *)
+
+(* Source : [WM88] *)
+
+(* Names : Problem 4 [WM88] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.50 v3.2.0, 0.57 v3.1.0, 0.67 v2.7.0, 0.55 v2.6.0, 0.17 v2.5.0, 0.25 v2.4.0, 0.00 v2.2.1, 0.56 v2.2.0, 0.57 v2.1.0, 1.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 6 ( 1 singleton) *)
+
+(* Maximal term depth : 5 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_u_combinator:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀apply:∀_:Univ.∀_:Univ.Univ.
+∀f:∀_:Univ.Univ.
+∀g:∀_:Univ.Univ.
+∀k:Univ.
+∀s:Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (apply (apply k X) Y) X.
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply s X) Y) Z) (apply (apply X Z) (apply Y Z)).∃Z:Univ.eq Univ (apply (apply Z (f Z)) (g Z)) (apply (g Z) (apply (apply (f Z) (f Z)) (g Z)))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#apply.
+#f.
+#g.
+#k.
+#s.
+#H0.
+#H1.
+napply ex_intro[
+nid2:
+nauto by H0,H1;
+nid|
+skip]
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL004-3.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL004-3 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Find combinator equivalent to U from S and K. *)
+
+(* Version : [WM88] (equality) axioms. *)
+
+(* Theorem formulation : The combination is provided and checked. *)
+
+(* English : Construct from S and K alone a combinator that behaves as the *)
+
+(* combinator U does, where ((Sx)y)z = (xz)(yz), (Kx)y = x, *)
+
+(* (Ux)y = y((xx)y). *)
+
+(* Refs : [WM88] Wos & McCune (1988), Challenge Problems Focusing on Eq *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.21 v3.1.0, 0.22 v2.7.0, 0.27 v2.6.0, 0.17 v2.5.0, 0.00 v2.2.1, 0.22 v2.2.0, 0.29 v2.1.0, 0.38 v2.0.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 1 singleton) *)
+
+(* Maximal term depth : 9 ( 4 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----This is the U equivalent *)
+ntheorem prove_u_combinator:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀apply:∀_:Univ.∀_:Univ.Univ.
+∀k:Univ.
+∀s:Univ.
+∀x:Univ.
+∀y:Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (apply (apply k X) Y) X.
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply s X) Y) Z) (apply (apply X Z) (apply Y Z)).eq Univ (apply (apply (apply (apply s (apply k (apply s (apply (apply s k) k)))) (apply (apply s (apply (apply s k) k)) (apply (apply s k) k))) x) y) (apply y (apply (apply x x) y))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#apply.
+#k.
+#s.
+#x.
+#y.
+#H0.
+#H1.
+nauto by H0,H1;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL005-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL005-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Find a model for S and W but not a weak fixed point *)
+
+(* Version : [WM88] (equality) axioms. *)
+
+(* English : The model one is seeking must satisfy S and W and fail *)
+
+(* to satisfy the weak fixed point property, where ((Sx)y)z *)
+
+(* = (xz)(yz), (Wx)y = (xy)y. *)
+
+(* Refs : [WM88] Wos & McCune (1988), Challenge Problems Focusing on Eq *)
+
+(* : [Zha92] Zhang (1992), Solution to an Open Question in Combinat *)
+
+(* : [Wos93] Wos (1993), The Kernel Strategy and Its Use for the St *)
+
+(* : [Pel98] Peltier (1998), A New Method for Automated Finite Mode *)
+
+(* Source : [WM88] *)
+
+(* Names : Problem 5 [WM88] *)
+
+(* : Question 15 [Wos93] *)
+
+(* : 4.2.5 (CL3) [Pel98] *)
+
+(* Status : Satisfiable *)
+
+(* Rating : 0.33 v3.2.0, 0.67 v3.1.0, 0.33 v2.7.0, 0.00 v2.6.0, 0.33 v2.4.0, 0.67 v2.2.1, 0.75 v2.2.0, 0.67 v2.1.0, 1.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 6 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_model:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀apply:∀_:Univ.∀_:Univ.Univ.
+∀combinator:Univ.
+∀s:Univ.
+∀w:Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (apply (apply w X) Y) (apply (apply X Y) Y).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply s X) Y) Z) (apply (apply X Z) (apply Y Z)).∃Y:Univ.eq Univ Y (apply combinator Y)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#apply.
+#combinator.
+#s.
+#w.
+#H0.
+#H1.
+napply ex_intro[
+nid2:
+nauto by H0,H1;
+nid|
+skip]
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL006-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL006-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Strong fixed point for S and K *)
+
+(* Version : [WM88] (equality) axioms. *)
+
+(* English : The strong fixed point property holds for the set *)
+
+(* P consisting of the combinators S and K alone, where *)
+
+(* ((Sx)y)z = (xz)(yz), (Kx)y = x. *)
+
+(* Refs : [WM88] Wos & McCune (1988), Challenge Problems Focusing on Eq *)
+
+(* Source : [WM88] *)
+
+(* Names : Problem 6 [WM88] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.57 v3.1.0, 0.78 v2.7.0, 0.64 v2.6.0, 0.67 v2.5.0, 0.50 v2.4.0, 0.67 v2.2.1, 0.67 v2.2.0, 0.57 v2.1.0, 1.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 6 ( 1 singleton) *)
+
+(* Maximal term depth : 4 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_fixed_point:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀apply:∀_:Univ.∀_:Univ.Univ.
+∀f:∀_:Univ.Univ.
+∀k:Univ.
+∀s:Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (apply (apply k X) Y) X.
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply s X) Y) Z) (apply (apply X Z) (apply Y Z)).∃Y:Univ.eq Univ (apply Y (f Y)) (apply (f Y) (apply Y (f Y)))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#apply.
+#f.
+#k.
+#s.
+#H0.
+#H1.
+napply ex_intro[
+nid2:
+nauto by H0,H1;
+nid|
+skip]
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL006-5.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL006-5 : TPTP v3.2.0. Released v2.1.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Strong fixed point for S and K *)
+
+(* Version : [WM88] (equality) axioms. *)
+
+(* Theorem formulation : The fixed point is provided and checked. *)
+
+(* English : The strong fixed point property holds for the set *)
+
+(* P consisting of the combinators S and K alone, where *)
+
+(* ((Sx)y)z = (xz)(yz), (Kx)y = x. *)
+
+(* Refs : [WM88] Wos & McCune (1988), Challenge Problems Focusing on Eq *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.64 v3.1.0, 0.56 v2.7.0, 0.91 v2.6.0, 0.33 v2.5.0, 0.00 v2.4.0, 0.67 v2.2.1, 0.88 v2.2.0, 0.80 v2.1.0 *)
+
+(* Syntax : Number of clauses : 4 ( 0 non-Horn; 4 unit; 2 RR) *)
+
+(* Number of atoms : 4 ( 4 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 1 singleton) *)
+
+(* Maximal term depth : 8 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_strong_fixed_point:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀apply:∀_:Univ.∀_:Univ.Univ.
+∀fixed_pt:Univ.
+∀k:Univ.
+∀s:Univ.
+∀strong_fixed_point:Univ.
+∀H0:eq Univ strong_fixed_point (apply (apply s (apply k (apply (apply s (apply (apply s k) k)) (apply (apply s k) k)))) (apply (apply s (apply k (apply (apply s s) (apply s k)))) (apply (apply s (apply k s)) k))).
+∀H1:∀X:Univ.∀Y:Univ.eq Univ (apply (apply k X) Y) X.
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply s X) Y) Z) (apply (apply X Z) (apply Y Z)).eq Univ (apply strong_fixed_point fixed_pt) (apply fixed_pt (apply strong_fixed_point fixed_pt))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#apply.
+#fixed_pt.
+#k.
+#s.
+#strong_fixed_point.
+#H0.
+#H1.
+#H2.
+nauto by H0,H1,H2;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL006-6.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL006-6 : TPTP v3.2.0. Released v2.1.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Strong fixed point for S and K *)
+
+(* Version : [WM88] (equality) axioms. *)
+
+(* Theorem formulation : The fixed point is provided and checked. *)
+
+(* English : The strong fixed point property holds for the set *)
+
+(* P consisting of the combinators S and K alone, where *)
+
+(* ((Sx)y)z = (xz)(yz), (Kx)y = x. *)
+
+(* Refs : [WM88] Wos & McCune (1988), Challenge Problems Focusing on Eq *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.64 v3.1.0, 0.78 v2.7.0, 0.73 v2.6.0, 0.50 v2.5.0, 0.75 v2.4.0, 0.67 v2.2.1, 0.88 v2.2.0, 0.80 v2.1.0 *)
+
+(* Syntax : Number of clauses : 4 ( 0 non-Horn; 4 unit; 2 RR) *)
+
+(* Number of atoms : 4 ( 4 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 1 singleton) *)
+
+(* Maximal term depth : 8 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_strong_fixed_point:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀apply:∀_:Univ.∀_:Univ.Univ.
+∀fixed_pt:Univ.
+∀k:Univ.
+∀s:Univ.
+∀strong_fixed_point:Univ.
+∀H0:eq Univ strong_fixed_point (apply (apply s (apply k (apply (apply s (apply (apply s k) k)) (apply (apply s k) k)))) (apply (apply s (apply (apply s (apply k s)) k)) (apply k (apply (apply s (apply (apply s k) k)) (apply (apply s k) k))))).
+∀H1:∀X:Univ.∀Y:Univ.eq Univ (apply (apply k X) Y) X.
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply s X) Y) Z) (apply (apply X Z) (apply Y Z)).eq Univ (apply strong_fixed_point fixed_pt) (apply fixed_pt (apply strong_fixed_point fixed_pt))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#apply.
+#fixed_pt.
+#k.
+#s.
+#strong_fixed_point.
+#H0.
+#H1.
+#H2.
+nauto by H0,H1,H2;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL006-7.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL006-7 : TPTP v3.2.0. Released v2.1.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Strong fixed point for S and K *)
+
+(* Version : [WM88] (equality) axioms. *)
+
+(* Theorem formulation : The fixed point is provided and checked. *)
+
+(* English : The strong fixed point property holds for the set *)
+
+(* P consisting of the combinators S and K alone, where *)
+
+(* ((Sx)y)z = (xz)(yz), (Kx)y = x. *)
+
+(* Refs : [WM88] Wos & McCune (1988), Challenge Problems Focusing on Eq *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.64 v3.1.0, 0.78 v2.7.0, 0.91 v2.6.0, 0.67 v2.5.0, 0.50 v2.4.0, 0.67 v2.2.1, 0.88 v2.2.0, 0.80 v2.1.0 *)
+
+(* Syntax : Number of clauses : 4 ( 0 non-Horn; 4 unit; 2 RR) *)
+
+(* Number of atoms : 4 ( 4 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 1 singleton) *)
+
+(* Maximal term depth : 8 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_strong_fixed_point:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀apply:∀_:Univ.∀_:Univ.Univ.
+∀fixed_pt:Univ.
+∀k:Univ.
+∀s:Univ.
+∀strong_fixed_point:Univ.
+∀H0:eq Univ strong_fixed_point (apply (apply s (apply k (apply (apply (apply s s) (apply (apply s k) k)) (apply (apply s s) (apply s k))))) (apply (apply s (apply k s)) k)).
+∀H1:∀X:Univ.∀Y:Univ.eq Univ (apply (apply k X) Y) X.
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply s X) Y) Z) (apply (apply X Z) (apply Y Z)).eq Univ (apply strong_fixed_point fixed_pt) (apply fixed_pt (apply strong_fixed_point fixed_pt))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#apply.
+#fixed_pt.
+#k.
+#s.
+#strong_fixed_point.
+#H0.
+#H1.
+#H2.
+nauto by H0,H1,H2;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL007-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL007-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Weak fixed point for L *)
+
+(* Version : [WM88] (equality) axioms. *)
+
+(* English : The weak fixed point property holds for the set P consisting *)
+
+(* of the combinator L, where (Lx)y = x(yy). *)
+
+(* Refs : [Smu85] Smullyan (1978), To Mock a Mocking Bird and Other Logi *)
+
+(* : [MW87] McCune & Wos (1987), A Case Study in Automated Theorem *)
+
+(* : [WM88] Wos & McCune (1988), Challenge Problems Focusing on Eq *)
+
+(* : [MW88] McCune & Wos (1988), Some Fixed Point Problems in Comb *)
+
+(* Source : [MW88] *)
+
+(* Names : - [MW88] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 3 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_fixed_point:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.
+∀apply:∀_:Univ.∀_:Univ.Univ.
+∀combinator:Univ.
+∀l:Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (apply (apply l X) Y) (apply X (apply Y Y)).∃Y:Univ.eq Univ Y (apply combinator Y)
+.
+#Univ.
+#X.
+#Y.
+#apply.
+#combinator.
+#l.
+#H0.
+napply ex_intro[
+nid2:
+nauto by H0;
+nid|
+skip]
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL008-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL008-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Weak fixed point for M and B *)
+
+(* Version : [WM88] (equality) axioms. *)
+
+(* English : The weak fixed point property holds for the set P consisting *)
+
+(* of the combinators M and B, where ((Bx)y)z = x(yz), Mx = xx. *)
+
+(* Refs : [Smu85] Smullyan (1978), To Mock a Mocking Bird and Other Logi *)
+
+(* : [MW87] McCune & Wos (1987), A Case Study in Automated Theorem *)
+
+(* : [WM88] Wos & McCune (1988), Challenge Problems Focusing on Eq *)
+
+(* : [MW88] McCune & Wos (1988), Some Fixed Point Problems in Comb *)
+
+(* : [Wos93] Wos (1993), The Kernel Strategy and Its Use for the St *)
+
+(* Source : [MW88] *)
+
+(* Names : - [MW88] *)
+
+(* : Question 13 [Wos93] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_fixed_point:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀apply:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀combinator:Univ.
+∀m:Univ.
+∀H0:∀X:Univ.eq Univ (apply m X) (apply X X).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply b X) Y) Z) (apply X (apply Y Z)).∃Y:Univ.eq Univ Y (apply combinator Y)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#apply.
+#b.
+#combinator.
+#m.
+#H0.
+#H1.
+napply ex_intro[
+nid2:
+nauto by H0,H1;
+nid|
+skip]
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL009-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL009-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Weak fixed point for B and L2 *)
+
+(* Version : [WM88] (equality) axioms. *)
+
+(* English : The weak fixed point property holds for the set P consisting *)
+
+(* of the combinators B and L2, where ((Bx)y)z = x(yz), (L2x)y *)
+
+(* = y(xx). *)
+
+(* Refs : [Smu85] Smullyan (1978), To Mock a Mocking Bird and Other Logi *)
+
+(* : [MW87] McCune & Wos (1987), A Case Study in Automated Theorem *)
+
+(* : [WM88] Wos & McCune (1988), Challenge Problems Focusing on Eq *)
+
+(* : [MW88] McCune & Wos (1988), Some Fixed Point Problems in Comb *)
+
+(* Source : [MW88] *)
+
+(* Names : - [MW88] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.07 v3.1.0, 0.11 v2.7.0, 0.00 v2.2.1, 0.33 v2.2.0, 0.14 v2.1.0, 0.38 v2.0.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 6 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_fixed_point:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀apply:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀combinator:Univ.
+∀l2:Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (apply (apply l2 X) Y) (apply Y (apply X X)).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply b X) Y) Z) (apply X (apply Y Z)).∃Y:Univ.eq Univ Y (apply combinator Y)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#apply.
+#b.
+#combinator.
+#l2.
+#H0.
+#H1.
+napply ex_intro[
+nid2:
+nauto by H0,H1;
+nid|
+skip]
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL010-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL010-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Weak fixed point for B and S2 *)
+
+(* Version : [WM88] (equality) axioms. *)
+
+(* English : The weak fixed point property holds for the set P consisting *)
+
+(* of the combinators B and S2, where ((Bx)y)z = x(yz), *)
+
+(* ((S2x)y)z = (xz)(yy). *)
+
+(* Refs : [Smu85] Smullyan (1978), To Mock a Mocking Bird and Other Logi *)
+
+(* : [MW87] McCune & Wos (1987), A Case Study in Automated Theorem *)
+
+(* : [WM88] Wos & McCune (1988), Challenge Problems Focusing on Eq *)
+
+(* : [MW88] McCune & Wos (1988), Some Fixed Point Problems in Comb *)
+
+(* Source : [MW88] *)
+
+(* Names : - [MW88] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.07 v3.1.0, 0.11 v2.7.0, 0.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_fixed_point:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀apply:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀combinator:Univ.
+∀s2:Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply s2 X) Y) Z) (apply (apply X Z) (apply Y Y)).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply b X) Y) Z) (apply X (apply Y Z)).∃Y:Univ.eq Univ Y (apply combinator Y)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#apply.
+#b.
+#combinator.
+#s2.
+#H0.
+#H1.
+napply ex_intro[
+nid2:
+nauto by H0,H1;
+nid|
+skip]
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL011-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL011-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Weak fixed point for O and Q1 *)
+
+(* Version : [WM88] (equality) axioms. *)
+
+(* English : The weak fixed point property holds for the set P consisting *)
+
+(* of the combinators O and Q1, where (Ox)y = y(xy), ((Q1x)y)z *)
+
+(* = x(zy). *)
+
+(* Refs : [Smu85] Smullyan (1978), To Mock a Mocking Bird and Other Logi *)
+
+(* : [MW87] McCune & Wos (1987), A Case Study in Automated Theorem *)
+
+(* : [WM88] Wos & McCune (1988), Challenge Problems Focusing on Eq *)
+
+(* : [MW88] McCune & Wos (1988), Some Fixed Point Problems in Comb *)
+
+(* Source : [MW88] *)
+
+(* Names : - [MW88] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.36 v3.1.0, 0.67 v2.7.0, 0.36 v2.6.0, 0.33 v2.5.0, 0.25 v2.4.0, 0.00 v2.2.1, 0.33 v2.2.0, 0.14 v2.1.0, 0.62 v2.0.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 6 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_fixed_point:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀apply:∀_:Univ.∀_:Univ.Univ.
+∀combinator:Univ.
+∀o:Univ.
+∀q1:Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply q1 X) Y) Z) (apply X (apply Z Y)).
+∀H1:∀X:Univ.∀Y:Univ.eq Univ (apply (apply o X) Y) (apply Y (apply X Y)).∃Y:Univ.eq Univ Y (apply combinator Y)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#apply.
+#combinator.
+#o.
+#q1.
+#H0.
+#H1.
+napply ex_intro[
+nid2:
+nauto by H0,H1;
+nid|
+skip]
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL012-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL012-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Weak fixed point for U *)
+
+(* Version : [WM88] (equality) axioms. *)
+
+(* English : The weak fixed point property holds for the set P consisting *)
+
+(* of the combinator U, where (Ux)y = y((xx)y). *)
+
+(* Refs : [Smu85] Smullyan (1978), To Mock a Mocking Bird and Other Logi *)
+
+(* : [MW87] McCune & Wos (1987), A Case Study in Automated Theorem *)
+
+(* : [WM88] Wos & McCune (1988), Challenge Problems Focusing on Eq *)
+
+(* : [MW88] McCune & Wos (1988), Some Fixed Point Problems in Comb *)
+
+(* Source : [MW88] *)
+
+(* Names : - [MW88] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 3 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_fixed_point:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.
+∀apply:∀_:Univ.∀_:Univ.Univ.
+∀combinator:Univ.
+∀u:Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (apply (apply u X) Y) (apply Y (apply (apply X X) Y)).∃Y:Univ.eq Univ Y (apply combinator Y)
+.
+#Univ.
+#X.
+#Y.
+#apply.
+#combinator.
+#u.
+#H0.
+napply ex_intro[
+nid2:
+nauto by H0;
+nid|
+skip]
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL013-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL013-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Weak fixed point for S and L *)
+
+(* Version : [WM88] (equality) axioms. *)
+
+(* English : The weak fixed point property holds for the set P consisting *)
+
+(* of the combinators S and L, where ((Sx)y)z = (xz)(yz), (Lx)y *)
+
+(* = x(yy). *)
+
+(* Refs : [Smu85] Smullyan (1978), To Mock a Mocking Bird and Other Logi *)
+
+(* : [MW87] McCune & Wos (1987), A Case Study in Automated Theorem *)
+
+(* : [WM88] Wos & McCune (1988), Challenge Problems Focusing on Eq *)
+
+(* : [MW88] McCune & Wos (1988), Some Fixed Point Problems in Comb *)
+
+(* Source : [MW88] *)
+
+(* Names : - [MW88] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 6 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_fixed_point:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀apply:∀_:Univ.∀_:Univ.Univ.
+∀combinator:Univ.
+∀l:Univ.
+∀s:Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (apply (apply l X) Y) (apply X (apply Y Y)).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply s X) Y) Z) (apply (apply X Z) (apply Y Z)).∃Y:Univ.eq Univ Y (apply combinator Y)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#apply.
+#combinator.
+#l.
+#s.
+#H0.
+#H1.
+napply ex_intro[
+nid2:
+nauto by H0,H1;
+nid|
+skip]
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL014-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL014-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Weak fixed point for L and O *)
+
+(* Version : [WM88] (equality) axioms. *)
+
+(* English : The weak fixed point property holds for the set P consisting *)
+
+(* of the combinators L and O, where (Lx)y = x(yy), (Ox)y *)
+
+(* = y(xy). *)
+
+(* Refs : [Smu85] Smullyan (1978), To Mock a Mocking Bird and Other Logi *)
+
+(* : [MW87] McCune & Wos (1987), A Case Study in Automated Theorem *)
+
+(* : [WM88] Wos & McCune (1988), Challenge Problems Focusing on Eq *)
+
+(* : [MW88] McCune & Wos (1988), Some Fixed Point Problems in Comb *)
+
+(* Source : [MW88] *)
+
+(* Names : - [MW88] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_fixed_point:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.
+∀apply:∀_:Univ.∀_:Univ.Univ.
+∀combinator:Univ.
+∀l:Univ.
+∀o:Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (apply (apply o X) Y) (apply Y (apply X Y)).
+∀H1:∀X:Univ.∀Y:Univ.eq Univ (apply (apply l X) Y) (apply X (apply Y Y)).∃Y:Univ.eq Univ Y (apply combinator Y)
+.
+#Univ.
+#X.
+#Y.
+#apply.
+#combinator.
+#l.
+#o.
+#H0.
+#H1.
+napply ex_intro[
+nid2:
+nauto by H0,H1;
+nid|
+skip]
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL015-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL015-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Weak fixed point for Q and M *)
+
+(* Version : [WM88] (equality) axioms. *)
+
+(* English : The weak fixed point property holds for the set P consisting *)
+
+(* of the combinators Q and M, where Mx = xx, ((Qx)y)z = y(xz). *)
+
+(* Refs : [Smu85] Smullyan (1978), To Mock a Mocking Bird and Other Logi *)
+
+(* : [MW87] McCune & Wos (1987), A Case Study in Automated Theorem *)
+
+(* : [WM88] Wos & McCune (1988), Challenge Problems Focusing on Eq *)
+
+(* : [MW88] McCune & Wos (1988), Some Fixed Point Problems in Comb *)
+
+(* Source : [MW88] *)
+
+(* Names : - [MW88] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_fixed_point:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀apply:∀_:Univ.∀_:Univ.Univ.
+∀combinator:Univ.
+∀m:Univ.
+∀q:Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply q X) Y) Z) (apply Y (apply X Z)).
+∀H1:∀X:Univ.eq Univ (apply m X) (apply X X).∃Y:Univ.eq Univ Y (apply combinator Y)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#apply.
+#combinator.
+#m.
+#q.
+#H0.
+#H1.
+napply ex_intro[
+nid2:
+nauto by H0,H1;
+nid|
+skip]
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL016-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL016-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Weak fixed point for B, M and L *)
+
+(* Version : [WM88] (equality) axioms. *)
+
+(* English : The weak fixed point property holds for the set P consisting *)
+
+(* of the combinators B, M and L, where ((Bx)y)z = x(yz), (Lx)y *)
+
+(* = x(yy), Mx = xx. *)
+
+(* Refs : [Smu85] Smullyan (1978), To Mock a Mocking Bird and Other Logi *)
+
+(* : [MW87] McCune & Wos (1987), A Case Study in Automated Theorem *)
+
+(* : [WM88] Wos & McCune (1988), Challenge Problems Focusing on Eq *)
+
+(* : [MW88] McCune & Wos (1988), Some Fixed Point Problems in Comb *)
+
+(* Source : [MW88] *)
+
+(* Names : - [MW88] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v3.1.0, 0.11 v2.7.0, 0.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 4 ( 0 non-Horn; 4 unit; 1 RR) *)
+
+(* Number of atoms : 4 ( 4 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_fixed_point:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀apply:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀combinator:Univ.
+∀l:Univ.
+∀m:Univ.
+∀H0:∀X:Univ.eq Univ (apply m X) (apply X X).
+∀H1:∀X:Univ.∀Y:Univ.eq Univ (apply (apply l X) Y) (apply X (apply Y Y)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply b X) Y) Z) (apply X (apply Y Z)).∃Y:Univ.eq Univ Y (apply combinator Y)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#apply.
+#b.
+#combinator.
+#l.
+#m.
+#H0.
+#H1.
+#H2.
+napply ex_intro[
+nid2:
+nauto by H0,H1,H2;
+nid|
+skip]
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL017-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL017-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Weak fixed point for B, M, and T *)
+
+(* Version : [WM88] (equality) axioms. *)
+
+(* English : The weak fixed point property holds for the set P consisting *)
+
+(* of the combinators B, M, and T, where ((Bx)y)z = x(yz), *)
+
+(* Mx = xx, (Tx)y = yx. *)
+
+(* Refs : [Smu85] Smullyan (1978), To Mock a Mocking Bird and Other Logi *)
+
+(* : [MW87] McCune & Wos (1987), A Case Study in Automated Theorem *)
+
+(* : [WM88] Wos & McCune (1988), Challenge Problems Focusing on Eq *)
+
+(* : [MW88] McCune & Wos (1988), Some Fixed Point Problems in Comb *)
+
+(* Source : [MW88] *)
+
+(* Names : - [MW88] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 4 ( 0 non-Horn; 4 unit; 1 RR) *)
+
+(* Number of atoms : 4 ( 4 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_fixed_point:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀apply:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀combinator:Univ.
+∀m:Univ.
+∀t:Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (apply (apply t X) Y) (apply Y X).
+∀H1:∀X:Univ.eq Univ (apply m X) (apply X X).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply b X) Y) Z) (apply X (apply Y Z)).∃Y:Univ.eq Univ Y (apply combinator Y)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#apply.
+#b.
+#combinator.
+#m.
+#t.
+#H0.
+#H1.
+#H2.
+napply ex_intro[
+nid2:
+nauto by H0,H1,H2;
+nid|
+skip]
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL018-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL018-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Weak fixed point for W, Q, and L *)
+
+(* Version : [WM88] (equality) axioms. *)
+
+(* English : The weak fixed point property holds for the set P consisting *)
+
+(* of the combinators W, Q, and L, where (Lx)y = x(yy), (Wx)y *)
+
+(* = (xy)y, ((Qx)y)z = y(xz). *)
+
+(* Refs : [Smu85] Smullyan (1978), To Mock a Mocking Bird and Other Logi *)
+
+(* : [MW87] McCune & Wos (1987), A Case Study in Automated Theorem *)
+
+(* : [WM88] Wos & McCune (1988), Challenge Problems Focusing on Eq *)
+
+(* : [MW88] McCune & Wos (1988), Some Fixed Point Problems in Comb *)
+
+(* Source : [MW88] *)
+
+(* Names : - [MW88] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 4 ( 0 non-Horn; 4 unit; 1 RR) *)
+
+(* Number of atoms : 4 ( 4 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 8 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_fixed_point:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀apply:∀_:Univ.∀_:Univ.Univ.
+∀combinator:Univ.
+∀l:Univ.
+∀q:Univ.
+∀w:Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply q X) Y) Z) (apply Y (apply X Z)).
+∀H1:∀X:Univ.∀Y:Univ.eq Univ (apply (apply w X) Y) (apply (apply X Y) Y).
+∀H2:∀X:Univ.∀Y:Univ.eq Univ (apply (apply l X) Y) (apply X (apply Y Y)).∃Y:Univ.eq Univ Y (apply combinator Y)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#apply.
+#combinator.
+#l.
+#q.
+#w.
+#H0.
+#H1.
+#H2.
+napply ex_intro[
+nid2:
+nauto by H0,H1,H2;
+nid|
+skip]
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL019-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL019-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Weak fixed point for B, S, and T *)
+
+(* Version : [WM88] (equality) axioms. *)
+
+(* English : The weak fixed point property holds for the set P consisting *)
+
+(* of the combinators B, S, and T, where ((Sx)y)z = (xz)(yz), *)
+
+(* ((Bx)y)z = x(yz), (Tx)y = yx. *)
+
+(* Refs : [Smu85] Smullyan (1978), To Mock a Mocking Bird and Other Logi *)
+
+(* : [MW87] McCune & Wos (1987), A Case Study in Automated Theorem *)
+
+(* : [WM88] Wos & McCune (1988), Challenge Problems Focusing on Eq *)
+
+(* : [MW88] McCune & Wos (1988), Some Fixed Point Problems in Comb *)
+
+(* Source : [MW88] *)
+
+(* Names : - [MW88] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.1.0, 0.13 v2.0.0 *)
+
+(* Syntax : Number of clauses : 4 ( 0 non-Horn; 4 unit; 1 RR) *)
+
+(* Number of atoms : 4 ( 4 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 9 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_fixed_point:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀apply:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀combinator:Univ.
+∀s:Univ.
+∀t:Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (apply (apply t X) Y) (apply Y X).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply b X) Y) Z) (apply X (apply Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply s X) Y) Z) (apply (apply X Z) (apply Y Z)).∃Y:Univ.eq Univ Y (apply combinator Y)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#apply.
+#b.
+#combinator.
+#s.
+#t.
+#H0.
+#H1.
+#H2.
+napply ex_intro[
+nid2:
+nauto by H0,H1,H2;
+nid|
+skip]
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL020-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL020-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Weak fixed point for B, S, and C *)
+
+(* Version : [WM88] (equality) axioms. *)
+
+(* English : The weak fixed point property holds for the set P consisting *)
+
+(* of the combinators B, S, and C, where ((Sx)y)z = (xz)(yz), *)
+
+(* ((Bx)y)z = x(yz), ((Cx)y)z = (xz)y. *)
+
+(* Refs : [Smu85] Smullyan (1978), To Mock a Mocking Bird and Other Logi *)
+
+(* : [MW87] McCune & Wos (1987), A Case Study in Automated Theorem *)
+
+(* : [WM88] Wos & McCune (1988), Challenge Problems Focusing on Eq *)
+
+(* : [MW88] McCune & Wos (1988), Some Fixed Point Problems in Comb *)
+
+(* Source : [MW88] *)
+
+(* Names : - [MW88] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.07 v3.1.0, 0.11 v2.7.0, 0.00 v2.6.0, 0.17 v2.5.0, 0.00 v2.4.0, 0.33 v2.3.0, 0.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 4 ( 0 non-Horn; 4 unit; 1 RR) *)
+
+(* Number of atoms : 4 ( 4 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 10 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_fixed_point:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀apply:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀c:Univ.
+∀combinator:Univ.
+∀s:Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply c X) Y) Z) (apply (apply X Z) Y).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply b X) Y) Z) (apply X (apply Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply s X) Y) Z) (apply (apply X Z) (apply Y Z)).∃Y:Univ.eq Univ Y (apply combinator Y)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#apply.
+#b.
+#c.
+#combinator.
+#s.
+#H0.
+#H1.
+#H2.
+napply ex_intro[
+nid2:
+nauto by H0,H1,H2;
+nid|
+skip]
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL021-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL021-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Weak fixed point for B, M, and V *)
+
+(* Version : [WM88] (equality) axioms. *)
+
+(* English : The weak fixed point property holds for the set P consisting *)
+
+(* of the combinators B, M, and V, where ((Bx)y)z = x(yz), *)
+
+(* Mx = xx, ((Vx)y)z = (zx)y. *)
+
+(* Refs : [Smu85] Smullyan (1978), To Mock a Mocking Bird and Other Logi *)
+
+(* : [MW87] McCune & Wos (1987), A Case Study in Automated Theorem *)
+
+(* : [WM88] Wos & McCune (1988), Challenge Problems Focusing on Eq *)
+
+(* : [MW88] McCune & Wos (1988), Some Fixed Point Problems in Comb *)
+
+(* Source : [MW88] *)
+
+(* Names : - [MW88] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 4 ( 0 non-Horn; 4 unit; 1 RR) *)
+
+(* Number of atoms : 4 ( 4 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 8 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_fixed_point:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀apply:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀combinator:Univ.
+∀m:Univ.
+∀v:Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply v X) Y) Z) (apply (apply Z X) Y).
+∀H1:∀X:Univ.eq Univ (apply m X) (apply X X).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply b X) Y) Z) (apply X (apply Y Z)).∃Y:Univ.eq Univ Y (apply combinator Y)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#apply.
+#b.
+#combinator.
+#m.
+#v.
+#H0.
+#H1.
+#H2.
+napply ex_intro[
+nid2:
+nauto by H0,H1,H2;
+nid|
+skip]
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL022-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL022-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Weak fixed point for B, O, and M *)
+
+(* Version : [WM88] (equality) axioms. *)
+
+(* English : The weak fixed point property holds for the set P consisting *)
+
+(* of the combinators B, O, and M, where ((Bx)y)z = x(yz), *)
+
+(* Mx = xx, (Ox)y = y(xy). *)
+
+(* Refs : [Smu85] Smullyan (1978), To Mock a Mocking Bird and Other Logi *)
+
+(* : [MW87] McCune & Wos (1987), A Case Study in Automated Theorem *)
+
+(* : [WM88] Wos & McCune (1988), Challenge Problems Focusing on Eq *)
+
+(* : [MW88] McCune & Wos (1988), Some Fixed Point Problems in Comb *)
+
+(* Source : [MW88] *)
+
+(* Names : - [MW88] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 4 ( 0 non-Horn; 4 unit; 1 RR) *)
+
+(* Number of atoms : 4 ( 4 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_fixed_point:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀apply:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀combinator:Univ.
+∀m:Univ.
+∀o:Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (apply (apply o X) Y) (apply Y (apply X Y)).
+∀H1:∀X:Univ.eq Univ (apply m X) (apply X X).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply b X) Y) Z) (apply X (apply Y Z)).∃Y:Univ.eq Univ Y (apply combinator Y)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#apply.
+#b.
+#combinator.
+#m.
+#o.
+#H0.
+#H1.
+#H2.
+napply ex_intro[
+nid2:
+nauto by H0,H1,H2;
+nid|
+skip]
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL023-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL023-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Weak fixed point for B and N *)
+
+(* Version : [WM88] (equality) axioms. *)
+
+(* English : The weak fixed point property holds for the set P consisting *)
+
+(* of the combinators B and N, where ((Bx)y)z = x(yz), ((Nx)y)z *)
+
+(* = ((xz)y)z. *)
+
+(* Refs : [Smu85] Smullyan (1978), To Mock a Mocking Bird and Other Logi *)
+
+(* : [MW87] McCune & Wos (1987), A Case Study in Automated Theorem *)
+
+(* : [WM88] Wos & McCune (1988), Challenge Problems Focusing on Eq *)
+
+(* : [MW88] McCune & Wos (1988), Some Fixed Point Problems in Comb *)
+
+(* Source : [MW88] *)
+
+(* Names : - [MW88] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_fixed_point:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀apply:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀combinator:Univ.
+∀n:Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply n X) Y) Z) (apply (apply (apply X Z) Y) Z).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply b X) Y) Z) (apply X (apply Y Z)).∃Y:Univ.eq Univ Y (apply combinator Y)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#apply.
+#b.
+#combinator.
+#n.
+#H0.
+#H1.
+napply ex_intro[
+nid2:
+nauto by H0,H1;
+nid|
+skip]
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL024-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL024-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Weak fixed point for B, M, and C *)
+
+(* Version : [WM88] (equality) axioms. *)
+
+(* English : The weak fixed point property holds for the set P consisting *)
+
+(* of the combinators B, M, and C, where ((Bx)y)z = x(yz), *)
+
+(* Mx = xx, ((Cx)y)z = (xz)y. *)
+
+(* Refs : [Smu85] Smullyan (1978), To Mock a Mocking Bird and Other Logi *)
+
+(* : [MW87] McCune & Wos (1987), A Case Study in Automated Theorem *)
+
+(* : [WM88] Wos & McCune (1988), Challenge Problems Focusing on Eq *)
+
+(* : [MW88] McCune & Wos (1988), Some Fixed Point Problems in Comb *)
+
+(* Source : [MW88] *)
+
+(* Names : - [MW88] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 4 ( 0 non-Horn; 4 unit; 1 RR) *)
+
+(* Number of atoms : 4 ( 4 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 8 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_fixed_point:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀apply:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀c:Univ.
+∀combinator:Univ.
+∀m:Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply c X) Y) Z) (apply (apply X Z) Y).
+∀H1:∀X:Univ.eq Univ (apply m X) (apply X X).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply b X) Y) Z) (apply X (apply Y Z)).∃Y:Univ.eq Univ Y (apply combinator Y)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#apply.
+#b.
+#c.
+#combinator.
+#m.
+#H0.
+#H1.
+#H2.
+napply ex_intro[
+nid2:
+nauto by H0,H1,H2;
+nid|
+skip]
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL025-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL025-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Weak fixed point for B and W *)
+
+(* Version : [WM88] (equality) axioms. *)
+
+(* English : The weak fixed point property holds for the set P consisting *)
+
+(* of the combinators B and W, where ((Bx)y)z = x(yz), (Wx)y *)
+
+(* = (xy)y. *)
+
+(* Refs : [Smu85] Smullyan (1978), To Mock a Mocking Bird and Other Logi *)
+
+(* : [MW87] McCune & Wos (1987), A Case Study in Automated Theorem *)
+
+(* : [WM88] Wos & McCune (1988), Challenge Problems Focusing on Eq *)
+
+(* : [MW88] McCune & Wos (1988), Some Fixed Point Problems in Comb *)
+
+(* Source : [MW88] *)
+
+(* Names : stage1.in & stage2.in [OTTER] *)
+
+(* : - [MW88] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 6 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_fixed_point:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀apply:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀combinator:Univ.
+∀w:Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (apply (apply w X) Y) (apply (apply X Y) Y).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply b X) Y) Z) (apply X (apply Y Z)).∃Y:Univ.eq Univ Y (apply combinator Y)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#apply.
+#b.
+#combinator.
+#w.
+#H0.
+#H1.
+napply ex_intro[
+nid2:
+nauto by H0,H1;
+nid|
+skip]
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL026-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL026-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Weak fixed point for B and W1 *)
+
+(* Version : [WM88] (equality) axioms. *)
+
+(* English : The weak fixed point property holds for the set P consisting *)
+
+(* of the combinators B and W1, where ((Bx)y)z = x(yz), (W1x)y *)
+
+(* = (yx)x. *)
+
+(* Refs : [Smu85] Smullyan (1978), To Mock a Mocking Bird and Other Logi *)
+
+(* : [MW87] McCune & Wos (1987), A Case Study in Automated Theorem *)
+
+(* : [WM88] Wos & McCune (1988), Challenge Problems Focusing on Eq *)
+
+(* : [MW88] McCune & Wos (1988), Some Fixed Point Problems in Comb *)
+
+(* Source : [MW88] *)
+
+(* Names : - [MW88] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.07 v3.1.0, 0.11 v2.7.0, 0.00 v2.1.0, 0.13 v2.0.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 6 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_fixed_point:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀apply:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀combinator:Univ.
+∀w1:Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (apply (apply w1 X) Y) (apply (apply Y X) X).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply b X) Y) Z) (apply X (apply Y Z)).∃Y:Univ.eq Univ Y (apply combinator Y)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#apply.
+#b.
+#combinator.
+#w1.
+#H0.
+#H1.
+napply ex_intro[
+nid2:
+nauto by H0,H1;
+nid|
+skip]
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL027-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL027-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Weak fixed point for B and H *)
+
+(* Version : [WM88] (equality) axioms. *)
+
+(* English : The weak fixed point property holds for the set P consisting *)
+
+(* of the combinators B and H, where ((Bx)y)z = x(yz), ((Hx)y)z *)
+
+(* = ((xy)z)y. *)
+
+(* Refs : [Smu85] Smullyan (1978), To Mock a Mocking Bird and Other Logi *)
+
+(* : [MW87] McCune & Wos (1987), A Case Study in Automated Theorem *)
+
+(* : [WM88] Wos & McCune (1988), Challenge Problems Focusing on Eq *)
+
+(* : [MW88] McCune & Wos (1988), Some Fixed Point Problems in Comb *)
+
+(* Source : [MW88] *)
+
+(* Names : - [MW88] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_fixed_point:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀apply:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀combinator:Univ.
+∀h:Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply h X) Y) Z) (apply (apply (apply X Y) Z) Y).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply b X) Y) Z) (apply X (apply Y Z)).∃Y:Univ.eq Univ Y (apply combinator Y)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#apply.
+#b.
+#combinator.
+#h.
+#H0.
+#H1.
+napply ex_intro[
+nid2:
+nauto by H0,H1;
+nid|
+skip]
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL029-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL029-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Strong fixed point for U *)
+
+(* Version : [WM88] (equality) axioms. *)
+
+(* English : The strong fixed point property holds for the set *)
+
+(* P consisting of the combinator U, where (Ux)y = y((xx)y). *)
+
+(* Refs : [Smu85] Smullyan (1978), To Mock a Mocking Bird and Other Logi *)
+
+(* : [MW87] McCune & Wos (1987), A Case Study in Automated Theorem *)
+
+(* : [WM88] Wos & McCune (1988), Challenge Problems Focusing on Eq *)
+
+(* : [MW88] McCune & Wos (1988), Some Fixed Point Problems in Comb *)
+
+(* : [Wos93] Wos (1993), The Kernel Strategy and Its Use for the St *)
+
+(* Source : [MW88] *)
+
+(* Names : - [MW88] *)
+
+(* : Question 1 [Wos93] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 3 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 4 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_fixed_point:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.
+∀apply:∀_:Univ.∀_:Univ.Univ.
+∀f:∀_:Univ.Univ.
+∀u:Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (apply (apply u X) Y) (apply Y (apply (apply X X) Y)).∃Y:Univ.eq Univ (apply Y (f Y)) (apply (f Y) (apply Y (f Y)))
+.
+#Univ.
+#X.
+#Y.
+#apply.
+#f.
+#u.
+#H0.
+napply ex_intro[
+nid2:
+nauto by H0;
+nid|
+skip]
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL030-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL030-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Strong fixed point for S and L *)
+
+(* Version : [WM88] (equality) axioms. *)
+
+(* English : The strong fixed point property holds for the set *)
+
+(* P consisting of the combinators S and L, where ((Sx)y)z *)
+
+(* = (xz)(yz), (Lx)y = x(yy). *)
+
+(* Refs : [Smu85] Smullyan (1978), To Mock a Mocking Bird and Other Logi *)
+
+(* : [MW87] McCune & Wos (1987), A Case Study in Automated Theorem *)
+
+(* : [WM88] Wos & McCune (1988), Challenge Problems Focusing on Eq *)
+
+(* : [MW88] McCune & Wos (1988), Some Fixed Point Problems in Comb *)
+
+(* Source : [MW88] *)
+
+(* Names : - [MW88] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.14 v3.1.0, 0.11 v2.7.0, 0.09 v2.6.0, 0.17 v2.5.0, 0.00 v2.2.1, 0.11 v2.2.0, 0.00 v2.1.0, 0.13 v2.0.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 6 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_fixed_point:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀apply:∀_:Univ.∀_:Univ.Univ.
+∀f:∀_:Univ.Univ.
+∀l:Univ.
+∀s:Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (apply (apply l X) Y) (apply X (apply Y Y)).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply s X) Y) Z) (apply (apply X Z) (apply Y Z)).∃Y:Univ.eq Univ (apply Y (f Y)) (apply (f Y) (apply Y (f Y)))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#apply.
+#f.
+#l.
+#s.
+#H0.
+#H1.
+napply ex_intro[
+nid2:
+nauto by H0,H1;
+nid|
+skip]
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL031-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL031-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Strong fixed point for L and O *)
+
+(* Version : [WM88] (equality) axioms. *)
+
+(* English : The strong fixed point property holds for the set *)
+
+(* P consisting of the combinators L and O, where (Lx)y = x(yy), *)
+
+(* (Ox)y = y(xy). *)
+
+(* Refs : [Smu85] Smullyan (1978), To Mock a Mocking Bird and Other Logi *)
+
+(* : [MW87] McCune & Wos (1987), A Case Study in Automated Theorem *)
+
+(* : [WM88] Wos & McCune (1988), Challenge Problems Focusing on Eq *)
+
+(* : [MW88] McCune & Wos (1988), Some Fixed Point Problems in Comb *)
+
+(* Source : [MW88] *)
+
+(* Names : - [MW88] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_fixed_point:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.
+∀apply:∀_:Univ.∀_:Univ.Univ.
+∀f:∀_:Univ.Univ.
+∀l:Univ.
+∀o:Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (apply (apply o X) Y) (apply Y (apply X Y)).
+∀H1:∀X:Univ.∀Y:Univ.eq Univ (apply (apply l X) Y) (apply X (apply Y Y)).∃Y:Univ.eq Univ (apply Y (f Y)) (apply (f Y) (apply Y (f Y)))
+.
+#Univ.
+#X.
+#Y.
+#apply.
+#f.
+#l.
+#o.
+#H0.
+#H1.
+napply ex_intro[
+nid2:
+nauto by H0,H1;
+nid|
+skip]
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL032-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL032-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Strong fixed point for Q and M *)
+
+(* Version : [WM88] (equality) axioms. *)
+
+(* English : The strong fixed point property holds for the set *)
+
+(* P consisting of the combinators Q and M, where Mx = xx, *)
+
+(* ((Qx)y)z = y(xz). *)
+
+(* Refs : [Smu85] Smullyan (1978), To Mock a Mocking Bird and Other Logi *)
+
+(* : [MW87] McCune & Wos (1987), A Case Study in Automated Theorem *)
+
+(* : [WM88] Wos & McCune (1988), Challenge Problems Focusing on Eq *)
+
+(* : [MW88] McCune & Wos (1988), Some Fixed Point Problems in Comb *)
+
+(* Source : [MW88] *)
+
+(* Names : - [MW88] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.14 v3.1.0, 0.22 v2.7.0, 0.00 v2.2.1, 0.11 v2.2.0, 0.00 v2.1.0, 0.13 v2.0.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_fixed_point:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀apply:∀_:Univ.∀_:Univ.Univ.
+∀f:∀_:Univ.Univ.
+∀m:Univ.
+∀q:Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply q X) Y) Z) (apply Y (apply X Z)).
+∀H1:∀X:Univ.eq Univ (apply m X) (apply X X).∃Y:Univ.eq Univ (apply Y (f Y)) (apply (f Y) (apply Y (f Y)))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#apply.
+#f.
+#m.
+#q.
+#H0.
+#H1.
+napply ex_intro[
+nid2:
+nauto by H0,H1;
+nid|
+skip]
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL033-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL033-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Strong fixed point for B, M and L *)
+
+(* Version : [WM88] (equality) axioms. *)
+
+(* English : The strong fixed point property holds for the set *)
+
+(* P consisting of the combinators B, M and L, where ((Bx)y)z *)
+
+(* = x(yz), (Lx)y = x(yy), Mx = xx. *)
+
+(* Refs : [Smu85] Smullyan (1978), To Mock a Mocking Bird and Other Logi *)
+
+(* : [MW87] McCune & Wos (1987), A Case Study in Automated Theorem *)
+
+(* : [WM88] Wos & McCune (1988), Challenge Problems Focusing on Eq *)
+
+(* : [MW88] McCune & Wos (1988), Some Fixed Point Problems in Comb *)
+
+(* Source : [MW88] *)
+
+(* Names : - [MW88] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.07 v3.1.0, 0.11 v2.7.0, 0.09 v2.6.0, 0.17 v2.5.0, 0.25 v2.4.0, 0.00 v2.2.1, 0.11 v2.2.0, 0.00 v2.1.0, 0.13 v2.0.0 *)
+
+(* Syntax : Number of clauses : 4 ( 0 non-Horn; 4 unit; 1 RR) *)
+
+(* Number of atoms : 4 ( 4 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_fixed_point:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀apply:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀f:∀_:Univ.Univ.
+∀l:Univ.
+∀m:Univ.
+∀H0:∀X:Univ.eq Univ (apply m X) (apply X X).
+∀H1:∀X:Univ.∀Y:Univ.eq Univ (apply (apply l X) Y) (apply X (apply Y Y)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply b X) Y) Z) (apply X (apply Y Z)).∃Y:Univ.eq Univ (apply Y (f Y)) (apply (f Y) (apply Y (f Y)))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#apply.
+#b.
+#f.
+#l.
+#m.
+#H0.
+#H1.
+#H2.
+napply ex_intro[
+nid2:
+nauto by H0,H1,H2;
+nid|
+skip]
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL034-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL034-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Strong fixed point for B, M, and T *)
+
+(* Version : [WM88] (equality) axioms. *)
+
+(* English : The strong fixed point property holds for the set *)
+
+(* P consisting of the combinators B, M, and T, where ((Bx)y)z *)
+
+(* = x(yz), Mx = xx, (Tx)y = yx. *)
+
+(* Refs : [Smu85] Smullyan (1978), To Mock a Mocking Bird and Other Logi *)
+
+(* : [MW87] McCune & Wos (1987), A Case Study in Automated Theorem *)
+
+(* : [WM88] Wos & McCune (1988), Challenge Problems Focusing on Eq *)
+
+(* : [MW88] McCune & Wos (1988), Some Fixed Point Problems in Comb *)
+
+(* Source : [MW88] *)
+
+(* Names : - [MW88] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.36 v3.2.0, 0.43 v3.1.0, 0.56 v2.7.0, 0.36 v2.6.0, 0.17 v2.5.0, 0.25 v2.4.0, 0.00 v2.2.1, 0.44 v2.2.0, 0.43 v2.1.0, 0.62 v2.0.0 *)
+
+(* Syntax : Number of clauses : 4 ( 0 non-Horn; 4 unit; 1 RR) *)
+
+(* Number of atoms : 4 ( 4 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_fixed_point:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀apply:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀f:∀_:Univ.Univ.
+∀m:Univ.
+∀t:Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (apply (apply t X) Y) (apply Y X).
+∀H1:∀X:Univ.eq Univ (apply m X) (apply X X).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply b X) Y) Z) (apply X (apply Y Z)).∃Y:Univ.eq Univ (apply Y (f Y)) (apply (f Y) (apply Y (f Y)))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#apply.
+#b.
+#f.
+#m.
+#t.
+#H0.
+#H1.
+#H2.
+napply ex_intro[
+nid2:
+nauto by H0,H1,H2;
+nid|
+skip]
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL035-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL035-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Strong fixed point for W, Q, and L *)
+
+(* Version : [WM88] (equality) axioms. *)
+
+(* English : The strong fixed point property holds for the set *)
+
+(* P consisting of the combinators W, Q, and L, where (Lx)y *)
+
+(* = x(yy), (Wx)y = (xy)y, ((Qx)y)z = y(xz). *)
+
+(* Refs : [Smu85] Smullyan (1978), To Mock a Mocking Bird and Other Logi *)
+
+(* : [MW87] McCune & Wos (1987), A Case Study in Automated Theorem *)
+
+(* : [WM88] Wos & McCune (1988), Challenge Problems Focusing on Eq *)
+
+(* : [MW88] McCune & Wos (1988), Some Fixed Point Problems in Comb *)
+
+(* Source : [MW88] *)
+
+(* Names : - [MW88] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.21 v3.2.0, 0.14 v3.1.0, 0.22 v2.7.0, 0.09 v2.6.0, 0.17 v2.5.0, 0.00 v2.2.1, 0.11 v2.2.0, 0.00 v2.1.0, 0.25 v2.0.0 *)
+
+(* Syntax : Number of clauses : 4 ( 0 non-Horn; 4 unit; 1 RR) *)
+
+(* Number of atoms : 4 ( 4 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 8 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_fixed_point:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀apply:∀_:Univ.∀_:Univ.Univ.
+∀f:∀_:Univ.Univ.
+∀l:Univ.
+∀q:Univ.
+∀w:Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply q X) Y) Z) (apply Y (apply X Z)).
+∀H1:∀X:Univ.∀Y:Univ.eq Univ (apply (apply w X) Y) (apply (apply X Y) Y).
+∀H2:∀X:Univ.∀Y:Univ.eq Univ (apply (apply l X) Y) (apply X (apply Y Y)).∃Y:Univ.eq Univ (apply Y (f Y)) (apply (f Y) (apply Y (f Y)))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#apply.
+#f.
+#l.
+#q.
+#w.
+#H0.
+#H1.
+#H2.
+napply ex_intro[
+nid2:
+nauto by H0,H1,H2;
+nid|
+skip]
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL036-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL036-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Strong fixed point for B, S, and T *)
+
+(* Version : [WM88] (equality) axioms. *)
+
+(* English : The strong fixed point property holds for the set *)
+
+(* P consisting of the combinators B, S, and T, where ((Sx)y)z *)
+
+(* = (xz)(yz), ((Bx)y)z = x(yz), (Tx)y = yx. *)
+
+(* Refs : [Smu85] Smullyan (1978), To Mock a Mocking Bird and Other Logi *)
+
+(* : [MW87] McCune & Wos (1987), A Case Study in Automated Theorem *)
+
+(* : [WM88] Wos & McCune (1988), Challenge Problems Focusing on Eq *)
+
+(* : [MW88] McCune & Wos (1988), Some Fixed Point Problems in Comb *)
+
+(* Source : [MW88] *)
+
+(* Names : - [MW88] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.50 v3.2.0, 0.57 v3.1.0, 0.67 v2.7.0, 0.64 v2.6.0, 0.33 v2.5.0, 0.00 v2.4.0, 0.00 v2.2.1, 0.22 v2.2.0, 0.14 v2.1.0, 0.75 v2.0.0 *)
+
+(* Syntax : Number of clauses : 4 ( 0 non-Horn; 4 unit; 1 RR) *)
+
+(* Number of atoms : 4 ( 4 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 9 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_fixed_point:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀apply:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀f:∀_:Univ.Univ.
+∀s:Univ.
+∀t:Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (apply (apply t X) Y) (apply Y X).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply b X) Y) Z) (apply X (apply Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply s X) Y) Z) (apply (apply X Z) (apply Y Z)).∃Y:Univ.eq Univ (apply Y (f Y)) (apply (f Y) (apply Y (f Y)))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#apply.
+#b.
+#f.
+#s.
+#t.
+#H0.
+#H1.
+#H2.
+napply ex_intro[
+nid2:
+nauto by H0,H1,H2;
+nid|
+skip]
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL037-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL037-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Strong fixed point for B, S, and C *)
+
+(* Version : [WM88] (equality) axioms. *)
+
+(* English : The strong fixed point property holds for the set *)
+
+(* P consisting of the combinators B, S, and C, where ((Sx)y)z *)
+
+(* = (xz)(yz), ((Bx)y)z = x(yz), ((Cx)y)z = (xz)y. *)
+
+(* Refs : [Smu85] Smullyan (1978), To Mock a Mocking Bird and Other Logi *)
+
+(* : [MW87] McCune & Wos (1987), A Case Study in Automated Theorem *)
+
+(* : [WM88] Wos & McCune (1988), Challenge Problems Focusing on Eq *)
+
+(* : [MW88] McCune & Wos (1988), Some Fixed Point Problems in Comb *)
+
+(* Source : [MW88] *)
+
+(* Names : - [MW88] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.29 v3.1.0, 0.44 v2.7.0, 0.36 v2.6.0, 0.33 v2.5.0, 0.25 v2.4.0, 0.33 v2.3.0, 0.00 v2.2.1, 0.22 v2.2.0, 0.14 v2.1.0, 0.88 v2.0.0 *)
+
+(* Syntax : Number of clauses : 4 ( 0 non-Horn; 4 unit; 1 RR) *)
+
+(* Number of atoms : 4 ( 4 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 10 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 4 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_fixed_point:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀apply:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀c:Univ.
+∀f:∀_:Univ.Univ.
+∀s:Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply c X) Y) Z) (apply (apply X Z) Y).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply b X) Y) Z) (apply X (apply Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply s X) Y) Z) (apply (apply X Z) (apply Y Z)).∃Y:Univ.eq Univ (apply Y (f Y)) (apply (f Y) (apply Y (f Y)))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#apply.
+#b.
+#c.
+#f.
+#s.
+#H0.
+#H1.
+#H2.
+napply ex_intro[
+nid2:
+nauto by H0,H1,H2;
+nid|
+skip]
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL038-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL038-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Strong fixed point for B, M, and V *)
+
+(* Version : [WM88] (equality) axioms. *)
+
+(* English : The strong fixed point property holds for the set *)
+
+(* P consisting of the combinators B, M, and V, where ((Bx)y)z *)
+
+(* = x(yz), Mx = xx, ((Vx)y)z = (zx)y. *)
+
+(* Refs : [Smu85] Smullyan (1978), To Mock a Mocking Bird and Other Logi *)
+
+(* : [MW87] McCune & Wos (1987), A Case Study in Automated Theorem *)
+
+(* : [WM88] Wos & McCune (1988), Challenge Problems Focusing on Eq *)
+
+(* : [MW88] McCune & Wos (1988), Some Fixed Point Problems in Comb *)
+
+(* Source : [MW88] *)
+
+(* Names : - [MW88] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.57 v3.2.0, 0.64 v3.1.0, 0.78 v2.7.0, 0.64 v2.6.0, 0.17 v2.5.0, 0.50 v2.4.0, 0.00 v2.2.1, 0.56 v2.2.0, 0.43 v2.1.0, 0.88 v2.0.0 *)
+
+(* Syntax : Number of clauses : 4 ( 0 non-Horn; 4 unit; 1 RR) *)
+
+(* Number of atoms : 4 ( 4 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 8 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_fixed_point:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀apply:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀f:∀_:Univ.Univ.
+∀m:Univ.
+∀v:Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply v X) Y) Z) (apply (apply Z X) Y).
+∀H1:∀X:Univ.eq Univ (apply m X) (apply X X).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply b X) Y) Z) (apply X (apply Y Z)).∃Y:Univ.eq Univ (apply Y (f Y)) (apply (f Y) (apply Y (f Y)))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#apply.
+#b.
+#f.
+#m.
+#v.
+#H0.
+#H1.
+#H2.
+napply ex_intro[
+nid2:
+nauto by H0,H1,H2;
+nid|
+skip]
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL039-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL039-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Strong fixed point for B, O, and M *)
+
+(* Version : [WM88] (equality) axioms. *)
+
+(* English : The strong fixed point property holds for the set *)
+
+(* P consisting of the combinators B, O, and M, where ((Bx)y)z *)
+
+(* = x(yz), Mx = xx, (Ox)y = y(xy). *)
+
+(* Refs : [Smu85] Smullyan (1978), To Mock a Mocking Bird and Other Logi *)
+
+(* : [MW87] McCune & Wos (1987), A Case Study in Automated Theorem *)
+
+(* : [WM88] Wos & McCune (1988), Challenge Problems Focusing on Eq *)
+
+(* : [MW88] McCune & Wos (1988), Some Fixed Point Problems in Comb *)
+
+(* Source : [MW88] *)
+
+(* Names : - [MW88] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v3.1.0, 0.22 v2.7.0, 0.09 v2.6.0, 0.00 v2.2.1, 0.11 v2.2.0, 0.00 v2.1.0, 0.25 v2.0.0 *)
+
+(* Syntax : Number of clauses : 4 ( 0 non-Horn; 4 unit; 1 RR) *)
+
+(* Number of atoms : 4 ( 4 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_fixed_point:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀apply:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀f:∀_:Univ.Univ.
+∀m:Univ.
+∀o:Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (apply (apply o X) Y) (apply Y (apply X Y)).
+∀H1:∀X:Univ.eq Univ (apply m X) (apply X X).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply b X) Y) Z) (apply X (apply Y Z)).∃Y:Univ.eq Univ (apply Y (f Y)) (apply (f Y) (apply Y (f Y)))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#apply.
+#b.
+#f.
+#m.
+#o.
+#H0.
+#H1.
+#H2.
+napply ex_intro[
+nid2:
+nauto by H0,H1,H2;
+nid|
+skip]
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL041-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL041-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Strong fixed point for B, M, and C *)
+
+(* Version : [WM88] (equality) axioms. *)
+
+(* English : The strong fixed point property holds for the set *)
+
+(* P consisting of the combinators B, M, and C, where ((Bx)y)z *)
+
+(* = x(yz), Mx = xx, ((Cx)y)z = (xz)y. *)
+
+(* Refs : [Smu85] Smullyan (1978), To Mock a Mocking Bird and Other Logi *)
+
+(* : [MW87] McCune & Wos (1987), A Case Study in Automated Theorem *)
+
+(* : [WM88] Wos & McCune (1988), Challenge Problems Focusing on Eq *)
+
+(* : [MW88] McCune & Wos (1988), Some Fixed Point Problems in Comb *)
+
+(* Source : [MW88] *)
+
+(* Names : - [MW88] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.29 v3.2.0, 0.36 v3.1.0, 0.56 v2.7.0, 0.27 v2.6.0, 0.17 v2.5.0, 0.00 v2.4.0, 0.00 v2.2.1, 0.22 v2.2.0, 0.14 v2.1.0, 0.62 v2.0.0 *)
+
+(* Syntax : Number of clauses : 4 ( 0 non-Horn; 4 unit; 1 RR) *)
+
+(* Number of atoms : 4 ( 4 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 8 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_fixed_point:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀apply:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀c:Univ.
+∀f:∀_:Univ.Univ.
+∀m:Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply c X) Y) Z) (apply (apply X Z) Y).
+∀H1:∀X:Univ.eq Univ (apply m X) (apply X X).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply b X) Y) Z) (apply X (apply Y Z)).∃Y:Univ.eq Univ (apply Y (f Y)) (apply (f Y) (apply Y (f Y)))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#apply.
+#b.
+#c.
+#f.
+#m.
+#H0.
+#H1.
+#H2.
+napply ex_intro[
+nid2:
+nauto by H0,H1,H2;
+nid|
+skip]
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL042-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL042-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Strong fixed point for B and W1 *)
+
+(* Version : [WM88] (equality) axioms. *)
+
+(* English : The strong fixed point property holds for the set *)
+
+(* P consisting of the combinators B and W1, where ((Bx)y)z *)
+
+(* = x(yz), (W1x)y = (yx)x. *)
+
+(* Refs : [Smu85] Smullyan (1978), To Mock a Mocking Bird and Other Logi *)
+
+(* : [MW87] McCune & Wos (1987), A Case Study in Automated Theorem *)
+
+(* : [WM88] Wos & McCune (1988), Challenge Problems Focusing on Eq *)
+
+(* : [MW88] McCune & Wos (1988), Some Fixed Point Problems in Comb *)
+
+(* : [Wos93] Wos (1993), The Kernel Strategy and Its Use for the St *)
+
+(* Source : [MW88] *)
+
+(* Names : - [MW88] *)
+
+(* : Question 5 [Wos93] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.86 v3.1.0, 0.89 v2.7.0, 0.91 v2.6.0, 0.83 v2.5.0, 0.75 v2.4.0, 0.33 v2.2.1, 0.89 v2.2.0, 1.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 6 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_fixed_point:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀apply:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀f:∀_:Univ.Univ.
+∀w1:Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (apply (apply w1 X) Y) (apply (apply Y X) X).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply b X) Y) Z) (apply X (apply Y Z)).∃Y:Univ.eq Univ (apply Y (f Y)) (apply (f Y) (apply Y (f Y)))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#apply.
+#b.
+#f.
+#w1.
+#H0.
+#H1.
+napply ex_intro[
+nid2:
+nauto by H0,H1;
+nid|
+skip]
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL042-6.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL042-6 : TPTP v3.2.0. Released v2.1.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Strong fixed point for B and W1 *)
+
+(* Version : [WM88] (equality) axioms. *)
+
+(* Theorem formulation : The fixed point is provided and checked. *)
+
+(* English : The strong fixed point property holds for the set *)
+
+(* P consisting of the combinators B and W1, where ((Bx)y)z *)
+
+(* = x(yz), (W1x)y = (yx)x. *)
+
+(* Refs : [WM88] Wos & McCune (1988), Challenge Problems Focusing on Eq *)
+
+(* : [Wos93] Wos (1993), The Kernel Strategy and Its Use for the St *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.14 v3.1.0, 0.11 v2.7.0, 0.18 v2.6.0, 0.00 v2.5.0, 0.25 v2.4.0, 0.00 v2.2.1, 0.38 v2.2.0, 0.60 v2.1.0 *)
+
+(* Syntax : Number of clauses : 4 ( 0 non-Horn; 4 unit; 2 RR) *)
+
+(* Number of atoms : 4 ( 4 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 8 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_strong_fixed_point:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀apply:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀fixed_pt:Univ.
+∀strong_fixed_point:Univ.
+∀w1:Univ.
+∀H0:eq Univ strong_fixed_point (apply (apply b (apply (apply b (apply (apply b (apply w1 w1)) (apply b w1))) b)) b).
+∀H1:∀X:Univ.∀Y:Univ.eq Univ (apply (apply w1 X) Y) (apply (apply Y X) X).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply b X) Y) Z) (apply X (apply Y Z)).eq Univ (apply strong_fixed_point fixed_pt) (apply fixed_pt (apply strong_fixed_point fixed_pt))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#apply.
+#b.
+#fixed_pt.
+#strong_fixed_point.
+#w1.
+#H0.
+#H1.
+#H2.
+nauto by H0,H1,H2;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL042-7.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL042-7 : TPTP v3.2.0. Released v2.1.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Strong fixed point for B and W1 *)
+
+(* Version : [WM88] (equality) axioms. *)
+
+(* Theorem formulation : The fixed point is provided and checked. *)
+
+(* English : The strong fixed point property holds for the set *)
+
+(* P consisting of the combinators B and W1, where ((Bx)y)z *)
+
+(* = x(yz), (W1x)y = (yx)x. *)
+
+(* Refs : [WM88] Wos & McCune (1988), Challenge Problems Focusing on Eq *)
+
+(* : [Wos93] Wos (1993), The Kernel Strategy and Its Use for the St *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.14 v3.1.0, 0.11 v2.7.0, 0.18 v2.6.0, 0.00 v2.5.0, 0.25 v2.4.0, 0.00 v2.2.1, 0.38 v2.2.0, 0.60 v2.1.0 *)
+
+(* Syntax : Number of clauses : 4 ( 0 non-Horn; 4 unit; 2 RR) *)
+
+(* Number of atoms : 4 ( 4 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_strong_fixed_point:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀apply:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀fixed_pt:Univ.
+∀strong_fixed_point:Univ.
+∀w1:Univ.
+∀H0:eq Univ strong_fixed_point (apply (apply b (apply (apply b (apply w1 w1)) (apply b w1))) (apply (apply b b) b)).
+∀H1:∀X:Univ.∀Y:Univ.eq Univ (apply (apply w1 X) Y) (apply (apply Y X) X).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply b X) Y) Z) (apply X (apply Y Z)).eq Univ (apply strong_fixed_point fixed_pt) (apply fixed_pt (apply strong_fixed_point fixed_pt))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#apply.
+#b.
+#fixed_pt.
+#strong_fixed_point.
+#w1.
+#H0.
+#H1.
+#H2.
+nauto by H0,H1,H2;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL042-8.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL042-8 : TPTP v3.2.0. Released v2.1.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Strong fixed point for B and W1 *)
+
+(* Version : [WM88] (equality) axioms. *)
+
+(* Theorem formulation : The fixed point is provided and checked. *)
+
+(* English : The strong fixed point property holds for the set *)
+
+(* P consisting of the combinators B and W1, where ((Bx)y)z *)
+
+(* = x(yz), (W1x)y = (yx)x. *)
+
+(* Refs : [WM88] Wos & McCune (1988), Challenge Problems Focusing on Eq *)
+
+(* : [Wos93] Wos (1993), The Kernel Strategy and Its Use for the St *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.14 v3.1.0, 0.11 v2.7.0, 0.18 v2.6.0, 0.00 v2.5.0, 0.25 v2.4.0, 0.00 v2.2.1, 0.50 v2.2.0, 0.80 v2.1.0 *)
+
+(* Syntax : Number of clauses : 4 ( 0 non-Horn; 4 unit; 2 RR) *)
+
+(* Number of atoms : 4 ( 4 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 7 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_strong_fixed_point:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀apply:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀fixed_pt:Univ.
+∀strong_fixed_point:Univ.
+∀w1:Univ.
+∀H0:eq Univ strong_fixed_point (apply (apply b (apply (apply b (apply w1 w1)) (apply (apply b (apply b w1)) b))) b).
+∀H1:∀X:Univ.∀Y:Univ.eq Univ (apply (apply w1 X) Y) (apply (apply Y X) X).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply b X) Y) Z) (apply X (apply Y Z)).eq Univ (apply strong_fixed_point fixed_pt) (apply fixed_pt (apply strong_fixed_point fixed_pt))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#apply.
+#b.
+#fixed_pt.
+#strong_fixed_point.
+#w1.
+#H0.
+#H1.
+#H2.
+nauto by H0,H1,H2;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL042-9.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL042-9 : TPTP v3.2.0. Released v2.1.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Strong fixed point for B and W1 *)
+
+(* Version : [WM88] (equality) axioms. *)
+
+(* Theorem formulation : The fixed point is provided and checked. *)
+
+(* English : The strong fixed point property holds for the set *)
+
+(* P consisting of the combinators B and W1, where ((Bx)y)z *)
+
+(* = x(yz), (W1x)y = (yx)x. *)
+
+(* Refs : [WM88] Wos & McCune (1988), Challenge Problems Focusing on Eq *)
+
+(* : [Wos93] Wos (1993), The Kernel Strategy and Its Use for the St *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.14 v3.1.0, 0.11 v2.7.0, 0.18 v2.6.0, 0.00 v2.5.0, 0.25 v2.4.0, 0.00 v2.2.1, 0.38 v2.2.0, 0.60 v2.1.0 *)
+
+(* Syntax : Number of clauses : 4 ( 0 non-Horn; 4 unit; 2 RR) *)
+
+(* Number of atoms : 4 ( 4 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 5 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_strong_fixed_point:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀apply:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀fixed_pt:Univ.
+∀strong_fixed_point:Univ.
+∀w1:Univ.
+∀H0:eq Univ strong_fixed_point (apply (apply b (apply w1 w1)) (apply (apply b (apply b w1)) (apply (apply b b) b))).
+∀H1:∀X:Univ.∀Y:Univ.eq Univ (apply (apply w1 X) Y) (apply (apply Y X) X).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply b X) Y) Z) (apply X (apply Y Z)).eq Univ (apply strong_fixed_point fixed_pt) (apply fixed_pt (apply strong_fixed_point fixed_pt))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#apply.
+#b.
+#fixed_pt.
+#strong_fixed_point.
+#w1.
+#H0.
+#H1.
+#H2.
+nauto by H0,H1,H2;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL043-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL043-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Strong fixed point for B and H *)
+
+(* Version : [WM88] (equality) axioms. *)
+
+(* English : The strong fixed point property holds for the set *)
+
+(* P consisting of the combinators B and H, where ((Bx)y)z *)
+
+(* = x(yz), ((Hx)y)z = ((xy)z)y. *)
+
+(* Refs : [Smu85] Smullyan (1978), To Mock a Mocking Bird and Other Logi *)
+
+(* : [MW87] McCune & Wos (1987), A Case Study in Automated Theorem *)
+
+(* : [WM88] Wos & McCune (1988), Challenge Problems Focusing on Eq *)
+
+(* : [MW88] McCune & Wos (1988), Some Fixed Point Problems in Comb *)
+
+(* : [LW92] Lusk & Wos (1992), Benchmark Problems in Which Equalit *)
+
+(* : [Wos93] Wos (1993), The Kernel Strategy and Its Use for the St *)
+
+(* Source : [MW88] *)
+
+(* Names : - [MW88] *)
+
+(* : CL4 [LW92] *)
+
+(* : Question 5 [Wos93] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.86 v3.1.0, 0.89 v2.7.0, 0.91 v2.6.0, 0.83 v2.5.0, 0.75 v2.4.0, 0.67 v2.3.0, 1.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 4 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_fixed_point:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀apply:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀f:∀_:Univ.Univ.
+∀h:Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply h X) Y) Z) (apply (apply (apply X Y) Z) Y).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply b X) Y) Z) (apply X (apply Y Z)).∃Y:Univ.eq Univ (apply Y (f Y)) (apply (f Y) (apply Y (f Y)))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#apply.
+#b.
+#f.
+#h.
+#H0.
+#H1.
+napply ex_intro[
+nid2:
+nauto by H0,H1;
+nid|
+skip]
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL043-3.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL043-3 : TPTP v3.2.0. Bugfixed v2.3.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Strong fixed point for B and H *)
+
+(* Version : [WM88] (equality) axioms. *)
+
+(* Theorem formulation : The fixed point is provided and checked. *)
+
+(* English : The strong fixed point property holds for the set *)
+
+(* P consisting of the combinators B and H, where ((Bx)y)z *)
+
+(* = x(yz), ((Hx)y)z = ((xy)z)y. *)
+
+(* Refs : [WM88] Wos & McCune (1988), Challenge Problems Focusing on Eq *)
+
+(* : [Wos93] Wos (1993), The Kernel Strategy and Its Use for the St *)
+
+(* Source : [TPTP] *)
+
+(* Names : - [Wos93] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.79 v3.2.0, 0.71 v3.1.0, 0.78 v2.7.0, 1.00 v2.6.0, 0.83 v2.5.0, 0.75 v2.4.0, 0.67 v2.3.0 *)
+
+(* Syntax : Number of clauses : 4 ( 0 non-Horn; 4 unit; 2 RR) *)
+
+(* Number of atoms : 4 ( 4 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 6 ( 0 singleton) *)
+
+(* Maximal term depth : 11 ( 4 average) *)
+
+(* Comments : *)
+
+(* Bugfixes : v2.3.0 - Clause strong_fixed_point fixed. *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_strong_fixed_point:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀apply:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀fixed_pt:Univ.
+∀h:Univ.
+∀strong_fixed_point:Univ.
+∀H0:eq Univ strong_fixed_point (apply (apply b (apply (apply b (apply (apply h (apply (apply b (apply (apply b h) (apply b b))) (apply h (apply (apply b h) (apply b b))))) h)) b)) b).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply h X) Y) Z) (apply (apply (apply X Y) Z) Y).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply b X) Y) Z) (apply X (apply Y Z)).eq Univ (apply strong_fixed_point fixed_pt) (apply fixed_pt (apply strong_fixed_point fixed_pt))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#apply.
+#b.
+#fixed_pt.
+#h.
+#strong_fixed_point.
+#H0.
+#H1.
+#H2.
+nauto by H0,H1,H2;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL044-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL044-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Strong fixed point for B and N *)
+
+(* Version : [WM88] (equality) axioms. *)
+
+(* English : The strong fixed point property holds for the set *)
+
+(* P consisting of the combinators B and N, where ((Bx)y)z *)
+
+(* = x(yz), ((Nx)y)z = ((xz)y)z. *)
+
+(* Refs : [Smu85] Smullyan (1978), To Mock a Mocking Bird and Other Logi *)
+
+(* : [MW87] McCune & Wos (1987), A Case Study in Automated Theorem *)
+
+(* : [WM88] Wos & McCune (1988), Challenge Problems Focusing on Eq *)
+
+(* : [MW88] McCune & Wos (1988), Some Fixed Point Problems in Comb *)
+
+(* : [LW92] Lusk & Wos (1992), Benchmark Problems in Which Equalit *)
+
+(* : [Wos93] Wos (1993), The Kernel Strategy and Its Use for the St *)
+
+(* Source : [MW88] *)
+
+(* Names : - [MW88] *)
+
+(* : CL3 [LW92] *)
+
+(* : Question 5 [Wos93] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.29 v3.2.0, 0.21 v3.1.0, 0.22 v2.7.0, 0.18 v2.6.0, 0.33 v2.5.0, 0.00 v2.2.1, 0.11 v2.2.0, 0.00 v2.1.0, 0.25 v2.0.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 4 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_fixed_point:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀apply:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀f:∀_:Univ.Univ.
+∀n:Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply n X) Y) Z) (apply (apply (apply X Z) Y) Z).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply b X) Y) Z) (apply X (apply Y Z)).∃Y:Univ.eq Univ (apply Y (f Y)) (apply (f Y) (apply Y (f Y)))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#apply.
+#b.
+#f.
+#n.
+#H0.
+#H1.
+napply ex_intro[
+nid2:
+nauto by H0,H1;
+nid|
+skip]
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL044-6.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL044-6 : TPTP v3.2.0. Released v2.1.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Strong fixed point for B and N *)
+
+(* Version : [WM88] (equality) axioms. *)
+
+(* Theorem formulation : The fixed point is provided and checked. *)
+
+(* English : The strong fixed point property holds for the set *)
+
+(* P consisting of the combinators B and N, where ((Bx)y)z *)
+
+(* = x(yz), ((Nx)y)z = ((xz)y)z. *)
+
+(* Refs : [WM88] Wos & McCune (1988), Challenge Problems Focusing on Eq *)
+
+(* : [Wos93] Wos (1993), The Kernel Strategy and Its Use for the St *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.71 v3.2.0, 0.64 v3.1.0, 0.56 v2.7.0, 0.82 v2.6.0, 0.67 v2.5.0, 0.50 v2.4.0, 0.67 v2.2.1, 0.88 v2.2.0, 0.80 v2.1.0 *)
+
+(* Syntax : Number of clauses : 4 ( 0 non-Horn; 4 unit; 2 RR) *)
+
+(* Number of atoms : 4 ( 4 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 6 ( 0 singleton) *)
+
+(* Maximal term depth : 12 ( 4 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_strong_fixed_point:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀apply:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀fixed_pt:Univ.
+∀n:Univ.
+∀strong_fixed_point:Univ.
+∀H0:eq Univ strong_fixed_point (apply (apply b (apply (apply b (apply (apply n (apply (apply b b) (apply (apply n (apply (apply b b) n)) n))) n)) b)) b).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply n X) Y) Z) (apply (apply (apply X Z) Y) Z).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply b X) Y) Z) (apply X (apply Y Z)).eq Univ (apply strong_fixed_point fixed_pt) (apply fixed_pt (apply strong_fixed_point fixed_pt))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#apply.
+#b.
+#fixed_pt.
+#n.
+#strong_fixed_point.
+#H0.
+#H1.
+#H2.
+nauto by H0,H1,H2;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL044-7.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL044-7 : TPTP v3.2.0. Released v2.1.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Strong fixed point for B and N *)
+
+(* Version : [WM88] (equality) axioms. *)
+
+(* Theorem formulation : The fixed point is provided and checked. *)
+
+(* English : The strong fixed point property holds for the set *)
+
+(* P consisting of the combinators B and N, where ((Bx)y)z *)
+
+(* = x(yz), ((Nx)y)z = ((xz)y)z. *)
+
+(* Refs : [WM88] Wos & McCune (1988), Challenge Problems Focusing on Eq *)
+
+(* : [Wos93] Wos (1993), The Kernel Strategy and Its Use for the St *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.57 v3.2.0, 0.50 v3.1.0, 0.44 v2.7.0, 0.82 v2.6.0, 0.67 v2.5.0, 0.50 v2.4.0, 0.67 v2.2.1, 0.88 v2.2.0, 0.80 v2.1.0 *)
+
+(* Syntax : Number of clauses : 4 ( 0 non-Horn; 4 unit; 2 RR) *)
+
+(* Number of atoms : 4 ( 4 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 6 ( 0 singleton) *)
+
+(* Maximal term depth : 12 ( 4 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_strong_fixed_point:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀apply:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀fixed_pt:Univ.
+∀n:Univ.
+∀strong_fixed_point:Univ.
+∀H0:eq Univ strong_fixed_point (apply (apply b (apply (apply b (apply (apply n (apply (apply b b) (apply (apply n (apply n (apply b b))) n))) n)) b)) b).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply n X) Y) Z) (apply (apply (apply X Z) Y) Z).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply b X) Y) Z) (apply X (apply Y Z)).eq Univ (apply strong_fixed_point fixed_pt) (apply fixed_pt (apply strong_fixed_point fixed_pt))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#apply.
+#b.
+#fixed_pt.
+#n.
+#strong_fixed_point.
+#H0.
+#H1.
+#H2.
+nauto by H0,H1,H2;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL044-8.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL044-8 : TPTP v3.2.0. Released v2.1.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Strong fixed point for B and N *)
+
+(* Version : [WM88] (equality) axioms. *)
+
+(* Theorem formulation : The fixed point is provided and checked. *)
+
+(* English : The strong fixed point property holds for the set *)
+
+(* P consisting of the combinators B and N, where ((Bx)y)z *)
+
+(* = x(yz), ((Nx)y)z = ((xz)y)z. *)
+
+(* Refs : [WM88] Wos & McCune (1988), Challenge Problems Focusing on Eq *)
+
+(* : [Wos93] Wos (1993), The Kernel Strategy and Its Use for the St *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.79 v3.2.0, 0.71 v3.1.0, 0.78 v2.7.0, 1.00 v2.6.0, 0.67 v2.5.0, 0.50 v2.4.0, 0.67 v2.2.1, 0.88 v2.2.0, 0.80 v2.1.0 *)
+
+(* Syntax : Number of clauses : 4 ( 0 non-Horn; 4 unit; 2 RR) *)
+
+(* Number of atoms : 4 ( 4 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 6 ( 0 singleton) *)
+
+(* Maximal term depth : 12 ( 4 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_strong_fixed_point:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀apply:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀fixed_pt:Univ.
+∀n:Univ.
+∀strong_fixed_point:Univ.
+∀H0:eq Univ strong_fixed_point (apply (apply b (apply (apply b (apply (apply n (apply n (apply (apply b (apply b b)) (apply n (apply (apply b b) n))))) n)) b)) b).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply n X) Y) Z) (apply (apply (apply X Z) Y) Z).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply b X) Y) Z) (apply X (apply Y Z)).eq Univ (apply strong_fixed_point fixed_pt) (apply fixed_pt (apply strong_fixed_point fixed_pt))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#apply.
+#b.
+#fixed_pt.
+#n.
+#strong_fixed_point.
+#H0.
+#H1.
+#H2.
+nauto by H0,H1,H2;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL044-9.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL044-9 : TPTP v3.2.0. Released v2.1.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Strong fixed point for B and N *)
+
+(* Version : [WM88] (equality) axioms. *)
+
+(* Theorem formulation : The fixed point is provided and checked. *)
+
+(* English : The strong fixed point property holds for the set *)
+
+(* P consisting of the combinators B and N, where ((Bx)y)z *)
+
+(* = x(yz), ((Nx)y)z = ((xz)y)z. *)
+
+(* Refs : [WM88] Wos & McCune (1988), Challenge Problems Focusing on Eq *)
+
+(* : [Wos93] Wos (1993), The Kernel Strategy and Its Use for the St *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.71 v3.2.0, 0.64 v3.1.0, 0.67 v2.7.0, 1.00 v2.6.0, 0.67 v2.5.0, 0.50 v2.4.0, 0.67 v2.2.1, 0.88 v2.2.0, 0.80 v2.1.0 *)
+
+(* Syntax : Number of clauses : 4 ( 0 non-Horn; 4 unit; 2 RR) *)
+
+(* Number of atoms : 4 ( 4 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 6 ( 0 singleton) *)
+
+(* Maximal term depth : 12 ( 4 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_strong_fixed_point:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀apply:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀fixed_pt:Univ.
+∀n:Univ.
+∀strong_fixed_point:Univ.
+∀H0:eq Univ strong_fixed_point (apply (apply b (apply (apply b (apply (apply n (apply n (apply (apply b (apply b b)) (apply n (apply n (apply b b)))))) n)) b)) b).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply n X) Y) Z) (apply (apply (apply X Z) Y) Z).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply b X) Y) Z) (apply X (apply Y Z)).eq Univ (apply strong_fixed_point fixed_pt) (apply fixed_pt (apply strong_fixed_point fixed_pt))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#apply.
+#b.
+#fixed_pt.
+#n.
+#strong_fixed_point.
+#H0.
+#H1.
+#H2.
+nauto by H0,H1,H2;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL045-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL045-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Weak fixed point for B, M and S *)
+
+(* Version : [WM88] (equality) axioms. *)
+
+(* English : The weak fixed point property holds for the set P consisting *)
+
+(* of the combinators B, M and S, where ((Sx)y)z = (xz)(yz), *)
+
+(* ((Bx)y)z = x(yz), Mx = xx. *)
+
+(* Refs : [Smu85] Smullyan (1978), To Mock a Mocking Bird and Other Logi *)
+
+(* : [MW87] McCune & Wos (1987), A Case Study in Automated Theorem *)
+
+(* : [WM88] Wos & McCune (1988), Challenge Problems Focusing on Eq *)
+
+(* : [Wos89] Wos (1989), A Challenge Problem and a Recent Workshop *)
+
+(* Source : [Wos89] *)
+
+(* Names : - [Wos89] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 4 ( 0 non-Horn; 4 unit; 1 RR) *)
+
+(* Number of atoms : 4 ( 4 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 8 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_fixed_point:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀apply:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀combinator:Univ.
+∀m:Univ.
+∀s:Univ.
+∀H0:∀X:Univ.eq Univ (apply m X) (apply X X).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply b X) Y) Z) (apply X (apply Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply s X) Y) Z) (apply (apply X Z) (apply Y Z)).∃Y:Univ.eq Univ Y (apply combinator Y)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#apply.
+#b.
+#combinator.
+#m.
+#s.
+#H0.
+#H1.
+#H2.
+napply ex_intro[
+nid2:
+nauto by H0,H1,H2;
+nid|
+skip]
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL046-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL046-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Strong fixed point for B, M and S *)
+
+(* Version : [WM88] (equality) axioms. *)
+
+(* English : The strong fixed point property holds for the set *)
+
+(* P consisting of the combinators B, M and S, where ((Sx)y)z *)
+
+(* = (xz)(yz), ((Bx)y)z = x(yz), Mx = xx. *)
+
+(* Refs : [Smu85] Smullyan (1978), To Mock a Mocking Bird and Other Logi *)
+
+(* : [MW87] McCune & Wos (1987), A Case Study in Automated Theorem *)
+
+(* : [WM88] Wos & McCune (1988), Challenge Problems Focusing on Eq *)
+
+(* : [Wos89] Wos (1989), A Challenge Problem and a Recent Workshop *)
+
+(* Source : [Wos89] *)
+
+(* Names : - [Wos89] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.57 v3.2.0, 0.64 v3.1.0, 0.67 v2.7.0, 0.55 v2.6.0, 0.17 v2.5.0, 0.25 v2.4.0, 0.00 v2.2.1, 0.67 v2.2.0, 0.71 v2.1.0, 1.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 4 ( 0 non-Horn; 4 unit; 1 RR) *)
+
+(* Number of atoms : 4 ( 4 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 8 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_fixed_point:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀apply:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀f:∀_:Univ.Univ.
+∀m:Univ.
+∀s:Univ.
+∀H0:∀X:Univ.eq Univ (apply m X) (apply X X).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply b X) Y) Z) (apply X (apply Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply s X) Y) Z) (apply (apply X Z) (apply Y Z)).∃Y:Univ.eq Univ (apply Y (f Y)) (apply (f Y) (apply Y (f Y)))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#apply.
+#b.
+#f.
+#m.
+#s.
+#H0.
+#H1.
+#H2.
+napply ex_intro[
+nid2:
+nauto by H0,H1,H2;
+nid|
+skip]
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL047-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL047-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Find a model for L and Q but not a strong fixed point *)
+
+(* Version : [WM88] (equality) axioms. *)
+
+(* English : The model one is seeking must satisfy L and Q and fail *)
+
+(* to satisfy the strong fixed point property, where (Lx)y *)
+
+(* = x(yy), ((Qx)y)z = y(xz). *)
+
+(* Refs : [Zha92] Zhang (1992), Solution to an Open Question in Combinat *)
+
+(* : [Wos93] Wos (1993), The Kernel Strategy and Its Use for the St *)
+
+(* : [Pel98] Peltier (1998), A New Method for Automated Finite Mode *)
+
+(* Source : [Zhang, 1992] *)
+
+(* Names : Question 7 [Wos93] *)
+
+(* : Question 17 [Wos93] *)
+
+(* : 4.2.5 (CL2) [Pel98] *)
+
+(* Status : Satisfiable *)
+
+(* Rating : 0.33 v3.2.0, 0.67 v3.1.0, 0.33 v2.4.0, 0.67 v2.2.1, 0.75 v2.2.0, 0.67 v2.1.0, 1.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 6 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_model:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀apply:∀_:Univ.∀_:Univ.Univ.
+∀f:∀_:Univ.Univ.
+∀l:Univ.
+∀q:Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply q X) Y) Z) (apply Y (apply X Z)).
+∀H1:∀X:Univ.∀Y:Univ.eq Univ (apply (apply l X) Y) (apply X (apply Y Y)).∃Y:Univ.eq Univ (apply Y (f Y)) (apply (f Y) (apply Y (f Y)))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#apply.
+#f.
+#l.
+#q.
+#H0.
+#H1.
+napply ex_intro[
+nid2:
+nauto by H0,H1;
+nid|
+skip]
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL048-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL048-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Weak fixed point for B, W, and M *)
+
+(* Version : [WM88] (equality) axioms. *)
+
+(* English : The weak fixed point property holds for the set P consisting *)
+
+(* of the combinators B, W, and M, where ((Bx)y)z = x(yz), (Wx)y *)
+
+(* = (xy)y, Mx = xx. *)
+
+(* Refs : [Smu85] Smullyan (1978), To Mock a Mocking Bird and Other Logi *)
+
+(* : [MW87] McCune & Wos (1987), A Case Study in Automated Theorem *)
+
+(* : [WM88] Wos & McCune (1988), Challenge Problems Focusing on Eq *)
+
+(* : [MW88] McCune & Wos (1988), Some Fixed Point Problems in Comb *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 4 ( 0 non-Horn; 4 unit; 1 RR) *)
+
+(* Number of atoms : 4 ( 4 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_fixed_point:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀apply:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀combinator:Univ.
+∀m:Univ.
+∀w:Univ.
+∀H0:∀X:Univ.eq Univ (apply m X) (apply X X).
+∀H1:∀X:Univ.∀Y:Univ.eq Univ (apply (apply w X) Y) (apply (apply X Y) Y).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply b X) Y) Z) (apply X (apply Y Z)).∃Y:Univ.eq Univ Y (apply combinator Y)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#apply.
+#b.
+#combinator.
+#m.
+#w.
+#H0.
+#H1.
+#H2.
+napply ex_intro[
+nid2:
+nauto by H0,H1,H2;
+nid|
+skip]
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL049-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL049-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Strong fixed point for B, W, and M *)
+
+(* Version : [WM88] (equality) axioms. *)
+
+(* English : The strong fixed point property holds for the set *)
+
+(* P consisting of the combinators B, W, and M, where ((Bx)y)z *)
+
+(* = x(yz), (Wx)y = (xy)y, Mx = xx. *)
+
+(* Refs : [Smu85] Smullyan (1978), To Mock a Mocking Bird and Other Logi *)
+
+(* : [MW87] McCune & Wos (1987), A Case Study in Automated Theorem *)
+
+(* : [WM88] Wos & McCune (1988), Challenge Problems Focusing on Eq *)
+
+(* : [Ove90] Overbeek (1990), ATP competition announced at CADE-10 *)
+
+(* : [LW92] Lusk & Wos (1992), Benchmark Problems in Which Equalit *)
+
+(* : [Wos93] Wos (1993), The Kernel Strategy and Its Use for the St *)
+
+(* : [Ove93] Overbeek (1993), The CADE-11 Competitions: A Personal *)
+
+(* : [LM93] Lusk & McCune (1993), Uniform Strategies: The CADE-11 *)
+
+(* : [Zha93] Zhang (1993), Automated Proofs of Equality Problems in *)
+
+(* Source : [Ove90] *)
+
+(* Names : Problem 2 [WM88] *)
+
+(* : CADE-11 Competition Eq-6 [Ove90] *)
+
+(* : CL1 [LW92] *)
+
+(* : THEOREM EQ-6 [LM93] *)
+
+(* : Question 2 [Wos93] *)
+
+(* : PROBLEM 6 [Zha93] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.29 v3.1.0, 0.44 v2.7.0, 0.27 v2.6.0, 0.17 v2.5.0, 0.25 v2.4.0, 0.00 v2.2.1, 0.22 v2.2.0, 0.14 v2.1.0, 0.62 v2.0.0 *)
+
+(* Syntax : Number of clauses : 4 ( 0 non-Horn; 4 unit; 1 RR) *)
+
+(* Number of atoms : 4 ( 4 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_strong_fixed_point:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀apply:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀f:∀_:Univ.Univ.
+∀m:Univ.
+∀w:Univ.
+∀H0:∀X:Univ.eq Univ (apply m X) (apply X X).
+∀H1:∀X:Univ.∀Y:Univ.eq Univ (apply (apply w X) Y) (apply (apply X Y) Y).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply b X) Y) Z) (apply X (apply Y Z)).∃Y:Univ.eq Univ (apply Y (f Y)) (apply (f Y) (apply Y (f Y)))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#apply.
+#b.
+#f.
+#m.
+#w.
+#H0.
+#H1.
+#H2.
+napply ex_intro[
+nid2:
+nauto by H0,H1,H2;
+nid|
+skip]
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL050-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL050-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : The Significance of the Mockingbird *)
+
+(* Version : Especial. *)
+
+(* English : There exists a mocking bird. For all birds x and y, there *)
+
+(* exists a bird z that composes x with y for all birds w. Prove *)
+
+(* that every bird is fond of at least one other bird. *)
+
+(* Refs : [Smu85] Smullyan (1978), To Mock a Mocking Bird and Other Logi *)
+
+(* Source : [ANL] *)
+
+(* Names : bird1.ver1.in [ANL] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ---- There exists a mocking bird (Mock). *)
+
+(* ---- TEx FAy [response(x,y) = response(y,y)]. *)
+
+(* ---- response(Mock,y) = response(y,y). *)
+
+(* ---- For all birds x and y, there exists a bird z that composes *)
+
+(* ---- x with y for all birds w. *)
+
+(* ---- FAx FAy TEz FAw [response(z,w) = response(x,response(y,w))] *)
+
+(* ---- response(comp(x,y),w) = response(x,response(y,w)). *)
+
+(* ---- Hypothesis: Every bird is fond of at least one other bird. *)
+
+(* ---- -FAx TEy [response(x,y) = y]. *)
+
+(* ---- TEx FAy -[response(x,y) = y]. *)
+
+(* ---- Letting A = x, *)
+
+(* ---- -[response(A,y) = y]. *)
+ntheorem prove_all_fond_of_another:
+ ∀Univ:Type.∀W:Univ.∀X:Univ.∀Y:Univ.
+∀a:Univ.
+∀compose:∀_:Univ.∀_:Univ.Univ.
+∀mocking_bird:Univ.
+∀response:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀W:Univ.∀X:Univ.∀Y:Univ.eq Univ (response (compose X Y) W) (response X (response Y W)).
+∀H1:∀Y:Univ.eq Univ (response mocking_bird Y) (response Y Y).∃Y:Univ.eq Univ (response a Y) Y
+.
+#Univ.
+#W.
+#X.
+#Y.
+#a.
+#compose.
+#mocking_bird.
+#response.
+#H0.
+#H1.
+napply ex_intro[
+nid2:
+nauto by H0,H1;
+nid|
+skip]
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL051-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL051-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Egocentric mocking bird? *)
+
+(* Version : Especial. *)
+
+(* English : There exists a mocking bird. For all birds x and y, there *)
+
+(* exists a bird z that composes x with y for all birds w. Prove *)
+
+(* that there exists a bird x that is fond of itself. *)
+
+(* Refs : [Smu85] Smullyan (1978), To Mock a Mocking Bird and Other Logi *)
+
+(* Source : [ANL] *)
+
+(* Names : bird2.ver1.in [ANL] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ---- There exists a mocking bird (Mock). *)
+
+(* ---- TEx FAy [response(x,y) = response(y,y)]. *)
+
+(* ---- response(Mock,y) = response(y,y). *)
+
+(* ---- For all birds x and y, there exists a bird z that composes *)
+
+(* ---- x with y for all birds w. *)
+
+(* ---- FAx FAy TEz FAw [response(z,w) = response(x,response(y,w))] *)
+
+(* ---- response(comp(x,y),w) = response(x,response(y,w)). *)
+
+(* ---- Hypothesis: There exists a bird x that is fond of itself. *)
+
+(* ---- -TEx [response(x,x) = x]. *)
+
+(* ---- FAx -[response(x,x) = x]. *)
+ntheorem prove_the_bird_exists:
+ ∀Univ:Type.∀W:Univ.∀X:Univ.∀Y:Univ.
+∀compose:∀_:Univ.∀_:Univ.Univ.
+∀mocking_bird:Univ.
+∀response:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀W:Univ.∀X:Univ.∀Y:Univ.eq Univ (response (compose X Y) W) (response X (response Y W)).
+∀H1:∀Y:Univ.eq Univ (response mocking_bird Y) (response Y Y).∃X:Univ.eq Univ (response X X) X
+.
+#Univ.
+#W.
+#X.
+#Y.
+#compose.
+#mocking_bird.
+#response.
+#H0.
+#H1.
+napply ex_intro[
+nid2:
+nauto by H0,H1;
+nid|
+skip]
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL052-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL052-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : A Question on Agreeable Birds *)
+
+(* Version : Especial. *)
+
+(* Theorem formulation : Implicit definition of agreeable. *)
+
+(* English : For all birds x and y, there exists a bird z that composes *)
+
+(* x with y for all birds w. Prove that if C is agreeable then *)
+
+(* A is agreeable. *)
+
+(* Refs : [Smu85] Smullyan (1978), To Mock a Mocking Bird and Other Logi *)
+
+(* Source : [ANL] *)
+
+(* Names : bird4.ver1.in [ANL] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 4 ( 0 non-Horn; 4 unit; 2 RR) *)
+
+(* Number of atoms : 4 ( 4 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For all birds x and y, there exists a bird z that composes x with *)
+
+(* ----y for all birds w. *)
+
+(* ---- FAx FAy TEz FAw [response(z,w) = response(x,response(y,w))]. *)
+
+(* ---- response(comp(x,y),w) = response(x,response(y,w)). *)
+
+(* ----Hypothesis: If C is agreeable then A is agreeable. *)
+
+(* ---- -[ If FAx TEy (response(C,y) = response(x,y)), *)
+
+(* ---- then FAw TEv (response(A,v) = response(w,v)) ]. *)
+
+(* ---- -[ TEx FAy -(response(C,y) = response(x,y)) | *)
+
+(* ---- FAw TEv (response(A,v) = response(w,v)) ]. *)
+
+(* ---- FAx TEy (response(C,y) = response(x,y)) and *)
+
+(* ---- TEw FAv -(response(A,v) = response(w,v). *)
+
+(* ---- response(C,commom_bird(x)) = response(x,common_bird(x)) and *)
+
+(* ---- -(response(A,v) = response(odd_bird,v)). *)
+ntheorem prove_a_is_agreeable:
+ ∀Univ:Type.∀V:Univ.∀W:Univ.∀X:Univ.∀Y:Univ.
+∀a:Univ.
+∀c:Univ.
+∀common_bird:∀_:Univ.Univ.
+∀compose:∀_:Univ.∀_:Univ.Univ.
+∀odd_bird:Univ.
+∀response:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.eq Univ (response c (common_bird X)) (response X (common_bird X)).
+∀H1:∀W:Univ.∀X:Univ.∀Y:Univ.eq Univ (response (compose X Y) W) (response X (response Y W)).∃V:Univ.eq Univ (response a V) (response odd_bird V)
+.
+#Univ.
+#V.
+#W.
+#X.
+#Y.
+#a.
+#c.
+#common_bird.
+#compose.
+#odd_bird.
+#response.
+#H0.
+#H1.
+napply ex_intro[
+nid2:
+nauto by H0,H1;
+nid|
+skip]
+nqed.
+
+(* ----C composes A with B. WHY is this here? *)
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL053-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL053-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : An Exercise in Composition *)
+
+(* Version : Especial. *)
+
+(* English : For all birds x and y, there exists a bird z that composes *)
+
+(* x with y for all birds w. Prove that for all birds x, y, and *)
+
+(* z, there exists a bird u such that for all w, uw = x(y(zw)). *)
+
+(* Refs : [Smu85] Smullyan (1978), To Mock a Mocking Bird and Other Logi *)
+
+(* Source : [ANL] *)
+
+(* Names : bird5.ver1.in [ANL] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 4 ( 0 singleton) *)
+
+(* Maximal term depth : 5 ( 4 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For all birds x and y, there exists a bird z that composes x with *)
+
+(* ----y for all birds w. *)
+
+(* ---- FAx FAy TEz FAw [response(z,w) = response(x,response(y,w))]. *)
+
+(* ---- response(comp(x,y),w) = response(x,response(y,w)). *)
+
+(* ----Hypothesis: For all birds x, y, and z, there exists a bird u such *)
+
+(* ----that for all w, uw = x(y(zw)). *)
+
+(* ----Finding clause (using xy to replace response(x,y)): *)
+
+(* ---- - (FAx FAy FAz TEu FAw (uw = x(y(zw)))). *)
+
+(* ---- TEx TEy TEz FAu TEw -(uw = x(y(zw))). *)
+
+(* ---- Letting w = f(u), x = A, y = B, and z = C, *)
+
+(* ---- -[(u)f(u) = A(B((C)f(u)))]. *)
+ntheorem prove_bird_exists:
+ ∀Univ:Type.∀U:Univ.∀W:Univ.∀X:Univ.∀Y:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀compose:∀_:Univ.∀_:Univ.Univ.
+∀f:∀_:Univ.Univ.
+∀response:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀W:Univ.∀X:Univ.∀Y:Univ.eq Univ (response (compose X Y) W) (response X (response Y W)).∃U:Univ.eq Univ (response U (f U)) (response a (response b (response c (f U))))
+.
+#Univ.
+#U.
+#W.
+#X.
+#Y.
+#a.
+#b.
+#c.
+#compose.
+#f.
+#response.
+#H0.
+napply ex_intro[
+nid2:
+nauto by H0;
+nid|
+skip]
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL056-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL056-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Normal Birds *)
+
+(* Version : Especial. *)
+
+(* English : For all birds x and y, there exists a bird z that composes *)
+
+(* x with y for all birds w. Prove that if there exists a happy *)
+
+(* bird then there exists a normal bird. *)
+
+(* Refs : [Smu85] Smullyan (1978), To Mock a Mocking Bird and Other Logi *)
+
+(* Source : [ANL] *)
+
+(* Names : bird8.ver1.in [ANL] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.07 v3.1.0, 0.00 v2.7.0, 0.09 v2.6.0, 0.17 v2.5.0, 0.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 4 ( 0 non-Horn; 4 unit; 3 RR) *)
+
+(* Number of atoms : 4 ( 4 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 1 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For all birds x and y, there exists a bird z that composes x with *)
+
+(* ----y for all birds w. *)
+
+(* ---- FAx FAy TEz FAw [response(z,w) = response(x,response(y,w))]. *)
+
+(* ---- response(comp(x,y),w) = response(x,response(y,w)). *)
+
+(* ----Hypothesis: If there exists a happy bird then there exists a normal *)
+
+(* ----bird. *)
+
+(* ----Finding clause (using xy to replace response(x,y)): *)
+
+(* ---- -[ If TEx TEy TEz (xy = z) and (xz = y) *)
+
+(* ---- then TEw TEv (wv = v) ]. *)
+
+(* ---- -[ FAx FAy FAz -((xy = z) and (xz = y)) | TEw TEv (wv = v) ] *)
+
+(* ---- TEx TEy TEz [(xy = z) and (xz = y)] and FAw FAv -(wv = v). *)
+
+(* ---- (AB = C) and (AC = B) and -(wv = v). *)
+ntheorem prove_there_exists_a_happy_bird:
+ ∀Univ:Type.∀V:Univ.∀W:Univ.∀X:Univ.∀Y:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀compose:∀_:Univ.∀_:Univ.Univ.
+∀response:∀_:Univ.∀_:Univ.Univ.
+∀H0:eq Univ (response a c) b.
+∀H1:eq Univ (response a b) c.
+∀H2:∀W:Univ.∀X:Univ.∀Y:Univ.eq Univ (response (compose X Y) W) (response X (response Y W)).∃V:Univ.∃W:Univ.eq Univ (response W V) V
+.
+#Univ.
+#V.
+#W.
+#X.
+#Y.
+#a.
+#b.
+#c.
+#compose.
+#response.
+#H0.
+#H1.
+#H2.
+napply ex_intro[
+nid2:
+napply ex_intro[
+nid2:
+nauto by H0,H1,H2;
+nid|
+skip]
+nid|
+skip]
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL057-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL057-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Strong fixed point for S, B, C, and I *)
+
+(* Version : [WM88] (equality) axioms. *)
+
+(* English : The strong fixed point property holds for the set *)
+
+(* P consisting of the combinators S, B, C, and I, where *)
+
+(* ((Sx)y)z = (xz)(yz), ((Bx)y)z = x(yz), ((Cx)y)z = (xz)y, and *)
+
+(* Ix = x. *)
+
+(* Refs : [LW92] Lusk & Wos (1992), Benchmark Problems in Which Equalit *)
+
+(* Source : [LW92] *)
+
+(* Names : CL5 [LW92] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.36 v3.1.0, 0.56 v2.7.0, 0.27 v2.6.0, 0.17 v2.5.0, 0.00 v2.1.0, 0.25 v2.0.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 1 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 11 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_strong_fixed_point:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀apply:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀c:Univ.
+∀f:∀_:Univ.Univ.
+∀i:Univ.
+∀s:Univ.
+∀H0:∀X:Univ.eq Univ (apply i X) X.
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply c X) Y) Z) (apply (apply X Z) Y).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply b X) Y) Z) (apply X (apply Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply s X) Y) Z) (apply (apply X Z) (apply Y Z)).∃Y:Univ.eq Univ (apply Y (f Y)) (apply (f Y) (apply Y (f Y)))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#apply.
+#b.
+#c.
+#f.
+#i.
+#s.
+#H0.
+#H1.
+#H2.
+#H3.
+napply ex_intro[
+nid2:
+nauto by H0,H1,H2,H3;
+nid|
+skip]
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL058-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL058-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : If there's a lark, then there's an egocentric bird. *)
+
+(* Version : Especial. *)
+
+(* English : Suppose we are given a forest that conrtains a lark, and *)
+
+(* we are not given any other information. Prove that at least *)
+
+(* one bird in the forest must be egocentric. *)
+
+(* Refs : [Smu85] Smullyan (1978), To Mock a Mocking Bird and Other Logi *)
+
+(* : [GO86] Glickfield & Overbeek (1986), A Foray into Combinatory *)
+
+(* Source : [GO86] *)
+
+(* Names : - [GO86] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 3 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ---- There exists a lark *)
+
+(* ---- Hypothesis: There exists a bird x that is fond of itself. *)
+ntheorem prove_the_bird_exists:
+ ∀Univ:Type.∀X:Univ.∀X1:Univ.∀X2:Univ.
+∀lark:Univ.
+∀response:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X1:Univ.∀X2:Univ.eq Univ (response (response lark X1) X2) (response X1 (response X2 X2)).∃X:Univ.eq Univ (response X X) X
+.
+#Univ.
+#X.
+#X1.
+#X2.
+#lark.
+#response.
+#H0.
+napply ex_intro[
+nid2:
+nauto by H0;
+nid|
+skip]
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL058-2.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL058-2 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : If there's a lark, then there's an egocentric bird. *)
+
+(* Version : Especial. *)
+
+(* Theorem formulation : The egocentric bird is provided and *)
+
+(* checked. *)
+
+(* English : Suppose we are given a forest that contains a lark, and *)
+
+(* we are not given any other information. Prove that at least *)
+
+(* one bird in the forest must be egocentric. *)
+
+(* Refs : [Smu85] Smullyan (1978), To Mock a Mocking Bird and Other Logi *)
+
+(* : [GO86] Glickfield & Overbeek (1986), A Foray into Combinatory *)
+
+(* Source : [GO86] *)
+
+(* Names : - [GO86] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.2.1, 0.11 v2.2.0, 0.14 v2.1.0, 0.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 2 ( 0 singleton) *)
+
+(* Maximal term depth : 7 ( 5 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ---- There exists a lark *)
+
+(* ---- Hypothesis: This bird is egocentric *)
+ntheorem prove_the_bird_exists:
+ ∀Univ:Type.∀X1:Univ.∀X2:Univ.
+∀lark:Univ.
+∀response:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X1:Univ.∀X2:Univ.eq Univ (response (response lark X1) X2) (response X1 (response X2 X2)).eq Univ (response (response (response lark (response (response lark (response lark lark)) (response lark (response lark lark)))) (response lark (response lark lark))) (response (response lark (response (response lark (response lark lark)) (response lark (response lark lark)))) (response lark (response lark lark)))) (response (response lark (response (response lark (response lark lark)) (response lark (response lark lark)))) (response lark (response lark lark)))
+.
+#Univ.
+#X1.
+#X2.
+#lark.
+#response.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL058-3.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL058-3 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : If there's a lark, then there's an egocentric bird. *)
+
+(* Version : Especial. *)
+
+(* Theorem formulation : The egocentric bird is provided and *)
+
+(* checked. *)
+
+(* English : Suppose we are given a forest that conrtains a lark, and *)
+
+(* we are not given any other information. Prove that at least *)
+
+(* one bird in the forest must be egocentric. *)
+
+(* Refs : [Smu85] Smullyan (1978), To Mock a Mocking Bird and Other Logi *)
+
+(* : [GO86] Glickfield & Overbeek (1986), A Foray into Combinatory *)
+
+(* Source : [GO86] *)
+
+(* Names : - [GO86] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.2.1, 0.11 v2.2.0, 0.14 v2.1.0, 0.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 2 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 4 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ---- There exists a lark *)
+
+(* ---- Hypothesis: This bird is egocentric *)
+ntheorem prove_the_bird_exists:
+ ∀Univ:Type.∀X1:Univ.∀X2:Univ.
+∀lark:Univ.
+∀response:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X1:Univ.∀X2:Univ.eq Univ (response (response lark X1) X2) (response X1 (response X2 X2)).eq Univ (response (response (response (response lark lark) (response lark (response lark lark))) (response lark (response lark lark))) (response (response (response lark lark) (response lark (response lark lark))) (response lark (response lark lark)))) (response (response (response lark lark) (response lark (response lark lark))) (response lark (response lark lark)))
+.
+#Univ.
+#X1.
+#X2.
+#lark.
+#response.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL059-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL059-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : L3 ((lark lark) lark) is not egocentric. *)
+
+(* Version : Especial. *)
+
+(* English : *)
+
+(* Refs : [Smu85] Smullyan (1978), To Mock a Mocking Bird and Other Logi *)
+
+(* : [GO86] Glickfield & Overbeek (1986), A Foray into Combinatory *)
+
+(* Source : [GO86] *)
+
+(* Names : - [GO86] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.2.1, 0.22 v2.2.0, 0.29 v2.1.0, 0.25 v2.0.0 *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 4 RR) *)
+
+(* Number of atoms : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 5 constant; 0-2 arity) *)
+
+(* Number of variables : 8 ( 2 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ---- There exists a kestrel *)
+
+(* ---- There exists a lark *)
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL060-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL060-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Find combinator equivalent to Q from B and T *)
+
+(* Version : [WM88] (equality) axioms. *)
+
+(* English : Construct from B and T alone a combinator that behaves as the *)
+
+(* combinator Q does, where ((Bx)y)z = x(yz), (Tx)y = yx, *)
+
+(* ((Qx)y)z = y(xz). *)
+
+(* Refs : [WM88] Wos & McCune (1988), Challenge Problems Focusing on Eq *)
+
+(* : [WW+90] Wos et al. (1990), Automated Reasoning Contributes to *)
+
+(* Source : [WW+90] *)
+
+(* Names : CL-1 [WW+90] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.36 v3.1.0, 0.33 v2.7.0, 0.36 v2.6.0, 0.17 v2.5.0, 0.00 v2.2.1, 0.11 v2.2.0, 0.14 v2.1.0, 0.62 v2.0.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 6 ( 0 singleton) *)
+
+(* Maximal term depth : 5 ( 4 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_q_combinator:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀apply:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀f:∀_:Univ.Univ.
+∀g:∀_:Univ.Univ.
+∀h:∀_:Univ.Univ.
+∀t:Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (apply (apply t X) Y) (apply Y X).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply b X) Y) Z) (apply X (apply Y Z)).∃X:Univ.eq Univ (apply (apply (apply X (f X)) (g X)) (h X)) (apply (g X) (apply (f X) (h X)))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#apply.
+#b.
+#f.
+#g.
+#h.
+#t.
+#H0.
+#H1.
+napply ex_intro[
+nid2:
+nauto by H0,H1;
+nid|
+skip]
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL060-2.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL060-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Find combinator equivalent to Q from B and T *)
+
+(* Version : [WM88] (equality) axioms. *)
+
+(* Theorem formulation : The combinator is provided and checked. *)
+
+(* English : Construct from B and T alone a combinator that behaves as the *)
+
+(* combinator Q does, where ((Bx)y)z = x(yz), (Tx)y = yx, *)
+
+(* ((Qx)y)z = y(xz). *)
+
+(* Refs : [WM88] Wos & McCune (1988), Challenge Problems Focusing on Eq *)
+
+(* : [WW+90] Wos et al. (1990), Automated Reasoning Contributes to *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.1.0, 0.29 v2.0.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 5 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 7 ( 4 average) *)
+
+(* Comments : *)
+
+(* Bugfixes : v1.2.0 : Redundant [fgh]_substitution axioms removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----This is the q equivalent *)
+ntheorem prove_q_combinator:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀apply:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀t:Univ.
+∀x:Univ.
+∀y:Univ.
+∀z:Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (apply (apply t X) Y) (apply Y X).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply b X) Y) Z) (apply X (apply Y Z)).eq Univ (apply (apply (apply (apply (apply b (apply t b)) (apply (apply b b) t)) x) y) z) (apply y (apply x z))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#apply.
+#b.
+#t.
+#x.
+#y.
+#z.
+#H0.
+#H1.
+nauto by H0,H1;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL060-3.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL060-3 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Find combinator equivalent to Q from B and T *)
+
+(* Version : [WM88] (equality) axioms. *)
+
+(* Theorem formulation : The combinator is provided and checked. *)
+
+(* English : Construct from B and T alone a combinator that behaves as the *)
+
+(* combinator Q does, where ((Bx)y)z = x(yz), (Tx)y = yx, *)
+
+(* ((Qx)y)z = y(xz). *)
+
+(* Refs : [WM88] Wos & McCune (1988), Challenge Problems Focusing on Eq *)
+
+(* : [WW+90] Wos et al. (1990), Automated Reasoning Contributes to *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.1.0, 0.29 v2.0.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 5 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 9 ( 4 average) *)
+
+(* Comments : *)
+
+(* Bugfixes : v1.2.0 : Redundant [fgh]_substitution axioms removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----This is the q equivalent *)
+ntheorem prove_q_combinator:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀apply:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀t:Univ.
+∀x:Univ.
+∀y:Univ.
+∀z:Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (apply (apply t X) Y) (apply Y X).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply b X) Y) Z) (apply X (apply Y Z)).eq Univ (apply (apply (apply (apply (apply b (apply (apply b (apply t b)) b)) t) x) y) z) (apply y (apply x z))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#apply.
+#b.
+#t.
+#x.
+#y.
+#z.
+#H0.
+#H1.
+nauto by H0,H1;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL061-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL061-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Find combinator equivalent to Q1 from B and T *)
+
+(* Version : [WM88] (equality) axioms. *)
+
+(* English : Construct from B and T alone a combinator that behaves as the *)
+
+(* combinator Q1 does, where ((Bx)y)z = x(yz), (Tx)y = yx, *)
+
+(* ((Q1x)y)z = x(zy). *)
+
+(* Refs : [WM88] Wos & McCune (1988), Challenge Problems Focusing on Eq *)
+
+(* : [WW+90] Wos et al. (1990), Automated Reasoning Contributes to *)
+
+(* Source : [WW+90] *)
+
+(* Names : CL-2 [WW+90] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.36 v3.1.0, 0.33 v2.7.0, 0.36 v2.6.0, 0.17 v2.5.0, 0.00 v2.2.1, 0.22 v2.2.0, 0.29 v2.1.0, 0.75 v2.0.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 6 ( 0 singleton) *)
+
+(* Maximal term depth : 5 ( 4 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_q1_combinator:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀apply:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀f:∀_:Univ.Univ.
+∀g:∀_:Univ.Univ.
+∀h:∀_:Univ.Univ.
+∀t:Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (apply (apply t X) Y) (apply Y X).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply b X) Y) Z) (apply X (apply Y Z)).∃X:Univ.eq Univ (apply (apply (apply X (f X)) (g X)) (h X)) (apply (f X) (apply (h X) (g X)))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#apply.
+#b.
+#f.
+#g.
+#h.
+#t.
+#H0.
+#H1.
+napply ex_intro[
+nid2:
+nauto by H0,H1;
+nid|
+skip]
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL061-2.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL061-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Find combinator equivalent to Q1 from B and T *)
+
+(* Version : [WM88] (equality) axioms. *)
+
+(* Theorem formulation : The combinator is provided and checked. *)
+
+(* English : Construct from B and T alone a combinator that behaves as the *)
+
+(* combinator Q1 does, where ((Bx)y)z = x(yz), (Tx)y = yx, *)
+
+(* ((Q1x)y)z = x(zy). *)
+
+(* Refs : [WM88] Wos & McCune (1988), Challenge Problems Focusing on Eq *)
+
+(* : [WW+90] Wos et al. (1990), Automated Reasoning Contributes to *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.1.0, 0.29 v2.0.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 5 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 7 ( 4 average) *)
+
+(* Comments : *)
+
+(* Bugfixes : v1.2.0 : Redundant [fgh]_substitution axioms removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----This is the Q1 equivalent *)
+ntheorem prove_q1_combinator:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀apply:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀t:Univ.
+∀x:Univ.
+∀y:Univ.
+∀z:Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (apply (apply t X) Y) (apply Y X).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply b X) Y) Z) (apply X (apply Y Z)).eq Univ (apply (apply (apply (apply (apply b (apply t t)) (apply (apply b b) b)) x) y) z) (apply x (apply z y))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#apply.
+#b.
+#t.
+#x.
+#y.
+#z.
+#H0.
+#H1.
+nauto by H0,H1;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL061-3.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL061-3 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Find combinator equivalent to Q1 from B and T *)
+
+(* Version : [WM88] (equality) axioms. *)
+
+(* Theorem formulation : The combinator is provided and checked. *)
+
+(* English : Construct from B and T alone a combinator that behaves as the *)
+
+(* combinator Q1 does, where ((Bx)y)z = x(yz), (Tx)y = yx, *)
+
+(* ((Q1x)y)z = x(zy). *)
+
+(* Refs : [WM88] Wos & McCune (1988), Challenge Problems Focusing on Eq *)
+
+(* : [WW+90] Wos et al. (1990), Automated Reasoning Contributes to *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.1.0, 0.29 v2.0.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 5 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 9 ( 4 average) *)
+
+(* Comments : *)
+
+(* Bugfixes : v1.2.0 : Redundant [fgh]_substitution axioms removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----This is the Q1 equivalent *)
+ntheorem prove_q1_combinator:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀apply:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀t:Univ.
+∀x:Univ.
+∀y:Univ.
+∀z:Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (apply (apply t X) Y) (apply Y X).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply b X) Y) Z) (apply X (apply Y Z)).eq Univ (apply (apply (apply (apply (apply b (apply (apply b (apply t t)) b)) b) x) y) z) (apply x (apply z y))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#apply.
+#b.
+#t.
+#x.
+#y.
+#z.
+#H0.
+#H1.
+nauto by H0,H1;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL062-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL062-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Find combinator equivalent to C from B and T *)
+
+(* Version : [WM88] (equality) axioms. *)
+
+(* English : Construct from B and T alone a combinator that behaves as the *)
+
+(* combinator C does, where ((Bx)y)z = x(yz), (Tx)y = yx, *)
+
+(* ((Cx)y)z = (xz)y *)
+
+(* Refs : [WM88] Wos & McCune (1988), Challenge Problems Focusing on Eq *)
+
+(* : [WW+90] Wos et al. (1990), Automated Reasoning Contributes to *)
+
+(* Source : [WW+90] *)
+
+(* Names : CL-3 [WW+90] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.50 v3.1.0, 0.44 v2.7.0, 0.36 v2.6.0, 0.17 v2.5.0, 0.00 v2.2.1, 0.44 v2.2.0, 0.57 v2.1.0, 1.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 6 ( 0 singleton) *)
+
+(* Maximal term depth : 5 ( 4 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_c_combinator:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀apply:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀f:∀_:Univ.Univ.
+∀g:∀_:Univ.Univ.
+∀h:∀_:Univ.Univ.
+∀t:Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (apply (apply t X) Y) (apply Y X).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply b X) Y) Z) (apply X (apply Y Z)).∃X:Univ.eq Univ (apply (apply (apply X (f X)) (g X)) (h X)) (apply (apply (f X) (h X)) (g X))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#apply.
+#b.
+#f.
+#g.
+#h.
+#t.
+#H0.
+#H1.
+napply ex_intro[
+nid2:
+nauto by H0,H1;
+nid|
+skip]
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL062-2.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL062-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Find combinator equivalent to C from B and T *)
+
+(* Version : [WM88] (equality) axioms. *)
+
+(* Theorem formulation : The combinator is provided and checked. *)
+
+(* English : Construct from B and T alone a combinator that behaves as the *)
+
+(* combinator C does, where ((Bx)y)z = x(yz), (Tx)y = yx, *)
+
+(* ((Cx)y)z = (xz)y *)
+
+(* Refs : [WM88] Wos & McCune (1988), Challenge Problems Focusing on Eq *)
+
+(* : [WW+90] Wos et al. (1990), Automated Reasoning Contributes to *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.2.1, 0.11 v2.2.0, 0.14 v2.1.0, 0.29 v2.0.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 5 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 9 ( 4 average) *)
+
+(* Comments : *)
+
+(* Bugfixes : v1.2.0 : Redundant [fgh]_substitution axioms removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----This is the C equivalent *)
+ntheorem prove_c_combinator:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀apply:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀t:Univ.
+∀x:Univ.
+∀y:Univ.
+∀z:Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (apply (apply t X) Y) (apply Y X).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply b X) Y) Z) (apply X (apply Y Z)).eq Univ (apply (apply (apply (apply (apply b (apply t (apply (apply b b) t))) (apply (apply b b) t)) x) y) z) (apply (apply x z) y)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#apply.
+#b.
+#t.
+#x.
+#y.
+#z.
+#H0.
+#H1.
+nauto by H0,H1;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL062-3.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL062-3 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Find combinator equivalent to C from B and T *)
+
+(* Version : [WM88] (equality) axioms. *)
+
+(* Theorem formulation : The combinator is provided and checked. *)
+
+(* English : Construct from B and T alone a combinator that behaves as the *)
+
+(* combinator C does, where ((Bx)y)z = x(yz), (Tx)y = yx, *)
+
+(* ((Cx)y)z = (xz)y *)
+
+(* Refs : [WM88] Wos & McCune (1988), Challenge Problems Focusing on Eq *)
+
+(* : [WW+90] Wos et al. (1990), Automated Reasoning Contributes to *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.2.1, 0.11 v2.2.0, 0.14 v2.1.0, 0.43 v2.0.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 5 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 11 ( 4 average) *)
+
+(* Comments : *)
+
+(* Bugfixes : v1.2.0 : Redundant [fgh]_substitution axioms removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----This is the C equivalent *)
+ntheorem prove_c_combinator:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀apply:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀t:Univ.
+∀x:Univ.
+∀y:Univ.
+∀z:Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (apply (apply t X) Y) (apply Y X).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply b X) Y) Z) (apply X (apply Y Z)).eq Univ (apply (apply (apply (apply (apply b (apply (apply b (apply t (apply (apply b b) t))) b)) t) x) y) z) (apply (apply x z) y)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#apply.
+#b.
+#t.
+#x.
+#y.
+#z.
+#H0.
+#H1.
+nauto by H0,H1;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL063-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL063-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Find combinator equivalent to F from B and T *)
+
+(* Version : [WM88] (equality) axioms. *)
+
+(* English : Construct from B and T alone a combinator that behaves as the *)
+
+(* combinator F does, where ((Bx)y)z = x(yz), (Tx)y = yx, *)
+
+(* ((Fx)y)z = (zy)x. *)
+
+(* Refs : [WM88] Wos & McCune (1988), Challenge Problems Focusing on Eq *)
+
+(* : [WW+90] Wos et al. (1990), Automated Reasoning Contributes to *)
+
+(* Source : [WW+90] *)
+
+(* Names : CL-4 [WW+90] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.50 v3.1.0, 0.44 v2.7.0, 0.36 v2.6.0, 0.17 v2.5.0, 0.00 v2.2.1, 0.44 v2.2.0, 0.57 v2.1.0, 1.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 6 ( 0 singleton) *)
+
+(* Maximal term depth : 5 ( 4 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_f_combinator:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀apply:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀f:∀_:Univ.Univ.
+∀g:∀_:Univ.Univ.
+∀h:∀_:Univ.Univ.
+∀t:Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (apply (apply t X) Y) (apply Y X).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply b X) Y) Z) (apply X (apply Y Z)).∃X:Univ.eq Univ (apply (apply (apply X (f X)) (g X)) (h X)) (apply (apply (h X) (g X)) (f X))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#apply.
+#b.
+#f.
+#g.
+#h.
+#t.
+#H0.
+#H1.
+napply ex_intro[
+nid2:
+nauto by H0,H1;
+nid|
+skip]
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL063-2.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL063-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Find combinator equivalent to F from B and T *)
+
+(* Version : [WM88] (equality) axioms. *)
+
+(* Theorem formulation : The combinator is provided and checked. *)
+
+(* English : Construct from B and T alone a combinator that behaves as the *)
+
+(* combinator F does, where ((Bx)y)z = x(yz), (Tx)y = yx, *)
+
+(* ((Fx)y)z = (zy)x. *)
+
+(* Refs : [WM88] Wos & McCune (1988), Challenge Problems Focusing on Eq *)
+
+(* : [WW+90] Wos et al. (1990), Automated Reasoning Contributes to *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.2.1, 0.11 v2.2.0, 0.14 v2.1.0, 0.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 5 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 8 ( 4 average) *)
+
+(* Comments : *)
+
+(* Bugfixes : v1.2.0 : Redundant [fgh]_substitution axioms removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----This is the F equivalent *)
+ntheorem prove_f_combinator:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀apply:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀t:Univ.
+∀x:Univ.
+∀y:Univ.
+∀z:Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (apply (apply t X) Y) (apply Y X).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply b X) Y) Z) (apply X (apply Y Z)).eq Univ (apply (apply (apply (apply (apply b (apply t t)) (apply (apply b b) (apply (apply b b) t))) x) y) z) (apply (apply z y) x)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#apply.
+#b.
+#t.
+#x.
+#y.
+#z.
+#H0.
+#H1.
+nauto by H0,H1;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL063-3.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL063-3 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Find combinator equivalent to F from B and T *)
+
+(* Version : [WM88] (equality) axioms. *)
+
+(* Theorem formulation : The combinator is provided and checked. *)
+
+(* English : Construct from B and T alone a combinator that behaves as the *)
+
+(* combinator F does, where ((Bx)y)z = x(yz), (Tx)y = yx, *)
+
+(* ((Fx)y)z = (zy)x. *)
+
+(* Refs : [WM88] Wos & McCune (1988), Challenge Problems Focusing on Eq *)
+
+(* : [WW+90] Wos et al. (1990), Automated Reasoning Contributes to *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.2.1, 0.11 v2.2.0, 0.14 v2.1.0, 0.29 v2.0.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 5 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 9 ( 4 average) *)
+
+(* Comments : *)
+
+(* Bugfixes : v1.2.0 : Redundant [fgh]_substitution axioms removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----This is the F equivalent *)
+ntheorem prove_f_combinator:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀apply:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀t:Univ.
+∀x:Univ.
+∀y:Univ.
+∀z:Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (apply (apply t X) Y) (apply Y X).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply b X) Y) Z) (apply X (apply Y Z)).eq Univ (apply (apply (apply (apply (apply b (apply (apply b (apply t t)) b)) (apply (apply b b) t)) x) y) z) (apply (apply z y) x)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#apply.
+#b.
+#t.
+#x.
+#y.
+#z.
+#H0.
+#H1.
+nauto by H0,H1;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL063-4.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL063-4 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Find combinator equivalent to F from B and T *)
+
+(* Version : [WM88] (equality) axioms. *)
+
+(* Theorem formulation : The combinator is provided and checked. *)
+
+(* English : Construct from B and T alone a combinator that behaves as the *)
+
+(* combinator F does, where ((Bx)y)z = x(yz), (Tx)y = yx, *)
+
+(* ((Fx)y)z = (zy)x. *)
+
+(* Refs : [WM88] Wos & McCune (1988), Challenge Problems Focusing on Eq *)
+
+(* : [WW+90] Wos et al. (1990), Automated Reasoning Contributes to *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.2.1, 0.11 v2.2.0, 0.14 v2.1.0, 0.43 v2.0.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 5 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 9 ( 4 average) *)
+
+(* Comments : *)
+
+(* Bugfixes : v1.2.0 : Redundant [fgh]_substitution axioms removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----This is the F equivalent *)
+ntheorem prove_f_combinator:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀apply:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀t:Univ.
+∀x:Univ.
+∀y:Univ.
+∀z:Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (apply (apply t X) Y) (apply Y X).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply b X) Y) Z) (apply X (apply Y Z)).eq Univ (apply (apply (apply (apply (apply b (apply t t)) (apply (apply b (apply (apply b b) b)) t)) x) y) z) (apply (apply z y) x)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#apply.
+#b.
+#t.
+#x.
+#y.
+#z.
+#H0.
+#H1.
+nauto by H0,H1;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL063-5.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL063-5 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Find combinator equivalent to F from B and T *)
+
+(* Version : [WM88] (equality) axioms. *)
+
+(* Theorem formulation : The combinator is provided and checked. *)
+
+(* English : Construct from B and T alone a combinator that behaves as the *)
+
+(* combinator F does, where ((Bx)y)z = x(yz), (Tx)y = yx, *)
+
+(* ((Fx)y)z = (zy)x. *)
+
+(* Refs : [WM88] Wos & McCune (1988), Challenge Problems Focusing on Eq *)
+
+(* : [WW+90] Wos et al. (1990), Automated Reasoning Contributes to *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.2.1, 0.11 v2.2.0, 0.14 v2.1.0, 0.43 v2.0.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 5 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 9 ( 4 average) *)
+
+(* Comments : *)
+
+(* Bugfixes : v1.2.0 : Redundant [fgh]_substitution axioms removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----This is the F equivalent *)
+ntheorem prove_f_combinator:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀apply:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀t:Univ.
+∀x:Univ.
+∀y:Univ.
+∀z:Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (apply (apply t X) Y) (apply Y X).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply b X) Y) Z) (apply X (apply Y Z)).eq Univ (apply (apply (apply (apply (apply b (apply (apply b (apply t t)) (apply (apply b b) b))) t) x) y) z) (apply (apply z y) x)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#apply.
+#b.
+#t.
+#x.
+#y.
+#z.
+#H0.
+#H1.
+nauto by H0,H1;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL063-6.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL063-6 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Find combinator equivalent to F from B and T *)
+
+(* Version : [WM88] (equality) axioms. *)
+
+(* Theorem formulation : The combinator is provided and checked. *)
+
+(* English : Construct from B and T alone a combinator that behaves as the *)
+
+(* combinator F does, where ((Bx)y)z = x(yz), (Tx)y = yx, *)
+
+(* ((Fx)y)z = (zy)x. *)
+
+(* Refs : [WM88] Wos & McCune (1988), Challenge Problems Focusing on Eq *)
+
+(* : [WW+90] Wos et al. (1990), Automated Reasoning Contributes to *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.1.0, 0.43 v2.0.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 5 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 11 ( 4 average) *)
+
+(* Comments : *)
+
+(* Bugfixes : v1.2.0 : Redundant [fgh]_substitution axioms removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----This is the F equivalent *)
+ntheorem prove_f_combinator:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀apply:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀t:Univ.
+∀x:Univ.
+∀y:Univ.
+∀z:Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (apply (apply t X) Y) (apply Y X).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply b X) Y) Z) (apply X (apply Y Z)).eq Univ (apply (apply (apply (apply (apply b (apply (apply b (apply (apply b (apply t t)) b)) b)) t) x) y) z) (apply (apply z y) x)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#apply.
+#b.
+#t.
+#x.
+#y.
+#z.
+#H0.
+#H1.
+nauto by H0,H1;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL064-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL064-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Find combinator equivalent to V from B and T *)
+
+(* Version : [WM88] (equality) axioms. *)
+
+(* English : Construct from B and T alone a combinator that behaves as the *)
+
+(* combinator V does, where ((Bx)y)z = x(yz), (Tx)y = yx, *)
+
+(* ((Vx)y)z = (zx)y. *)
+
+(* Refs : [WM88] Wos & McCune (1988), Challenge Problems Focusing on Eq *)
+
+(* : [WW+90] Wos et al. (1990), Automated Reasoning Contributes to *)
+
+(* Source : [WW+90] *)
+
+(* Names : CL-5 [WW+90] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.64 v3.1.0, 0.44 v2.7.0, 0.45 v2.6.0, 0.17 v2.5.0, 0.00 v2.4.0, 0.00 v2.2.1, 0.78 v2.2.0, 0.86 v2.1.0, 1.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 6 ( 0 singleton) *)
+
+(* Maximal term depth : 5 ( 4 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_v_combinator:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀apply:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀f:∀_:Univ.Univ.
+∀g:∀_:Univ.Univ.
+∀h:∀_:Univ.Univ.
+∀t:Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (apply (apply t X) Y) (apply Y X).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply b X) Y) Z) (apply X (apply Y Z)).∃X:Univ.eq Univ (apply (apply (apply X (f X)) (g X)) (h X)) (apply (apply (h X) (f X)) (g X))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#apply.
+#b.
+#f.
+#g.
+#h.
+#t.
+#H0.
+#H1.
+napply ex_intro[
+nid2:
+nauto by H0,H1;
+nid|
+skip]
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL064-2.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL064-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Find combinator equivalent to V from B and T *)
+
+(* Version : [WM88] (equality) axioms. *)
+
+(* Theorem formulation : The combinator is provided and checked. *)
+
+(* English : Construct from B and T alone a combinator that behaves as the *)
+
+(* combinator V does, where ((Bx)y)z = x(yz), (Tx)y = yx, *)
+
+(* ((Vx)y)z = (zx)y. *)
+
+(* Refs : [WM88] Wos & McCune (1988), Challenge Problems Focusing on Eq *)
+
+(* : [WW+90] Wos et al. (1990), Automated Reasoning Contributes to *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.07 v3.1.0, 0.11 v2.7.0, 0.00 v2.2.1, 0.22 v2.2.0, 0.29 v2.1.0, 0.57 v2.0.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 5 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 9 ( 4 average) *)
+
+(* Comments : *)
+
+(* Bugfixes : v1.2.0 : Redundant [fgh]_substitution axioms removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----This is the V equivalent *)
+ntheorem prove_v_combinator:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀apply:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀t:Univ.
+∀x:Univ.
+∀y:Univ.
+∀z:Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (apply (apply t X) Y) (apply Y X).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply b X) Y) Z) (apply X (apply Y Z)).eq Univ (apply (apply (apply (apply (apply b (apply t (apply (apply b b) t))) (apply (apply b b) (apply (apply b b) t))) x) y) z) (apply (apply z x) y)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#apply.
+#b.
+#t.
+#x.
+#y.
+#z.
+#H0.
+#H1.
+nauto by H0,H1;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL064-3.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL064-3 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Find combinator equivalent to V from B and T *)
+
+(* Version : [WM88] (equality) axioms. *)
+
+(* Theorem formulation : The combinator is provided and checked. *)
+
+(* English : Construct from B and T alone a combinator that behaves as the *)
+
+(* combinator V does, where ((Bx)y)z = x(yz), (Tx)y = yx, *)
+
+(* ((Vx)y)z = (zx)y. *)
+
+(* Refs : [WM88] Wos & McCune (1988), Challenge Problems Focusing on Eq *)
+
+(* : [WW+90] Wos et al. (1990), Automated Reasoning Contributes to *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.2.1, 0.22 v2.2.0, 0.29 v2.1.0, 0.43 v2.0.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 5 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 11 ( 4 average) *)
+
+(* Comments : *)
+
+(* Bugfixes : v1.2.0 : Redundant [fgh]_substitution axioms removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----This is the V equivalent *)
+ntheorem prove_v_combinator:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀apply:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀t:Univ.
+∀x:Univ.
+∀y:Univ.
+∀z:Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (apply (apply t X) Y) (apply Y X).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply b X) Y) Z) (apply X (apply Y Z)).eq Univ (apply (apply (apply (apply (apply b (apply (apply b (apply t (apply (apply b b) t))) b)) (apply (apply b b) t)) x) y) z) (apply (apply z x) y)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#apply.
+#b.
+#t.
+#x.
+#y.
+#z.
+#H0.
+#H1.
+nauto by H0,H1;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL064-4.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL064-4 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Find combinator equivalent to V from B and T *)
+
+(* Version : [WM88] (equality) axioms. *)
+
+(* Theorem formulation : The combinator is provided and checked. *)
+
+(* English : Construct from B and T alone a combinator that behaves as the *)
+
+(* combinator V does, where ((Bx)y)z = x(yz), (Tx)y = yx, *)
+
+(* ((Vx)y)z = (zx)y. *)
+
+(* Refs : [WM88] Wos & McCune (1988), Challenge Problems Focusing on Eq *)
+
+(* : [WW+90] Wos et al. (1990), Automated Reasoning Contributes to *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.07 v3.1.0, 0.11 v2.7.0, 0.00 v2.2.1, 0.22 v2.2.0, 0.29 v2.1.0, 0.57 v2.0.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 5 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 9 ( 4 average) *)
+
+(* Comments : *)
+
+(* Bugfixes : v1.2.0 : Redundant [fgh]_substitution axioms removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----This is the V equivalent *)
+ntheorem prove_v_combinator:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀apply:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀t:Univ.
+∀x:Univ.
+∀y:Univ.
+∀z:Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (apply (apply t X) Y) (apply Y X).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply b X) Y) Z) (apply X (apply Y Z)).eq Univ (apply (apply (apply (apply (apply b (apply t (apply (apply b b) t))) (apply (apply b (apply (apply b b) b)) t)) x) y) z) (apply (apply z x) y)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#apply.
+#b.
+#t.
+#x.
+#y.
+#z.
+#H0.
+#H1.
+nauto by H0,H1;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL064-5.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL064-5 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Find combinator equivalent to V from B and T *)
+
+(* Version : [WM88] (equality) axioms. *)
+
+(* Theorem formulation : The combinator is provided and checked. *)
+
+(* English : Construct from B and T alone a combinator that behaves as the *)
+
+(* combinator V does, where ((Bx)y)z = x(yz), (Tx)y = yx, *)
+
+(* ((Vx)y)z = (zx)y. *)
+
+(* Refs : [WM88] Wos & McCune (1988), Challenge Problems Focusing on Eq *)
+
+(* : [WW+90] Wos et al. (1990), Automated Reasoning Contributes to *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.07 v3.1.0, 0.11 v2.7.0, 0.00 v2.2.1, 0.22 v2.2.0, 0.29 v2.1.0, 0.57 v2.0.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 5 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 11 ( 4 average) *)
+
+(* Comments : *)
+
+(* Bugfixes : v1.2.0 : Redundant [fgh]_substitution axioms removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----This is the V equivalent *)
+ntheorem prove_v_combinator:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀apply:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀t:Univ.
+∀x:Univ.
+∀y:Univ.
+∀z:Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (apply (apply t X) Y) (apply Y X).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply b X) Y) Z) (apply X (apply Y Z)).eq Univ (apply (apply (apply (apply (apply b (apply (apply b (apply t (apply (apply b b) t))) (apply (apply b b) b))) t) x) y) z) (apply (apply z x) y)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#apply.
+#b.
+#t.
+#x.
+#y.
+#z.
+#H0.
+#H1.
+nauto by H0,H1;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL064-6.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL064-6 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Find combinator equivalent to V from B and T *)
+
+(* Version : [WM88] (equality) axioms. *)
+
+(* Theorem formulation : The combinator is provided and checked. *)
+
+(* English : Construct from B and T alone a combinator that behaves as the *)
+
+(* combinator V does, where ((Bx)y)z = x(yz), (Tx)y = yx, *)
+
+(* ((Vx)y)z = (zx)y. *)
+
+(* Refs : [WM88] Wos & McCune (1988), Challenge Problems Focusing on Eq *)
+
+(* : [WW+90] Wos et al. (1990), Automated Reasoning Contributes to *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.2.1, 0.11 v2.2.0, 0.14 v2.1.0, 0.43 v2.0.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 5 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 13 ( 5 average) *)
+
+(* Comments : *)
+
+(* Bugfixes : v1.2.0 : Redundant [fgh]_substitution axioms removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----This is the V equivalent *)
+ntheorem prove_v_combinator:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀apply:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀t:Univ.
+∀x:Univ.
+∀y:Univ.
+∀z:Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (apply (apply t X) Y) (apply Y X).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply b X) Y) Z) (apply X (apply Y Z)).eq Univ (apply (apply (apply (apply (apply b (apply (apply b (apply (apply b (apply t (apply (apply b b) t))) b)) b)) t) x) y) z) (apply (apply z x) y)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#apply.
+#b.
+#t.
+#x.
+#y.
+#z.
+#H0.
+#H1.
+nauto by H0,H1;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL064-7.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL064-7 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Find combinator equivalent to V from B and T *)
+
+(* Version : [WM88] (equality) axioms. *)
+
+(* Theorem formulation : The combinator is provided and checked. *)
+
+(* English : Construct from B and T alone a combinator that behaves as the *)
+
+(* combinator V does, where ((Bx)y)z = x(yz), (Tx)y = yx, *)
+
+(* ((Vx)y)z = (zx)y. *)
+
+(* Refs : [WM88] Wos & McCune (1988), Challenge Problems Focusing on Eq *)
+
+(* : [WW+90] Wos et al. (1990), Automated Reasoning Contributes to *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.07 v3.1.0, 0.11 v2.7.0, 0.00 v2.2.1, 0.22 v2.2.0, 0.29 v2.1.0, 0.71 v2.0.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 5 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 9 ( 4 average) *)
+
+(* Comments : *)
+
+(* Bugfixes : v1.2.0 : Redundant [fgh]_substitution axioms removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----This is the V equivalent *)
+ntheorem prove_v_combinator:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀apply:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀t:Univ.
+∀x:Univ.
+∀y:Univ.
+∀z:Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (apply (apply t X) Y) (apply Y X).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply b X) Y) Z) (apply X (apply Y Z)).eq Univ (apply (apply (apply (apply (apply b (apply t (apply (apply b b) t))) (apply (apply b b) (apply (apply b t) t))) x) y) z) (apply (apply z x) y)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#apply.
+#b.
+#t.
+#x.
+#y.
+#z.
+#H0.
+#H1.
+nauto by H0,H1;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL064-8.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL064-8 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Find combinator equivalent to V from B and T *)
+
+(* Version : [WM88] (equality) axioms. *)
+
+(* Theorem formulation : The combinator is provided and checked. *)
+
+(* English : Construct from B and T alone a combinator that behaves as the *)
+
+(* combinator V does, where ((Bx)y)z = x(yz), (Tx)y = yx, *)
+
+(* ((Vx)y)z = (zx)y. *)
+
+(* Refs : [WM88] Wos & McCune (1988), Challenge Problems Focusing on Eq *)
+
+(* : [WW+90] Wos et al. (1990), Automated Reasoning Contributes to *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.2.1, 0.22 v2.2.0, 0.29 v2.1.0, 0.71 v2.0.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 5 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 11 ( 4 average) *)
+
+(* Comments : *)
+
+(* Bugfixes : v1.2.0 : Redundant [fgh]_substitution axioms removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----This is the V equivalent *)
+ntheorem prove_v_combinator:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀apply:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀t:Univ.
+∀x:Univ.
+∀y:Univ.
+∀z:Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (apply (apply t X) Y) (apply Y X).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply b X) Y) Z) (apply X (apply Y Z)).eq Univ (apply (apply (apply (apply (apply b (apply (apply b (apply t (apply (apply b b) t))) b)) (apply (apply b t) t)) x) y) z) (apply (apply z x) y)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#apply.
+#b.
+#t.
+#x.
+#y.
+#z.
+#H0.
+#H1.
+nauto by H0,H1;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL064-9.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL064-9 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Find combinator equivalent to V from B and T *)
+
+(* Version : [WM88] (equality) axioms. *)
+
+(* Theorem formulation : The combinator is provided and checked. *)
+
+(* English : Construct from B and T alone a combinator that behaves as the *)
+
+(* combinator V does, where ((Bx)y)z = x(yz), (Tx)y = yx, *)
+
+(* ((Vx)y)z = (zx)y. *)
+
+(* Refs : [WM88] Wos & McCune (1988), Challenge Problems Focusing on Eq *)
+
+(* : [WW+90] Wos et al. (1990), Automated Reasoning Contributes to *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.07 v3.1.0, 0.11 v2.7.0, 0.00 v2.2.1, 0.22 v2.2.0, 0.29 v2.1.0, 0.71 v2.0.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 5 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 9 ( 4 average) *)
+
+(* Comments : *)
+
+(* Bugfixes : v1.2.0 : Redundant [fgh]_substitution axioms removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----This is the V equivalent *)
+ntheorem prove_v_combinator:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀apply:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀t:Univ.
+∀x:Univ.
+∀y:Univ.
+∀z:Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (apply (apply t X) Y) (apply Y X).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply b X) Y) Z) (apply X (apply Y Z)).eq Univ (apply (apply (apply (apply (apply b (apply t (apply (apply b b) t))) (apply (apply b (apply (apply b b) t)) t)) x) y) z) (apply (apply z x) y)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#apply.
+#b.
+#t.
+#x.
+#y.
+#z.
+#H0.
+#H1.
+nauto by H0,H1;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL065-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL065-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Find combinator equivalent to G from B and T *)
+
+(* Version : [WM88] (equality) axioms. *)
+
+(* English : Construct from B and T alone a combinator that behaves as the *)
+
+(* combinator G does, where ((Bx)y)z = x(yz), (Tx)y = yx, *)
+
+(* (((Gx)y)z)w = (xw)(yz) *)
+
+(* Refs : [WM88] Wos & McCune (1988), Challenge Problems Focusing on Eq *)
+
+(* : [WW+90] Wos et al. (1990), Automated Reasoning Contributes to *)
+
+(* Source : [WW+90] *)
+
+(* Names : CL-6 [WW+90] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.64 v3.2.0, 0.71 v3.1.0, 0.56 v2.7.0, 0.45 v2.6.0, 0.33 v2.5.0, 0.00 v2.4.0, 0.00 v2.2.1, 0.89 v2.2.0, 0.86 v2.1.0, 1.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 6 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 4 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_g_combinator:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀apply:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀f:∀_:Univ.Univ.
+∀g:∀_:Univ.Univ.
+∀h:∀_:Univ.Univ.
+∀i:∀_:Univ.Univ.
+∀t:Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (apply (apply t X) Y) (apply Y X).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply b X) Y) Z) (apply X (apply Y Z)).∃X:Univ.eq Univ (apply (apply (apply (apply X (f X)) (g X)) (h X)) (i X)) (apply (apply (f X) (i X)) (apply (g X) (h X)))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#apply.
+#b.
+#f.
+#g.
+#h.
+#i.
+#t.
+#H0.
+#H1.
+napply ex_intro[
+nid2:
+nauto by H0,H1;
+nid|
+skip]
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL066-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL066-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Find combinator equivalent to P from B, Q and W *)
+
+(* Version : [WM88] (equality) axioms. *)
+
+(* English : Construct from B, Q and W alone a combinator that behaves as *)
+
+(* the combinator P does, where ((Bx)y)z = x(yz), ((Qx)y)z = *)
+
+(* y(xz), (Wx)y = (xy)y, (((Px)y)y)z = (xy)((xy)z) *)
+
+(* Refs : [WM88] Wos & McCune (1988), Challenge Problems Focusing on Eq *)
+
+(* : [WW+90] Wos et al. (1990), Automated Reasoning Contributes to *)
+
+(* Source : [WW+90] *)
+
+(* Names : CL-7 [WW+90] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.93 v3.1.0, 0.89 v2.7.0, 0.82 v2.6.0, 0.67 v2.5.0, 0.25 v2.4.0, 0.00 v2.3.0, 0.33 v2.2.1, 0.89 v2.2.0, 0.86 v2.1.0, 1.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 4 ( 0 non-Horn; 4 unit; 1 RR) *)
+
+(* Number of atoms : 4 ( 4 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 9 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 4 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_p_combinator:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀apply:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀f:∀_:Univ.Univ.
+∀g:∀_:Univ.Univ.
+∀h:∀_:Univ.Univ.
+∀q:Univ.
+∀w:Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (apply (apply w X) Y) (apply (apply X Y) Y).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply q X) Y) Z) (apply Y (apply X Z)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply b X) Y) Z) (apply X (apply Y Z)).∃X:Univ.eq Univ (apply (apply (apply (apply X (f X)) (g X)) (g X)) (h X)) (apply (apply (f X) (g X)) (apply (apply (f X) (g X)) (h X)))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#apply.
+#b.
+#f.
+#g.
+#h.
+#q.
+#w.
+#H0.
+#H1.
+#H2.
+napply ex_intro[
+nid2:
+nauto by H0,H1,H2;
+nid|
+skip]
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL066-2.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL066-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Find combinator equivalent to P from B, Q and W *)
+
+(* Version : [WM88] (equality) axioms. *)
+
+(* Theorem formulation : The combinator is provided and checked. *)
+
+(* English : Construct from B, Q and W alone a combinator that behaves as *)
+
+(* the combinator P does, where ((Bx)y)z = x(yz), ((Qx)y)z = *)
+
+(* y(xz), (Wx)y = (xy)y, (((Px)y)y)z = (xy)((xy)z) *)
+
+(* Refs : [WM88] Wos & McCune (1988), Challenge Problems Focusing on Eq *)
+
+(* : [WW+90] Wos et al. (1990), Automated Reasoning Contributes to *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.7.0, 0.09 v2.6.0, 0.00 v2.1.0, 0.29 v2.0.0 *)
+
+(* Syntax : Number of clauses : 4 ( 0 non-Horn; 4 unit; 1 RR) *)
+
+(* Number of atoms : 4 ( 4 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 6 constant; 0-2 arity) *)
+
+(* Number of variables : 8 ( 0 singleton) *)
+
+(* Maximal term depth : 9 ( 4 average) *)
+
+(* Comments : *)
+
+(* Bugfixes : v1.2.0 : Redundant [fgh]_substitution axioms removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----This is the P equivalent *)
+ntheorem prove_p_combinator:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀apply:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀q:Univ.
+∀w:Univ.
+∀x:Univ.
+∀y:Univ.
+∀z:Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (apply (apply w X) Y) (apply (apply X Y) Y).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply q X) Y) Z) (apply Y (apply X Z)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply b X) Y) Z) (apply X (apply Y Z)).eq Univ (apply (apply (apply (apply (apply (apply q q) (apply w (apply q (apply q q)))) x) y) y) z) (apply (apply x y) (apply (apply x y) z))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#apply.
+#b.
+#q.
+#w.
+#x.
+#y.
+#z.
+#H0.
+#H1.
+#H2.
+nauto by H0,H1,H2;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL066-3.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL066-3 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Find combinator equivalent to P from B, Q and W *)
+
+(* Version : [WM88] (equality) axioms. *)
+
+(* Theorem formulation : The combinator is provided and checked. *)
+
+(* English : Construct from B, Q and W alone a combinator that behaves as *)
+
+(* the combinator P does, where ((Bx)y)z = x(yz), ((Qx)y)z = *)
+
+(* y(xz), (Wx)y = (xy)y, (((Px)y)y)z = (xy)((xy)z) *)
+
+(* Refs : [WM88] Wos & McCune (1988), Challenge Problems Focusing on Eq *)
+
+(* : [WW+90] Wos et al. (1990), Automated Reasoning Contributes to *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v3.1.0, 0.11 v2.7.0, 0.00 v2.1.0, 0.29 v2.0.0 *)
+
+(* Syntax : Number of clauses : 4 ( 0 non-Horn; 4 unit; 1 RR) *)
+
+(* Number of atoms : 4 ( 4 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 6 constant; 0-2 arity) *)
+
+(* Number of variables : 8 ( 0 singleton) *)
+
+(* Maximal term depth : 10 ( 4 average) *)
+
+(* Comments : *)
+
+(* Bugfixes : v1.2.0 : Redundant [fgh]_substitution axioms removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----This is the P equivalent *)
+ntheorem prove_p_combinator:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀apply:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀q:Univ.
+∀w:Univ.
+∀x:Univ.
+∀y:Univ.
+∀z:Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (apply (apply w X) Y) (apply (apply X Y) Y).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply q X) Y) Z) (apply Y (apply X Z)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply b X) Y) Z) (apply X (apply Y Z)).eq Univ (apply (apply (apply (apply (apply (apply b (apply w (apply q (apply q q)))) q) x) y) y) z) (apply (apply x y) (apply (apply x y) z))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#apply.
+#b.
+#q.
+#w.
+#x.
+#y.
+#z.
+#H0.
+#H1.
+#H2.
+nauto by H0,H1,H2;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL067-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL067-1 : TPTP v3.2.0. Released v1.2.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Strong fixed point for B and S *)
+
+(* Version : [WM88] (equality) axioms. *)
+
+(* English : The strong fixed point property holds for the set *)
+
+(* P consisting of the combinators B and S, where ((Sx)y)z *)
+
+(* = (xz)(yz), ((Bx)y)z = x(yz). *)
+
+(* Refs : [WM88] Wos & McCune (1988), Challenge Problems Focusing on Eq *)
+
+(* : [Wos93] Wos (1993), The Kernel Strategy and Its Use for the St *)
+
+(* Source : [Wos93] *)
+
+(* Names : Question 4 [Wos93] *)
+
+(* : Question 5 [Wos93] *)
+
+(* Status : Unknown *)
+
+(* Rating : 1.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 4 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_fixed_point:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀apply:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀f:∀_:Univ.Univ.
+∀s:Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply b X) Y) Z) (apply X (apply Y Z)).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply s X) Y) Z) (apply (apply X Z) (apply Y Z)).∃Y:Univ.eq Univ (apply Y (f Y)) (apply (f Y) (apply Y (f Y)))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#apply.
+#b.
+#f.
+#s.
+#H0.
+#H1.
+napply ex_intro[
+nid2:
+nauto by H0,H1;
+nid|
+skip]
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL068-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL068-1 : TPTP v3.2.0. Released v1.2.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Weak fixed point for B and S *)
+
+(* Version : [WM88] (equality) axioms. *)
+
+(* English : The weak fixed point property holds for the set P consisting *)
+
+(* of the combinators B and S, where ((Sx)y)z (xz)(yz), ((Bx)y)z *)
+
+(* = x(yz). *)
+
+(* Refs : [WM88] Wos & McCune (1988), Challenge Problems Focusing on Eq *)
+
+(* : [Wos93] Wos (1993), The Kernel Strategy and Its Use for the St *)
+
+(* Source : [Wos93] *)
+
+(* Names : Question 11 [Wos93] *)
+
+(* Status : Unknown *)
+
+(* Rating : 1.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_fixed_point:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀apply:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀combinator:Univ.
+∀s:Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply b X) Y) Z) (apply X (apply Y Z)).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply s X) Y) Z) (apply (apply X Z) (apply Y Z)).∃Y:Univ.eq Univ Y (apply combinator Y)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#apply.
+#b.
+#combinator.
+#s.
+#H0.
+#H1.
+napply ex_intro[
+nid2:
+nauto by H0,H1;
+nid|
+skip]
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL069-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL069-1 : TPTP v3.2.0. Released v1.2.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Strong fixed point for B and L *)
+
+(* Version : [WM88] (equality) axioms. *)
+
+(* English : The strong fixed point property holds for the set *)
+
+(* P consisting of the combinators B and L, where ((Bx)y)z *)
+
+(* = x(yz), (Lx)y = x(yy). *)
+
+(* Refs : [WM88] Wos & McCune (1988), Challenge Problems Focusing on Eq *)
+
+(* : [Wos93] Wos (1993), The Kernel Strategy and Its Use for the St *)
+
+(* Source : [Wos93] *)
+
+(* Names : Question 6 [Wos93] *)
+
+(* : Question 16 [Wos93] *)
+
+(* Status : Satisfiable *)
+
+(* Rating : 1.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 6 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_fixed_point:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀apply:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀f:∀_:Univ.Univ.
+∀l:Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (apply (apply l X) Y) (apply X (apply Y Y)).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply b X) Y) Z) (apply X (apply Y Z)).∃Y:Univ.eq Univ (apply Y (f Y)) (apply (f Y) (apply Y (f Y)))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#apply.
+#b.
+#f.
+#l.
+#H0.
+#H1.
+napply ex_intro[
+nid2:
+nauto by H0,H1;
+nid|
+skip]
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL070-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL070-1 : TPTP v3.2.0. Released v1.2.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Weak fixed point for B and N1 *)
+
+(* Version : [WM88] (equality) axioms. *)
+
+(* English : The weak fixed point property holds for the set P consisting *)
+
+(* of the combinators B and N1, where N1xyz = xyyz, ((Bx)y)z *)
+
+(* = x(yz). *)
+
+(* Refs : [WM88] Wos & McCune (1988), Challenge Problems Focusing on Eq *)
+
+(* : [Wos93] Wos (1993), The Kernel Strategy and Its Use for the St *)
+
+(* : [Zha94] Zhang (1994), Solution to Another Open Question in Com *)
+
+(* Source : [Wos93] *)
+
+(* Names : Question 12 [Wos93] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.07 v3.1.0, 0.11 v2.7.0, 0.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_fixed_point:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀apply:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀combinator:Univ.
+∀n1:Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply b X) Y) Z) (apply X (apply Y Z)).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply n1 X) Y) Z) (apply (apply (apply X Y) Y) Z).∃Y:Univ.eq Univ Y (apply combinator Y)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#apply.
+#b.
+#combinator.
+#n1.
+#H0.
+#H1.
+napply ex_intro[
+nid2:
+nauto by H0,H1;
+nid|
+skip]
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL071-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL071-1 : TPTP v3.2.0. Released v1.2.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Strong fixed point for N and Q *)
+
+(* Version : [WM88] (equality) axioms. *)
+
+(* English : The strong fixed point property holds for the set *)
+
+(* P consisting of the combinators N and Q, where ((Nx)y)z *)
+
+(* = ((xz)y)z, ((Qx)y)z = y(xz). *)
+
+(* Refs : [WM88] Wos & McCune (1988), Challenge Problems Focusing on Eq *)
+
+(* : [Wos93] Wos (1993), The Kernel Strategy and Its Use for the St *)
+
+(* : [Zha95] Zhang (1995), Email to G. Sutcliffe *)
+
+(* Source : [Wos93] *)
+
+(* Names : Question 14 [Wos93] *)
+
+(* Status : Satisfiable *)
+
+(* Rating : 0.33 v3.2.0, 0.67 v3.1.0, 0.33 v2.4.0, 1.00 v2.3.0, 0.67 v2.2.1, 0.75 v2.2.0, 1.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 4 average) *)
+
+(* Comments : [Zha95] provided a 4 element model of these clauses. *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_fixed_point:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀apply:∀_:Univ.∀_:Univ.Univ.
+∀f:∀_:Univ.Univ.
+∀n:Univ.
+∀q:Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply q X) Y) Z) (apply Y (apply X Z)).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply n X) Y) Z) (apply (apply (apply X Z) Y) Z).∃Y:Univ.eq Univ (apply Y (f Y)) (apply (f Y) (apply Y (f Y)))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#apply.
+#f.
+#n.
+#q.
+#H0.
+#H1.
+napply ex_intro[
+nid2:
+nauto by H0,H1;
+nid|
+skip]
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL073-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL073-1 : TPTP v3.2.0. Released v1.2.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Strong fixed point for B and N1 *)
+
+(* Version : [WM88] (equality) axioms. *)
+
+(* English : The strong fixed point property holds for the set *)
+
+(* P consisting of the combinators B and N1, where N1xyz = xyyz, *)
+
+(* ((Bx)y)z = x(yz). *)
+
+(* Refs : [WM88] Wos & McCune (1988), Challenge Problems Focusing on Eq *)
+
+(* : [Wos93] Wos (1993), The Kernel Strategy and Its Use for the St *)
+
+(* : [Zha94] Zhang (1994), Solution to Another Open Question in Com *)
+
+(* : [Pel98] Peltier (1998), A New Method for Automated Finite Mode *)
+
+(* Source : [Wos93] *)
+
+(* Names : Question 18 [Wos93] *)
+
+(* : 4.2.5 (CL1) [Pel98] *)
+
+(* Status : Satisfiable *)
+
+(* Rating : 0.33 v3.2.0, 0.67 v3.1.0, 0.33 v2.4.0, 0.67 v2.3.0, 1.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 4 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_strong_fixed_point:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀apply:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀f:∀_:Univ.Univ.
+∀n1:Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply b X) Y) Z) (apply X (apply Y Z)).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply n1 X) Y) Z) (apply (apply (apply X Y) Y) Z).∃Y:Univ.eq Univ (apply Y (f Y)) (apply (f Y) (apply Y (f Y)))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#apply.
+#b.
+#f.
+#n1.
+#H0.
+#H1.
+napply ex_intro[
+nid2:
+nauto by H0,H1;
+nid|
+skip]
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL075-2.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL075-2 : TPTP v3.2.0. Released v1.2.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Lemma 1 for showing the unsatisfiable variant of TRC *)
+
+(* Version : [Jec95] (equality) axioms : Reduced > Incomplete. *)
+
+(* English : Searching for a diagonal combinator F with the property *)
+
+(* f X Y = X X. *)
+
+(* Refs : [Jec95] Jech (1995), Otter Experiments in a System of Combinat *)
+
+(* Source : [Jec95] *)
+
+(* Names : - [Jec95] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 6 ( 1 singleton) *)
+
+(* Maximal term depth : 4 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Don't include axioms of Type-respecting combinators *)
+
+(* include('Axioms/COL001-0.ax'). *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Replace k function by k combinator *)
+
+(* input_clause(k_definition,axiom, *)
+
+(* [++equal(apply(k(X),Y),X)]). *)
+
+(* ----Replace k function by k combinator *)
+
+(* input_clause(abstraction,axiom, *)
+
+(* [++equal(apply(apply(apply(abstraction,X),Y),Z),apply(apply(X,k(Z)), *)
+
+(* apply(Y,Z)))]). *)
+
+(* ----Subsitution axioms *)
+ntheorem prove_diagonal_combinator:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀abstraction:Univ.
+∀apply:∀_:Univ.∀_:Univ.Univ.
+∀b:∀_:Univ.Univ.
+∀c:∀_:Univ.Univ.
+∀k:Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply abstraction X) Y) Z) (apply (apply X (apply k Z)) (apply Y Z)).
+∀H1:∀X:Univ.∀Y:Univ.eq Univ (apply (apply k X) Y) X.∃Y:Univ.eq Univ (apply (apply Y (b Y)) (c Y)) (apply (b Y) (b Y))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#abstraction.
+#apply.
+#b.
+#c.
+#k.
+#H0.
+#H1.
+napply ex_intro[
+nid2:
+nauto by H0,H1;
+nid|
+skip]
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL083-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL083-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Compatible Birds, part 1 *)
+
+(* Version : Especial. *)
+
+(* English : *)
+
+(* Refs : [Smu85] Smullyan (1978), To Mock a Mocking Bird and Other Logi *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 6 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : A UEQ part of COL054-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_birds_are_compatible_1:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a:Univ.
+∀compose:∀_:Univ.∀_:Univ.Univ.
+∀mocking_bird:Univ.
+∀response:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (response (compose A B) C) (response A (response B C)).
+∀H1:∀A:Univ.eq Univ (response mocking_bird A) (response A A).∃A:Univ.∃B:Univ.eq Univ (response a A) B
+.
+#Univ.
+#A.
+#B.
+#C.
+#a.
+#compose.
+#mocking_bird.
+#response.
+#H0.
+#H1.
+napply ex_intro[
+nid2:
+napply ex_intro[
+nid2:
+nauto by H0,H1;
+nid|
+skip]
+nid|
+skip]
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL084-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL084-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Compatible Birds, part 2 *)
+
+(* Version : Especial. *)
+
+(* English : *)
+
+(* Refs : [Smu85] Smullyan (1978), To Mock a Mocking Bird and Other Logi *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 6 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : A UEQ part of COL054-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_birds_are_compatible_2:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀b:Univ.
+∀compose:∀_:Univ.∀_:Univ.Univ.
+∀mocking_bird:Univ.
+∀response:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (response (compose A B) C) (response A (response B C)).
+∀H1:∀A:Univ.eq Univ (response mocking_bird A) (response A A).∃A:Univ.∃B:Univ.eq Univ (response b B) A
+.
+#Univ.
+#A.
+#B.
+#C.
+#b.
+#compose.
+#mocking_bird.
+#response.
+#H0.
+#H1.
+napply ex_intro[
+nid2:
+napply ex_intro[
+nid2:
+nauto by H0,H1;
+nid|
+skip]
+nid|
+skip]
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL085-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL085-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Happy Birds, part 1 *)
+
+(* Version : Especial. *)
+
+(* English : *)
+
+(* Refs : [Smu85] Smullyan (1978), To Mock a Mocking Bird and Other Logi *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 2 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 2 ( 2 singleton) *)
+
+(* Maximal term depth : 2 ( 2 average) *)
+
+(* Comments : A UEQ part of COL055-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_happiness_1:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.
+∀a:Univ.
+∀b:Univ.
+∀response:∀_:Univ.∀_:Univ.Univ.
+∀H0:eq Univ (response a b) b.∃A:Univ.∃B:Univ.eq Univ (response a A) B
+.
+#Univ.
+#A.
+#B.
+#a.
+#b.
+#response.
+#H0.
+napply ex_intro[
+nid2:
+napply ex_intro[
+nid2:
+nauto by H0;
+nid|
+skip]
+nid|
+skip]
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL086-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL086-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Happy Birds, part 2 *)
+
+(* Version : Especial. *)
+
+(* English : *)
+
+(* Refs : [Smu85] Smullyan (1978), To Mock a Mocking Bird and Other Logi *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 2 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 2 ( 2 singleton) *)
+
+(* Maximal term depth : 2 ( 2 average) *)
+
+(* Comments : A UEQ part of COL055-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_happiness_2:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.
+∀a:Univ.
+∀b:Univ.
+∀response:∀_:Univ.∀_:Univ.Univ.
+∀H0:eq Univ (response a b) b.∃A:Univ.∃B:Univ.eq Univ (response a B) A
+.
+#Univ.
+#A.
+#B.
+#a.
+#b.
+#response.
+#H0.
+napply ex_intro[
+nid2:
+napply ex_intro[
+nid2:
+nauto by H0;
+nid|
+skip]
+nid|
+skip]
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL087-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL087-1 : TPTP v3.2.0. Released v2.7.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Strong fixed point for B and M *)
+
+(* Version : [Cla03] axioms. *)
+
+(* English : The strong fixed point property holds for the set with the *)
+
+(* combinators B and M as a basis, where Bxyz = x(yz) and *)
+
+(* Mx = xx. *)
+
+(* Refs : [WM88] Wos & McCune (1988), Challenge Problems Focusing on Eq *)
+
+(* : [Cla03] Claessen (2003), Email to G. Sutcliffe *)
+
+(* Source : [Cla03] *)
+
+(* Names : *)
+
+(* Status : Open *)
+
+(* Rating : 1.00 v2.7.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem strong_fixpoint:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀apply:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀f:∀_:Univ.Univ.
+∀m:Univ.
+∀H0:∀X:Univ.eq Univ (apply m X) (apply X X).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply b X) Y) Z) (apply X (apply Y Z)).∃Y:Univ.eq Univ (apply Y (f Y)) (apply (f Y) (apply Y (f Y)))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#apply.
+#b.
+#f.
+#m.
+#H0.
+#H1.
+napply ex_intro[
+nid2:
+nauto by H0,H1;
+nid|
+skip]
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP001-2.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP001-2 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : X^2 = identity => commutativity *)
+
+(* Version : [MOW76] (equality) axioms : Augmented. *)
+
+(* English : If the square of every element is the identity, the system *)
+
+(* is commutative. *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [LO85] Lusk & Overbeek (1985), Reasoning about Equality *)
+
+(* : [LW92] Lusk & Wos (1992), Benchmark Problems in Which Equalit *)
+
+(* Source : [ANL] *)
+
+(* Names : GP1 [MOW76] *)
+
+(* : Problem 1 [LO85] *)
+
+(* : GT1 [LW92] *)
+
+(* : xsquared.ver2.in [ANL] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.2.1, 0.22 v2.2.0, 0.29 v2.1.0, 0.25 v2.0.0 *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 2 RR) *)
+
+(* Number of atoms : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 8 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Redundant two axioms *)
+ntheorem prove_b_times_a_is_c:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:eq Univ (multiply a b) c.
+∀H1:∀X:Univ.eq Univ (multiply X X) identity.
+∀H2:∀X:Univ.eq Univ (multiply X (inverse X)) identity.
+∀H3:∀X:Univ.eq Univ (multiply X identity) X.
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H5:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H6:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (multiply b a) c
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#identity.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+nauto by H0,H1,H2,H3,H4,H5,H6;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP001-4.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP001-4 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : X^2 = identity => commutativity *)
+
+(* Version : [Wos65] (equality) axioms : Incomplete. *)
+
+(* English : If the square of every element is the identity, the system *)
+
+(* is commutative. *)
+
+(* Refs : [Wos65] Wos (1965), Unpublished Note *)
+
+(* : [Pel86] Pelletier (1986), Seventy-five Problems for Testing Au *)
+
+(* Source : [Pel86] *)
+
+(* Names : Pelletier 65 [Pel86] *)
+
+(* : x2_quant.in [OTTER] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.2.1, 0.22 v2.2.0, 0.29 v2.1.0, 0.25 v2.0.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 2 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [Pel86] says "... problems, published I think, by Larry Wos *)
+
+(* (but I cannot locate where)." *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The operation '*' is associative *)
+
+(* ----There exists an identity element 'e' defined below. *)
+ntheorem prove_b_times_a_is_c:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀identity:Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:eq Univ (multiply a b) c.
+∀H1:∀X:Univ.eq Univ (multiply X X) identity.
+∀H2:∀X:Univ.eq Univ (multiply identity X) X.
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).eq Univ (multiply b a) c
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#identity.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP002-2.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP002-2 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Commutator equals identity in groups of order 3 *)
+
+(* Version : [MOW76] (equality) axioms. *)
+
+(* English : In a group, if (for all x) the cube of x is the identity *)
+
+(* (i.e. a group of order 3), then the equation [[x,y],y]= *)
+
+(* identity holds, where [x,y] is the product of x, y, the *)
+
+(* inverse of x and the inverse of y (i.e. the commutator *)
+
+(* of x and y). *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* Source : [ANL] *)
+
+(* Names : commutator.ver2.in [ANL] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.2.1, 0.22 v2.2.0, 0.29 v2.1.0, 0.29 v2.0.0 *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 6 RR) *)
+
+(* Number of atoms : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 10 ( 8 constant; 0-2 arity) *)
+
+(* Number of variables : 8 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* Bugfixes : v1.2.1 - Clause x_cubed_is_identity fixed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Redundant two axioms, but established in standard axiomatizations. *)
+
+(* ----This hypothesis is omitted in the ANL source version *)
+ntheorem prove_k_times_inverse_b_is_e:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀d:Univ.
+∀h:Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀j:Univ.
+∀k:Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:eq Univ (multiply j (inverse h)) k.
+∀H1:eq Univ (multiply h b) j.
+∀H2:eq Univ (multiply d (inverse b)) h.
+∀H3:eq Univ (multiply c (inverse a)) d.
+∀H4:eq Univ (multiply a b) c.
+∀H5:∀X:Univ.eq Univ (multiply X (multiply X X)) identity.
+∀H6:∀X:Univ.eq Univ (multiply X (inverse X)) identity.
+∀H7:∀X:Univ.eq Univ (multiply X identity) X.
+∀H8:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H9:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H10:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (multiply k (inverse b)) identity
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#d.
+#h.
+#identity.
+#inverse.
+#j.
+#k.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP002-3.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP002-3 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Commutator equals identity in groups of order 3 *)
+
+(* Version : [Ove90] (equality) axioms. *)
+
+(* English : In a group, if (for all x) the cube of x is the identity *)
+
+(* (i.e. a group of order 3), then the equation [[x,y],y]= *)
+
+(* identity holds, where [x,y] is the product of x, y, the *)
+
+(* inverse of x and the inverse of y (i.e. the commutator *)
+
+(* of x and y). *)
+
+(* Refs : [Ove93] Overbeek (1993), The CADE-11 Competitions: A Personal *)
+
+(* : [LM93] Lusk & McCune (1993), Uniform Strategies: The CADE-11 *)
+
+(* : [Zha93] Zhang (1993), Automated Proofs of Equality Problems in *)
+
+(* : [Ove90] Overbeek (1990), ATP competition announced at CADE-10 *)
+
+(* : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* Source : [Ove90] *)
+
+(* Names : CADE-11 Competition Eq-1 [Ove90] *)
+
+(* : THEOREM EQ-1 [LM93] *)
+
+(* : PROBLEM 1 [Zha93] *)
+
+(* : comm.in [OTTER] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v3.1.0, 0.11 v2.7.0, 0.00 v2.2.1, 0.33 v2.2.0, 0.43 v2.1.0, 0.25 v2.0.0 *)
+
+(* Syntax : Number of clauses : 6 ( 0 non-Horn; 6 unit; 1 RR) *)
+
+(* Number of atoms : 6 ( 6 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 8 ( 0 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : Uses an explicit formulation of the commutator. *)
+
+(* : Same axioms as [MOW76] (equality) axioms. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Definition of the commutator *)
+ntheorem prove_commutator:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀commutator:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.eq Univ (multiply X (multiply X X)) identity.
+∀H1:∀X:Univ.∀Y:Univ.eq Univ (commutator X Y) (multiply X (multiply Y (multiply (inverse X) (inverse Y)))).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H3:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H4:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (commutator (commutator a b) b) identity
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#commutator.
+#identity.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+nauto by H0,H1,H2,H3,H4;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP002-4.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP002-4 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Commutator equals identity in groups of order 3 *)
+
+(* Version : [MOW76] (equality) axioms. *)
+
+(* Theorem formulation : Explicit formulation of the commutator. *)
+
+(* English : In a group, if (for all x) the cube of x is the identity *)
+
+(* (i.e. a group of order 3), then the equation [[x,y],y]= *)
+
+(* identity holds, where [x,y] is the product of x, y, the *)
+
+(* inverse of x and the inverse of y (i.e. the commutator *)
+
+(* of x and y). *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [LO85] Lusk & Overbeek (1985), Reasoning about Equality *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* : [LW92] Lusk & Wos (1992), Benchmark Problems in Which Equalit *)
+
+(* Source : [TPTP] *)
+
+(* Names : Problem 4 [LO85] *)
+
+(* : Test Problem 2 [Wos88] *)
+
+(* : Commutator Theorem [Wos88] *)
+
+(* : GT3 [LW92] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v3.1.0, 0.11 v2.7.0, 0.00 v2.2.1, 0.33 v2.2.0, 0.43 v2.1.0, 0.25 v2.0.0 *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 1 RR) *)
+
+(* Number of atoms : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 10 ( 0 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Redundant two axioms, but used in established axiomatizations. *)
+
+(* ----Definition of the commutator *)
+ntheorem prove_commutator:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀commutator:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.eq Univ (multiply X (multiply X X)) identity.
+∀H1:∀X:Univ.∀Y:Univ.eq Univ (commutator X Y) (multiply X (multiply Y (multiply (inverse X) (inverse Y)))).
+∀H2:∀X:Univ.eq Univ (multiply X (inverse X)) identity.
+∀H3:∀X:Univ.eq Univ (multiply X identity) X.
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H5:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H6:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (commutator (commutator a b) b) identity
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#commutator.
+#identity.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+nauto by H0,H1,H2,H3,H4,H5,H6;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP010-4.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP010-4 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Inverse is a symmetric relationship *)
+
+(* Version : [Wos65] (equality) axioms : Incomplete. *)
+
+(* English : If a is an inverse of b then b is an inverse of a. *)
+
+(* Refs : [Wos65] Wos (1965), Unpublished Note *)
+
+(* : [Pel86] Pelletier (1986), Seventy-five Problems for Testing Au *)
+
+(* Source : [Pel86] *)
+
+(* Names : Pelletier 64 [Pel86] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.1.0, 0.13 v2.0.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 2 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [Pel86] says "... problems, published I think, by Larry Wos *)
+
+(* (but I cannot locate where)." *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The operation '*' is associative *)
+
+(* ----There exists an identity element 'e' defined below. *)
+ntheorem prove_b_times_c_is_e:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀b:Univ.
+∀c:Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:eq Univ (multiply c b) identity.
+∀H1:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H2:∀X:Univ.eq Univ (multiply identity X) X.
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).eq Univ (multiply b c) identity
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#b.
+#c.
+#identity.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP011-4.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP011-4 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Left cancellation *)
+
+(* Version : [Wos65] (equality) axioms : Incomplete. *)
+
+(* English : *)
+
+(* Refs : [Wos65] Wos (1965), Unpublished Note *)
+
+(* : [Pel86] Pelletier (1986), Seventy-five Problems for Testing Au *)
+
+(* Source : [Pel86] *)
+
+(* Names : Pelletier 63 [Pel86] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.2.1, 0.22 v2.2.0, 0.29 v2.1.0, 0.25 v2.0.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 2 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [Pel86] says "... problems, published I think, by Larry Wos *)
+
+(* (but I cannot locate where)." *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The operation '*' is associative *)
+
+(* ----There exists an identity element *)
+ntheorem prove_left_cancellation:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀b:Univ.
+∀c:Univ.
+∀d:Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:eq Univ (multiply b c) (multiply d c).
+∀H1:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H2:∀X:Univ.eq Univ (multiply identity X) X.
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).eq Univ b d
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#b.
+#c.
+#d.
+#identity.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP012-4.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP012-4 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Inverse of products = Product of inverses *)
+
+(* Version : [MOW76] (equality) axioms : Augmented. *)
+
+(* English : The inverse of products equals the product of the inverse, *)
+
+(* in opposite order *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* Source : [ANL] *)
+
+(* Names : - [ANL] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.2.1, 0.22 v2.2.0, 0.29 v2.1.0, 0.25 v2.0.0 *)
+
+(* Syntax : Number of clauses : 6 ( 0 non-Horn; 6 unit; 1 RR) *)
+
+(* Number of atoms : 6 ( 6 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : In Lemmas.eq.clauses of [ANL] *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Redundant two axioms *)
+ntheorem prove_inverse_of_product_is_product_of_inverses:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.eq Univ (multiply X (inverse X)) identity.
+∀H1:∀X:Univ.eq Univ (multiply X identity) X.
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H3:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H4:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (inverse (multiply a b)) (multiply (inverse b) (inverse a))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#identity.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+nauto by H0,H1,H2,H3,H4;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP014-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP014-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Product is associative in this group theory *)
+
+(* Version : [Ove90] (equality) axioms : Incomplete. *)
+
+(* English : The group theory specified by the axiom given implies the *)
+
+(* associativity of multiply. *)
+
+(* Refs : [Ove90] Overbeek (1990), ATP competition announced at CADE-10 *)
+
+(* : [Ove93] Overbeek (1993), The CADE-11 Competitions: A Personal *)
+
+(* : [LM93] Lusk & McCune (1993), Uniform Strategies: The CADE-11 *)
+
+(* : [Zha93] Zhang (1993), Automated Proofs of Equality Problems in *)
+
+(* Source : [Ove90] *)
+
+(* Names : CADE-11 Competition Eq-4 [Ove90] *)
+
+(* : THEOREM EQ-4 [LM93] *)
+
+(* : PROBLEM 4 [Zha93] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.07 v3.2.0, 0.14 v3.1.0, 0.11 v2.7.0, 0.18 v2.6.0, 0.00 v2.2.1, 0.33 v2.2.0, 0.43 v2.1.0, 0.50 v2.0.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 4 ( 0 singleton) *)
+
+(* Maximal term depth : 9 ( 4 average) *)
+
+(* Comments : The group_axiom is in fact a single axiom for group theory *)
+
+(* [LM93]. *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_associativity:
+ ∀Univ:Type.∀W:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀W:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (inverse (multiply (multiply (inverse (multiply (inverse Y) (multiply (inverse X) W))) Z) (inverse (multiply Y Z))))) W.eq Univ (multiply a (multiply b c)) (multiply (multiply a b) c)
+.
+#Univ.
+#W.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#inverse.
+#multiply.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP022-2.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP022-2 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Inverse is an involution *)
+
+(* Version : [MOW76] (equality) axioms : Augmented. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [LO85] Lusk & Overbeek (1985), Reasoning about Equality *)
+
+(* Source : [TPTP] *)
+
+(* Names : Established lemma [MOW76] *)
+
+(* : Problem 2 [LO85] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 6 ( 0 non-Horn; 6 unit; 1 RR) *)
+
+(* Number of atoms : 6 ( 6 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Redundant two axioms *)
+ntheorem prove_inverse_of_inverse_is_original:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.eq Univ (multiply X (inverse X)) identity.
+∀H1:∀X:Univ.eq Univ (multiply X identity) X.
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H3:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H4:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (inverse (inverse a)) a
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#identity.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+nauto by H0,H1,H2,H3,H4;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP023-2.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP023-2 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : The inverse of the identity is the identity *)
+
+(* Version : [MOW76] (equality) axioms : Augmented. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* Source : [TPTP] *)
+
+(* Names : Established lemma [MOW76] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 6 ( 0 non-Horn; 6 unit; 1 RR) *)
+
+(* Number of atoms : 6 ( 6 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Redundant two axioms *)
+ntheorem prove_inverse_of_id_is_id:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.eq Univ (multiply X (inverse X)) identity.
+∀H1:∀X:Univ.eq Univ (multiply X identity) X.
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H3:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H4:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (inverse identity) identity
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#identity.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+nauto by H0,H1,H2,H3,H4;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP024-5.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP024-5 : TPTP v3.2.0. Released v2.2.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Levi commutator problem. *)
+
+(* Version : [McC98] (equality) axioms. *)
+
+(* English : In group theory, if the commutator [x,y] is associative, *)
+
+(* then x*[y,z] = [y,z]*x. *)
+
+(* Refs : [McC98] McCune (1998), Email to G. Sutcliffe *)
+
+(* : [ML92] McCune & Lusk (1992), A Challenging Theorem of Levi *)
+
+(* : [Kur56] Kurosh (1956), The Theory of Groups *)
+
+(* Source : [McC98] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.57 v3.2.0, 0.50 v3.1.0, 0.44 v2.7.0, 0.64 v2.6.0, 0.33 v2.5.0, 0.00 v2.4.0, 0.33 v2.3.0, 0.67 v2.2.1 *)
+
+(* Syntax : Number of clauses : 6 ( 0 non-Horn; 6 unit; 1 RR) *)
+
+(* Number of atoms : 6 ( 6 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 10 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Definition of commutator: *)
+
+(* ----Theorem: commutator is associative implies x*[y,z] = [y,z]*x. *)
+
+(* ----Hypothesis: *)
+
+(* ----Denial of conclusion: *)
+ntheorem prove_center:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀commutator:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (commutator (commutator X Y) Z) (commutator X (commutator Y Z)).
+∀H1:∀X:Univ.∀Y:Univ.eq Univ (commutator X Y) (multiply (inverse X) (multiply (inverse Y) (multiply X Y))).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H3:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H4:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (multiply a (commutator b c)) (multiply (commutator b c) a)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#commutator.
+#identity.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+nauto by H0,H1,H2,H3,H4;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP114-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP114-1 : TPTP v3.2.0. Released v1.2.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Product of positive and negative parts of X equals X *)
+
+(* Version : [MOW76] (equality) axioms : Augmented. *)
+
+(* English : Prove that for each element X in a group, X is equal to the *)
+
+(* product of its positive part (the union with the identity) *)
+
+(* and its negative part (the intersection with the identity). *)
+
+(* Refs : [Wos94] Wos (1994), Challenge in Group Theory *)
+
+(* Source : [Wos94] *)
+
+(* Names : - [Wos94] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.29 v3.1.0, 0.22 v2.7.0, 0.36 v2.6.0, 0.17 v2.5.0, 0.00 v2.4.0, 0.00 v2.2.1, 0.44 v2.2.0, 0.57 v2.1.0, 0.86 v2.0.0 *)
+
+(* Syntax : Number of clauses : 21 ( 0 non-Horn; 21 unit; 2 RR) *)
+
+(* Number of atoms : 21 ( 21 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 8 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 38 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : I know some of the axioms are redundant, and have put comments *)
+
+(* to that effect. However, I don't know how to make a complete *)
+
+(* standard axiomatisation for the union and intersection axioms. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include the axioms for named groups *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----This axiom is a lemma *)
+
+(* ----This axiom is a lemma *)
+
+(* ----This axiom is a lemma *)
+ntheorem prove_product:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀identity:Univ.
+∀intersection:∀_:Univ.∀_:Univ.Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀negative_part:∀_:Univ.Univ.
+∀positive_part:∀_:Univ.Univ.
+∀union:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.eq Univ (negative_part X) (intersection X identity).
+∀H1:∀X:Univ.eq Univ (positive_part X) (union X identity).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (intersection Y Z) X) (intersection (multiply Y X) (multiply Z X)).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (union Y Z) X) (union (multiply Y X) (multiply Z X)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (intersection Y Z)) (intersection (multiply X Y) (multiply X Z)).
+∀H5:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (union Y Z)) (union (multiply X Y) (multiply X Z)).
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (intersection (union X Y) Y) Y.
+∀H7:∀X:Univ.∀Y:Univ.eq Univ (union (intersection X Y) Y) Y.
+∀H8:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (union X (union Y Z)) (union (union X Y) Z).
+∀H9:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (intersection X (intersection Y Z)) (intersection (intersection X Y) Z).
+∀H10:∀X:Univ.∀Y:Univ.eq Univ (union X Y) (union Y X).
+∀H11:∀X:Univ.∀Y:Univ.eq Univ (intersection X Y) (intersection Y X).
+∀H12:∀X:Univ.eq Univ (union X X) X.
+∀H13:∀X:Univ.eq Univ (intersection X X) X.
+∀H14:∀X:Univ.∀Y:Univ.eq Univ (inverse (multiply X Y)) (multiply (inverse Y) (inverse X)).
+∀H15:∀X:Univ.eq Univ (inverse (inverse X)) X.
+∀H16:eq Univ (inverse identity) identity.
+∀H17:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H18:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H19:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (multiply (positive_part a) (negative_part a)) a
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#identity.
+#intersection.
+#inverse.
+#multiply.
+#negative_part.
+#positive_part.
+#union.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+#H16.
+#H17.
+#H18.
+#H19.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15,H16,H17,H18,H19;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP115-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP115-1 : TPTP v3.2.0. Released v1.2.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Derive order 3 from a single axiom for groups order 3 *)
+
+(* Version : [Wos96] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Wos96] Wos (1996), The Automation of Reasoning: An Experiment *)
+
+(* Source : [OTTER] *)
+
+(* Names : groups.exp3.in part 1 [OTTER] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.2.1, 0.22 v2.2.0, 0.29 v2.1.0, 0.14 v2.0.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 3 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_order3:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀identity:Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (multiply (multiply X (multiply (multiply X Y) Z)) (multiply identity (multiply Z Z)))) Y.eq Univ (multiply a (multiply a a)) identity
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#identity.
+#multiply.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP116-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP116-1 : TPTP v3.2.0. Released v1.2.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Derive left identity from a single axiom for groups order 3 *)
+
+(* Version : [Wos96] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Wos96] Wos (1996), The Automation of Reasoning: An Experiment *)
+
+(* Source : [OTTER] *)
+
+(* Names : groups.exp3.in part 2 [OTTER] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.2.1, 0.22 v2.2.0, 0.29 v2.1.0, 0.14 v2.0.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 3 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_order3:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀identity:Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (multiply (multiply X (multiply (multiply X Y) Z)) (multiply identity (multiply Z Z)))) Y.eq Univ (multiply identity a) a
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#identity.
+#multiply.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP117-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP117-1 : TPTP v3.2.0. Released v1.2.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Derive right identity from a single axiom for groups order 3 *)
+
+(* Version : [Wos96] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Wos96] Wos (1996), The Automation of Reasoning: An Experiment *)
+
+(* Source : [OTTER] *)
+
+(* Names : groups.exp3.in part 3 [OTTER] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.2.1, 0.22 v2.2.0, 0.29 v2.1.0, 0.14 v2.0.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 3 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_order3:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀identity:Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (multiply (multiply X (multiply (multiply X Y) Z)) (multiply identity (multiply Z Z)))) Y.eq Univ (multiply a identity) a
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#identity.
+#multiply.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP118-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP118-1 : TPTP v3.2.0. Released v1.2.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Derive associativity from a single axiom for groups order 3 *)
+
+(* Version : [Wos96] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Wos96] Wos (1996), The Automation of Reasoning: An Experiment *)
+
+(* Source : [OTTER] *)
+
+(* Names : groups.exp3.in part 4 [OTTER] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.2.1, 0.22 v2.2.0, 0.29 v2.1.0, 0.14 v2.0.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 3 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_order3:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀identity:Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (multiply (multiply X (multiply (multiply X Y) Z)) (multiply identity (multiply Z Z)))) Y.eq Univ (multiply (multiply a b) c) (multiply a (multiply b c))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#identity.
+#multiply.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP119-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP119-1 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Derive order 4 from a single axiom for groups order 4 *)
+
+(* Version : [Wos96] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Wos96] Wos (1996), The Automation of Reasoning: An Experiment *)
+
+(* Source : [OTTER] *)
+
+(* Names : groups.exp4.in part 1 [OTTER] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v3.1.0, 0.11 v2.7.0, 0.00 v2.2.1, 0.44 v2.2.0, 0.57 v2.1.0, 0.29 v2.0.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 2 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 3 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 2 average) *)
+
+(* Comments : *)
+
+(* Bugfixes : v1.2.1 - Clause prove_order4 fixed. *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_order4:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀identity:Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:eq Univ (multiply identity identity) identity.
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply Y (multiply (multiply Y (multiply (multiply Y Y) (multiply X Z))) (multiply Z (multiply Z Z)))) X.eq Univ (multiply a (multiply a (multiply a a))) identity
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#identity.
+#multiply.
+#H0.
+#H1.
+nauto by H0,H1;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP120-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP120-1 : TPTP v3.2.0. Released v1.2.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Derive left identity from a single axiom for groups order 4 *)
+
+(* Version : [Wos96] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Wos96] Wos (1996), The Automation of Reasoning: An Experiment *)
+
+(* Source : [OTTER] *)
+
+(* Names : groups.exp4.in part 2 [OTTER] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v3.1.0, 0.11 v2.7.0, 0.00 v2.2.1, 0.44 v2.2.0, 0.57 v2.1.0, 0.43 v2.0.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 2 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 3 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_order3:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀identity:Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:eq Univ (multiply identity identity) identity.
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply Y (multiply (multiply Y (multiply (multiply Y Y) (multiply X Z))) (multiply Z (multiply Z Z)))) X.eq Univ (multiply identity a) a
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#identity.
+#multiply.
+#H0.
+#H1.
+nauto by H0,H1;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP121-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP121-1 : TPTP v3.2.0. Released v1.2.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Derive right identity from a single axiom for groups order 4 *)
+
+(* Version : [Wos96] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Wos96] Wos (1996), The Automation of Reasoning: An Experiment *)
+
+(* Source : [OTTER] *)
+
+(* Names : groups.exp4.in part 3 [OTTER] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v3.1.0, 0.11 v2.7.0, 0.00 v2.2.1, 0.44 v2.2.0, 0.57 v2.1.0, 0.43 v2.0.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 2 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 3 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_order3:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀identity:Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:eq Univ (multiply identity identity) identity.
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply Y (multiply (multiply Y (multiply (multiply Y Y) (multiply X Z))) (multiply Z (multiply Z Z)))) X.eq Univ (multiply a identity) a
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#identity.
+#multiply.
+#H0.
+#H1.
+nauto by H0,H1;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP122-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP122-1 : TPTP v3.2.0. Released v1.2.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Derive associativity from a single axiom for groups order 4 *)
+
+(* Version : [Wos96] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Wos96] Wos (1996), The Automation of Reasoning: An Experiment *)
+
+(* Source : [OTTER] *)
+
+(* Names : groups.exp4.in part 4 [OTTER] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.14 v3.2.0, 0.07 v3.1.0, 0.22 v2.7.0, 0.00 v2.2.1, 0.56 v2.2.0, 0.57 v2.1.0, 0.43 v2.0.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 2 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 3 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_order3:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀identity:Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:eq Univ (multiply identity identity) identity.
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply Y (multiply (multiply Y (multiply (multiply Y Y) (multiply X Z))) (multiply Z (multiply Z Z)))) X.eq Univ (multiply (multiply a b) c) (multiply a (multiply b c))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#identity.
+#multiply.
+#H0.
+#H1.
+nauto by H0,H1;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP136-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP136-1 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : Prove anti-symmetry axiom using the LUB transformation *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : This problem proves the original anti-symmetry axiom from the *)
+
+(* equational axiomatization. *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : ax_antisyma [Sch95] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 18 ( 0 non-Horn; 18 unit; 3 RR) *)
+
+(* Number of atoms : 18 ( 18 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 33 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : ORDERING LPO inverse > product > greatest_lower_bound > *)
+
+(* least_upper_bound > identity > a > b *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_ax_antisyma:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:eq Univ (least_upper_bound a b) a.
+∀H1:eq Univ (least_upper_bound a b) b.
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H5:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H7:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H8:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H9:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H10:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H11:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H12:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H13:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H14:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H15:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H16:∀X:Univ.eq Univ (multiply identity X) X.eq Univ a b
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+#H16.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15,H16;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP137-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP137-1 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : Prove anti-symmetry axiom using the GLB transformation *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : This problem proves the original anti-symmetry axiom from the *)
+
+(* equational axiomatization. *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : ax_antisymb [Sch95] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 18 ( 0 non-Horn; 18 unit; 3 RR) *)
+
+(* Number of atoms : 18 ( 18 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 33 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : ORDERING LPO inverse > product > greatest_lower_bound > *)
+
+(* least_upper_bound > identity > a > b *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_ax_antisymb:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:eq Univ (greatest_lower_bound a b) b.
+∀H1:eq Univ (greatest_lower_bound a b) a.
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H5:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H7:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H8:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H9:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H10:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H11:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H12:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H13:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H14:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H15:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H16:∀X:Univ.eq Univ (multiply identity X) X.eq Univ a b
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+#H16.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15,H16;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP138-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP138-1 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : Prove greatest lower-bound axiom using the LUB transformation *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : This problem proves the original greatest lower-bound axiom *)
+
+(* from the equational axiomatization. *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : ax_glb1a [Sch95] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.2.1, 0.33 v2.2.0, 0.43 v2.1.0, 0.43 v2.0.0 *)
+
+(* Syntax : Number of clauses : 18 ( 0 non-Horn; 18 unit; 3 RR) *)
+
+(* Number of atoms : 18 ( 18 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 8 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 33 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : ORDERING LPO inverse > product > greatest_lower_bound > *)
+
+(* least_upper_bound > identity > a > b > c *)
+
+(* : ORDERING LPO greatest_lower_bound > least_upper_bound > *)
+
+(* inverse > product > identity > a > b > c *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_ax_glb1a:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:eq Univ (least_upper_bound b c) b.
+∀H1:eq Univ (least_upper_bound a c) a.
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H5:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H7:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H8:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H9:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H10:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H11:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H12:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H13:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H14:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H15:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H16:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (least_upper_bound (greatest_lower_bound a b) c) (greatest_lower_bound a b)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+#H16.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15,H16;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP139-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP139-1 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : Prove greatest lower-bound axiom using the GLB transformation *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : This problem proves the original axiom of anti-symmetry from *)
+
+(* the equational axiomatization. *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : ax_glb1b [Sch95] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 18 ( 0 non-Horn; 18 unit; 3 RR) *)
+
+(* Number of atoms : 18 ( 18 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 8 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 33 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : ORDERING LPO inverse > product > greatest_lower_bound > *)
+
+(* least_upper_bound > identity > a > b > c *)
+
+(* : ORDERING LPO greatest_lower_bound > least_upper_bound > *)
+
+(* inverse > product > identity > a > b > c *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_ax_glb1b:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:eq Univ (greatest_lower_bound b c) c.
+∀H1:eq Univ (greatest_lower_bound a c) c.
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H5:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H7:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H8:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H9:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H10:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H11:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H12:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H13:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H14:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H15:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H16:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (greatest_lower_bound (greatest_lower_bound a b) c) c
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+#H16.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15,H16;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP140-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP140-1 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : Prove greatest lower-bound axiom using a transformation *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : This problem proves the original greatest lower-bound axiom *)
+
+(* from the equational axiomatization. *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : ax_glb1c [Sch95] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.1.0, 0.43 v2.0.0 *)
+
+(* Syntax : Number of clauses : 18 ( 0 non-Horn; 18 unit; 3 RR) *)
+
+(* Number of atoms : 18 ( 18 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 8 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 33 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : ORDERING LPO inverse > product > greatest_lower_bound > *)
+
+(* least_upper_bound > identity > a > b > c *)
+
+(* : ORDERING LPO greatest_lower_bound > least_upper_bound > *)
+
+(* inverse > product > identity > a > b > c *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_ax_glb1c:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:eq Univ (greatest_lower_bound b c) c.
+∀H1:eq Univ (greatest_lower_bound a c) c.
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H5:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H7:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H8:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H9:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H10:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H11:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H12:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H13:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H14:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H15:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H16:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (least_upper_bound (greatest_lower_bound a b) c) (greatest_lower_bound a b)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+#H16.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15,H16;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP141-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP141-1 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : Prove greatest lower-bound axiom using a transformation *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : This problem proves the original greatest lower-bound axiom *)
+
+(* from the equational axiomatization. *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : ax_glb1d [Sch95] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.2.1, 0.22 v2.2.0, 0.29 v2.1.0, 0.14 v2.0.0 *)
+
+(* Syntax : Number of clauses : 18 ( 0 non-Horn; 18 unit; 3 RR) *)
+
+(* Number of atoms : 18 ( 18 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 8 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 33 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : ORDERING LPO inverse > product > greatest_lower_bound > *)
+
+(* least_upper_bound > identity > a > b > c *)
+
+(* : ORDERING LPO greatest_lower_bound > least_upper_bound > *)
+
+(* inverse > product > identity > a > b > c *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_ax_glb1d:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:eq Univ (least_upper_bound b c) b.
+∀H1:eq Univ (least_upper_bound a c) a.
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H5:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H7:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H8:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H9:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H10:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H11:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H12:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H13:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H14:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H15:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H16:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (greatest_lower_bound (greatest_lower_bound a b) c) c
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+#H16.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15,H16;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP142-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP142-1 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : Prove greatest lower-bound axiom using the LUB transformation *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : This problem proves the original greatest lower-bound axiom *)
+
+(* from the equational axiomatization. *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : ax_glb2a [Sch95] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 16 ( 0 non-Horn; 16 unit; 1 RR) *)
+
+(* Number of atoms : 16 ( 16 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 33 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : ORDERING LPO inverse > product > greatest_lower_bound > *)
+
+(* least_upper_bound > identity > a > b *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_ax_glb2a:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H6:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H7:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H8:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H9:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H10:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H11:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H12:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H13:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H14:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (least_upper_bound (greatest_lower_bound a b) a) a
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP143-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP143-1 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : Prove greatest lower-bound axiom using the GLB transformation *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : This problem proves the original greatest lower-bound axiom *)
+
+(* from the equational axiomatization. *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : ax_glb2b [Sch95] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 16 ( 0 non-Horn; 16 unit; 1 RR) *)
+
+(* Number of atoms : 16 ( 16 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 33 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : ORDERING LPO inverse > product > greatest_lower_bound > *)
+
+(* least_upper_bound > identity > a > b *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_ax_glb2b:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H6:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H7:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H8:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H9:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H10:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H11:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H12:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H13:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H14:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (greatest_lower_bound (greatest_lower_bound a b) a) (greatest_lower_bound a b)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP144-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP144-1 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : Prove greatest lower-bound axiom using the LUB transformation *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : This problem proves the original greatest lower-bound axiom *)
+
+(* from the equational axiomatization. *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : ax_glb3a [Sch95] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 16 ( 0 non-Horn; 16 unit; 1 RR) *)
+
+(* Number of atoms : 16 ( 16 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 33 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : ORDERING LPO inverse > product > greatest_lower_bound > *)
+
+(* least_upper_bound > identity > a > b *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_ax_glb3a:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H6:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H7:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H8:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H9:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H10:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H11:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H12:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H13:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H14:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (least_upper_bound (greatest_lower_bound a b) b) b
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP145-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP145-1 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : Prove greatest lower-bound axiom using the GLB transformation *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : This problem proves the original greatest lower-bound axiom *)
+
+(* from the equational axiomatization. *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : ax_glb3b [Sch95] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 16 ( 0 non-Horn; 16 unit; 1 RR) *)
+
+(* Number of atoms : 16 ( 16 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 33 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : ORDERING LPO inverse > product > greatest_lower_bound > *)
+
+(* least_upper_bound > identity > a > b *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_ax_glb3b:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H6:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H7:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H8:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H9:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H10:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H11:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H12:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H13:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H14:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (greatest_lower_bound (greatest_lower_bound a b) b) (greatest_lower_bound a b)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP146-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP146-1 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : Prove least upper-bound axiom using the LUB transformation *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : This problem proves the original least upper-bound axiom from *)
+
+(* the equational axiomatization. *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : ax_lub1a [Sch95] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 18 ( 0 non-Horn; 18 unit; 3 RR) *)
+
+(* Number of atoms : 18 ( 18 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 8 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 33 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : ORDERING LPO inverse > product > greatest_lower_bound > *)
+
+(* least_upper_bound > identity > a > b > c *)
+
+(* : ORDERING LPO greatest_lower_bound > least_upper_bound > *)
+
+(* inverse > product > identity > a > b > c *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_ax_lub1a:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:eq Univ (least_upper_bound b c) c.
+∀H1:eq Univ (least_upper_bound a c) c.
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H5:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H7:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H8:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H9:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H10:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H11:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H12:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H13:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H14:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H15:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H16:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (least_upper_bound (least_upper_bound a b) c) c
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+#H16.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15,H16;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP147-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP147-1 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : Prove least upper-bound axiom using the GLB transformation *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : This problem proves the original least upper-bound axiom from *)
+
+(* the equational axiomatization. *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : ax_lub1b [Sch95] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.2.1, 0.33 v2.2.0, 0.43 v2.1.0, 0.43 v2.0.0 *)
+
+(* Syntax : Number of clauses : 18 ( 0 non-Horn; 18 unit; 3 RR) *)
+
+(* Number of atoms : 18 ( 18 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 8 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 33 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : ORDERING LPO inverse > product > greatest_lower_bound > *)
+
+(* least_upper_bound > identity > a > b > c *)
+
+(* : ORDERING LPO greatest_lower_bound > least_upper_bound > *)
+
+(* inverse > product > identity > a > b > c *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_ax_lub1b:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:eq Univ (greatest_lower_bound b c) b.
+∀H1:eq Univ (greatest_lower_bound a c) a.
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H5:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H7:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H8:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H9:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H10:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H11:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H12:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H13:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H14:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H15:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H16:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (greatest_lower_bound (least_upper_bound a b) c) (least_upper_bound a b)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+#H16.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15,H16;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP148-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP148-1 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : Prove least upper-bound axiom using a transformation *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : This problem proves the original least upper-bound axiom from *)
+
+(* the equational axiomatization. *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : ax_lub1c [Sch95] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.1.0, 0.43 v2.0.0 *)
+
+(* Syntax : Number of clauses : 18 ( 0 non-Horn; 18 unit; 3 RR) *)
+
+(* Number of atoms : 18 ( 18 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 8 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 33 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : ORDERING LPO inverse > product > greatest_lower_bound > *)
+
+(* least_upper_bound > identity > a > b > c *)
+
+(* : ORDERING LPO greatest_lower_bound > least_upper_bound > *)
+
+(* inverse > product > identity > a > b > c *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_ax_lub1c:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:eq Univ (least_upper_bound b c) c.
+∀H1:eq Univ (least_upper_bound a c) c.
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H5:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H7:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H8:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H9:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H10:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H11:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H12:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H13:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H14:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H15:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H16:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (greatest_lower_bound (least_upper_bound a b) c) (least_upper_bound a b)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+#H16.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15,H16;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP149-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP149-1 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : Prove least upper-bound axiom using a transformation *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : This problem proves the original least upper-bound axiom from *)
+
+(* the equational axiomatization. *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : ax_lub1d [Sch95] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.2.1, 0.22 v2.2.0, 0.29 v2.1.0, 0.14 v2.0.0 *)
+
+(* Syntax : Number of clauses : 18 ( 0 non-Horn; 18 unit; 3 RR) *)
+
+(* Number of atoms : 18 ( 18 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 8 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 33 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : ORDERING LPO inverse > product > greatest_lower_bound > *)
+
+(* least_upper_bound > identity > a > b > c *)
+
+(* : ORDERING LPO greatest_lower_bound > least_upper_bound > *)
+
+(* inverse > product > identity > a > b > c *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_ax_lub1d:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:eq Univ (greatest_lower_bound b c) b.
+∀H1:eq Univ (greatest_lower_bound a c) a.
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H5:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H7:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H8:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H9:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H10:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H11:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H12:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H13:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H14:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H15:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H16:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (least_upper_bound (least_upper_bound a b) c) c
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+#H16.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15,H16;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP150-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP150-1 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : Prove least upper-bound axiom using the LUB transformation *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : This problem proves the original least upper-bound axiom from *)
+
+(* the equational axiomatization. *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : ax_lub2a [Sch95] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 16 ( 0 non-Horn; 16 unit; 1 RR) *)
+
+(* Number of atoms : 16 ( 16 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 33 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : ORDERING LPO inverse > product > greatest_lower_bound > *)
+
+(* least_upper_bound > identity > a > b *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_ax_lub2a:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H6:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H7:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H8:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H9:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H10:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H11:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H12:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H13:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H14:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (least_upper_bound a (least_upper_bound a b)) (least_upper_bound a b)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP151-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP151-1 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : Prove least upper-bound axiom using the GLB transformation *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : This problem proves the original least upper-bound axiom from *)
+
+(* the equational axiomatization. *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : ax_lub2b [Sch95] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 16 ( 0 non-Horn; 16 unit; 1 RR) *)
+
+(* Number of atoms : 16 ( 16 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 33 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : ORDERING LPO inverse > product > greatest_lower_bound > *)
+
+(* least_upper_bound > identity > a > b *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_ax_lub2b:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H6:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H7:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H8:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H9:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H10:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H11:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H12:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H13:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H14:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (greatest_lower_bound a (least_upper_bound a b)) a
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP152-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP152-1 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : Prove least upper-bound axiom using the LUB transformation *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : This problem proves the original least upper-bound axiom from *)
+
+(* the equational axiomatization. *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : ax_lub3a [Sch95] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 16 ( 0 non-Horn; 16 unit; 1 RR) *)
+
+(* Number of atoms : 16 ( 16 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 33 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : ORDERING LPO inverse > product > greatest_lower_bound > *)
+
+(* least_upper_bound > identity > a > b *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_ax_lub3a:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H6:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H7:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H8:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H9:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H10:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H11:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H12:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H13:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H14:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (least_upper_bound b (least_upper_bound a b)) (least_upper_bound a b)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP153-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP153-1 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : Prove least upper-bound axiom using the GLB transformation *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : This problem proves the original least upper-bound axiom from *)
+
+(* the equational axiomatization. *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : ax_lub3b [Sch95] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 16 ( 0 non-Horn; 16 unit; 1 RR) *)
+
+(* Number of atoms : 16 ( 16 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 33 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : ORDERING LPO inverse > product > greatest_lower_bound > *)
+
+(* least_upper_bound > identity > a > b *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_ax_lub3b:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H6:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H7:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H8:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H9:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H10:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H11:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H12:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H13:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H14:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (greatest_lower_bound b (least_upper_bound a b)) b
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP154-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP154-1 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : Prove monotonicity axiom using the LUB transformation *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : This problem proves the original mononicity axiom from the *)
+
+(* equational axiomatization. *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : ax_mono1a [Sch95] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 17 ( 0 non-Horn; 17 unit; 2 RR) *)
+
+(* Number of atoms : 17 ( 17 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 8 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 33 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : ORDERING LPO inverse > product > greatest_lower_bound > *)
+
+(* least_upper_bound > identity > a > b > c *)
+
+(* : ORDERING LPO greatest_lower_bound > least_upper_bound > *)
+
+(* inverse > product > identity > a > b > c *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_ax_mono1a:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:eq Univ (least_upper_bound a b) b.
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H7:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H8:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H9:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H10:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H11:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H12:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H13:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H14:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H15:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (least_upper_bound (multiply a c) (multiply b c)) (multiply b c)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP155-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP155-1 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : Prove monotonicity axiom using the GLB transformation *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : This problem proves the original monotonicity axiom from the *)
+
+(* equational axiomatization. *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : ax_mono1b [Sch95] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 17 ( 0 non-Horn; 17 unit; 2 RR) *)
+
+(* Number of atoms : 17 ( 17 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 8 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 33 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : ORDERING LPO inverse > product > greatest_lower_bound > *)
+
+(* least_upper_bound > identity > a > b > c *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_ax_mono1b:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:eq Univ (greatest_lower_bound a b) a.
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H7:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H8:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H9:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H10:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H11:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H12:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H13:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H14:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H15:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (greatest_lower_bound (multiply a c) (multiply b c)) (multiply a c)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP156-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP156-1 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : Prove monotonicity axiom using a transformation *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : This problem proves the original monotonicity axiom from the *)
+
+(* equational axiomatization. *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : ax_mono1c [Sch95] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 17 ( 0 non-Horn; 17 unit; 2 RR) *)
+
+(* Number of atoms : 17 ( 17 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 8 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 33 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : ORDERING LPO inverse > product > greatest_lower_bound > *)
+
+(* least_upper_bound > identity > a > b > c *)
+
+(* : ORDERING LPO greatest_lower_bound > least_upper_bound > *)
+
+(* inverse > product > identity > a > b > c *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_ax_mono1c:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:eq Univ (least_upper_bound a b) b.
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H7:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H8:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H9:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H10:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H11:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H12:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H13:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H14:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H15:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (greatest_lower_bound (multiply a c) (multiply b c)) (multiply a c)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP157-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP157-1 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : Prove monotonicity axiom using the LUB transformation *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : This problem proves the original monotonicity axiom from the *)
+
+(* equational axiomatization. *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : ax_mono2a [Sch95] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 17 ( 0 non-Horn; 17 unit; 2 RR) *)
+
+(* Number of atoms : 17 ( 17 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 8 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 33 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : ORDERING LPO inverse > product > greatest_lower_bound > *)
+
+(* least_upper_bound > identity > a > b > c *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_ax_mono2a:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:eq Univ (least_upper_bound a b) b.
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H7:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H8:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H9:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H10:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H11:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H12:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H13:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H14:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H15:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (least_upper_bound (multiply c a) (multiply c b)) (multiply c b)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP158-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP158-1 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : Prove monotonicity axiom using the GLB transformation *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : This problem proves the original monotonicity axiom from the *)
+
+(* equational axiomatization. *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : ax_mono2b [Sch95] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 17 ( 0 non-Horn; 17 unit; 2 RR) *)
+
+(* Number of atoms : 17 ( 17 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 8 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 33 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : ORDERING LPO inverse > product > greatest_lower_bound > *)
+
+(* least_upper_bound > identity > a > b > c *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_ax_mono2b:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:eq Univ (greatest_lower_bound a b) a.
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H7:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H8:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H9:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H10:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H11:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H12:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H13:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H14:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H15:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (greatest_lower_bound (multiply c a) (multiply c b)) (multiply c a)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP159-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP159-1 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : Prove monotonicity axiom using a transformation *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : This problem proves the original monotonicity axiom from the *)
+
+(* equational axiomatization. *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : ax_mono2c [Sch95] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.1.0, 0.14 v2.0.0 *)
+
+(* Syntax : Number of clauses : 17 ( 0 non-Horn; 17 unit; 2 RR) *)
+
+(* Number of atoms : 17 ( 17 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 8 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 33 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : ORDERING LPO inverse > product > greatest_lower_bound > *)
+
+(* least_upper_bound > identity > a > b > c *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_ax_mono2c:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:eq Univ (greatest_lower_bound a b) a.
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H7:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H8:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H9:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H10:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H11:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H12:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H13:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H14:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H15:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (least_upper_bound (multiply c a) (multiply c b)) (multiply c b)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP160-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP160-1 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : Prove reflexivity axiom using the LUB transformation *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : This problem proves the original reflexivity axiom from the *)
+
+(* equational axiomatization. *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : ax_refla [Sch95] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 16 ( 0 non-Horn; 16 unit; 1 RR) *)
+
+(* Number of atoms : 16 ( 16 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 33 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : ORDERING LPO inverse > product > greatest_lower_bound > *)
+
+(* least_upper_bound > identity > a *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_ax_refla:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H6:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H7:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H8:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H9:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H10:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H11:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H12:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H13:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H14:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (least_upper_bound a a) a
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP161-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP161-1 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : Prove reflexivity axiom using the GLB transformation *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : This problem proves the original reflexivity axiom from the *)
+
+(* equational axiomatization. *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : ax_reflb [Sch95] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 16 ( 0 non-Horn; 16 unit; 1 RR) *)
+
+(* Number of atoms : 16 ( 16 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 33 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : ORDERING LPO inverse > product > greatest_lower_bound > *)
+
+(* least_upper_bound > identity > a *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_ax_reflb:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H6:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H7:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H8:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H9:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H10:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H11:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H12:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H13:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H14:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (greatest_lower_bound a a) a
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP162-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP162-1 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : Prove transitivity axiom using the LUB transformation *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : This problem proves the original transitivity axiom from the *)
+
+(* equational axiomatization. *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : ax_transa [Sch95] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 18 ( 0 non-Horn; 18 unit; 3 RR) *)
+
+(* Number of atoms : 18 ( 18 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 8 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 33 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : ORDERING LPO inverse > product > greatest_lower_bound > *)
+
+(* least_upper_bound > identity > a > b > c *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_ax_transa:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:eq Univ (least_upper_bound b c) c.
+∀H1:eq Univ (least_upper_bound a b) b.
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H5:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H7:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H8:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H9:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H10:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H11:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H12:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H13:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H14:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H15:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H16:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (least_upper_bound a c) c
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+#H16.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15,H16;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP163-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP163-1 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : Prove transitivity axiom using the GLB transformation *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : This problem proves the original transitivity axiom from *)
+
+(* equational axiomatization. *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : ax_transb [Sch95] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 18 ( 0 non-Horn; 18 unit; 3 RR) *)
+
+(* Number of atoms : 18 ( 18 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 8 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 33 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : ORDERING LPO inverse > product > greatest_lower_bound > *)
+
+(* least_upper_bound > identity > a > b > c *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_ax_transb:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:eq Univ (greatest_lower_bound b c) b.
+∀H1:eq Univ (greatest_lower_bound a b) a.
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H5:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H7:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H8:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H9:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H10:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H11:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H12:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H13:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H14:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H15:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H16:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (greatest_lower_bound a c) a
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+#H16.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15,H16;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP164-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP164-1 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : The lattice of each LOG is distributive *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* : [Dah95] Dahn (1995), Email to G. Sutcliffe *)
+
+(* Source : [Sch95] *)
+
+(* Names : distrnu [Sch95] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.93 v3.1.0, 0.89 v2.7.0, 0.91 v2.6.0, 0.83 v2.5.0, 0.75 v2.4.0, 0.67 v2.2.1, 1.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 16 ( 0 non-Horn; 16 unit; 1 RR) *)
+
+(* Number of atoms : 16 ( 16 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 8 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 33 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : ORDERING LPO inverse > product > greatest_lower_bound > *)
+
+(* least_upper_bound > identity > a > b > c *)
+
+(* : [Dah95] says "The proof is very complex". *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_distrnu:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H6:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H7:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H8:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H9:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H10:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H11:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H12:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H13:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H14:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (least_upper_bound a (greatest_lower_bound b c)) (greatest_lower_bound (least_upper_bound a b) (least_upper_bound a c))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP164-2.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP164-2 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : The lattice of each LOG is distributive *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* Theorem formulation : Dual. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : distrun [Sch95] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.93 v3.1.0, 0.89 v2.7.0, 0.91 v2.6.0, 0.83 v2.5.0, 0.75 v2.4.0, 0.67 v2.2.1, 1.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 16 ( 0 non-Horn; 16 unit; 1 RR) *)
+
+(* Number of atoms : 16 ( 16 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 8 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 33 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : ORDERING LPO inverse > product > greatest_lower_bound > *)
+
+(* least_upper_bound > identity > a > b > c *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_distrun:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H6:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H7:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H8:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H9:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H10:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H11:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H12:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H13:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H14:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (greatest_lower_bound a (least_upper_bound b c)) (least_upper_bound (greatest_lower_bound a b) (greatest_lower_bound a c))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP165-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP165-1 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : An application of monotonicity *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : Essentially a simple application of monotonicity, more *)
+
+(* difficult when proved from the equations replacing *)
+
+(* monotonicity. *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : lat1a [Sch95] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 17 ( 0 non-Horn; 17 unit; 2 RR) *)
+
+(* Number of atoms : 17 ( 17 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 33 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : ORDERING LPO inverse > product > greatest_lower_bound > *)
+
+(* least_upper_bound > identity > a *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_lat1a:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:eq Univ (least_upper_bound a identity) a.
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H7:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H8:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H9:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H10:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H11:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H12:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H13:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H14:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H15:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (least_upper_bound a (multiply a a)) (multiply a a)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP165-2.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP165-2 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : An application of monotonicity *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* Theorem formulation : Dual. *)
+
+(* English : Essentially a simple application of monotonicity, more *)
+
+(* difficult when proved from the equations replacing *)
+
+(* monotonicity. *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : lat1b [Sch95] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.1.0, 0.14 v2.0.0 *)
+
+(* Syntax : Number of clauses : 17 ( 0 non-Horn; 17 unit; 2 RR) *)
+
+(* Number of atoms : 17 ( 17 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 33 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : ORDERING LPO inverse > product > greatest_lower_bound > *)
+
+(* least_upper_bound > identity > a *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_lat1b:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:eq Univ (greatest_lower_bound a identity) identity.
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H7:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H8:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H9:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H10:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H11:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H12:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H13:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H14:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H15:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (greatest_lower_bound a (multiply a a)) a
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP166-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP166-1 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : Multiplication with a positive element increases a value *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* : [Dah95] Dahn (1995), Email to G. Sutcliffe *)
+
+(* Source : [Sch95] *)
+
+(* Names : lat2a [Sch95] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.2.1, 0.22 v2.2.0, 0.29 v2.1.0, 0.29 v2.0.0 *)
+
+(* Syntax : Number of clauses : 18 ( 0 non-Horn; 18 unit; 3 RR) *)
+
+(* Number of atoms : 18 ( 18 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 33 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : ORDERING LPO inverse > product > greatest_lower_bound > *)
+
+(* least_upper_bound > identity > a > b *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----[Dah95] says this is redundant. *)
+ntheorem prove_lat2a:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:eq Univ (least_upper_bound b identity) b.
+∀H1:eq Univ (least_upper_bound a identity) a.
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H5:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H7:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H8:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H9:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H10:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H11:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H12:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H13:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H14:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H15:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H16:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (least_upper_bound a (multiply a b)) (multiply a b)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+#H16.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15,H16;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP166-2.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP166-2 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : Multiplication with a positive element increases a value *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* Theorem formulation : Dual. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : lat2b [Sch95] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.2.1, 0.22 v2.2.0, 0.29 v2.1.0, 0.29 v2.0.0 *)
+
+(* Syntax : Number of clauses : 18 ( 0 non-Horn; 18 unit; 3 RR) *)
+
+(* Number of atoms : 18 ( 18 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 33 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : ORDERING LPO inverse > product > greatest_lower_bound > *)
+
+(* least_upper_bound > identity > a > b *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_lat2b:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:eq Univ (greatest_lower_bound b identity) identity.
+∀H1:eq Univ (greatest_lower_bound a identity) identity.
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H5:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H7:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H8:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H9:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H10:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H11:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H12:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H13:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H14:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H15:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H16:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (greatest_lower_bound a (multiply a b)) a
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+#H16.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15,H16;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP166-3.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP166-3 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : Multiplication with a positive element increases a value *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* Theorem formulation : Uses a different positive element. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : lat3a [Sch95] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 18 ( 0 non-Horn; 18 unit; 3 RR) *)
+
+(* Number of atoms : 18 ( 18 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 33 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : ORDERING LPO inverse > product > greatest_lower_bound > *)
+
+(* least_upper_bound > identity > a > b *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_lat3a:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:eq Univ (least_upper_bound b identity) b.
+∀H1:eq Univ (least_upper_bound a identity) a.
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H5:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H7:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H8:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H9:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H10:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H11:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H12:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H13:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H14:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H15:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H16:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (least_upper_bound a (multiply b a)) (multiply b a)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+#H16.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15,H16;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP166-4.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP166-4 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : Multiplication with a positive element increases a value *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* Theorem formulation : Dual, and uses a different positive *)
+
+(* element. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : lat3b [Sch95] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.1.0, 0.14 v2.0.0 *)
+
+(* Syntax : Number of clauses : 18 ( 0 non-Horn; 18 unit; 3 RR) *)
+
+(* Number of atoms : 18 ( 18 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 33 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : ORDERING LPO inverse > product > greatest_lower_bound > *)
+
+(* least_upper_bound > identity > a > b *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_lat3b:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:eq Univ (greatest_lower_bound b identity) identity.
+∀H1:eq Univ (greatest_lower_bound a identity) identity.
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H5:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H7:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H8:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H9:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H10:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H11:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H12:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H13:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H14:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H15:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H16:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (greatest_lower_bound a (multiply b a)) a
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+#H16.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15,H16;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP167-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP167-1 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : Product of positive and negative parts *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : Each element in a lattice ordered group can be stated as a *)
+
+(* product of it's positive and it's negative part. *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* : [Dah95] Dahn (1995), Email to G. Sutcliffe *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.21 v3.1.0, 0.11 v2.7.0, 0.18 v2.6.0, 0.17 v2.5.0, 0.00 v2.4.0, 0.00 v2.2.1, 0.33 v2.2.0, 0.43 v2.1.0, 0.86 v2.0.0 *)
+
+(* Syntax : Number of clauses : 20 ( 0 non-Horn; 20 unit; 1 RR) *)
+
+(* Number of atoms : 20 ( 20 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 8 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 41 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : ORDERING LPO inverse > greatest_lower_bound > *)
+
+(* least_upper_bound > product > negative_part > positive_part > *)
+
+(* identity > a *)
+
+(* : This is a standardized version of the problem that appears in *)
+
+(* [Sch95]. *)
+
+(* : [Dah95] says "This is crucial for reducing some problems *)
+
+(* on arbitrary elements to problems on positive elements. The *)
+
+(* proof is relatively difficult. It is non-obvious to humans *)
+
+(* since the standard tactics (unfold definitions - use *)
+
+(* distributivity - simplify) is not useful." *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_lat4:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀negative_part:∀_:Univ.Univ.
+∀positive_part:∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (least_upper_bound Y Z)) (least_upper_bound (greatest_lower_bound X Y) (greatest_lower_bound X Z)).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (least_upper_bound X Y) (least_upper_bound X Z)).
+∀H2:∀X:Univ.eq Univ (negative_part X) (greatest_lower_bound X identity).
+∀H3:∀X:Univ.eq Univ (positive_part X) (least_upper_bound X identity).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H5:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H6:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H7:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H8:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H9:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H10:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H11:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H12:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H13:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H14:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H15:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H16:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H17:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H18:∀X:Univ.eq Univ (multiply identity X) X.eq Univ a (multiply (positive_part a) (negative_part a))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#negative_part.
+#positive_part.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+#H16.
+#H17.
+#H18.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15,H16,H17,H18;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP167-2.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP167-2 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : Product of positive and negative parts *)
+
+(* Version : [Fuc94] (equality) axioms : Augmented. *)
+
+(* English : Each element in a lattice ordered group can be stated as a *)
+
+(* product of it's positive and it's negative part. *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : lat4 [Sch95] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.21 v3.1.0, 0.11 v2.7.0, 0.09 v2.6.0, 0.17 v2.5.0, 0.00 v2.4.0, 0.00 v2.2.1, 0.33 v2.2.0, 0.57 v2.1.0, 0.71 v2.0.0 *)
+
+(* Syntax : Number of clauses : 23 ( 0 non-Horn; 23 unit; 2 RR) *)
+
+(* Number of atoms : 23 ( 23 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 8 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 44 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : ORDERING LPO inverse > greatest_lower_bound > *)
+
+(* least_upper_bound > product > negative_part > positive_part > *)
+
+(* identity > a *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_lat4:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀negative_part:∀_:Univ.Univ.
+∀positive_part:∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (least_upper_bound Y Z)) (least_upper_bound (greatest_lower_bound X Y) (greatest_lower_bound X Z)).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (least_upper_bound X Y) (least_upper_bound X Z)).
+∀H2:∀X:Univ.eq Univ (negative_part X) (greatest_lower_bound X identity).
+∀H3:∀X:Univ.eq Univ (positive_part X) (least_upper_bound X identity).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (inverse (multiply X Y)) (multiply (inverse Y) (inverse X)).
+∀H5:∀X:Univ.eq Univ (inverse (inverse X)) X.
+∀H6:eq Univ (inverse identity) identity.
+∀H7:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H8:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H9:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H10:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H11:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H12:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H13:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H14:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H15:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H16:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H17:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H18:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H19:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H20:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H21:∀X:Univ.eq Univ (multiply identity X) X.eq Univ a (multiply (positive_part a) (negative_part a))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#negative_part.
+#positive_part.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+#H16.
+#H17.
+#H18.
+#H19.
+#H20.
+#H21.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15,H16,H17,H18,H19,H20,H21;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP167-3.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP167-3 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : Product of positive and negative parts *)
+
+(* Version : [Fuc94] (equality) axioms : Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.50 v3.1.0, 0.44 v2.7.0, 0.45 v2.6.0, 0.50 v2.5.0, 0.25 v2.4.0, 0.00 v2.2.1, 0.67 v2.2.0, 0.71 v2.1.0, 0.86 v2.0.0 *)
+
+(* Syntax : Number of clauses : 16 ( 0 non-Horn; 16 unit; 1 RR) *)
+
+(* Number of atoms : 16 ( 16 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 33 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : ORDERING LPO inverse > product > greatest_lower_bound > *)
+
+(* least_upper_bound > identity > a *)
+
+(* : ORDERING LPO greatest_lower_bound > least_upper_bound > *)
+
+(* inverse > product > identity > a *)
+
+(* : This is a standardized version of the problem that appears in *)
+
+(* [Sch95]. *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_p19:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H6:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H7:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H8:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H9:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H10:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H11:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H12:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H13:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H14:∀X:Univ.eq Univ (multiply identity X) X.eq Univ a (multiply (least_upper_bound a identity) (greatest_lower_bound a identity))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP167-4.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP167-4 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : Product of positive and negative parts *)
+
+(* Version : [Fuc94] (equality) axioms : Augmented. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : p19 [Sch95] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.43 v3.1.0, 0.33 v2.7.0, 0.45 v2.6.0, 0.50 v2.5.0, 0.25 v2.4.0, 0.00 v2.2.1, 0.67 v2.2.0, 0.71 v2.1.0, 0.86 v2.0.0 *)
+
+(* Syntax : Number of clauses : 19 ( 0 non-Horn; 19 unit; 2 RR) *)
+
+(* Number of atoms : 19 ( 19 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 36 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : ORDERING LPO inverse > product > greatest_lower_bound > *)
+
+(* least_upper_bound > identity > a *)
+
+(* : ORDERING LPO greatest_lower_bound > least_upper_bound > *)
+
+(* inverse > product > identity > a *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_p19:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (inverse (multiply X Y)) (multiply (inverse Y) (inverse X)).
+∀H1:∀X:Univ.eq Univ (inverse (inverse X)) X.
+∀H2:eq Univ (inverse identity) identity.
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H5:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H6:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H7:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H8:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H9:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H10:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H11:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H12:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H13:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H14:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H15:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H16:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H17:∀X:Univ.eq Univ (multiply identity X) X.eq Univ a (multiply (least_upper_bound a identity) (greatest_lower_bound a identity))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+#H16.
+#H17.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15,H16,H17;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP167-5.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP167-5 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : Product of positive and negative parts *)
+
+(* Version : [Fuc94] (equality) axioms : Augmented. *)
+
+(* English : Each element in a lattice ordered group can be stated as a *)
+
+(* product of it's positive and it's negative part. *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* : [Dah95] Dahn (1995), Email to G. Sutcliffe *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.14 v3.1.0, 0.11 v2.7.0, 0.09 v2.6.0, 0.17 v2.5.0, 0.00 v2.2.1, 0.33 v2.2.0, 0.43 v2.1.0, 0.20 v2.0.0 *)
+
+(* Syntax : Number of clauses : 21 ( 0 non-Horn; 21 unit; 1 RR) *)
+
+(* Number of atoms : 21 ( 21 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 8 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 43 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : ORDERING LPO inverse > greatest_lower_bound > *)
+
+(* least_upper_bound > product > negative_part > positive_part > *)
+
+(* identity > a *)
+
+(* : This is a standardized version of the problem that appears in *)
+
+(* [Sch95]. *)
+
+(* : [Dah95] suggested the addition of p10 as a useful lemma. *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* : v1.2.1 - Clause p10 fixed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Extra lemma *)
+ntheorem prove_lat4:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀negative_part:∀_:Univ.Univ.
+∀positive_part:∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (least_upper_bound Y Z)) (least_upper_bound (greatest_lower_bound X Y) (greatest_lower_bound X Z)).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (least_upper_bound X Y) (least_upper_bound X Z)).
+∀H2:∀X:Univ.eq Univ (negative_part X) (greatest_lower_bound X identity).
+∀H3:∀X:Univ.eq Univ (positive_part X) (least_upper_bound X identity).
+∀H4:∀A:Univ.∀B:Univ.eq Univ (inverse (least_upper_bound A B)) (greatest_lower_bound (inverse A) (inverse B)).
+∀H5:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H6:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H7:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H8:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H9:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H10:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H11:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H12:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H13:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H14:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H15:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H16:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H17:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H18:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H19:∀X:Univ.eq Univ (multiply identity X) X.eq Univ a (multiply (positive_part a) (negative_part a))
+.
+#Univ.
+#A.
+#B.
+#X.
+#Y.
+#Z.
+#a.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#negative_part.
+#positive_part.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+#H16.
+#H17.
+#H18.
+#H19.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15,H16,H17,H18,H19;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP168-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP168-1 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : Inner group automorphisms are order preserving *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* : [Dah95] Dahn (1995), Email to G. Sutcliffe *)
+
+(* Source : [Sch95] *)
+
+(* Names : p01a [Sch95] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.14 v3.2.0, 0.07 v3.1.0, 0.11 v2.7.0, 0.00 v2.1.0, 0.14 v2.0.0 *)
+
+(* Syntax : Number of clauses : 17 ( 0 non-Horn; 17 unit; 2 RR) *)
+
+(* Number of atoms : 17 ( 17 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 8 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 33 ( 2 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : ORDERING LPO inverse > product > greatest_lower_bound > *)
+
+(* least_upper_bound > identity > a > b > c *)
+
+(* : ORDERING LPO greatest_lower_bound > least_upper_bound > *)
+
+(* inverse > product > identity > a > b > c *)
+
+(* : [Dah95] says "Not difficult by monotony. Sometimes useful *)
+
+(* for transforming inequalities." *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_p01a:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:eq Univ (least_upper_bound a b) b.
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H7:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H8:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H9:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H10:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H11:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H12:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H13:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H14:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H15:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (least_upper_bound (multiply (inverse c) (multiply a c)) (multiply (inverse c) (multiply b c))) (multiply (inverse c) (multiply b c))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP168-2.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP168-2 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : Inner group automorphisms are order preserving *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* Theorem formulation : Dual. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : p01b [Sch95] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.14 v3.2.0, 0.07 v3.1.0, 0.11 v2.7.0, 0.00 v2.1.0, 0.14 v2.0.0 *)
+
+(* Syntax : Number of clauses : 17 ( 0 non-Horn; 17 unit; 2 RR) *)
+
+(* Number of atoms : 17 ( 17 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 8 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 33 ( 2 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : ORDERING LPO inverse > product > greatest_lower_bound > *)
+
+(* least_upper_bound > identity > a > b > c *)
+
+(* : ORDERING LPO greatest_lower_bound > least_upper_bound > *)
+
+(* inverse > product > identity > a > b > c *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_p01b:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:eq Univ (greatest_lower_bound a b) a.
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H7:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H8:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H9:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H10:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H11:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H12:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H13:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H14:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H15:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (greatest_lower_bound (multiply (inverse c) (multiply a c)) (multiply (inverse c) (multiply b c))) (multiply (inverse c) (multiply a c))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP169-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP169-1 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : Inverses reverse inequalities *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* : [Dah95] Dahn (1995), Email to G. Sutcliffe *)
+
+(* Source : [Sch95] *)
+
+(* Names : p02a [Sch95] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.2.1, 0.22 v2.2.0, 0.29 v2.1.0, 0.43 v2.0.0 *)
+
+(* Syntax : Number of clauses : 17 ( 0 non-Horn; 17 unit; 2 RR) *)
+
+(* Number of atoms : 17 ( 17 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 33 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : ORDERING LPO inverse > product > greatest_lower_bound > *)
+
+(* least_upper_bound > identity > a > b *)
+
+(* : ORDERING LPO greatest_lower_bound > least_upper_bound > *)
+
+(* inverse > product > identity > a > b *)
+
+(* : [Dah95] says "The proof has to introduce more complex terms." *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_p02a:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:eq Univ (least_upper_bound (inverse a) (inverse b)) (inverse b).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H7:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H8:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H9:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H10:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H11:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H12:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H13:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H14:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H15:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (least_upper_bound a b) a
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP169-2.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP169-2 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : Inverses reverse inequalities *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* Theorem formulation : Dual. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : p02b [Sch95] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.2.1, 0.22 v2.2.0, 0.29 v2.1.0, 0.43 v2.0.0 *)
+
+(* Syntax : Number of clauses : 17 ( 0 non-Horn; 17 unit; 2 RR) *)
+
+(* Number of atoms : 17 ( 17 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 33 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : ORDERING LPO inverse > product > greatest_lower_bound > *)
+
+(* least_upper_bound > identity > a > b *)
+
+(* : ORDERING LPO greatest_lower_bound > least_upper_bound > *)
+
+(* inverse > product > identity > a > b *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_p02b:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:eq Univ (greatest_lower_bound (inverse a) (inverse b)) (inverse a).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H7:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H8:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H9:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H10:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H11:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H12:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H13:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H14:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H15:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (greatest_lower_bound a b) b
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP170-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP170-1 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : General form of monotonicity *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : p03a [Sch95] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.07 v3.2.0, 0.00 v2.2.1, 0.56 v2.2.0, 0.57 v2.1.0, 0.57 v2.0.0 *)
+
+(* Syntax : Number of clauses : 18 ( 0 non-Horn; 18 unit; 3 RR) *)
+
+(* Number of atoms : 18 ( 18 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 9 ( 5 constant; 0-2 arity) *)
+
+(* Number of variables : 33 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : ORDERING LPO inverse > product > greatest_lower_bound > *)
+
+(* least_upper_bound > identity > a > b > c > d *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_p03a:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀d:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:eq Univ (least_upper_bound c d) d.
+∀H1:eq Univ (least_upper_bound a b) b.
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H5:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H7:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H8:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H9:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H10:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H11:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H12:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H13:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H14:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H15:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H16:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (least_upper_bound (multiply a c) (multiply b d)) (multiply b d)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#d.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+#H16.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15,H16;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP170-2.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP170-2 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : General form of monotonicity *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* Theorem formulation : Dual. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : p03b [Sch95] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.07 v3.2.0, 0.00 v2.2.1, 0.56 v2.2.0, 0.57 v2.1.0, 0.57 v2.0.0 *)
+
+(* Syntax : Number of clauses : 18 ( 0 non-Horn; 18 unit; 3 RR) *)
+
+(* Number of atoms : 18 ( 18 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 9 ( 5 constant; 0-2 arity) *)
+
+(* Number of variables : 33 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : ORDERING LPO inverse > product > greatest_lower_bound > *)
+
+(* least_upper_bound > identity > a > b > c > d *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_p03b:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀d:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:eq Univ (greatest_lower_bound c d) c.
+∀H1:eq Univ (greatest_lower_bound a b) a.
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H5:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H7:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H8:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H9:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H10:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H11:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H12:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H13:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H14:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H15:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H16:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (greatest_lower_bound (multiply a c) (multiply b d)) (multiply a c)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#d.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+#H16.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15,H16;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP170-3.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP170-3 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : General form of monotonicity *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* Theorem formulation : Using different definitions for =<. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : p03c [Sch95] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.07 v3.2.0, 0.00 v2.2.1, 0.56 v2.2.0, 0.57 v2.1.0, 0.57 v2.0.0 *)
+
+(* Syntax : Number of clauses : 18 ( 0 non-Horn; 18 unit; 3 RR) *)
+
+(* Number of atoms : 18 ( 18 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 9 ( 5 constant; 0-2 arity) *)
+
+(* Number of variables : 33 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : ORDERING LPO inverse > product > greatest_lower_bound > *)
+
+(* least_upper_bound > iden ity > a > b > c > d *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_p03c:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀d:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:eq Univ (least_upper_bound c d) d.
+∀H1:eq Univ (least_upper_bound a b) b.
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H5:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H7:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H8:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H9:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H10:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H11:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H12:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H13:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H14:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H15:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H16:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (greatest_lower_bound (multiply a c) (multiply b d)) (multiply a c)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#d.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+#H16.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15,H16;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP170-4.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP170-4 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : General form of monotonicity *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* Theorem formulation : Dual and using different definitions *)
+
+(* for =<. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : p03d [Sch95] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.07 v3.2.0, 0.00 v2.2.1, 0.67 v2.2.0, 0.71 v2.1.0, 0.57 v2.0.0 *)
+
+(* Syntax : Number of clauses : 18 ( 0 non-Horn; 18 unit; 3 RR) *)
+
+(* Number of atoms : 18 ( 18 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 9 ( 5 constant; 0-2 arity) *)
+
+(* Number of variables : 33 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : ORDERING LPO inverse > product > greatest_lower_bound > *)
+
+(* least_upper_bound > identity > a > b > c > d *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_p03d:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀d:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:eq Univ (greatest_lower_bound c d) c.
+∀H1:eq Univ (least_upper_bound a b) b.
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H5:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H7:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H8:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H9:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H10:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H11:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H12:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H13:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H14:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H15:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H16:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (least_upper_bound (multiply a c) (multiply b d)) (multiply b d)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#d.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+#H16.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15,H16;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP171-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP171-1 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : Positive elements form a semigroup *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : p04a [Sch95] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.2.1, 0.11 v2.2.0, 0.14 v2.1.0, 0.14 v2.0.0 *)
+
+(* Syntax : Number of clauses : 18 ( 0 non-Horn; 18 unit; 3 RR) *)
+
+(* Number of atoms : 18 ( 18 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 33 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : ORDERING LPO inverse > product > greatest_lower_bound > *)
+
+(* least_upper_bound > identity > a > b *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_p04a:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:eq Univ (least_upper_bound identity b) b.
+∀H1:eq Univ (least_upper_bound identity a) a.
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H5:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H7:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H8:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H9:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H10:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H11:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H12:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H13:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H14:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H15:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H16:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (least_upper_bound identity (multiply a b)) (multiply a b)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+#H16.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15,H16;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP171-2.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP171-2 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : Positive elements form a semigroup *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* Theorem formulation : Using different definitions for =<. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : p04c [Sch95] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.2.1, 0.11 v2.2.0, 0.14 v2.1.0, 0.14 v2.0.0 *)
+
+(* Syntax : Number of clauses : 18 ( 0 non-Horn; 18 unit; 3 RR) *)
+
+(* Number of atoms : 18 ( 18 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 33 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : ORDERING LPO inverse > product > greatest_lower_bound > *)
+
+(* least_upper_bound > identity > a > b *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_p04c:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:eq Univ (least_upper_bound identity b) b.
+∀H1:eq Univ (least_upper_bound identity a) a.
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H5:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H7:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H8:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H9:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H10:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H11:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H12:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H13:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H14:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H15:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H16:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (greatest_lower_bound identity (multiply a b)) identity
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+#H16.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15,H16;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP172-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP172-1 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : Negative elements form a semigroup *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : p04b [Sch95] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.2.1, 0.11 v2.2.0, 0.14 v2.1.0, 0.14 v2.0.0 *)
+
+(* Syntax : Number of clauses : 18 ( 0 non-Horn; 18 unit; 3 RR) *)
+
+(* Number of atoms : 18 ( 18 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 33 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : ORDERING LPO inverse > product > greatest_lower_bound > *)
+
+(* least_upper_bound > identity > a > b *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_p04b:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:eq Univ (greatest_lower_bound identity b) identity.
+∀H1:eq Univ (greatest_lower_bound identity a) identity.
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H5:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H7:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H8:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H9:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H10:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H11:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H12:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H13:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H14:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H15:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H16:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (greatest_lower_bound identity (multiply a b)) identity
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+#H16.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15,H16;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP172-2.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP172-2 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : Negative elements form a semigroup *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* Theorem formulation : Using different definitions for =<. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : p04d [Sch95] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.2.1, 0.11 v2.2.0, 0.14 v2.1.0, 0.14 v2.0.0 *)
+
+(* Syntax : Number of clauses : 18 ( 0 non-Horn; 18 unit; 3 RR) *)
+
+(* Number of atoms : 18 ( 18 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 33 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : ORDERING LPO inverse > product > greatest_lower_bound > *)
+
+(* least_upper_bound > identity > a > b *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_p04d:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:eq Univ (greatest_lower_bound identity b) identity.
+∀H1:eq Univ (greatest_lower_bound identity a) identity.
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H5:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H7:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H8:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H9:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H10:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H11:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H12:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H13:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H14:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H15:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H16:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (least_upper_bound identity (multiply a b)) (multiply a b)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+#H16.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15,H16;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP173-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP173-1 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : Each subgroup of negative elements is trivial *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* : [Dah95] Dahn (1995), Email to G. Sutcliffe *)
+
+(* Source : [Sch95] *)
+
+(* Names : p05a [Sch95] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.2.1, 0.22 v2.2.0, 0.29 v2.1.0, 0.14 v2.0.0 *)
+
+(* Syntax : Number of clauses : 18 ( 0 non-Horn; 18 unit; 3 RR) *)
+
+(* Number of atoms : 18 ( 18 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 33 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : ORDERING LPO greatest_lower_bound > least_upper_bound > *)
+
+(* inverse > product > identity > a *)
+
+(* : ORDERING LPO inverse > product > greatest_lower_bound > *)
+
+(* least_upper_bound > identity > a *)
+
+(* : [Dah95] says "The proof is not difficult but combines group *)
+
+(* theory, lattice theory and monotonicity." *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_p05a:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:eq Univ (least_upper_bound identity (inverse a)) identity.
+∀H1:eq Univ (least_upper_bound identity a) identity.
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H5:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H7:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H8:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H9:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H10:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H11:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H12:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H13:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H14:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H15:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H16:∀X:Univ.eq Univ (multiply identity X) X.eq Univ identity a
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+#H16.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15,H16;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP174-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP174-1 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : Each subgroup of positive elements is trivial *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : p05b [Sch95] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.2.1, 0.11 v2.2.0, 0.14 v2.1.0, 0.14 v2.0.0 *)
+
+(* Syntax : Number of clauses : 18 ( 0 non-Horn; 18 unit; 3 RR) *)
+
+(* Number of atoms : 18 ( 18 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 33 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : ORDERING LPO greatest_lower_bound > least_upper_bound > *)
+
+(* inverse > product > identity > a *)
+
+(* : ORDERING LPO inverse > product > greatest_lower_bound > *)
+
+(* least_upper_bound > identity > a *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_p05b:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:eq Univ (greatest_lower_bound identity (inverse a)) (inverse a).
+∀H1:eq Univ (greatest_lower_bound identity a) a.
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H5:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H7:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H8:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H9:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H10:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H11:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H12:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H13:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H14:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H15:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H16:∀X:Univ.eq Univ (multiply identity X) X.eq Univ identity a
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+#H16.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15,H16;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP175-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP175-1 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : Positivity is preserved under inner automorphisms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : p06a [Sch95] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.07 v3.2.0, 0.00 v3.1.0, 0.11 v2.7.0, 0.00 v2.2.1, 0.33 v2.2.0, 0.29 v2.1.0, 0.29 v2.0.0 *)
+
+(* Syntax : Number of clauses : 17 ( 0 non-Horn; 17 unit; 2 RR) *)
+
+(* Number of atoms : 17 ( 17 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 33 ( 2 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : ORDERING LPO inverse > product > greatest_lower_bound > *)
+
+(* least_upper_bound > identity > a > b *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_p06a:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:eq Univ (least_upper_bound identity b) b.
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H7:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H8:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H9:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H10:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H11:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H12:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H13:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H14:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H15:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (least_upper_bound identity (multiply (inverse a) (multiply b a))) (multiply (inverse a) (multiply b a))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP175-2.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP175-2 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : Positivity is preserved under inner automorphisms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* Theorem formulation : Using different definitions for =<. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : p06b [Sch95] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v3.1.0, 0.11 v2.7.0, 0.00 v2.2.1, 0.11 v2.2.0, 0.14 v2.1.0, 0.14 v2.0.0 *)
+
+(* Syntax : Number of clauses : 17 ( 0 non-Horn; 17 unit; 2 RR) *)
+
+(* Number of atoms : 17 ( 17 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 33 ( 2 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : ORDERING LPO inverse > product > greatest_lower_bound > *)
+
+(* least_upper_bound > identity > a > b *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_p06b:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:eq Univ (greatest_lower_bound identity b) identity.
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H7:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H8:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H9:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H10:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H11:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H12:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H13:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H14:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H15:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (greatest_lower_bound identity (multiply (inverse a) (multiply b a))) identity
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP175-3.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP175-3 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : Positivity is preserved under inner automorphisms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* Theorem formulation : Variant, and using different definitions *)
+
+(* for =<. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : p06c [Sch95] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v3.1.0, 0.11 v2.7.0, 0.00 v2.2.1, 0.22 v2.2.0, 0.14 v2.1.0, 0.14 v2.0.0 *)
+
+(* Syntax : Number of clauses : 17 ( 0 non-Horn; 17 unit; 2 RR) *)
+
+(* Number of atoms : 17 ( 17 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 33 ( 2 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : ORDERING LPO inverse > product > greatest_lower_bound > *)
+
+(* least_upper_bound > identity > a > b *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_p06c:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:eq Univ (least_upper_bound identity b) b.
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H7:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H8:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H9:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H10:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H11:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H12:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H13:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H14:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H15:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (greatest_lower_bound identity (multiply (inverse a) (multiply b a))) identity
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP175-4.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP175-4 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : Positivity is preserved under inner automorphisms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* Theorem formulation : Variant, and using different definitions *)
+
+(* for =<. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : p06d [Sch95] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.07 v3.2.0, 0.00 v3.1.0, 0.11 v2.7.0, 0.00 v2.2.1, 0.44 v2.2.0, 0.43 v2.1.0, 0.29 v2.0.0 *)
+
+(* Syntax : Number of clauses : 17 ( 0 non-Horn; 17 unit; 2 RR) *)
+
+(* Number of atoms : 17 ( 17 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 33 ( 2 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : ORDERING LPO inverse > product > greatest_lower_bound > *)
+
+(* least_upper_bound > identity > a > b *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_p06d:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:eq Univ (greatest_lower_bound identity b) identity.
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H7:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H8:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H9:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H10:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H11:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H12:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H13:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H14:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H15:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (least_upper_bound identity (multiply (inverse a) (multiply b a))) (multiply (inverse a) (multiply b a))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP176-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP176-1 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : General form of distributivity *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* : [Dah95] Dahn (1995), Email to G. Sutcliffe *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 16 ( 0 non-Horn; 16 unit; 1 RR) *)
+
+(* Number of atoms : 16 ( 16 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 9 ( 5 constant; 0-2 arity) *)
+
+(* Number of variables : 33 ( 2 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : ORDERING LPO inverse > product > greatest_lower_bound > *)
+
+(* least_upper_bound > identity > a > b > c > d *)
+
+(* : ORDERING LPO greatest_lower_bound > least_upper_bound > *)
+
+(* inverse > product > identity > a > b > c > d *)
+
+(* : This is a standardized version of the problem that appears in *)
+
+(* [Sch95]. *)
+
+(* : [Dah95] says "Easy from equational axioms, More difficult from *)
+
+(* monotonicity. The assumtion is a consequence of group theory." *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_p07:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀d:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H6:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H7:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H8:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H9:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H10:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H11:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H12:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H13:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H14:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (multiply c (multiply (least_upper_bound a b) d)) (least_upper_bound (multiply c (multiply a d)) (multiply c (multiply b d)))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#d.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP176-2.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP176-2 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : General form of distributivity *)
+
+(* Version : [Fuc94] (equality) axioms : Augmented. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : p07 [Sch95] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 17 ( 0 non-Horn; 17 unit; 1 RR) *)
+
+(* Number of atoms : 17 ( 17 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 9 ( 5 constant; 0-2 arity) *)
+
+(* Number of variables : 35 ( 2 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : ORDERING LPO inverse > product > greatest_lower_bound > *)
+
+(* least_upper_bound > identity > a > b > c > d *)
+
+(* : ORDERING LPO greatest_lower_bound > least_upper_bound > *)
+
+(* inverse > product > identity > a > b > c > d *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_p07:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀d:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (inverse (multiply X Y)) (multiply (inverse Y) (inverse X)).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H7:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H8:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H9:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H10:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H11:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H12:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H13:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H14:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H15:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (multiply c (multiply (least_upper_bound a b) d)) (least_upper_bound (multiply c (multiply a d)) (multiply c (multiply b d)))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#d.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP177-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP177-1 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : A consequence of monotonicity *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : p08a [Sch95] *)
+
+(* Status : Unknown *)
+
+(* Rating : 1.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 19 ( 0 non-Horn; 19 unit; 4 RR) *)
+
+(* Number of atoms : 19 ( 19 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 8 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 33 ( 2 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : ORDERING LPO inverse > product > greatest_lower_bound > *)
+
+(* least_upper_bound > identity > a > b > c *)
+
+(* : ORDERING LPO greatest_lower_bound > least_upper_bound > *)
+
+(* inverse > product > identity > a > b > c *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_p08a:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:eq Univ (least_upper_bound identity c) c.
+∀H1:eq Univ (least_upper_bound identity b) b.
+∀H2:eq Univ (least_upper_bound identity a) a.
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H5:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H6:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H7:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H8:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H9:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H10:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H11:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H12:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H13:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H14:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H15:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H16:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H17:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (least_upper_bound (greatest_lower_bound a (multiply b c)) (multiply (greatest_lower_bound a b) (greatest_lower_bound a c))) (multiply (greatest_lower_bound a b) (greatest_lower_bound a c))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+#H16.
+#H17.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15,H16,H17;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP177-2.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP177-2 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : A consequence of monotonicity *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* Theorem formulation : Dual. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : p08b [Sch95] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.21 v3.1.0, 0.33 v2.7.0, 0.36 v2.6.0, 0.50 v2.5.0, 0.75 v2.4.0, 0.67 v2.2.1, 0.78 v2.2.0, 0.71 v2.1.0, 0.86 v2.0.0 *)
+
+(* Syntax : Number of clauses : 19 ( 0 non-Horn; 19 unit; 4 RR) *)
+
+(* Number of atoms : 19 ( 19 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 8 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 33 ( 2 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : ORDERING LPO inverse > product > greatest_lower_bound > *)
+
+(* least_upper_bound > identity > a > b > c *)
+
+(* : ORDERING LPO greatest_lower_bound > least_upper_bound > *)
+
+(* inverse > product > identity > a > b > c *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_p08b:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:eq Univ (greatest_lower_bound identity c) identity.
+∀H1:eq Univ (greatest_lower_bound identity b) identity.
+∀H2:eq Univ (greatest_lower_bound identity a) identity.
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H5:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H6:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H7:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H8:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H9:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H10:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H11:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H12:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H13:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H14:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H15:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H16:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H17:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (greatest_lower_bound (greatest_lower_bound a (multiply b c)) (multiply (greatest_lower_bound a b) (greatest_lower_bound a c))) (greatest_lower_bound a (multiply b c))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+#H16.
+#H17.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15,H16,H17;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP178-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP178-1 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : A consequence of monotonicity *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : p09a [Sch95] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.21 v3.2.0, 0.14 v3.1.0, 0.11 v2.7.0, 0.45 v2.6.0, 0.50 v2.5.0, 0.25 v2.4.0, 0.00 v2.2.1, 0.56 v2.2.0, 0.57 v2.1.0, 0.43 v2.0.0 *)
+
+(* Syntax : Number of clauses : 20 ( 0 non-Horn; 20 unit; 5 RR) *)
+
+(* Number of atoms : 20 ( 20 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 8 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 33 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : ORDERING LPO inverse > product > greatest_lower_bound > *)
+
+(* least_upper_bound > identity > a > b > c *)
+
+(* : ORDERING LPO greatest_lower_bound > least_upper_bound > *)
+
+(* inverse > product > identity > a > b > c *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_p09a:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:eq Univ (greatest_lower_bound a b) identity.
+∀H1:eq Univ (least_upper_bound identity c) c.
+∀H2:eq Univ (least_upper_bound identity b) b.
+∀H3:eq Univ (least_upper_bound identity a) a.
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H5:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H6:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H7:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H8:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H9:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H10:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H11:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H12:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H13:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H14:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H15:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H16:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H17:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H18:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (greatest_lower_bound a (multiply b c)) (greatest_lower_bound a c)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+#H16.
+#H17.
+#H18.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15,H16,H17,H18;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP178-2.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP178-2 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : A consequence of monotonicity *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* Theorem formulation : Dual. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : p09b [Sch95] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.21 v3.2.0, 0.14 v3.1.0, 0.22 v2.7.0, 0.45 v2.6.0, 0.33 v2.5.0, 0.25 v2.4.0, 0.00 v2.2.1, 0.67 v2.2.0, 0.71 v2.1.0, 0.43 v2.0.0 *)
+
+(* Syntax : Number of clauses : 20 ( 0 non-Horn; 20 unit; 5 RR) *)
+
+(* Number of atoms : 20 ( 20 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 8 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 33 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : ORDERING LPO inverse > product > greatest_lower_bound > *)
+
+(* least_upper_bound > identity > a > b > c *)
+
+(* : ORDERING LPO greatest_lower_bound > least_upper_bound > *)
+
+(* inverse > product > identity > a > b > c *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_p09b:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:eq Univ (greatest_lower_bound a b) identity.
+∀H1:eq Univ (greatest_lower_bound identity c) identity.
+∀H2:eq Univ (greatest_lower_bound identity b) identity.
+∀H3:eq Univ (greatest_lower_bound identity a) identity.
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H5:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H6:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H7:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H8:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H9:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H10:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H11:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H12:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H13:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H14:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H15:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H16:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H17:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H18:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (greatest_lower_bound a (multiply b c)) (greatest_lower_bound a c)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+#H16.
+#H17.
+#H18.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15,H16,H17,H18;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP179-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP179-1 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : For converting between GLB and LUB *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* : [Dah95] Dahn (1995), Email to G. Sutcliffe *)
+
+(* Source : [Sch95] *)
+
+(* Names : p10 [Sch95] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.50 v3.1.0, 0.44 v2.7.0, 0.45 v2.6.0, 0.50 v2.5.0, 0.25 v2.4.0, 0.00 v2.2.1, 0.78 v2.2.0, 0.86 v2.1.0, 0.86 v2.0.0 *)
+
+(* Syntax : Number of clauses : 16 ( 0 non-Horn; 16 unit; 1 RR) *)
+
+(* Number of atoms : 16 ( 16 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 33 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : ORDERING LPO inverse > product > greatest_lower_bound > *)
+
+(* least_upper_bound > identity > a > b *)
+
+(* : [Dah95] says this is "non-obvious". *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_p10:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H6:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H7:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H8:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H9:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H10:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H11:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H12:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H13:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H14:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (inverse (least_upper_bound a b)) (greatest_lower_bound (inverse a) (inverse b))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP179-2.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP179-2 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : For converting between GLB and LUB *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* Theorem formulation : Special case. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.50 v3.1.0, 0.44 v2.7.0, 0.45 v2.6.0, 0.50 v2.5.0, 0.25 v2.4.0, 0.00 v2.2.1, 0.67 v2.2.0, 0.71 v2.1.0, 0.86 v2.0.0 *)
+
+(* Syntax : Number of clauses : 16 ( 0 non-Horn; 16 unit; 1 RR) *)
+
+(* Number of atoms : 16 ( 16 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 33 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : ORDERING LPO inverse > product > greatest_lower_bound > *)
+
+(* least_upper_bound > identity > a *)
+
+(* : ORDERING LPO greatest_lower_bound > least_upper_bound > *)
+
+(* inverse > product > identity > a *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_p18:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H6:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H7:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H8:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H9:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H10:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H11:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H12:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H13:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H14:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (least_upper_bound (inverse a) identity) (inverse (greatest_lower_bound a identity))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP179-3.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP179-3 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : For converting between GLB and LUB *)
+
+(* Version : [Fuc94] (equality) axioms : Augmented. *)
+
+(* Theorem formulation : Special case. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : p18 [Sch95] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.36 v3.1.0, 0.33 v2.7.0, 0.45 v2.6.0, 0.50 v2.5.0, 0.25 v2.4.0, 0.00 v2.2.1, 0.67 v2.2.0, 0.71 v2.1.0, 0.86 v2.0.0 *)
+
+(* Syntax : Number of clauses : 19 ( 0 non-Horn; 19 unit; 2 RR) *)
+
+(* Number of atoms : 19 ( 19 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 36 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : ORDERING LPO inverse > product > greatest_lower_bound > *)
+
+(* least_upper_bound > identity > a *)
+
+(* : ORDERING LPO greatest_lower_bound > least_upper_bound > *)
+
+(* inverse > product > identity > a *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_p18:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (inverse (multiply X Y)) (multiply (inverse Y) (inverse X)).
+∀H1:∀X:Univ.eq Univ (inverse (inverse X)) X.
+∀H2:eq Univ (inverse identity) identity.
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H5:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H6:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H7:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H8:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H9:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H10:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H11:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H12:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H13:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H14:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H15:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H16:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H17:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (least_upper_bound (inverse a) identity) (inverse (greatest_lower_bound a identity))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+#H16.
+#H17.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15,H16,H17;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP180-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP180-1 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : Consequence of converting between GLB and LUB *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.50 v3.1.0, 0.44 v2.7.0, 0.55 v2.6.0, 0.50 v2.5.0, 0.25 v2.4.0, 0.00 v2.2.1, 0.78 v2.2.0, 0.86 v2.1.0, 0.86 v2.0.0 *)
+
+(* Syntax : Number of clauses : 16 ( 0 non-Horn; 16 unit; 1 RR) *)
+
+(* Number of atoms : 16 ( 16 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 33 ( 2 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : ORDERING LPO inverse > product > greatest_lower_bound > *)
+
+(* least_upper_bound > identity > a > b *)
+
+(* : ORDERING LPO greatest_lower_bound > least_upper_bound > *)
+
+(* inverse > product > identity > a > b *)
+
+(* : This is a standardized version of the problem that appears in *)
+
+(* [Sch95]. *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_p11:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H6:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H7:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H8:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H9:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H10:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H11:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H12:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H13:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H14:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (multiply a (multiply (inverse (greatest_lower_bound a b)) b)) (least_upper_bound a b)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP180-2.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP180-2 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : Consequence of converting between GLB and LUB *)
+
+(* Version : [Fuc94] (equality) axioms : Augmented. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : p11 [Sch95] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.43 v3.1.0, 0.33 v2.7.0, 0.55 v2.6.0, 0.50 v2.5.0, 0.25 v2.4.0, 0.33 v2.2.1, 0.78 v2.2.0, 0.86 v2.1.0, 0.86 v2.0.0 *)
+
+(* Syntax : Number of clauses : 19 ( 0 non-Horn; 19 unit; 2 RR) *)
+
+(* Number of atoms : 19 ( 19 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 36 ( 2 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : ORDERING LPO inverse > product > greatest_lower_bound > *)
+
+(* least_upper_bound > identity > a > b *)
+
+(* : ORDERING LPO greatest_lower_bound > least_upper_bound > *)
+
+(* inverse > product > identity > a > b *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_p11:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (inverse (multiply X Y)) (multiply (inverse Y) (inverse X)).
+∀H1:∀X:Univ.eq Univ (inverse (inverse X)) X.
+∀H2:eq Univ (inverse identity) identity.
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H5:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H6:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H7:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H8:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H9:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H10:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H11:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H12:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H13:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H14:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H15:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H16:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H17:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (multiply a (multiply (inverse (greatest_lower_bound a b)) b)) (least_upper_bound a b)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+#H16.
+#H17.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15,H16,H17;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP181-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP181-1 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : Distributivity of a lattice *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.71 v3.1.0, 0.56 v2.7.0, 0.55 v2.6.0, 0.67 v2.5.0, 0.50 v2.4.0, 0.00 v2.2.1, 0.89 v2.2.0, 0.86 v2.1.0, 1.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 18 ( 0 non-Horn; 18 unit; 3 RR) *)
+
+(* Number of atoms : 18 ( 18 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 8 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 33 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : ORDERING LPO inverse > product > greatest_lower_bound > *)
+
+(* least_upper_bound > identity > a > b > c *)
+
+(* : ORDERING LPO greatest_lower_bound > least_upper_bound > *)
+
+(* inverse > product > identity > a > b > c *)
+
+(* : This is a standardized version of the problem that appears in *)
+
+(* [Sch95]. *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_p12:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:eq Univ (least_upper_bound a c) (least_upper_bound b c).
+∀H1:eq Univ (greatest_lower_bound a c) (greatest_lower_bound b c).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H5:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H7:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H8:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H9:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H10:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H11:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H12:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H13:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H14:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H15:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H16:∀X:Univ.eq Univ (multiply identity X) X.eq Univ a b
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+#H16.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15,H16;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP181-2.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP181-2 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : Distributivity of a lattice *)
+
+(* Version : [Fuc94] (equality) axioms : Augmented. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : p12 [Sch95] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.71 v3.1.0, 0.56 v2.7.0, 0.55 v2.6.0, 0.50 v2.5.0, 0.25 v2.4.0, 0.00 v2.2.1, 0.89 v2.2.0, 0.86 v2.1.0, 1.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 21 ( 0 non-Horn; 21 unit; 4 RR) *)
+
+(* Number of atoms : 21 ( 21 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 8 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 36 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : ORDERING LPO inverse > product > greatest_lower_bound > *)
+
+(* least_upper_bound > identity > a > b > c *)
+
+(* : ORDERING LPO greatest_lower_bound > least_upper_bound > *)
+
+(* inverse > product > identity > a > b > c *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_p12:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:eq Univ (least_upper_bound a c) (least_upper_bound b c).
+∀H1:eq Univ (greatest_lower_bound a c) (greatest_lower_bound b c).
+∀H2:∀X:Univ.∀Y:Univ.eq Univ (inverse (multiply X Y)) (multiply (inverse Y) (inverse X)).
+∀H3:∀X:Univ.eq Univ (inverse (inverse X)) X.
+∀H4:eq Univ (inverse identity) identity.
+∀H5:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H6:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H7:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H8:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H9:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H10:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H11:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H12:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H13:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H14:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H15:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H16:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H17:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H18:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H19:∀X:Univ.eq Univ (multiply identity X) X.eq Univ a b
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+#H16.
+#H17.
+#H18.
+#H19.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15,H16,H17,H18,H19;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP181-3.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP181-3 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : Distributivity of a lattice *)
+
+(* Version : [Fuc94] (equality) axioms : Augmented. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.43 v3.1.0, 0.44 v2.7.0, 0.45 v2.6.0, 0.17 v2.5.0, 0.50 v2.4.0, 0.00 v2.2.1, 0.56 v2.2.0, 0.57 v2.1.0, 1.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 20 ( 0 non-Horn; 20 unit; 3 RR) *)
+
+(* Number of atoms : 20 ( 20 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 8 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 37 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : ORDERING LPO inverse > product > greatest_lower_bound > *)
+
+(* least_upper_bound > identity > a > b > c *)
+
+(* : ORDERING LPO greatest_lower_bound > least_upper_bound > *)
+
+(* inverse > product > identity > a > b > c *)
+
+(* : This is a standardized version of the problem that appears in *)
+
+(* [Sch95]. *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_p12x:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (inverse (least_upper_bound X Y)) (greatest_lower_bound (inverse X) (inverse Y)).
+∀H1:∀X:Univ.∀Y:Univ.eq Univ (inverse (greatest_lower_bound X Y)) (least_upper_bound (inverse X) (inverse Y)).
+∀H2:eq Univ (least_upper_bound a c) (least_upper_bound b c).
+∀H3:eq Univ (greatest_lower_bound a c) (greatest_lower_bound b c).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H5:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H6:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H7:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H8:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H9:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H10:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H11:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H12:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H13:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H14:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H15:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H16:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H17:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H18:∀X:Univ.eq Univ (multiply identity X) X.eq Univ a b
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+#H16.
+#H17.
+#H18.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15,H16,H17,H18;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP181-4.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP181-4 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : Distributivity of a lattice *)
+
+(* Version : [Fuc94] (equality) axioms : Augmented. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : p12x [Sch95] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.43 v3.1.0, 0.44 v2.7.0, 0.36 v2.6.0, 0.00 v2.2.1, 0.56 v2.2.0, 0.57 v2.1.0, 1.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 23 ( 0 non-Horn; 23 unit; 4 RR) *)
+
+(* Number of atoms : 23 ( 23 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 8 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 40 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : ORDERING LPO inverse > product > greatest_lower_bound > *)
+
+(* least_upper_bound > identity > a > b > c *)
+
+(* : ORDERING LPO greatest_lower_bound > least_upper_bound > *)
+
+(* inverse > product > identity > a > b > c *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_p12x:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (inverse (least_upper_bound X Y)) (greatest_lower_bound (inverse X) (inverse Y)).
+∀H1:∀X:Univ.∀Y:Univ.eq Univ (inverse (greatest_lower_bound X Y)) (least_upper_bound (inverse X) (inverse Y)).
+∀H2:eq Univ (least_upper_bound a c) (least_upper_bound b c).
+∀H3:eq Univ (greatest_lower_bound a c) (greatest_lower_bound b c).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (inverse (multiply X Y)) (multiply (inverse Y) (inverse X)).
+∀H5:∀X:Univ.eq Univ (inverse (inverse X)) X.
+∀H6:eq Univ (inverse identity) identity.
+∀H7:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H8:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H9:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H10:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H11:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H12:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H13:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H14:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H15:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H16:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H17:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H18:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H19:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H20:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H21:∀X:Univ.eq Univ (multiply identity X) X.eq Univ a b
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+#H16.
+#H17.
+#H18.
+#H19.
+#H20.
+#H21.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15,H16,H17,H18,H19,H20,H21;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP182-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP182-1 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : Positive part of the negative part is identity *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* : [Dah95] Dahn (1995), Email to G. Sutcliffe *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 16 ( 0 non-Horn; 16 unit; 1 RR) *)
+
+(* Number of atoms : 16 ( 16 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 33 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : ORDERING LPO inverse > product > greatest_lower_bound > *)
+
+(* least_upper_bound > identity > a *)
+
+(* : ORDERING LPO greatest_lower_bound > least_upper_bound > *)
+
+(* inverse > product > identity > a *)
+
+(* : This is a standardized version of the problem that appears in *)
+
+(* [Sch95]. *)
+
+(* : The ntheorem clause has been modified according to instructions *)
+
+(* in [Dah95]. *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----This is Schulz's clause *)
+
+(* input_clause(prove_p17a,negated_conjecture, *)
+
+(* [--equal(least_upper_bound(identity,least_upper_bound(a,identity)), *)
+
+(* least_upper_bound(a,identity))]). *)
+
+(* ----This is Dahn's clause *)
+ntheorem prove_p17a:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H6:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H7:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H8:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H9:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H10:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H11:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H12:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H13:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H14:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (least_upper_bound identity (greatest_lower_bound a identity)) identity
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP182-2.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP182-2 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : Positive part of the negative part is identity *)
+
+(* Version : [Fuc94] (equality) axioms : Augmented. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* : [Dah95] Dahn (1995), Email to G. Sutcliffe *)
+
+(* Source : [Sch95] *)
+
+(* Names : p17a [Sch95] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 19 ( 0 non-Horn; 19 unit; 2 RR) *)
+
+(* Number of atoms : 19 ( 19 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 36 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : ORDERING LPO inverse > product > greatest_lower_bound > *)
+
+(* least_upper_bound > identity > a *)
+
+(* : ORDERING LPO greatest_lower_bound > least_upper_bound > *)
+
+(* inverse > product > identity > a *)
+
+(* : The ntheorem clause has been modified according to instructions *)
+
+(* in [Dah95]. *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----This is Schulz's clause *)
+
+(* input_clause(prove_p17a,negated_conjecture, *)
+
+(* [--equal(least_upper_bound(identity,least_upper_bound(a,identity)), *)
+
+(* least_upper_bound(a,identity))]). *)
+
+(* ----This is Dahn's clause *)
+ntheorem prove_p17a:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (inverse (multiply X Y)) (multiply (inverse Y) (inverse X)).
+∀H1:∀X:Univ.eq Univ (inverse (inverse X)) X.
+∀H2:eq Univ (inverse identity) identity.
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H5:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H6:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H7:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H8:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H9:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H10:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H11:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H12:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H13:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H14:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H15:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H16:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H17:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (least_upper_bound identity (greatest_lower_bound a identity)) identity
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+#H16.
+#H17.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15,H16,H17;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP182-3.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP182-3 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : Positive part of the negative part is identity *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* Theorem formulation : Dual. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 16 ( 0 non-Horn; 16 unit; 1 RR) *)
+
+(* Number of atoms : 16 ( 16 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 33 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : ORDERING LPO inverse > product > greatest_lower_bound > *)
+
+(* least_upper_bound > identity > a *)
+
+(* : ORDERING LPO greatest_lower_bound > least_upper_bound > *)
+
+(* inverse > product > identity > a *)
+
+(* : This is a standardized version of the problem that appears in *)
+
+(* [Sch95]. *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_p17b:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H6:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H7:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H8:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H9:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H10:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H11:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H12:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H13:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H14:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (greatest_lower_bound identity (least_upper_bound a identity)) identity
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP182-4.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP182-4 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : Positive part of the negative part is identity *)
+
+(* Version : [Fuc94] (equality) axioms : Augmented. *)
+
+(* Theorem formulation : Dual. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : p17b [Sch95] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 19 ( 0 non-Horn; 19 unit; 2 RR) *)
+
+(* Number of atoms : 19 ( 19 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 36 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : ORDERING LPO inverse > product > greatest_lower_bound > *)
+
+(* least_upper_bound > identity > a *)
+
+(* : ORDERING LPO greatest_lower_bound > least_upper_bound > *)
+
+(* inverse > product > identity > a *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_p17b:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (inverse (multiply X Y)) (multiply (inverse Y) (inverse X)).
+∀H1:∀X:Univ.eq Univ (inverse (inverse X)) X.
+∀H2:eq Univ (inverse identity) identity.
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H5:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H6:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H7:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H8:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H9:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H10:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H11:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H12:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H13:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H14:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H15:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H16:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H17:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (greatest_lower_bound identity (least_upper_bound a identity)) identity
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+#H16.
+#H17.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15,H16,H17;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP183-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP183-1 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : Orthogonal elements form a subgroup with orthogonal parts *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : For each X {Y: X orth Y} is a subgroup. Moreover, pp(a) *)
+
+(* is orthogonal to np(a). *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.43 v3.1.0, 0.33 v2.7.0, 0.36 v2.6.0, 0.33 v2.5.0, 0.25 v2.4.0, 0.00 v2.2.1, 0.67 v2.2.0, 0.71 v2.1.0, 0.86 v2.0.0 *)
+
+(* Syntax : Number of clauses : 16 ( 0 non-Horn; 16 unit; 1 RR) *)
+
+(* Number of atoms : 16 ( 16 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 33 ( 2 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : ORDERING LPO inverse > product > greatest_lower_bound > *)
+
+(* least_upper_bound > identity > a *)
+
+(* : This is a standardized version of the problem that appears in *)
+
+(* [Sch95]. *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_p20:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H6:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H7:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H8:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H9:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H10:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H11:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H12:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H13:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H14:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (greatest_lower_bound (least_upper_bound a identity) (inverse (greatest_lower_bound a identity))) identity
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP183-2.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP183-2 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : Orthogonal elements form a subgroup with orthogonal parts *)
+
+(* Version : [Fuc94] (equality) axioms : Augmented. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : p20 [Sch95] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.43 v3.1.0, 0.44 v2.7.0, 0.45 v2.6.0, 0.50 v2.5.0, 0.25 v2.4.0, 0.00 v2.2.1, 0.67 v2.2.0, 0.71 v2.1.0, 0.86 v2.0.0 *)
+
+(* Syntax : Number of clauses : 19 ( 0 non-Horn; 19 unit; 2 RR) *)
+
+(* Number of atoms : 19 ( 19 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 36 ( 2 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : ORDERING LPO inverse > product > greatest_lower_bound > *)
+
+(* least_upper_bound > identity > a *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_p20:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (inverse (multiply X Y)) (multiply (inverse Y) (inverse X)).
+∀H1:∀X:Univ.eq Univ (inverse (inverse X)) X.
+∀H2:eq Univ (inverse identity) identity.
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H5:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H6:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H7:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H8:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H9:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H10:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H11:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H12:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H13:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H14:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H15:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H16:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H17:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (greatest_lower_bound (least_upper_bound a identity) (inverse (greatest_lower_bound a identity))) identity
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+#H16.
+#H17.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15,H16,H17;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP183-3.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP183-3 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : Orthogonal elements form a subgroup with orthogonal parts *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* Theorem formulation : Variant. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.43 v3.1.0, 0.33 v2.7.0, 0.36 v2.6.0, 0.33 v2.5.0, 0.25 v2.4.0, 0.00 v2.2.1, 0.67 v2.2.0, 0.71 v2.1.0, 0.86 v2.0.0 *)
+
+(* Syntax : Number of clauses : 16 ( 0 non-Horn; 16 unit; 1 RR) *)
+
+(* Number of atoms : 16 ( 16 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 33 ( 2 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : ORDERING LPO inverse > product > greatest_lower_bound > *)
+
+(* least_upper_bound > identity > a *)
+
+(* : This is a standardized version of the problem that appears in *)
+
+(* [Sch95]. *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_20x:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H6:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H7:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H8:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H9:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H10:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H11:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H12:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H13:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H14:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (greatest_lower_bound (least_upper_bound a identity) (least_upper_bound (inverse a) identity)) identity
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP183-4.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP183-4 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : Orthogonal elements form a subgroup with orthogonal parts *)
+
+(* Version : [Fuc94] (equality) axioms : Augmented. *)
+
+(* Theorem formulation : Variant. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : p20x [Sch95] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.36 v3.1.0, 0.33 v2.7.0, 0.45 v2.6.0, 0.50 v2.5.0, 0.25 v2.4.0, 0.00 v2.2.1, 0.67 v2.2.0, 0.71 v2.1.0, 0.86 v2.0.0 *)
+
+(* Syntax : Number of clauses : 19 ( 0 non-Horn; 19 unit; 2 RR) *)
+
+(* Number of atoms : 19 ( 19 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 36 ( 2 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : ORDERING LPO inverse > product > greatest_lower_bound > *)
+
+(* least_upper_bound > identity > a *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_20x:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (inverse (multiply X Y)) (multiply (inverse Y) (inverse X)).
+∀H1:∀X:Univ.eq Univ (inverse (inverse X)) X.
+∀H2:eq Univ (inverse identity) identity.
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H5:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H6:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H7:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H8:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H9:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H10:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H11:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H12:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H13:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H14:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H15:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H16:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H17:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (greatest_lower_bound (least_upper_bound a identity) (least_upper_bound (inverse a) identity)) identity
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+#H16.
+#H17.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15,H16,H17;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP184-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP184-1 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : Orthogonal elements commute and form a subgroup *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : For each X {Y: X orth Y} is a subgroup. X orthogonal to Y *)
+
+(* implies that X and Y commute. Moreover, pp(a) orthogonal to *)
+
+(* np(a). *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.29 v3.1.0, 0.00 v2.7.0, 0.36 v2.6.0, 0.33 v2.5.0, 0.00 v2.4.0, 0.00 v2.2.1, 0.67 v2.2.0, 0.71 v2.1.0, 0.86 v2.0.0 *)
+
+(* Syntax : Number of clauses : 16 ( 0 non-Horn; 16 unit; 1 RR) *)
+
+(* Number of atoms : 16 ( 16 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 33 ( 2 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : ORDERING LPO inverse > product > greatest_lower_bound > *)
+
+(* least_upper_bound > identity > a *)
+
+(* : This is a standardized version of the problem that appears in *)
+
+(* [Sch95]. *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_p21:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H6:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H7:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H8:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H9:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H10:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H11:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H12:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H13:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H14:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (multiply (least_upper_bound a identity) (inverse (greatest_lower_bound a identity))) (multiply (inverse (greatest_lower_bound a identity)) (least_upper_bound a identity))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP184-2.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP184-2 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : Orthogonal elements commute and form a subgroup *)
+
+(* Version : [Fuc94] (equality) axioms : Augmented. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : p21 [Sch95] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.14 v3.1.0, 0.00 v2.7.0, 0.18 v2.6.0, 0.17 v2.5.0, 0.00 v2.4.0, 0.00 v2.2.1, 0.56 v2.2.0, 0.57 v2.1.0, 0.71 v2.0.0 *)
+
+(* Syntax : Number of clauses : 19 ( 0 non-Horn; 19 unit; 2 RR) *)
+
+(* Number of atoms : 19 ( 19 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 36 ( 2 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : ORDERING LPO inverse > product > greatest_lower_bound > *)
+
+(* least_upper_bound > identity > a *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_p21:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (inverse (multiply X Y)) (multiply (inverse Y) (inverse X)).
+∀H1:∀X:Univ.eq Univ (inverse (inverse X)) X.
+∀H2:eq Univ (inverse identity) identity.
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H5:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H6:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H7:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H8:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H9:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H10:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H11:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H12:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H13:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H14:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H15:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H16:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H17:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (multiply (least_upper_bound a identity) (inverse (greatest_lower_bound a identity))) (multiply (inverse (greatest_lower_bound a identity)) (least_upper_bound a identity))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+#H16.
+#H17.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15,H16,H17;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP184-3.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP184-3 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : Orthogonal elements commute and form a subgroup *)
+
+(* Version : [Fuc94] (equality) axioms : Augmented. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.21 v3.1.0, 0.00 v2.7.0, 0.45 v2.6.0, 0.33 v2.5.0, 0.00 v2.4.0, 0.00 v2.2.1, 0.67 v2.2.0, 0.71 v2.1.0, 0.86 v2.0.0 *)
+
+(* Syntax : Number of clauses : 16 ( 0 non-Horn; 16 unit; 1 RR) *)
+
+(* Number of atoms : 16 ( 16 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 33 ( 2 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : ORDERING LPO inverse > product > greatest_lower_bound > *)
+
+(* least_upper_bound > identity > a *)
+
+(* : This is a standardized version of the problem that appears in *)
+
+(* [Sch95]. *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_p21x:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H6:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H7:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H8:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H9:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H10:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H11:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H12:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H13:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H14:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (multiply (least_upper_bound a identity) (inverse (greatest_lower_bound a identity))) (multiply (inverse (greatest_lower_bound a identity)) (least_upper_bound a identity))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP184-4.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP184-4 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : Orthogonal elements commute and form a subgroup *)
+
+(* Version : [Fuc94] (equality) axioms : Augmented. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : p21x [Sch95] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.7.0, 0.09 v2.6.0, 0.00 v2.2.1, 0.56 v2.2.0, 0.57 v2.1.0, 0.43 v2.0.0 *)
+
+(* Syntax : Number of clauses : 21 ( 0 non-Horn; 21 unit; 2 RR) *)
+
+(* Number of atoms : 21 ( 21 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 40 ( 2 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : ORDERING LPO inverse > product > greatest_lower_bound > *)
+
+(* least_upper_bound > identity > a *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_p21x:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (inverse (least_upper_bound X Y)) (greatest_lower_bound (inverse X) (inverse Y)).
+∀H1:∀X:Univ.∀Y:Univ.eq Univ (inverse (greatest_lower_bound X Y)) (least_upper_bound (inverse X) (inverse Y)).
+∀H2:∀X:Univ.∀Y:Univ.eq Univ (inverse (multiply X Y)) (multiply (inverse Y) (inverse X)).
+∀H3:∀X:Univ.eq Univ (inverse (inverse X)) X.
+∀H4:eq Univ (inverse identity) identity.
+∀H5:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H6:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H7:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H8:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H9:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H10:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H11:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H12:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H13:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H14:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H15:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H16:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H17:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H18:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H19:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (multiply (least_upper_bound a identity) (inverse (greatest_lower_bound a identity))) (multiply (inverse (greatest_lower_bound a identity)) (least_upper_bound a identity))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+#H16.
+#H17.
+#H18.
+#H19.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15,H16,H17,H18,H19;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP185-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP185-1 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : Application of monotonicity and distributivity *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.21 v3.2.0, 0.07 v3.1.0, 0.11 v2.7.0, 0.27 v2.6.0, 0.17 v2.5.0, 0.25 v2.4.0, 0.67 v2.2.1, 0.67 v2.2.0, 0.57 v2.1.0, 0.29 v2.0.0 *)
+
+(* Syntax : Number of clauses : 16 ( 0 non-Horn; 16 unit; 1 RR) *)
+
+(* Number of atoms : 16 ( 16 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 33 ( 2 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : ORDERING LPO inverse > product > greatest_lower_bound > *)
+
+(* least_upper_bound > identity > a > b *)
+
+(* : This is a standardized version of the problem that appears in *)
+
+(* [Sch95]. *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_p22a:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H6:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H7:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H8:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H9:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H10:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H11:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H12:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H13:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H14:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (least_upper_bound (least_upper_bound (multiply a b) identity) (multiply (least_upper_bound a identity) (least_upper_bound b identity))) (multiply (least_upper_bound a identity) (least_upper_bound b identity))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP185-2.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP185-2 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : Application of monotonicity and distributivity *)
+
+(* Version : [Fuc94] (equality) axioms : Augmented. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : p22a [Sch95] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.21 v3.2.0, 0.07 v3.1.0, 0.11 v2.7.0, 0.27 v2.6.0, 0.17 v2.5.0, 0.25 v2.4.0, 0.67 v2.2.1, 0.78 v2.2.0, 0.71 v2.1.0, 0.29 v2.0.0 *)
+
+(* Syntax : Number of clauses : 19 ( 0 non-Horn; 19 unit; 2 RR) *)
+
+(* Number of atoms : 19 ( 19 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 36 ( 2 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : ORDERING LPO inverse > product > greatest_lower_bound > *)
+
+(* least_upper_bound > identity > a > b *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_p22a:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (inverse (multiply X Y)) (multiply (inverse Y) (inverse X)).
+∀H1:∀X:Univ.eq Univ (inverse (inverse X)) X.
+∀H2:eq Univ (inverse identity) identity.
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H5:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H6:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H7:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H8:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H9:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H10:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H11:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H12:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H13:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H14:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H15:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H16:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H17:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (least_upper_bound (least_upper_bound (multiply a b) identity) (multiply (least_upper_bound a identity) (least_upper_bound b identity))) (multiply (least_upper_bound a identity) (least_upper_bound b identity))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+#H16.
+#H17.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15,H16,H17;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP185-3.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP185-3 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : Application of monotonicity and distributivity *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* Theorem formulation : Using a dual definition of =<. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.21 v3.1.0, 0.33 v2.7.0, 0.36 v2.6.0, 0.67 v2.5.0, 0.75 v2.4.0, 0.33 v2.2.1, 0.56 v2.2.0, 0.43 v2.1.0, 0.43 v2.0.0 *)
+
+(* Syntax : Number of clauses : 16 ( 0 non-Horn; 16 unit; 1 RR) *)
+
+(* Number of atoms : 16 ( 16 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 33 ( 2 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : ORDERING LPO inverse > product > greatest_lower_bound > *)
+
+(* least_upper_bound > identity > a > b *)
+
+(* : This is a standardized version of the problem that appears in *)
+
+(* [Sch95]. *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_p22b:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H6:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H7:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H8:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H9:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H10:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H11:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H12:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H13:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H14:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (greatest_lower_bound (least_upper_bound (multiply a b) identity) (multiply (least_upper_bound a identity) (least_upper_bound b identity))) (least_upper_bound (multiply a b) identity)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP185-4.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP185-4 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : Application of monotonicity and distributivity *)
+
+(* Version : [Fuc94] (equality) axioms : Augmented. *)
+
+(* Theorem formulation : Using a dual definition of =<. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : p22b [Sch95] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.21 v3.1.0, 0.33 v2.7.0, 0.36 v2.6.0, 0.67 v2.5.0, 0.75 v2.4.0, 0.67 v2.2.1, 0.67 v2.2.0, 0.57 v2.1.0, 0.43 v2.0.0 *)
+
+(* Syntax : Number of clauses : 19 ( 0 non-Horn; 19 unit; 2 RR) *)
+
+(* Number of atoms : 19 ( 19 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 36 ( 2 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : ORDERING LPO inverse > product > greatest_lower_bound > *)
+
+(* least_upper_bound > identity > a > b *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_p22b:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (inverse (multiply X Y)) (multiply (inverse Y) (inverse X)).
+∀H1:∀X:Univ.eq Univ (inverse (inverse X)) X.
+∀H2:eq Univ (inverse identity) identity.
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H5:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H6:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H7:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H8:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H9:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H10:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H11:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H12:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H13:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H14:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H15:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H16:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H17:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (greatest_lower_bound (least_upper_bound (multiply a b) identity) (multiply (least_upper_bound a identity) (least_upper_bound b identity))) (least_upper_bound (multiply a b) identity)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+#H16.
+#H17.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15,H16,H17;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP186-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP186-1 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : Application of distributivity and group theory *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.57 v3.1.0, 0.44 v2.7.0, 0.55 v2.6.0, 0.50 v2.5.0, 0.25 v2.4.0, 0.00 v2.2.1, 0.78 v2.2.0, 0.86 v2.1.0, 0.86 v2.0.0 *)
+
+(* Syntax : Number of clauses : 16 ( 0 non-Horn; 16 unit; 1 RR) *)
+
+(* Number of atoms : 16 ( 16 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 33 ( 2 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : ORDERING LPO inverse > product > greatest_lower_bound > *)
+
+(* least_upper_bound > identity > a > b *)
+
+(* : This is a standardized version of the problem that appears in *)
+
+(* [Sch95]. *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_p23:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H6:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H7:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H8:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H9:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H10:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H11:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H12:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H13:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H14:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (least_upper_bound (multiply a b) identity) (multiply a (inverse (greatest_lower_bound a (inverse b))))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP186-2.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP186-2 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : Application of distributivity and group theory *)
+
+(* Version : [Fuc94] (equality) axioms : Augmented. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : p23 [Sch95] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.36 v3.1.0, 0.33 v2.7.0, 0.55 v2.6.0, 0.50 v2.5.0, 0.25 v2.4.0, 0.00 v2.2.1, 0.78 v2.2.0, 0.86 v2.1.0, 0.86 v2.0.0 *)
+
+(* Syntax : Number of clauses : 19 ( 0 non-Horn; 19 unit; 2 RR) *)
+
+(* Number of atoms : 19 ( 19 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 36 ( 2 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : ORDERING LPO inverse > product > greatest_lower_bound > *)
+
+(* least_upper_bound > identity > a > b *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_p23:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (inverse (multiply X Y)) (multiply (inverse Y) (inverse X)).
+∀H1:∀X:Univ.eq Univ (inverse (inverse X)) X.
+∀H2:eq Univ (inverse identity) identity.
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H5:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H6:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H7:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H8:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H9:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H10:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H11:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H12:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H13:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H14:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H15:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H16:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H17:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (least_upper_bound (multiply a b) identity) (multiply a (inverse (greatest_lower_bound a (inverse b))))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+#H16.
+#H17.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15,H16,H17;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP186-3.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP186-3 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : Application of distributivity and group theory *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* Theorem formulation : Switched from GLB to LUB. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.2.1, 0.11 v2.2.0, 0.14 v2.1.0, 0.14 v2.0.0 *)
+
+(* Syntax : Number of clauses : 16 ( 0 non-Horn; 16 unit; 1 RR) *)
+
+(* Number of atoms : 16 ( 16 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 33 ( 2 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : ORDERING LPO inverse > product > greatest_lower_bound > *)
+
+(* least_upper_bound > identity > a > b *)
+
+(* : This is a standardized version of the problem that appears in *)
+
+(* [Sch95]. *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_p23x:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H6:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H7:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H8:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H9:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H10:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H11:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H12:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H13:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H14:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (least_upper_bound (multiply a b) identity) (multiply a (least_upper_bound (inverse a) b))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP186-4.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP186-4 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : Application of distributivity and group theory *)
+
+(* Version : [Fuc94] (equality) axioms : Augmented. *)
+
+(* Theorem formulation : Switched from GLB to LUB. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : p23x [Sch95] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.1.0, 0.14 v2.0.0 *)
+
+(* Syntax : Number of clauses : 19 ( 0 non-Horn; 19 unit; 2 RR) *)
+
+(* Number of atoms : 19 ( 19 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 36 ( 2 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : ORDERING LPO inverse > product > greatest_lower_bound > *)
+
+(* least_upper_bound > identity > a > b *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_p23x:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (inverse (multiply X Y)) (multiply (inverse Y) (inverse X)).
+∀H1:∀X:Univ.eq Univ (inverse (inverse X)) X.
+∀H2:eq Univ (inverse identity) identity.
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H5:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H6:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H7:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H8:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H9:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H10:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H11:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H12:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H13:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H14:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H15:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H16:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H17:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (least_upper_bound (multiply a b) identity) (multiply a (least_upper_bound (inverse a) b))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+#H16.
+#H17.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15,H16,H17;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP187-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP187-1 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : Orthogonal elements commute *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* : [Dah95] Dahn (1995), Email to G. Sutcliffe *)
+
+(* Source : [Sch95] *)
+
+(* Names : p33 [Sch95] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.64 v3.1.0, 0.56 v2.7.0, 0.55 v2.6.0, 0.50 v2.5.0, 0.75 v2.4.0, 0.67 v2.2.1, 1.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 17 ( 0 non-Horn; 17 unit; 2 RR) *)
+
+(* Number of atoms : 17 ( 17 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 33 ( 2 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : ORDERING LPO inverse > product > greatest_lower_bound > *)
+
+(* least_upper_bound > identity > a > b *)
+
+(* : [Dah95] says "Non-obvious. Usually proved using lat4." *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_p33:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:eq Univ (greatest_lower_bound (least_upper_bound a (inverse a)) (least_upper_bound b (inverse b))) identity.
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H7:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H8:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H9:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H10:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H11:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H12:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H13:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H14:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H15:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (multiply a b) (multiply b a)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP188-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP188-1 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : Consequence of lattice theory *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 16 ( 0 non-Horn; 16 unit; 1 RR) *)
+
+(* Number of atoms : 16 ( 16 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 33 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : ORDERING LPO greatest_lower_bound > least_upper_bound > *)
+
+(* inverse > product > identity > a > b *)
+
+(* : This is a standardized version of the problem that appears in *)
+
+(* [Sch95]. *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_p38a:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H6:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H7:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H8:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H9:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H10:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H11:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H12:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H13:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H14:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (least_upper_bound b (least_upper_bound a b)) (least_upper_bound a b)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP188-2.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP188-2 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : Consequence of lattice theory *)
+
+(* Version : [Fuc94] (equality) axioms : Augmented. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : p38a [Sch95] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 19 ( 0 non-Horn; 19 unit; 2 RR) *)
+
+(* Number of atoms : 19 ( 19 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 36 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : ORDERING LPO greatest_lower_bound > least_upper_bound > *)
+
+(* inverse > product > identity > a > b *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_p38a:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (inverse (multiply X Y)) (multiply (inverse Y) (inverse X)).
+∀H1:∀X:Univ.eq Univ (inverse (inverse X)) X.
+∀H2:eq Univ (inverse identity) identity.
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H5:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H6:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H7:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H8:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H9:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H10:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H11:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H12:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H13:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H14:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H15:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H16:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H17:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (least_upper_bound b (least_upper_bound a b)) (least_upper_bound a b)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+#H16.
+#H17.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15,H16,H17;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP189-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP189-1 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : Consequence of lattice theory *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 16 ( 0 non-Horn; 16 unit; 1 RR) *)
+
+(* Number of atoms : 16 ( 16 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 33 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : ORDERING LPO greatest_lower_bound > least_upper_bound > *)
+
+(* inverse > product > identity > a > b *)
+
+(* : This is a standardized version of the problem that appears in *)
+
+(* [Sch95]. *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_p38b:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H6:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H7:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H8:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H9:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H10:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H11:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H12:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H13:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H14:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (greatest_lower_bound b (least_upper_bound a b)) b
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP189-2.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP189-2 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : Consequence of lattice theory *)
+
+(* Version : [Fuc94] (equality) axioms : Augmented. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : p38b [Sch95] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 19 ( 0 non-Horn; 19 unit; 2 RR) *)
+
+(* Number of atoms : 19 ( 19 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 36 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : ORDERING LPO greatest_lower_bound > least_upper_bound > *)
+
+(* inverse > product > identity > a > b *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_p38b:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (inverse (multiply X Y)) (multiply (inverse Y) (inverse X)).
+∀H1:∀X:Univ.eq Univ (inverse (inverse X)) X.
+∀H2:eq Univ (inverse identity) identity.
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H5:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H6:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H7:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H8:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H9:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H10:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H11:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H12:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H13:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H14:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H15:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H16:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H17:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (greatest_lower_bound b (least_upper_bound a b)) b
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+#H16.
+#H17.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15,H16,H17;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP190-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP190-1 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : Something useful for estimations *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : p39a [Sch95] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.2.1, 0.22 v2.2.0, 0.29 v2.1.0, 0.29 v2.0.0 *)
+
+(* Syntax : Number of clauses : 17 ( 0 non-Horn; 17 unit; 2 RR) *)
+
+(* Number of atoms : 17 ( 17 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 33 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : ORDERING LPO inverse > product > greatest_lower_bound > *)
+
+(* least_upper_bound > identity > a > b *)
+
+(* : ORDERING LPO greatest_lower_bound > least_upper_bound > *)
+
+(* inverse > product > identity > a > b *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_p39a:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:eq Univ (least_upper_bound a b) a.
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H7:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H8:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H9:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H10:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H11:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H12:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H13:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H14:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H15:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (least_upper_bound (inverse a) (inverse b)) (inverse b)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP190-2.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP190-2 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : Something useful for estimations *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* Theorem formulation : Using a dual definition of =<. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : p39c [Sch95] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.2.1, 0.22 v2.2.0, 0.29 v2.1.0, 0.29 v2.0.0 *)
+
+(* Syntax : Number of clauses : 17 ( 0 non-Horn; 17 unit; 2 RR) *)
+
+(* Number of atoms : 17 ( 17 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 33 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : ORDERING LPO inverse > product > greatest_lower_bound > *)
+
+(* least_upper_bound > identity > a > b *)
+
+(* : ORDERING LPO greatest_lower_bound > least_upper_bound > *)
+
+(* inverse > product > identity > a > b *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_p39c:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:eq Univ (least_upper_bound a b) a.
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H7:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H8:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H9:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H10:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H11:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H12:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H13:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H14:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H15:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (greatest_lower_bound (inverse a) (inverse b)) (inverse a)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP191-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP191-1 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : Something useful for estimations *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : p39b [Sch95] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.2.1, 0.22 v2.2.0, 0.29 v2.1.0, 0.29 v2.0.0 *)
+
+(* Syntax : Number of clauses : 17 ( 0 non-Horn; 17 unit; 2 RR) *)
+
+(* Number of atoms : 17 ( 17 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 33 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : ORDERING LPO inverse > product > greatest_lower_bound > *)
+
+(* least_upper_bound > identity > a > b *)
+
+(* : ORDERING LPO greatest_lower_bound > least_upper_bound > *)
+
+(* inverse > product > identity > a > b *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_p39b:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:eq Univ (greatest_lower_bound a b) b.
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H7:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H8:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H9:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H10:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H11:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H12:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H13:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H14:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H15:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (greatest_lower_bound (inverse a) (inverse b)) (inverse a)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP191-2.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP191-2 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : Something useful for estimations *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* Theorem formulation : Using a dual definition of =<. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : p39d [Sch95] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.2.1, 0.22 v2.2.0, 0.29 v2.1.0, 0.29 v2.0.0 *)
+
+(* Syntax : Number of clauses : 17 ( 0 non-Horn; 17 unit; 2 RR) *)
+
+(* Number of atoms : 17 ( 17 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 33 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : ORDERING LPO inverse > product > greatest_lower_bound > *)
+
+(* least_upper_bound > identity > a > b *)
+
+(* : ORDERING LPO greatest_lower_bound > least_upper_bound > *)
+
+(* inverse > product > identity > a > b *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_p39d:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:eq Univ (greatest_lower_bound a b) b.
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H7:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H8:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H9:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H10:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H11:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H12:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H13:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H14:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H15:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (least_upper_bound (inverse a) (inverse b)) (inverse b)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP192-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP192-1 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : Even elements implies trivial group *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : The assumption all(X,1 =< X) even implies that the group is *)
+
+(* trivial, i.e., all(X, X = 1). *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : p40a [Sch95] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.07 v3.1.0, 0.11 v2.7.0, 0.00 v2.2.1, 0.33 v2.2.0, 0.43 v2.1.0, 0.14 v2.0.0 *)
+
+(* Syntax : Number of clauses : 17 ( 0 non-Horn; 17 unit; 1 RR) *)
+
+(* Number of atoms : 17 ( 17 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 34 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : ORDERING LPO inverse > product > greatest_lower_bound > *)
+
+(* least_upper_bound > identity > a > b *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_p40a:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.eq Univ (least_upper_bound identity X) X.
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H7:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H8:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H9:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H10:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H11:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H12:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H13:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H14:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H15:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (multiply a b) (multiply b a)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP193-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP193-1 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : A combination of distributivity and monotonicity *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : p8_9a [Sch95] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.2.1, 0.44 v2.2.0, 0.43 v2.1.0, 0.14 v2.0.0 *)
+
+(* Syntax : Number of clauses : 21 ( 0 non-Horn; 21 unit; 6 RR) *)
+
+(* Number of atoms : 21 ( 21 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 8 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 33 ( 2 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : ORDERING LPO greatest_lower_bound > least_upper_bound > *)
+
+(* inverse > product > identity > a > b > c *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_p8_9a:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:eq Univ (least_upper_bound (greatest_lower_bound a (multiply b c)) (multiply (greatest_lower_bound a b) (greatest_lower_bound a c))) (multiply (greatest_lower_bound a b) (greatest_lower_bound a c)).
+∀H1:eq Univ (greatest_lower_bound a b) identity.
+∀H2:eq Univ (least_upper_bound identity c) c.
+∀H3:eq Univ (least_upper_bound identity b) b.
+∀H4:eq Univ (least_upper_bound identity a) a.
+∀H5:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H6:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H7:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H8:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H9:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H10:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H11:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H12:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H13:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H14:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H15:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H16:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H17:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H18:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H19:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (greatest_lower_bound a (multiply b c)) (greatest_lower_bound a c)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+#H16.
+#H17.
+#H18.
+#H19.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15,H16,H17,H18,H19;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP193-2.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP193-2 : TPTP v3.2.0. Bugfixed v1.2.1. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : A combination of distributivity and monotonicity *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* Theorem formulation : Using a dual definition of =<. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : p8_9b [Sch95] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.2.1, 0.44 v2.2.0, 0.43 v2.1.0, 0.14 v2.0.0 *)
+
+(* Syntax : Number of clauses : 21 ( 0 non-Horn; 21 unit; 6 RR) *)
+
+(* Number of atoms : 21 ( 21 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 8 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 33 ( 2 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : ORDERING LPO greatest_lower_bound > least_upper_bound > *)
+
+(* inverse > product > identity > a > b > c *)
+
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include equality group theory axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_p8_9b:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀greatest_lower_bound:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀least_upper_bound:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:eq Univ (greatest_lower_bound (greatest_lower_bound a (multiply b c)) (multiply (greatest_lower_bound a b) (greatest_lower_bound a c))) (greatest_lower_bound a (multiply b c)).
+∀H1:eq Univ (greatest_lower_bound a b) identity.
+∀H2:eq Univ (greatest_lower_bound identity c) identity.
+∀H3:eq Univ (greatest_lower_bound identity b) identity.
+∀H4:eq Univ (greatest_lower_bound identity a) identity.
+∀H5:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+∀H6:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+∀H7:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+∀H8:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+∀H9:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+∀H10:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+∀H11:∀X:Univ.eq Univ (greatest_lower_bound X X) X.
+∀H12:∀X:Univ.eq Univ (least_upper_bound X X) X.
+∀H13:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+∀H14:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+∀H15:∀X:Univ.∀Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+∀H16:∀X:Univ.∀Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+∀H17:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H18:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H19:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (greatest_lower_bound a (multiply b c)) (greatest_lower_bound a c)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#greatest_lower_bound.
+#identity.
+#inverse.
+#least_upper_bound.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+#H16.
+#H17.
+#H18.
+#H19.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15,H16,H17,H18,H19;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP195-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP195-1 : TPTP v3.2.0. Released v2.2.0. *)
+
+(* Domain : Group Theory (Semigroups) *)
+
+(* Problem : In semigroups, xyy=yyx -> (uv)^4 = u^4v^4. *)
+
+(* Version : [MP96] (equality) axioms. *)
+
+(* English : In semigroups, xyy=yyx -> uvuvuvuuv=uuuuvvvv. *)
+
+(* Refs : [McC98] McCune (1998), Email to G. Sutcliffe *)
+
+(* : [MP96] McCune & Padmanabhan (1996), Automated Deduction in Eq *)
+
+(* Source : [McC98] *)
+
+(* Names : CS-2 [MP96] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.7.0, 0.09 v2.6.0, 0.17 v2.5.0, 0.00 v2.4.0, 0.00 v2.2.1 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 8 ( 5 average) *)
+
+(* Comments : The problem was originally posed for cancellative semigroups, *)
+
+(* but Otter discovered that cancellation is not necessary. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include semigroups axioms *)
+
+(* Inclusion of: Axioms/GRP008-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP008-0 : TPTP v3.2.0. Released v2.2.0. *)
+
+(* Domain : Group Theory (Semigroups) *)
+
+(* Axioms : Semigroups axioms *)
+
+(* Version : [MP96] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC98] McCune (1998), Email to G. Sutcliffe *)
+
+(* : [MP96] McCune & Padmanabhan (1996), Automated Deduction in Eq *)
+
+(* Source : [McC98] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 1 ( 0 non-Horn; 1 unit; 0 RR) *)
+
+(* Number of literals : 1 ( 1 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 1 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 3 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Associativity: *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Hypothesis: *)
+
+(* ----Denial of conclusion: *)
+ntheorem prove_this:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (multiply X (multiply Y Y)) (multiply Y (multiply Y X)).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).eq Univ (multiply a (multiply b (multiply a (multiply b (multiply a (multiply b (multiply a b))))))) (multiply a (multiply a (multiply a (multiply a (multiply b (multiply b (multiply b b)))))))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#multiply.
+#H0.
+#H1.
+nauto by H0,H1;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP196-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP196-1 : TPTP v3.2.0. Released v2.2.0. *)
+
+(* Domain : Group Theory (Semigroups) *)
+
+(* Problem : In semigroups, xyyy=yyyx -> (uy)^9 = u^9v^9. *)
+
+(* Version : [MP96] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC98] McCune (1998), Email to G. Sutcliffe *)
+
+(* : [MP96] McCune & Padmanabhan (1996), Automated Deduction in Eq *)
+
+(* : [McC95] McCune (1995), Four Challenge Problems in Equational L *)
+
+(* Source : [McC98] *)
+
+(* Names : CS-3 [MP96] *)
+
+(* : Problem B [McC95] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.93 v3.1.0, 1.00 v2.2.1 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 18 ( 8 average) *)
+
+(* Comments : The problem was originally posed for cancellative semigroups, *)
+
+(* Otter does this with a nonstandard representation [MP96]. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include semigroups axioms *)
+
+(* Inclusion of: Axioms/GRP008-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP008-0 : TPTP v3.2.0. Released v2.2.0. *)
+
+(* Domain : Group Theory (Semigroups) *)
+
+(* Axioms : Semigroups axioms *)
+
+(* Version : [MP96] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC98] McCune (1998), Email to G. Sutcliffe *)
+
+(* : [MP96] McCune & Padmanabhan (1996), Automated Deduction in Eq *)
+
+(* Source : [McC98] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 1 ( 0 non-Horn; 1 unit; 0 RR) *)
+
+(* Number of literals : 1 ( 1 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 1 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 3 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Associativity: *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Hypothesis: *)
+
+(* ----Denial of conclusion: *)
+ntheorem prove_this:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (multiply X (multiply Y (multiply Y Y))) (multiply Y (multiply Y (multiply Y X))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).eq Univ (multiply a (multiply b (multiply a (multiply b (multiply a (multiply b (multiply a (multiply b (multiply a (multiply b (multiply a (multiply b (multiply a (multiply b (multiply a (multiply b (multiply a b))))))))))))))))) (multiply a (multiply a (multiply a (multiply a (multiply a (multiply a (multiply a (multiply a (multiply a (multiply b (multiply b (multiply b (multiply b (multiply b (multiply b (multiply b (multiply b b)))))))))))))))))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#multiply.
+#H0.
+#H1.
+nauto by H0,H1;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP200-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP200-1 : TPTP v3.2.0. Released v2.2.0. *)
+
+(* Domain : Group Theory (Loops) *)
+
+(* Problem : In Loops, Moufang-1 => Moufang-2. *)
+
+(* Version : [MP96] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC98] McCune (1998), Email to G. Sutcliffe *)
+
+(* : [Wos96] Wos (1996), OTTER and the Moufang Identity Problem *)
+
+(* : [MP96] McCune & Padmanabhan (1996), Automated Deduction in Eq *)
+
+(* Source : [McC98] *)
+
+(* Names : MFL-1 [MP96] *)
+
+(* : - [Wos96] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.29 v3.1.0, 0.33 v2.7.0, 0.27 v2.6.0, 0.00 v2.2.1 *)
+
+(* Syntax : Number of clauses : 10 ( 0 non-Horn; 10 unit; 1 RR) *)
+
+(* Number of atoms : 10 ( 10 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 9 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 15 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Loop axioms: *)
+
+(* ----Moufang-1: *)
+
+(* ----Denial of Moufang-2: *)
+ntheorem prove_moufang2:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀identity:Univ.
+∀left_division:∀_:Univ.∀_:Univ.Univ.
+∀left_inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀right_division:∀_:Univ.∀_:Univ.Univ.
+∀right_inverse:∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X (multiply Y Z)) X) (multiply (multiply X Y) (multiply Z X)).
+∀H1:∀X:Univ.eq Univ (multiply (left_inverse X) X) identity.
+∀H2:∀X:Univ.eq Univ (multiply X (right_inverse X)) identity.
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (right_division (multiply X Y) Y) X.
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (multiply (right_division X Y) Y) X.
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (left_division X (multiply X Y)) Y.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (multiply X (left_division X Y)) Y.
+∀H7:∀X:Univ.eq Univ (multiply X identity) X.
+∀H8:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (multiply (multiply (multiply a b) c) b) (multiply a (multiply b (multiply c b)))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#identity.
+#left_division.
+#left_inverse.
+#multiply.
+#right_division.
+#right_inverse.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP201-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP201-1 : TPTP v3.2.0. Released v2.2.0. *)
+
+(* Domain : Group Theory (Loops) *)
+
+(* Problem : In Loops, Moufang-2 => Moufang-3. *)
+
+(* Version : [MP96] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC98] McCune (1998), Email to G. Sutcliffe *)
+
+(* : [Wos96] Wos (1996), OTTER and the Moufang Identity Problem *)
+
+(* : [MP96] McCune & Padmanabhan (1996), Automated Deduction in Eq *)
+
+(* Source : [McC98] *)
+
+(* Names : MFL-2 [MP96] *)
+
+(* : - [Wos96] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.36 v3.1.0, 0.44 v2.7.0, 0.36 v2.6.0, 0.00 v2.2.1 *)
+
+(* Syntax : Number of clauses : 10 ( 0 non-Horn; 10 unit; 1 RR) *)
+
+(* Number of atoms : 10 ( 10 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 9 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 15 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Loop axioms: *)
+
+(* ----Moufang-2: *)
+
+(* ----Denial of Moufang-3: *)
+ntheorem prove_moufang3:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀identity:Univ.
+∀left_division:∀_:Univ.∀_:Univ.Univ.
+∀left_inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀right_division:∀_:Univ.∀_:Univ.Univ.
+∀right_inverse:∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply (multiply X Y) Z) Y) (multiply X (multiply Y (multiply Z Y))).
+∀H1:∀X:Univ.eq Univ (multiply (left_inverse X) X) identity.
+∀H2:∀X:Univ.eq Univ (multiply X (right_inverse X)) identity.
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (right_division (multiply X Y) Y) X.
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (multiply (right_division X Y) Y) X.
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (left_division X (multiply X Y)) Y.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (multiply X (left_division X Y)) Y.
+∀H7:∀X:Univ.eq Univ (multiply X identity) X.
+∀H8:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (multiply (multiply (multiply a b) a) c) (multiply a (multiply b (multiply a c)))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#identity.
+#left_division.
+#left_inverse.
+#multiply.
+#right_division.
+#right_inverse.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP202-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP202-1 : TPTP v3.2.0. Released v2.2.0. *)
+
+(* Domain : Group Theory (Loops) *)
+
+(* Problem : In Loops, Moufang-3 => Moufang-1. *)
+
+(* Version : [MP96] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC98] McCune (1998), Email to G. Sutcliffe *)
+
+(* : [Wos96] Wos (1996), OTTER and the Moufang Identity Problem *)
+
+(* : [MP96] McCune & Padmanabhan (1996), Automated Deduction in Eq *)
+
+(* Source : [McC98] *)
+
+(* Names : MFL-3 [MP96] *)
+
+(* : - [Wos96] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.43 v3.1.0, 0.44 v2.7.0, 0.36 v2.6.0, 0.00 v2.2.1 *)
+
+(* Syntax : Number of clauses : 10 ( 0 non-Horn; 10 unit; 1 RR) *)
+
+(* Number of atoms : 10 ( 10 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 9 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 15 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Loop axioms: *)
+
+(* ----Moufang-3 *)
+
+(* ----Denial of Moufang-1 *)
+ntheorem prove_moufang1:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀identity:Univ.
+∀left_division:∀_:Univ.∀_:Univ.Univ.
+∀left_inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀right_division:∀_:Univ.∀_:Univ.Univ.
+∀right_inverse:∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply (multiply X Y) X) Z) (multiply X (multiply Y (multiply X Z))).
+∀H1:∀X:Univ.eq Univ (multiply (left_inverse X) X) identity.
+∀H2:∀X:Univ.eq Univ (multiply X (right_inverse X)) identity.
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (right_division (multiply X Y) Y) X.
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (multiply (right_division X Y) Y) X.
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (left_division X (multiply X Y)) Y.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (multiply X (left_division X Y)) Y.
+∀H7:∀X:Univ.eq Univ (multiply X identity) X.
+∀H8:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (multiply (multiply a (multiply b c)) a) (multiply (multiply a b) (multiply c a))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#identity.
+#left_division.
+#left_inverse.
+#multiply.
+#right_division.
+#right_inverse.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP203-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP203-1 : TPTP v3.2.0. Released v2.2.0. *)
+
+(* Domain : Group Theory (Loops) *)
+
+(* Problem : Left identity, left inverse, Moufang-3 => Moufang-2 *)
+
+(* Version : [MP96] (equality) axioms : Especial. *)
+
+(* English : *)
+
+(* Refs : [McC98] McCune (1998), Email to G. Sutcliffe *)
+
+(* : [MP96] McCune & Padmanabhan (1996), Automated Deduction in Eq *)
+
+(* Source : [McC98] *)
+
+(* Names : MFL-7 [MP96] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.14 v3.2.0, 0.21 v3.1.0, 0.11 v2.7.0, 0.18 v2.6.0, 0.00 v2.2.1 *)
+
+(* Syntax : Number of clauses : 4 ( 0 non-Horn; 4 unit; 1 RR) *)
+
+(* Number of atoms : 4 ( 4 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 3 average) *)
+
+(* Comments : Given left identity and left inverse, Moufang-2 and Moufang-3 *)
+
+(* are equivalent, but Moufang-1 is weaker (see MFL-8). *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Left identity and left inverse: *)
+
+(* ----Moufang-3: *)
+
+(* ----Denial of Moufang-2: *)
+ntheorem prove_moufang2:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀identity:Univ.
+∀left_inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply (multiply X Y) X) Z) (multiply X (multiply Y (multiply X Z))).
+∀H1:∀X:Univ.eq Univ (multiply (left_inverse X) X) identity.
+∀H2:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (multiply (multiply (multiply a b) c) b) (multiply a (multiply b (multiply c b)))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#identity.
+#left_inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+nauto by H0,H1,H2;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP204-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP204-1 : TPTP v3.2.0. Released v2.2.0. *)
+
+(* Domain : Group Theory (Loops) *)
+
+(* Problem : A non-basis for Moufang loops. *)
+
+(* Version : [MP96] (equality) axioms : Especial. *)
+
+(* English : Left identity, left inverse, Moufang-1 do not imply Moufang-2; *)
+
+(* that is, is not a basis for Moufang loops. *)
+
+(* Refs : [McC98] McCune (1998), Email to G. Sutcliffe *)
+
+(* : [MP96] McCune & Padmanabhan (1996), Automated Deduction in Eq *)
+
+(* Source : [McC98] *)
+
+(* Names : MFL-8 [MP96] *)
+
+(* Status : Satisfiable *)
+
+(* Rating : 0.33 v3.2.0, 0.67 v3.1.0, 0.33 v2.4.0, 0.67 v2.3.0, 1.00 v2.2.1 *)
+
+(* Syntax : Number of clauses : 4 ( 0 non-Horn; 4 unit; 1 RR) *)
+
+(* Number of atoms : 4 ( 4 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 3 average) *)
+
+(* Comments : The smallest model has 3 elements. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Left identity and left inverse: *)
+
+(* ----Moufang-1: *)
+
+(* ----Denial of Moufang-2: *)
+ntheorem prove_moufang2:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀identity:Univ.
+∀left_inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X (multiply Y Z)) X) (multiply (multiply X Y) (multiply Z X)).
+∀H1:∀X:Univ.eq Univ (multiply (left_inverse X) X) identity.
+∀H2:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (multiply (multiply (multiply a b) c) b) (multiply a (multiply b (multiply c b)))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#identity.
+#left_inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+nauto by H0,H1,H2;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP205-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP205-1 : TPTP v3.2.0. Released v2.3.0. *)
+
+(* Domain : Group Theory (Loops) *)
+
+(* Problem : In Loops, Moufang-3 => Moufang-4. *)
+
+(* Version : [MP96] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Wos96] Wos (1996), OTTER and the Moufang Identity Problem *)
+
+(* Source : [Wos96] *)
+
+(* Names : - [Wos96] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.43 v3.1.0, 0.33 v2.7.0, 0.36 v2.6.0, 0.00 v2.4.0, 0.33 v2.3.0 *)
+
+(* Syntax : Number of clauses : 10 ( 0 non-Horn; 10 unit; 1 RR) *)
+
+(* Number of atoms : 10 ( 10 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 9 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 15 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Loop axioms: *)
+
+(* ----Moufang-3 *)
+
+(* ----Denial of Moufang-4 *)
+ntheorem prove_moufang4:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀identity:Univ.
+∀left_division:∀_:Univ.∀_:Univ.Univ.
+∀left_inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀right_division:∀_:Univ.∀_:Univ.Univ.
+∀right_inverse:∀_:Univ.Univ.
+∀x:Univ.
+∀y:Univ.
+∀z:Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply (multiply X Y) X) Z) (multiply X (multiply Y (multiply X Z))).
+∀H1:∀X:Univ.eq Univ (multiply (left_inverse X) X) identity.
+∀H2:∀X:Univ.eq Univ (multiply X (right_inverse X)) identity.
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (right_division (multiply X Y) Y) X.
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (multiply (right_division X Y) Y) X.
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (left_division X (multiply X Y)) Y.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (multiply X (left_division X Y)) Y.
+∀H7:∀X:Univ.eq Univ (multiply X identity) X.
+∀H8:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (multiply x (multiply (multiply y z) x)) (multiply (multiply x y) (multiply z x))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#identity.
+#left_division.
+#left_inverse.
+#multiply.
+#right_division.
+#right_inverse.
+#x.
+#y.
+#z.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP206-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP206-1 : TPTP v3.2.0. Released v2.3.0. *)
+
+(* Domain : Group Theory (Loops) *)
+
+(* Problem : In Loops, Moufang-4 => Moufang-1. *)
+
+(* Version : [MP96] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Wos96] Wos (1996), OTTER and the Moufang Identity Problem *)
+
+(* Source : [Wos96] *)
+
+(* Names : - [Wos96] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.3.0 *)
+
+(* Syntax : Number of clauses : 10 ( 0 non-Horn; 10 unit; 1 RR) *)
+
+(* Number of atoms : 10 ( 10 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 9 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 15 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Loop axioms: *)
+
+(* ----Moufang-4 *)
+
+(* ----Denial of Moufang-1 *)
+ntheorem prove_moufang1:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀identity:Univ.
+∀left_division:∀_:Univ.∀_:Univ.Univ.
+∀left_inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀right_division:∀_:Univ.∀_:Univ.Univ.
+∀right_inverse:∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (multiply (multiply Y Z) X)) (multiply (multiply X Y) (multiply Z X)).
+∀H1:∀X:Univ.eq Univ (multiply (left_inverse X) X) identity.
+∀H2:∀X:Univ.eq Univ (multiply X (right_inverse X)) identity.
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (right_division (multiply X Y) Y) X.
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (multiply (right_division X Y) Y) X.
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (left_division X (multiply X Y)) Y.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (multiply X (left_division X Y)) Y.
+∀H7:∀X:Univ.eq Univ (multiply X identity) X.
+∀H8:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (multiply (multiply a (multiply b c)) a) (multiply (multiply a b) (multiply c a))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#identity.
+#left_division.
+#left_inverse.
+#multiply.
+#right_division.
+#right_inverse.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP207-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP207-1 : TPTP v3.2.0. Released v2.4.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Single non-axiom for group theory, in product & inverse *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : This is a single axiom for group theory, in terms of product *)
+
+(* and inverse. *)
+
+(* Refs : [Pel98] Peltier (1998), A New Method for Automated Finite Mode *)
+
+(* : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [Pel98] *)
+
+(* Names : 4.2.2 [Pel98] *)
+
+(* Status : Satisfiable *)
+
+(* Rating : 0.33 v3.2.0, 0.67 v3.1.0, 0.33 v2.4.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 3 ( 0 singleton) *)
+
+(* Maximal term depth : 8 ( 4 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem try_prove_this_axiom:
+ ∀Univ:Type.∀U:Univ.∀Y:Univ.∀Z:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀u:Univ.
+∀x:Univ.
+∀y:Univ.
+∀z:Univ.
+∀H0:∀U:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply U (inverse (multiply Y (multiply (multiply (multiply Z (inverse Z)) (inverse (multiply U Y))) U)))) U.eq Univ (multiply x (inverse (multiply y (multiply (multiply (multiply z (inverse z)) (inverse (multiply u y))) x)))) u
+.
+#Univ.
+#U.
+#Y.
+#Z.
+#inverse.
+#multiply.
+#u.
+#x.
+#y.
+#z.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP393-2.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP393-2 : TPTP v3.2.0. Released v2.5.0. *)
+
+(* Domain : Group Theory (Semigroups) *)
+
+(* Problem : Semigroups axioms *)
+
+(* Version : [MP96] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC98] McCune (1998), Email to G. Sutcliffe *)
+
+(* : [MP96] McCune & Padmanabhan (1996), Automated Deduction in Eq *)
+
+(* Source : [McC98] *)
+
+(* Names : *)
+
+(* Status : Satisfiable *)
+
+(* Rating : 0.00 v3.2.0, 0.33 v3.1.0, 0.00 v2.5.0 *)
+
+(* Syntax : Number of clauses : 1 ( 0 non-Horn; 1 unit; 0 RR) *)
+
+(* Number of atoms : 1 ( 1 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 1 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 3 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Semigroups axioms *)
+
+(* Inclusion of: Axioms/GRP008-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP008-0 : TPTP v3.2.0. Released v2.2.0. *)
+
+(* Domain : Group Theory (Semigroups) *)
+
+(* Axioms : Semigroups axioms *)
+
+(* Version : [MP96] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC98] McCune (1998), Email to G. Sutcliffe *)
+
+(* : [MP96] McCune & Padmanabhan (1996), Automated Deduction in Eq *)
+
+(* Source : [McC98] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 1 ( 0 non-Horn; 1 unit; 0 RR) *)
+
+(* Number of literals : 1 ( 1 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 1 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 3 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Associativity: *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP394-3.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP394-3 : TPTP v3.2.0. Released v2.5.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : Satisfiable *)
+
+(* Rating : 0.00 v3.2.0, 0.33 v3.1.0, 0.00 v2.5.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Group theory (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP399-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP399-1 : TPTP v3.2.0. Released v2.5.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Problem : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : Satisfiable *)
+
+(* Rating : 0.33 v2.5.0 *)
+
+(* Syntax : Number of clauses : 15 ( 0 non-Horn; 15 unit; 0 RR) *)
+
+(* Number of atoms : 15 ( 15 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 33 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Group theory (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Group Theory *)
+
+(* Axioms : Group theory (equality) axioms *)
+
+(* Version : [MOW76] (equality) axioms : *)
+
+(* Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : [MOW76] also contains redundant right_identity and *)
+
+(* right_inverse axioms. *)
+
+(* : These axioms are also used in [Wos88] p.186, also with *)
+
+(* right_identity and right_inverse. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+
+(* ----is needed here since this is an instance of reflexivity *)
+
+(* ----There exists an identity element *)
+
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+
+(* ----= identity. *)
+
+(* ----The operation '*' is associative *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice ordered group (equality) axioms *)
+
+(* Inclusion of: Axioms/GRP004-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP004-2 : TPTP v3.2.0. Bugfixed v1.2.0. *)
+
+(* Domain : Group Theory (Lattice Ordered) *)
+
+(* Axioms : Lattice ordered group (equality) axioms *)
+
+(* Version : [Fuc94] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+
+(* Source : [Sch95] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+
+(* Number of literals : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Requires GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Specification of the least upper bound and greatest lower bound *)
+
+(* ----Monotony of multiply *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP403-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP403-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Axiom for group theory, in product & inverse, part 1 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Kun92] Kunen (1992), Single Axioms for Groups *)
+
+(* : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.07 v3.1.0, 0.11 v2.7.0, 0.09 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 3 ( 0 singleton) *)
+
+(* Maximal term depth : 8 ( 4 average) *)
+
+(* Comments : A UEQ part of GRP049-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_1:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a1:Univ.
+∀b1:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (multiply A (inverse (multiply (inverse (multiply (inverse (multiply A B)) C)) (inverse (multiply B (multiply (inverse B) B)))))) C.eq Univ (multiply (inverse a1) a1) (multiply (inverse b1) b1)
+.
+#Univ.
+#A.
+#B.
+#C.
+#a1.
+#b1.
+#inverse.
+#multiply.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP404-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP404-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Axiom for group theory, in product & inverse, part 2 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Kun92] Kunen (1992), Single Axioms for Groups *)
+
+(* : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.29 v3.2.0, 0.21 v3.1.0, 0.33 v2.7.0, 0.36 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 3 ( 0 singleton) *)
+
+(* Maximal term depth : 8 ( 4 average) *)
+
+(* Comments : A UEQ part of GRP049-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_2:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a2:Univ.
+∀b2:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (multiply A (inverse (multiply (inverse (multiply (inverse (multiply A B)) C)) (inverse (multiply B (multiply (inverse B) B)))))) C.eq Univ (multiply (multiply (inverse b2) b2) a2) a2
+.
+#Univ.
+#A.
+#B.
+#C.
+#a2.
+#b2.
+#inverse.
+#multiply.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP405-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP405-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Axiom for group theory, in product & inverse, part 3 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Kun92] Kunen (1992), Single Axioms for Groups *)
+
+(* : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.29 v3.2.0, 0.21 v3.1.0, 0.33 v2.7.0, 0.27 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 3 ( 0 singleton) *)
+
+(* Maximal term depth : 8 ( 4 average) *)
+
+(* Comments : A UEQ part of GRP049-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_3:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a3:Univ.
+∀b3:Univ.
+∀c3:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (multiply A (inverse (multiply (inverse (multiply (inverse (multiply A B)) C)) (inverse (multiply B (multiply (inverse B) B)))))) C.eq Univ (multiply (multiply a3 b3) c3) (multiply a3 (multiply b3 c3))
+.
+#Univ.
+#A.
+#B.
+#C.
+#a3.
+#b3.
+#c3.
+#inverse.
+#multiply.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP406-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP406-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Axiom for group theory, in product & inverse, part 1 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Kun92] Kunen (1992), Single Axioms for Groups *)
+
+(* : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 3 ( 0 singleton) *)
+
+(* Maximal term depth : 8 ( 4 average) *)
+
+(* Comments : A UEQ part of GRP050-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_1:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a1:Univ.
+∀b1:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (multiply A (inverse (multiply (inverse (multiply (inverse (multiply A B)) C)) (multiply (inverse B) (multiply (inverse B) B))))) C.eq Univ (multiply (inverse a1) a1) (multiply (inverse b1) b1)
+.
+#Univ.
+#A.
+#B.
+#C.
+#a1.
+#b1.
+#inverse.
+#multiply.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP407-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP407-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Axiom for group theory, in product & inverse, part 2 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Kun92] Kunen (1992), Single Axioms for Groups *)
+
+(* : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v3.1.0, 0.11 v2.7.0, 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 3 ( 0 singleton) *)
+
+(* Maximal term depth : 8 ( 4 average) *)
+
+(* Comments : A UEQ part of GRP050-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_2:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a2:Univ.
+∀b2:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (multiply A (inverse (multiply (inverse (multiply (inverse (multiply A B)) C)) (multiply (inverse B) (multiply (inverse B) B))))) C.eq Univ (multiply (multiply (inverse b2) b2) a2) a2
+.
+#Univ.
+#A.
+#B.
+#C.
+#a2.
+#b2.
+#inverse.
+#multiply.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP408-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP408-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Axiom for group theory, in product & inverse, part 3 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Kun92] Kunen (1992), Single Axioms for Groups *)
+
+(* : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v3.1.0, 0.11 v2.7.0, 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 3 ( 0 singleton) *)
+
+(* Maximal term depth : 8 ( 4 average) *)
+
+(* Comments : A UEQ part of GRP050-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_3:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a3:Univ.
+∀b3:Univ.
+∀c3:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (multiply A (inverse (multiply (inverse (multiply (inverse (multiply A B)) C)) (multiply (inverse B) (multiply (inverse B) B))))) C.eq Univ (multiply (multiply a3 b3) c3) (multiply a3 (multiply b3 c3))
+.
+#Univ.
+#A.
+#B.
+#C.
+#a3.
+#b3.
+#c3.
+#inverse.
+#multiply.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP409-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP409-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Axiom for group theory, in product & inverse, part 1 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Kun92] Kunen (1992), Single Axioms for Groups *)
+
+(* : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 3 ( 0 singleton) *)
+
+(* Maximal term depth : 7 ( 4 average) *)
+
+(* Comments : A UEQ part of GRP051-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_1:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a1:Univ.
+∀b1:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (multiply (multiply (inverse (multiply A (inverse (multiply B C)))) (multiply A (inverse C))) (inverse (multiply (inverse C) C))) B.eq Univ (multiply (inverse a1) a1) (multiply (inverse b1) b1)
+.
+#Univ.
+#A.
+#B.
+#C.
+#a1.
+#b1.
+#inverse.
+#multiply.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP410-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP410-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Axiom for group theory, in product & inverse, part 2 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Kun92] Kunen (1992), Single Axioms for Groups *)
+
+(* : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.21 v3.1.0, 0.22 v2.7.0, 0.18 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 3 ( 0 singleton) *)
+
+(* Maximal term depth : 7 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP051-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_2:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a2:Univ.
+∀b2:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (multiply (multiply (inverse (multiply A (inverse (multiply B C)))) (multiply A (inverse C))) (inverse (multiply (inverse C) C))) B.eq Univ (multiply (multiply (inverse b2) b2) a2) a2
+.
+#Univ.
+#A.
+#B.
+#C.
+#a2.
+#b2.
+#inverse.
+#multiply.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP411-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP411-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Axiom for group theory, in product & inverse, part 3 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Kun92] Kunen (1992), Single Axioms for Groups *)
+
+(* : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.21 v3.1.0, 0.11 v2.7.0, 0.45 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 3 ( 0 singleton) *)
+
+(* Maximal term depth : 7 ( 4 average) *)
+
+(* Comments : A UEQ part of GRP051-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_3:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a3:Univ.
+∀b3:Univ.
+∀c3:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (multiply (multiply (inverse (multiply A (inverse (multiply B C)))) (multiply A (inverse C))) (inverse (multiply (inverse C) C))) B.eq Univ (multiply (multiply a3 b3) c3) (multiply a3 (multiply b3 c3))
+.
+#Univ.
+#A.
+#B.
+#C.
+#a3.
+#b3.
+#c3.
+#inverse.
+#multiply.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP412-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP412-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Axiom for group theory, in product & inverse, part 1 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Kun92] Kunen (1992), Single Axioms for Groups *)
+
+(* : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 3 ( 0 singleton) *)
+
+(* Maximal term depth : 10 ( 4 average) *)
+
+(* Comments : A UEQ part of GRP052-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_1:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a1:Univ.
+∀b1:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (multiply A (inverse (multiply (multiply (multiply (inverse B) B) (inverse (multiply (inverse (multiply A (inverse B))) C))) B))) C.eq Univ (multiply (inverse a1) a1) (multiply (inverse b1) b1)
+.
+#Univ.
+#A.
+#B.
+#C.
+#a1.
+#b1.
+#inverse.
+#multiply.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP413-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP413-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Axiom for group theory, in product & inverse, part 2 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Kun92] Kunen (1992), Single Axioms for Groups *)
+
+(* : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v3.2.0, 0.07 v3.1.0, 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 3 ( 0 singleton) *)
+
+(* Maximal term depth : 10 ( 4 average) *)
+
+(* Comments : A UEQ part of GRP052-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_2:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a2:Univ.
+∀b2:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (multiply A (inverse (multiply (multiply (multiply (inverse B) B) (inverse (multiply (inverse (multiply A (inverse B))) C))) B))) C.eq Univ (multiply (multiply (inverse b2) b2) a2) a2
+.
+#Univ.
+#A.
+#B.
+#C.
+#a2.
+#b2.
+#inverse.
+#multiply.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP414-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP414-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Axiom for group theory, in product & inverse, part 3 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Kun92] Kunen (1992), Single Axioms for Groups *)
+
+(* : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.07 v3.1.0, 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 3 ( 0 singleton) *)
+
+(* Maximal term depth : 10 ( 4 average) *)
+
+(* Comments : A UEQ part of GRP052-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_3:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a3:Univ.
+∀b3:Univ.
+∀c3:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (multiply A (inverse (multiply (multiply (multiply (inverse B) B) (inverse (multiply (inverse (multiply A (inverse B))) C))) B))) C.eq Univ (multiply (multiply a3 b3) c3) (multiply a3 (multiply b3 c3))
+.
+#Univ.
+#A.
+#B.
+#C.
+#a3.
+#b3.
+#c3.
+#inverse.
+#multiply.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP415-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP415-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Axiom for group theory, in product & inverse, part 1 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Kun92] Kunen (1992), Single Axioms for Groups *)
+
+(* : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v3.1.0, 0.11 v2.7.0, 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 3 ( 0 singleton) *)
+
+(* Maximal term depth : 9 ( 4 average) *)
+
+(* Comments : A UEQ part of GRP053-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_1:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a1:Univ.
+∀b1:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (inverse (multiply A (inverse (multiply (inverse (multiply (inverse (multiply B A)) (multiply B (inverse C)))) (inverse (multiply (inverse A) A)))))) C.eq Univ (multiply (inverse a1) a1) (multiply (inverse b1) b1)
+.
+#Univ.
+#A.
+#B.
+#C.
+#a1.
+#b1.
+#inverse.
+#multiply.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP416-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP416-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Axiom for group theory, in product & inverse, part 2 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Kun92] Kunen (1992), Single Axioms for Groups *)
+
+(* : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.07 v3.1.0, 0.11 v2.7.0, 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 3 ( 0 singleton) *)
+
+(* Maximal term depth : 9 ( 4 average) *)
+
+(* Comments : A UEQ part of GRP053-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_2:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a2:Univ.
+∀b2:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (inverse (multiply A (inverse (multiply (inverse (multiply (inverse (multiply B A)) (multiply B (inverse C)))) (inverse (multiply (inverse A) A)))))) C.eq Univ (multiply (multiply (inverse b2) b2) a2) a2
+.
+#Univ.
+#A.
+#B.
+#C.
+#a2.
+#b2.
+#inverse.
+#multiply.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP417-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP417-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Axiom for group theory, in product & inverse, part 3 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Kun92] Kunen (1992), Single Axioms for Groups *)
+
+(* : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.07 v3.2.0, 0.14 v3.1.0, 0.11 v2.7.0, 0.09 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 3 ( 0 singleton) *)
+
+(* Maximal term depth : 9 ( 4 average) *)
+
+(* Comments : A UEQ part of GRP053-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_3:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a3:Univ.
+∀b3:Univ.
+∀c3:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (inverse (multiply A (inverse (multiply (inverse (multiply (inverse (multiply B A)) (multiply B (inverse C)))) (inverse (multiply (inverse A) A)))))) C.eq Univ (multiply (multiply a3 b3) c3) (multiply a3 (multiply b3 c3))
+.
+#Univ.
+#A.
+#B.
+#C.
+#a3.
+#b3.
+#c3.
+#inverse.
+#multiply.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP418-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP418-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Axiom for group theory, in product & inverse, part 1 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Kun92] Kunen (1992), Single Axioms for Groups *)
+
+(* : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.07 v3.1.0, 0.11 v2.7.0, 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 3 ( 0 singleton) *)
+
+(* Maximal term depth : 12 ( 5 average) *)
+
+(* Comments : A UEQ part of GRP054-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_1:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a1:Univ.
+∀b1:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (inverse (multiply (inverse (multiply A (inverse (multiply (inverse B) (inverse (multiply C (inverse (multiply (inverse C) C)))))))) (multiply A C))) B.eq Univ (multiply (inverse a1) a1) (multiply (inverse b1) b1)
+.
+#Univ.
+#A.
+#B.
+#C.
+#a1.
+#b1.
+#inverse.
+#multiply.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP419-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP419-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Axiom for group theory, in product & inverse, part 2 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Kun92] Kunen (1992), Single Axioms for Groups *)
+
+(* : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.29 v3.1.0, 0.11 v2.7.0, 0.18 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 3 ( 0 singleton) *)
+
+(* Maximal term depth : 12 ( 4 average) *)
+
+(* Comments : A UEQ part of GRP054-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_2:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a2:Univ.
+∀b2:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (inverse (multiply (inverse (multiply A (inverse (multiply (inverse B) (inverse (multiply C (inverse (multiply (inverse C) C)))))))) (multiply A C))) B.eq Univ (multiply (multiply (inverse b2) b2) a2) a2
+.
+#Univ.
+#A.
+#B.
+#C.
+#a2.
+#b2.
+#inverse.
+#multiply.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP420-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP420-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Axiom for group theory, in product & inverse, part 3 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Kun92] Kunen (1992), Single Axioms for Groups *)
+
+(* : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.36 v3.1.0, 0.22 v2.7.0, 0.36 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 3 ( 0 singleton) *)
+
+(* Maximal term depth : 12 ( 5 average) *)
+
+(* Comments : A UEQ part of GRP054-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_3:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a3:Univ.
+∀b3:Univ.
+∀c3:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (inverse (multiply (inverse (multiply A (inverse (multiply (inverse B) (inverse (multiply C (inverse (multiply (inverse C) C)))))))) (multiply A C))) B.eq Univ (multiply (multiply a3 b3) c3) (multiply a3 (multiply b3 c3))
+.
+#Univ.
+#A.
+#B.
+#C.
+#a3.
+#b3.
+#c3.
+#inverse.
+#multiply.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP421-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP421-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Axiom for group theory, in product & inverse, part 1 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Kun92] Kunen (1992), Single Axioms for Groups *)
+
+(* : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.14 v3.1.0, 0.22 v2.7.0, 0.09 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 3 ( 0 singleton) *)
+
+(* Maximal term depth : 11 ( 4 average) *)
+
+(* Comments : A UEQ part of GRP055-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_1:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a1:Univ.
+∀b1:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (inverse (multiply (inverse (multiply A (inverse (multiply (inverse B) (multiply (inverse C) (inverse (multiply (inverse C) C))))))) (multiply A C))) B.eq Univ (multiply (inverse a1) a1) (multiply (inverse b1) b1)
+.
+#Univ.
+#A.
+#B.
+#C.
+#a1.
+#b1.
+#inverse.
+#multiply.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP422-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP422-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Axiom for group theory, in product & inverse, part 2 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Kun92] Kunen (1992), Single Axioms for Groups *)
+
+(* : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.29 v3.1.0, 0.44 v2.7.0, 0.27 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 3 ( 0 singleton) *)
+
+(* Maximal term depth : 11 ( 4 average) *)
+
+(* Comments : A UEQ part of GRP055-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_2:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a2:Univ.
+∀b2:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (inverse (multiply (inverse (multiply A (inverse (multiply (inverse B) (multiply (inverse C) (inverse (multiply (inverse C) C))))))) (multiply A C))) B.eq Univ (multiply (multiply (inverse b2) b2) a2) a2
+.
+#Univ.
+#A.
+#B.
+#C.
+#a2.
+#b2.
+#inverse.
+#multiply.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP423-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP423-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Axiom for group theory, in product & inverse, part 3 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Kun92] Kunen (1992), Single Axioms for Groups *)
+
+(* : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.36 v3.1.0, 0.22 v2.7.0, 0.36 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 3 ( 0 singleton) *)
+
+(* Maximal term depth : 11 ( 4 average) *)
+
+(* Comments : A UEQ part of GRP055-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_3:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a3:Univ.
+∀b3:Univ.
+∀c3:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (inverse (multiply (inverse (multiply A (inverse (multiply (inverse B) (multiply (inverse C) (inverse (multiply (inverse C) C))))))) (multiply A C))) B.eq Univ (multiply (multiply a3 b3) c3) (multiply a3 (multiply b3 c3))
+.
+#Univ.
+#A.
+#B.
+#C.
+#a3.
+#b3.
+#c3.
+#inverse.
+#multiply.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP424-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP424-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Axiom for group theory, in product & inverse, part 1 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Kun92] Kunen (1992), Single Axioms for Groups *)
+
+(* : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 3 ( 0 singleton) *)
+
+(* Maximal term depth : 9 ( 4 average) *)
+
+(* Comments : A UEQ part of GRP056-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_1:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a1:Univ.
+∀b1:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (multiply (inverse (multiply (inverse (multiply A (inverse (multiply (inverse B) C)))) (multiply A (inverse C)))) (inverse (multiply (inverse C) C))) B.eq Univ (multiply (inverse a1) a1) (multiply (inverse b1) b1)
+.
+#Univ.
+#A.
+#B.
+#C.
+#a1.
+#b1.
+#inverse.
+#multiply.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP425-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP425-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Axiom for group theory, in product & inverse, part 2 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Kun92] Kunen (1992), Single Axioms for Groups *)
+
+(* : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.14 v3.2.0, 0.07 v3.1.0, 0.11 v2.7.0, 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 3 ( 0 singleton) *)
+
+(* Maximal term depth : 9 ( 4 average) *)
+
+(* Comments : A UEQ part of GRP056-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_2:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a2:Univ.
+∀b2:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (multiply (inverse (multiply (inverse (multiply A (inverse (multiply (inverse B) C)))) (multiply A (inverse C)))) (inverse (multiply (inverse C) C))) B.eq Univ (multiply (multiply (inverse b2) b2) a2) a2
+.
+#Univ.
+#A.
+#B.
+#C.
+#a2.
+#b2.
+#inverse.
+#multiply.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP426-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP426-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Axiom for group theory, in product & inverse, part 3 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Kun92] Kunen (1992), Single Axioms for Groups *)
+
+(* : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.14 v3.1.0, 0.11 v2.7.0, 0.18 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 3 ( 0 singleton) *)
+
+(* Maximal term depth : 9 ( 4 average) *)
+
+(* Comments : A UEQ part of GRP056-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_3:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a3:Univ.
+∀b3:Univ.
+∀c3:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (multiply (inverse (multiply (inverse (multiply A (inverse (multiply (inverse B) C)))) (multiply A (inverse C)))) (inverse (multiply (inverse C) C))) B.eq Univ (multiply (multiply a3 b3) c3) (multiply a3 (multiply b3 c3))
+.
+#Univ.
+#A.
+#B.
+#C.
+#a3.
+#b3.
+#c3.
+#inverse.
+#multiply.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP427-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP427-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Axiom for group theory, in product & inverse, part 1 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Neu81] Neumann (1981), Another Single Law for Groups *)
+
+(* : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 4 ( 0 singleton) *)
+
+(* Maximal term depth : 9 ( 4 average) *)
+
+(* Comments : A UEQ part of GRP057-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_1:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.
+∀a1:Univ.
+∀b1:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.eq Univ (multiply A (inverse (multiply (multiply (inverse (multiply (inverse B) (multiply (inverse A) C))) D) (inverse (multiply B D))))) C.eq Univ (multiply (inverse a1) a1) (multiply (inverse b1) b1)
+.
+#Univ.
+#A.
+#B.
+#C.
+#D.
+#a1.
+#b1.
+#inverse.
+#multiply.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP428-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP428-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Axiom for group theory, in product & inverse, part 2 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Neu81] Neumann (1981), Another Single Law for Groups *)
+
+(* : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 4 ( 0 singleton) *)
+
+(* Maximal term depth : 9 ( 4 average) *)
+
+(* Comments : A UEQ part of GRP057-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_2:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.
+∀a2:Univ.
+∀b2:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.eq Univ (multiply A (inverse (multiply (multiply (inverse (multiply (inverse B) (multiply (inverse A) C))) D) (inverse (multiply B D))))) C.eq Univ (multiply (multiply (inverse b2) b2) a2) a2
+.
+#Univ.
+#A.
+#B.
+#C.
+#D.
+#a2.
+#b2.
+#inverse.
+#multiply.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP429-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP429-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Axiom for group theory, in product & inverse, part 3 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Neu81] Neumann (1981), Another Single Law for Groups *)
+
+(* : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.07 v3.2.0, 0.14 v3.1.0, 0.11 v2.7.0, 0.18 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 4 ( 0 singleton) *)
+
+(* Maximal term depth : 9 ( 4 average) *)
+
+(* Comments : A UEQ part of GRP057-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_3:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.
+∀a3:Univ.
+∀b3:Univ.
+∀c3:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.eq Univ (multiply A (inverse (multiply (multiply (inverse (multiply (inverse B) (multiply (inverse A) C))) D) (inverse (multiply B D))))) C.eq Univ (multiply (multiply a3 b3) c3) (multiply a3 (multiply b3 c3))
+.
+#Univ.
+#A.
+#B.
+#C.
+#D.
+#a3.
+#b3.
+#c3.
+#inverse.
+#multiply.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP430-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP430-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Axiom for group theory, in product & inverse, part 1 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [LW92] Lusk & Wos (1992), Benchmark Problems in Which Equalit *)
+
+(* : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 4 ( 0 singleton) *)
+
+(* Maximal term depth : 8 ( 4 average) *)
+
+(* Comments : A UEQ part of GRP058-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_1:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.
+∀a1:Univ.
+∀b1:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.eq Univ (multiply A (inverse (multiply B (multiply (multiply (multiply C (inverse C)) (inverse (multiply D B))) A)))) D.eq Univ (multiply (inverse a1) a1) (multiply (inverse b1) b1)
+.
+#Univ.
+#A.
+#B.
+#C.
+#D.
+#a1.
+#b1.
+#inverse.
+#multiply.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP431-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP431-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Axiom for group theory, in product & inverse, part 2 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [LW92] Lusk & Wos (1992), Benchmark Problems in Which Equalit *)
+
+(* : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 4 ( 0 singleton) *)
+
+(* Maximal term depth : 8 ( 4 average) *)
+
+(* Comments : A UEQ part of GRP058-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_2:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.
+∀a2:Univ.
+∀b2:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.eq Univ (multiply A (inverse (multiply B (multiply (multiply (multiply C (inverse C)) (inverse (multiply D B))) A)))) D.eq Univ (multiply (multiply (inverse b2) b2) a2) a2
+.
+#Univ.
+#A.
+#B.
+#C.
+#D.
+#a2.
+#b2.
+#inverse.
+#multiply.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP432-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP432-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Axiom for group theory, in product & inverse, part 3 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [LW92] Lusk & Wos (1992), Benchmark Problems in Which Equalit *)
+
+(* : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 4 ( 0 singleton) *)
+
+(* Maximal term depth : 8 ( 4 average) *)
+
+(* Comments : A UEQ part of GRP058-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_3:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.
+∀a3:Univ.
+∀b3:Univ.
+∀c3:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.eq Univ (multiply A (inverse (multiply B (multiply (multiply (multiply C (inverse C)) (inverse (multiply D B))) A)))) D.eq Univ (multiply (multiply a3 b3) c3) (multiply a3 (multiply b3 c3))
+.
+#Univ.
+#A.
+#B.
+#C.
+#D.
+#a3.
+#b3.
+#c3.
+#inverse.
+#multiply.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP433-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP433-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Axiom for group theory, in product & inverse, part 1 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 4 ( 0 singleton) *)
+
+(* Maximal term depth : 8 ( 4 average) *)
+
+(* Comments : A UEQ part of GRP059-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_1:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.
+∀a1:Univ.
+∀b1:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.eq Univ (inverse (multiply (multiply (multiply (inverse (multiply (multiply A B) C)) A) B) (multiply D (inverse D)))) C.eq Univ (multiply (inverse a1) a1) (multiply (inverse b1) b1)
+.
+#Univ.
+#A.
+#B.
+#C.
+#D.
+#a1.
+#b1.
+#inverse.
+#multiply.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP434-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP434-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Axiom for group theory, in product & inverse, part 2 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 4 ( 0 singleton) *)
+
+(* Maximal term depth : 8 ( 4 average) *)
+
+(* Comments : A UEQ part of GRP059-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_2:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.
+∀a2:Univ.
+∀b2:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.eq Univ (inverse (multiply (multiply (multiply (inverse (multiply (multiply A B) C)) A) B) (multiply D (inverse D)))) C.eq Univ (multiply (multiply (inverse b2) b2) a2) a2
+.
+#Univ.
+#A.
+#B.
+#C.
+#D.
+#a2.
+#b2.
+#inverse.
+#multiply.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP435-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP435-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Axiom for group theory, in product & inverse, part 3 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.07 v3.2.0, 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 4 ( 0 singleton) *)
+
+(* Maximal term depth : 8 ( 4 average) *)
+
+(* Comments : A UEQ part of GRP059-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_3:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.
+∀a3:Univ.
+∀b3:Univ.
+∀c3:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.eq Univ (inverse (multiply (multiply (multiply (inverse (multiply (multiply A B) C)) A) B) (multiply D (inverse D)))) C.eq Univ (multiply (multiply a3 b3) c3) (multiply a3 (multiply b3 c3))
+.
+#Univ.
+#A.
+#B.
+#C.
+#D.
+#a3.
+#b3.
+#c3.
+#inverse.
+#multiply.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP436-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP436-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Axiom for group theory, in product & inverse, part 1 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.07 v3.1.0, 0.11 v2.7.0, 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 4 ( 0 singleton) *)
+
+(* Maximal term depth : 9 ( 4 average) *)
+
+(* Comments : A UEQ part of GRP060-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_1:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.
+∀a1:Univ.
+∀b1:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.eq Univ (multiply A (inverse (multiply B (multiply C (multiply (multiply (inverse C) (inverse (multiply D B))) A))))) D.eq Univ (multiply (inverse a1) a1) (multiply (inverse b1) b1)
+.
+#Univ.
+#A.
+#B.
+#C.
+#D.
+#a1.
+#b1.
+#inverse.
+#multiply.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP437-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP437-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Axiom for group theory, in product & inverse, part 2 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.07 v3.1.0, 0.11 v2.7.0, 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 4 ( 0 singleton) *)
+
+(* Maximal term depth : 9 ( 4 average) *)
+
+(* Comments : A UEQ part of GRP060-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_2:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.
+∀a2:Univ.
+∀b2:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.eq Univ (multiply A (inverse (multiply B (multiply C (multiply (multiply (inverse C) (inverse (multiply D B))) A))))) D.eq Univ (multiply (multiply (inverse b2) b2) a2) a2
+.
+#Univ.
+#A.
+#B.
+#C.
+#D.
+#a2.
+#b2.
+#inverse.
+#multiply.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP438-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP438-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Axiom for group theory, in product & inverse, part 3 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.07 v3.1.0, 0.11 v2.7.0, 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 4 ( 0 singleton) *)
+
+(* Maximal term depth : 9 ( 4 average) *)
+
+(* Comments : A UEQ part of GRP060-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_3:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.
+∀a3:Univ.
+∀b3:Univ.
+∀c3:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.eq Univ (multiply A (inverse (multiply B (multiply C (multiply (multiply (inverse C) (inverse (multiply D B))) A))))) D.eq Univ (multiply (multiply a3 b3) c3) (multiply a3 (multiply b3 c3))
+.
+#Univ.
+#A.
+#B.
+#C.
+#D.
+#a3.
+#b3.
+#c3.
+#inverse.
+#multiply.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP439-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP439-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Axiom for group theory, in product & inverse, part 1 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.14 v3.2.0, 0.07 v3.1.0, 0.00 v2.7.0, 0.09 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 4 ( 0 singleton) *)
+
+(* Maximal term depth : 8 ( 4 average) *)
+
+(* Comments : A UEQ part of GRP061-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_1:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.
+∀a1:Univ.
+∀b1:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.eq Univ (inverse (multiply A (multiply B (multiply (multiply (inverse B) C) (inverse (multiply D (multiply A C))))))) D.eq Univ (multiply (inverse a1) a1) (multiply (inverse b1) b1)
+.
+#Univ.
+#A.
+#B.
+#C.
+#D.
+#a1.
+#b1.
+#inverse.
+#multiply.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP440-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP440-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Axiom for group theory, in product & inverse, part 2 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.21 v3.2.0, 0.14 v3.1.0, 0.11 v2.7.0, 0.09 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 4 ( 0 singleton) *)
+
+(* Maximal term depth : 8 ( 4 average) *)
+
+(* Comments : A UEQ part of GRP061-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_2:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.
+∀a2:Univ.
+∀b2:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.eq Univ (inverse (multiply A (multiply B (multiply (multiply (inverse B) C) (inverse (multiply D (multiply A C))))))) D.eq Univ (multiply (multiply (inverse b2) b2) a2) a2
+.
+#Univ.
+#A.
+#B.
+#C.
+#D.
+#a2.
+#b2.
+#inverse.
+#multiply.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP441-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP441-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Axiom for group theory, in product & inverse, part 3 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.21 v3.1.0, 0.22 v2.7.0, 0.18 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 4 ( 0 singleton) *)
+
+(* Maximal term depth : 8 ( 4 average) *)
+
+(* Comments : A UEQ part of GRP061-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_3:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.
+∀a3:Univ.
+∀b3:Univ.
+∀c3:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.eq Univ (inverse (multiply A (multiply B (multiply (multiply (inverse B) C) (inverse (multiply D (multiply A C))))))) D.eq Univ (multiply (multiply a3 b3) c3) (multiply a3 (multiply b3 c3))
+.
+#Univ.
+#A.
+#B.
+#C.
+#D.
+#a3.
+#b3.
+#c3.
+#inverse.
+#multiply.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP442-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP442-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Axiom for group theory, in product & inverse, part 1 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.14 v3.1.0, 0.00 v2.7.0, 0.18 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 4 ( 0 singleton) *)
+
+(* Maximal term depth : 8 ( 4 average) *)
+
+(* Comments : A UEQ part of GRP062-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_1:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.
+∀a1:Univ.
+∀b1:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.eq Univ (inverse (multiply A (multiply B (multiply (multiply C (inverse C)) (inverse (multiply D (multiply A B))))))) D.eq Univ (multiply (inverse a1) a1) (multiply (inverse b1) b1)
+.
+#Univ.
+#A.
+#B.
+#C.
+#D.
+#a1.
+#b1.
+#inverse.
+#multiply.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP443-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP443-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Axiom for group theory, in product & inverse, part 2 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.07 v3.1.0, 0.00 v2.7.0, 0.09 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 4 ( 0 singleton) *)
+
+(* Maximal term depth : 8 ( 4 average) *)
+
+(* Comments : A UEQ part of GRP062-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_2:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.
+∀a2:Univ.
+∀b2:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.eq Univ (inverse (multiply A (multiply B (multiply (multiply C (inverse C)) (inverse (multiply D (multiply A B))))))) D.eq Univ (multiply (multiply (inverse b2) b2) a2) a2
+.
+#Univ.
+#A.
+#B.
+#C.
+#D.
+#a2.
+#b2.
+#inverse.
+#multiply.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP444-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP444-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Axiom for group theory, in product & inverse, part 3 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.21 v3.1.0, 0.22 v2.7.0, 0.27 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 4 ( 0 singleton) *)
+
+(* Maximal term depth : 8 ( 4 average) *)
+
+(* Comments : A UEQ part of GRP062-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_3:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.
+∀a3:Univ.
+∀b3:Univ.
+∀c3:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.eq Univ (inverse (multiply A (multiply B (multiply (multiply C (inverse C)) (inverse (multiply D (multiply A B))))))) D.eq Univ (multiply (multiply a3 b3) c3) (multiply a3 (multiply b3 c3))
+.
+#Univ.
+#A.
+#B.
+#C.
+#D.
+#a3.
+#b3.
+#c3.
+#inverse.
+#multiply.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP445-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP445-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Axiom for group theory, in division, part 1 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [HN52] Higman & Neumann (1952), Groups as Groupoids with One *)
+
+(* : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 4 ( 0 non-Horn; 4 unit; 1 RR) *)
+
+(* Number of atoms : 4 ( 4 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 8 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP063-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_1:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a1:Univ.
+∀b1:Univ.
+∀divide:∀_:Univ.∀_:Univ.Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.eq Univ (inverse A) (divide (divide B B) A).
+∀H1:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (multiply A B) (divide A (divide (divide C C) B)).
+∀H2:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (divide A (divide (divide (divide (divide A A) B) C) (divide (divide (divide A A) A) C))) B.eq Univ (multiply (inverse a1) a1) (multiply (inverse b1) b1)
+.
+#Univ.
+#A.
+#B.
+#C.
+#a1.
+#b1.
+#divide.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+nauto by H0,H1,H2;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP446-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP446-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Axiom for group theory, in division, part 2 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [HN52] Higman & Neumann (1952), Groups as Groupoids with One *)
+
+(* : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 4 ( 0 non-Horn; 4 unit; 1 RR) *)
+
+(* Number of atoms : 4 ( 4 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 8 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP063-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_2:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a2:Univ.
+∀b2:Univ.
+∀divide:∀_:Univ.∀_:Univ.Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.eq Univ (inverse A) (divide (divide B B) A).
+∀H1:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (multiply A B) (divide A (divide (divide C C) B)).
+∀H2:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (divide A (divide (divide (divide (divide A A) B) C) (divide (divide (divide A A) A) C))) B.eq Univ (multiply (multiply (inverse b2) b2) a2) a2
+.
+#Univ.
+#A.
+#B.
+#C.
+#a2.
+#b2.
+#divide.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+nauto by H0,H1,H2;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP447-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP447-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Axiom for group theory, in division, part 3 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [HN52] Higman & Neumann (1952), Groups as Groupoids with One *)
+
+(* : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.14 v3.1.0, 0.00 v2.7.0, 0.09 v2.6.0 *)
+
+(* Syntax : Number of clauses : 4 ( 0 non-Horn; 4 unit; 1 RR) *)
+
+(* Number of atoms : 4 ( 4 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 8 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP063-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_3:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a3:Univ.
+∀b3:Univ.
+∀c3:Univ.
+∀divide:∀_:Univ.∀_:Univ.Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.eq Univ (inverse A) (divide (divide B B) A).
+∀H1:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (multiply A B) (divide A (divide (divide C C) B)).
+∀H2:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (divide A (divide (divide (divide (divide A A) B) C) (divide (divide (divide A A) A) C))) B.eq Univ (multiply (multiply a3 b3) c3) (multiply a3 (multiply b3 c3))
+.
+#Univ.
+#A.
+#B.
+#C.
+#a3.
+#b3.
+#c3.
+#divide.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+nauto by H0,H1,H2;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP448-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP448-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Axiom for group theory, in division, part 1 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 4 ( 0 non-Horn; 4 unit; 1 RR) *)
+
+(* Number of atoms : 4 ( 4 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 8 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP064-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_1:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a1:Univ.
+∀b1:Univ.
+∀divide:∀_:Univ.∀_:Univ.Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.eq Univ (inverse A) (divide (divide B B) A).
+∀H1:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (multiply A B) (divide A (divide (divide C C) B)).
+∀H2:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (divide A (divide (divide (divide (divide B B) B) C) (divide (divide (divide B B) A) C))) B.eq Univ (multiply (inverse a1) a1) (multiply (inverse b1) b1)
+.
+#Univ.
+#A.
+#B.
+#C.
+#a1.
+#b1.
+#divide.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+nauto by H0,H1,H2;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP449-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP449-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Axiom for group theory, in division, part 2 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 4 ( 0 non-Horn; 4 unit; 1 RR) *)
+
+(* Number of atoms : 4 ( 4 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 8 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP064-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_2:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a2:Univ.
+∀b2:Univ.
+∀divide:∀_:Univ.∀_:Univ.Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.eq Univ (inverse A) (divide (divide B B) A).
+∀H1:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (multiply A B) (divide A (divide (divide C C) B)).
+∀H2:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (divide A (divide (divide (divide (divide B B) B) C) (divide (divide (divide B B) A) C))) B.eq Univ (multiply (multiply (inverse b2) b2) a2) a2
+.
+#Univ.
+#A.
+#B.
+#C.
+#a2.
+#b2.
+#divide.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+nauto by H0,H1,H2;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP450-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP450-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Axiom for group theory, in division, part 3 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.07 v3.1.0, 0.11 v2.7.0, 0.09 v2.6.0 *)
+
+(* Syntax : Number of clauses : 4 ( 0 non-Horn; 4 unit; 1 RR) *)
+
+(* Number of atoms : 4 ( 4 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 8 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP064-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_3:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a3:Univ.
+∀b3:Univ.
+∀c3:Univ.
+∀divide:∀_:Univ.∀_:Univ.Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.eq Univ (inverse A) (divide (divide B B) A).
+∀H1:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (multiply A B) (divide A (divide (divide C C) B)).
+∀H2:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (divide A (divide (divide (divide (divide B B) B) C) (divide (divide (divide B B) A) C))) B.eq Univ (multiply (multiply a3 b3) c3) (multiply a3 (multiply b3 c3))
+.
+#Univ.
+#A.
+#B.
+#C.
+#a3.
+#b3.
+#c3.
+#divide.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+nauto by H0,H1,H2;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP451-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP451-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Axiom for group theory, in division, part 1 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 4 ( 0 non-Horn; 4 unit; 1 RR) *)
+
+(* Number of atoms : 4 ( 4 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 8 ( 0 singleton) *)
+
+(* Maximal term depth : 8 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP065-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_1:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a1:Univ.
+∀b1:Univ.
+∀divide:∀_:Univ.∀_:Univ.Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.eq Univ (inverse A) (divide (divide B B) A).
+∀H1:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (multiply A B) (divide A (divide (divide C C) B)).
+∀H2:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (divide (divide (divide A A) (divide A (divide B (divide (divide (divide A A) A) C)))) C) B.eq Univ (multiply (inverse a1) a1) (multiply (inverse b1) b1)
+.
+#Univ.
+#A.
+#B.
+#C.
+#a1.
+#b1.
+#divide.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+nauto by H0,H1,H2;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP452-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP452-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Axiom for group theory, in division, part 2 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.21 v3.2.0, 0.14 v3.1.0, 0.11 v2.7.0, 0.18 v2.6.0 *)
+
+(* Syntax : Number of clauses : 4 ( 0 non-Horn; 4 unit; 1 RR) *)
+
+(* Number of atoms : 4 ( 4 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 8 ( 0 singleton) *)
+
+(* Maximal term depth : 8 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP065-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_2:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a2:Univ.
+∀b2:Univ.
+∀divide:∀_:Univ.∀_:Univ.Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.eq Univ (inverse A) (divide (divide B B) A).
+∀H1:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (multiply A B) (divide A (divide (divide C C) B)).
+∀H2:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (divide (divide (divide A A) (divide A (divide B (divide (divide (divide A A) A) C)))) C) B.eq Univ (multiply (multiply (inverse b2) b2) a2) a2
+.
+#Univ.
+#A.
+#B.
+#C.
+#a2.
+#b2.
+#divide.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+nauto by H0,H1,H2;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP453-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP453-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Axiom for group theory, in division, part 3 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.21 v3.1.0, 0.11 v2.7.0, 0.27 v2.6.0 *)
+
+(* Syntax : Number of clauses : 4 ( 0 non-Horn; 4 unit; 1 RR) *)
+
+(* Number of atoms : 4 ( 4 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 8 ( 0 singleton) *)
+
+(* Maximal term depth : 8 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP065-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_3:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a3:Univ.
+∀b3:Univ.
+∀c3:Univ.
+∀divide:∀_:Univ.∀_:Univ.Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.eq Univ (inverse A) (divide (divide B B) A).
+∀H1:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (multiply A B) (divide A (divide (divide C C) B)).
+∀H2:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (divide (divide (divide A A) (divide A (divide B (divide (divide (divide A A) A) C)))) C) B.eq Univ (multiply (multiply a3 b3) c3) (multiply a3 (multiply b3 c3))
+.
+#Univ.
+#A.
+#B.
+#C.
+#a3.
+#b3.
+#c3.
+#divide.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+nauto by H0,H1,H2;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP454-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP454-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Axiom for group theory, in division and identity, part 1 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 1 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 8 ( 2 average) *)
+
+(* Comments : A UEQ part of GRP066-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_1:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a1:Univ.
+∀divide:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.eq Univ identity (divide A A).
+∀H1:∀A:Univ.eq Univ (inverse A) (divide identity A).
+∀H2:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (divide A (divide identity B)).
+∀H3:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (divide (divide identity (divide A (divide B (divide (divide (divide A A) A) C)))) C) B.eq Univ (multiply (inverse a1) a1) identity
+.
+#Univ.
+#A.
+#B.
+#C.
+#a1.
+#divide.
+#identity.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP455-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP455-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Axiom for group theory, in division and identity, part 2 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 1 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 8 ( 2 average) *)
+
+(* Comments : A UEQ part of GRP066-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_2:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a2:Univ.
+∀divide:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.eq Univ identity (divide A A).
+∀H1:∀A:Univ.eq Univ (inverse A) (divide identity A).
+∀H2:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (divide A (divide identity B)).
+∀H3:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (divide (divide identity (divide A (divide B (divide (divide (divide A A) A) C)))) C) B.eq Univ (multiply identity a2) a2
+.
+#Univ.
+#A.
+#B.
+#C.
+#a2.
+#divide.
+#identity.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP456-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP456-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Axiom for group theory, in division and identity, part 3 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.07 v3.1.0, 0.11 v2.7.0, 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 1 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 8 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP066-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_3:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a3:Univ.
+∀b3:Univ.
+∀c3:Univ.
+∀divide:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.eq Univ identity (divide A A).
+∀H1:∀A:Univ.eq Univ (inverse A) (divide identity A).
+∀H2:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (divide A (divide identity B)).
+∀H3:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (divide (divide identity (divide A (divide B (divide (divide (divide A A) A) C)))) C) B.eq Univ (multiply (multiply a3 b3) c3) (multiply a3 (multiply b3 c3))
+.
+#Univ.
+#A.
+#B.
+#C.
+#a3.
+#b3.
+#c3.
+#divide.
+#identity.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP457-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP457-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Axiom for group theory, in division and identity, part 1 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 1 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 7 ( 2 average) *)
+
+(* Comments : A UEQ part of GRP067-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_1:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a1:Univ.
+∀divide:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.eq Univ identity (divide A A).
+∀H1:∀A:Univ.eq Univ (inverse A) (divide identity A).
+∀H2:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (divide A (divide identity B)).
+∀H3:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (divide (divide (divide A A) (divide A (divide B (divide (divide identity A) C)))) C) B.eq Univ (multiply (inverse a1) a1) identity
+.
+#Univ.
+#A.
+#B.
+#C.
+#a1.
+#divide.
+#identity.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP458-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP458-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Axiom for group theory, in division and identity, part 2 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 1 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 7 ( 2 average) *)
+
+(* Comments : A UEQ part of GRP067-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_2:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a2:Univ.
+∀divide:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.eq Univ identity (divide A A).
+∀H1:∀A:Univ.eq Univ (inverse A) (divide identity A).
+∀H2:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (divide A (divide identity B)).
+∀H3:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (divide (divide (divide A A) (divide A (divide B (divide (divide identity A) C)))) C) B.eq Univ (multiply identity a2) a2
+.
+#Univ.
+#A.
+#B.
+#C.
+#a2.
+#divide.
+#identity.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP459-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP459-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Axiom for group theory, in division and identity, part 3 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 1 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 7 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP067-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_3:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a3:Univ.
+∀b3:Univ.
+∀c3:Univ.
+∀divide:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.eq Univ identity (divide A A).
+∀H1:∀A:Univ.eq Univ (inverse A) (divide identity A).
+∀H2:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (divide A (divide identity B)).
+∀H3:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (divide (divide (divide A A) (divide A (divide B (divide (divide identity A) C)))) C) B.eq Univ (multiply (multiply a3 b3) c3) (multiply a3 (multiply b3 c3))
+.
+#Univ.
+#A.
+#B.
+#C.
+#a3.
+#b3.
+#c3.
+#divide.
+#identity.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP460-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP460-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Axiom for group theory, in division and identity, part 1 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 1 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 2 average) *)
+
+(* Comments : A UEQ part of GRP068-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_1:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a1:Univ.
+∀divide:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.eq Univ identity (divide A A).
+∀H1:∀A:Univ.eq Univ (inverse A) (divide identity A).
+∀H2:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (divide A (divide identity B)).
+∀H3:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (divide A (divide (divide (divide identity B) C) (divide (divide (divide A A) A) C))) B.eq Univ (multiply (inverse a1) a1) identity
+.
+#Univ.
+#A.
+#B.
+#C.
+#a1.
+#divide.
+#identity.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP461-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP461-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Axiom for group theory, in division and identity, part 2 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 1 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 2 average) *)
+
+(* Comments : A UEQ part of GRP068-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_2:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a2:Univ.
+∀divide:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.eq Univ identity (divide A A).
+∀H1:∀A:Univ.eq Univ (inverse A) (divide identity A).
+∀H2:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (divide A (divide identity B)).
+∀H3:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (divide A (divide (divide (divide identity B) C) (divide (divide (divide A A) A) C))) B.eq Univ (multiply identity a2) a2
+.
+#Univ.
+#A.
+#B.
+#C.
+#a2.
+#divide.
+#identity.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP462-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP462-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Axiom for group theory, in division and identity, part 3 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 1 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 2 average) *)
+
+(* Comments : A UEQ part of GRP068-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_3:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a3:Univ.
+∀b3:Univ.
+∀c3:Univ.
+∀divide:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.eq Univ identity (divide A A).
+∀H1:∀A:Univ.eq Univ (inverse A) (divide identity A).
+∀H2:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (divide A (divide identity B)).
+∀H3:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (divide A (divide (divide (divide identity B) C) (divide (divide (divide A A) A) C))) B.eq Univ (multiply (multiply a3 b3) c3) (multiply a3 (multiply b3 c3))
+.
+#Univ.
+#A.
+#B.
+#C.
+#a3.
+#b3.
+#c3.
+#divide.
+#identity.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP463-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP463-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Axiom for group theory, in division and identity, part 1 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 1 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 2 average) *)
+
+(* Comments : A UEQ part of GRP069-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_1:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a1:Univ.
+∀divide:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.eq Univ identity (divide A A).
+∀H1:∀A:Univ.eq Univ (inverse A) (divide identity A).
+∀H2:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (divide A (divide identity B)).
+∀H3:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (divide A (divide (divide (divide (divide A A) B) C) (divide (divide identity A) C))) B.eq Univ (multiply (inverse a1) a1) identity
+.
+#Univ.
+#A.
+#B.
+#C.
+#a1.
+#divide.
+#identity.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP464-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP464-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Axiom for group theory, in division and identity, part 2 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 1 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 2 average) *)
+
+(* Comments : A UEQ part of GRP069-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_2:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a2:Univ.
+∀divide:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.eq Univ identity (divide A A).
+∀H1:∀A:Univ.eq Univ (inverse A) (divide identity A).
+∀H2:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (divide A (divide identity B)).
+∀H3:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (divide A (divide (divide (divide (divide A A) B) C) (divide (divide identity A) C))) B.eq Univ (multiply identity a2) a2
+.
+#Univ.
+#A.
+#B.
+#C.
+#a2.
+#divide.
+#identity.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP465-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP465-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Axiom for group theory, in division and identity, part 3 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 1 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 2 average) *)
+
+(* Comments : A UEQ part of GRP069-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_3:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a3:Univ.
+∀b3:Univ.
+∀c3:Univ.
+∀divide:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.eq Univ identity (divide A A).
+∀H1:∀A:Univ.eq Univ (inverse A) (divide identity A).
+∀H2:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (divide A (divide identity B)).
+∀H3:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (divide A (divide (divide (divide (divide A A) B) C) (divide (divide identity A) C))) B.eq Univ (multiply (multiply a3 b3) c3) (multiply a3 (multiply b3 c3))
+.
+#Univ.
+#A.
+#B.
+#C.
+#a3.
+#b3.
+#c3.
+#divide.
+#identity.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP466-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP466-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Axiom for group theory, in division and inverse, part 1 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 6 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP070-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_1:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.
+∀a1:Univ.
+∀b1:Univ.
+∀divide:∀_:Univ.∀_:Univ.Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (divide A (inverse B)).
+∀H1:∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.eq Univ (divide (divide A A) (divide B (divide (divide C (divide D B)) (inverse D)))) C.eq Univ (multiply (inverse a1) a1) (multiply (inverse b1) b1)
+.
+#Univ.
+#A.
+#B.
+#C.
+#D.
+#a1.
+#b1.
+#divide.
+#inverse.
+#multiply.
+#H0.
+#H1.
+nauto by H0,H1;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP467-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP467-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Axiom for group theory, in division and inverse, part 2 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 6 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP070-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_2:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.
+∀a2:Univ.
+∀b2:Univ.
+∀divide:∀_:Univ.∀_:Univ.Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (divide A (inverse B)).
+∀H1:∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.eq Univ (divide (divide A A) (divide B (divide (divide C (divide D B)) (inverse D)))) C.eq Univ (multiply (multiply (inverse b2) b2) a2) a2
+.
+#Univ.
+#A.
+#B.
+#C.
+#D.
+#a2.
+#b2.
+#divide.
+#inverse.
+#multiply.
+#H0.
+#H1.
+nauto by H0,H1;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP468-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP468-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Axiom for group theory, in division and inverse, part 3 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.07 v3.2.0, 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 6 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP070-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_3:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.
+∀a3:Univ.
+∀b3:Univ.
+∀c3:Univ.
+∀divide:∀_:Univ.∀_:Univ.Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (divide A (inverse B)).
+∀H1:∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.eq Univ (divide (divide A A) (divide B (divide (divide C (divide D B)) (inverse D)))) C.eq Univ (multiply (multiply a3 b3) c3) (multiply a3 (multiply b3 c3))
+.
+#Univ.
+#A.
+#B.
+#C.
+#D.
+#a3.
+#b3.
+#c3.
+#divide.
+#inverse.
+#multiply.
+#H0.
+#H1.
+nauto by H0,H1;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP469-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP469-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Axiom for group theory, in division and inverse, part 1 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.29 v3.2.0, 0.21 v3.1.0, 0.11 v2.7.0, 0.18 v2.6.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 6 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP071-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_1:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.
+∀a1:Univ.
+∀b1:Univ.
+∀divide:∀_:Univ.∀_:Univ.Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (divide A (inverse B)).
+∀H1:∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.eq Univ (divide (inverse (divide A (divide B (divide C D)))) (divide (divide D C) A)) B.eq Univ (multiply (inverse a1) a1) (multiply (inverse b1) b1)
+.
+#Univ.
+#A.
+#B.
+#C.
+#D.
+#a1.
+#b1.
+#divide.
+#inverse.
+#multiply.
+#H0.
+#H1.
+nauto by H0,H1;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP470-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP470-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Axiom for group theory, in division and inverse, part 2 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.29 v3.2.0, 0.21 v3.1.0, 0.11 v2.7.0, 0.18 v2.6.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 6 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP071-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_2:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.
+∀a2:Univ.
+∀b2:Univ.
+∀divide:∀_:Univ.∀_:Univ.Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (divide A (inverse B)).
+∀H1:∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.eq Univ (divide (inverse (divide A (divide B (divide C D)))) (divide (divide D C) A)) B.eq Univ (multiply (multiply (inverse b2) b2) a2) a2
+.
+#Univ.
+#A.
+#B.
+#C.
+#D.
+#a2.
+#b2.
+#divide.
+#inverse.
+#multiply.
+#H0.
+#H1.
+nauto by H0,H1;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP471-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP471-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Axiom for group theory, in division and inverse, part 3 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.29 v3.2.0, 0.21 v3.1.0, 0.11 v2.7.0, 0.18 v2.6.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 6 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP071-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_3:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.
+∀a3:Univ.
+∀b3:Univ.
+∀c3:Univ.
+∀divide:∀_:Univ.∀_:Univ.Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (divide A (inverse B)).
+∀H1:∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.eq Univ (divide (inverse (divide A (divide B (divide C D)))) (divide (divide D C) A)) B.eq Univ (multiply (multiply a3 b3) c3) (multiply a3 (multiply b3 c3))
+.
+#Univ.
+#A.
+#B.
+#C.
+#D.
+#a3.
+#b3.
+#c3.
+#divide.
+#inverse.
+#multiply.
+#H0.
+#H1.
+nauto by H0,H1;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP472-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP472-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Axiom for group theory, in division and inverse, part 1 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.07 v3.1.0, 0.11 v2.7.0, 0.18 v2.6.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 6 ( 0 singleton) *)
+
+(* Maximal term depth : 5 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP072-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_1:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.
+∀a1:Univ.
+∀b1:Univ.
+∀divide:∀_:Univ.∀_:Univ.Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (divide A (inverse B)).
+∀H1:∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.eq Univ (divide (divide (inverse (divide A B)) (divide (divide C D) A)) (divide D C)) B.eq Univ (multiply (inverse a1) a1) (multiply (inverse b1) b1)
+.
+#Univ.
+#A.
+#B.
+#C.
+#D.
+#a1.
+#b1.
+#divide.
+#inverse.
+#multiply.
+#H0.
+#H1.
+nauto by H0,H1;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP473-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP473-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Axiom for group theory, in division and inverse, part 2 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.07 v3.1.0, 0.11 v2.7.0, 0.18 v2.6.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 6 ( 0 singleton) *)
+
+(* Maximal term depth : 5 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP072-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_2:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.
+∀a2:Univ.
+∀b2:Univ.
+∀divide:∀_:Univ.∀_:Univ.Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (divide A (inverse B)).
+∀H1:∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.eq Univ (divide (divide (inverse (divide A B)) (divide (divide C D) A)) (divide D C)) B.eq Univ (multiply (multiply (inverse b2) b2) a2) a2
+.
+#Univ.
+#A.
+#B.
+#C.
+#D.
+#a2.
+#b2.
+#divide.
+#inverse.
+#multiply.
+#H0.
+#H1.
+nauto by H0,H1;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP474-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP474-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Axiom for group theory, in division and inverse, part 3 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.14 v3.2.0, 0.07 v3.1.0, 0.11 v2.7.0, 0.18 v2.6.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 6 ( 0 singleton) *)
+
+(* Maximal term depth : 5 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP072-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_3:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.
+∀a3:Univ.
+∀b3:Univ.
+∀c3:Univ.
+∀divide:∀_:Univ.∀_:Univ.Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (divide A (inverse B)).
+∀H1:∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.eq Univ (divide (divide (inverse (divide A B)) (divide (divide C D) A)) (divide D C)) B.eq Univ (multiply (multiply a3 b3) c3) (multiply a3 (multiply b3 c3))
+.
+#Univ.
+#A.
+#B.
+#C.
+#D.
+#a3.
+#b3.
+#c3.
+#divide.
+#inverse.
+#multiply.
+#H0.
+#H1.
+nauto by H0,H1;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP475-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP475-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Axiom for group theory, in division and inverse, part 1 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.21 v3.1.0, 0.00 v2.7.0, 0.27 v2.6.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 6 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP073-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_1:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.
+∀a1:Univ.
+∀b1:Univ.
+∀divide:∀_:Univ.∀_:Univ.Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (divide A (inverse B)).
+∀H1:∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.eq Univ (divide (inverse (divide (divide (divide A B) C) (divide D C))) (divide B A)) D.eq Univ (multiply (inverse a1) a1) (multiply (inverse b1) b1)
+.
+#Univ.
+#A.
+#B.
+#C.
+#D.
+#a1.
+#b1.
+#divide.
+#inverse.
+#multiply.
+#H0.
+#H1.
+nauto by H0,H1;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP476-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP476-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Axiom for group theory, in division and inverse, part 2 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.21 v3.1.0, 0.00 v2.7.0, 0.27 v2.6.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 6 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP073-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_2:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.
+∀a2:Univ.
+∀b2:Univ.
+∀divide:∀_:Univ.∀_:Univ.Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (divide A (inverse B)).
+∀H1:∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.eq Univ (divide (inverse (divide (divide (divide A B) C) (divide D C))) (divide B A)) D.eq Univ (multiply (multiply (inverse b2) b2) a2) a2
+.
+#Univ.
+#A.
+#B.
+#C.
+#D.
+#a2.
+#b2.
+#divide.
+#inverse.
+#multiply.
+#H0.
+#H1.
+nauto by H0,H1;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP477-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP477-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Axiom for group theory, in division and inverse, part 3 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.21 v3.1.0, 0.00 v2.7.0, 0.27 v2.6.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 6 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP073-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_3:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.
+∀a3:Univ.
+∀b3:Univ.
+∀c3:Univ.
+∀divide:∀_:Univ.∀_:Univ.Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (divide A (inverse B)).
+∀H1:∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.eq Univ (divide (inverse (divide (divide (divide A B) C) (divide D C))) (divide B A)) D.eq Univ (multiply (multiply a3 b3) c3) (multiply a3 (multiply b3 c3))
+.
+#Univ.
+#A.
+#B.
+#C.
+#D.
+#a3.
+#b3.
+#c3.
+#divide.
+#inverse.
+#multiply.
+#H0.
+#H1.
+nauto by H0,H1;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP478-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP478-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Axiom for group theory, in division and inverse, part 1 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.21 v3.2.0, 0.14 v3.1.0, 0.11 v2.7.0, 0.09 v2.6.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 6 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP074-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_1:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.
+∀a1:Univ.
+∀b1:Univ.
+∀divide:∀_:Univ.∀_:Univ.Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (divide A (inverse B)).
+∀H1:∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.eq Univ (divide (inverse (divide (divide (divide A A) B) (divide C (divide B D)))) D) C.eq Univ (multiply (inverse a1) a1) (multiply (inverse b1) b1)
+.
+#Univ.
+#A.
+#B.
+#C.
+#D.
+#a1.
+#b1.
+#divide.
+#inverse.
+#multiply.
+#H0.
+#H1.
+nauto by H0,H1;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP479-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP479-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Axiom for group theory, in division and inverse, part 2 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.21 v3.2.0, 0.14 v3.1.0, 0.11 v2.7.0, 0.09 v2.6.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 6 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP074-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_2:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.
+∀a2:Univ.
+∀b2:Univ.
+∀divide:∀_:Univ.∀_:Univ.Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (divide A (inverse B)).
+∀H1:∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.eq Univ (divide (inverse (divide (divide (divide A A) B) (divide C (divide B D)))) D) C.eq Univ (multiply (multiply (inverse b2) b2) a2) a2
+.
+#Univ.
+#A.
+#B.
+#C.
+#D.
+#a2.
+#b2.
+#divide.
+#inverse.
+#multiply.
+#H0.
+#H1.
+nauto by H0,H1;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP480-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP480-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Axiom for group theory, in division and inverse, part 3 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.21 v3.1.0, 0.11 v2.7.0, 0.27 v2.6.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 6 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP074-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_3:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.
+∀a3:Univ.
+∀b3:Univ.
+∀c3:Univ.
+∀divide:∀_:Univ.∀_:Univ.Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (divide A (inverse B)).
+∀H1:∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.eq Univ (divide (inverse (divide (divide (divide A A) B) (divide C (divide B D)))) D) C.eq Univ (multiply (multiply a3 b3) c3) (multiply a3 (multiply b3 c3))
+.
+#Univ.
+#A.
+#B.
+#C.
+#D.
+#a3.
+#b3.
+#c3.
+#divide.
+#inverse.
+#multiply.
+#H0.
+#H1.
+nauto by H0,H1;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP481-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP481-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Axiom for group theory, in double division and identity, part 1 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Neu86] Neumann (1986), Yet Another Single Law for Groups *)
+
+(* : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 1 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 8 ( 0 singleton) *)
+
+(* Maximal term depth : 7 ( 2 average) *)
+
+(* Comments : A UEQ part of GRP075-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_1:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.
+∀a1:Univ.
+∀double_divide:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.eq Univ identity (double_divide A (inverse A)).
+∀H1:∀A:Univ.eq Univ (inverse A) (double_divide A identity).
+∀H2:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (double_divide (double_divide B A) identity).
+∀H3:∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.eq Univ (double_divide (double_divide (double_divide A (double_divide B identity)) (double_divide (double_divide C (double_divide D (double_divide D identity))) (double_divide A identity))) B) C.eq Univ (multiply (inverse a1) a1) identity
+.
+#Univ.
+#A.
+#B.
+#C.
+#D.
+#a1.
+#double_divide.
+#identity.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP482-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP482-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Axiom for group theory, in double division and identity, part 2 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Neu86] Neumann (1986), Yet Another Single Law for Groups *)
+
+(* : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 1 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 8 ( 0 singleton) *)
+
+(* Maximal term depth : 7 ( 2 average) *)
+
+(* Comments : A UEQ part of GRP075-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_2:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.
+∀a2:Univ.
+∀double_divide:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.eq Univ identity (double_divide A (inverse A)).
+∀H1:∀A:Univ.eq Univ (inverse A) (double_divide A identity).
+∀H2:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (double_divide (double_divide B A) identity).
+∀H3:∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.eq Univ (double_divide (double_divide (double_divide A (double_divide B identity)) (double_divide (double_divide C (double_divide D (double_divide D identity))) (double_divide A identity))) B) C.eq Univ (multiply identity a2) a2
+.
+#Univ.
+#A.
+#B.
+#C.
+#D.
+#a2.
+#double_divide.
+#identity.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP483-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP483-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Axiom for group theory, in double division and identity, part 3 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Neu86] Neumann (1986), Yet Another Single Law for Groups *)
+
+(* : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 1 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 8 ( 0 singleton) *)
+
+(* Maximal term depth : 7 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP075-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_3:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.
+∀a3:Univ.
+∀b3:Univ.
+∀c3:Univ.
+∀double_divide:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.eq Univ identity (double_divide A (inverse A)).
+∀H1:∀A:Univ.eq Univ (inverse A) (double_divide A identity).
+∀H2:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (double_divide (double_divide B A) identity).
+∀H3:∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.eq Univ (double_divide (double_divide (double_divide A (double_divide B identity)) (double_divide (double_divide C (double_divide D (double_divide D identity))) (double_divide A identity))) B) C.eq Univ (multiply (multiply a3 b3) c3) (multiply a3 (multiply b3 c3))
+.
+#Univ.
+#A.
+#B.
+#C.
+#D.
+#a3.
+#b3.
+#c3.
+#double_divide.
+#identity.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP484-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP484-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Axiom for group theory, in double division and identity, part 1 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 1 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 2 average) *)
+
+(* Comments : A UEQ part of GRP076-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_1:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a1:Univ.
+∀double_divide:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.eq Univ identity (double_divide A (inverse A)).
+∀H1:∀A:Univ.eq Univ (inverse A) (double_divide A identity).
+∀H2:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (double_divide (double_divide B A) identity).
+∀H3:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (double_divide (double_divide A (double_divide (double_divide (double_divide A B) C) (double_divide B identity))) (double_divide identity identity)) C.eq Univ (multiply (inverse a1) a1) identity
+.
+#Univ.
+#A.
+#B.
+#C.
+#a1.
+#double_divide.
+#identity.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP485-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP485-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Axiom for group theory, in double division and identity, part 2 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 1 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 2 average) *)
+
+(* Comments : A UEQ part of GRP076-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_2:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a2:Univ.
+∀double_divide:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.eq Univ identity (double_divide A (inverse A)).
+∀H1:∀A:Univ.eq Univ (inverse A) (double_divide A identity).
+∀H2:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (double_divide (double_divide B A) identity).
+∀H3:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (double_divide (double_divide A (double_divide (double_divide (double_divide A B) C) (double_divide B identity))) (double_divide identity identity)) C.eq Univ (multiply identity a2) a2
+.
+#Univ.
+#A.
+#B.
+#C.
+#a2.
+#double_divide.
+#identity.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP486-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP486-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Axiom for group theory, in double division and identity, part 3 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 1 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP076-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_3:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a3:Univ.
+∀b3:Univ.
+∀c3:Univ.
+∀double_divide:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.eq Univ identity (double_divide A (inverse A)).
+∀H1:∀A:Univ.eq Univ (inverse A) (double_divide A identity).
+∀H2:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (double_divide (double_divide B A) identity).
+∀H3:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (double_divide (double_divide A (double_divide (double_divide (double_divide A B) C) (double_divide B identity))) (double_divide identity identity)) C.eq Univ (multiply (multiply a3 b3) c3) (multiply a3 (multiply b3 c3))
+.
+#Univ.
+#A.
+#B.
+#C.
+#a3.
+#b3.
+#c3.
+#double_divide.
+#identity.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP487-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP487-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Axiom for group theory, in double division and identity, part 1 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 1 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 7 ( 2 average) *)
+
+(* Comments : A UEQ part of GRP077-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_1:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a1:Univ.
+∀double_divide:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.eq Univ identity (double_divide A (inverse A)).
+∀H1:∀A:Univ.eq Univ (inverse A) (double_divide A identity).
+∀H2:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (double_divide (double_divide B A) identity).
+∀H3:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (double_divide A (double_divide (double_divide (double_divide identity (double_divide (double_divide A identity) (double_divide B C))) B) identity)) C.eq Univ (multiply (inverse a1) a1) identity
+.
+#Univ.
+#A.
+#B.
+#C.
+#a1.
+#double_divide.
+#identity.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP488-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP488-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Axiom for group theory, in double division and identity, part 2 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 1 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 7 ( 2 average) *)
+
+(* Comments : A UEQ part of GRP077-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_2:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a2:Univ.
+∀double_divide:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.eq Univ identity (double_divide A (inverse A)).
+∀H1:∀A:Univ.eq Univ (inverse A) (double_divide A identity).
+∀H2:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (double_divide (double_divide B A) identity).
+∀H3:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (double_divide A (double_divide (double_divide (double_divide identity (double_divide (double_divide A identity) (double_divide B C))) B) identity)) C.eq Univ (multiply identity a2) a2
+.
+#Univ.
+#A.
+#B.
+#C.
+#a2.
+#double_divide.
+#identity.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP489-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP489-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Axiom for group theory, in double division and identity, part 3 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 1 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 7 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP077-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_3:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a3:Univ.
+∀b3:Univ.
+∀c3:Univ.
+∀double_divide:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.eq Univ identity (double_divide A (inverse A)).
+∀H1:∀A:Univ.eq Univ (inverse A) (double_divide A identity).
+∀H2:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (double_divide (double_divide B A) identity).
+∀H3:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (double_divide A (double_divide (double_divide (double_divide identity (double_divide (double_divide A identity) (double_divide B C))) B) identity)) C.eq Univ (multiply (multiply a3 b3) c3) (multiply a3 (multiply b3 c3))
+.
+#Univ.
+#A.
+#B.
+#C.
+#a3.
+#b3.
+#c3.
+#double_divide.
+#identity.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP490-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP490-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Axiom for group theory, in double division and identity, part 1 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 1 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 2 average) *)
+
+(* Comments : A UEQ part of GRP078-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_1:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a1:Univ.
+∀double_divide:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.eq Univ identity (double_divide A (inverse A)).
+∀H1:∀A:Univ.eq Univ (inverse A) (double_divide A identity).
+∀H2:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (double_divide (double_divide B A) identity).
+∀H3:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (double_divide (double_divide identity A) (double_divide identity (double_divide (double_divide (double_divide A B) identity) (double_divide C B)))) C.eq Univ (multiply (inverse a1) a1) identity
+.
+#Univ.
+#A.
+#B.
+#C.
+#a1.
+#double_divide.
+#identity.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP491-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP491-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Axiom for group theory, in double division and identity, part 2 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 1 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 2 average) *)
+
+(* Comments : A UEQ part of GRP078-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_2:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a2:Univ.
+∀double_divide:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.eq Univ identity (double_divide A (inverse A)).
+∀H1:∀A:Univ.eq Univ (inverse A) (double_divide A identity).
+∀H2:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (double_divide (double_divide B A) identity).
+∀H3:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (double_divide (double_divide identity A) (double_divide identity (double_divide (double_divide (double_divide A B) identity) (double_divide C B)))) C.eq Univ (multiply identity a2) a2
+.
+#Univ.
+#A.
+#B.
+#C.
+#a2.
+#double_divide.
+#identity.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP492-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP492-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Axiom for group theory, in double division and identity, part 3 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 1 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP078-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_3:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a3:Univ.
+∀b3:Univ.
+∀c3:Univ.
+∀double_divide:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.eq Univ identity (double_divide A (inverse A)).
+∀H1:∀A:Univ.eq Univ (inverse A) (double_divide A identity).
+∀H2:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (double_divide (double_divide B A) identity).
+∀H3:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (double_divide (double_divide identity A) (double_divide identity (double_divide (double_divide (double_divide A B) identity) (double_divide C B)))) C.eq Univ (multiply (multiply a3 b3) c3) (multiply a3 (multiply b3 c3))
+.
+#Univ.
+#A.
+#B.
+#C.
+#a3.
+#b3.
+#c3.
+#double_divide.
+#identity.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP493-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP493-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Axiom for group theory, in double division and identity, part 1 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 1 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : A UEQ part of GRP079-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_1:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a1:Univ.
+∀double_divide:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.eq Univ identity (double_divide A (inverse A)).
+∀H1:∀A:Univ.eq Univ (inverse A) (double_divide A identity).
+∀H2:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (double_divide (double_divide B A) identity).
+∀H3:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (double_divide (double_divide identity A) (double_divide (double_divide (double_divide B C) (double_divide identity identity)) (double_divide A C))) B.eq Univ (multiply (inverse a1) a1) identity
+.
+#Univ.
+#A.
+#B.
+#C.
+#a1.
+#double_divide.
+#identity.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP494-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP494-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Axiom for group theory, in double division and identity, part 2 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 1 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : A UEQ part of GRP079-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_2:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a2:Univ.
+∀double_divide:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.eq Univ identity (double_divide A (inverse A)).
+∀H1:∀A:Univ.eq Univ (inverse A) (double_divide A identity).
+∀H2:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (double_divide (double_divide B A) identity).
+∀H3:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (double_divide (double_divide identity A) (double_divide (double_divide (double_divide B C) (double_divide identity identity)) (double_divide A C))) B.eq Univ (multiply identity a2) a2
+.
+#Univ.
+#A.
+#B.
+#C.
+#a2.
+#double_divide.
+#identity.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP495-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP495-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Axiom for group theory, in double division and identity, part 3 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 1 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : A UEQ part of GRP079-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_3:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a3:Univ.
+∀b3:Univ.
+∀c3:Univ.
+∀double_divide:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.eq Univ identity (double_divide A (inverse A)).
+∀H1:∀A:Univ.eq Univ (inverse A) (double_divide A identity).
+∀H2:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (double_divide (double_divide B A) identity).
+∀H3:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (double_divide (double_divide identity A) (double_divide (double_divide (double_divide B C) (double_divide identity identity)) (double_divide A C))) B.eq Univ (multiply (multiply a3 b3) c3) (multiply a3 (multiply b3 c3))
+.
+#Univ.
+#A.
+#B.
+#C.
+#a3.
+#b3.
+#c3.
+#double_divide.
+#identity.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP496-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP496-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Axiom for group theory, in double division and identity, part 1 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 1 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : A UEQ part of GRP080-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_1:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a1:Univ.
+∀double_divide:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.eq Univ identity (double_divide A (inverse A)).
+∀H1:∀A:Univ.eq Univ (inverse A) (double_divide A identity).
+∀H2:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (double_divide (double_divide B A) identity).
+∀H3:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (double_divide (double_divide identity (double_divide A (double_divide B identity))) (double_divide (double_divide B (double_divide C A)) identity)) C.eq Univ (multiply (inverse a1) a1) identity
+.
+#Univ.
+#A.
+#B.
+#C.
+#a1.
+#double_divide.
+#identity.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP497-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP497-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Axiom for group theory, in double division and identity, part 2 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 1 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : A UEQ part of GRP080-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_2:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a2:Univ.
+∀double_divide:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.eq Univ identity (double_divide A (inverse A)).
+∀H1:∀A:Univ.eq Univ (inverse A) (double_divide A identity).
+∀H2:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (double_divide (double_divide B A) identity).
+∀H3:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (double_divide (double_divide identity (double_divide A (double_divide B identity))) (double_divide (double_divide B (double_divide C A)) identity)) C.eq Univ (multiply identity a2) a2
+.
+#Univ.
+#A.
+#B.
+#C.
+#a2.
+#double_divide.
+#identity.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP498-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP498-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Axiom for group theory, in double division and identity, part 3 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 1 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : A UEQ part of GRP080-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_3:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a3:Univ.
+∀b3:Univ.
+∀c3:Univ.
+∀double_divide:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.eq Univ identity (double_divide A (inverse A)).
+∀H1:∀A:Univ.eq Univ (inverse A) (double_divide A identity).
+∀H2:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (double_divide (double_divide B A) identity).
+∀H3:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (double_divide (double_divide identity (double_divide A (double_divide B identity))) (double_divide (double_divide B (double_divide C A)) identity)) C.eq Univ (multiply (multiply a3 b3) c3) (multiply a3 (multiply b3 c3))
+.
+#Univ.
+#A.
+#B.
+#C.
+#a3.
+#b3.
+#c3.
+#double_divide.
+#identity.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP499-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP499-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Axiom for group theory, in double division and inverse, part 1 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 6 ( 0 singleton) *)
+
+(* Maximal term depth : 7 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP082-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_1:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.
+∀a1:Univ.
+∀b1:Univ.
+∀double_divide:∀_:Univ.∀_:Univ.Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (inverse (double_divide B A)).
+∀H1:∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.eq Univ (double_divide (inverse A) (inverse (double_divide (inverse (double_divide A (double_divide B C))) (double_divide D (double_divide B D))))) C.eq Univ (multiply (inverse a1) a1) (multiply (inverse b1) b1)
+.
+#Univ.
+#A.
+#B.
+#C.
+#D.
+#a1.
+#b1.
+#double_divide.
+#inverse.
+#multiply.
+#H0.
+#H1.
+nauto by H0,H1;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP500-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP500-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Axiom for group theory, in double division and inverse, part 2 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.07 v3.1.0, 0.11 v2.7.0, 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 6 ( 0 singleton) *)
+
+(* Maximal term depth : 7 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP082-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_2:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.
+∀a2:Univ.
+∀b2:Univ.
+∀double_divide:∀_:Univ.∀_:Univ.Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (inverse (double_divide B A)).
+∀H1:∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.eq Univ (double_divide (inverse A) (inverse (double_divide (inverse (double_divide A (double_divide B C))) (double_divide D (double_divide B D))))) C.eq Univ (multiply (multiply (inverse b2) b2) a2) a2
+.
+#Univ.
+#A.
+#B.
+#C.
+#D.
+#a2.
+#b2.
+#double_divide.
+#inverse.
+#multiply.
+#H0.
+#H1.
+nauto by H0,H1;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP501-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP501-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Axiom for group theory, in double division and inverse, part 3 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.14 v3.1.0, 0.11 v2.7.0, 0.09 v2.6.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 6 ( 0 singleton) *)
+
+(* Maximal term depth : 7 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP082-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_3:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.
+∀a3:Univ.
+∀b3:Univ.
+∀c3:Univ.
+∀double_divide:∀_:Univ.∀_:Univ.Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (inverse (double_divide B A)).
+∀H1:∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.eq Univ (double_divide (inverse A) (inverse (double_divide (inverse (double_divide A (double_divide B C))) (double_divide D (double_divide B D))))) C.eq Univ (multiply (multiply a3 b3) c3) (multiply a3 (multiply b3 c3))
+.
+#Univ.
+#A.
+#B.
+#C.
+#D.
+#a3.
+#b3.
+#c3.
+#double_divide.
+#inverse.
+#multiply.
+#H0.
+#H1.
+nauto by H0,H1;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP502-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP502-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Axiom for group theory, in double division and inverse, part 1 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.14 v3.1.0, 0.11 v2.7.0, 0.09 v2.6.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 6 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP083-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_1:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.
+∀a1:Univ.
+∀b1:Univ.
+∀double_divide:∀_:Univ.∀_:Univ.Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (inverse (double_divide B A)).
+∀H1:∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.eq Univ (double_divide (double_divide A (inverse (double_divide B C))) (double_divide (inverse B) (inverse (double_divide D (double_divide A D))))) C.eq Univ (multiply (inverse a1) a1) (multiply (inverse b1) b1)
+.
+#Univ.
+#A.
+#B.
+#C.
+#D.
+#a1.
+#b1.
+#double_divide.
+#inverse.
+#multiply.
+#H0.
+#H1.
+nauto by H0,H1;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP503-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP503-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Axiom for group theory, in double division and inverse, part 2 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.14 v3.1.0, 0.11 v2.7.0, 0.09 v2.6.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 6 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP083-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_2:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.
+∀a2:Univ.
+∀b2:Univ.
+∀double_divide:∀_:Univ.∀_:Univ.Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (inverse (double_divide B A)).
+∀H1:∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.eq Univ (double_divide (double_divide A (inverse (double_divide B C))) (double_divide (inverse B) (inverse (double_divide D (double_divide A D))))) C.eq Univ (multiply (multiply (inverse b2) b2) a2) a2
+.
+#Univ.
+#A.
+#B.
+#C.
+#D.
+#a2.
+#b2.
+#double_divide.
+#inverse.
+#multiply.
+#H0.
+#H1.
+nauto by H0,H1;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP504-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP504-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory *)
+
+(* Problem : Axiom for group theory, in double division and inverse, part 3 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.21 v3.2.0, 0.14 v3.1.0, 0.11 v2.7.0, 0.09 v2.6.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 6 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP083-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_3:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.
+∀a3:Univ.
+∀b3:Univ.
+∀c3:Univ.
+∀double_divide:∀_:Univ.∀_:Univ.Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (inverse (double_divide B A)).
+∀H1:∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.eq Univ (double_divide (double_divide A (inverse (double_divide B C))) (double_divide (inverse B) (inverse (double_divide D (double_divide A D))))) C.eq Univ (multiply (multiply a3 b3) c3) (multiply a3 (multiply b3 c3))
+.
+#Univ.
+#A.
+#B.
+#C.
+#D.
+#a3.
+#b3.
+#c3.
+#double_divide.
+#inverse.
+#multiply.
+#H0.
+#H1.
+nauto by H0,H1;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP505-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP505-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in product and inverse, part 1 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Neu81] Neumann (1981), Another Single Law for Groups *)
+
+(* : [LW92] Lusk & Wos (1992), Benchmark Problems in Which Equalit *)
+
+(* : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.57 v3.2.0, 0.50 v3.1.0, 0.44 v2.7.0, 0.64 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 6 ( 0 singleton) *)
+
+(* Maximal term depth : 10 ( 4 average) *)
+
+(* Comments : A UEQ part of GRP084-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_1:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.∀E:Univ.∀F:Univ.
+∀a1:Univ.
+∀b1:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.∀E:Univ.∀F:Univ.eq Univ (multiply (inverse (multiply (inverse (multiply (inverse (multiply A B)) (multiply B A))) (multiply (inverse (multiply C D)) (multiply C (inverse (multiply (multiply E (inverse F)) (inverse D))))))) F) E.eq Univ (multiply (inverse a1) a1) (multiply (inverse b1) b1)
+.
+#Univ.
+#A.
+#B.
+#C.
+#D.
+#E.
+#F.
+#a1.
+#b1.
+#inverse.
+#multiply.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP506-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP506-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in product and inverse, part 2 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Neu81] Neumann (1981), Another Single Law for Groups *)
+
+(* : [LW92] Lusk & Wos (1992), Benchmark Problems in Which Equalit *)
+
+(* : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.71 v3.2.0, 0.64 v3.1.0, 0.67 v2.7.0, 0.73 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 6 ( 0 singleton) *)
+
+(* Maximal term depth : 10 ( 4 average) *)
+
+(* Comments : A UEQ part of GRP084-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_2:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.∀E:Univ.∀F:Univ.
+∀a2:Univ.
+∀b2:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.∀E:Univ.∀F:Univ.eq Univ (multiply (inverse (multiply (inverse (multiply (inverse (multiply A B)) (multiply B A))) (multiply (inverse (multiply C D)) (multiply C (inverse (multiply (multiply E (inverse F)) (inverse D))))))) F) E.eq Univ (multiply (multiply (inverse b2) b2) a2) a2
+.
+#Univ.
+#A.
+#B.
+#C.
+#D.
+#E.
+#F.
+#a2.
+#b2.
+#inverse.
+#multiply.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP507-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP507-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in product and inverse, part 3 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Neu81] Neumann (1981), Another Single Law for Groups *)
+
+(* : [LW92] Lusk & Wos (1992), Benchmark Problems in Which Equalit *)
+
+(* : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.64 v3.2.0, 0.57 v3.1.0, 0.56 v2.7.0, 0.64 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 6 ( 0 singleton) *)
+
+(* Maximal term depth : 10 ( 4 average) *)
+
+(* Comments : A UEQ part of GRP084-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_3:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.∀E:Univ.∀F:Univ.
+∀a3:Univ.
+∀b3:Univ.
+∀c3:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.∀E:Univ.∀F:Univ.eq Univ (multiply (inverse (multiply (inverse (multiply (inverse (multiply A B)) (multiply B A))) (multiply (inverse (multiply C D)) (multiply C (inverse (multiply (multiply E (inverse F)) (inverse D))))))) F) E.eq Univ (multiply (multiply a3 b3) c3) (multiply a3 (multiply b3 c3))
+.
+#Univ.
+#A.
+#B.
+#C.
+#D.
+#E.
+#F.
+#a3.
+#b3.
+#c3.
+#inverse.
+#multiply.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP508-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP508-1 : TPTP v3.2.0. Bugfixed v2.7.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in product and inverse, part 4 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Neu81] Neumann (1981), Another Single Law for Groups *)
+
+(* : [LW92] Lusk & Wos (1992), Benchmark Problems in Which Equalit *)
+
+(* : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.64 v3.2.0, 0.57 v3.1.0, 0.56 v2.7.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 6 ( 0 singleton) *)
+
+(* Maximal term depth : 10 ( 4 average) *)
+
+(* Comments : A UEQ part of GRP084-1 *)
+
+(* Bugfixes : v2.7.0 - Grounded conjecture *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_4:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.∀E:Univ.∀F:Univ.
+∀a:Univ.
+∀b:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.∀E:Univ.∀F:Univ.eq Univ (multiply (inverse (multiply (inverse (multiply (inverse (multiply A B)) (multiply B A))) (multiply (inverse (multiply C D)) (multiply C (inverse (multiply (multiply E (inverse F)) (inverse D))))))) F) E.eq Univ (multiply a b) (multiply b a)
+.
+#Univ.
+#A.
+#B.
+#C.
+#D.
+#E.
+#F.
+#a.
+#b.
+#inverse.
+#multiply.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP509-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP509-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in product and inverse, part 1 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [LW92] Lusk & Wos (1992), Benchmark Problems in Which Equalit *)
+
+(* : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 3 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP085-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_1:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a1:Univ.
+∀b1:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (multiply (multiply (multiply A B) C) (inverse (multiply A C))) B.eq Univ (multiply (inverse a1) a1) (multiply (inverse b1) b1)
+.
+#Univ.
+#A.
+#B.
+#C.
+#a1.
+#b1.
+#inverse.
+#multiply.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP510-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP510-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in product and inverse, part 2 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [LW92] Lusk & Wos (1992), Benchmark Problems in Which Equalit *)
+
+(* : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 3 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : A UEQ part of GRP085-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_2:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a2:Univ.
+∀b2:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (multiply (multiply (multiply A B) C) (inverse (multiply A C))) B.eq Univ (multiply (multiply (inverse b2) b2) a2) a2
+.
+#Univ.
+#A.
+#B.
+#C.
+#a2.
+#b2.
+#inverse.
+#multiply.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP511-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP511-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in product and inverse, part 3 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [LW92] Lusk & Wos (1992), Benchmark Problems in Which Equalit *)
+
+(* : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 3 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP085-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_3:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a3:Univ.
+∀b3:Univ.
+∀c3:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (multiply (multiply (multiply A B) C) (inverse (multiply A C))) B.eq Univ (multiply (multiply a3 b3) c3) (multiply a3 (multiply b3 c3))
+.
+#Univ.
+#A.
+#B.
+#C.
+#a3.
+#b3.
+#c3.
+#inverse.
+#multiply.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP512-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP512-1 : TPTP v3.2.0. Bugfixed v2.7.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in product and inverse, part 4 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [LW92] Lusk & Wos (1992), Benchmark Problems in Which Equalit *)
+
+(* : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.7.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 3 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : A UEQ part of GRP085-1 *)
+
+(* Bugfixes : v2.7.0 - Grounded conjecture *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_4:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a:Univ.
+∀b:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (multiply (multiply (multiply A B) C) (inverse (multiply A C))) B.eq Univ (multiply a b) (multiply b a)
+.
+#Univ.
+#A.
+#B.
+#C.
+#a.
+#b.
+#inverse.
+#multiply.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP513-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP513-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in product and inverse, part 1 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.7.0, 0.09 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 3 ( 0 singleton) *)
+
+(* Maximal term depth : 5 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP086-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_1:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a1:Univ.
+∀b1:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (multiply A (multiply (multiply B C) (inverse (multiply A C)))) B.eq Univ (multiply (inverse a1) a1) (multiply (inverse b1) b1)
+.
+#Univ.
+#A.
+#B.
+#C.
+#a1.
+#b1.
+#inverse.
+#multiply.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP514-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP514-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in product and inverse, part 2 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 3 ( 0 singleton) *)
+
+(* Maximal term depth : 5 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP086-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_2:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a2:Univ.
+∀b2:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (multiply A (multiply (multiply B C) (inverse (multiply A C)))) B.eq Univ (multiply (multiply (inverse b2) b2) a2) a2
+.
+#Univ.
+#A.
+#B.
+#C.
+#a2.
+#b2.
+#inverse.
+#multiply.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP515-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP515-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in product and inverse, part 3 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v3.1.0, 0.11 v2.7.0, 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 3 ( 0 singleton) *)
+
+(* Maximal term depth : 5 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP086-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_3:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a3:Univ.
+∀b3:Univ.
+∀c3:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (multiply A (multiply (multiply B C) (inverse (multiply A C)))) B.eq Univ (multiply (multiply a3 b3) c3) (multiply a3 (multiply b3 c3))
+.
+#Univ.
+#A.
+#B.
+#C.
+#a3.
+#b3.
+#c3.
+#inverse.
+#multiply.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP516-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP516-1 : TPTP v3.2.0. Bugfixed v2.7.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in product and inverse, part 4 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.7.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 3 ( 0 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : A UEQ part of GRP086-1 *)
+
+(* Bugfixes : v2.7.0 - Grounded conjecture *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_4:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a:Univ.
+∀b:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (multiply A (multiply (multiply B C) (inverse (multiply A C)))) B.eq Univ (multiply a b) (multiply b a)
+.
+#Univ.
+#A.
+#B.
+#C.
+#a.
+#b.
+#inverse.
+#multiply.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP517-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP517-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in product and inverse, part 1 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 3 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP087-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_1:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a1:Univ.
+∀b1:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (multiply A (multiply (multiply (inverse (multiply A B)) C) B)) C.eq Univ (multiply (inverse a1) a1) (multiply (inverse b1) b1)
+.
+#Univ.
+#A.
+#B.
+#C.
+#a1.
+#b1.
+#inverse.
+#multiply.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP518-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP518-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in product and inverse, part 2 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 3 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP087-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_2:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a2:Univ.
+∀b2:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (multiply A (multiply (multiply (inverse (multiply A B)) C) B)) C.eq Univ (multiply (multiply (inverse b2) b2) a2) a2
+.
+#Univ.
+#A.
+#B.
+#C.
+#a2.
+#b2.
+#inverse.
+#multiply.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP519-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP519-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in product and inverse, part 3 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 3 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP087-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_3:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a3:Univ.
+∀b3:Univ.
+∀c3:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (multiply A (multiply (multiply (inverse (multiply A B)) C) B)) C.eq Univ (multiply (multiply a3 b3) c3) (multiply a3 (multiply b3 c3))
+.
+#Univ.
+#A.
+#B.
+#C.
+#a3.
+#b3.
+#c3.
+#inverse.
+#multiply.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP520-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP520-1 : TPTP v3.2.0. Bugfixed v2.7.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in product and inverse, part 4 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.7.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 3 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP087-1 *)
+
+(* Bugfixes : v2.7.0 - Grounded conjecture *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_4:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a:Univ.
+∀b:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (multiply A (multiply (multiply (inverse (multiply A B)) C) B)) C.eq Univ (multiply a b) (multiply b a)
+.
+#Univ.
+#A.
+#B.
+#C.
+#a.
+#b.
+#inverse.
+#multiply.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP521-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP521-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in division, part 1 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Tar38] Tarski (1938), Ein Beitrag zur Axiomatik der Abelschen *)
+
+(* : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 4 ( 0 non-Horn; 4 unit; 1 RR) *)
+
+(* Number of atoms : 4 ( 4 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 8 ( 0 singleton) *)
+
+(* Maximal term depth : 5 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP088-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_1:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a1:Univ.
+∀b1:Univ.
+∀divide:∀_:Univ.∀_:Univ.Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.eq Univ (inverse A) (divide (divide B B) A).
+∀H1:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (multiply A B) (divide A (divide (divide C C) B)).
+∀H2:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (divide A (divide B (divide C (divide A B)))) C.eq Univ (multiply (inverse a1) a1) (multiply (inverse b1) b1)
+.
+#Univ.
+#A.
+#B.
+#C.
+#a1.
+#b1.
+#divide.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+nauto by H0,H1,H2;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP522-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP522-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in division, part 2 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Tar38] Tarski (1938), Ein Beitrag zur Axiomatik der Abelschen *)
+
+(* : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 4 ( 0 non-Horn; 4 unit; 1 RR) *)
+
+(* Number of atoms : 4 ( 4 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 8 ( 0 singleton) *)
+
+(* Maximal term depth : 5 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP088-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_2:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a2:Univ.
+∀b2:Univ.
+∀divide:∀_:Univ.∀_:Univ.Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.eq Univ (inverse A) (divide (divide B B) A).
+∀H1:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (multiply A B) (divide A (divide (divide C C) B)).
+∀H2:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (divide A (divide B (divide C (divide A B)))) C.eq Univ (multiply (multiply (inverse b2) b2) a2) a2
+.
+#Univ.
+#A.
+#B.
+#C.
+#a2.
+#b2.
+#divide.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+nauto by H0,H1,H2;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP523-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP523-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in division, part 3 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Tar38] Tarski (1938), Ein Beitrag zur Axiomatik der Abelschen *)
+
+(* : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 4 ( 0 non-Horn; 4 unit; 1 RR) *)
+
+(* Number of atoms : 4 ( 4 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 8 ( 0 singleton) *)
+
+(* Maximal term depth : 5 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP088-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_3:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a3:Univ.
+∀b3:Univ.
+∀c3:Univ.
+∀divide:∀_:Univ.∀_:Univ.Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.eq Univ (inverse A) (divide (divide B B) A).
+∀H1:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (multiply A B) (divide A (divide (divide C C) B)).
+∀H2:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (divide A (divide B (divide C (divide A B)))) C.eq Univ (multiply (multiply a3 b3) c3) (multiply a3 (multiply b3 c3))
+.
+#Univ.
+#A.
+#B.
+#C.
+#a3.
+#b3.
+#c3.
+#divide.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+nauto by H0,H1,H2;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP524-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP524-1 : TPTP v3.2.0. Bugfixed v2.7.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in division, part 4 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Tar38] Tarski (1938), Ein Beitrag zur Axiomatik der Abelschen *)
+
+(* : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.7.0 *)
+
+(* Syntax : Number of clauses : 4 ( 0 non-Horn; 4 unit; 1 RR) *)
+
+(* Number of atoms : 4 ( 4 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 8 ( 0 singleton) *)
+
+(* Maximal term depth : 5 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP088-1 *)
+
+(* Bugfixes : v2.7.0 - Grounded conjecture *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_4:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a:Univ.
+∀b:Univ.
+∀divide:∀_:Univ.∀_:Univ.Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.eq Univ (inverse A) (divide (divide B B) A).
+∀H1:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (multiply A B) (divide A (divide (divide C C) B)).
+∀H2:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (divide A (divide B (divide C (divide A B)))) C.eq Univ (multiply a b) (multiply b a)
+.
+#Univ.
+#A.
+#B.
+#C.
+#a.
+#b.
+#divide.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+nauto by H0,H1,H2;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP525-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP525-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in division, part 1 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 4 ( 0 non-Horn; 4 unit; 1 RR) *)
+
+(* Number of atoms : 4 ( 4 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 8 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP089-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_1:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a1:Univ.
+∀b1:Univ.
+∀divide:∀_:Univ.∀_:Univ.Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.eq Univ (inverse A) (divide (divide B B) A).
+∀H1:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (multiply A B) (divide A (divide (divide C C) B)).
+∀H2:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (divide A (divide (divide A B) (divide C B))) C.eq Univ (multiply (inverse a1) a1) (multiply (inverse b1) b1)
+.
+#Univ.
+#A.
+#B.
+#C.
+#a1.
+#b1.
+#divide.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+nauto by H0,H1,H2;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP526-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP526-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in division, part 2 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 4 ( 0 non-Horn; 4 unit; 1 RR) *)
+
+(* Number of atoms : 4 ( 4 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 8 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP089-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_2:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a2:Univ.
+∀b2:Univ.
+∀divide:∀_:Univ.∀_:Univ.Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.eq Univ (inverse A) (divide (divide B B) A).
+∀H1:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (multiply A B) (divide A (divide (divide C C) B)).
+∀H2:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (divide A (divide (divide A B) (divide C B))) C.eq Univ (multiply (multiply (inverse b2) b2) a2) a2
+.
+#Univ.
+#A.
+#B.
+#C.
+#a2.
+#b2.
+#divide.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+nauto by H0,H1,H2;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP527-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP527-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in division, part 3 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 4 ( 0 non-Horn; 4 unit; 1 RR) *)
+
+(* Number of atoms : 4 ( 4 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 8 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP089-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_3:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a3:Univ.
+∀b3:Univ.
+∀c3:Univ.
+∀divide:∀_:Univ.∀_:Univ.Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.eq Univ (inverse A) (divide (divide B B) A).
+∀H1:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (multiply A B) (divide A (divide (divide C C) B)).
+∀H2:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (divide A (divide (divide A B) (divide C B))) C.eq Univ (multiply (multiply a3 b3) c3) (multiply a3 (multiply b3 c3))
+.
+#Univ.
+#A.
+#B.
+#C.
+#a3.
+#b3.
+#c3.
+#divide.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+nauto by H0,H1,H2;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP528-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP528-1 : TPTP v3.2.0. Bugfixed v2.7.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in division, part 4 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.7.0 *)
+
+(* Syntax : Number of clauses : 4 ( 0 non-Horn; 4 unit; 1 RR) *)
+
+(* Number of atoms : 4 ( 4 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 8 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : A UEQ part of GRP089-1 *)
+
+(* Bugfixes : v2.7.0 - Grounded conjecture *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_4:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a:Univ.
+∀b:Univ.
+∀divide:∀_:Univ.∀_:Univ.Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.eq Univ (inverse A) (divide (divide B B) A).
+∀H1:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (multiply A B) (divide A (divide (divide C C) B)).
+∀H2:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (divide A (divide (divide A B) (divide C B))) C.eq Univ (multiply a b) (multiply b a)
+.
+#Univ.
+#A.
+#B.
+#C.
+#a.
+#b.
+#divide.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+nauto by H0,H1,H2;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP529-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP529-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in division, part 1 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.7.0, 0.09 v2.6.0 *)
+
+(* Syntax : Number of clauses : 4 ( 0 non-Horn; 4 unit; 1 RR) *)
+
+(* Number of atoms : 4 ( 4 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 8 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP090-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_1:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a1:Univ.
+∀b1:Univ.
+∀divide:∀_:Univ.∀_:Univ.Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.eq Univ (inverse A) (divide (divide B B) A).
+∀H1:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (multiply A B) (divide A (divide (divide C C) B)).
+∀H2:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (divide (divide A (divide B C)) (divide A B)) C.eq Univ (multiply (inverse a1) a1) (multiply (inverse b1) b1)
+.
+#Univ.
+#A.
+#B.
+#C.
+#a1.
+#b1.
+#divide.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+nauto by H0,H1,H2;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP530-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP530-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in division, part 2 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 4 ( 0 non-Horn; 4 unit; 1 RR) *)
+
+(* Number of atoms : 4 ( 4 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 8 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP090-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_2:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a2:Univ.
+∀b2:Univ.
+∀divide:∀_:Univ.∀_:Univ.Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.eq Univ (inverse A) (divide (divide B B) A).
+∀H1:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (multiply A B) (divide A (divide (divide C C) B)).
+∀H2:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (divide (divide A (divide B C)) (divide A B)) C.eq Univ (multiply (multiply (inverse b2) b2) a2) a2
+.
+#Univ.
+#A.
+#B.
+#C.
+#a2.
+#b2.
+#divide.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+nauto by H0,H1,H2;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP531-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP531-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in division, part 3 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 4 ( 0 non-Horn; 4 unit; 1 RR) *)
+
+(* Number of atoms : 4 ( 4 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 8 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP090-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_3:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a3:Univ.
+∀b3:Univ.
+∀c3:Univ.
+∀divide:∀_:Univ.∀_:Univ.Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.eq Univ (inverse A) (divide (divide B B) A).
+∀H1:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (multiply A B) (divide A (divide (divide C C) B)).
+∀H2:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (divide (divide A (divide B C)) (divide A B)) C.eq Univ (multiply (multiply a3 b3) c3) (multiply a3 (multiply b3 c3))
+.
+#Univ.
+#A.
+#B.
+#C.
+#a3.
+#b3.
+#c3.
+#divide.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+nauto by H0,H1,H2;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP532-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP532-1 : TPTP v3.2.0. Bugfixed v2.7.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in division, part 4 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.7.0 *)
+
+(* Syntax : Number of clauses : 4 ( 0 non-Horn; 4 unit; 1 RR) *)
+
+(* Number of atoms : 4 ( 4 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 8 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : A UEQ part of GRP090-1 *)
+
+(* Bugfixes : v2.7.0 - Grounded conjecture *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_4:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a:Univ.
+∀b:Univ.
+∀divide:∀_:Univ.∀_:Univ.Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.eq Univ (inverse A) (divide (divide B B) A).
+∀H1:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (multiply A B) (divide A (divide (divide C C) B)).
+∀H2:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (divide (divide A (divide B C)) (divide A B)) C.eq Univ (multiply a b) (multiply b a)
+.
+#Univ.
+#A.
+#B.
+#C.
+#a.
+#b.
+#divide.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+nauto by H0,H1,H2;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP533-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP533-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in division, part 1 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 1 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 9 ( 0 singleton) *)
+
+(* Maximal term depth : 5 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP091-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_1:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a1:Univ.
+∀b1:Univ.
+∀divide:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.eq Univ identity (divide A A).
+∀H1:∀A:Univ.∀B:Univ.eq Univ (inverse A) (divide (divide B B) A).
+∀H2:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (multiply A B) (divide A (divide (divide C C) B)).
+∀H3:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (divide (divide A (divide (divide A B) C)) B) C.eq Univ (multiply (inverse a1) a1) (multiply (inverse b1) b1)
+.
+#Univ.
+#A.
+#B.
+#C.
+#a1.
+#b1.
+#divide.
+#identity.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP534-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP534-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in division, part 2 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 1 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 9 ( 0 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : A UEQ part of GRP091-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_2:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a2:Univ.
+∀b2:Univ.
+∀divide:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.eq Univ identity (divide A A).
+∀H1:∀A:Univ.∀B:Univ.eq Univ (inverse A) (divide (divide B B) A).
+∀H2:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (multiply A B) (divide A (divide (divide C C) B)).
+∀H3:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (divide (divide A (divide (divide A B) C)) B) C.eq Univ (multiply (multiply (inverse b2) b2) a2) a2
+.
+#Univ.
+#A.
+#B.
+#C.
+#a2.
+#b2.
+#divide.
+#identity.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP535-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP535-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in division, part 3 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 1 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 9 ( 0 singleton) *)
+
+(* Maximal term depth : 5 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP091-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_3:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a3:Univ.
+∀b3:Univ.
+∀c3:Univ.
+∀divide:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.eq Univ identity (divide A A).
+∀H1:∀A:Univ.∀B:Univ.eq Univ (inverse A) (divide (divide B B) A).
+∀H2:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (multiply A B) (divide A (divide (divide C C) B)).
+∀H3:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (divide (divide A (divide (divide A B) C)) B) C.eq Univ (multiply (multiply a3 b3) c3) (multiply a3 (multiply b3 c3))
+.
+#Univ.
+#A.
+#B.
+#C.
+#a3.
+#b3.
+#c3.
+#divide.
+#identity.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP536-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP536-1 : TPTP v3.2.0. Bugfixed v2.7.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in division, part 4 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.7.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 1 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 9 ( 0 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : A UEQ part of GRP091-1 *)
+
+(* Bugfixes : v2.7.0 - Grounded conjecture *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_4:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a:Univ.
+∀b:Univ.
+∀divide:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.eq Univ identity (divide A A).
+∀H1:∀A:Univ.∀B:Univ.eq Univ (inverse A) (divide (divide B B) A).
+∀H2:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (multiply A B) (divide A (divide (divide C C) B)).
+∀H3:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (divide (divide A (divide (divide A B) C)) B) C.eq Univ (multiply a b) (multiply b a)
+.
+#Univ.
+#A.
+#B.
+#C.
+#a.
+#b.
+#divide.
+#identity.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP537-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP537-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in division, part 1 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 1 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 9 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : A UEQ part of GRP092-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_1:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a1:Univ.
+∀b1:Univ.
+∀divide:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.eq Univ identity (divide A A).
+∀H1:∀A:Univ.∀B:Univ.eq Univ (inverse A) (divide (divide B B) A).
+∀H2:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (multiply A B) (divide A (divide (divide C C) B)).
+∀H3:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (divide (divide A B) (divide (divide A C) B)) C.eq Univ (multiply (inverse a1) a1) (multiply (inverse b1) b1)
+.
+#Univ.
+#A.
+#B.
+#C.
+#a1.
+#b1.
+#divide.
+#identity.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP538-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP538-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in division, part 2 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 1 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 9 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : A UEQ part of GRP092-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_2:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a2:Univ.
+∀b2:Univ.
+∀divide:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.eq Univ identity (divide A A).
+∀H1:∀A:Univ.∀B:Univ.eq Univ (inverse A) (divide (divide B B) A).
+∀H2:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (multiply A B) (divide A (divide (divide C C) B)).
+∀H3:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (divide (divide A B) (divide (divide A C) B)) C.eq Univ (multiply (multiply (inverse b2) b2) a2) a2
+.
+#Univ.
+#A.
+#B.
+#C.
+#a2.
+#b2.
+#divide.
+#identity.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP539-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP539-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in division, part 3 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 1 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 9 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : A UEQ part of GRP092-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_3:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a3:Univ.
+∀b3:Univ.
+∀c3:Univ.
+∀divide:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.eq Univ identity (divide A A).
+∀H1:∀A:Univ.∀B:Univ.eq Univ (inverse A) (divide (divide B B) A).
+∀H2:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (multiply A B) (divide A (divide (divide C C) B)).
+∀H3:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (divide (divide A B) (divide (divide A C) B)) C.eq Univ (multiply (multiply a3 b3) c3) (multiply a3 (multiply b3 c3))
+.
+#Univ.
+#A.
+#B.
+#C.
+#a3.
+#b3.
+#c3.
+#divide.
+#identity.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP540-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP540-1 : TPTP v3.2.0. Bugfixed v2.7.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in division, part 4 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.7.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 1 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 9 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : A UEQ part of GRP092-1 *)
+
+(* Bugfixes : v2.7.0 - Grounded conjecture *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_4:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a:Univ.
+∀b:Univ.
+∀divide:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.eq Univ identity (divide A A).
+∀H1:∀A:Univ.∀B:Univ.eq Univ (inverse A) (divide (divide B B) A).
+∀H2:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (multiply A B) (divide A (divide (divide C C) B)).
+∀H3:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (divide (divide A B) (divide (divide A C) B)) C.eq Univ (multiply a b) (multiply b a)
+.
+#Univ.
+#A.
+#B.
+#C.
+#a.
+#b.
+#divide.
+#identity.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP541-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP541-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in division and identity, part 1 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 1 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 2 average) *)
+
+(* Comments : A UEQ part of GRP093-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_1:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a1:Univ.
+∀b1:Univ.
+∀divide:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.eq Univ identity (divide A A).
+∀H1:∀A:Univ.eq Univ (inverse A) (divide identity A).
+∀H2:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (divide A (divide identity B)).
+∀H3:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (divide (divide identity (divide (divide (divide A B) C) A)) C) B.eq Univ (multiply (inverse a1) a1) (multiply (inverse b1) b1)
+.
+#Univ.
+#A.
+#B.
+#C.
+#a1.
+#b1.
+#divide.
+#identity.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP542-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP542-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in division and identity, part 2 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 1 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 2 average) *)
+
+(* Comments : A UEQ part of GRP093-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_2:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a2:Univ.
+∀b2:Univ.
+∀divide:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.eq Univ identity (divide A A).
+∀H1:∀A:Univ.eq Univ (inverse A) (divide identity A).
+∀H2:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (divide A (divide identity B)).
+∀H3:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (divide (divide identity (divide (divide (divide A B) C) A)) C) B.eq Univ (multiply (multiply (inverse b2) b2) a2) a2
+.
+#Univ.
+#A.
+#B.
+#C.
+#a2.
+#b2.
+#divide.
+#identity.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP543-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP543-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in division and identity, part 3 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 1 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 2 average) *)
+
+(* Comments : A UEQ part of GRP093-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_3:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a3:Univ.
+∀b3:Univ.
+∀c3:Univ.
+∀divide:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.eq Univ identity (divide A A).
+∀H1:∀A:Univ.eq Univ (inverse A) (divide identity A).
+∀H2:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (divide A (divide identity B)).
+∀H3:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (divide (divide identity (divide (divide (divide A B) C) A)) C) B.eq Univ (multiply (multiply a3 b3) c3) (multiply a3 (multiply b3 c3))
+.
+#Univ.
+#A.
+#B.
+#C.
+#a3.
+#b3.
+#c3.
+#divide.
+#identity.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP544-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP544-1 : TPTP v3.2.0. Bugfixed v2.7.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in division and identity, part 4 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.7.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 1 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 2 average) *)
+
+(* Comments : A UEQ part of GRP093-1 *)
+
+(* Bugfixes : v2.7.0 - Grounded conjecture *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_4:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a:Univ.
+∀b:Univ.
+∀divide:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.eq Univ identity (divide A A).
+∀H1:∀A:Univ.eq Univ (inverse A) (divide identity A).
+∀H2:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (divide A (divide identity B)).
+∀H3:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (divide (divide identity (divide (divide (divide A B) C) A)) C) B.eq Univ (multiply a b) (multiply b a)
+.
+#Univ.
+#A.
+#B.
+#C.
+#a.
+#b.
+#divide.
+#identity.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP545-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP545-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in division and identity, part 1 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 1 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : A UEQ part of GRP094-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_1:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a1:Univ.
+∀b1:Univ.
+∀divide:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.eq Univ identity (divide A A).
+∀H1:∀A:Univ.eq Univ (inverse A) (divide identity A).
+∀H2:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (divide A (divide identity B)).
+∀H3:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (divide (divide identity (divide A B)) (divide (divide B C) A)) C.eq Univ (multiply (inverse a1) a1) (multiply (inverse b1) b1)
+.
+#Univ.
+#A.
+#B.
+#C.
+#a1.
+#b1.
+#divide.
+#identity.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP546-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP546-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in division and identity, part 2 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 1 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : A UEQ part of GRP094-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_2:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a2:Univ.
+∀b2:Univ.
+∀divide:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.eq Univ identity (divide A A).
+∀H1:∀A:Univ.eq Univ (inverse A) (divide identity A).
+∀H2:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (divide A (divide identity B)).
+∀H3:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (divide (divide identity (divide A B)) (divide (divide B C) A)) C.eq Univ (multiply (multiply (inverse b2) b2) a2) a2
+.
+#Univ.
+#A.
+#B.
+#C.
+#a2.
+#b2.
+#divide.
+#identity.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP547-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP547-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in division and identity, part 3 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 1 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : A UEQ part of GRP094-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_3:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a3:Univ.
+∀b3:Univ.
+∀c3:Univ.
+∀divide:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.eq Univ identity (divide A A).
+∀H1:∀A:Univ.eq Univ (inverse A) (divide identity A).
+∀H2:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (divide A (divide identity B)).
+∀H3:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (divide (divide identity (divide A B)) (divide (divide B C) A)) C.eq Univ (multiply (multiply a3 b3) c3) (multiply a3 (multiply b3 c3))
+.
+#Univ.
+#A.
+#B.
+#C.
+#a3.
+#b3.
+#c3.
+#divide.
+#identity.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP548-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP548-1 : TPTP v3.2.0. Bugfixed v2.7.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in division and identity, part 4 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.7.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 1 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : A UEQ part of GRP094-1 *)
+
+(* Bugfixes : v2.7.0 - Grounded conjecture *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_4:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a:Univ.
+∀b:Univ.
+∀divide:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.eq Univ identity (divide A A).
+∀H1:∀A:Univ.eq Univ (inverse A) (divide identity A).
+∀H2:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (divide A (divide identity B)).
+∀H3:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (divide (divide identity (divide A B)) (divide (divide B C) A)) C.eq Univ (multiply a b) (multiply b a)
+.
+#Univ.
+#A.
+#B.
+#C.
+#a.
+#b.
+#divide.
+#identity.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP549-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP549-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in division and identity, part 1 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 1 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : A UEQ part of GRP095-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_1:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a1:Univ.
+∀b1:Univ.
+∀divide:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.eq Univ identity (divide A A).
+∀H1:∀A:Univ.eq Univ (inverse A) (divide identity A).
+∀H2:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (divide A (divide identity B)).
+∀H3:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (divide (divide identity A) (divide (divide (divide B A) C) B)) C.eq Univ (multiply (inverse a1) a1) (multiply (inverse b1) b1)
+.
+#Univ.
+#A.
+#B.
+#C.
+#a1.
+#b1.
+#divide.
+#identity.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP550-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP550-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in division and identity, part 2 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.07 v3.1.0, 0.11 v2.7.0, 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 1 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : A UEQ part of GRP095-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_2:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a2:Univ.
+∀b2:Univ.
+∀divide:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.eq Univ identity (divide A A).
+∀H1:∀A:Univ.eq Univ (inverse A) (divide identity A).
+∀H2:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (divide A (divide identity B)).
+∀H3:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (divide (divide identity A) (divide (divide (divide B A) C) B)) C.eq Univ (multiply (multiply (inverse b2) b2) a2) a2
+.
+#Univ.
+#A.
+#B.
+#C.
+#a2.
+#b2.
+#divide.
+#identity.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP551-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP551-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in division and identity, part 3 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 1 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : A UEQ part of GRP095-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_3:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a3:Univ.
+∀b3:Univ.
+∀c3:Univ.
+∀divide:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.eq Univ identity (divide A A).
+∀H1:∀A:Univ.eq Univ (inverse A) (divide identity A).
+∀H2:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (divide A (divide identity B)).
+∀H3:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (divide (divide identity A) (divide (divide (divide B A) C) B)) C.eq Univ (multiply (multiply a3 b3) c3) (multiply a3 (multiply b3 c3))
+.
+#Univ.
+#A.
+#B.
+#C.
+#a3.
+#b3.
+#c3.
+#divide.
+#identity.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP552-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP552-1 : TPTP v3.2.0. Bugfixed v2.7.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in division and identity, part 4 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.7.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 1 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : A UEQ part of GRP095-1 *)
+
+(* Bugfixes : v2.7.0 - Grounded conjecture *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_4:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a:Univ.
+∀b:Univ.
+∀divide:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.eq Univ identity (divide A A).
+∀H1:∀A:Univ.eq Univ (inverse A) (divide identity A).
+∀H2:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (divide A (divide identity B)).
+∀H3:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (divide (divide identity A) (divide (divide (divide B A) C) B)) C.eq Univ (multiply a b) (multiply b a)
+.
+#Univ.
+#A.
+#B.
+#C.
+#a.
+#b.
+#divide.
+#identity.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP553-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP553-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in division and inverse, part 1 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP096-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_1:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a1:Univ.
+∀b1:Univ.
+∀divide:∀_:Univ.∀_:Univ.Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (divide A (inverse B)).
+∀H1:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (divide (divide A (inverse (divide B (divide A C)))) C) B.eq Univ (multiply (inverse a1) a1) (multiply (inverse b1) b1)
+.
+#Univ.
+#A.
+#B.
+#C.
+#a1.
+#b1.
+#divide.
+#inverse.
+#multiply.
+#H0.
+#H1.
+nauto by H0,H1;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP554-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP554-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in division and inverse, part 2 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP096-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_2:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a2:Univ.
+∀b2:Univ.
+∀divide:∀_:Univ.∀_:Univ.Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (divide A (inverse B)).
+∀H1:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (divide (divide A (inverse (divide B (divide A C)))) C) B.eq Univ (multiply (multiply (inverse b2) b2) a2) a2
+.
+#Univ.
+#A.
+#B.
+#C.
+#a2.
+#b2.
+#divide.
+#inverse.
+#multiply.
+#H0.
+#H1.
+nauto by H0,H1;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP555-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP555-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in division and inverse, part 3 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP096-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_3:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a3:Univ.
+∀b3:Univ.
+∀c3:Univ.
+∀divide:∀_:Univ.∀_:Univ.Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (divide A (inverse B)).
+∀H1:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (divide (divide A (inverse (divide B (divide A C)))) C) B.eq Univ (multiply (multiply a3 b3) c3) (multiply a3 (multiply b3 c3))
+.
+#Univ.
+#A.
+#B.
+#C.
+#a3.
+#b3.
+#c3.
+#divide.
+#inverse.
+#multiply.
+#H0.
+#H1.
+nauto by H0,H1;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP556-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP556-1 : TPTP v3.2.0. Bugfixed v2.7.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in division and inverse, part 4 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.7.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP096-1 *)
+
+(* Bugfixes : v2.7.0 - Grounded conjecture *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_4:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a:Univ.
+∀b:Univ.
+∀divide:∀_:Univ.∀_:Univ.Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (divide A (inverse B)).
+∀H1:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (divide (divide A (inverse (divide B (divide A C)))) C) B.eq Univ (multiply a b) (multiply b a)
+.
+#Univ.
+#A.
+#B.
+#C.
+#a.
+#b.
+#divide.
+#inverse.
+#multiply.
+#H0.
+#H1.
+nauto by H0,H1;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP557-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP557-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in division and inverse, part 1 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 5 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP097-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_1:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a1:Univ.
+∀b1:Univ.
+∀divide:∀_:Univ.∀_:Univ.Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (divide A (inverse B)).
+∀H1:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (divide A (inverse (divide (divide B C) (divide A C)))) B.eq Univ (multiply (inverse a1) a1) (multiply (inverse b1) b1)
+.
+#Univ.
+#A.
+#B.
+#C.
+#a1.
+#b1.
+#divide.
+#inverse.
+#multiply.
+#H0.
+#H1.
+nauto by H0,H1;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP558-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP558-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in division and inverse, part 2 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 5 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP097-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_2:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a2:Univ.
+∀b2:Univ.
+∀divide:∀_:Univ.∀_:Univ.Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (divide A (inverse B)).
+∀H1:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (divide A (inverse (divide (divide B C) (divide A C)))) B.eq Univ (multiply (multiply (inverse b2) b2) a2) a2
+.
+#Univ.
+#A.
+#B.
+#C.
+#a2.
+#b2.
+#divide.
+#inverse.
+#multiply.
+#H0.
+#H1.
+nauto by H0,H1;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP559-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP559-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in division and inverse, part 3 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 5 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP097-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_3:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a3:Univ.
+∀b3:Univ.
+∀c3:Univ.
+∀divide:∀_:Univ.∀_:Univ.Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (divide A (inverse B)).
+∀H1:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (divide A (inverse (divide (divide B C) (divide A C)))) B.eq Univ (multiply (multiply a3 b3) c3) (multiply a3 (multiply b3 c3))
+.
+#Univ.
+#A.
+#B.
+#C.
+#a3.
+#b3.
+#c3.
+#divide.
+#inverse.
+#multiply.
+#H0.
+#H1.
+nauto by H0,H1;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP560-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP560-1 : TPTP v3.2.0. Bugfixed v2.7.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in division and inverse, part 4 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.7.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : A UEQ part of GRP097-1 *)
+
+(* Bugfixes : v2.7.0 - Grounded conjecture *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_4:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a:Univ.
+∀b:Univ.
+∀divide:∀_:Univ.∀_:Univ.Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (divide A (inverse B)).
+∀H1:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (divide A (inverse (divide (divide B C) (divide A C)))) B.eq Univ (multiply a b) (multiply b a)
+.
+#Univ.
+#A.
+#B.
+#C.
+#a.
+#b.
+#divide.
+#inverse.
+#multiply.
+#H0.
+#H1.
+nauto by H0,H1;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP561-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP561-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in division and inverse, part 1 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.7.0, 0.09 v2.6.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 5 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP098-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_1:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a1:Univ.
+∀b1:Univ.
+∀divide:∀_:Univ.∀_:Univ.Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (divide A (inverse B)).
+∀H1:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (divide (divide (divide A (inverse B)) C) (divide A C)) B.eq Univ (multiply (inverse a1) a1) (multiply (inverse b1) b1)
+.
+#Univ.
+#A.
+#B.
+#C.
+#a1.
+#b1.
+#divide.
+#inverse.
+#multiply.
+#H0.
+#H1.
+nauto by H0,H1;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP562-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP562-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in division and inverse, part 2 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 5 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP098-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_2:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a2:Univ.
+∀b2:Univ.
+∀divide:∀_:Univ.∀_:Univ.Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (divide A (inverse B)).
+∀H1:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (divide (divide (divide A (inverse B)) C) (divide A C)) B.eq Univ (multiply (multiply (inverse b2) b2) a2) a2
+.
+#Univ.
+#A.
+#B.
+#C.
+#a2.
+#b2.
+#divide.
+#inverse.
+#multiply.
+#H0.
+#H1.
+nauto by H0,H1;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP563-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP563-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in division and inverse, part 3 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 5 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP098-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_3:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a3:Univ.
+∀b3:Univ.
+∀c3:Univ.
+∀divide:∀_:Univ.∀_:Univ.Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (divide A (inverse B)).
+∀H1:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (divide (divide (divide A (inverse B)) C) (divide A C)) B.eq Univ (multiply (multiply a3 b3) c3) (multiply a3 (multiply b3 c3))
+.
+#Univ.
+#A.
+#B.
+#C.
+#a3.
+#b3.
+#c3.
+#divide.
+#inverse.
+#multiply.
+#H0.
+#H1.
+nauto by H0,H1;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP564-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP564-1 : TPTP v3.2.0. Bugfixed v2.7.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in division and inverse, part 4 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.7.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : A UEQ part of GRP098-1 *)
+
+(* Bugfixes : v2.7.0 - Grounded conjecture *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_4:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a:Univ.
+∀b:Univ.
+∀divide:∀_:Univ.∀_:Univ.Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (divide A (inverse B)).
+∀H1:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (divide (divide (divide A (inverse B)) C) (divide A C)) B.eq Univ (multiply a b) (multiply b a)
+.
+#Univ.
+#A.
+#B.
+#C.
+#a.
+#b.
+#divide.
+#inverse.
+#multiply.
+#H0.
+#H1.
+nauto by H0,H1;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP565-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP565-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in double div and id, part 1 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 1 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 2 average) *)
+
+(* Comments : A UEQ part of GRP099-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_1:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a1:Univ.
+∀double_divide:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.eq Univ identity (double_divide A (inverse A)).
+∀H1:∀A:Univ.eq Univ (inverse A) (double_divide A identity).
+∀H2:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (double_divide (double_divide B A) identity).
+∀H3:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (double_divide (double_divide A (double_divide (double_divide B (double_divide A C)) (double_divide identity C))) (double_divide identity identity)) B.eq Univ (multiply (inverse a1) a1) identity
+.
+#Univ.
+#A.
+#B.
+#C.
+#a1.
+#double_divide.
+#identity.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP566-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP566-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in double div and id, part 2 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 1 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 2 average) *)
+
+(* Comments : A UEQ part of GRP099-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_2:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a2:Univ.
+∀double_divide:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.eq Univ identity (double_divide A (inverse A)).
+∀H1:∀A:Univ.eq Univ (inverse A) (double_divide A identity).
+∀H2:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (double_divide (double_divide B A) identity).
+∀H3:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (double_divide (double_divide A (double_divide (double_divide B (double_divide A C)) (double_divide identity C))) (double_divide identity identity)) B.eq Univ (multiply identity a2) a2
+.
+#Univ.
+#A.
+#B.
+#C.
+#a2.
+#double_divide.
+#identity.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP567-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP567-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in double div and id, part 3 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.14 v3.1.0, 0.11 v2.7.0, 0.09 v2.6.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 1 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP099-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_3:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a3:Univ.
+∀b3:Univ.
+∀c3:Univ.
+∀double_divide:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.eq Univ identity (double_divide A (inverse A)).
+∀H1:∀A:Univ.eq Univ (inverse A) (double_divide A identity).
+∀H2:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (double_divide (double_divide B A) identity).
+∀H3:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (double_divide (double_divide A (double_divide (double_divide B (double_divide A C)) (double_divide identity C))) (double_divide identity identity)) B.eq Univ (multiply (multiply a3 b3) c3) (multiply a3 (multiply b3 c3))
+.
+#Univ.
+#A.
+#B.
+#C.
+#a3.
+#b3.
+#c3.
+#double_divide.
+#identity.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP568-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP568-1 : TPTP v3.2.0. Bugfixed v2.7.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in double div and id, part 4 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.14 v3.1.0, 0.11 v2.7.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 1 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 2 average) *)
+
+(* Comments : A UEQ part of GRP099-1 *)
+
+(* Bugfixes : v2.7.0 - Grounded conjecture *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_4:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a:Univ.
+∀b:Univ.
+∀double_divide:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.eq Univ identity (double_divide A (inverse A)).
+∀H1:∀A:Univ.eq Univ (inverse A) (double_divide A identity).
+∀H2:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (double_divide (double_divide B A) identity).
+∀H3:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (double_divide (double_divide A (double_divide (double_divide B (double_divide A C)) (double_divide identity C))) (double_divide identity identity)) B.eq Univ (multiply a b) (multiply b a)
+.
+#Univ.
+#A.
+#B.
+#C.
+#a.
+#b.
+#double_divide.
+#identity.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP569-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP569-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in double div and id, part 1 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 1 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 2 average) *)
+
+(* Comments : A UEQ part of GRP100-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_1:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a1:Univ.
+∀double_divide:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.eq Univ identity (double_divide A (inverse A)).
+∀H1:∀A:Univ.eq Univ (inverse A) (double_divide A identity).
+∀H2:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (double_divide (double_divide B A) identity).
+∀H3:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (double_divide (double_divide A (double_divide (double_divide B (double_divide A C)) (double_divide C identity))) (double_divide identity identity)) B.eq Univ (multiply (inverse a1) a1) identity
+.
+#Univ.
+#A.
+#B.
+#C.
+#a1.
+#double_divide.
+#identity.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP570-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP570-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in double div and id, part 2 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 1 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 2 average) *)
+
+(* Comments : A UEQ part of GRP100-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_2:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a2:Univ.
+∀double_divide:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.eq Univ identity (double_divide A (inverse A)).
+∀H1:∀A:Univ.eq Univ (inverse A) (double_divide A identity).
+∀H2:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (double_divide (double_divide B A) identity).
+∀H3:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (double_divide (double_divide A (double_divide (double_divide B (double_divide A C)) (double_divide C identity))) (double_divide identity identity)) B.eq Univ (multiply identity a2) a2
+.
+#Univ.
+#A.
+#B.
+#C.
+#a2.
+#double_divide.
+#identity.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP571-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP571-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in double div and id, part 3 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.14 v3.1.0, 0.11 v2.7.0, 0.09 v2.6.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 1 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP100-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_3:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a3:Univ.
+∀b3:Univ.
+∀c3:Univ.
+∀double_divide:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.eq Univ identity (double_divide A (inverse A)).
+∀H1:∀A:Univ.eq Univ (inverse A) (double_divide A identity).
+∀H2:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (double_divide (double_divide B A) identity).
+∀H3:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (double_divide (double_divide A (double_divide (double_divide B (double_divide A C)) (double_divide C identity))) (double_divide identity identity)) B.eq Univ (multiply (multiply a3 b3) c3) (multiply a3 (multiply b3 c3))
+.
+#Univ.
+#A.
+#B.
+#C.
+#a3.
+#b3.
+#c3.
+#double_divide.
+#identity.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP572-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP572-1 : TPTP v3.2.0. Bugfixed v2.7.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in double div and id, part 4 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.07 v3.1.0, 0.11 v2.7.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 1 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 2 average) *)
+
+(* Comments : A UEQ part of GRP100-1 *)
+
+(* Bugfixes : v2.7.0 - Grounded conjecture *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_4:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a:Univ.
+∀b:Univ.
+∀double_divide:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.eq Univ identity (double_divide A (inverse A)).
+∀H1:∀A:Univ.eq Univ (inverse A) (double_divide A identity).
+∀H2:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (double_divide (double_divide B A) identity).
+∀H3:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (double_divide (double_divide A (double_divide (double_divide B (double_divide A C)) (double_divide C identity))) (double_divide identity identity)) B.eq Univ (multiply a b) (multiply b a)
+.
+#Univ.
+#A.
+#B.
+#C.
+#a.
+#b.
+#double_divide.
+#identity.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP573-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP573-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in double div and id, part 1 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 1 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 2 average) *)
+
+(* Comments : A UEQ part of GRP101-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_1:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a1:Univ.
+∀double_divide:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.eq Univ identity (double_divide A (inverse A)).
+∀H1:∀A:Univ.eq Univ (inverse A) (double_divide A identity).
+∀H2:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (double_divide (double_divide B A) identity).
+∀H3:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (double_divide (double_divide A (double_divide (double_divide B (double_divide C A)) (double_divide C identity))) (double_divide identity identity)) B.eq Univ (multiply (inverse a1) a1) identity
+.
+#Univ.
+#A.
+#B.
+#C.
+#a1.
+#double_divide.
+#identity.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP574-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP574-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in double div and id, part 2 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 1 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 2 average) *)
+
+(* Comments : A UEQ part of GRP101-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_2:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a2:Univ.
+∀double_divide:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.eq Univ identity (double_divide A (inverse A)).
+∀H1:∀A:Univ.eq Univ (inverse A) (double_divide A identity).
+∀H2:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (double_divide (double_divide B A) identity).
+∀H3:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (double_divide (double_divide A (double_divide (double_divide B (double_divide C A)) (double_divide C identity))) (double_divide identity identity)) B.eq Univ (multiply identity a2) a2
+.
+#Univ.
+#A.
+#B.
+#C.
+#a2.
+#double_divide.
+#identity.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP575-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP575-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in double div and id, part 3 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.14 v3.1.0, 0.11 v2.7.0, 0.09 v2.6.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 1 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP101-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_3:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a3:Univ.
+∀b3:Univ.
+∀c3:Univ.
+∀double_divide:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.eq Univ identity (double_divide A (inverse A)).
+∀H1:∀A:Univ.eq Univ (inverse A) (double_divide A identity).
+∀H2:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (double_divide (double_divide B A) identity).
+∀H3:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (double_divide (double_divide A (double_divide (double_divide B (double_divide C A)) (double_divide C identity))) (double_divide identity identity)) B.eq Univ (multiply (multiply a3 b3) c3) (multiply a3 (multiply b3 c3))
+.
+#Univ.
+#A.
+#B.
+#C.
+#a3.
+#b3.
+#c3.
+#double_divide.
+#identity.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP576-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP576-1 : TPTP v3.2.0. Bugfixed v2.7.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in double div and id, part 4 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.7.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 1 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 2 average) *)
+
+(* Comments : A UEQ part of GRP101-1 *)
+
+(* Bugfixes : v2.7.0 - Grounded conjecture *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_4:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a:Univ.
+∀b:Univ.
+∀double_divide:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.eq Univ identity (double_divide A (inverse A)).
+∀H1:∀A:Univ.eq Univ (inverse A) (double_divide A identity).
+∀H2:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (double_divide (double_divide B A) identity).
+∀H3:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (double_divide (double_divide A (double_divide (double_divide B (double_divide C A)) (double_divide C identity))) (double_divide identity identity)) B.eq Univ (multiply a b) (multiply b a)
+.
+#Univ.
+#A.
+#B.
+#C.
+#a.
+#b.
+#double_divide.
+#identity.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP577-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP577-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in double div and id, part 1 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 1 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 2 average) *)
+
+(* Comments : A UEQ part of GRP102-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_1:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a1:Univ.
+∀double_divide:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.eq Univ identity (double_divide A (inverse A)).
+∀H1:∀A:Univ.eq Univ (inverse A) (double_divide A identity).
+∀H2:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (double_divide (double_divide B A) identity).
+∀H3:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (double_divide (double_divide A (double_divide (double_divide (double_divide B A) C) (double_divide B identity))) (double_divide identity identity)) C.eq Univ (multiply (inverse a1) a1) identity
+.
+#Univ.
+#A.
+#B.
+#C.
+#a1.
+#double_divide.
+#identity.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP578-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP578-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in double div and id, part 2 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 1 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 2 average) *)
+
+(* Comments : A UEQ part of GRP102-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_2:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a2:Univ.
+∀double_divide:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.eq Univ identity (double_divide A (inverse A)).
+∀H1:∀A:Univ.eq Univ (inverse A) (double_divide A identity).
+∀H2:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (double_divide (double_divide B A) identity).
+∀H3:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (double_divide (double_divide A (double_divide (double_divide (double_divide B A) C) (double_divide B identity))) (double_divide identity identity)) C.eq Univ (multiply identity a2) a2
+.
+#Univ.
+#A.
+#B.
+#C.
+#a2.
+#double_divide.
+#identity.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP579-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP579-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in double div and id, part 3 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.07 v3.1.0, 0.11 v2.7.0, 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 1 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP102-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_3:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a3:Univ.
+∀b3:Univ.
+∀c3:Univ.
+∀double_divide:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.eq Univ identity (double_divide A (inverse A)).
+∀H1:∀A:Univ.eq Univ (inverse A) (double_divide A identity).
+∀H2:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (double_divide (double_divide B A) identity).
+∀H3:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (double_divide (double_divide A (double_divide (double_divide (double_divide B A) C) (double_divide B identity))) (double_divide identity identity)) C.eq Univ (multiply (multiply a3 b3) c3) (multiply a3 (multiply b3 c3))
+.
+#Univ.
+#A.
+#B.
+#C.
+#a3.
+#b3.
+#c3.
+#double_divide.
+#identity.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP580-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP580-1 : TPTP v3.2.0. Bugfixed v2.7.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in double div and id, part 4 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.7.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 1 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 2 average) *)
+
+(* Comments : A UEQ part of GRP102-1 *)
+
+(* Bugfixes : v2.7.0 - Grounded conjecture *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_4:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a:Univ.
+∀b:Univ.
+∀double_divide:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.eq Univ identity (double_divide A (inverse A)).
+∀H1:∀A:Univ.eq Univ (inverse A) (double_divide A identity).
+∀H2:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (double_divide (double_divide B A) identity).
+∀H3:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (double_divide (double_divide A (double_divide (double_divide (double_divide B A) C) (double_divide B identity))) (double_divide identity identity)) C.eq Univ (multiply a b) (multiply b a)
+.
+#Univ.
+#A.
+#B.
+#C.
+#a.
+#b.
+#double_divide.
+#identity.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP581-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP581-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in double div and id, part 1 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 1 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 2 average) *)
+
+(* Comments : A UEQ part of GRP103-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_1:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a1:Univ.
+∀double_divide:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.eq Univ identity (double_divide A (inverse A)).
+∀H1:∀A:Univ.eq Univ (inverse A) (double_divide A identity).
+∀H2:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (double_divide (double_divide B A) identity).
+∀H3:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (double_divide (double_divide A (double_divide (double_divide identity B) (double_divide C (double_divide B A)))) (double_divide identity identity)) C.eq Univ (multiply (inverse a1) a1) identity
+.
+#Univ.
+#A.
+#B.
+#C.
+#a1.
+#double_divide.
+#identity.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP582-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP582-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in double div and id, part 2 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 1 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 2 average) *)
+
+(* Comments : A UEQ part of GRP103-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_2:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a2:Univ.
+∀double_divide:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.eq Univ identity (double_divide A (inverse A)).
+∀H1:∀A:Univ.eq Univ (inverse A) (double_divide A identity).
+∀H2:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (double_divide (double_divide B A) identity).
+∀H3:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (double_divide (double_divide A (double_divide (double_divide identity B) (double_divide C (double_divide B A)))) (double_divide identity identity)) C.eq Univ (multiply identity a2) a2
+.
+#Univ.
+#A.
+#B.
+#C.
+#a2.
+#double_divide.
+#identity.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP583-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP583-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in double div and id, part 3 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.07 v3.1.0, 0.11 v2.7.0, 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 1 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP103-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_3:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a3:Univ.
+∀b3:Univ.
+∀c3:Univ.
+∀double_divide:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.eq Univ identity (double_divide A (inverse A)).
+∀H1:∀A:Univ.eq Univ (inverse A) (double_divide A identity).
+∀H2:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (double_divide (double_divide B A) identity).
+∀H3:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (double_divide (double_divide A (double_divide (double_divide identity B) (double_divide C (double_divide B A)))) (double_divide identity identity)) C.eq Univ (multiply (multiply a3 b3) c3) (multiply a3 (multiply b3 c3))
+.
+#Univ.
+#A.
+#B.
+#C.
+#a3.
+#b3.
+#c3.
+#double_divide.
+#identity.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP584-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP584-1 : TPTP v3.2.0. Bugfixed v2.7.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in double div and id, part 4 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.7.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 1 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 2 average) *)
+
+(* Comments : A UEQ part of GRP103-1 *)
+
+(* Bugfixes : v2.7.0 - Grounded conjecture *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_4:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a:Univ.
+∀b:Univ.
+∀double_divide:∀_:Univ.∀_:Univ.Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.eq Univ identity (double_divide A (inverse A)).
+∀H1:∀A:Univ.eq Univ (inverse A) (double_divide A identity).
+∀H2:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (double_divide (double_divide B A) identity).
+∀H3:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (double_divide (double_divide A (double_divide (double_divide identity B) (double_divide C (double_divide B A)))) (double_divide identity identity)) C.eq Univ (multiply a b) (multiply b a)
+.
+#Univ.
+#A.
+#B.
+#C.
+#a.
+#b.
+#double_divide.
+#identity.
+#inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP585-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP585-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in double div and inv, part 1 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 7 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP104-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_1:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a1:Univ.
+∀b1:Univ.
+∀double_divide:∀_:Univ.∀_:Univ.Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (inverse (double_divide B A)).
+∀H1:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (double_divide A (inverse (double_divide (inverse (double_divide (double_divide A B) (inverse C))) B))) C.eq Univ (multiply (inverse a1) a1) (multiply (inverse b1) b1)
+.
+#Univ.
+#A.
+#B.
+#C.
+#a1.
+#b1.
+#double_divide.
+#inverse.
+#multiply.
+#H0.
+#H1.
+nauto by H0,H1;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP586-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP586-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in double div and inv, part 2 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 7 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP104-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_2:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a2:Univ.
+∀b2:Univ.
+∀double_divide:∀_:Univ.∀_:Univ.Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (inverse (double_divide B A)).
+∀H1:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (double_divide A (inverse (double_divide (inverse (double_divide (double_divide A B) (inverse C))) B))) C.eq Univ (multiply (multiply (inverse b2) b2) a2) a2
+.
+#Univ.
+#A.
+#B.
+#C.
+#a2.
+#b2.
+#double_divide.
+#inverse.
+#multiply.
+#H0.
+#H1.
+nauto by H0,H1;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP587-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP587-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in double div and inv, part 3 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.07 v3.2.0, 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 7 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP104-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_3:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a3:Univ.
+∀b3:Univ.
+∀c3:Univ.
+∀double_divide:∀_:Univ.∀_:Univ.Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (inverse (double_divide B A)).
+∀H1:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (double_divide A (inverse (double_divide (inverse (double_divide (double_divide A B) (inverse C))) B))) C.eq Univ (multiply (multiply a3 b3) c3) (multiply a3 (multiply b3 c3))
+.
+#Univ.
+#A.
+#B.
+#C.
+#a3.
+#b3.
+#c3.
+#double_divide.
+#inverse.
+#multiply.
+#H0.
+#H1.
+nauto by H0,H1;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP588-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP588-1 : TPTP v3.2.0. Bugfixed v2.7.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in double div and inv, part 4 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.7.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 7 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP104-1 *)
+
+(* Bugfixes : v2.7.0 - Grounded conjecture *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_4:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a:Univ.
+∀b:Univ.
+∀double_divide:∀_:Univ.∀_:Univ.Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (inverse (double_divide B A)).
+∀H1:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (double_divide A (inverse (double_divide (inverse (double_divide (double_divide A B) (inverse C))) B))) C.eq Univ (multiply a b) (multiply b a)
+.
+#Univ.
+#A.
+#B.
+#C.
+#a.
+#b.
+#double_divide.
+#inverse.
+#multiply.
+#H0.
+#H1.
+nauto by H0,H1;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP589-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP589-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in double div and inv, part 1 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 7 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP105-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_1:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a1:Univ.
+∀b1:Univ.
+∀double_divide:∀_:Univ.∀_:Univ.Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (inverse (double_divide B A)).
+∀H1:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (double_divide (inverse (double_divide (double_divide A B) (inverse (double_divide A (inverse C))))) B) C.eq Univ (multiply (inverse a1) a1) (multiply (inverse b1) b1)
+.
+#Univ.
+#A.
+#B.
+#C.
+#a1.
+#b1.
+#double_divide.
+#inverse.
+#multiply.
+#H0.
+#H1.
+nauto by H0,H1;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP590-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP590-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in double div and inv, part 2 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 7 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP105-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_2:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a2:Univ.
+∀b2:Univ.
+∀double_divide:∀_:Univ.∀_:Univ.Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (inverse (double_divide B A)).
+∀H1:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (double_divide (inverse (double_divide (double_divide A B) (inverse (double_divide A (inverse C))))) B) C.eq Univ (multiply (multiply (inverse b2) b2) a2) a2
+.
+#Univ.
+#A.
+#B.
+#C.
+#a2.
+#b2.
+#double_divide.
+#inverse.
+#multiply.
+#H0.
+#H1.
+nauto by H0,H1;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP591-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP591-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in double div and inv, part 3 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.14 v3.2.0, 0.07 v3.1.0, 0.11 v2.7.0, 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 7 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP105-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_3:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a3:Univ.
+∀b3:Univ.
+∀c3:Univ.
+∀double_divide:∀_:Univ.∀_:Univ.Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (inverse (double_divide B A)).
+∀H1:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (double_divide (inverse (double_divide (double_divide A B) (inverse (double_divide A (inverse C))))) B) C.eq Univ (multiply (multiply a3 b3) c3) (multiply a3 (multiply b3 c3))
+.
+#Univ.
+#A.
+#B.
+#C.
+#a3.
+#b3.
+#c3.
+#double_divide.
+#inverse.
+#multiply.
+#H0.
+#H1.
+nauto by H0,H1;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP592-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP592-1 : TPTP v3.2.0. Bugfixed v2.7.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in double div and inv, part 4 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.07 v3.1.0, 0.11 v2.7.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 7 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP105-1 *)
+
+(* Bugfixes : v2.7.0 - Grounded conjecture *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_4:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a:Univ.
+∀b:Univ.
+∀double_divide:∀_:Univ.∀_:Univ.Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (inverse (double_divide B A)).
+∀H1:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (double_divide (inverse (double_divide (double_divide A B) (inverse (double_divide A (inverse C))))) B) C.eq Univ (multiply a b) (multiply b a)
+.
+#Univ.
+#A.
+#B.
+#C.
+#a.
+#b.
+#double_divide.
+#inverse.
+#multiply.
+#H0.
+#H1.
+nauto by H0,H1;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP593-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP593-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in double div and inv, part 1 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 7 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP106-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_1:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a1:Univ.
+∀b1:Univ.
+∀double_divide:∀_:Univ.∀_:Univ.Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (inverse (double_divide B A)).
+∀H1:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (inverse (double_divide (double_divide A B) (inverse (double_divide A (inverse (double_divide C B)))))) C.eq Univ (multiply (inverse a1) a1) (multiply (inverse b1) b1)
+.
+#Univ.
+#A.
+#B.
+#C.
+#a1.
+#b1.
+#double_divide.
+#inverse.
+#multiply.
+#H0.
+#H1.
+nauto by H0,H1;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP594-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP594-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in double div and inv, part 2 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 7 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP106-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_2:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a2:Univ.
+∀b2:Univ.
+∀double_divide:∀_:Univ.∀_:Univ.Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (inverse (double_divide B A)).
+∀H1:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (inverse (double_divide (double_divide A B) (inverse (double_divide A (inverse (double_divide C B)))))) C.eq Univ (multiply (multiply (inverse b2) b2) a2) a2
+.
+#Univ.
+#A.
+#B.
+#C.
+#a2.
+#b2.
+#double_divide.
+#inverse.
+#multiply.
+#H0.
+#H1.
+nauto by H0,H1;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP595-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP595-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in double div and inv, part 3 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.07 v3.1.0, 0.11 v2.7.0, 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 7 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP106-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_3:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a3:Univ.
+∀b3:Univ.
+∀c3:Univ.
+∀double_divide:∀_:Univ.∀_:Univ.Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (inverse (double_divide B A)).
+∀H1:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (inverse (double_divide (double_divide A B) (inverse (double_divide A (inverse (double_divide C B)))))) C.eq Univ (multiply (multiply a3 b3) c3) (multiply a3 (multiply b3 c3))
+.
+#Univ.
+#A.
+#B.
+#C.
+#a3.
+#b3.
+#c3.
+#double_divide.
+#inverse.
+#multiply.
+#H0.
+#H1.
+nauto by H0,H1;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP596-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP596-1 : TPTP v3.2.0. Bugfixed v2.7.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in double div and inv, part 4 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.07 v3.1.0, 0.11 v2.7.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 7 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP106-1 *)
+
+(* Bugfixes : v2.7.0 - Grounded conjecture *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_4:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a:Univ.
+∀b:Univ.
+∀double_divide:∀_:Univ.∀_:Univ.Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (inverse (double_divide B A)).
+∀H1:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (inverse (double_divide (double_divide A B) (inverse (double_divide A (inverse (double_divide C B)))))) C.eq Univ (multiply a b) (multiply b a)
+.
+#Univ.
+#A.
+#B.
+#C.
+#a.
+#b.
+#double_divide.
+#inverse.
+#multiply.
+#H0.
+#H1.
+nauto by H0,H1;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP597-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP597-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in double div and inv, part 1 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 7 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP107-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_1:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a1:Univ.
+∀b1:Univ.
+∀double_divide:∀_:Univ.∀_:Univ.Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (inverse (double_divide B A)).
+∀H1:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (double_divide (double_divide A B) (inverse (double_divide A (inverse (double_divide (inverse C) B))))) C.eq Univ (multiply (inverse a1) a1) (multiply (inverse b1) b1)
+.
+#Univ.
+#A.
+#B.
+#C.
+#a1.
+#b1.
+#double_divide.
+#inverse.
+#multiply.
+#H0.
+#H1.
+nauto by H0,H1;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP598-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP598-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in double div and inv, part 2 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.07 v3.1.0, 0.11 v2.7.0, 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 7 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP107-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_2:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a2:Univ.
+∀b2:Univ.
+∀double_divide:∀_:Univ.∀_:Univ.Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (inverse (double_divide B A)).
+∀H1:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (double_divide (double_divide A B) (inverse (double_divide A (inverse (double_divide (inverse C) B))))) C.eq Univ (multiply (multiply (inverse b2) b2) a2) a2
+.
+#Univ.
+#A.
+#B.
+#C.
+#a2.
+#b2.
+#double_divide.
+#inverse.
+#multiply.
+#H0.
+#H1.
+nauto by H0,H1;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP599-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP599-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in double div and inv, part 3 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.14 v3.2.0, 0.07 v3.1.0, 0.11 v2.7.0, 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 7 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP107-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_3:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a3:Univ.
+∀b3:Univ.
+∀c3:Univ.
+∀double_divide:∀_:Univ.∀_:Univ.Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (inverse (double_divide B A)).
+∀H1:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (double_divide (double_divide A B) (inverse (double_divide A (inverse (double_divide (inverse C) B))))) C.eq Univ (multiply (multiply a3 b3) c3) (multiply a3 (multiply b3 c3))
+.
+#Univ.
+#A.
+#B.
+#C.
+#a3.
+#b3.
+#c3.
+#double_divide.
+#inverse.
+#multiply.
+#H0.
+#H1.
+nauto by H0,H1;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP600-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP600-1 : TPTP v3.2.0. Bugfixed v2.7.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in double div and inv, part 4 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.07 v3.1.0, 0.11 v2.7.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 7 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP107-1 *)
+
+(* Bugfixes : v2.7.0 - Grounded conjecture *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_4:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a:Univ.
+∀b:Univ.
+∀double_divide:∀_:Univ.∀_:Univ.Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (inverse (double_divide B A)).
+∀H1:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (double_divide (double_divide A B) (inverse (double_divide A (inverse (double_divide (inverse C) B))))) C.eq Univ (multiply a b) (multiply b a)
+.
+#Univ.
+#A.
+#B.
+#C.
+#a.
+#b.
+#double_divide.
+#inverse.
+#multiply.
+#H0.
+#H1.
+nauto by H0,H1;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP601-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP601-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in double div and inv, part 1 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.07 v3.1.0, 0.11 v2.7.0, 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 8 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP108-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_1:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a1:Univ.
+∀b1:Univ.
+∀double_divide:∀_:Univ.∀_:Univ.Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (inverse (double_divide B A)).
+∀H1:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (inverse (double_divide (inverse (double_divide A (inverse (double_divide B (double_divide A C))))) C)) B.eq Univ (multiply (inverse a1) a1) (multiply (inverse b1) b1)
+.
+#Univ.
+#A.
+#B.
+#C.
+#a1.
+#b1.
+#double_divide.
+#inverse.
+#multiply.
+#H0.
+#H1.
+nauto by H0,H1;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP602-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP602-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in double div and inv, part 2 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.07 v3.1.0, 0.11 v2.7.0, 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 8 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP108-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_2:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a2:Univ.
+∀b2:Univ.
+∀double_divide:∀_:Univ.∀_:Univ.Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (inverse (double_divide B A)).
+∀H1:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (inverse (double_divide (inverse (double_divide A (inverse (double_divide B (double_divide A C))))) C)) B.eq Univ (multiply (multiply (inverse b2) b2) a2) a2
+.
+#Univ.
+#A.
+#B.
+#C.
+#a2.
+#b2.
+#double_divide.
+#inverse.
+#multiply.
+#H0.
+#H1.
+nauto by H0,H1;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP603-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP603-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in double div and inv, part 3 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.07 v3.1.0, 0.11 v2.7.0, 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 8 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP108-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_3:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a3:Univ.
+∀b3:Univ.
+∀c3:Univ.
+∀double_divide:∀_:Univ.∀_:Univ.Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (inverse (double_divide B A)).
+∀H1:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (inverse (double_divide (inverse (double_divide A (inverse (double_divide B (double_divide A C))))) C)) B.eq Univ (multiply (multiply a3 b3) c3) (multiply a3 (multiply b3 c3))
+.
+#Univ.
+#A.
+#B.
+#C.
+#a3.
+#b3.
+#c3.
+#double_divide.
+#inverse.
+#multiply.
+#H0.
+#H1.
+nauto by H0,H1;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP604-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP604-1 : TPTP v3.2.0. Bugfixed v2.7.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in double div and inv, part 4 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.07 v3.1.0, 0.11 v2.7.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 8 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP108-1 *)
+
+(* Bugfixes : v2.7.0 - Grounded conjecture *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_4:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a:Univ.
+∀b:Univ.
+∀double_divide:∀_:Univ.∀_:Univ.Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (inverse (double_divide B A)).
+∀H1:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (inverse (double_divide (inverse (double_divide A (inverse (double_divide B (double_divide A C))))) C)) B.eq Univ (multiply a b) (multiply b a)
+.
+#Univ.
+#A.
+#B.
+#C.
+#a.
+#b.
+#double_divide.
+#inverse.
+#multiply.
+#H0.
+#H1.
+nauto by H0,H1;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP605-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP605-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in double div and inv, part 1 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 7 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP109-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_1:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a1:Univ.
+∀b1:Univ.
+∀double_divide:∀_:Univ.∀_:Univ.Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (inverse (double_divide B A)).
+∀H1:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (double_divide (inverse (double_divide A (inverse (double_divide (inverse B) (double_divide A C))))) C) B.eq Univ (multiply (inverse a1) a1) (multiply (inverse b1) b1)
+.
+#Univ.
+#A.
+#B.
+#C.
+#a1.
+#b1.
+#double_divide.
+#inverse.
+#multiply.
+#H0.
+#H1.
+nauto by H0,H1;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP606-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP606-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in double div and inv, part 2 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 7 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP109-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_2:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a2:Univ.
+∀b2:Univ.
+∀double_divide:∀_:Univ.∀_:Univ.Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (inverse (double_divide B A)).
+∀H1:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (double_divide (inverse (double_divide A (inverse (double_divide (inverse B) (double_divide A C))))) C) B.eq Univ (multiply (multiply (inverse b2) b2) a2) a2
+.
+#Univ.
+#A.
+#B.
+#C.
+#a2.
+#b2.
+#double_divide.
+#inverse.
+#multiply.
+#H0.
+#H1.
+nauto by H0,H1;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP607-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP607-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in double div and inv, part 3 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.07 v3.2.0, 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 7 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP109-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_3:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a3:Univ.
+∀b3:Univ.
+∀c3:Univ.
+∀double_divide:∀_:Univ.∀_:Univ.Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (inverse (double_divide B A)).
+∀H1:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (double_divide (inverse (double_divide A (inverse (double_divide (inverse B) (double_divide A C))))) C) B.eq Univ (multiply (multiply a3 b3) c3) (multiply a3 (multiply b3 c3))
+.
+#Univ.
+#A.
+#B.
+#C.
+#a3.
+#b3.
+#c3.
+#double_divide.
+#inverse.
+#multiply.
+#H0.
+#H1.
+nauto by H0,H1;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP608-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP608-1 : TPTP v3.2.0. Bugfixed v2.7.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in double div and inv, part 4 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.7.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 7 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP109-1 *)
+
+(* Bugfixes : v2.7.0 - Grounded conjecture *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_4:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a:Univ.
+∀b:Univ.
+∀double_divide:∀_:Univ.∀_:Univ.Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (inverse (double_divide B A)).
+∀H1:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (double_divide (inverse (double_divide A (inverse (double_divide (inverse B) (double_divide A C))))) C) B.eq Univ (multiply a b) (multiply b a)
+.
+#Univ.
+#A.
+#B.
+#C.
+#a.
+#b.
+#double_divide.
+#inverse.
+#multiply.
+#H0.
+#H1.
+nauto by H0,H1;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP609-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP609-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in double div and inv, part 1 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.07 v3.1.0, 0.11 v2.7.0, 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 7 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP110-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_1:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a1:Univ.
+∀b1:Univ.
+∀double_divide:∀_:Univ.∀_:Univ.Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (inverse (double_divide B A)).
+∀H1:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (inverse (double_divide (inverse (double_divide (inverse (double_divide A B)) C)) (double_divide A C))) B.eq Univ (multiply (inverse a1) a1) (multiply (inverse b1) b1)
+.
+#Univ.
+#A.
+#B.
+#C.
+#a1.
+#b1.
+#double_divide.
+#inverse.
+#multiply.
+#H0.
+#H1.
+nauto by H0,H1;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP610-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP610-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in double div and inv, part 2 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.07 v3.1.0, 0.11 v2.7.0, 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 7 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP110-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_2:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a2:Univ.
+∀b2:Univ.
+∀double_divide:∀_:Univ.∀_:Univ.Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (inverse (double_divide B A)).
+∀H1:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (inverse (double_divide (inverse (double_divide (inverse (double_divide A B)) C)) (double_divide A C))) B.eq Univ (multiply (multiply (inverse b2) b2) a2) a2
+.
+#Univ.
+#A.
+#B.
+#C.
+#a2.
+#b2.
+#double_divide.
+#inverse.
+#multiply.
+#H0.
+#H1.
+nauto by H0,H1;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP611-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP611-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in double div and inv, part 3 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 7 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP110-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_3:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a3:Univ.
+∀b3:Univ.
+∀c3:Univ.
+∀double_divide:∀_:Univ.∀_:Univ.Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (inverse (double_divide B A)).
+∀H1:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (inverse (double_divide (inverse (double_divide (inverse (double_divide A B)) C)) (double_divide A C))) B.eq Univ (multiply (multiply a3 b3) c3) (multiply a3 (multiply b3 c3))
+.
+#Univ.
+#A.
+#B.
+#C.
+#a3.
+#b3.
+#c3.
+#double_divide.
+#inverse.
+#multiply.
+#H0.
+#H1.
+nauto by H0,H1;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP612-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP612-1 : TPTP v3.2.0. Bugfixed v2.7.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in double div and inv, part 4 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.07 v3.1.0, 0.11 v2.7.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 7 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP110-1 *)
+
+(* Bugfixes : v2.7.0 - Grounded conjecture *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_4:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a:Univ.
+∀b:Univ.
+∀double_divide:∀_:Univ.∀_:Univ.Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (inverse (double_divide B A)).
+∀H1:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (inverse (double_divide (inverse (double_divide (inverse (double_divide A B)) C)) (double_divide A C))) B.eq Univ (multiply a b) (multiply b a)
+.
+#Univ.
+#A.
+#B.
+#C.
+#a.
+#b.
+#double_divide.
+#inverse.
+#multiply.
+#H0.
+#H1.
+nauto by H0,H1;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP613-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP613-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in double div and inv, part 1 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 7 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP111-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_1:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a1:Univ.
+∀b1:Univ.
+∀double_divide:∀_:Univ.∀_:Univ.Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (inverse (double_divide B A)).
+∀H1:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (double_divide (inverse (double_divide (inverse (double_divide A (inverse B))) C)) (double_divide A C)) B.eq Univ (multiply (inverse a1) a1) (multiply (inverse b1) b1)
+.
+#Univ.
+#A.
+#B.
+#C.
+#a1.
+#b1.
+#double_divide.
+#inverse.
+#multiply.
+#H0.
+#H1.
+nauto by H0,H1;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP614-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP614-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in double div and inv, part 2 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 7 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP111-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_2:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a2:Univ.
+∀b2:Univ.
+∀double_divide:∀_:Univ.∀_:Univ.Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (inverse (double_divide B A)).
+∀H1:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (double_divide (inverse (double_divide (inverse (double_divide A (inverse B))) C)) (double_divide A C)) B.eq Univ (multiply (multiply (inverse b2) b2) a2) a2
+.
+#Univ.
+#A.
+#B.
+#C.
+#a2.
+#b2.
+#double_divide.
+#inverse.
+#multiply.
+#H0.
+#H1.
+nauto by H0,H1;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP615-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP615-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in double div and inv, part 3 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.14 v3.2.0, 0.07 v3.1.0, 0.11 v2.7.0, 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 7 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP111-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_3:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a3:Univ.
+∀b3:Univ.
+∀c3:Univ.
+∀double_divide:∀_:Univ.∀_:Univ.Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (inverse (double_divide B A)).
+∀H1:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (double_divide (inverse (double_divide (inverse (double_divide A (inverse B))) C)) (double_divide A C)) B.eq Univ (multiply (multiply a3 b3) c3) (multiply a3 (multiply b3 c3))
+.
+#Univ.
+#A.
+#B.
+#C.
+#a3.
+#b3.
+#c3.
+#double_divide.
+#inverse.
+#multiply.
+#H0.
+#H1.
+nauto by H0,H1;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP616-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP616-1 : TPTP v3.2.0. Bugfixed v2.7.0. *)
+
+(* Domain : Group Theory (Abelian) *)
+
+(* Problem : Axiom for Abelian group theory, in double div and inv, part 4 *)
+
+(* Version : [McC93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.7.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 0 singleton) *)
+
+(* Maximal term depth : 7 ( 3 average) *)
+
+(* Comments : A UEQ part of GRP111-1 *)
+
+(* Bugfixes : v2.7.0 - Grounded conjecture *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_these_axioms_4:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a:Univ.
+∀b:Univ.
+∀double_divide:∀_:Univ.∀_:Univ.Univ.
+∀inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.eq Univ (multiply A B) (inverse (double_divide B A)).
+∀H1:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (double_divide (inverse (double_divide (inverse (double_divide A (inverse B))) C)) (double_divide A C)) B.eq Univ (multiply a b) (multiply b a)
+.
+#Univ.
+#A.
+#B.
+#C.
+#a.
+#b.
+#double_divide.
+#inverse.
+#multiply.
+#H0.
+#H1.
+nauto by H0,H1;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: HWC004-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : HWC004-1 : TPTP v3.2.0. Released v2.5.0. *)
+
+(* Domain : Hardware Creation *)
+
+(* Problem : Definitions of AND, OR and NOT *)
+
+(* Version : [WO+92] axioms. *)
+
+(* Axiom formulation : Ground axioms. *)
+
+(* English : *)
+
+(* Refs : [WO+92] Wos et al. (1992), Automated Reasoning: Introduction a *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : Satisfiable *)
+
+(* Rating : 0.33 v3.2.0, 0.17 v3.1.0, 0.00 v2.5.0 *)
+
+(* Syntax : Number of clauses : 10 ( 0 non-Horn; 10 unit; 10 RR) *)
+
+(* Number of atoms : 10 ( 10 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 0 ( 0 singleton) *)
+
+(* Maximal term depth : 2 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Definitions of AND, OR and NOT *)
+
+(* Inclusion of: Axioms/HWC001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : HWC001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Hardware Creation *)
+
+(* Axioms : Definitions of AND, OR and NOT *)
+
+(* Version : [WO+92] axioms. *)
+
+(* Axiom formulation : Ground axioms. *)
+
+(* English : *)
+
+(* Refs : [WO+92] Wos et al. (1992), Automated Reasoning: Introduction a *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 10 ( 0 non-Horn; 10 unit; 10 RR) *)
+
+(* Number of literals : 10 ( 10 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 0 ( 0 singleton) *)
+
+(* Maximal term depth : 2 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: HWC004-2.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : HWC004-2 : TPTP v3.2.0. Released v2.5.0. *)
+
+(* Domain : Hardware Creation *)
+
+(* Problem : Definitions of AND, OR and NOT *)
+
+(* Version : [WO+92] axioms. *)
+
+(* Axiom formulation : Non-ground axioms. *)
+
+(* English : *)
+
+(* Refs : [WO+92] Wos et al. (1992), Automated Reasoning: Introduction a *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : Satisfiable *)
+
+(* Rating : 0.00 v3.2.0, 0.33 v3.1.0, 0.00 v2.5.0 *)
+
+(* Syntax : Number of clauses : 6 ( 0 non-Horn; 6 unit; 2 RR) *)
+
+(* Number of atoms : 6 ( 6 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 4 ( 2 singleton) *)
+
+(* Maximal term depth : 2 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Definitions of AND, OR and NOT *)
+
+(* Inclusion of: Axioms/HWC002-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : HWC002-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Hardware Creation *)
+
+(* Axioms : Definitions of AND, OR and NOT *)
+
+(* Version : [WO+92] axioms. *)
+
+(* Axiom formulation : Non-ground axioms. *)
+
+(* English : *)
+
+(* Refs : [WO+92] Wos et al. (1992), Automated Reasoning: Introduction a *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 6 ( 0 non-Horn; 6 unit; 2 RR) *)
+
+(* Number of literals : 6 ( 6 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 4 ( 2 singleton) *)
+
+(* Maximal term depth : 2 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT006-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT006-1 : TPTP v3.2.0. Released v2.2.0. *)
+
+(* Domain : Lattice Theory (Distributive lattices) *)
+
+(* Problem : Sholander's basis for distributive lattices, part 2 (of 6). *)
+
+(* Version : [MP96] (equality) axioms. *)
+
+(* English : This is part of the proof that Sholanders 2-basis for *)
+
+(* distributive lattices is correct. Here we prove associativity *)
+
+(* of meet. *)
+
+(* Refs : [McC98] McCune (1998), Email to G. Sutcliffe *)
+
+(* : [MP96] McCune & Padmanabhan (1996), Automated Deduction in Eq *)
+
+(* Source : [McC98] *)
+
+(* Names : LT-3-b [MP96] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.21 v3.2.0, 0.14 v3.1.0, 0.22 v2.7.0, 0.00 v2.2.1 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 1 singleton) *)
+
+(* Maximal term depth : 3 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Sholander's 2-basis for distributive lattices: *)
+
+(* ----Denial of the conclusion: *)
+ntheorem prove_associativity_of_meet:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet X (join Y Z)) (join (meet Z X) (meet Y X)).
+∀H1:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.eq Univ (meet (meet a b) c) (meet a (meet b c))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#join.
+#meet.
+#H0.
+#H1.
+nauto by H0,H1;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT007-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT007-1 : TPTP v3.2.0. Released v2.2.0. *)
+
+(* Domain : Lattice Theory (Distributive lattices) *)
+
+(* Problem : Sholander's basis for distributive lattices, part 5 (of 6). *)
+
+(* Version : [MP96] (equality) axioms. *)
+
+(* English : This is part of the proof that Sholanders 2-basis for *)
+
+(* distributive lattices is correct. Here we prove associativity *)
+
+(* of join. *)
+
+(* Refs : [McC98] McCune (1998), Email to G. Sutcliffe *)
+
+(* : [MP96] McCune & Padmanabhan (1996), Automated Deduction in Eq *)
+
+(* Source : [McC98] *)
+
+(* Names : LT-3-e [MP96] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.36 v3.1.0, 0.33 v2.7.0, 0.27 v2.6.0, 0.17 v2.5.0, 0.00 v2.2.1 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 1 singleton) *)
+
+(* Maximal term depth : 3 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Sholander's 2-basis for distributive lattices: *)
+
+(* ----Denial of the conclusion: *)
+ntheorem prove_associativity_of_join:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet X (join Y Z)) (join (meet Z X) (meet Y X)).
+∀H1:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.eq Univ (join (join a b) c) (join a (join b c))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#join.
+#meet.
+#H0.
+#H1.
+nauto by H0,H1;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT008-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT008-1 : TPTP v3.2.0. Released v2.2.0. *)
+
+(* Domain : Lattice Theory (Distributive lattices) *)
+
+(* Problem : Sholander's basis for distributive lattices, part 5 (of 6). *)
+
+(* Version : [MP96] (equality) axioms. *)
+
+(* English : This is part of the proof that Sholanders 2-basis for *)
+
+(* distributive lattices is correct. Here we prove the absorption *)
+
+(* law x v (x ^ y) = x. *)
+
+(* Refs : [McC98] McCune (1998), Email to G. Sutcliffe *)
+
+(* : [MP96] McCune & Padmanabhan (1996), Automated Deduction in Eq *)
+
+(* Source : [McC98] *)
+
+(* Names : LT-3-f [MP96] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.07 v3.1.0, 0.11 v2.7.0, 0.00 v2.2.1 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 5 ( 1 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Sholander's 2-basis for distributive lattices: *)
+
+(* ----Denial of the conclusion: *)
+ntheorem prove_absorbtion_dual:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet X (join Y Z)) (join (meet Z X) (meet Y X)).
+∀H1:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.eq Univ (join a (meet a b)) a
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#join.
+#meet.
+#H0.
+#H1.
+nauto by H0,H1;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT009-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT009-1 : TPTP v3.2.0. Released v2.2.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : A self-dual form of distributivity for lattice theory. *)
+
+(* Version : [MP96] (equality) axioms : Especial. *)
+
+(* English : From lattice theory axioms and a self-dual form of *)
+
+(* distributivity, we prove ordinary distributivity. *)
+
+(* Refs : [McC98] McCune (1998), Email to G. Sutcliffe *)
+
+(* : [MP96] McCune & Padmanabhan (1996), Automated Deduction in Eq *)
+
+(* Source : [McC98] *)
+
+(* Names : LT-5 [MP96] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.2.1 *)
+
+(* Syntax : Number of clauses : 10 ( 0 non-Horn; 10 unit; 1 RR) *)
+
+(* Number of atoms : 10 ( 10 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 19 ( 2 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include lattice axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----A self-dual form of distributivity for lattice theory. *)
+
+(* ----Denial of ordinary distributivity. *)
+ntheorem prove_distributivity:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (meet (join (meet X Y) Z) Y) (meet Z X)) (meet (join (meet (join X Y) Z) Y) (join Z X)).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H7:∀X:Univ.eq Univ (join X X) X.
+∀H8:∀X:Univ.eq Univ (meet X X) X.eq Univ (join a (meet b c)) (meet (join a b) (join a c))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#join.
+#meet.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT010-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT010-1 : TPTP v3.2.0. Released v2.2.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : McKenzie's basis for the variety generated by N5. *)
+
+(* Version : [MP96] (equality) axioms : Especial. *)
+
+(* English : McKenzie's basis for the variety generated by N5. *)
+
+(* Refs : [McC98] McCune (1998), Email to G. Sutcliffe *)
+
+(* : [MP96] McCune & Padmanabhan (1996), Automated Deduction in Eq *)
+
+(* Source : [McC98] *)
+
+(* Names : LT-6 [MP96] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.07 v3.2.0, 0.00 v2.5.0, 0.25 v2.4.0, 0.00 v2.2.1 *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 1 RR) *)
+
+(* Number of atoms : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 27 ( 2 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include lattice axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Hypotheses: *)
+
+(* ----Denial of the conclusion: *)
+ntheorem prove_this:
+ ∀Univ:Type.∀U:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀d:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (join X (meet Y Z)) (join Z (meet X Y))) (join (meet Z (join X (meet Y Z))) (meet X (join Y Z))).
+∀H1:∀U:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join X (meet Y (join Z (meet X U)))) (meet (join X (meet Y (join X Z))) (join X (meet (join X Y) (join Z U)))).
+∀H2:∀U:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet X (join Y (meet Z (join X U)))) (join (meet X (join Y (meet X Z))) (meet X (join (meet X Y) (meet Z U)))).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H7:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H8:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H9:∀X:Univ.eq Univ (join X X) X.
+∀H10:∀X:Univ.eq Univ (meet X X) X.eq Univ (meet a (meet (join b c) (join b d))) (meet (meet a (meet (join b c) (join b d))) (join (meet a (join b (meet c d))) (join (meet a c) (meet a d))))
+.
+#Univ.
+#U.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#d.
+#join.
+#meet.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT011-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT011-1 : TPTP v3.2.0. Released v2.2.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Uniqueness of meet (dually join) in lattice theory *)
+
+(* Version : [MP96] (equality) axioms : Especial. *)
+
+(* English : Let's say we have a lattice with two meet operations, say *)
+
+(* meet1 and meet2. In other words, {join,meet1} is a lattice, *)
+
+(* and {join,meet2} is a lattice. Then, we can prove that the *)
+
+(* two meet operations are really the same. *)
+
+(* Refs : [McC98] McCune (1998), Email to G. Sutcliffe *)
+
+(* : [MP96] McCune & Padmanabhan (1996), Automated Deduction in Eq *)
+
+(* Source : [McC98] *)
+
+(* Names : LT-8 [MP96] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.07 v3.1.0, 0.00 v2.7.0, 0.09 v2.6.0, 0.00 v2.2.1 *)
+
+(* Syntax : Number of clauses : 14 ( 0 non-Horn; 14 unit; 1 RR) *)
+
+(* Number of atoms : 14 ( 14 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 26 ( 4 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : For quasilattice, meet (dually join) is not necessarily unique. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include lattice axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----{join,meet2} is a lattice: *)
+
+(* ----Denial that meet1 and meet2 are the same: *)
+ntheorem prove_meets_are_same:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀meet2:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet2 (meet2 X Y) Z) (meet2 X (meet2 Y Z)).
+∀H1:∀X:Univ.∀Y:Univ.eq Univ (join X (meet2 X Y)) X.
+∀H2:∀X:Univ.∀Y:Univ.eq Univ (meet2 X (join X Y)) X.
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (meet2 X Y) (meet2 Y X).
+∀H4:∀X:Univ.eq Univ (meet2 X X) X.
+∀H5:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H6:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H7:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H8:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H9:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H10:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H11:∀X:Univ.eq Univ (join X X) X.
+∀H12:∀X:Univ.eq Univ (meet X X) X.eq Univ (meet a b) (meet2 a b)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#join.
+#meet.
+#meet2.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT012-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT012-1 : TPTP v3.2.0. Released v2.2.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : McKenzie's 4-basis for lattice theory, part 1 (of 3) *)
+
+(* Version : [MP96] (equality) axioms. *)
+
+(* English : This is part of a proof that McKenzie's 4-basis axiomatizes *)
+
+(* lattice theory. We prove half of the standard basis. *)
+
+(* The other half follows by duality. In this part we prove *)
+
+(* commutativity of meet. *)
+
+(* Refs : [McC98] McCune (1998), Email to G. Sutcliffe *)
+
+(* : [MP96] McCune & Padmanabhan (1996), Automated Deduction in Eq *)
+
+(* Source : [McC98] *)
+
+(* Names : LT-9-a [MP96] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v3.1.0, 0.11 v2.7.0, 0.00 v2.2.1 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 1 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 12 ( 8 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----McKenzie's self-dual (independent) absorptive 4-basis for lattice theory. *)
+
+(* ----Denial of conclusion: *)
+ntheorem prove_commutativity_of_meet:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet (join X Y) (join Y Z)) Y) Y.
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join (meet X Y) (meet Y Z)) Y) Y.
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet X (join Y (join X Z))) X.
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join X (meet Y (meet X Z))) X.eq Univ (meet b a) (meet a b)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#join.
+#meet.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT013-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT013-1 : TPTP v3.2.0. Released v2.2.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : McKenzie's 4-basis for lattice theory, part 2 (of 3) *)
+
+(* Version : [MP96] (equality) axioms. *)
+
+(* English : This is part of a proof that McKenzie's 4-basis axiomatizes *)
+
+(* lattice theory. We prove half of the standard basis. *)
+
+(* The other half follows by duality. In this part we prove *)
+
+(* associativity of meet. *)
+
+(* Refs : [McC98] McCune (1998), Email to G. Sutcliffe *)
+
+(* : [MP96] McCune & Padmanabhan (1996), Automated Deduction in Eq *)
+
+(* Source : [McC98] *)
+
+(* Names : LT-9-b [MP96] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.21 v3.2.0, 0.14 v3.1.0, 0.22 v2.7.0, 0.18 v2.6.0, 0.17 v2.5.0, 0.00 v2.2.1 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 1 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 12 ( 8 singleton) *)
+
+(* Maximal term depth : 4 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----McKenzie's self-dual (independent) absorptive 4-basis for lattice theory. *)
+
+(* ----Denial of conclusion: *)
+ntheorem prove_associativity_of_meet:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet (join X Y) (join Y Z)) Y) Y.
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join (meet X Y) (meet Y Z)) Y) Y.
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet X (join Y (join X Z))) X.
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join X (meet Y (meet X Z))) X.eq Univ (meet (meet a b) c) (meet a (meet b c))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#join.
+#meet.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT014-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT014-1 : TPTP v3.2.0. Released v2.2.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : McKenzie's 4-basis for lattice theory, part 3 (of 3) *)
+
+(* Version : [MP96] (equality) axioms. *)
+
+(* English : This is part of a proof that McKenzie's 4-basis axiomatizes *)
+
+(* lattice theory. We prove half of the standard basis. *)
+
+(* The other half follows by duality. In this part we prove *)
+
+(* absorbtion of meet. *)
+
+(* Refs : [McC98] McCune (1998), Email to G. Sutcliffe *)
+
+(* : [MP96] McCune & Padmanabhan (1996), Automated Deduction in Eq *)
+
+(* Source : [McC98] *)
+
+(* Names : LT-9 [MP96] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.2.1 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 1 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 12 ( 8 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----McKenzie's self-dual (independent) absorptive 4-basis for lattice theory. *)
+
+(* ----Denial of conclusion: *)
+ntheorem prove_absorbtion:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet (join X Y) (join Y Z)) Y) Y.
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join (meet X Y) (meet Y Z)) Y) Y.
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet X (join Y (join X Z))) X.
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join X (meet Y (meet X Z))) X.eq Univ (meet a (join a b)) a
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#join.
+#meet.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT016-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT016-1 : TPTP v3.2.0. Bugfixed v2.2.1. *)
+
+(* Domain : Lattice Theory (Ortholattices) *)
+
+(* Problem : E1 fails for Ortholattices. *)
+
+(* Version : [McC98b] (equality) axioms. *)
+
+(* English : Show that Ortholattices do not necessarily satisfy equation E1. *)
+
+(* Refs : [McC98a] McCune (1998), Automatic Proofs and Counterexamples f *)
+
+(* : [McC98b] McCune (1998), Email to G. Sutcliffe *)
+
+(* Source : [McC98b] *)
+
+(* Names : OL-1 [McC98b] *)
+
+(* Status : Satisfiable *)
+
+(* Rating : 0.33 v2.6.0, 0.67 v2.5.0, 1.00 v2.2.1 *)
+
+(* Syntax : Number of clauses : 11 ( 0 non-Horn; 11 unit; 1 RR) *)
+
+(* Number of atoms : 11 ( 11 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 19 ( 2 singleton) *)
+
+(* Maximal term depth : 9 ( 3 average) *)
+
+(* Comments : Ortholattice lemmas are included in McCunes original, but have *)
+
+(* been removed here. *)
+
+(* : The smallest model has 10 elements. *)
+
+(* Bugfixes : v2.2.1 - Bugfix in LAT003-0.ax. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include ortholattice axioms *)
+
+(* Inclusion of: Axioms/LAT003-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT003-0 : TPTP v3.2.0. Bugfixed v2.2.1. *)
+
+(* Domain : Lattice Theory (Ortholattices) *)
+
+(* Axioms : Ortholattice theory (equality) axioms *)
+
+(* Version : [McC98b] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC98a] McCune (1998), Automatic Proofs and Counterexamples f *)
+
+(* : [McC98b] McCune (1998), Email to G. Sutcliffe *)
+
+(* Source : [McC98b] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 10 ( 0 non-Horn; 10 unit; 0 RR) *)
+
+(* Number of literals : 10 ( 10 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 19 ( 2 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : *)
+
+(* Bugfixes : v2.2.1 - Added clauses top and bottom. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Axioms for an Ortholattice: *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Denial of equation E1 *)
+ntheorem prove_e1:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀complement:∀_:Univ.Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀n0:Univ.
+∀n1:Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (complement (join (complement X) (complement Y))).
+∀H1:∀X:Univ.∀Y:Univ.eq Univ (join X (join Y (complement Y))) (join Y (complement Y)).
+∀H2:∀X:Univ.eq Univ (complement (complement X)) X.
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H7:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H8:∀X:Univ.eq Univ (meet (complement X) X) n0.
+∀H9:∀X:Univ.eq Univ (join (complement X) X) n1.eq Univ (join (complement (join (meet a (complement b)) (complement a))) (join (meet a (complement b)) (join (meet (complement a) (meet (join a (complement b)) (join a b))) (meet (complement a) (complement (meet (join a (complement b)) (join a b))))))) n1
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#complement.
+#join.
+#meet.
+#n0.
+#n1.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT017-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT017-1 : TPTP v3.2.0. Bugfixed v2.2.1. *)
+
+(* Domain : Lattice Theory (Ortholattices) *)
+
+(* Problem : E2 holds in Ortholattices. *)
+
+(* Version : [McC98b] (equality) axioms. *)
+
+(* English : Prove that from ortholattice axioms, one can derive equation E2. *)
+
+(* Refs : [McC98a] McCune (1998), Automatic Proofs and Counterexamples f *)
+
+(* : [McC98b] McCune (1998), Email to G. Sutcliffe *)
+
+(* Source : [McC98b] *)
+
+(* Names : OL-2 [McC98b] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.64 v3.2.0, 0.71 v3.1.0, 0.56 v2.7.0, 0.73 v2.6.0, 0.33 v2.5.0, 0.25 v2.4.0, 0.67 v2.2.1 *)
+
+(* Syntax : Number of clauses : 11 ( 0 non-Horn; 11 unit; 1 RR) *)
+
+(* Number of atoms : 11 ( 11 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 19 ( 2 singleton) *)
+
+(* Maximal term depth : 7 ( 3 average) *)
+
+(* Comments : Ortholattice lemmas are included in McCunes original, but have *)
+
+(* been removed here. *)
+
+(* Bugfixes : v2.2.1 - Bugfix in LAT003-0.ax. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include ortholattice axioms *)
+
+(* Inclusion of: Axioms/LAT003-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT003-0 : TPTP v3.2.0. Bugfixed v2.2.1. *)
+
+(* Domain : Lattice Theory (Ortholattices) *)
+
+(* Axioms : Ortholattice theory (equality) axioms *)
+
+(* Version : [McC98b] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC98a] McCune (1998), Automatic Proofs and Counterexamples f *)
+
+(* : [McC98b] McCune (1998), Email to G. Sutcliffe *)
+
+(* Source : [McC98b] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 10 ( 0 non-Horn; 10 unit; 0 RR) *)
+
+(* Number of literals : 10 ( 10 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 19 ( 2 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : *)
+
+(* Bugfixes : v2.2.1 - Added clauses top and bottom. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Axioms for an Ortholattice: *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Denial of equation E2 *)
+ntheorem prove_e2:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀complement:∀_:Univ.Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀n0:Univ.
+∀n1:Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (complement (join (complement X) (complement Y))).
+∀H1:∀X:Univ.∀Y:Univ.eq Univ (join X (join Y (complement Y))) (join Y (complement Y)).
+∀H2:∀X:Univ.eq Univ (complement (complement X)) X.
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H7:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H8:∀X:Univ.eq Univ (meet (complement X) X) n0.
+∀H9:∀X:Univ.eq Univ (join (complement X) X) n1.eq Univ (join a (join (meet (complement a) (meet (join a (complement b)) (join a b))) (meet (complement a) (join (meet (complement a) b) (meet (complement a) (complement b)))))) n1
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#complement.
+#join.
+#meet.
+#n0.
+#n1.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT018-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT018-1 : TPTP v3.2.0. Bugfixed v2.2.1. *)
+
+(* Domain : Lattice Theory (Ortholattices) *)
+
+(* Problem : E3 holds in Ortholattices. *)
+
+(* Version : [McC98b] (equality) axioms. *)
+
+(* English : Prove that from ortholattice axioms, one can derive equation E3. *)
+
+(* Refs : [McC98a] McCune (1998), Automatic Proofs and Counterexamples f *)
+
+(* : [McC98b] McCune (1998), Email to G. Sutcliffe *)
+
+(* Source : [McC98b] *)
+
+(* Names : OL-3 [McC98b] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.93 v3.1.0, 1.00 v2.2.1 *)
+
+(* Syntax : Number of clauses : 11 ( 0 non-Horn; 11 unit; 1 RR) *)
+
+(* Number of atoms : 11 ( 11 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 19 ( 2 singleton) *)
+
+(* Maximal term depth : 7 ( 3 average) *)
+
+(* Comments : Ortholattice lemmas are included in McCunes original, but have *)
+
+(* been removed here. *)
+
+(* Bugfixes : v2.2.1 - Bugfix in LAT003-0.ax. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include ortholattice axioms *)
+
+(* Inclusion of: Axioms/LAT003-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT003-0 : TPTP v3.2.0. Bugfixed v2.2.1. *)
+
+(* Domain : Lattice Theory (Ortholattices) *)
+
+(* Axioms : Ortholattice theory (equality) axioms *)
+
+(* Version : [McC98b] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC98a] McCune (1998), Automatic Proofs and Counterexamples f *)
+
+(* : [McC98b] McCune (1998), Email to G. Sutcliffe *)
+
+(* Source : [McC98b] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 10 ( 0 non-Horn; 10 unit; 0 RR) *)
+
+(* Number of literals : 10 ( 10 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 19 ( 2 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : *)
+
+(* Bugfixes : v2.2.1 - Added clauses top and bottom. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Axioms for an Ortholattice: *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Denial of equation E3 *)
+ntheorem prove_e3:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀complement:∀_:Univ.Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀n0:Univ.
+∀n1:Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (complement (join (complement X) (complement Y))).
+∀H1:∀X:Univ.∀Y:Univ.eq Univ (join X (join Y (complement Y))) (join Y (complement Y)).
+∀H2:∀X:Univ.eq Univ (complement (complement X)) X.
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H7:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H8:∀X:Univ.eq Univ (meet (complement X) X) n0.
+∀H9:∀X:Univ.eq Univ (join (complement X) X) n1.eq Univ (join (complement (join (join (meet (complement a) b) (meet (complement a) (complement b))) (meet a (join (complement a) b)))) (join (complement a) b)) n1
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#complement.
+#join.
+#meet.
+#n0.
+#n1.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT019-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT019-1 : TPTP v3.2.0. Released v2.2.0. *)
+
+(* Domain : Lattice Theory (Quasilattices) *)
+
+(* Problem : In quasilattices, a distributive law implies its dual. *)
+
+(* Version : [MP96] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC98] McCune (1998), Email to G. Sutcliffe *)
+
+(* : [MP96] McCune & Padmanabhan (1996), Automated Deduction in Eq *)
+
+(* Source : [McC98] *)
+
+(* Names : QLT-2 [MP96] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.5.0, 0.25 v2.4.0, 0.00 v2.2.1 *)
+
+(* Syntax : Number of clauses : 10 ( 0 non-Horn; 10 unit; 1 RR) *)
+
+(* Number of atoms : 10 ( 10 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 21 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Quasilattice theory (equality) axioms *)
+
+(* Inclusion of: Axioms/LAT004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT004-0 : TPTP v3.2.0. Released v2.2.0. *)
+
+(* Domain : Lattice Theory (Quasilattices) *)
+
+(* Axioms : Quasilattice theory (equality) axioms *)
+
+(* Version : [McC98b] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC98] McCune (1998), Email to G. Sutcliffe *)
+
+(* : [MP96] McCune & Padmanabhan (1996), Automated Deduction in Eq *)
+
+(* Source : [McC98] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 18 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Quasilattice theory: *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----A distributivity law: *)
+
+(* ----Denial of the corresponding dual distributivity law: *)
+ntheorem prove_distributivity_law_dual:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet X (join Y Z)) (join (meet X Y) (meet X Z)).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (join X (meet Y Z)) (join X Y)) (join X (meet Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (meet X (join Y Z)) (meet X Y)) (meet X (join Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H7:∀X:Univ.eq Univ (join X X) X.
+∀H8:∀X:Univ.eq Univ (meet X X) X.eq Univ (join a (meet b c)) (meet (join a b) (join a c))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#join.
+#meet.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT020-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT020-1 : TPTP v3.2.0. Released v2.2.0. *)
+
+(* Domain : Lattice Theory (Quasilattices) *)
+
+(* Problem : Self-dual distributivity for quasilattices. *)
+
+(* Version : [MP96] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC98] McCune (1998), Email to G. Sutcliffe *)
+
+(* : [MP96] McCune & Padmanabhan (1996), Automated Deduction in Eq *)
+
+(* Source : [McC98] *)
+
+(* Names : QLT-3 [MP96] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.50 v3.1.0, 0.44 v2.7.0, 0.55 v2.6.0, 0.33 v2.5.0, 0.25 v2.4.0, 0.33 v2.3.0, 0.67 v2.2.1 *)
+
+(* Syntax : Number of clauses : 10 ( 0 non-Horn; 10 unit; 1 RR) *)
+
+(* Number of atoms : 10 ( 10 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 21 ( 0 singleton) *)
+
+(* Maximal term depth : 5 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Quasilattice theory (equality) axioms *)
+
+(* Inclusion of: Axioms/LAT004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT004-0 : TPTP v3.2.0. Released v2.2.0. *)
+
+(* Domain : Lattice Theory (Quasilattices) *)
+
+(* Axioms : Quasilattice theory (equality) axioms *)
+
+(* Version : [McC98b] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC98] McCune (1998), Email to G. Sutcliffe *)
+
+(* : [MP96] McCune & Padmanabhan (1996), Automated Deduction in Eq *)
+
+(* Source : [McC98] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 18 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Quasilattice theory: *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Self-dual distributivity: *)
+
+(* ----Denial of ordinary distributivity: *)
+ntheorem prove_distributivity:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (meet (join (meet X Y) Z) Y) (meet Z X)) (meet (join (meet (join X Y) Z) Y) (join Z X)).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (join X (meet Y Z)) (join X Y)) (join X (meet Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (meet X (join Y Z)) (meet X Y)) (meet X (join Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H7:∀X:Univ.eq Univ (join X X) X.
+∀H8:∀X:Univ.eq Univ (meet X X) X.eq Univ (meet a (join b c)) (join (meet a b) (meet a c))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#join.
+#meet.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT021-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT021-1 : TPTP v3.2.0. Released v2.2.0. *)
+
+(* Domain : Lattice Theory (Quasilattices) *)
+
+(* Problem : Bowden's inequality gives distributivity in lattice theory. *)
+
+(* Version : [MP96] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC98] McCune (1998), Email to G. Sutcliffe *)
+
+(* : [MP96] McCune & Padmanabhan (1996), Automated Deduction in Eq *)
+
+(* Source : [McC98] *)
+
+(* Names : QLT-4 [MP96] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.5.0, 0.25 v2.4.0, 0.00 v2.2.1 *)
+
+(* Syntax : Number of clauses : 10 ( 0 non-Horn; 10 unit; 1 RR) *)
+
+(* Number of atoms : 10 ( 10 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 21 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 3 average) *)
+
+(* Comments : Bowden's inequality is written as an equation. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Quasilattice theory (equality) axioms *)
+
+(* Inclusion of: Axioms/LAT004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT004-0 : TPTP v3.2.0. Released v2.2.0. *)
+
+(* Domain : Lattice Theory (Quasilattices) *)
+
+(* Axioms : Quasilattice theory (equality) axioms *)
+
+(* Version : [McC98b] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC98] McCune (1998), Email to G. Sutcliffe *)
+
+(* : [MP96] McCune & Padmanabhan (1996), Automated Deduction in Eq *)
+
+(* Source : [McC98] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 18 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Quasilattice theory: *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Bowden's inequality (as an equation): *)
+
+(* ----Denial of distributivity: *)
+ntheorem prove_distributivity:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X (meet Y Z)) (meet (join X Y) Z)) (join X (meet Y Z)).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (join X (meet Y Z)) (join X Y)) (join X (meet Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (meet X (join Y Z)) (meet X Y)) (meet X (join Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H7:∀X:Univ.eq Univ (join X X) X.
+∀H8:∀X:Univ.eq Univ (meet X X) X.eq Univ (meet a (join b c)) (join (meet a b) (meet a c))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#join.
+#meet.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT022-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT022-1 : TPTP v3.2.0. Released v2.2.0. *)
+
+(* Domain : Lattice Theory (Quasilattices) *)
+
+(* Problem : Self-dual modularity for quasilattices. *)
+
+(* Version : [MP96] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC98] McCune (1998), Email to G. Sutcliffe *)
+
+(* : [MP96] McCune & Padmanabhan (1996), Automated Deduction in Eq *)
+
+(* : [McC95] McCune (1995), Four Challenge Problems in Equational L *)
+
+(* Source : [McC98] *)
+
+(* Names : QLT-5 [MP96] *)
+
+(* : Problem C [McC95] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.21 v3.1.0, 0.22 v2.7.0, 0.09 v2.6.0, 0.00 v2.5.0, 0.25 v2.4.0, 0.33 v2.3.0, 0.67 v2.2.1 *)
+
+(* Syntax : Number of clauses : 10 ( 0 non-Horn; 10 unit; 1 RR) *)
+
+(* Number of atoms : 10 ( 10 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 21 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Quasilattice theory (equality) axioms *)
+
+(* Inclusion of: Axioms/LAT004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT004-0 : TPTP v3.2.0. Released v2.2.0. *)
+
+(* Domain : Lattice Theory (Quasilattices) *)
+
+(* Axioms : Quasilattice theory (equality) axioms *)
+
+(* Version : [McC98b] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC98] McCune (1998), Email to G. Sutcliffe *)
+
+(* : [MP96] McCune & Padmanabhan (1996), Automated Deduction in Eq *)
+
+(* Source : [McC98] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 18 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Quasilattice theory: *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Self-dual modularity: *)
+
+(* ----Denial of ordinary equational modularity: *)
+ntheorem prove_modularity:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (meet X Y) (meet Z (join X Y))) (meet (join X Y) (join Z (meet X Y))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (join X (meet Y Z)) (join X Y)) (join X (meet Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (meet X (join Y Z)) (meet X Y)) (meet X (join Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H7:∀X:Univ.eq Univ (join X X) X.
+∀H8:∀X:Univ.eq Univ (meet X X) X.eq Univ (meet a (join b (meet a c))) (join (meet a b) (meet a c))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#join.
+#meet.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT023-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT023-1 : TPTP v3.2.0. Released v2.2.0. *)
+
+(* Domain : Lattice Theory (Quasilattices) *)
+
+(* Problem : Yet another modularity equation for quasilattices. *)
+
+(* Version : [MP96] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC98] McCune (1998), Email to G. Sutcliffe *)
+
+(* : [MP96] McCune & Padmanabhan (1996), Automated Deduction in Eq *)
+
+(* Source : [McC98] *)
+
+(* Names : QLT-6 [MP96] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.21 v3.1.0, 0.22 v2.7.0, 0.09 v2.6.0, 0.00 v2.5.0, 0.25 v2.4.0, 0.33 v2.3.0, 0.67 v2.2.1 *)
+
+(* Syntax : Number of clauses : 10 ( 0 non-Horn; 10 unit; 1 RR) *)
+
+(* Number of atoms : 10 ( 10 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 21 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Quasilattice theory (equality) axioms *)
+
+(* Inclusion of: Axioms/LAT004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT004-0 : TPTP v3.2.0. Released v2.2.0. *)
+
+(* Domain : Lattice Theory (Quasilattices) *)
+
+(* Axioms : Quasilattice theory (equality) axioms *)
+
+(* Version : [McC98b] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC98] McCune (1998), Email to G. Sutcliffe *)
+
+(* : [MP96] McCune & Padmanabhan (1996), Automated Deduction in Eq *)
+
+(* Source : [McC98] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 18 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Quasilattice theory: *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Yet another modularity axoim: *)
+
+(* ----Denial of ordinary equational modularity: *)
+ntheorem prove_modularity:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (meet (join X Y) Z) Y) (join (meet (join Z Y) X) Y).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (join X (meet Y Z)) (join X Y)) (join X (meet Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (meet X (join Y Z)) (meet X Y)) (meet X (join Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H7:∀X:Univ.eq Univ (join X X) X.
+∀H8:∀X:Univ.eq Univ (meet X X) X.eq Univ (meet a (join b (meet a c))) (join (meet a b) (meet a c))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#join.
+#meet.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT024-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT024-1 : TPTP v3.2.0. Released v2.2.0. *)
+
+(* Domain : Lattice Theory (Quasilattices) *)
+
+(* Problem : Meet (dually join) is not necessarily unique for quasilattices. *)
+
+(* Version : [MP96] (equality) axioms. *)
+
+(* English : Let's say we have a quasilattice with two meet operations, say *)
+
+(* meet1 and meet2. In other words, {join,meet1} is a lattice, *)
+
+(* and {join,meet2} is a lattice. Then, we can show that the *)
+
+(* two meet operations not necessarily the same. *)
+
+(* Refs : [McC98] McCune (1998), Email to G. Sutcliffe *)
+
+(* : [MP96] McCune & Padmanabhan (1996), Automated Deduction in Eq *)
+
+(* Source : [McC98] *)
+
+(* Names : QLT-7 [MP96] *)
+
+(* Status : Satisfiable *)
+
+(* Rating : 0.33 v3.2.0, 0.67 v3.1.0, 0.33 v2.4.0, 0.67 v2.3.0, 1.00 v2.2.1 *)
+
+(* Syntax : Number of clauses : 14 ( 0 non-Horn; 14 unit; 1 RR) *)
+
+(* Number of atoms : 14 ( 14 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 30 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 3 average) *)
+
+(* Comments : There is a 2-element model. *)
+
+(* : For lattices meet (dually join) is unique. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Quasilattice theory (equality) axioms *)
+
+(* Inclusion of: Axioms/LAT004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT004-0 : TPTP v3.2.0. Released v2.2.0. *)
+
+(* Domain : Lattice Theory (Quasilattices) *)
+
+(* Axioms : Quasilattice theory (equality) axioms *)
+
+(* Version : [McC98b] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC98] McCune (1998), Email to G. Sutcliffe *)
+
+(* : [MP96] McCune & Padmanabhan (1996), Automated Deduction in Eq *)
+
+(* Source : [McC98] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 18 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Quasilattice theory: *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----{join,meet2} is a quasilattice: *)
+
+(* ----Denial that meet1 and meet2 are the same: *)
+ntheorem prove_meets_equal:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀meet2:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet2 (join X (meet2 Y Z)) (join X Y)) (join X (meet2 Y Z)).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (meet2 X (join Y Z)) (meet2 X Y)) (meet2 X (join Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet2 (meet2 X Y) Z) (meet2 X (meet2 Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (meet2 X Y) (meet2 Y X).
+∀H4:∀X:Univ.eq Univ (meet2 X X) X.
+∀H5:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (join X (meet Y Z)) (join X Y)) (join X (meet Y Z)).
+∀H6:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (meet X (join Y Z)) (meet X Y)) (meet X (join Y Z)).
+∀H7:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H8:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H9:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H10:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H11:∀X:Univ.eq Univ (join X X) X.
+∀H12:∀X:Univ.eq Univ (meet X X) X.eq Univ (meet a b) (meet2 a b)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#join.
+#meet.
+#meet2.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT025-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT025-1 : TPTP v3.2.0. Released v2.2.0. *)
+
+(* Domain : Lattice Theory (Ternary Near Lattices) *)
+
+(* Problem : Non-uniqueness of meet (dually join) in TNL *)
+
+(* Version : [MP96] (equality) axioms. *)
+
+(* English : Let's say we have a ternary near-lattice (TNL) with two meet *)
+
+(* operations, say meet1 and meet2. In other words, {join,meet1} *)
+
+(* and {join,meet2} are TNLs. Are the two meets necessarily *)
+
+(* the same? No, they aren't. Here is a counterexample. *)
+
+(* Refs : [McC98] McCune (1998), Email to G. Sutcliffe *)
+
+(* : [MP96] McCune & Padmanabhan (1996), Automated Deduction in Eq *)
+
+(* Source : [McC98] *)
+
+(* Names : TNL-2 [MP96] *)
+
+(* Status : Satisfiable *)
+
+(* Rating : 0.33 v3.2.0, 0.67 v3.1.0, 0.33 v2.4.0, 0.67 v2.3.0, 1.00 v2.2.1 *)
+
+(* Syntax : Number of clauses : 15 ( 0 non-Horn; 15 unit; 1 RR) *)
+
+(* Number of atoms : 15 ( 15 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 29 ( 12 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : The smallest model has 5 elements. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----{join,meet} is a TNL: *)
+
+(* ----{join,meet2} is a TNL: *)
+
+(* ----Denial of meet=meet2. *)
+ntheorem prove_meets_equal:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀meet2:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet2 X (join Y (join X Z))) X.
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join X (meet2 Y (meet2 X Z))) X.
+∀H2:∀X:Univ.∀Y:Univ.eq Univ (meet2 X Y) (meet2 Y X).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (join X (meet2 X Y)) X.
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (meet2 X (join X Y)) X.
+∀H5:∀X:Univ.eq Univ (meet2 X X) X.
+∀H6:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet X (join Y (join X Z))) X.
+∀H7:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join X (meet Y (meet X Z))) X.
+∀H8:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H9:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H10:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H11:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H12:∀X:Univ.eq Univ (join X X) X.
+∀H13:∀X:Univ.eq Univ (meet X X) X.eq Univ (meet a b) (meet2 a b)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#join.
+#meet.
+#meet2.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT026-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT026-1 : TPTP v3.2.0. Released v2.2.0. *)
+
+(* Domain : Lattice Theory (Weakly Associative Lattices) *)
+
+(* Problem : WAL + absorption gives LT, part 1. *)
+
+(* Version : [MP96] (equality) axioms. *)
+
+(* English : A Weakly associative lattice (WAL) satisfying an absorption *)
+
+(* law is associative, and therefore a full lattice, part 1. *)
+
+(* Refs : [McC98] McCune (1998), Email to G. Sutcliffe *)
+
+(* : [MP96] McCune & Padmanabhan (1996), Automated Deduction in Eq *)
+
+(* Source : [McC98] *)
+
+(* Names : WAL-1-a [MP96] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.14 v3.1.0, 0.11 v2.7.0, 0.09 v2.6.0, 0.00 v2.4.0, 0.00 v2.2.1 *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 1 RR) *)
+
+(* Number of atoms : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 15 ( 6 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Weakly Associative Lattices theory (equality) axioms *)
+
+(* Inclusion of: Axioms/LAT005-0.ax *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* File : LAT005-0 : TPTP v3.2.0. Released v2.2.0. *)
+
+(* Domain : Lattice Theory (Weakly Associative Lattices) *)
+
+(* Axioms : Weakly Associative Lattices theory (equality) axioms *)
+
+(* Version : [McC98b] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC98] McCune (1998), Email to G. Sutcliffe *)
+
+(* : [MP96] McCune & Padmanabhan (1996), Automated Deduction in Eq *)
+
+(* Source : [McC98] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 6 ( 0 non-Horn; 6 unit; 0 RR) *)
+
+(* Number of literals : 6 ( 6 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 12 ( 4 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* ----Axioms for a weakly associative lattice: *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----An absorption law. *)
+
+(* ----Denial of associativity of meet: *)
+ntheorem prove_associativity_of_meet:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet X (join Y (join X Z))) X.
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join (meet X Y) (meet Z Y)) Y) Y.
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet (join X Y) (join Z Y)) Y) Y.
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H5:∀X:Univ.eq Univ (join X X) X.
+∀H6:∀X:Univ.eq Univ (meet X X) X.eq Univ (meet (meet a b) c) (meet a (meet b c))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#join.
+#meet.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+nauto by H0,H1,H2,H3,H4,H5,H6;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT027-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT027-1 : TPTP v3.2.0. Released v2.2.0. *)
+
+(* Domain : Lattice Theory (Weakly Associative Lattices) *)
+
+(* Problem : WAL + absorption gives LT, part 2. *)
+
+(* Version : [MP96] (equality) axioms. *)
+
+(* English : A Weakly associative lattice (WAL) satisfying an absorption *)
+
+(* law is associative, and therefore a full lattice, part 2. *)
+
+(* Refs : [McC98] McCune (1998), Email to G. Sutcliffe *)
+
+(* : [MP96] McCune & Padmanabhan (1996), Automated Deduction in Eq *)
+
+(* Source : [McC98] *)
+
+(* Names : WAL-1-b [MP96] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.07 v3.1.0, 0.00 v2.7.0, 0.09 v2.6.0, 0.00 v2.2.1 *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 1 RR) *)
+
+(* Number of atoms : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 15 ( 6 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Weakly Associative Lattices theory (equality) axioms *)
+
+(* Inclusion of: Axioms/LAT005-0.ax *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* File : LAT005-0 : TPTP v3.2.0. Released v2.2.0. *)
+
+(* Domain : Lattice Theory (Weakly Associative Lattices) *)
+
+(* Axioms : Weakly Associative Lattices theory (equality) axioms *)
+
+(* Version : [McC98b] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC98] McCune (1998), Email to G. Sutcliffe *)
+
+(* : [MP96] McCune & Padmanabhan (1996), Automated Deduction in Eq *)
+
+(* Source : [McC98] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 6 ( 0 non-Horn; 6 unit; 0 RR) *)
+
+(* Number of literals : 6 ( 6 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 12 ( 4 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* ----Axioms for a weakly associative lattice: *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----An absorption law. *)
+
+(* ----Denial of associativity of join: *)
+ntheorem prove_associativity_of_join:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet X (join Y (join X Z))) X.
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join (meet X Y) (meet Z Y)) Y) Y.
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet (join X Y) (join Z Y)) Y) Y.
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H5:∀X:Univ.eq Univ (join X X) X.
+∀H6:∀X:Univ.eq Univ (meet X X) X.eq Univ (join (join a b) c) (join a (join b c))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#join.
+#meet.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+nauto by H0,H1,H2,H3,H4,H5,H6;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT028-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT028-1 : TPTP v3.2.0. Released v2.2.0. *)
+
+(* Domain : Lattice Theory (Weakly Associative Lattices) *)
+
+(* Problem : Uniqueness of meet (dually join) in WAL *)
+
+(* Version : [MP96] (equality) axioms. *)
+
+(* English : Let's say we have a weakly-associative lattice (WAL) with 2 meet *)
+
+(* operations, say meet1 and meet2. In other words, {join,meet1} *)
+
+(* is a WAL, and {join,meet2} is a WAL. Then, we can prove that the *)
+
+(* two meet operations are really the same. *)
+
+(* Refs : [McC98] McCune (1998), Email to G. Sutcliffe *)
+
+(* : [MP96] McCune & Padmanabhan (1996), Automated Deduction in Eq *)
+
+(* Source : [McC98] *)
+
+(* Names : WAL-2 [MP96] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.2.1 *)
+
+(* Syntax : Number of clauses : 11 ( 0 non-Horn; 11 unit; 1 RR) *)
+
+(* Number of atoms : 11 ( 11 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 21 ( 8 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Weakly Associative Lattices theory (equality) axioms *)
+
+(* Inclusion of: Axioms/LAT005-0.ax *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* File : LAT005-0 : TPTP v3.2.0. Released v2.2.0. *)
+
+(* Domain : Lattice Theory (Weakly Associative Lattices) *)
+
+(* Axioms : Weakly Associative Lattices theory (equality) axioms *)
+
+(* Version : [McC98b] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC98] McCune (1998), Email to G. Sutcliffe *)
+
+(* : [MP96] McCune & Padmanabhan (1996), Automated Deduction in Eq *)
+
+(* Source : [McC98] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 6 ( 0 non-Horn; 6 unit; 0 RR) *)
+
+(* Number of literals : 6 ( 6 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 12 ( 4 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* ----Axioms for a weakly associative lattice: *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----{join,meet2} is a weakly-associative lattice: *)
+
+(* ----Denial of meet=meet2: *)
+ntheorem name:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀meet2:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join (meet2 X Y) (meet2 Z Y)) Y) Y.
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet2 (meet2 (join X Y) (join Z Y)) Y) Y.
+∀H2:∀X:Univ.∀Y:Univ.eq Univ (meet2 X Y) (meet2 Y X).
+∀H3:∀X:Univ.eq Univ (meet2 X X) X.
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join (meet X Y) (meet Z Y)) Y) Y.
+∀H5:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet (join X Y) (join Z Y)) Y) Y.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H7:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H8:∀X:Univ.eq Univ (join X X) X.
+∀H9:∀X:Univ.eq Univ (meet X X) X.eq Univ (meet a b) (meet2 a b)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#join.
+#meet.
+#meet2.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT031-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT031-1 : TPTP v3.2.0. Released v2.4.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Distributivity of meet implies distributivity of join *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [DeN00] DeNivelle (2000), Email to G. Sutcliffe *)
+
+(* [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* Source : [DeN00] *)
+
+(* Names : dist_join [DeN00] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.4.0 *)
+
+(* Syntax : Number of clauses : 10 ( 0 non-Horn; 10 unit; 1 RR) *)
+
+(* Number of atoms : 10 ( 10 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 19 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include lattice theory axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem dist_join:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀xx:Univ.
+∀yy:Univ.
+∀zz:Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet X (join Y Z)) (join (meet X Y) (meet X Z)).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H7:∀X:Univ.eq Univ (join X X) X.
+∀H8:∀X:Univ.eq Univ (meet X X) X.eq Univ (join xx (meet yy zz)) (meet (join xx yy) (join xx zz))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#join.
+#meet.
+#xx.
+#yy.
+#zz.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT032-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT032-1 : TPTP v3.2.0. Released v2.4.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Distributivity of join implies distributivity of meet *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [DeN00] DeNivelle (2000), Email to G. Sutcliffe *)
+
+(* [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* Source : [DeN00] *)
+
+(* Names : dist_meet [DeN00] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.4.0 *)
+
+(* Syntax : Number of clauses : 10 ( 0 non-Horn; 10 unit; 1 RR) *)
+
+(* Number of atoms : 10 ( 10 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 19 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include lattice theory axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem dist_meet:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀xx:Univ.
+∀yy:Univ.
+∀zz:Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join X (meet Y Z)) (meet (join X Y) (join X Z)).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H7:∀X:Univ.eq Univ (join X X) X.
+∀H8:∀X:Univ.eq Univ (meet X X) X.eq Univ (meet xx (join yy zz)) (join (meet xx yy) (meet xx zz))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#join.
+#meet.
+#xx.
+#yy.
+#zz.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT033-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT033-1 : TPTP v3.2.0. Bugfixed v2.5.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Idempotency of join *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [DeN00] DeNivelle (2000), Email to G. Sutcliffe *)
+
+(* [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* Source : [DeN00] *)
+
+(* Names : idemp_of_join [DeN00] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.5.0 *)
+
+(* Syntax : Number of clauses : 7 ( 0 non-Horn; 7 unit; 1 RR) *)
+
+(* Number of atoms : 7 ( 7 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 14 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* Bugfixes : v2.5.0 - Used axioms without the conjecture *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include lattice theory axioms *)
+
+(* include('Axioms/LAT001-0.ax'). *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem idempotence_of_join:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀xx:Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.eq Univ (join xx xx) xx
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#join.
+#meet.
+#xx.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+nauto by H0,H1,H2,H3,H4,H5;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT034-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT034-1 : TPTP v3.2.0. Bugfixed v2.5.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Idempotency of meet *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [DeN00] DeNivelle (2000), Email to G. Sutcliffe *)
+
+(* [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* Source : [DeN00] *)
+
+(* Names : idemp_of_meet [DeN00] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.5.0 *)
+
+(* Syntax : Number of clauses : 7 ( 0 non-Horn; 7 unit; 1 RR) *)
+
+(* Number of atoms : 7 ( 7 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 14 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* Bugfixes : v2.5.0 - Used axioms without the conjecture *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include lattice theory axioms *)
+
+(* include('Axioms/LAT001-0.ax'). *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem idempotence_of_meet:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀xx:Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.eq Univ (meet xx xx) xx
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#join.
+#meet.
+#xx.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+nauto by H0,H1,H2,H3,H4,H5;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT038-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT038-1 : TPTP v3.2.0. Released v2.4.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Simplification rule in a distributive lattice *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : In a distributive lattice, the following simplification rule *)
+
+(* holds: *)
+
+(* forall a, b, c, d: *)
+
+(* if f(a v b, d) = f(c v b, d) and *)
+
+(* f(a, d) & f(b, d) = f(c, d) & f(b, d) *)
+
+(* then f(a,d) = f(c,d). *)
+
+(* Refs : [DeN00] DeNivelle (2000), Email to G. Sutcliffe *)
+
+(* [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* Source : [DeN00] *)
+
+(* Names : lattice-hemi-simplif [DeN00] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.43 v3.1.0, 0.44 v2.7.0, 0.36 v2.6.0, 0.17 v2.5.0, 0.25 v2.4.0 *)
+
+(* Syntax : Number of clauses : 17 ( 0 non-Horn; 17 unit; 3 RR) *)
+
+(* Number of atoms : 17 ( 17 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 8 ( 5 constant; 0-2 arity) *)
+
+(* Number of variables : 30 ( 4 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include lattice theory axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem rhs:
+ ∀Univ:Type.∀U:Univ.∀V:Univ.∀W:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀aa:Univ.
+∀bb:Univ.
+∀cc:Univ.
+∀dd:Univ.
+∀f:∀_:Univ.∀_:Univ.Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀n0:Univ.
+∀H0:eq Univ (meet (f aa dd) (f bb dd)) (meet (f cc dd) (f bb dd)).
+∀H1:eq Univ (f (join aa bb) dd) (f (join cc bb) dd).
+∀H2:∀W:Univ.eq Univ (f W n0) n0.
+∀H3:∀U:Univ.∀V:Univ.∀W:Univ.eq Univ (f W (join U V)) (join (f W U) (f W V)).
+∀H4:∀W:Univ.eq Univ (f n0 W) n0.
+∀H5:∀U:Univ.∀V:Univ.∀W:Univ.eq Univ (f (join U V) W) (join (f U W) (f V W)).
+∀H6:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet X (join Y Z)) (join (meet X Y) (meet X Z)).
+∀H7:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join X (meet Y Z)) (meet (join X Y) (join X Z)).
+∀H8:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H9:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H10:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H11:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H12:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H13:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H14:∀X:Univ.eq Univ (join X X) X.
+∀H15:∀X:Univ.eq Univ (meet X X) X.eq Univ (f aa dd) (f cc dd)
+.
+#Univ.
+#U.
+#V.
+#W.
+#X.
+#Y.
+#Z.
+#aa.
+#bb.
+#cc.
+#dd.
+#f.
+#join.
+#meet.
+#n0.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT039-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT039-1 : TPTP v3.2.0. Released v2.4.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Every distributive lattice is modular *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* Theorem formulation : Modularity is expressed by: *)
+
+(* x <= y -> x v (y & z) = y & (x v z) *)
+
+(* English : *)
+
+(* Refs : [DeN00] DeNivelle (2000), Email to G. Sutcliffe *)
+
+(* [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* Source : [DeN00] *)
+
+(* Names : lattice-mod-2 [DeN00] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.4.0 *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 2 RR) *)
+
+(* Number of atoms : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 22 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include lattice theory axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem rhs:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀xx:Univ.
+∀yy:Univ.
+∀zz:Univ.
+∀H0:eq Univ (join xx yy) yy.
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet X (join Y Z)) (join (meet X Y) (meet X Z)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join X (meet Y Z)) (meet (join X Y) (join X Z)).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H7:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H8:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H9:∀X:Univ.eq Univ (join X X) X.
+∀H10:∀X:Univ.eq Univ (meet X X) X.eq Univ (join xx (meet yy zz)) (meet yy (join xx zz))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#join.
+#meet.
+#xx.
+#yy.
+#zz.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT039-2.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT039-2 : TPTP v3.2.0. Released v2.4.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Every distributive lattice is modular *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : Theorem formulation : Modularity is expressed by: *)
+
+(* x <= y -> x v (y & z) = (x v y) & (x v z) *)
+
+(* Refs : [DeN00] DeNivelle (2000), Email to G. Sutcliffe *)
+
+(* [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* Source : [DeN00] *)
+
+(* Names : lattice-mod-3 [DeN00] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.4.0 *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 2 RR) *)
+
+(* Number of atoms : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 22 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include lattice theory axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem rhs:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀xx:Univ.
+∀yy:Univ.
+∀zz:Univ.
+∀H0:eq Univ (join xx yy) yy.
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet X (join Y Z)) (join (meet X Y) (meet X Z)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join X (meet Y Z)) (meet (join X Y) (join X Z)).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H7:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H8:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H9:∀X:Univ.eq Univ (join X X) X.
+∀H10:∀X:Univ.eq Univ (meet X X) X.eq Univ (join xx (meet yy zz)) (meet (join xx yy) (join xx zz))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#join.
+#meet.
+#xx.
+#yy.
+#zz.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT040-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT040-1 : TPTP v3.2.0. Released v2.4.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Another simplification rule for distributive lattices *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : In every distributive lattice the simplification rule holds: *)
+
+(* forall x, y, z: (x v y = x v z, x & y = x & z -> y = z ). *)
+
+(* Refs : [DeN00] DeNivelle (2000), Email to G. Sutcliffe *)
+
+(* [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* Source : [DeN00] *)
+
+(* Names : lattice-simpl [DeN00] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.4.0 *)
+
+(* Syntax : Number of clauses : 13 ( 0 non-Horn; 13 unit; 3 RR) *)
+
+(* Number of atoms : 13 ( 13 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 22 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include lattice theory axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem rhs:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀xx:Univ.
+∀yy:Univ.
+∀zz:Univ.
+∀H0:eq Univ (meet xx yy) (meet xx zz).
+∀H1:eq Univ (join xx yy) (join xx zz).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet X (join Y Z)) (join (meet X Y) (meet X Z)).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join X (meet Y Z)) (meet (join X Y) (join X Z)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H5:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H7:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H8:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H9:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H10:∀X:Univ.eq Univ (join X X) X.
+∀H11:∀X:Univ.eq Univ (meet X X) X.eq Univ yy zz
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#join.
+#meet.
+#xx.
+#yy.
+#zz.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT042-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT042-1 : TPTP v3.2.0. Released v2.5.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Lattice modularity from Boolean algebra *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [RW01] Rose & Wilkinson (2001), Application of Model Search *)
+
+(* Source : [RW01] *)
+
+(* Names : eqp-a1.in [RW01] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.07 v3.2.0, 0.00 v2.5.0 *)
+
+(* Syntax : Number of clauses : 13 ( 0 non-Horn; 13 unit; 1 RR) *)
+
+(* Number of atoms : 13 ( 13 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 8 ( 5 constant; 0-2 arity) *)
+
+(* Number of variables : 22 ( 2 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include lattice axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* Distributivity (4) *)
+
+(* Invertability (5) *)
+
+(* ----Preceding gives us Boolean Algebra *)
+
+(* ----Denial of modular law: *)
+ntheorem prove_modular_law:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀complement:∀_:Univ.Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀n0:Univ.
+∀n1:Univ.
+∀H0:∀X:Univ.eq Univ (complement (complement X)) X.
+∀H1:∀X:Univ.eq Univ (meet (complement X) X) n0.
+∀H2:∀X:Univ.eq Univ (join (complement X) X) n1.
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet X (join Y Z)) (join (meet X Y) (meet X Z)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H5:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H7:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H8:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H9:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H10:∀X:Univ.eq Univ (join X X) X.
+∀H11:∀X:Univ.eq Univ (meet X X) X.eq Univ (join a (meet b (join a c))) (meet (join a b) (join a c))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#complement.
+#join.
+#meet.
+#n0.
+#n1.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT043-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT043-1 : TPTP v3.2.0. Released v2.5.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Lattice compatability from Boolean algebra *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [RW01] Rose & Wilkinson (2001), Application of Model Search *)
+
+(* Source : [RW01] *)
+
+(* Names : eqp-a2.in [RW01] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.5.0 *)
+
+(* Syntax : Number of clauses : 13 ( 0 non-Horn; 13 unit; 1 RR) *)
+
+(* Number of atoms : 13 ( 13 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 22 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include lattice axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Distributivity (4) *)
+
+(* ----Invertability (5) *)
+
+(* ----Preceding gives us Boolean Algebra *)
+
+(* ----Denial of compatability *)
+ntheorem prove_compatability_law:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀c:Univ.
+∀complement:∀_:Univ.Univ.
+∀d:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀n0:Univ.
+∀n1:Univ.
+∀H0:∀X:Univ.eq Univ (complement (complement X)) X.
+∀H1:∀X:Univ.eq Univ (meet (complement X) X) n0.
+∀H2:∀X:Univ.eq Univ (join (complement X) X) n1.
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet X (join Y Z)) (join (meet X Y) (meet X Z)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H5:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H7:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H8:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H9:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H10:∀X:Univ.eq Univ (join X X) X.
+∀H11:∀X:Univ.eq Univ (meet X X) X.eq Univ (complement (join c d)) (meet (complement c) (complement d))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#c.
+#complement.
+#d.
+#join.
+#meet.
+#n0.
+#n1.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT044-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT044-1 : TPTP v3.2.0. Released v2.5.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Lattice weak orthomodular law from orthomodular lattice *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [RW01] Rose & Wilkinson (2001), Application of Model Search *)
+
+(* Source : [RW01] *)
+
+(* Names : eqp-c.in [RW01] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.21 v3.2.0, 0.14 v3.1.0, 0.00 v2.7.0, 0.27 v2.6.0, 0.00 v2.5.0 *)
+
+(* Syntax : Number of clauses : 15 ( 0 non-Horn; 15 unit; 1 RR) *)
+
+(* Number of atoms : 15 ( 15 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 25 ( 2 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include lattice axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Compatibility (6) *)
+
+(* ----Invertability (5) *)
+
+(* ----Orthomodular law (8) *)
+
+(* ----Denial of weak orthomodular law (10) *)
+ntheorem prove_weak_orthomodular_law:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀complement:∀_:Univ.Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀n0:Univ.
+∀n1:Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (join X (meet (complement X) (join X Y))) (join X Y).
+∀H1:∀X:Univ.eq Univ (complement (complement X)) X.
+∀H2:∀X:Univ.eq Univ (meet (complement X) X) n0.
+∀H3:∀X:Univ.eq Univ (join (complement X) X) n1.
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (complement (meet X Y)) (join (complement X) (complement Y)).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (complement (join X Y)) (meet (complement X) (complement Y)).
+∀H6:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H7:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H8:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H9:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H10:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H11:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H12:∀X:Univ.eq Univ (join X X) X.
+∀H13:∀X:Univ.eq Univ (meet X X) X.eq Univ (join (meet (complement a) (join a b)) (join (complement b) (meet a b))) n1
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#complement.
+#join.
+#meet.
+#n0.
+#n1.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT045-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT045-1 : TPTP v3.2.0. Released v2.5.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Lattice orthomodular law from modular lattice *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [RW01] Rose & Wilkinson (2001), Application of Model Search *)
+
+(* Source : [RW01] *)
+
+(* Names : eqp-f.in [RW01] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.07 v3.1.0, 0.11 v2.7.0, 0.00 v2.5.0 *)
+
+(* Syntax : Number of clauses : 15 ( 0 non-Horn; 15 unit; 1 RR) *)
+
+(* Number of atoms : 15 ( 15 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 26 ( 2 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include lattice axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Compatibility (6) *)
+
+(* ----Invertability (5) *)
+
+(* ----Modular law (7) *)
+
+(* ----Denial of orthomodular law (8) *)
+ntheorem prove_orthomodular_law:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀complement:∀_:Univ.Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀n0:Univ.
+∀n1:Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join X (meet Y (join X Z))) (meet (join X Y) (join X Z)).
+∀H1:∀X:Univ.eq Univ (complement (complement X)) X.
+∀H2:∀X:Univ.eq Univ (meet (complement X) X) n0.
+∀H3:∀X:Univ.eq Univ (join (complement X) X) n1.
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (complement (meet X Y)) (join (complement X) (complement Y)).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (complement (join X Y)) (meet (complement X) (complement Y)).
+∀H6:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H7:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H8:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H9:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H10:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H11:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H12:∀X:Univ.eq Univ (join X X) X.
+∀H13:∀X:Univ.eq Univ (meet X X) X.eq Univ (join a (meet (complement a) (join a b))) (join a b)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#complement.
+#join.
+#meet.
+#n0.
+#n1.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT046-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT046-1 : TPTP v3.2.0. Released v2.5.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Modular ortholattice is not Boolean algebra *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [RW01] Rose & Wilkinson (2001), Application of Model Search *)
+
+(* Source : [RW01] *)
+
+(* Names : mace-a.in [RW01] *)
+
+(* Status : Satisfiable *)
+
+(* Rating : 0.33 v3.2.0, 0.67 v3.1.0, 0.33 v2.5.0 *)
+
+(* Syntax : Number of clauses : 15 ( 0 non-Horn; 15 unit; 1 RR) *)
+
+(* Number of atoms : 15 ( 15 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 8 ( 5 constant; 0-2 arity) *)
+
+(* Number of variables : 26 ( 2 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include lattice axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Compatibility (6) *)
+
+(* ----Invertability (5) *)
+
+(* ----Modular law (7) *)
+
+(* ----Denial of distributivity (4) *)
+ntheorem prove_distributivity:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀complement:∀_:Univ.Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀n0:Univ.
+∀n1:Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join X (meet Y (join X Z))) (meet (join X Y) (join X Z)).
+∀H1:∀X:Univ.eq Univ (complement (complement X)) X.
+∀H2:∀X:Univ.eq Univ (meet (complement X) X) n0.
+∀H3:∀X:Univ.eq Univ (join (complement X) X) n1.
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (complement (meet X Y)) (join (complement X) (complement Y)).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (complement (join X Y)) (meet (complement X) (complement Y)).
+∀H6:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H7:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H8:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H9:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H10:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H11:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H12:∀X:Univ.eq Univ (join X X) X.
+∀H13:∀X:Univ.eq Univ (meet X X) X.eq Univ (meet a (join b c)) (join (meet a b) (meet a c))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#complement.
+#join.
+#meet.
+#n0.
+#n1.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT047-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT047-1 : TPTP v3.2.0. Released v2.5.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Lattice is not modular lattice *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [RW01] Rose & Wilkinson (2001), Application of Model Search *)
+
+(* Source : [RW01] *)
+
+(* Names : mace-b.in [RW01] *)
+
+(* Status : Satisfiable *)
+
+(* Rating : 0.33 v3.2.0, 0.67 v3.1.0, 0.33 v2.5.0 *)
+
+(* Syntax : Number of clauses : 9 ( 0 non-Horn; 9 unit; 1 RR) *)
+
+(* Number of atoms : 9 ( 9 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include lattice axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Denial of modularity (7) *)
+ntheorem prove_modularity:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H6:∀X:Univ.eq Univ (join X X) X.
+∀H7:∀X:Univ.eq Univ (meet X X) X.eq Univ (join a (meet b (join a c))) (meet (join a b) (join a c))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#join.
+#meet.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT048-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT048-1 : TPTP v3.2.0. Released v2.5.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Weakly orthomodular lattice is not orthomodular lattice *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [RW01] Rose & Wilkinson (2001), Application of Model Search *)
+
+(* Source : [RW01] *)
+
+(* Names : mace-c.in [RW01] *)
+
+(* Status : Satisfiable *)
+
+(* Rating : 0.33 v2.6.0, 0.67 v2.5.0 *)
+
+(* Syntax : Number of clauses : 15 ( 0 non-Horn; 15 unit; 1 RR) *)
+
+(* Number of atoms : 15 ( 15 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 25 ( 2 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include lattice axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Compatibility (6) *)
+
+(* ----Invertability (5) *)
+
+(* ----Weak orthomodular law (10) *)
+
+(* ----Denial of orthomodular law (8) *)
+ntheorem prove_orthomodular_law:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀complement:∀_:Univ.Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀n0:Univ.
+∀n1:Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (join (meet (complement X) (join X Y)) (join (complement Y) (meet X Y))) n1.
+∀H1:∀X:Univ.eq Univ (complement (complement X)) X.
+∀H2:∀X:Univ.eq Univ (meet (complement X) X) n0.
+∀H3:∀X:Univ.eq Univ (join (complement X) X) n1.
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (complement (meet X Y)) (join (complement X) (complement Y)).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (complement (join X Y)) (meet (complement X) (complement Y)).
+∀H6:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H7:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H8:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H9:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H10:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H11:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H12:∀X:Univ.eq Univ (join X X) X.
+∀H13:∀X:Univ.eq Univ (meet X X) X.eq Univ (join a (meet (complement a) (join a b))) (join a b)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#complement.
+#join.
+#meet.
+#n0.
+#n1.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT049-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT049-1 : TPTP v3.2.0. Released v2.5.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Ortholattice is not weakly orthomodular lattice *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [RW01] Rose & Wilkinson (2001), Application of Model Search *)
+
+(* Source : [RW01] *)
+
+(* Names : mace-d.in [RW01] *)
+
+(* Status : Satisfiable *)
+
+(* Rating : 0.33 v2.5.0 *)
+
+(* Syntax : Number of clauses : 14 ( 0 non-Horn; 14 unit; 1 RR) *)
+
+(* Number of atoms : 14 ( 14 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 23 ( 2 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include lattice axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Compatibility (6) *)
+
+(* ----Invertability (5) *)
+
+(* ----Denial of weak orthomodular law (10) *)
+ntheorem prove_weak_orthomodular_law:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀complement:∀_:Univ.Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀n0:Univ.
+∀n1:Univ.
+∀H0:∀X:Univ.eq Univ (complement (complement X)) X.
+∀H1:∀X:Univ.eq Univ (meet (complement X) X) n0.
+∀H2:∀X:Univ.eq Univ (join (complement X) X) n1.
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (complement (meet X Y)) (join (complement X) (complement Y)).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (complement (join X Y)) (meet (complement X) (complement Y)).
+∀H5:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H6:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H7:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H8:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H9:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H10:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H11:∀X:Univ.eq Univ (join X X) X.
+∀H12:∀X:Univ.eq Univ (meet X X) X.eq Univ (join (meet (complement a) (join a b)) (join (complement b) (meet a b))) n1
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#complement.
+#join.
+#meet.
+#n0.
+#n1.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT050-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT050-1 : TPTP v3.2.0. Released v2.5.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Orthomodular lattice is not modular lattice *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [RW01] Rose & Wilkinson (2001), Application of Model Search *)
+
+(* : [EF+02] Ernst et al. (2002), More First-order Test Problems in *)
+
+(* Source : [RW01] *)
+
+(* Names : mace-f.in [RW01] *)
+
+(* : oml-mod [EF+02] *)
+
+(* Status : Satisfiable *)
+
+(* Rating : 0.33 v3.2.0, 0.67 v3.1.0, 0.33 v2.6.0, 0.67 v2.5.0 *)
+
+(* Syntax : Number of clauses : 15 ( 0 non-Horn; 15 unit; 1 RR) *)
+
+(* Number of atoms : 15 ( 15 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 8 ( 5 constant; 0-2 arity) *)
+
+(* Number of variables : 25 ( 2 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : This is well known, but it is a good test problem for finite *)
+
+(* model search. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include lattice axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Compatibility (6) *)
+
+(* ----Invertability (5) *)
+
+(* ----Orthomodular law (8) *)
+
+(* ----Denial of modular law: *)
+ntheorem prove_modular_law:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀complement:∀_:Univ.Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀n0:Univ.
+∀n1:Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (join X (meet (complement X) (join X Y))) (join X Y).
+∀H1:∀X:Univ.eq Univ (complement (complement X)) X.
+∀H2:∀X:Univ.eq Univ (meet (complement X) X) n0.
+∀H3:∀X:Univ.eq Univ (join (complement X) X) n1.
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (complement (meet X Y)) (join (complement X) (complement Y)).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (complement (join X Y)) (meet (complement X) (complement Y)).
+∀H6:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H7:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H8:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H9:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H10:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H11:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H12:∀X:Univ.eq Univ (join X X) X.
+∀H13:∀X:Univ.eq Univ (meet X X) X.eq Univ (join a (meet b (join a c))) (meet (join a b) (join a c))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#complement.
+#join.
+#meet.
+#n0.
+#n1.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT051-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT051-1 : TPTP v3.2.0. Released v2.5.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Lattice is not ortholattice *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [RW01] Rose & Wilkinson (2001), Application of Model Search *)
+
+(* Source : [RW01] *)
+
+(* Names : mace-e1.in [RW01] *)
+
+(* Status : Satisfiable *)
+
+(* Rating : 0.33 v3.2.0, 0.67 v3.1.0, 0.33 v2.5.0 *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 1 RR) *)
+
+(* Number of atoms : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 19 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : First part of the problem. The second part, mace-e2.in, requires *)
+
+(* MACE specific input. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include lattice axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Invertability (5) *)
+
+(* ----Denial of compatibility (6) *)
+ntheorem prove_compatibility_law:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀complement:∀_:Univ.Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀n0:Univ.
+∀n1:Univ.
+∀H0:∀X:Univ.eq Univ (complement (complement X)) X.
+∀H1:∀X:Univ.eq Univ (meet (complement X) X) n0.
+∀H2:∀X:Univ.eq Univ (join (complement X) X) n1.
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H7:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H8:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H9:∀X:Univ.eq Univ (join X X) X.
+∀H10:∀X:Univ.eq Univ (meet X X) X.eq Univ (complement (join a b)) (meet (complement a) (complement b))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#complement.
+#join.
+#meet.
+#n0.
+#n1.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT052-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT052-1 : TPTP v3.2.0. Released v2.5.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Modular lattice is not modular ortholattice *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [RW01] Rose & Wilkinson (2001), Application of Model Search *)
+
+(* Source : [RW01] *)
+
+(* Names : mace-g1.in [RW01] *)
+
+(* Status : Satisfiable *)
+
+(* Rating : 0.33 v3.2.0, 0.67 v3.1.0, 0.33 v2.6.0, 0.67 v2.5.0 *)
+
+(* Syntax : Number of clauses : 13 ( 0 non-Horn; 13 unit; 1 RR) *)
+
+(* Number of atoms : 13 ( 13 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 22 ( 2 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : First part of the problem. The second part, mace-g2.in, requires *)
+
+(* MACE specific input. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include lattice axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Invertability (5) *)
+
+(* ----Modular law (7) *)
+
+(* ----Denial of compatibility (6) *)
+ntheorem prove_compatibility_law:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀complement:∀_:Univ.Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀n0:Univ.
+∀n1:Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join X (meet Y (join X Z))) (meet (join X Y) (join X Z)).
+∀H1:∀X:Univ.eq Univ (complement (complement X)) X.
+∀H2:∀X:Univ.eq Univ (meet (complement X) X) n0.
+∀H3:∀X:Univ.eq Univ (join (complement X) X) n1.
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H5:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H7:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H8:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H9:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H10:∀X:Univ.eq Univ (join X X) X.
+∀H11:∀X:Univ.eq Univ (meet X X) X.eq Univ (complement (join a b)) (meet (complement a) (complement b))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#complement.
+#join.
+#meet.
+#n0.
+#n1.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT053-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT053-1 : TPTP v3.2.0. Released v2.5.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Countermodel for Megill equation for weakly orthomodular lattices *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Meg00] Megill & Pavicic (2000), Equations and State and Latti *)
+
+(* : [RW01] Rose & Wilkinson (2001), Application of Model Search *)
+
+(* : [EF+02] Ernst et al. (2002), More First-order Test Problems in *)
+
+(* Source : [RW01] *)
+
+(* Names : sem-rw-1.in [RW01] *)
+
+(* : ol-rw1 [EF+02] *)
+
+(* Status : Satisfiable *)
+
+(* Rating : 0.67 v2.7.0, 0.33 v2.6.0, 0.67 v2.5.0 *)
+
+(* Syntax : Number of clauses : 15 ( 0 non-Horn; 15 unit; 1 RR) *)
+
+(* Number of atoms : 15 ( 15 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 25 ( 2 singleton) *)
+
+(* Maximal term depth : 6 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include lattice axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Compatibility (6) *)
+
+(* ----Invertability (5) *)
+
+(* ----Megill equation *)
+
+(* ----Denial of equation in question *)
+ntheorem prove_this:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀complement:∀_:Univ.Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀n0:Univ.
+∀n1:Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (join (meet (complement X) (join X Y)) (join (complement Y) (meet X Y))) n1.
+∀H1:∀X:Univ.eq Univ (complement (complement X)) X.
+∀H2:∀X:Univ.eq Univ (meet (complement X) X) n0.
+∀H3:∀X:Univ.eq Univ (join (complement X) X) n1.
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (complement (meet X Y)) (join (complement X) (complement Y)).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (complement (join X Y)) (meet (complement X) (complement Y)).
+∀H6:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H7:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H8:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H9:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H10:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H11:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H12:∀X:Univ.eq Univ (join X X) X.
+∀H13:∀X:Univ.eq Univ (meet X X) X.eq Univ (meet a (join b (meet a (join (complement a) (meet a b))))) (meet a (join (complement a) (meet a b)))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#complement.
+#join.
+#meet.
+#n0.
+#n1.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT054-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT054-1 : TPTP v3.2.0. Released v2.5.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Countermodel for Megill equation for ortholattices *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Meg00] Megill & Pavicic (2000), Equations and State and Latti *)
+
+(* : [RW01] Rose & Wilkinson (2001), Application of Model Search *)
+
+(* : [EF+02] Ernst et al. (2002), More First-order Test Problems in *)
+
+(* Source : [RW01] *)
+
+(* Names : sem-rw-2.in [RW01] *)
+
+(* : ol-rw2 [EF+02] *)
+
+(* Status : Satisfiable *)
+
+(* Rating : 0.33 v3.2.0, 0.67 v3.1.0, 0.33 v2.6.0, 0.67 v2.5.0 *)
+
+(* Syntax : Number of clauses : 14 ( 0 non-Horn; 14 unit; 1 RR) *)
+
+(* Number of atoms : 14 ( 14 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 23 ( 2 singleton) *)
+
+(* Maximal term depth : 10 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include lattice axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Compatibility (6) *)
+
+(* ----Invertability (5) *)
+
+(* ----Denial of equation in question *)
+ntheorem prove_this:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀complement:∀_:Univ.Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀n0:Univ.
+∀n1:Univ.
+∀H0:∀X:Univ.eq Univ (complement (complement X)) X.
+∀H1:∀X:Univ.eq Univ (meet (complement X) X) n0.
+∀H2:∀X:Univ.eq Univ (join (complement X) X) n1.
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (complement (meet X Y)) (join (complement X) (complement Y)).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (complement (join X Y)) (meet (complement X) (complement Y)).
+∀H5:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H6:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H7:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H8:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H9:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H10:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H11:∀X:Univ.eq Univ (join X X) X.
+∀H12:∀X:Univ.eq Univ (meet X X) X.eq Univ (join a (meet (complement b) (join (complement a) (meet (complement b) (join a (meet (complement b) (complement a))))))) (join a (meet (complement b) (join (complement a) (meet (complement b) (join a (meet (complement b) (join (complement a) (meet (complement b) a))))))))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#complement.
+#join.
+#meet.
+#n0.
+#n1.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT055-2.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT055-2 : TPTP v3.2.0. Released v2.5.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : Satisfiable *)
+
+(* Rating : 0.00 v3.2.0, 0.33 v3.1.0, 0.00 v2.5.0 *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of atoms : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Lattice theory (equality) axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT059-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT059-1 : TPTP v3.2.0. Released v2.5.0. *)
+
+(* Domain : Lattice Theory (Ortholattices) *)
+
+(* Problem : Ortholattice theory (equality) axioms *)
+
+(* Version : [McC98b] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC98a] McCune (1998), Automatic Proofs and Counterexamples f *)
+
+(* : [McC98b] McCune (1998), Email to G. Sutcliffe *)
+
+(* Source : [McC98b] *)
+
+(* Names : *)
+
+(* Status : Satisfiable *)
+
+(* Rating : 0.00 v3.2.0, 0.33 v3.1.0, 0.00 v2.5.0 *)
+
+(* Syntax : Number of clauses : 10 ( 0 non-Horn; 10 unit; 0 RR) *)
+
+(* Number of atoms : 10 ( 10 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 19 ( 2 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Ortholattice theory (equality) axioms *)
+
+(* Inclusion of: Axioms/LAT003-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT003-0 : TPTP v3.2.0. Bugfixed v2.2.1. *)
+
+(* Domain : Lattice Theory (Ortholattices) *)
+
+(* Axioms : Ortholattice theory (equality) axioms *)
+
+(* Version : [McC98b] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC98a] McCune (1998), Automatic Proofs and Counterexamples f *)
+
+(* : [McC98b] McCune (1998), Email to G. Sutcliffe *)
+
+(* Source : [McC98b] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 10 ( 0 non-Horn; 10 unit; 0 RR) *)
+
+(* Number of literals : 10 ( 10 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 19 ( 2 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : *)
+
+(* Bugfixes : v2.2.1 - Added clauses top and bottom. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Axioms for an Ortholattice: *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT060-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT060-1 : TPTP v3.2.0. Released v2.5.0. *)
+
+(* Domain : Lattice Theory (Quasilattices) *)
+
+(* Problem : Quasilattice theory (equality) axioms *)
+
+(* Version : [McC98b] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC98] McCune (1998), Email to G. Sutcliffe *)
+
+(* : [MP96] McCune & Padmanabhan (1996), Automated Deduction in Eq *)
+
+(* Source : [McC98] *)
+
+(* Names : *)
+
+(* Status : Satisfiable *)
+
+(* Rating : 0.00 v3.2.0, 0.33 v3.1.0, 0.00 v2.5.0 *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of atoms : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 18 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Quasilattice theory (equality) axioms *)
+
+(* Inclusion of: Axioms/LAT004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT004-0 : TPTP v3.2.0. Released v2.2.0. *)
+
+(* Domain : Lattice Theory (Quasilattices) *)
+
+(* Axioms : Quasilattice theory (equality) axioms *)
+
+(* Version : [McC98b] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC98] McCune (1998), Email to G. Sutcliffe *)
+
+(* : [MP96] McCune & Padmanabhan (1996), Automated Deduction in Eq *)
+
+(* Source : [McC98] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 18 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Quasilattice theory: *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT061-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT061-1 : TPTP v3.2.0. Released v2.5.0. *)
+
+(* Domain : Lattice Theory (Weakly Associative Lattices) *)
+
+(* Problem : Weakly Associative Lattices theory (equality) axioms *)
+
+(* Version : [McC98b] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC98] McCune (1998), Email to G. Sutcliffe *)
+
+(* : [MP96] McCune & Padmanabhan (1996), Automated Deduction in Eq *)
+
+(* Source : [McC98] *)
+
+(* Names : *)
+
+(* Status : Satisfiable *)
+
+(* Rating : 0.00 v3.2.0, 0.33 v3.1.0, 0.00 v2.5.0 *)
+
+(* Syntax : Number of clauses : 6 ( 0 non-Horn; 6 unit; 0 RR) *)
+
+(* Number of atoms : 6 ( 6 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 12 ( 4 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Weakly Associative Lattices theory (equality) axioms *)
+
+(* Inclusion of: Axioms/LAT005-0.ax *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* File : LAT005-0 : TPTP v3.2.0. Released v2.2.0. *)
+
+(* Domain : Lattice Theory (Weakly Associative Lattices) *)
+
+(* Axioms : Weakly Associative Lattices theory (equality) axioms *)
+
+(* Version : [McC98b] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC98] McCune (1998), Email to G. Sutcliffe *)
+
+(* : [MP96] McCune & Padmanabhan (1996), Automated Deduction in Eq *)
+
+(* Source : [McC98] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 6 ( 0 non-Horn; 6 unit; 0 RR) *)
+
+(* Number of literals : 6 ( 6 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 12 ( 4 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* ----Axioms for a weakly associative lattice: *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT062-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT062-1 : TPTP v3.2.0. Released v2.5.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : E51 does not necessarily hold in ortholattices *)
+
+(* Version : [EF+02] axioms. *)
+
+(* English : *)
+
+(* Refs : [EF+02] Ernst et al. (2002), More First-order Test Problems in *)
+
+(* Source : [EF+02] *)
+
+(* Names : ol-e51 [EF+02] *)
+
+(* Status : Satisfiable *)
+
+(* Rating : 0.33 v3.2.0, 0.67 v3.1.0, 0.33 v2.6.0, 0.67 v2.5.0 *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 1 RR) *)
+
+(* Number of atoms : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 20 ( 2 singleton) *)
+
+(* Maximal term depth : 6 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include lattice axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Ortholattice axioms *)
+
+(* ----Denial of E51 *)
+ntheorem prove_e51:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀complement:∀_:Univ.Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀n0:Univ.
+∀n1:Univ.
+∀H0:∀A:Univ.∀B:Univ.eq Univ (meet A B) (complement (join (complement A) (complement B))).
+∀H1:∀A:Univ.eq Univ (meet (complement A) A) n0.
+∀H2:∀A:Univ.eq Univ (join (complement A) A) n1.
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H7:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H8:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H9:∀X:Univ.eq Univ (join X X) X.
+∀H10:∀X:Univ.eq Univ (meet X X) X.eq Univ (meet (join a (complement b)) (join (join (meet a b) (meet (complement a) b)) (meet (complement a) (complement b)))) (join (meet a b) (meet (complement a) (complement b)))
+.
+#Univ.
+#A.
+#B.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#complement.
+#join.
+#meet.
+#n0.
+#n1.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT063-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT063-1 : TPTP v3.2.0. Released v2.5.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : E62 does not necessarily hold in ortholattices *)
+
+(* Version : [EF+02] axioms. *)
+
+(* English : *)
+
+(* Refs : [EF+02] Ernst et al. (2002), More First-order Test Problems in *)
+
+(* Source : [EF+02] *)
+
+(* Names : ol-e62 [EF+02] *)
+
+(* Status : Satisfiable *)
+
+(* Rating : 0.33 v3.2.0, 0.67 v3.1.0, 0.33 v2.6.0, 0.67 v2.5.0 *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 1 RR) *)
+
+(* Number of atoms : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 20 ( 2 singleton) *)
+
+(* Maximal term depth : 6 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include lattice axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Ortholattice axioms *)
+
+(* ----Denial of E62 *)
+ntheorem prove_e62:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀complement:∀_:Univ.Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀n0:Univ.
+∀n1:Univ.
+∀H0:∀A:Univ.∀B:Univ.eq Univ (meet A B) (complement (join (complement A) (complement B))).
+∀H1:∀A:Univ.eq Univ (meet (complement A) A) n0.
+∀H2:∀A:Univ.eq Univ (join (complement A) A) n1.
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H7:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H8:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H9:∀X:Univ.eq Univ (join X X) X.
+∀H10:∀X:Univ.eq Univ (meet X X) X.eq Univ (meet a (join b (meet a (join (complement a) (meet a b))))) (meet a (join (complement a) (meet a b)))
+.
+#Univ.
+#A.
+#B.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#complement.
+#join.
+#meet.
+#n0.
+#n1.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT070-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT070-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Lattice Theory (Ortholattices) *)
+
+(* Problem : Given single axiom OL-23A, prove associativity *)
+
+(* Version : [MRV03] (equality) axioms. *)
+
+(* English : Given a single axiom OL-23A for ortholattices (OL) in terms of *)
+
+(* the Sheffer Stroke, prove a Sheffer stroke form of associativity. *)
+
+(* Refs : [MRV03] McCune et al. (2003), Sheffer Stroke Bases for Ortholatt *)
+
+(* Source : [MRV03] *)
+
+(* Names : OL-23A-associativity [MRV03] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.93 v3.1.0, 1.00 v2.7.0, 0.91 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 4 ( 1 singleton) *)
+
+(* Maximal term depth : 7 ( 4 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Single axiom OL-23A *)
+
+(* ----Denial of Sheffer stroke associativity *)
+ntheorem associativity:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀f:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.eq Univ (f (f (f (f B A) (f A C)) D) (f A (f (f A (f (f B B) B)) C))) A.eq Univ (f a (f (f b c) (f b c))) (f c (f (f b a) (f b a)))
+.
+#Univ.
+#A.
+#B.
+#C.
+#D.
+#a.
+#b.
+#c.
+#f.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT071-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT071-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Lattice Theory (Orthomodularlattices) *)
+
+(* Problem : Given single axiom OML-21C, prove associativity *)
+
+(* Version : [MRV03] (equality) axioms. *)
+
+(* English : Given a single axiom candidate OML-21C for orthomodular lattices *)
+
+(* (OML) in terms of the Sheffer Stroke, prove a Sheffer stroke form *)
+
+(* of associativity. *)
+
+(* Refs : [MRV03] McCune et al. (2003), Sheffer Stroke Bases for Ortholatt *)
+
+(* Source : [MRV03] *)
+
+(* Names : OML-21C-associativity [MRV03] *)
+
+(* Status : Open *)
+
+(* Rating : 1.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 4 ( 2 singleton) *)
+
+(* Maximal term depth : 6 ( 4 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Single axiom OML-21C *)
+
+(* ----Denial of Sheffer stroke associativity *)
+ntheorem associativity:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀f:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.eq Univ (f (f B A) (f (f (f (f B A) A) (f C A)) (f (f A A) D))) A.eq Univ (f a (f (f b c) (f b c))) (f c (f (f b a) (f b a)))
+.
+#Univ.
+#A.
+#B.
+#C.
+#D.
+#a.
+#b.
+#c.
+#f.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT072-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT072-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Lattice Theory (Ortholattices) *)
+
+(* Problem : Given single axiom OML-23A, prove associativity *)
+
+(* Version : [MRV03] (equality) axioms. *)
+
+(* English : Given a single axiom candidate OML-23A for orthomodular lattices *)
+
+(* (OML) in terms of the Sheffer Stroke, prove a Sheffer stroke form *)
+
+(* of associativity. *)
+
+(* Refs : [MRV03] McCune et al. (2003), Sheffer Stroke Bases for Ortholatt *)
+
+(* Source : [MRV03] *)
+
+(* Names : OML-23A-associativity [MRV03] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 1.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 4 ( 2 singleton) *)
+
+(* Maximal term depth : 7 ( 4 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Single axiom OML-23A *)
+
+(* ----Denial of Sheffer stroke associativity *)
+ntheorem associativity:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀f:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.eq Univ (f (f (f (f B A) (f A C)) D) (f A (f (f C (f (f A A) C)) C))) A.eq Univ (f a (f (f b c) (f b c))) (f c (f (f b a) (f b a)))
+.
+#Univ.
+#A.
+#B.
+#C.
+#D.
+#a.
+#b.
+#c.
+#f.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT073-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT073-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Lattice Theory (Ortholattices) *)
+
+(* Problem : Given single axiom MOL-23C, prove modularity *)
+
+(* Version : [MRV03] (equality) axioms. *)
+
+(* English : Given a single axiom candidate MOL-23C for modular ortholattices *)
+
+(* (MOL) in terms of the Sheffer Stroke, prove a Sheffer stroke form *)
+
+(* of modularity. *)
+
+(* Refs : [MRV03] McCune et al. (2003), Sheffer Stroke Bases for Ortholatt *)
+
+(* Source : [MRV03] *)
+
+(* Names : MOL-23C-modularity [MRV03] *)
+
+(* Status : Open *)
+
+(* Rating : 1.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 4 ( 1 singleton) *)
+
+(* Maximal term depth : 7 ( 4 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Single axiom MOL-23C *)
+
+(* ----Denial of Sheffer stroke modularity *)
+ntheorem modularity:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀f:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.eq Univ (f (f (f B (f A B)) B) (f A (f C (f (f A B) (f (f C C) D))))) A.eq Univ (f a (f b (f a (f c c)))) (f a (f c (f a (f b b))))
+.
+#Univ.
+#A.
+#B.
+#C.
+#D.
+#a.
+#b.
+#c.
+#f.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT074-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT074-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Lattice Theory (Ortholattices) *)
+
+(* Problem : Given single axiom MOL-25A, prove associativity *)
+
+(* Version : [MRV03] (equality) axioms. *)
+
+(* English : Given a single axiom candidate MOL-25A for modular ortholattices *)
+
+(* (MOL) in terms of the Sheffer Stroke, prove a Sheffer stroke form *)
+
+(* of associativity. *)
+
+(* Refs : [MRV03] McCune et al. (2003), Sheffer Stroke Bases for Ortholatt *)
+
+(* Source : [MRV03] *)
+
+(* Names : MOL-25A-associativity [MRV03] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 1.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 4 ( 1 singleton) *)
+
+(* Maximal term depth : 8 ( 4 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Single axiom MOL-25A *)
+
+(* ----Denial of Sheffer stroke associativity *)
+ntheorem associativity:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀f:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.eq Univ (f (f B A) (f (f (f A A) C) (f (f (f (f (f A B) C) C) A) (f A D)))) A.eq Univ (f a (f (f b c) (f b c))) (f c (f (f b a) (f b a)))
+.
+#Univ.
+#A.
+#B.
+#C.
+#D.
+#a.
+#b.
+#c.
+#f.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT075-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT075-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Lattice Theory (Ortholattices) *)
+
+(* Problem : Given single axiom MOL-25A, prove modularity *)
+
+(* Version : [MRV03] (equality) axioms. *)
+
+(* English : Given a single axiom candidate MOL-25A for modular ortholattices *)
+
+(* (MOL) in terms of the Sheffer Stroke, prove a Sheffer stroke form *)
+
+(* of modularity. *)
+
+(* Refs : [MRV03] McCune et al. (2003), Sheffer Stroke Bases for Ortholatt *)
+
+(* Source : [MRV03] *)
+
+(* Names : MOL-25A-modularity [MRV03] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 1.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 4 ( 1 singleton) *)
+
+(* Maximal term depth : 8 ( 5 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Single axiom MOL-25A *)
+
+(* ----Denial of Sheffer stroke modularity *)
+ntheorem modularity:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀f:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.eq Univ (f (f B A) (f (f (f A A) C) (f (f (f (f (f A B) C) C) A) (f A D)))) A.eq Univ (f a (f b (f a (f c c)))) (f a (f c (f a (f b b))))
+.
+#Univ.
+#A.
+#B.
+#C.
+#D.
+#a.
+#b.
+#c.
+#f.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT076-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT076-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Lattice Theory (Ortholattices) *)
+
+(* Problem : Given single axiom MOL-27B1, prove associativity *)
+
+(* Version : [MRV03] (equality) axioms. *)
+
+(* English : Given a single axiom candidate MOL-27B1 for modular ortholattices *)
+
+(* (MOL) in terms of the Sheffer Stroke, prove a Sheffer stroke form *)
+
+(* of associativity. *)
+
+(* Refs : [MRV03] McCune et al. (2003), Sheffer Stroke Bases for Ortholatt *)
+
+(* Source : [MRV03] *)
+
+(* Names : MOL-27B1-associativity [MRV03] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 1.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 4 ( 1 singleton) *)
+
+(* Maximal term depth : 9 ( 4 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Single axiom MOL-27B1 *)
+
+(* ----Denial of Sheffer stroke associativity *)
+ntheorem associativity:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀f:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.eq Univ (f (f (f (f B A) (f C A)) D) (f A (f (f (f (f (f (f B B) A) C) C) A) B))) A.eq Univ (f a (f (f b c) (f b c))) (f c (f (f b a) (f b a)))
+.
+#Univ.
+#A.
+#B.
+#C.
+#D.
+#a.
+#b.
+#c.
+#f.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT077-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT077-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Lattice Theory (Ortholattices) *)
+
+(* Problem : Given single axiom MOL-27B1, prove modularity *)
+
+(* Version : [MRV03] (equality) axioms. *)
+
+(* English : Given a single axiom candidate MOL-27B1 for modular ortholattices *)
+
+(* (MOL) in terms of the Sheffer Stroke, prove a Sheffer stroke form *)
+
+(* of modularity. *)
+
+(* Refs : [MRV03] McCune et al. (2003), Sheffer Stroke Bases for Ortholatt *)
+
+(* Source : [MRV03] *)
+
+(* Names : MOL-27B1-modularity [MRV03] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 1.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 4 ( 1 singleton) *)
+
+(* Maximal term depth : 9 ( 5 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Single axiom MOL-27B1 *)
+
+(* ----Denial of Sheffer stroke modularity *)
+ntheorem modularity:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀f:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.eq Univ (f (f (f (f B A) (f C A)) D) (f A (f (f (f (f (f (f B B) A) C) C) A) B))) A.eq Univ (f a (f b (f a (f c c)))) (f a (f c (f a (f b b))))
+.
+#Univ.
+#A.
+#B.
+#C.
+#D.
+#a.
+#b.
+#c.
+#f.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT078-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT078-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Lattice Theory (Ortholattices) *)
+
+(* Problem : Given single axiom MOL-27B2, prove associativity *)
+
+(* Version : [MRV03] (equality) axioms. *)
+
+(* English : Given a single axiom candidate MOL-27B2 for modular ortholattices *)
+
+(* (MOL) in terms of the Sheffer Stroke, prove a Sheffer stroke form *)
+
+(* of associativity. *)
+
+(* Refs : [MRV03] McCune et al. (2003), Sheffer Stroke Bases for Ortholatt *)
+
+(* Source : [MRV03] *)
+
+(* Names : MOL-27B2-associativity [MRV03] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 1.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 4 ( 1 singleton) *)
+
+(* Maximal term depth : 9 ( 4 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Single axiom MOL-27B2 *)
+
+(* ----Denial of Sheffer stroke associativity *)
+ntheorem associativity:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀f:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.eq Univ (f (f (f (f B A) (f A C)) D) (f A (f (f (f B (f B (f (f C C) A))) A) C))) A.eq Univ (f a (f (f b c) (f b c))) (f c (f (f b a) (f b a)))
+.
+#Univ.
+#A.
+#B.
+#C.
+#D.
+#a.
+#b.
+#c.
+#f.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT079-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT079-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Lattice Theory (Ortholattices) *)
+
+(* Problem : Given single axiom MOL-27B2, prove modularity *)
+
+(* Version : [MRV03] (equality) axioms. *)
+
+(* English : Given a single axiom candidate MOL-27B2 for modular ortholattices *)
+
+(* (MOL) in terms of the Sheffer Stroke, prove a Sheffer stroke form *)
+
+(* of modularity. *)
+
+(* Refs : [MRV03] McCune et al. (2003), Sheffer Stroke Bases for Ortholatt *)
+
+(* Source : [MRV03] *)
+
+(* Names : MOL-27B2-modularity [MRV03] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 1.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 4 ( 1 singleton) *)
+
+(* Maximal term depth : 9 ( 5 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Single axiom MOL-27B2 *)
+
+(* ----Denial of Sheffer stroke modularity *)
+ntheorem modularity:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀f:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.eq Univ (f (f (f (f B A) (f A C)) D) (f A (f (f (f B (f B (f (f C C) A))) A) C))) A.eq Univ (f a (f b (f a (f c c)))) (f a (f c (f a (f b b))))
+.
+#Univ.
+#A.
+#B.
+#C.
+#D.
+#a.
+#b.
+#c.
+#f.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT080-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT080-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Axiom for lattice theory, part 1 *)
+
+(* Version : [MP96] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC98] McCune (1998), Email to G. Sutcliffe *)
+
+(* : [MP96] McCune & Padmanabhan (1996), Automated Deduction in Eq *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.36 v3.1.0, 0.11 v2.7.0, 0.55 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 7 ( 1 singleton) *)
+
+(* Maximal term depth : 12 ( 4 average) *)
+
+(* Comments : A UEQ part of LAT015-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_normal_axioms_1:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.∀E:Univ.∀F:Univ.∀G:Univ.
+∀a:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.∀E:Univ.∀F:Univ.∀G:Univ.eq Univ (join (meet (join (meet A B) (meet B (join A B))) C) (meet (join (meet A (join (join (meet D B) (meet B E)) B)) (meet (join (meet B (meet (meet (join D (join B E)) (join F B)) B)) (meet G (join B (meet (meet (join D (join B E)) (join F B)) B)))) (join A (join (join (meet D B) (meet B E)) B)))) (join (join (meet A B) (meet B (join A B))) C))) B.eq Univ (meet a a) a
+.
+#Univ.
+#A.
+#B.
+#C.
+#D.
+#E.
+#F.
+#G.
+#a.
+#join.
+#meet.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT081-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT081-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Axiom for lattice theory, part 2 *)
+
+(* Version : [MP96] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC98] McCune (1998), Email to G. Sutcliffe *)
+
+(* : [MP96] McCune & Padmanabhan (1996), Automated Deduction in Eq *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.57 v3.1.0, 0.44 v2.7.0, 0.55 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 7 ( 1 singleton) *)
+
+(* Maximal term depth : 12 ( 4 average) *)
+
+(* Comments : A UEQ part of LAT015-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_normal_axioms_2:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.∀E:Univ.∀F:Univ.∀G:Univ.
+∀a:Univ.
+∀b:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.∀E:Univ.∀F:Univ.∀G:Univ.eq Univ (join (meet (join (meet A B) (meet B (join A B))) C) (meet (join (meet A (join (join (meet D B) (meet B E)) B)) (meet (join (meet B (meet (meet (join D (join B E)) (join F B)) B)) (meet G (join B (meet (meet (join D (join B E)) (join F B)) B)))) (join A (join (join (meet D B) (meet B E)) B)))) (join (join (meet A B) (meet B (join A B))) C))) B.eq Univ (meet a b) (meet b a)
+.
+#Univ.
+#A.
+#B.
+#C.
+#D.
+#E.
+#F.
+#G.
+#a.
+#b.
+#join.
+#meet.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT082-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT082-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Axiom for lattice theory, part 3 *)
+
+(* Version : [MP96] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC98] McCune (1998), Email to G. Sutcliffe *)
+
+(* : [MP96] McCune & Padmanabhan (1996), Automated Deduction in Eq *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.64 v3.1.0, 0.56 v2.7.0, 0.64 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 7 ( 1 singleton) *)
+
+(* Maximal term depth : 12 ( 5 average) *)
+
+(* Comments : A UEQ part of LAT015-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_normal_axioms_3:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.∀E:Univ.∀F:Univ.∀G:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.∀E:Univ.∀F:Univ.∀G:Univ.eq Univ (join (meet (join (meet A B) (meet B (join A B))) C) (meet (join (meet A (join (join (meet D B) (meet B E)) B)) (meet (join (meet B (meet (meet (join D (join B E)) (join F B)) B)) (meet G (join B (meet (meet (join D (join B E)) (join F B)) B)))) (join A (join (join (meet D B) (meet B E)) B)))) (join (join (meet A B) (meet B (join A B))) C))) B.eq Univ (meet (meet a b) c) (meet a (meet b c))
+.
+#Univ.
+#A.
+#B.
+#C.
+#D.
+#E.
+#F.
+#G.
+#a.
+#b.
+#c.
+#join.
+#meet.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT083-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT083-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Axiom for lattice theory, part 4 *)
+
+(* Version : [MP96] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC98] McCune (1998), Email to G. Sutcliffe *)
+
+(* : [MP96] McCune & Padmanabhan (1996), Automated Deduction in Eq *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.36 v3.1.0, 0.11 v2.7.0, 0.55 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 7 ( 1 singleton) *)
+
+(* Maximal term depth : 12 ( 4 average) *)
+
+(* Comments : A UEQ part of LAT015-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_normal_axioms_4:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.∀E:Univ.∀F:Univ.∀G:Univ.
+∀a:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.∀E:Univ.∀F:Univ.∀G:Univ.eq Univ (join (meet (join (meet A B) (meet B (join A B))) C) (meet (join (meet A (join (join (meet D B) (meet B E)) B)) (meet (join (meet B (meet (meet (join D (join B E)) (join F B)) B)) (meet G (join B (meet (meet (join D (join B E)) (join F B)) B)))) (join A (join (join (meet D B) (meet B E)) B)))) (join (join (meet A B) (meet B (join A B))) C))) B.eq Univ (join a a) a
+.
+#Univ.
+#A.
+#B.
+#C.
+#D.
+#E.
+#F.
+#G.
+#a.
+#join.
+#meet.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT084-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT084-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Axiom for lattice theory, part 5 *)
+
+(* Version : [MP96] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC98] McCune (1998), Email to G. Sutcliffe *)
+
+(* : [MP96] McCune & Padmanabhan (1996), Automated Deduction in Eq *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.50 v3.1.0, 0.56 v2.7.0, 0.64 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 7 ( 1 singleton) *)
+
+(* Maximal term depth : 12 ( 4 average) *)
+
+(* Comments : A UEQ part of LAT015-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_normal_axioms_5:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.∀E:Univ.∀F:Univ.∀G:Univ.
+∀a:Univ.
+∀b:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.∀E:Univ.∀F:Univ.∀G:Univ.eq Univ (join (meet (join (meet A B) (meet B (join A B))) C) (meet (join (meet A (join (join (meet D B) (meet B E)) B)) (meet (join (meet B (meet (meet (join D (join B E)) (join F B)) B)) (meet G (join B (meet (meet (join D (join B E)) (join F B)) B)))) (join A (join (join (meet D B) (meet B E)) B)))) (join (join (meet A B) (meet B (join A B))) C))) B.eq Univ (join a b) (join b a)
+.
+#Univ.
+#A.
+#B.
+#C.
+#D.
+#E.
+#F.
+#G.
+#a.
+#b.
+#join.
+#meet.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT085-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT085-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Axiom for lattice theory, part 6 *)
+
+(* Version : [MP96] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC98] McCune (1998), Email to G. Sutcliffe *)
+
+(* : [MP96] McCune & Padmanabhan (1996), Automated Deduction in Eq *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.57 v3.1.0, 0.67 v2.7.0, 0.64 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 7 ( 1 singleton) *)
+
+(* Maximal term depth : 12 ( 5 average) *)
+
+(* Comments : A UEQ part of LAT015-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_normal_axioms_6:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.∀E:Univ.∀F:Univ.∀G:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.∀E:Univ.∀F:Univ.∀G:Univ.eq Univ (join (meet (join (meet A B) (meet B (join A B))) C) (meet (join (meet A (join (join (meet D B) (meet B E)) B)) (meet (join (meet B (meet (meet (join D (join B E)) (join F B)) B)) (meet G (join B (meet (meet (join D (join B E)) (join F B)) B)))) (join A (join (join (meet D B) (meet B E)) B)))) (join (join (meet A B) (meet B (join A B))) C))) B.eq Univ (join (join a b) c) (join a (join b c))
+.
+#Univ.
+#A.
+#B.
+#C.
+#D.
+#E.
+#F.
+#G.
+#a.
+#b.
+#c.
+#join.
+#meet.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT086-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT086-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Axiom for lattice theory, part 7 *)
+
+(* Version : [MP96] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC98] McCune (1998), Email to G. Sutcliffe *)
+
+(* : [MP96] McCune & Padmanabhan (1996), Automated Deduction in Eq *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.50 v3.1.0, 0.33 v2.7.0, 0.55 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 7 ( 1 singleton) *)
+
+(* Maximal term depth : 12 ( 4 average) *)
+
+(* Comments : A UEQ part of LAT015-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_normal_axioms_7:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.∀E:Univ.∀F:Univ.∀G:Univ.
+∀a:Univ.
+∀b:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.∀E:Univ.∀F:Univ.∀G:Univ.eq Univ (join (meet (join (meet A B) (meet B (join A B))) C) (meet (join (meet A (join (join (meet D B) (meet B E)) B)) (meet (join (meet B (meet (meet (join D (join B E)) (join F B)) B)) (meet G (join B (meet (meet (join D (join B E)) (join F B)) B)))) (join A (join (join (meet D B) (meet B E)) B)))) (join (join (meet A B) (meet B (join A B))) C))) B.eq Univ (meet a (join a b)) a
+.
+#Univ.
+#A.
+#B.
+#C.
+#D.
+#E.
+#F.
+#G.
+#a.
+#b.
+#join.
+#meet.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT087-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT087-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Axiom for lattice theory, part 8 *)
+
+(* Version : [MP96] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [McC98] McCune (1998), Email to G. Sutcliffe *)
+
+(* : [MP96] McCune & Padmanabhan (1996), Automated Deduction in Eq *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.43 v3.1.0, 0.22 v2.7.0, 0.55 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 7 ( 1 singleton) *)
+
+(* Maximal term depth : 12 ( 4 average) *)
+
+(* Comments : A UEQ part of LAT015-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_normal_axioms_8:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.∀E:Univ.∀F:Univ.∀G:Univ.
+∀a:Univ.
+∀b:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.∀E:Univ.∀F:Univ.∀G:Univ.eq Univ (join (meet (join (meet A B) (meet B (join A B))) C) (meet (join (meet A (join (join (meet D B) (meet B E)) B)) (meet (join (meet B (meet (meet (join D (join B E)) (join F B)) B)) (meet G (join B (meet (meet (join D (join B E)) (join F B)) B)))) (join A (join (join (meet D B) (meet B E)) B)))) (join (join (meet A B) (meet B (join A B))) C))) B.eq Univ (join a (meet a b)) a
+.
+#Univ.
+#A.
+#B.
+#C.
+#D.
+#E.
+#F.
+#G.
+#a.
+#b.
+#join.
+#meet.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT088-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT088-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Lattice Theory (Weakly Associative Lattices) *)
+
+(* Problem : Absorption basis for WAL, part 1 *)
+
+(* Version : [MP96] (equality) axioms : Especial. *)
+
+(* English : *)
+
+(* Refs : [McC98] McCune (1998), Email to G. Sutcliffe *)
+
+(* : [MP96] McCune & Padmanabhan (1996), Automated Deduction in Eq *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 6 ( 0 non-Horn; 6 unit; 1 RR) *)
+
+(* Number of atoms : 6 ( 6 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 12 ( 5 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : A UEQ part of LAT029-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_normal_axioms_1:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (join (join (meet A B) (meet C A)) A) A.
+∀H1:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (meet (meet (join A B) (join C A)) A) A.
+∀H2:∀A:Univ.∀B:Univ.eq Univ (join (meet A B) (meet B (join A B))) B.
+∀H3:∀A:Univ.∀B:Univ.eq Univ (join (meet A A) (meet B (join A A))) A.
+∀H4:∀A:Univ.∀B:Univ.eq Univ (join (meet A B) (meet A (join A B))) A.eq Univ (meet a a) a
+.
+#Univ.
+#A.
+#B.
+#C.
+#a.
+#join.
+#meet.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+nauto by H0,H1,H2,H3,H4;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT089-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT089-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Lattice Theory (Weakly Associative Lattices) *)
+
+(* Problem : Absorption basis for WAL, part 2 *)
+
+(* Version : [MP96] (equality) axioms : Especial. *)
+
+(* English : *)
+
+(* Refs : [McC98] McCune (1998), Email to G. Sutcliffe *)
+
+(* : [MP96] McCune & Padmanabhan (1996), Automated Deduction in Eq *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 6 ( 0 non-Horn; 6 unit; 1 RR) *)
+
+(* Number of atoms : 6 ( 6 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 12 ( 5 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : A UEQ part of LAT029-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_normal_axioms_2:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a:Univ.
+∀b:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (join (join (meet A B) (meet C A)) A) A.
+∀H1:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (meet (meet (join A B) (join C A)) A) A.
+∀H2:∀A:Univ.∀B:Univ.eq Univ (join (meet A B) (meet B (join A B))) B.
+∀H3:∀A:Univ.∀B:Univ.eq Univ (join (meet A A) (meet B (join A A))) A.
+∀H4:∀A:Univ.∀B:Univ.eq Univ (join (meet A B) (meet A (join A B))) A.eq Univ (meet b a) (meet a b)
+.
+#Univ.
+#A.
+#B.
+#C.
+#a.
+#b.
+#join.
+#meet.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+nauto by H0,H1,H2,H3,H4;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT090-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT090-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Lattice Theory (Weakly Associative Lattices) *)
+
+(* Problem : Absorption basis for WAL, part 3 *)
+
+(* Version : [MP96] (equality) axioms : Especial. *)
+
+(* English : *)
+
+(* Refs : [McC98] McCune (1998), Email to G. Sutcliffe *)
+
+(* : [MP96] McCune & Padmanabhan (1996), Automated Deduction in Eq *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 6 ( 0 non-Horn; 6 unit; 1 RR) *)
+
+(* Number of atoms : 6 ( 6 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 12 ( 5 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : A UEQ part of LAT029-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_normal_axioms_3:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (join (join (meet A B) (meet C A)) A) A.
+∀H1:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (meet (meet (join A B) (join C A)) A) A.
+∀H2:∀A:Univ.∀B:Univ.eq Univ (join (meet A B) (meet B (join A B))) B.
+∀H3:∀A:Univ.∀B:Univ.eq Univ (join (meet A A) (meet B (join A A))) A.
+∀H4:∀A:Univ.∀B:Univ.eq Univ (join (meet A B) (meet A (join A B))) A.eq Univ (join a a) a
+.
+#Univ.
+#A.
+#B.
+#C.
+#a.
+#join.
+#meet.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+nauto by H0,H1,H2,H3,H4;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT091-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT091-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Lattice Theory (Weakly Associative Lattices) *)
+
+(* Problem : Absorption basis for WAL, part 4 *)
+
+(* Version : [MP96] (equality) axioms : Especial. *)
+
+(* English : *)
+
+(* Refs : [McC98] McCune (1998), Email to G. Sutcliffe *)
+
+(* : [MP96] McCune & Padmanabhan (1996), Automated Deduction in Eq *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.07 v3.1.0, 0.11 v2.7.0, 0.00 v2.6.0 *)
+
+(* Syntax : Number of clauses : 6 ( 0 non-Horn; 6 unit; 1 RR) *)
+
+(* Number of atoms : 6 ( 6 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 12 ( 5 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : A UEQ part of LAT029-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_normal_axioms_4:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.
+∀a:Univ.
+∀b:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (join (join (meet A B) (meet C A)) A) A.
+∀H1:∀A:Univ.∀B:Univ.∀C:Univ.eq Univ (meet (meet (join A B) (join C A)) A) A.
+∀H2:∀A:Univ.∀B:Univ.eq Univ (join (meet A B) (meet B (join A B))) B.
+∀H3:∀A:Univ.∀B:Univ.eq Univ (join (meet A A) (meet B (join A A))) A.
+∀H4:∀A:Univ.∀B:Univ.eq Univ (join (meet A B) (meet A (join A B))) A.eq Univ (join b a) (join a b)
+.
+#Univ.
+#A.
+#B.
+#C.
+#a.
+#b.
+#join.
+#meet.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+nauto by H0,H1,H2,H3,H4;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT092-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT092-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Lattice Theory (Weakly Associative Lattices) *)
+
+(* Problem : Axiom for weakly associative lattices (WAL), part 1 *)
+
+(* Version : [MP96] (equality) axioms : Especial. *)
+
+(* English : *)
+
+(* Refs : [McC98] McCune (1998), Email to G. Sutcliffe *)
+
+(* : [MP96] McCune & Padmanabhan (1996), Automated Deduction in Eq *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.36 v3.1.0, 0.11 v2.7.0, 0.45 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 6 ( 1 singleton) *)
+
+(* Maximal term depth : 11 ( 4 average) *)
+
+(* Comments : A UEQ part of LAT030-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_wal_axioms_1:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.∀E:Univ.∀F:Univ.
+∀a:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.∀E:Univ.∀F:Univ.eq Univ (join (meet (join (meet A B) (meet B (join A B))) C) (meet (join (meet A (join (join (meet B D) (meet E B)) B)) (meet (join (meet B (meet (meet (join B D) (join E B)) B)) (meet F (join B (meet (meet (join B D) (join E B)) B)))) (join A (join (join (meet B D) (meet E B)) B)))) (join (join (meet A B) (meet B (join A B))) C))) B.eq Univ (meet a a) a
+.
+#Univ.
+#A.
+#B.
+#C.
+#D.
+#E.
+#F.
+#a.
+#join.
+#meet.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT093-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT093-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Lattice Theory (Weakly Associative Lattices) *)
+
+(* Problem : Axiom for weakly associative lattices (WAL), part 2 *)
+
+(* Version : [MP96] (equality) axioms : Especial. *)
+
+(* English : *)
+
+(* Refs : [McC98] McCune (1998), Email to G. Sutcliffe *)
+
+(* : [MP96] McCune & Padmanabhan (1996), Automated Deduction in Eq *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.43 v3.1.0, 0.22 v2.7.0, 0.45 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 6 ( 1 singleton) *)
+
+(* Maximal term depth : 11 ( 4 average) *)
+
+(* Comments : A UEQ part of LAT030-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_wal_axioms_2:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.∀E:Univ.∀F:Univ.
+∀a:Univ.
+∀b:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.∀E:Univ.∀F:Univ.eq Univ (join (meet (join (meet A B) (meet B (join A B))) C) (meet (join (meet A (join (join (meet B D) (meet E B)) B)) (meet (join (meet B (meet (meet (join B D) (join E B)) B)) (meet F (join B (meet (meet (join B D) (join E B)) B)))) (join A (join (join (meet B D) (meet E B)) B)))) (join (join (meet A B) (meet B (join A B))) C))) B.eq Univ (meet b a) (meet a b)
+.
+#Univ.
+#A.
+#B.
+#C.
+#D.
+#E.
+#F.
+#a.
+#b.
+#join.
+#meet.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT094-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT094-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Lattice Theory (Weakly Associative Lattices) *)
+
+(* Problem : Axiom for weakly associative lattices (WAL), part 3 *)
+
+(* Version : [MP96] (equality) axioms : Especial. *)
+
+(* English : *)
+
+(* Refs : [McC98] McCune (1998), Email to G. Sutcliffe *)
+
+(* : [MP96] McCune & Padmanabhan (1996), Automated Deduction in Eq *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.36 v3.1.0, 0.11 v2.7.0, 0.45 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 6 ( 1 singleton) *)
+
+(* Maximal term depth : 11 ( 4 average) *)
+
+(* Comments : A UEQ part of LAT030-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_wal_axioms_3:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.∀E:Univ.∀F:Univ.
+∀a:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.∀E:Univ.∀F:Univ.eq Univ (join (meet (join (meet A B) (meet B (join A B))) C) (meet (join (meet A (join (join (meet B D) (meet E B)) B)) (meet (join (meet B (meet (meet (join B D) (join E B)) B)) (meet F (join B (meet (meet (join B D) (join E B)) B)))) (join A (join (join (meet B D) (meet E B)) B)))) (join (join (meet A B) (meet B (join A B))) C))) B.eq Univ (join a a) a
+.
+#Univ.
+#A.
+#B.
+#C.
+#D.
+#E.
+#F.
+#a.
+#join.
+#meet.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT095-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT095-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Lattice Theory (Weakly Associative Lattices) *)
+
+(* Problem : Axiom for weakly associative lattices (WAL), part 4 *)
+
+(* Version : [MP96] (equality) axioms : Especial. *)
+
+(* English : *)
+
+(* Refs : [McC98] McCune (1998), Email to G. Sutcliffe *)
+
+(* : [MP96] McCune & Padmanabhan (1996), Automated Deduction in Eq *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.43 v3.1.0, 0.22 v2.7.0, 0.45 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 6 ( 1 singleton) *)
+
+(* Maximal term depth : 11 ( 4 average) *)
+
+(* Comments : A UEQ part of LAT030-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_wal_axioms_4:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.∀E:Univ.∀F:Univ.
+∀a:Univ.
+∀b:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.∀E:Univ.∀F:Univ.eq Univ (join (meet (join (meet A B) (meet B (join A B))) C) (meet (join (meet A (join (join (meet B D) (meet E B)) B)) (meet (join (meet B (meet (meet (join B D) (join E B)) B)) (meet F (join B (meet (meet (join B D) (join E B)) B)))) (join A (join (join (meet B D) (meet E B)) B)))) (join (join (meet A B) (meet B (join A B))) C))) B.eq Univ (join b a) (join a b)
+.
+#Univ.
+#A.
+#B.
+#C.
+#D.
+#E.
+#F.
+#a.
+#b.
+#join.
+#meet.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT096-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT096-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Lattice Theory (Weakly Associative Lattices) *)
+
+(* Problem : Axiom for weakly associative lattices (WAL), part 5 *)
+
+(* Version : [MP96] (equality) axioms : Especial. *)
+
+(* English : *)
+
+(* Refs : [McC98] McCune (1998), Email to G. Sutcliffe *)
+
+(* : [MP96] McCune & Padmanabhan (1996), Automated Deduction in Eq *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.36 v3.1.0, 0.11 v2.7.0, 0.45 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 6 ( 1 singleton) *)
+
+(* Maximal term depth : 11 ( 4 average) *)
+
+(* Comments : A UEQ part of LAT030-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_wal_axioms_5:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.∀E:Univ.∀F:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.∀E:Univ.∀F:Univ.eq Univ (join (meet (join (meet A B) (meet B (join A B))) C) (meet (join (meet A (join (join (meet B D) (meet E B)) B)) (meet (join (meet B (meet (meet (join B D) (join E B)) B)) (meet F (join B (meet (meet (join B D) (join E B)) B)))) (join A (join (join (meet B D) (meet E B)) B)))) (join (join (meet A B) (meet B (join A B))) C))) B.eq Univ (meet (meet (join a b) (join c b)) b) b
+.
+#Univ.
+#A.
+#B.
+#C.
+#D.
+#E.
+#F.
+#a.
+#b.
+#c.
+#join.
+#meet.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT097-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT097-1 : TPTP v3.2.0. Released v2.6.0. *)
+
+(* Domain : Lattice Theory (Weakly Associative Lattices) *)
+
+(* Problem : Single axiom for weakly associative lattices (WAL), part 6 *)
+
+(* Version : [MP96] (equality) axioms : Especial. *)
+
+(* English : *)
+
+(* Refs : [McC98] McCune (1998), Email to G. Sutcliffe *)
+
+(* : [MP96] McCune & Padmanabhan (1996), Automated Deduction in Eq *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.36 v3.1.0, 0.11 v2.7.0, 0.45 v2.6.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 6 ( 1 singleton) *)
+
+(* Maximal term depth : 11 ( 4 average) *)
+
+(* Comments : A UEQ part of LAT030-1 *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_wal_axioms_6:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.∀E:Univ.∀F:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀A:Univ.∀B:Univ.∀C:Univ.∀D:Univ.∀E:Univ.∀F:Univ.eq Univ (join (meet (join (meet A B) (meet B (join A B))) C) (meet (join (meet A (join (join (meet B D) (meet E B)) B)) (meet (join (meet B (meet (meet (join B D) (join E B)) B)) (meet F (join B (meet (meet (join B D) (join E B)) B)))) (join A (join (join (meet B D) (meet E B)) B)))) (join (join (meet A B) (meet B (join A B))) C))) B.eq Univ (join (join (meet a b) (meet c b)) b) b
+.
+#Univ.
+#A.
+#B.
+#C.
+#D.
+#E.
+#F.
+#a.
+#b.
+#c.
+#join.
+#meet.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT098-1.p *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* File : LAT098-1 : TPTP v3.2.0. Released v3.1.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Huntington equation H3 is independent of H2 *)
+
+(* Version : [McC05] (equality) axioms : Especial. *)
+
+(* English : Show that Huntington equation H2 does not imply Huntington *)
+
+(* equation H3 in lattice theory. *)
+
+(* Refs : [McC05] McCune (2005), Email to Geoff Sutcliffe *)
+
+(* Source : [McC05] *)
+
+(* Names : *)
+
+(* Status : Satisfiable *)
+
+(* Rating : 0.33 v3.2.0, 0.67 v3.1.0 *)
+
+(* Syntax : Number of clauses : 10 ( 0 non-Horn; 10 unit; 1 RR) *)
+
+(* Number of atoms : 10 ( 10 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 19 ( 2 singleton) *)
+
+(* Maximal term depth : 8 ( 3 average) *)
+
+(* Comments : *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* ----Include Lattice theory (equality) axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ------------------------------------------------------------------------------ *)
+ntheorem prove_H3:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet X (join Y (meet X Z))) (meet X (join Y (meet Z (join (meet X (join Y Z)) (meet Y Z))))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H7:∀X:Univ.eq Univ (join X X) X.
+∀H8:∀X:Univ.eq Univ (meet X X) X.eq Univ (meet a (join b (meet a c))) (meet a (join b (meet c (join b (meet a (join c (meet a b)))))))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#join.
+#meet.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8;
+nqed.
+
+(* ------------------------------------------------------------------------------ *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT099-1.p *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* File : LAT099-1 : TPTP v3.2.0. Released v3.1.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Huntington equation H2 is independent of H3 *)
+
+(* Version : [McC05] (equality) axioms : Especial. *)
+
+(* English : Show that Huntington equation H3 does not imply Huntington *)
+
+(* equation H2 in lattice theory. *)
+
+(* Refs : [McC05] McCune (2005), Email to Geoff Sutcliffe *)
+
+(* Source : [McC05] *)
+
+(* Names : *)
+
+(* Status : Satisfiable *)
+
+(* Rating : 0.33 v3.2.0, 1.00 v3.1.0 *)
+
+(* Syntax : Number of clauses : 10 ( 0 non-Horn; 10 unit; 1 RR) *)
+
+(* Number of atoms : 10 ( 10 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 19 ( 2 singleton) *)
+
+(* Maximal term depth : 8 ( 3 average) *)
+
+(* Comments : *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* ----Include Lattice theory (equality) axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ------------------------------------------------------------------------------ *)
+ntheorem prove_H2:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet X (join Y (meet X Z))) (meet X (join Y (meet Z (join Y (meet X (join Z (meet X Y))))))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H7:∀X:Univ.eq Univ (join X X) X.
+∀H8:∀X:Univ.eq Univ (meet X X) X.eq Univ (meet a (join b (meet a c))) (meet a (join b (meet c (join (meet a (join b c)) (meet b c)))))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#join.
+#meet.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8;
+nqed.
+
+(* ------------------------------------------------------------------------------ *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT100-1.p *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* File : LAT100-1 : TPTP v3.2.0. Released v3.1.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Huntington equation H4 is independent of H6 *)
+
+(* Version : [McC05] (equality) axioms : Especial. *)
+
+(* English : Show that Huntington equation H6 does not imply Huntington *)
+
+(* equation H4 in lattice theory. *)
+
+(* Refs : [McC05] McCune (2005), Email to Geoff Sutcliffe *)
+
+(* Source : [McC05] *)
+
+(* Names : *)
+
+(* Status : Satisfiable *)
+
+(* Rating : 0.33 v3.2.0, 0.67 v3.1.0 *)
+
+(* Syntax : Number of clauses : 10 ( 0 non-Horn; 10 unit; 1 RR) *)
+
+(* Number of atoms : 10 ( 10 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 19 ( 2 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* ----Include Lattice theory (equality) axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ------------------------------------------------------------------------------ *)
+ntheorem prove_H4:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀d:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet X (join Y (meet X Z))) (meet X (join (meet X (join Y (meet X Z))) (meet Z (join X Y)))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H7:∀X:Univ.eq Univ (join X X) X.
+∀H8:∀X:Univ.eq Univ (meet X X) X.eq Univ (meet a (join b (meet a (join c d)))) (meet a (join b (meet (join a (meet b d)) (join c d))))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#d.
+#join.
+#meet.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8;
+nqed.
+
+(* ------------------------------------------------------------------------------ *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT101-1.p *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* File : LAT101-1 : TPTP v3.2.0. Released v3.1.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Huntington equation H10 is independent of H6 *)
+
+(* Version : [McC05] (equality) axioms : Especial. *)
+
+(* English : Show that Huntington equation H6 does not imply Huntington *)
+
+(* equation H10 in lattice theory. *)
+
+(* Refs : [McC05] McCune (2005), Email to Geoff Sutcliffe *)
+
+(* Source : [McC05] *)
+
+(* Names : *)
+
+(* Status : Satisfiable *)
+
+(* Rating : 0.33 v3.2.0, 0.67 v3.1.0 *)
+
+(* Syntax : Number of clauses : 10 ( 0 non-Horn; 10 unit; 1 RR) *)
+
+(* Number of atoms : 10 ( 10 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 19 ( 2 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* ----Include Lattice theory (equality) axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ------------------------------------------------------------------------------ *)
+ntheorem prove_H10:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet X (join Y (meet X Z))) (meet X (join (meet X (join Y (meet X Z))) (meet Z (join X Y)))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H7:∀X:Univ.eq Univ (join X X) X.
+∀H8:∀X:Univ.eq Univ (meet X X) X.eq Univ (meet a (join b (meet a c))) (meet a (join b (meet c (join a (meet b c)))))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#join.
+#meet.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8;
+nqed.
+
+(* ------------------------------------------------------------------------------ *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT102-1.p *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* File : LAT102-1 : TPTP v3.2.0. Released v3.1.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Huntington equation H4 is independent of H7 *)
+
+(* Version : [McC05] (equality) axioms : Especial. *)
+
+(* English : Show that Huntington equation H7 does not imply Huntington *)
+
+(* equation H4 in lattice theory. *)
+
+(* Refs : [McC05] McCune (2005), Email to Geoff Sutcliffe *)
+
+(* Source : [McC05] *)
+
+(* Names : *)
+
+(* Status : Satisfiable *)
+
+(* Rating : 0.33 v3.2.0, 0.67 v3.1.0 *)
+
+(* Syntax : Number of clauses : 10 ( 0 non-Horn; 10 unit; 1 RR) *)
+
+(* Number of atoms : 10 ( 10 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 19 ( 2 singleton) *)
+
+(* Maximal term depth : 7 ( 3 average) *)
+
+(* Comments : *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* ----Include Lattice theory (equality) axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ------------------------------------------------------------------------------ *)
+ntheorem prove_H4:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀d:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet X (join Y (meet X Z))) (meet X (join Y (meet X (join (meet X Y) (meet Z (join X Y)))))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H7:∀X:Univ.eq Univ (join X X) X.
+∀H8:∀X:Univ.eq Univ (meet X X) X.eq Univ (meet a (join b (meet a (join c d)))) (meet a (join b (meet (join a (meet b d)) (join c d))))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#d.
+#join.
+#meet.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8;
+nqed.
+
+(* ------------------------------------------------------------------------------ *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT103-1.p *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* File : LAT103-1 : TPTP v3.2.0. Released v3.1.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Huntington equation H6 is independent of H10 *)
+
+(* Version : [McC05] (equality) axioms : Especial. *)
+
+(* English : Show that Huntington equation H10 does not imply Huntington *)
+
+(* equation H6 in lattice theory. *)
+
+(* Refs : [McC05] McCune (2005), Email to Geoff Sutcliffe *)
+
+(* Source : [McC05] *)
+
+(* Names : *)
+
+(* Status : Satisfiable *)
+
+(* Rating : 0.33 v3.2.0, 0.67 v3.1.0 *)
+
+(* Syntax : Number of clauses : 10 ( 0 non-Horn; 10 unit; 1 RR) *)
+
+(* Number of atoms : 10 ( 10 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 19 ( 2 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* ----Include Lattice theory (equality) axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ------------------------------------------------------------------------------ *)
+ntheorem prove_H6:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet X (join Y (meet X Z))) (meet X (join Y (meet Z (join X (meet Y Z))))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H7:∀X:Univ.eq Univ (join X X) X.
+∀H8:∀X:Univ.eq Univ (meet X X) X.eq Univ (meet a (join b (meet a c))) (meet a (join (meet a (join b (meet a c))) (meet c (join a b))))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#join.
+#meet.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8;
+nqed.
+
+(* ------------------------------------------------------------------------------ *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT104-1.p *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* File : LAT104-1 : TPTP v3.2.0. Released v3.1.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Huntington equation H3 is independent of H21 *)
+
+(* Version : [McC05] (equality) axioms : Especial. *)
+
+(* English : Show that Huntington equation H21 does not imply Huntington *)
+
+(* equation H3 in lattice theory. *)
+
+(* Refs : [McC05] McCune (2005), Email to Geoff Sutcliffe *)
+
+(* Source : [McC05] *)
+
+(* Names : *)
+
+(* Status : Satisfiable *)
+
+(* Rating : 0.33 v3.2.0, 0.67 v3.1.0 *)
+
+(* Syntax : Number of clauses : 10 ( 0 non-Horn; 10 unit; 1 RR) *)
+
+(* Number of atoms : 10 ( 10 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 19 ( 2 singleton) *)
+
+(* Maximal term depth : 8 ( 3 average) *)
+
+(* Comments : *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* ----Include Lattice theory (equality) axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ------------------------------------------------------------------------------ *)
+ntheorem prove_H3:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (meet X Y) (meet X Z)) (meet X (join (meet Y (join X (meet Y Z))) (meet Z (join X Y)))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H7:∀X:Univ.eq Univ (join X X) X.
+∀H8:∀X:Univ.eq Univ (meet X X) X.eq Univ (meet a (join b (meet a c))) (meet a (join b (meet c (join b (meet a (join c (meet a b)))))))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#join.
+#meet.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8;
+nqed.
+
+(* ------------------------------------------------------------------------------ *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT105-1.p *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* File : LAT105-1 : TPTP v3.2.0. Released v3.1.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Huntington equation H10 is independent of H21 *)
+
+(* Version : [McC05] (equality) axioms : Especial. *)
+
+(* English : Show that Huntington equation H21 does not imply Huntington *)
+
+(* equation H10 in lattice theory. *)
+
+(* Refs : [McC05] McCune (2005), Email to Geoff Sutcliffe *)
+
+(* Source : [McC05] *)
+
+(* Names : *)
+
+(* Status : Satisfiable *)
+
+(* Rating : 0.33 v3.2.0, 0.67 v3.1.0 *)
+
+(* Syntax : Number of clauses : 10 ( 0 non-Horn; 10 unit; 1 RR) *)
+
+(* Number of atoms : 10 ( 10 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 19 ( 2 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* ----Include Lattice theory (equality) axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ------------------------------------------------------------------------------ *)
+ntheorem prove_H10:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (meet X Y) (meet X Z)) (meet X (join (meet Y (join X (meet Y Z))) (meet Z (join X Y)))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H7:∀X:Univ.eq Univ (join X X) X.
+∀H8:∀X:Univ.eq Univ (meet X X) X.eq Univ (meet a (join b (meet a c))) (meet a (join b (meet c (join a (meet b c)))))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#join.
+#meet.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8;
+nqed.
+
+(* ------------------------------------------------------------------------------ *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT106-1.p *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* File : LAT106-1 : TPTP v3.2.0. Released v3.1.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Huntington equation H3 is independent of H22 *)
+
+(* Version : [McC05] (equality) axioms : Especial. *)
+
+(* English : Show that Huntington equation H22 does not imply Huntington *)
+
+(* equation H3 in lattice theory. *)
+
+(* Refs : [McC05] McCune (2005), Email to Geoff Sutcliffe *)
+
+(* Source : [McC05] *)
+
+(* Names : *)
+
+(* Status : Satisfiable *)
+
+(* Rating : 0.67 v3.2.0, 1.00 v3.1.0 *)
+
+(* Syntax : Number of clauses : 10 ( 0 non-Horn; 10 unit; 1 RR) *)
+
+(* Number of atoms : 10 ( 10 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 19 ( 2 singleton) *)
+
+(* Maximal term depth : 8 ( 3 average) *)
+
+(* Comments : *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* ----Include Lattice theory (equality) axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ------------------------------------------------------------------------------ *)
+ntheorem prove_H3:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (meet X Y) (meet X Z)) (meet X (join (meet Y (join Z (meet X Y))) (meet Z (join X Y)))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H7:∀X:Univ.eq Univ (join X X) X.
+∀H8:∀X:Univ.eq Univ (meet X X) X.eq Univ (meet a (join b (meet a c))) (meet a (join b (meet c (join b (meet a (join c (meet a b)))))))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#join.
+#meet.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8;
+nqed.
+
+(* ------------------------------------------------------------------------------ *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT107-1.p *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* File : LAT107-1 : TPTP v3.2.0. Released v3.1.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Huntington equation H17 is independent of H22 *)
+
+(* Version : [McC05] (equality) axioms : Especial. *)
+
+(* English : Show that Huntington equation H22 does not imply Huntington *)
+
+(* equation H17 in lattice theory. *)
+
+(* Refs : [McC05] McCune (2005), Email to Geoff Sutcliffe *)
+
+(* Source : [McC05] *)
+
+(* Names : *)
+
+(* Status : Satisfiable *)
+
+(* Rating : 0.67 v3.2.0, 1.00 v3.1.0 *)
+
+(* Syntax : Number of clauses : 10 ( 0 non-Horn; 10 unit; 1 RR) *)
+
+(* Number of atoms : 10 ( 10 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 19 ( 2 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* ----Include Lattice theory (equality) axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ------------------------------------------------------------------------------ *)
+ntheorem prove_H17:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (meet X Y) (meet X Z)) (meet X (join (meet Y (join Z (meet X Y))) (meet Z (join X Y)))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H7:∀X:Univ.eq Univ (join X X) X.
+∀H8:∀X:Univ.eq Univ (meet X X) X.eq Univ (meet a (join (meet a b) (meet a c))) (meet a (join (meet b (join a (meet b c))) (meet c (join a b))))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#join.
+#meet.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8;
+nqed.
+
+(* ------------------------------------------------------------------------------ *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT108-1.p *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* File : LAT108-1 : TPTP v3.2.0. Released v3.1.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Huntington equation H42 is independent of H31 *)
+
+(* Version : [McC05] (equality) axioms : Especial. *)
+
+(* English : Show that Huntington equation H31 does not imply Huntington *)
+
+(* equation H42 in lattice theory. *)
+
+(* Refs : [McC05] McCune (2005), Email to Geoff Sutcliffe *)
+
+(* Source : [McC05] *)
+
+(* Names : *)
+
+(* Status : Satisfiable *)
+
+(* Rating : 0.33 v3.2.0, 1.00 v3.1.0 *)
+
+(* Syntax : Number of clauses : 10 ( 0 non-Horn; 10 unit; 1 RR) *)
+
+(* Number of atoms : 10 ( 10 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 20 ( 2 singleton) *)
+
+(* Maximal term depth : 7 ( 3 average) *)
+
+(* Comments : *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* ----Include Lattice theory (equality) axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ------------------------------------------------------------------------------ *)
+ntheorem prove_H42:
+ ∀Univ:Type.∀U:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀d:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀U:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet X (join Y (meet X (meet Z U)))) (meet X (join Y (meet Z (meet U (join Y (meet X Z)))))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H7:∀X:Univ.eq Univ (join X X) X.
+∀H8:∀X:Univ.eq Univ (meet X X) X.eq Univ (meet a (join b (meet c (join a d)))) (meet a (join b (meet c (join b (join d (meet a c))))))
+.
+#Univ.
+#U.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#d.
+#join.
+#meet.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8;
+nqed.
+
+(* ------------------------------------------------------------------------------ *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT109-1.p *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* File : LAT109-1 : TPTP v3.2.0. Released v3.1.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Huntington equation H40 is independent of H37 *)
+
+(* Version : [McC05] (equality) axioms : Especial. *)
+
+(* English : Show that Huntington equation H37 does not imply Huntington *)
+
+(* equation H40 in lattice theory. *)
+
+(* Refs : [McC05] McCune (2005), Email to Geoff Sutcliffe *)
+
+(* Source : [McC05] *)
+
+(* Names : *)
+
+(* Status : Satisfiable *)
+
+(* Rating : 0.33 v3.2.0, 0.67 v3.1.0 *)
+
+(* Syntax : Number of clauses : 10 ( 0 non-Horn; 10 unit; 1 RR) *)
+
+(* Number of atoms : 10 ( 10 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 20 ( 2 singleton) *)
+
+(* Maximal term depth : 7 ( 3 average) *)
+
+(* Comments : *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* ----Include Lattice theory (equality) axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ------------------------------------------------------------------------------ *)
+ntheorem prove_H40:
+ ∀Univ:Type.∀U:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀d:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀U:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet X (join Y (join Z (meet X U)))) (meet X (join Y (join Z (meet U (join X (meet Y Z)))))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H7:∀X:Univ.eq Univ (join X X) X.
+∀H8:∀X:Univ.eq Univ (meet X X) X.eq Univ (meet a (join b (meet c (join a d)))) (meet a (join b (meet c (join d (meet c (join a b))))))
+.
+#Univ.
+#U.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#d.
+#join.
+#meet.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8;
+nqed.
+
+(* ------------------------------------------------------------------------------ *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT110-1.p *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* File : LAT110-1 : TPTP v3.2.0. Released v3.1.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Huntington equation H42 is independent of H37 *)
+
+(* Version : [McC05] (equality) axioms : Especial. *)
+
+(* English : Show that Huntington equation H37 does not imply Huntington *)
+
+(* equation H42 in lattice theory. *)
+
+(* Refs : [McC05] McCune (2005), Email to Geoff Sutcliffe *)
+
+(* Source : [McC05] *)
+
+(* Names : *)
+
+(* Status : Satisfiable *)
+
+(* Rating : 0.33 v3.2.0, 1.00 v3.1.0 *)
+
+(* Syntax : Number of clauses : 10 ( 0 non-Horn; 10 unit; 1 RR) *)
+
+(* Number of atoms : 10 ( 10 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 20 ( 2 singleton) *)
+
+(* Maximal term depth : 7 ( 3 average) *)
+
+(* Comments : *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* ----Include Lattice theory (equality) axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ------------------------------------------------------------------------------ *)
+ntheorem prove_H42:
+ ∀Univ:Type.∀U:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀d:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀U:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet X (join Y (join Z (meet X U)))) (meet X (join Y (join Z (meet U (join X (meet Y Z)))))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H7:∀X:Univ.eq Univ (join X X) X.
+∀H8:∀X:Univ.eq Univ (meet X X) X.eq Univ (meet a (join b (meet c (join a d)))) (meet a (join b (meet c (join b (join d (meet a c))))))
+.
+#Univ.
+#U.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#d.
+#join.
+#meet.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8;
+nqed.
+
+(* ------------------------------------------------------------------------------ *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT111-1.p *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* File : LAT111-1 : TPTP v3.2.0. Released v3.1.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Huntington equation H40 is independent of H45 *)
+
+(* Version : [McC05] (equality) axioms : Especial. *)
+
+(* English : Show that Huntington equation H45 does not imply Huntington *)
+
+(* equation H40 in lattice theory. *)
+
+(* Refs : [McC05] McCune (2005), Email to Geoff Sutcliffe *)
+
+(* Source : [McC05] *)
+
+(* Names : *)
+
+(* Status : Satisfiable *)
+
+(* Rating : 0.33 v3.2.0, 0.67 v3.1.0 *)
+
+(* Syntax : Number of clauses : 10 ( 0 non-Horn; 10 unit; 1 RR) *)
+
+(* Number of atoms : 10 ( 10 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 20 ( 2 singleton) *)
+
+(* Maximal term depth : 7 ( 3 average) *)
+
+(* Comments : *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* ----Include Lattice theory (equality) axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ------------------------------------------------------------------------------ *)
+ntheorem prove_H40:
+ ∀Univ:Type.∀U:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀d:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀U:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet X (meet Y (join Z (meet X U)))) (meet X (meet Y (join Z (meet U (join X (meet Y Z)))))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H7:∀X:Univ.eq Univ (join X X) X.
+∀H8:∀X:Univ.eq Univ (meet X X) X.eq Univ (meet a (join b (meet c (join a d)))) (meet a (join b (meet c (join d (meet c (join a b))))))
+.
+#Univ.
+#U.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#d.
+#join.
+#meet.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8;
+nqed.
+
+(* ------------------------------------------------------------------------------ *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT112-1.p *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* File : LAT112-1 : TPTP v3.2.0. Released v3.1.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Huntington equation H42 is independent of H47 *)
+
+(* Version : [McC05] (equality) axioms : Especial. *)
+
+(* English : Show that Huntington equation H47 does not imply Huntington *)
+
+(* equation H42 in lattice theory. *)
+
+(* Refs : [McC05] McCune (2005), Email to Geoff Sutcliffe *)
+
+(* Source : [McC05] *)
+
+(* Names : *)
+
+(* Status : Satisfiable *)
+
+(* Rating : 0.33 v3.2.0, 1.00 v3.1.0 *)
+
+(* Syntax : Number of clauses : 10 ( 0 non-Horn; 10 unit; 1 RR) *)
+
+(* Number of atoms : 10 ( 10 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 20 ( 2 singleton) *)
+
+(* Maximal term depth : 7 ( 3 average) *)
+
+(* Comments : *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* ----Include Lattice theory (equality) axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ------------------------------------------------------------------------------ *)
+ntheorem prove_H42:
+ ∀Univ:Type.∀U:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀d:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀U:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet X (meet Y (join Z (meet Y U)))) (meet X (meet Y (join Z (meet U (join Y (meet X Z)))))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H7:∀X:Univ.eq Univ (join X X) X.
+∀H8:∀X:Univ.eq Univ (meet X X) X.eq Univ (meet a (join b (meet c (join a d)))) (meet a (join b (meet c (join b (join d (meet a c))))))
+.
+#Univ.
+#U.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#d.
+#join.
+#meet.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8;
+nqed.
+
+(* ------------------------------------------------------------------------------ *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT113-1.p *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* File : LAT113-1 : TPTP v3.2.0. Released v3.1.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Huntington equation H40 is independent of H50 *)
+
+(* Version : [McC05] (equality) axioms : Especial. *)
+
+(* English : Show that Huntington equation H50 does not imply Huntington *)
+
+(* equation H40 in lattice theory. *)
+
+(* Refs : [McC05] McCune (2005), Email to Geoff Sutcliffe *)
+
+(* Source : [McC05] *)
+
+(* Names : *)
+
+(* Status : Satisfiable *)
+
+(* Rating : 0.33 v3.2.0, 0.67 v3.1.0 *)
+
+(* Syntax : Number of clauses : 10 ( 0 non-Horn; 10 unit; 1 RR) *)
+
+(* Number of atoms : 10 ( 10 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 20 ( 2 singleton) *)
+
+(* Maximal term depth : 7 ( 3 average) *)
+
+(* Comments : *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* ----Include Lattice theory (equality) axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ------------------------------------------------------------------------------ *)
+ntheorem prove_H40:
+ ∀Univ:Type.∀U:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀d:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀U:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet X (join Y (meet Z (join X U)))) (meet X (join Y (meet Z (join X (meet Z (join Y U)))))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H7:∀X:Univ.eq Univ (join X X) X.
+∀H8:∀X:Univ.eq Univ (meet X X) X.eq Univ (meet a (join b (meet c (join a d)))) (meet a (join b (meet c (join d (meet c (join a b))))))
+.
+#Univ.
+#U.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#d.
+#join.
+#meet.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8;
+nqed.
+
+(* ------------------------------------------------------------------------------ *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT114-1.p *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* File : LAT114-1 : TPTP v3.2.0. Released v3.1.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Huntington equation H56 is independent of H55 *)
+
+(* Version : [McC05] (equality) axioms : Especial. *)
+
+(* English : Show that Huntington equation H55 does not imply Huntington *)
+
+(* equation H56 in lattice theory. *)
+
+(* Refs : [McC05] McCune (2005), Email to Geoff Sutcliffe *)
+
+(* Source : [McC05] *)
+
+(* Names : *)
+
+(* Status : Satisfiable *)
+
+(* Rating : 0.33 v3.2.0, 0.67 v3.1.0 *)
+
+(* Syntax : Number of clauses : 10 ( 0 non-Horn; 10 unit; 1 RR) *)
+
+(* Number of atoms : 10 ( 10 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 19 ( 2 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* ----Include Lattice theory (equality) axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ------------------------------------------------------------------------------ *)
+ntheorem prove_H56:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join X (meet Y (join X Z))) (join X (meet Y (join Z (meet X (join Z Y))))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H7:∀X:Univ.eq Univ (join X X) X.
+∀H8:∀X:Univ.eq Univ (meet X X) X.eq Univ (join (meet a b) (meet a (join b c))) (meet a (join b (meet (join a b) (join c (meet a b)))))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#join.
+#meet.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8;
+nqed.
+
+(* ------------------------------------------------------------------------------ *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT115-1.p *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* File : LAT115-1 : TPTP v3.2.0. Released v3.1.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Huntington equation H59 is independent of H55 *)
+
+(* Version : [McC05] (equality) axioms : Especial. *)
+
+(* English : Show that Huntington equation H55 does not imply Huntington *)
+
+(* equation H59 in lattice theory. *)
+
+(* Refs : [McC05] McCune (2005), Email to Geoff Sutcliffe *)
+
+(* Source : [McC05] *)
+
+(* Names : *)
+
+(* Status : Satisfiable *)
+
+(* Rating : 0.33 v3.2.0, 0.67 v3.1.0 *)
+
+(* Syntax : Number of clauses : 10 ( 0 non-Horn; 10 unit; 1 RR) *)
+
+(* Number of atoms : 10 ( 10 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 19 ( 2 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* ----Include Lattice theory (equality) axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ------------------------------------------------------------------------------ *)
+ntheorem prove_H59:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀d:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join X (meet Y (join X Z))) (join X (meet Y (join Z (meet X (join Z Y))))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H7:∀X:Univ.eq Univ (join X X) X.
+∀H8:∀X:Univ.eq Univ (meet X X) X.eq Univ (meet a (meet (join b c) (join b d))) (meet a (join b (meet (join b d) (join c (meet a b)))))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#d.
+#join.
+#meet.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8;
+nqed.
+
+(* ------------------------------------------------------------------------------ *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT116-1.p *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* File : LAT116-1 : TPTP v3.2.0. Released v3.1.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Huntington equation H60 is independent of H55 *)
+
+(* Version : [McC05] (equality) axioms : Especial. *)
+
+(* English : Show that Huntington equation H55 does not imply Huntington *)
+
+(* equation H60 in lattice theory. *)
+
+(* Refs : [McC05] McCune (2005), Email to Geoff Sutcliffe *)
+
+(* Source : [McC05] *)
+
+(* Names : *)
+
+(* Status : Satisfiable *)
+
+(* Rating : 0.33 v3.2.0, 0.67 v3.1.0 *)
+
+(* Syntax : Number of clauses : 10 ( 0 non-Horn; 10 unit; 1 RR) *)
+
+(* Number of atoms : 10 ( 10 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 19 ( 2 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* ----Include Lattice theory (equality) axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ------------------------------------------------------------------------------ *)
+ntheorem prove_H60:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀d:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join X (meet Y (join X Z))) (join X (meet Y (join Z (meet X (join Z Y))))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H7:∀X:Univ.eq Univ (join X X) X.
+∀H8:∀X:Univ.eq Univ (meet X X) X.eq Univ (meet a (meet (join b c) (join b d))) (meet a (join b (meet (join b c) (join d (meet a b)))))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#d.
+#join.
+#meet.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8;
+nqed.
+
+(* ------------------------------------------------------------------------------ *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT117-1.p *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* File : LAT117-1 : TPTP v3.2.0. Released v3.1.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Huntington equation H69 is independent of H65 *)
+
+(* Version : [McC05] (equality) axioms : Especial. *)
+
+(* English : Show that Huntington equation H65 does not imply Huntington *)
+
+(* equation H69 in lattice theory. *)
+
+(* Refs : [McC05] McCune (2005), Email to Geoff Sutcliffe *)
+
+(* Source : [McC05] *)
+
+(* Names : *)
+
+(* Status : Satisfiable *)
+
+(* Rating : 0.67 v3.2.0, 1.00 v3.1.0 *)
+
+(* Syntax : Number of clauses : 10 ( 0 non-Horn; 10 unit; 1 RR) *)
+
+(* Number of atoms : 10 ( 10 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 20 ( 2 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* ----Include Lattice theory (equality) axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ------------------------------------------------------------------------------ *)
+ntheorem prove_H69:
+ ∀Univ:Type.∀U:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀U:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet X (join Y (meet Z U))) (meet X (join Y (meet X (join (meet X Y) (meet Z U))))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H7:∀X:Univ.eq Univ (join X X) X.
+∀H8:∀X:Univ.eq Univ (meet X X) X.eq Univ (meet a (join b c)) (join (meet a (join c (meet a b))) (meet a (join b (meet a c))))
+.
+#Univ.
+#U.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#join.
+#meet.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8;
+nqed.
+
+(* ------------------------------------------------------------------------------ *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT118-1.p *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* File : LAT118-1 : TPTP v3.2.0. Released v3.1.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Huntington equation H69 is independent of H79 *)
+
+(* Version : [McC05] (equality) axioms : Especial. *)
+
+(* English : Show that Huntington equation H79 does not imply Huntington *)
+
+(* equation H69 in lattice theory. *)
+
+(* Refs : [McC05] McCune (2005), Email to Geoff Sutcliffe *)
+
+(* Source : [McC05] *)
+
+(* Names : *)
+
+(* Status : Satisfiable *)
+
+(* Rating : 0.67 v3.2.0, 1.00 v3.1.0 *)
+
+(* Syntax : Number of clauses : 10 ( 0 non-Horn; 10 unit; 1 RR) *)
+
+(* Number of atoms : 10 ( 10 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 20 ( 2 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* ----Include Lattice theory (equality) axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ------------------------------------------------------------------------------ *)
+ntheorem prove_H69:
+ ∀Univ:Type.∀U:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀U:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet X (join Y (meet Z (join X U)))) (meet X (join (meet X (join Y (meet X Z))) (meet Z U))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H7:∀X:Univ.eq Univ (join X X) X.
+∀H8:∀X:Univ.eq Univ (meet X X) X.eq Univ (meet a (join b c)) (join (meet a (join c (meet a b))) (meet a (join b (meet a c))))
+.
+#Univ.
+#U.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#join.
+#meet.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8;
+nqed.
+
+(* ------------------------------------------------------------------------------ *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT119-1.p *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* File : LAT119-1 : TPTP v3.2.0. Released v3.1.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Huntington equation H3 is independent of H82 *)
+
+(* Version : [McC05] (equality) axioms : Especial. *)
+
+(* English : Show that Huntington equation H82 does not imply Huntington *)
+
+(* equation H3 in lattice theory. *)
+
+(* Refs : [McC05] McCune (2005), Email to Geoff Sutcliffe *)
+
+(* Source : [McC05] *)
+
+(* Names : *)
+
+(* Status : Satisfiable *)
+
+(* Rating : 0.33 v3.2.0, 0.67 v3.1.0 *)
+
+(* Syntax : Number of clauses : 10 ( 0 non-Horn; 10 unit; 1 RR) *)
+
+(* Number of atoms : 10 ( 10 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 19 ( 2 singleton) *)
+
+(* Maximal term depth : 8 ( 3 average) *)
+
+(* Comments : *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* ----Include Lattice theory (equality) axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ------------------------------------------------------------------------------ *)
+ntheorem prove_H3:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet X (join (meet Y (join X Z)) (meet Z (join X Y)))) (join (meet X Y) (meet X Z)).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H7:∀X:Univ.eq Univ (join X X) X.
+∀H8:∀X:Univ.eq Univ (meet X X) X.eq Univ (meet a (join b (meet a c))) (meet a (join b (meet c (join b (meet a (join c (meet a b)))))))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#join.
+#meet.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8;
+nqed.
+
+(* ------------------------------------------------------------------------------ *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT120-1.p *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* File : LAT120-1 : TPTP v3.2.0. Released v3.1.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Huntington equation H58 is independent of H10_dual *)
+
+(* Version : [McC05] (equality) axioms : Especial. *)
+
+(* English : Show that Huntington equation H10_dual does not imply Huntington *)
+
+(* equation H58 in lattice theory. *)
+
+(* Refs : [McC05] McCune (2005), Email to Geoff Sutcliffe *)
+
+(* Source : [McC05] *)
+
+(* Names : *)
+
+(* Status : Satisfiable *)
+
+(* Rating : 0.33 v3.2.0, 0.67 v3.1.0 *)
+
+(* Syntax : Number of clauses : 10 ( 0 non-Horn; 10 unit; 1 RR) *)
+
+(* Number of atoms : 10 ( 10 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 19 ( 2 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* ----Include Lattice theory (equality) axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ------------------------------------------------------------------------------ *)
+ntheorem prove_H58:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join X (meet Y (join X Z))) (join X (meet Y (join Z (meet X (join Y Z))))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H7:∀X:Univ.eq Univ (join X X) X.
+∀H8:∀X:Univ.eq Univ (meet X X) X.eq Univ (meet a (join b c)) (meet a (join b (meet (join a b) (join c (meet a b)))))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#join.
+#meet.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8;
+nqed.
+
+(* ------------------------------------------------------------------------------ *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT121-1.p *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* File : LAT121-1 : TPTP v3.2.0. Released v3.1.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Huntington equation H55 is independent of H18_dual *)
+
+(* Version : [McC05] (equality) axioms : Especial. *)
+
+(* English : Show that Huntington equation H18_dual does not imply Huntington *)
+
+(* equation H55 in lattice theory. *)
+
+(* Refs : [McC05] McCune (2005), Email to Geoff Sutcliffe *)
+
+(* Source : [McC05] *)
+
+(* Names : *)
+
+(* Status : Satisfiable *)
+
+(* Rating : 0.33 v3.2.0, 0.67 v3.1.0 *)
+
+(* Syntax : Number of clauses : 10 ( 0 non-Horn; 10 unit; 1 RR) *)
+
+(* Number of atoms : 10 ( 10 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 19 ( 2 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* ----Include Lattice theory (equality) axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ------------------------------------------------------------------------------ *)
+ntheorem prove_H55:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (join X Y) (join X Z)) (join X (meet (join X Y) (meet (join X Z) (join Y (meet X Z))))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H7:∀X:Univ.eq Univ (join X X) X.
+∀H8:∀X:Univ.eq Univ (meet X X) X.eq Univ (join a (meet b (join a c))) (join a (meet b (join c (meet a (join c b)))))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#join.
+#meet.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8;
+nqed.
+
+(* ------------------------------------------------------------------------------ *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT122-1.p *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* File : LAT122-1 : TPTP v3.2.0. Released v3.1.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Huntington equation H55 is independent of H21_dual *)
+
+(* Version : [McC05] (equality) axioms : Especial. *)
+
+(* English : Show that Huntington equation H21_dual does not imply Huntington *)
+
+(* equation H55 in lattice theory. *)
+
+(* Refs : [McC05] McCune (2005), Email to Geoff Sutcliffe *)
+
+(* Source : [McC05] *)
+
+(* Names : *)
+
+(* Status : Satisfiable *)
+
+(* Rating : 0.33 v3.2.0, 0.67 v3.1.0 *)
+
+(* Syntax : Number of clauses : 10 ( 0 non-Horn; 10 unit; 1 RR) *)
+
+(* Number of atoms : 10 ( 10 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 19 ( 2 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* ----Include Lattice theory (equality) axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ------------------------------------------------------------------------------ *)
+ntheorem prove_H55:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (join X Y) (join X Z)) (join X (meet (join Y (meet X (join Y Z))) (join Z (meet X Y)))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H7:∀X:Univ.eq Univ (join X X) X.
+∀H8:∀X:Univ.eq Univ (meet X X) X.eq Univ (join a (meet b (join a c))) (join a (meet b (join c (meet a (join c b)))))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#join.
+#meet.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8;
+nqed.
+
+(* ------------------------------------------------------------------------------ *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT123-1.p *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* File : LAT123-1 : TPTP v3.2.0. Released v3.1.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Huntington equation H55 is independent of H22_dual *)
+
+(* Version : [McC05] (equality) axioms : Especial. *)
+
+(* English : Show that Huntington equation H22_dual does not imply Huntington *)
+
+(* equation H55 in lattice theory. *)
+
+(* Refs : [McC05] McCune (2005), Email to Geoff Sutcliffe *)
+
+(* Source : [McC05] *)
+
+(* Names : *)
+
+(* Status : Satisfiable *)
+
+(* Rating : 0.67 v3.2.0, 1.00 v3.1.0 *)
+
+(* Syntax : Number of clauses : 10 ( 0 non-Horn; 10 unit; 1 RR) *)
+
+(* Number of atoms : 10 ( 10 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 19 ( 2 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* ----Include Lattice theory (equality) axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ------------------------------------------------------------------------------ *)
+ntheorem prove_H55:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (join X Y) (join X Z)) (join X (meet (join Y (meet Z (join X Y))) (join Z (meet X Y)))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H7:∀X:Univ.eq Univ (join X X) X.
+∀H8:∀X:Univ.eq Univ (meet X X) X.eq Univ (join a (meet b (join a c))) (join a (meet b (join c (meet a (join c b)))))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#join.
+#meet.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8;
+nqed.
+
+(* ------------------------------------------------------------------------------ *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT124-1.p *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* File : LAT124-1 : TPTP v3.2.0. Released v3.1.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Huntington equation H69 is independent of H32_dual *)
+
+(* Version : [McC05] (equality) axioms : Especial. *)
+
+(* English : Show that Huntington equation H32_dual does not imply Huntington *)
+
+(* equation H69 in lattice theory. *)
+
+(* Refs : [McC05] McCune (2005), Email to Geoff Sutcliffe *)
+
+(* Source : [McC05] *)
+
+(* Names : *)
+
+(* Status : Satisfiable *)
+
+(* Rating : 0.67 v3.2.0, 1.00 v3.1.0 *)
+
+(* Syntax : Number of clauses : 10 ( 0 non-Horn; 10 unit; 1 RR) *)
+
+(* Number of atoms : 10 ( 10 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 20 ( 2 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* ----Include Lattice theory (equality) axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ------------------------------------------------------------------------------ *)
+ntheorem prove_H69:
+ ∀Univ:Type.∀U:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀U:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join X (meet Y (join X (join Z U)))) (join X (meet Y (join Z (meet (join X U) (join Y U))))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H7:∀X:Univ.eq Univ (join X X) X.
+∀H8:∀X:Univ.eq Univ (meet X X) X.eq Univ (meet a (join b c)) (join (meet a (join c (meet a b))) (meet a (join b (meet a c))))
+.
+#Univ.
+#U.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#join.
+#meet.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8;
+nqed.
+
+(* ------------------------------------------------------------------------------ *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT125-1.p *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* File : LAT125-1 : TPTP v3.2.0. Released v3.1.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Huntington equation H69 is independent of H34_dual *)
+
+(* Version : [McC05] (equality) axioms : Especial. *)
+
+(* English : Show that Huntington equation H34_dual does not imply Huntington *)
+
+(* equation H69 in lattice theory. *)
+
+(* Refs : [McC05] McCune (2005), Email to Geoff Sutcliffe *)
+
+(* Source : [McC05] *)
+
+(* Names : *)
+
+(* Status : Satisfiable *)
+
+(* Rating : 0.67 v3.2.0, 1.00 v3.1.0 *)
+
+(* Syntax : Number of clauses : 10 ( 0 non-Horn; 10 unit; 1 RR) *)
+
+(* Number of atoms : 10 ( 10 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 20 ( 2 singleton) *)
+
+(* Maximal term depth : 7 ( 3 average) *)
+
+(* Comments : *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* ----Include Lattice theory (equality) axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ------------------------------------------------------------------------------ *)
+ntheorem prove_H69:
+ ∀Univ:Type.∀U:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀U:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join X (meet Y (join Z U))) (join X (meet Y (join Z (meet Y (join U (meet Y Z)))))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H7:∀X:Univ.eq Univ (join X X) X.
+∀H8:∀X:Univ.eq Univ (meet X X) X.eq Univ (meet a (join b c)) (join (meet a (join c (meet a b))) (meet a (join b (meet a c))))
+.
+#Univ.
+#U.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#join.
+#meet.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8;
+nqed.
+
+(* ------------------------------------------------------------------------------ *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT126-1.p *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* File : LAT126-1 : TPTP v3.2.0. Released v3.1.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Huntington equation H69 is independent of H39_dual *)
+
+(* Version : [McC05] (equality) axioms : Especial. *)
+
+(* English : Show that Huntington equation H39_dual does not imply Huntington *)
+
+(* equation H69 in lattice theory. *)
+
+(* Refs : [McC05] McCune (2005), Email to Geoff Sutcliffe *)
+
+(* Source : [McC05] *)
+
+(* Names : *)
+
+(* Status : Satisfiable *)
+
+(* Rating : 0.33 v3.2.0, 0.67 v3.1.0 *)
+
+(* Syntax : Number of clauses : 10 ( 0 non-Horn; 10 unit; 1 RR) *)
+
+(* Number of atoms : 10 ( 10 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 20 ( 2 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* ----Include Lattice theory (equality) axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ------------------------------------------------------------------------------ *)
+ntheorem prove_H69:
+ ∀Univ:Type.∀U:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀U:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join X (meet Y (join Z (meet X U)))) (join X (meet Y (join Z (meet U (join X Z))))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H7:∀X:Univ.eq Univ (join X X) X.
+∀H8:∀X:Univ.eq Univ (meet X X) X.eq Univ (meet a (join b c)) (join (meet a (join c (meet a b))) (meet a (join b (meet a c))))
+.
+#Univ.
+#U.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#join.
+#meet.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8;
+nqed.
+
+(* ------------------------------------------------------------------------------ *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT127-1.p *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* File : LAT127-1 : TPTP v3.2.0. Released v3.1.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Huntington equation H6 is independent of H55_dual *)
+
+(* Version : [McC05] (equality) axioms : Especial. *)
+
+(* English : Show that Huntington equation H55_dual does not imply Huntington *)
+
+(* equation H6 in lattice theory. *)
+
+(* Refs : [McC05] McCune (2005), Email to Geoff Sutcliffe *)
+
+(* Source : [McC05] *)
+
+(* Names : *)
+
+(* Status : Satisfiable *)
+
+(* Rating : 0.33 v3.2.0, 0.67 v3.1.0 *)
+
+(* Syntax : Number of clauses : 10 ( 0 non-Horn; 10 unit; 1 RR) *)
+
+(* Number of atoms : 10 ( 10 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 19 ( 2 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* ----Include Lattice theory (equality) axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ------------------------------------------------------------------------------ *)
+ntheorem prove_H6:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet X (join Y (meet X Z))) (meet X (join Y (meet Z (join X (meet Z Y))))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H7:∀X:Univ.eq Univ (join X X) X.
+∀H8:∀X:Univ.eq Univ (meet X X) X.eq Univ (meet a (join b (meet a c))) (meet a (join (meet a (join b (meet a c))) (meet c (join a b))))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#join.
+#meet.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8;
+nqed.
+
+(* ------------------------------------------------------------------------------ *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT128-1.p *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* File : LAT128-1 : TPTP v3.2.0. Released v3.1.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Huntington equation H3 is independent of H58_dual *)
+
+(* Version : [McC05] (equality) axioms : Especial. *)
+
+(* English : Show that Huntington equation H58_dual does not imply Huntington *)
+
+(* equation H3 in lattice theory. *)
+
+(* Refs : [McC05] McCune (2005), Email to Geoff Sutcliffe *)
+
+(* Source : [McC05] *)
+
+(* Names : *)
+
+(* Status : Satisfiable *)
+
+(* Rating : 0.33 v3.2.0, 0.67 v3.1.0 *)
+
+(* Syntax : Number of clauses : 10 ( 0 non-Horn; 10 unit; 1 RR) *)
+
+(* Number of atoms : 10 ( 10 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 19 ( 2 singleton) *)
+
+(* Maximal term depth : 8 ( 3 average) *)
+
+(* Comments : *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* ----Include Lattice theory (equality) axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ------------------------------------------------------------------------------ *)
+ntheorem prove_H3:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join X (meet Y Z)) (join X (meet Y (join (meet X Y) (meet Z (join X Y))))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H7:∀X:Univ.eq Univ (join X X) X.
+∀H8:∀X:Univ.eq Univ (meet X X) X.eq Univ (meet a (join b (meet a c))) (meet a (join b (meet c (join b (meet a (join c (meet a b)))))))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#join.
+#meet.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8;
+nqed.
+
+(* ------------------------------------------------------------------------------ *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT129-1.p *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* File : LAT129-1 : TPTP v3.2.0. Released v3.1.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Huntington equation H10 is independent of H58_dual *)
+
+(* Version : [McC05] (equality) axioms : Especial. *)
+
+(* English : Show that Huntington equation H58_dual does not imply Huntington *)
+
+(* equation H10 in lattice theory. *)
+
+(* Refs : [McC05] McCune (2005), Email to Geoff Sutcliffe *)
+
+(* Source : [McC05] *)
+
+(* Names : *)
+
+(* Status : Satisfiable *)
+
+(* Rating : 0.33 v3.2.0, 0.67 v3.1.0 *)
+
+(* Syntax : Number of clauses : 10 ( 0 non-Horn; 10 unit; 1 RR) *)
+
+(* Number of atoms : 10 ( 10 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 19 ( 2 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* ----Include Lattice theory (equality) axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ------------------------------------------------------------------------------ *)
+ntheorem prove_H10:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join X (meet Y Z)) (join X (meet Y (join (meet X Y) (meet Z (join X Y))))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H7:∀X:Univ.eq Univ (join X X) X.
+∀H8:∀X:Univ.eq Univ (meet X X) X.eq Univ (meet a (join b (meet a c))) (meet a (join b (meet c (join a (meet b c)))))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#join.
+#meet.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8;
+nqed.
+
+(* ------------------------------------------------------------------------------ *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT130-1.p *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* File : LAT130-1 : TPTP v3.2.0. Released v3.1.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Huntington equation H39 is independent of H68_dual *)
+
+(* Version : [McC05] (equality) axioms : Especial. *)
+
+(* English : Show that Huntington equation H68_dual does not imply Huntington *)
+
+(* equation H39 in lattice theory. *)
+
+(* Refs : [McC05] McCune (2005), Email to Geoff Sutcliffe *)
+
+(* Source : [McC05] *)
+
+(* Names : *)
+
+(* Status : Satisfiable *)
+
+(* Rating : 0.33 v3.2.0, 0.67 v3.1.0 *)
+
+(* Syntax : Number of clauses : 10 ( 0 non-Horn; 10 unit; 1 RR) *)
+
+(* Number of atoms : 10 ( 10 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 19 ( 2 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* ----Include Lattice theory (equality) axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ------------------------------------------------------------------------------ *)
+ntheorem prove_H39:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀d:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join X (meet Y Z)) (join X (meet Y (join X (meet Z (join X Y))))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H7:∀X:Univ.eq Univ (join X X) X.
+∀H8:∀X:Univ.eq Univ (meet X X) X.eq Univ (meet a (join b (meet c (join a d)))) (meet a (join b (meet c (join d (meet a c)))))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#d.
+#join.
+#meet.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8;
+nqed.
+
+(* ------------------------------------------------------------------------------ *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT131-1.p *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* File : LAT131-1 : TPTP v3.2.0. Released v3.1.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Huntington equation H42 is independent of H68_dual *)
+
+(* Version : [McC05] (equality) axioms : Especial. *)
+
+(* English : Show that Huntington equation H68_dual does not imply Huntington *)
+
+(* equation H42 in lattice theory. *)
+
+(* Refs : [McC05] McCune (2005), Email to Geoff Sutcliffe *)
+
+(* Source : [McC05] *)
+
+(* Names : *)
+
+(* Status : Satisfiable *)
+
+(* Rating : 0.33 v3.2.0, 0.67 v3.1.0 *)
+
+(* Syntax : Number of clauses : 10 ( 0 non-Horn; 10 unit; 1 RR) *)
+
+(* Number of atoms : 10 ( 10 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 19 ( 2 singleton) *)
+
+(* Maximal term depth : 7 ( 3 average) *)
+
+(* Comments : *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* ----Include Lattice theory (equality) axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ------------------------------------------------------------------------------ *)
+ntheorem prove_H42:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀d:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join X (meet Y Z)) (join X (meet Y (join X (meet Z (join X Y))))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H7:∀X:Univ.eq Univ (join X X) X.
+∀H8:∀X:Univ.eq Univ (meet X X) X.eq Univ (meet a (join b (meet c (join a d)))) (meet a (join b (meet c (join b (join d (meet a c))))))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#d.
+#join.
+#meet.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8;
+nqed.
+
+(* ------------------------------------------------------------------------------ *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT132-1.p *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* File : LAT132-1 : TPTP v3.2.0. Released v3.1.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Huntington equation H42 is independent of H69_dual *)
+
+(* Version : [McC05] (equality) axioms : Especial. *)
+
+(* English : Show that Huntington equation H69_dual does not imply Huntington *)
+
+(* equation H42 in lattice theory. *)
+
+(* Refs : [McC05] McCune (2005), Email to Geoff Sutcliffe *)
+
+(* Source : [McC05] *)
+
+(* Names : *)
+
+(* Status : Satisfiable *)
+
+(* Rating : 0.33 v3.2.0, 0.67 v3.1.0 *)
+
+(* Syntax : Number of clauses : 10 ( 0 non-Horn; 10 unit; 1 RR) *)
+
+(* Number of atoms : 10 ( 10 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 19 ( 2 singleton) *)
+
+(* Maximal term depth : 7 ( 3 average) *)
+
+(* Comments : *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* ----Include Lattice theory (equality) axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ------------------------------------------------------------------------------ *)
+ntheorem prove_H42:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀d:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join X (meet Y Z)) (meet (join X (meet Z (join X Y))) (join X (meet Y (join X Z)))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H7:∀X:Univ.eq Univ (join X X) X.
+∀H8:∀X:Univ.eq Univ (meet X X) X.eq Univ (meet a (join b (meet c (join a d)))) (meet a (join b (meet c (join b (join d (meet a c))))))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#d.
+#join.
+#meet.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8;
+nqed.
+
+(* ------------------------------------------------------------------------------ *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT133-1.p *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* File : LAT133-1 : TPTP v3.2.0. Released v3.1.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Huntington equation H6_dual is independent of H55 *)
+
+(* Version : [McC05] (equality) axioms : Especial. *)
+
+(* English : Show that Huntington equation H55 does not imply Huntington *)
+
+(* equation H6_dual in lattice theory. *)
+
+(* Refs : [McC05] McCune (2005), Email to Geoff Sutcliffe *)
+
+(* Source : [McC05] *)
+
+(* Names : *)
+
+(* Status : Satisfiable *)
+
+(* Rating : 0.33 v3.2.0, 0.67 v3.1.0 *)
+
+(* Syntax : Number of clauses : 10 ( 0 non-Horn; 10 unit; 1 RR) *)
+
+(* Number of atoms : 10 ( 10 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 19 ( 2 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* ----Include Lattice theory (equality) axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ------------------------------------------------------------------------------ *)
+ntheorem prove_H6_dual:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join X (meet Y (join X Z))) (join X (meet Y (join Z (meet X (join Z Y))))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H7:∀X:Univ.eq Univ (join X X) X.
+∀H8:∀X:Univ.eq Univ (meet X X) X.eq Univ (join a (meet b (join a c))) (join a (meet (join a (meet b (join a c))) (join c (meet a b))))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#join.
+#meet.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8;
+nqed.
+
+(* ------------------------------------------------------------------------------ *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT134-1.p *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* File : LAT134-1 : TPTP v3.2.0. Released v3.1.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Huntington equation H22_dual is independent of H61 *)
+
+(* Version : [McC05] (equality) axioms : Especial. *)
+
+(* English : Show that Huntington equation H61 does not imply Huntington *)
+
+(* equation H22_dual in lattice theory. *)
+
+(* Refs : [McC05] McCune (2005), Email to Geoff Sutcliffe *)
+
+(* Source : [McC05] *)
+
+(* Names : *)
+
+(* Status : Satisfiable *)
+
+(* Rating : 0.33 v3.2.0, 0.67 v3.1.0 *)
+
+(* Syntax : Number of clauses : 10 ( 0 non-Horn; 10 unit; 1 RR) *)
+
+(* Number of atoms : 10 ( 10 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 19 ( 2 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* ----Include Lattice theory (equality) axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ------------------------------------------------------------------------------ *)
+ntheorem prove_H22_dual:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (join X Y) (join X Z)) (join X (meet (join X Y) (join (meet X Y) Z))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H7:∀X:Univ.eq Univ (join X X) X.
+∀H8:∀X:Univ.eq Univ (meet X X) X.eq Univ (meet (join a b) (join a c)) (join a (meet (join b (meet c (join a b))) (join c (meet a b))))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#join.
+#meet.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8;
+nqed.
+
+(* ------------------------------------------------------------------------------ *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT135-1.p *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* File : LAT135-1 : TPTP v3.2.0. Released v3.1.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Huntington equation H39_dual is independent of H68 *)
+
+(* Version : [McC05] (equality) axioms : Especial. *)
+
+(* English : Show that Huntington equation H68 does not imply Huntington *)
+
+(* equation H39_dual in lattice theory. *)
+
+(* Refs : [McC05] McCune (2005), Email to Geoff Sutcliffe *)
+
+(* Source : [McC05] *)
+
+(* Names : *)
+
+(* Status : Satisfiable *)
+
+(* Rating : 0.33 v3.2.0, 0.67 v3.1.0 *)
+
+(* Syntax : Number of clauses : 10 ( 0 non-Horn; 10 unit; 1 RR) *)
+
+(* Number of atoms : 10 ( 10 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 19 ( 2 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* ----Include Lattice theory (equality) axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ------------------------------------------------------------------------------ *)
+ntheorem prove_H39_dual:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀d:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet X (join Y Z)) (meet X (join Y (meet X (join Z (meet X Y))))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H7:∀X:Univ.eq Univ (join X X) X.
+∀H8:∀X:Univ.eq Univ (meet X X) X.eq Univ (join a (meet b (join c (meet a d)))) (join a (meet b (join c (meet d (join a c)))))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#d.
+#join.
+#meet.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8;
+nqed.
+
+(* ------------------------------------------------------------------------------ *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT136-1.p *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* File : LAT136-1 : TPTP v3.2.0. Released v3.1.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Huntington equation H39_dual is independent of H69 *)
+
+(* Version : [McC05] (equality) axioms : Especial. *)
+
+(* English : Show that Huntington equation H69 does not imply Huntington *)
+
+(* equation H39_dual in lattice theory. *)
+
+(* Refs : [McC05] McCune (2005), Email to Geoff Sutcliffe *)
+
+(* Source : [McC05] *)
+
+(* Names : *)
+
+(* Status : Satisfiable *)
+
+(* Rating : 0.33 v3.2.0, 0.67 v3.1.0 *)
+
+(* Syntax : Number of clauses : 10 ( 0 non-Horn; 10 unit; 1 RR) *)
+
+(* Number of atoms : 10 ( 10 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 19 ( 2 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* ----Include Lattice theory (equality) axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ------------------------------------------------------------------------------ *)
+ntheorem prove_H39_dual:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀d:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet X (join Y Z)) (join (meet X (join Z (meet X Y))) (meet X (join Y (meet X Z)))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H7:∀X:Univ.eq Univ (join X X) X.
+∀H8:∀X:Univ.eq Univ (meet X X) X.eq Univ (join a (meet b (join c (meet a d)))) (join a (meet b (join c (meet d (join a c)))))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#d.
+#join.
+#meet.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8;
+nqed.
+
+(* ------------------------------------------------------------------------------ *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT137-1.p *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* File : LAT137-1 : TPTP v3.2.0. Released v3.1.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Huntington equation H40_dual is independent of H69 *)
+
+(* Version : [McC05] (equality) axioms : Especial. *)
+
+(* English : Show that Huntington equation H69 does not imply Huntington *)
+
+(* equation H40_dual in lattice theory. *)
+
+(* Refs : [McC05] McCune (2005), Email to Geoff Sutcliffe *)
+
+(* Source : [McC05] *)
+
+(* Names : *)
+
+(* Status : Satisfiable *)
+
+(* Rating : 0.33 v3.2.0, 0.67 v3.1.0 *)
+
+(* Syntax : Number of clauses : 10 ( 0 non-Horn; 10 unit; 1 RR) *)
+
+(* Number of atoms : 10 ( 10 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 19 ( 2 singleton) *)
+
+(* Maximal term depth : 7 ( 3 average) *)
+
+(* Comments : *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* ----Include Lattice theory (equality) axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ------------------------------------------------------------------------------ *)
+ntheorem prove_H40_dual:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀d:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet X (join Y Z)) (join (meet X (join Z (meet X Y))) (meet X (join Y (meet X Z)))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H7:∀X:Univ.eq Univ (join X X) X.
+∀H8:∀X:Univ.eq Univ (meet X X) X.eq Univ (join a (meet b (join c (meet a d)))) (join a (meet b (join c (meet d (join c (meet a b))))))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#d.
+#join.
+#meet.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8;
+nqed.
+
+(* ------------------------------------------------------------------------------ *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT138-1.p *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* File : LAT138-1 : TPTP v3.2.0. Released v3.1.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Huntington equation H7 implies H6 *)
+
+(* Version : [McC05] (equality) axioms : Especial. *)
+
+(* English : *)
+
+(* Refs : [McC05] McCune (2005), Email to Geoff Sutcliffe *)
+
+(* Source : [McC05] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 1.00 v3.1.0 *)
+
+(* Syntax : Number of clauses : 10 ( 0 non-Horn; 10 unit; 1 RR) *)
+
+(* Number of atoms : 10 ( 10 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 19 ( 2 singleton) *)
+
+(* Maximal term depth : 7 ( 3 average) *)
+
+(* Comments : *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* ----Include Lattice theory (equality) axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ------------------------------------------------------------------------------ *)
+ntheorem prove_H6:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet X (join Y (meet X Z))) (meet X (join Y (meet X (join (meet X Y) (meet Z (join X Y)))))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H7:∀X:Univ.eq Univ (join X X) X.
+∀H8:∀X:Univ.eq Univ (meet X X) X.eq Univ (meet a (join b (meet a c))) (meet a (join (meet a (join b (meet a c))) (meet c (join a b))))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#join.
+#meet.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8;
+nqed.
+
+(* ------------------------------------------------------------------------------ *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT139-1.p *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* File : LAT139-1 : TPTP v3.2.0. Released v3.1.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Huntington equation H11 implies H10 *)
+
+(* Version : [McC05] (equality) axioms : Especial. *)
+
+(* English : *)
+
+(* Refs : [McC05] McCune (2005), Email to Geoff Sutcliffe *)
+
+(* Source : [McC05] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 1.00 v3.1.0 *)
+
+(* Syntax : Number of clauses : 10 ( 0 non-Horn; 10 unit; 1 RR) *)
+
+(* Number of atoms : 10 ( 10 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 19 ( 2 singleton) *)
+
+(* Maximal term depth : 8 ( 3 average) *)
+
+(* Comments : *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* ----Include Lattice theory (equality) axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ------------------------------------------------------------------------------ *)
+ntheorem prove_H10:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet X (join Y (meet X Z))) (meet X (join Y (meet Z (join X (meet Y (join Z (meet X Y))))))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H7:∀X:Univ.eq Univ (join X X) X.
+∀H8:∀X:Univ.eq Univ (meet X X) X.eq Univ (meet a (join b (meet a c))) (meet a (join b (meet c (join a (meet b c)))))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#join.
+#meet.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8;
+nqed.
+
+(* ------------------------------------------------------------------------------ *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT140-1.p *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* File : LAT140-1 : TPTP v3.2.0. Released v3.1.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Huntington equation H21 implies H2 *)
+
+(* Version : [McC05] (equality) axioms : Especial. *)
+
+(* English : *)
+
+(* Refs : [McC05] McCune (2005), Email to Geoff Sutcliffe *)
+
+(* Source : [McC05] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 1.00 v3.1.0 *)
+
+(* Syntax : Number of clauses : 10 ( 0 non-Horn; 10 unit; 1 RR) *)
+
+(* Number of atoms : 10 ( 10 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 19 ( 2 singleton) *)
+
+(* Maximal term depth : 7 ( 3 average) *)
+
+(* Comments : *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* ----Include Lattice theory (equality) axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ------------------------------------------------------------------------------ *)
+ntheorem prove_H2:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (meet X Y) (meet X Z)) (meet X (join (meet Y (join X (meet Y Z))) (meet Z (join X Y)))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H7:∀X:Univ.eq Univ (join X X) X.
+∀H8:∀X:Univ.eq Univ (meet X X) X.eq Univ (meet a (join b (meet a c))) (meet a (join b (meet c (join (meet a (join b c)) (meet b c)))))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#join.
+#meet.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8;
+nqed.
+
+(* ------------------------------------------------------------------------------ *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT141-1.p *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* File : LAT141-1 : TPTP v3.2.0. Released v3.1.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Huntington equation H21 implies H6 *)
+
+(* Version : [McC05] (equality) axioms : Especial. *)
+
+(* English : *)
+
+(* Refs : [McC05] McCune (2005), Email to Geoff Sutcliffe *)
+
+(* Source : [McC05] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 1.00 v3.1.0 *)
+
+(* Syntax : Number of clauses : 10 ( 0 non-Horn; 10 unit; 1 RR) *)
+
+(* Number of atoms : 10 ( 10 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 19 ( 2 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* ----Include Lattice theory (equality) axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ------------------------------------------------------------------------------ *)
+ntheorem prove_H6:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (meet X Y) (meet X Z)) (meet X (join (meet Y (join X (meet Y Z))) (meet Z (join X Y)))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H7:∀X:Univ.eq Univ (join X X) X.
+∀H8:∀X:Univ.eq Univ (meet X X) X.eq Univ (meet a (join b (meet a c))) (meet a (join (meet a (join b (meet a c))) (meet c (join a b))))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#join.
+#meet.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8;
+nqed.
+
+(* ------------------------------------------------------------------------------ *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT142-1.p *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* File : LAT142-1 : TPTP v3.2.0. Released v3.1.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Huntington equation H22 implies H6 *)
+
+(* Version : [McC05] (equality) axioms : Especial. *)
+
+(* English : *)
+
+(* Refs : [McC05] McCune (2005), Email to Geoff Sutcliffe *)
+
+(* Source : [McC05] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.71 v3.2.0, 0.64 v3.1.0 *)
+
+(* Syntax : Number of clauses : 10 ( 0 non-Horn; 10 unit; 1 RR) *)
+
+(* Number of atoms : 10 ( 10 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 19 ( 2 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* ----Include Lattice theory (equality) axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ------------------------------------------------------------------------------ *)
+ntheorem prove_H6:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (meet X Y) (meet X Z)) (meet X (join (meet Y (join Z (meet X Y))) (meet Z (join X Y)))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H7:∀X:Univ.eq Univ (join X X) X.
+∀H8:∀X:Univ.eq Univ (meet X X) X.eq Univ (meet a (join b (meet a c))) (meet a (join (meet a (join b (meet a c))) (meet c (join a b))))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#join.
+#meet.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8;
+nqed.
+
+(* ------------------------------------------------------------------------------ *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT143-1.p *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* File : LAT143-1 : TPTP v3.2.0. Released v3.1.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Huntington equation H24 implies H15 *)
+
+(* Version : [McC05] (equality) axioms : Especial. *)
+
+(* English : *)
+
+(* Refs : [McC05] McCune (2005), Email to Geoff Sutcliffe *)
+
+(* Source : [McC05] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.21 v3.2.0, 0.14 v3.1.0 *)
+
+(* Syntax : Number of clauses : 10 ( 0 non-Horn; 10 unit; 1 RR) *)
+
+(* Number of atoms : 10 ( 10 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 19 ( 2 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* ----Include Lattice theory (equality) axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ------------------------------------------------------------------------------ *)
+ntheorem prove_H15:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (meet X Y) (meet Y Z)) (join (meet X Y) (meet Y (join (meet X Y) (meet Z (join X Y))))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H7:∀X:Univ.eq Univ (join X X) X.
+∀H8:∀X:Univ.eq Univ (meet X X) X.eq Univ (meet a (join (meet a b) (meet a c))) (meet a (join (meet a b) (join (meet a c) (meet c (join a b)))))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#join.
+#meet.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8;
+nqed.
+
+(* ------------------------------------------------------------------------------ *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT144-1.p *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* File : LAT144-1 : TPTP v3.2.0. Released v3.1.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Huntington equation H32 implies H2 *)
+
+(* Version : [McC05] (equality) axioms : Especial. *)
+
+(* English : *)
+
+(* Refs : [McC05] McCune (2005), Email to Geoff Sutcliffe *)
+
+(* Source : [McC05] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.86 v3.1.0 *)
+
+(* Syntax : Number of clauses : 10 ( 0 non-Horn; 10 unit; 1 RR) *)
+
+(* Number of atoms : 10 ( 10 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 20 ( 2 singleton) *)
+
+(* Maximal term depth : 7 ( 3 average) *)
+
+(* Comments : *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* ----Include Lattice theory (equality) axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ------------------------------------------------------------------------------ *)
+ntheorem prove_H2:
+ ∀Univ:Type.∀U:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀U:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet X (join Y (meet X (meet Z U)))) (meet X (join Y (meet Z (join (meet X U) (meet Y U))))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H7:∀X:Univ.eq Univ (join X X) X.
+∀H8:∀X:Univ.eq Univ (meet X X) X.eq Univ (meet a (join b (meet a c))) (meet a (join b (meet c (join (meet a (join b c)) (meet b c)))))
+.
+#Univ.
+#U.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#join.
+#meet.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8;
+nqed.
+
+(* ------------------------------------------------------------------------------ *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT145-1.p *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* File : LAT145-1 : TPTP v3.2.0. Released v3.1.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Huntington equation H32 implies H6 *)
+
+(* Version : [McC05] (equality) axioms : Especial. *)
+
+(* English : *)
+
+(* Refs : [McC05] McCune (2005), Email to Geoff Sutcliffe *)
+
+(* Source : [McC05] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.93 v3.1.0 *)
+
+(* Syntax : Number of clauses : 10 ( 0 non-Horn; 10 unit; 1 RR) *)
+
+(* Number of atoms : 10 ( 10 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 20 ( 2 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* ----Include Lattice theory (equality) axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ------------------------------------------------------------------------------ *)
+ntheorem prove_H6:
+ ∀Univ:Type.∀U:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀U:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet X (join Y (meet X (meet Z U)))) (meet X (join Y (meet Z (join (meet X U) (meet Y U))))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H7:∀X:Univ.eq Univ (join X X) X.
+∀H8:∀X:Univ.eq Univ (meet X X) X.eq Univ (meet a (join b (meet a c))) (meet a (join (meet a (join b (meet a c))) (meet c (join a b))))
+.
+#Univ.
+#U.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#join.
+#meet.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8;
+nqed.
+
+(* ------------------------------------------------------------------------------ *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT146-1.p *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* File : LAT146-1 : TPTP v3.2.0. Released v3.1.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Huntington equation H34 implies H28 *)
+
+(* Version : [McC05] (equality) axioms : Especial. *)
+
+(* English : *)
+
+(* Refs : [McC05] McCune (2005), Email to Geoff Sutcliffe *)
+
+(* Source : [McC05] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.79 v3.1.0 *)
+
+(* Syntax : Number of clauses : 10 ( 0 non-Horn; 10 unit; 1 RR) *)
+
+(* Number of atoms : 10 ( 10 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 20 ( 2 singleton) *)
+
+(* Maximal term depth : 7 ( 3 average) *)
+
+(* Comments : *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* ----Include Lattice theory (equality) axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ------------------------------------------------------------------------------ *)
+ntheorem prove_H28:
+ ∀Univ:Type.∀U:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀d:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀U:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet X (join Y (meet Z U))) (meet X (join Y (meet Z (join Y (meet U (join Y Z)))))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H7:∀X:Univ.eq Univ (join X X) X.
+∀H8:∀X:Univ.eq Univ (meet X X) X.eq Univ (meet a (join b (meet a (meet c d)))) (meet a (join b (meet c (meet d (join a (meet b d))))))
+.
+#Univ.
+#U.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#d.
+#join.
+#meet.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8;
+nqed.
+
+(* ------------------------------------------------------------------------------ *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT147-1.p *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* File : LAT147-1 : TPTP v3.2.0. Released v3.1.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Huntington equation H34 implies H45 *)
+
+(* Version : [McC05] (equality) axioms : Especial. *)
+
+(* English : *)
+
+(* Refs : [McC05] McCune (2005), Email to Geoff Sutcliffe *)
+
+(* Source : [McC05] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.79 v3.1.0 *)
+
+(* Syntax : Number of clauses : 10 ( 0 non-Horn; 10 unit; 1 RR) *)
+
+(* Number of atoms : 10 ( 10 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 20 ( 2 singleton) *)
+
+(* Maximal term depth : 7 ( 3 average) *)
+
+(* Comments : *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* ----Include Lattice theory (equality) axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ------------------------------------------------------------------------------ *)
+ntheorem prove_H45:
+ ∀Univ:Type.∀U:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀d:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀U:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet X (join Y (meet Z U))) (meet X (join Y (meet Z (join Y (meet U (join Y Z)))))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H7:∀X:Univ.eq Univ (join X X) X.
+∀H8:∀X:Univ.eq Univ (meet X X) X.eq Univ (meet a (meet b (join c (meet a d)))) (meet a (meet b (join c (meet d (join a (meet b c))))))
+.
+#Univ.
+#U.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#d.
+#join.
+#meet.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8;
+nqed.
+
+(* ------------------------------------------------------------------------------ *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT148-1.p *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* File : LAT148-1 : TPTP v3.2.0. Released v3.1.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Huntington equation H34 implies H7 *)
+
+(* Version : [McC05] (equality) axioms : Especial. *)
+
+(* English : *)
+
+(* Refs : [McC05] McCune (2005), Email to Geoff Sutcliffe *)
+
+(* Source : [McC05] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.71 v3.1.0 *)
+
+(* Syntax : Number of clauses : 10 ( 0 non-Horn; 10 unit; 1 RR) *)
+
+(* Number of atoms : 10 ( 10 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 20 ( 2 singleton) *)
+
+(* Maximal term depth : 7 ( 3 average) *)
+
+(* Comments : *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* ----Include Lattice theory (equality) axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ------------------------------------------------------------------------------ *)
+ntheorem prove_H7:
+ ∀Univ:Type.∀U:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀U:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet X (join Y (meet Z U))) (meet X (join Y (meet Z (join Y (meet U (join Y Z)))))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H7:∀X:Univ.eq Univ (join X X) X.
+∀H8:∀X:Univ.eq Univ (meet X X) X.eq Univ (meet a (join b (meet a c))) (meet a (join b (meet a (join (meet a b) (meet c (join a b))))))
+.
+#Univ.
+#U.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#join.
+#meet.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8;
+nqed.
+
+(* ------------------------------------------------------------------------------ *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT149-1.p *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* File : LAT149-1 : TPTP v3.2.0. Released v3.1.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Huntington equation H37 implies H43 *)
+
+(* Version : [McC05] (equality) axioms : Especial. *)
+
+(* English : *)
+
+(* Refs : [McC05] McCune (2005), Email to Geoff Sutcliffe *)
+
+(* Source : [McC05] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.93 v3.1.0 *)
+
+(* Syntax : Number of clauses : 10 ( 0 non-Horn; 10 unit; 1 RR) *)
+
+(* Number of atoms : 10 ( 10 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 20 ( 2 singleton) *)
+
+(* Maximal term depth : 7 ( 3 average) *)
+
+(* Comments : *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* ----Include Lattice theory (equality) axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ------------------------------------------------------------------------------ *)
+ntheorem prove_H43:
+ ∀Univ:Type.∀U:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀d:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀U:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet X (join Y (join Z (meet X U)))) (meet X (join Y (join Z (meet U (join X (meet Y Z)))))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H7:∀X:Univ.eq Univ (join X X) X.
+∀H8:∀X:Univ.eq Univ (meet X X) X.eq Univ (meet a (join b (meet c (join b d)))) (meet a (join b (meet c (join d (meet a (join b d))))))
+.
+#Univ.
+#U.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#d.
+#join.
+#meet.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8;
+nqed.
+
+(* ------------------------------------------------------------------------------ *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT150-1.p *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* File : LAT150-1 : TPTP v3.2.0. Released v3.1.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Huntington equation H39 implies H40 *)
+
+(* Version : [McC05] (equality) axioms : Especial. *)
+
+(* English : *)
+
+(* Refs : [McC05] McCune (2005), Email to Geoff Sutcliffe *)
+
+(* Source : [McC05] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.86 v3.1.0 *)
+
+(* Syntax : Number of clauses : 10 ( 0 non-Horn; 10 unit; 1 RR) *)
+
+(* Number of atoms : 10 ( 10 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 20 ( 2 singleton) *)
+
+(* Maximal term depth : 7 ( 3 average) *)
+
+(* Comments : *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* ----Include Lattice theory (equality) axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ------------------------------------------------------------------------------ *)
+ntheorem prove_H40:
+ ∀Univ:Type.∀U:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀d:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀U:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet X (join Y (meet Z (join X U)))) (meet X (join Y (meet Z (join U (meet X Z))))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H7:∀X:Univ.eq Univ (join X X) X.
+∀H8:∀X:Univ.eq Univ (meet X X) X.eq Univ (meet a (join b (meet c (join a d)))) (meet a (join b (meet c (join d (meet c (join a b))))))
+.
+#Univ.
+#U.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#d.
+#join.
+#meet.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8;
+nqed.
+
+(* ------------------------------------------------------------------------------ *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT151-1.p *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* File : LAT151-1 : TPTP v3.2.0. Released v3.1.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Huntington equation H39 implies H42 *)
+
+(* Version : [McC05] (equality) axioms : Especial. *)
+
+(* English : *)
+
+(* Refs : [McC05] McCune (2005), Email to Geoff Sutcliffe *)
+
+(* Source : [McC05] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.86 v3.1.0 *)
+
+(* Syntax : Number of clauses : 10 ( 0 non-Horn; 10 unit; 1 RR) *)
+
+(* Number of atoms : 10 ( 10 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 20 ( 2 singleton) *)
+
+(* Maximal term depth : 7 ( 3 average) *)
+
+(* Comments : *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* ----Include Lattice theory (equality) axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ------------------------------------------------------------------------------ *)
+ntheorem prove_H42:
+ ∀Univ:Type.∀U:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀d:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀U:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet X (join Y (meet Z (join X U)))) (meet X (join Y (meet Z (join U (meet X Z))))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H7:∀X:Univ.eq Univ (join X X) X.
+∀H8:∀X:Univ.eq Univ (meet X X) X.eq Univ (meet a (join b (meet c (join a d)))) (meet a (join b (meet c (join b (join d (meet a c))))))
+.
+#Univ.
+#U.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#d.
+#join.
+#meet.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8;
+nqed.
+
+(* ------------------------------------------------------------------------------ *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT152-1.p *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* File : LAT152-1 : TPTP v3.2.0. Released v3.1.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Huntington equation H40 implies H6 *)
+
+(* Version : [McC05] (equality) axioms : Especial. *)
+
+(* English : *)
+
+(* Refs : [McC05] McCune (2005), Email to Geoff Sutcliffe *)
+
+(* Source : [McC05] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.86 v3.2.0, 0.79 v3.1.0 *)
+
+(* Syntax : Number of clauses : 10 ( 0 non-Horn; 10 unit; 1 RR) *)
+
+(* Number of atoms : 10 ( 10 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 20 ( 2 singleton) *)
+
+(* Maximal term depth : 7 ( 3 average) *)
+
+(* Comments : *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* ----Include Lattice theory (equality) axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ------------------------------------------------------------------------------ *)
+ntheorem prove_H6:
+ ∀Univ:Type.∀U:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀U:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet X (join Y (meet Z (join X U)))) (meet X (join Y (meet Z (join U (meet Z (join X Y)))))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H7:∀X:Univ.eq Univ (join X X) X.
+∀H8:∀X:Univ.eq Univ (meet X X) X.eq Univ (meet a (join b (meet a c))) (meet a (join (meet a (join b (meet a c))) (meet c (join a b))))
+.
+#Univ.
+#U.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#join.
+#meet.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8;
+nqed.
+
+(* ------------------------------------------------------------------------------ *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT153-1.p *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* File : LAT153-1 : TPTP v3.2.0. Released v3.1.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Huntington equation H40 implies H7 *)
+
+(* Version : [McC05] (equality) axioms : Especial. *)
+
+(* English : *)
+
+(* Refs : [McC05] McCune (2005), Email to Geoff Sutcliffe *)
+
+(* Source : [McC05] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.93 v3.1.0 *)
+
+(* Syntax : Number of clauses : 10 ( 0 non-Horn; 10 unit; 1 RR) *)
+
+(* Number of atoms : 10 ( 10 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 20 ( 2 singleton) *)
+
+(* Maximal term depth : 7 ( 3 average) *)
+
+(* Comments : *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* ----Include Lattice theory (equality) axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ------------------------------------------------------------------------------ *)
+ntheorem prove_H7:
+ ∀Univ:Type.∀U:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀U:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet X (join Y (meet Z (join X U)))) (meet X (join Y (meet Z (join U (meet Z (join X Y)))))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H7:∀X:Univ.eq Univ (join X X) X.
+∀H8:∀X:Univ.eq Univ (meet X X) X.eq Univ (meet a (join b (meet a c))) (meet a (join b (meet a (join (meet a b) (meet c (join a b))))))
+.
+#Univ.
+#U.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#join.
+#meet.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8;
+nqed.
+
+(* ------------------------------------------------------------------------------ *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT154-1.p *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* File : LAT154-1 : TPTP v3.2.0. Released v3.1.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Huntington equation H42 implies H6 *)
+
+(* Version : [McC05] (equality) axioms : Especial. *)
+
+(* English : *)
+
+(* Refs : [McC05] McCune (2005), Email to Geoff Sutcliffe *)
+
+(* Source : [McC05] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.86 v3.2.0, 0.79 v3.1.0 *)
+
+(* Syntax : Number of clauses : 10 ( 0 non-Horn; 10 unit; 1 RR) *)
+
+(* Number of atoms : 10 ( 10 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 20 ( 2 singleton) *)
+
+(* Maximal term depth : 7 ( 3 average) *)
+
+(* Comments : *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* ----Include Lattice theory (equality) axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ------------------------------------------------------------------------------ *)
+ntheorem prove_H6:
+ ∀Univ:Type.∀U:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀U:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet X (join Y (meet Z (join X U)))) (meet X (join Y (meet Z (join Y (join U (meet X Z)))))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H7:∀X:Univ.eq Univ (join X X) X.
+∀H8:∀X:Univ.eq Univ (meet X X) X.eq Univ (meet a (join b (meet a c))) (meet a (join (meet a (join b (meet a c))) (meet c (join a b))))
+.
+#Univ.
+#U.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#join.
+#meet.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8;
+nqed.
+
+(* ------------------------------------------------------------------------------ *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT155-1.p *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* File : LAT155-1 : TPTP v3.2.0. Released v3.1.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Huntington equation H49 implies H2 *)
+
+(* Version : [McC05] (equality) axioms : Especial. *)
+
+(* English : *)
+
+(* Refs : [McC05] McCune (2005), Email to Geoff Sutcliffe *)
+
+(* Source : [McC05] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.86 v3.1.0 *)
+
+(* Syntax : Number of clauses : 10 ( 0 non-Horn; 10 unit; 1 RR) *)
+
+(* Number of atoms : 10 ( 10 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 20 ( 2 singleton) *)
+
+(* Maximal term depth : 7 ( 3 average) *)
+
+(* Comments : *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* ----Include Lattice theory (equality) axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ------------------------------------------------------------------------------ *)
+ntheorem prove_H2:
+ ∀Univ:Type.∀U:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀U:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet X (join Y (meet Z (join X U)))) (meet X (join Y (join (meet X Z) (meet Z (join Y U))))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H7:∀X:Univ.eq Univ (join X X) X.
+∀H8:∀X:Univ.eq Univ (meet X X) X.eq Univ (meet a (join b (meet a c))) (meet a (join b (meet c (join (meet a (join b c)) (meet b c)))))
+.
+#Univ.
+#U.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#join.
+#meet.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8;
+nqed.
+
+(* ------------------------------------------------------------------------------ *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT156-1.p *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* File : LAT156-1 : TPTP v3.2.0. Released v3.1.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Huntington equation H49 implies H6 *)
+
+(* Version : [McC05] (equality) axioms : Especial. *)
+
+(* English : *)
+
+(* Refs : [McC05] McCune (2005), Email to Geoff Sutcliffe *)
+
+(* Source : [McC05] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.71 v3.2.0, 0.64 v3.1.0 *)
+
+(* Syntax : Number of clauses : 10 ( 0 non-Horn; 10 unit; 1 RR) *)
+
+(* Number of atoms : 10 ( 10 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 20 ( 2 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* ----Include Lattice theory (equality) axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ------------------------------------------------------------------------------ *)
+ntheorem prove_H6:
+ ∀Univ:Type.∀U:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀U:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet X (join Y (meet Z (join X U)))) (meet X (join Y (join (meet X Z) (meet Z (join Y U))))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H7:∀X:Univ.eq Univ (join X X) X.
+∀H8:∀X:Univ.eq Univ (meet X X) X.eq Univ (meet a (join b (meet a c))) (meet a (join (meet a (join b (meet a c))) (meet c (join a b))))
+.
+#Univ.
+#U.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#join.
+#meet.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8;
+nqed.
+
+(* ------------------------------------------------------------------------------ *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT157-1.p *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* File : LAT157-1 : TPTP v3.2.0. Released v3.1.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Huntington equation H50 implies H2 *)
+
+(* Version : [McC05] (equality) axioms : Especial. *)
+
+(* English : *)
+
+(* Refs : [McC05] McCune (2005), Email to Geoff Sutcliffe *)
+
+(* Source : [McC05] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.86 v3.1.0 *)
+
+(* Syntax : Number of clauses : 10 ( 0 non-Horn; 10 unit; 1 RR) *)
+
+(* Number of atoms : 10 ( 10 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 20 ( 2 singleton) *)
+
+(* Maximal term depth : 7 ( 3 average) *)
+
+(* Comments : *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* ----Include Lattice theory (equality) axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ------------------------------------------------------------------------------ *)
+ntheorem prove_H2:
+ ∀Univ:Type.∀U:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀U:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet X (join Y (meet Z (join X U)))) (meet X (join Y (meet Z (join X (meet Z (join Y U)))))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H7:∀X:Univ.eq Univ (join X X) X.
+∀H8:∀X:Univ.eq Univ (meet X X) X.eq Univ (meet a (join b (meet a c))) (meet a (join b (meet c (join (meet a (join b c)) (meet b c)))))
+.
+#Univ.
+#U.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#join.
+#meet.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8;
+nqed.
+
+(* ------------------------------------------------------------------------------ *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT158-1.p *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* File : LAT158-1 : TPTP v3.2.0. Released v3.1.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Huntington equation H50 implies H49 *)
+
+(* Version : [McC05] (equality) axioms : Especial. *)
+
+(* English : *)
+
+(* Refs : [McC05] McCune (2005), Email to Geoff Sutcliffe *)
+
+(* Source : [McC05] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.93 v3.1.0 *)
+
+(* Syntax : Number of clauses : 10 ( 0 non-Horn; 10 unit; 1 RR) *)
+
+(* Number of atoms : 10 ( 10 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 20 ( 2 singleton) *)
+
+(* Maximal term depth : 7 ( 3 average) *)
+
+(* Comments : *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* ----Include Lattice theory (equality) axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ------------------------------------------------------------------------------ *)
+ntheorem prove_H49:
+ ∀Univ:Type.∀U:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀d:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀U:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet X (join Y (meet Z (join X U)))) (meet X (join Y (meet Z (join X (meet Z (join Y U)))))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H7:∀X:Univ.eq Univ (join X X) X.
+∀H8:∀X:Univ.eq Univ (meet X X) X.eq Univ (meet a (join b (meet c (join a d)))) (meet a (join b (join (meet a c) (meet c (join b d)))))
+.
+#Univ.
+#U.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#d.
+#join.
+#meet.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8;
+nqed.
+
+(* ------------------------------------------------------------------------------ *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT159-1.p *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* File : LAT159-1 : TPTP v3.2.0. Released v3.1.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Huntington equation H50 implies H7 *)
+
+(* Version : [McC05] (equality) axioms : Especial. *)
+
+(* English : *)
+
+(* Refs : [McC05] McCune (2005), Email to Geoff Sutcliffe *)
+
+(* Source : [McC05] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.86 v3.1.0 *)
+
+(* Syntax : Number of clauses : 10 ( 0 non-Horn; 10 unit; 1 RR) *)
+
+(* Number of atoms : 10 ( 10 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 20 ( 2 singleton) *)
+
+(* Maximal term depth : 7 ( 3 average) *)
+
+(* Comments : *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* ----Include Lattice theory (equality) axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ------------------------------------------------------------------------------ *)
+ntheorem prove_H7:
+ ∀Univ:Type.∀U:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀U:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet X (join Y (meet Z (join X U)))) (meet X (join Y (meet Z (join X (meet Z (join Y U)))))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H7:∀X:Univ.eq Univ (join X X) X.
+∀H8:∀X:Univ.eq Univ (meet X X) X.eq Univ (meet a (join b (meet a c))) (meet a (join b (meet a (join (meet a b) (meet c (join a b))))))
+.
+#Univ.
+#U.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#join.
+#meet.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8;
+nqed.
+
+(* ------------------------------------------------------------------------------ *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT160-1.p *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* File : LAT160-1 : TPTP v3.2.0. Released v3.1.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Huntington equation H52 implies H51 *)
+
+(* Version : [McC05] (equality) axioms : Especial. *)
+
+(* English : *)
+
+(* Refs : [McC05] McCune (2005), Email to Geoff Sutcliffe *)
+
+(* Source : [McC05] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.64 v3.1.0 *)
+
+(* Syntax : Number of clauses : 10 ( 0 non-Horn; 10 unit; 1 RR) *)
+
+(* Number of atoms : 10 ( 10 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 20 ( 2 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* ----Include Lattice theory (equality) axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ------------------------------------------------------------------------------ *)
+ntheorem prove_H51:
+ ∀Univ:Type.∀U:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀d:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀U:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet X (join Y (meet Z (join X U)))) (meet X (join Y (join (meet Z U) (meet Z (join X Y))))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H7:∀X:Univ.eq Univ (join X X) X.
+∀H8:∀X:Univ.eq Univ (meet X X) X.eq Univ (meet a (join b (meet c (join a d)))) (meet a (join b (join (meet a c) (meet c d))))
+.
+#Univ.
+#U.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#d.
+#join.
+#meet.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8;
+nqed.
+
+(* ------------------------------------------------------------------------------ *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT161-1.p *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* File : LAT161-1 : TPTP v3.2.0. Released v3.1.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Huntington equation H58 implies H59 *)
+
+(* Version : [McC05] (equality) axioms : Especial. *)
+
+(* English : *)
+
+(* Refs : [McC05] McCune (2005), Email to Geoff Sutcliffe *)
+
+(* Source : [McC05] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 1.00 v3.1.0 *)
+
+(* Syntax : Number of clauses : 10 ( 0 non-Horn; 10 unit; 1 RR) *)
+
+(* Number of atoms : 10 ( 10 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 19 ( 2 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* ----Include Lattice theory (equality) axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ------------------------------------------------------------------------------ *)
+ntheorem prove_H59:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀d:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet X (join Y Z)) (meet X (join Y (meet (join X Y) (join Z (meet X Y))))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H7:∀X:Univ.eq Univ (join X X) X.
+∀H8:∀X:Univ.eq Univ (meet X X) X.eq Univ (meet a (meet (join b c) (join b d))) (meet a (join b (meet (join b d) (join c (meet a b)))))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#d.
+#join.
+#meet.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8;
+nqed.
+
+(* ------------------------------------------------------------------------------ *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT162-1.p *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* File : LAT162-1 : TPTP v3.2.0. Released v3.1.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Huntington equation H68 implies H73 *)
+
+(* Version : [McC05] (equality) axioms : Especial. *)
+
+(* English : *)
+
+(* Refs : [McC05] McCune (2005), Email to Geoff Sutcliffe *)
+
+(* Source : [McC05] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.86 v3.1.0 *)
+
+(* Syntax : Number of clauses : 10 ( 0 non-Horn; 10 unit; 1 RR) *)
+
+(* Number of atoms : 10 ( 10 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 19 ( 2 singleton) *)
+
+(* Maximal term depth : 7 ( 3 average) *)
+
+(* Comments : *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* ----Include Lattice theory (equality) axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ------------------------------------------------------------------------------ *)
+ntheorem prove_H73:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀d:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet X (join Y Z)) (meet X (join Y (meet X (join Z (meet X Y))))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H7:∀X:Univ.eq Univ (join X X) X.
+∀H8:∀X:Univ.eq Univ (meet X X) X.eq Univ (meet a (meet b (join c d))) (meet a (meet b (join c (meet a (join d (meet b c))))))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#d.
+#join.
+#meet.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8;
+nqed.
+
+(* ------------------------------------------------------------------------------ *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT163-1.p *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* File : LAT163-1 : TPTP v3.2.0. Released v3.1.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Huntington equation H76 implies H32 *)
+
+(* Version : [McC05] (equality) axioms : Especial. *)
+
+(* English : *)
+
+(* Refs : [McC05] McCune (2005), Email to Geoff Sutcliffe *)
+
+(* Source : [McC05] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.93 v3.1.0 *)
+
+(* Syntax : Number of clauses : 10 ( 0 non-Horn; 10 unit; 1 RR) *)
+
+(* Number of atoms : 10 ( 10 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 20 ( 2 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* ----Include Lattice theory (equality) axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ------------------------------------------------------------------------------ *)
+ntheorem prove_H32:
+ ∀Univ:Type.∀U:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀d:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀U:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet X (join Y (meet Z (join Y U)))) (meet X (join Y (meet Z (join U (meet X Y))))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H7:∀X:Univ.eq Univ (join X X) X.
+∀H8:∀X:Univ.eq Univ (meet X X) X.eq Univ (meet a (join b (meet a (meet c d)))) (meet a (join b (meet c (join (meet a d) (meet b d)))))
+.
+#Univ.
+#U.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#d.
+#join.
+#meet.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8;
+nqed.
+
+(* ------------------------------------------------------------------------------ *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT164-1.p *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* File : LAT164-1 : TPTP v3.2.0. Released v3.1.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Huntington equation H76 implies H6 *)
+
+(* Version : [McC05] (equality) axioms : Especial. *)
+
+(* English : *)
+
+(* Refs : [McC05] McCune (2005), Email to Geoff Sutcliffe *)
+
+(* Source : [McC05] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.93 v3.1.0 *)
+
+(* Syntax : Number of clauses : 10 ( 0 non-Horn; 10 unit; 1 RR) *)
+
+(* Number of atoms : 10 ( 10 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 20 ( 2 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* ----Include Lattice theory (equality) axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ------------------------------------------------------------------------------ *)
+ntheorem prove_H6:
+ ∀Univ:Type.∀U:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀U:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet X (join Y (meet Z (join Y U)))) (meet X (join Y (meet Z (join U (meet X Y))))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H7:∀X:Univ.eq Univ (join X X) X.
+∀H8:∀X:Univ.eq Univ (meet X X) X.eq Univ (meet a (join b (meet a c))) (meet a (join (meet a (join b (meet a c))) (meet c (join a b))))
+.
+#Univ.
+#U.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#join.
+#meet.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8;
+nqed.
+
+(* ------------------------------------------------------------------------------ *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT165-1.p *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* File : LAT165-1 : TPTP v3.2.0. Released v3.1.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Huntington equation H76 implies H77 *)
+
+(* Version : [McC05] (equality) axioms : Especial. *)
+
+(* English : *)
+
+(* Refs : [McC05] McCune (2005), Email to Geoff Sutcliffe *)
+
+(* Source : [McC05] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.86 v3.2.0, 0.93 v3.1.0 *)
+
+(* Syntax : Number of clauses : 10 ( 0 non-Horn; 10 unit; 1 RR) *)
+
+(* Number of atoms : 10 ( 10 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 20 ( 2 singleton) *)
+
+(* Maximal term depth : 7 ( 3 average) *)
+
+(* Comments : *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* ----Include Lattice theory (equality) axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ------------------------------------------------------------------------------ *)
+ntheorem prove_H77:
+ ∀Univ:Type.∀U:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀d:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀U:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet X (join Y (meet Z (join Y U)))) (meet X (join Y (meet Z (join U (meet X Y))))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H7:∀X:Univ.eq Univ (join X X) X.
+∀H8:∀X:Univ.eq Univ (meet X X) X.eq Univ (meet a (join b (meet c (join b d)))) (meet a (join b (meet c (join d (meet a (meet b c))))))
+.
+#Univ.
+#U.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#d.
+#join.
+#meet.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8;
+nqed.
+
+(* ------------------------------------------------------------------------------ *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT166-1.p *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* File : LAT166-1 : TPTP v3.2.0. Released v3.1.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Huntington equation H77 implies H78 *)
+
+(* Version : [McC05] (equality) axioms : Especial. *)
+
+(* English : *)
+
+(* Refs : [McC05] McCune (2005), Email to Geoff Sutcliffe *)
+
+(* Source : [McC05] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.93 v3.1.0 *)
+
+(* Syntax : Number of clauses : 10 ( 0 non-Horn; 10 unit; 1 RR) *)
+
+(* Number of atoms : 10 ( 10 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 20 ( 2 singleton) *)
+
+(* Maximal term depth : 7 ( 3 average) *)
+
+(* Comments : *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* ----Include Lattice theory (equality) axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ------------------------------------------------------------------------------ *)
+ntheorem prove_H78:
+ ∀Univ:Type.∀U:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀d:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀U:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet X (join Y (meet Z (join Y U)))) (meet X (join Y (meet Z (join U (meet X (meet Y Z)))))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H7:∀X:Univ.eq Univ (join X X) X.
+∀H8:∀X:Univ.eq Univ (meet X X) X.eq Univ (meet a (join b (meet c (join b d)))) (meet a (join b (meet c (join d (meet b (join a d))))))
+.
+#Univ.
+#U.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#d.
+#join.
+#meet.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8;
+nqed.
+
+(* ------------------------------------------------------------------------------ *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT167-1.p *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* File : LAT167-1 : TPTP v3.2.0. Released v3.1.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Huntington equation H78 implies H77 *)
+
+(* Version : [McC05] (equality) axioms : Especial. *)
+
+(* English : *)
+
+(* Refs : [McC05] McCune (2005), Email to Geoff Sutcliffe *)
+
+(* Source : [McC05] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.93 v3.1.0 *)
+
+(* Syntax : Number of clauses : 10 ( 0 non-Horn; 10 unit; 1 RR) *)
+
+(* Number of atoms : 10 ( 10 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 20 ( 2 singleton) *)
+
+(* Maximal term depth : 7 ( 3 average) *)
+
+(* Comments : *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* ----Include Lattice theory (equality) axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ------------------------------------------------------------------------------ *)
+ntheorem prove_H77:
+ ∀Univ:Type.∀U:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀d:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀U:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet X (join Y (meet Z (join Y U)))) (meet X (join Y (meet Z (join U (meet Y (join X U)))))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H7:∀X:Univ.eq Univ (join X X) X.
+∀H8:∀X:Univ.eq Univ (meet X X) X.eq Univ (meet a (join b (meet c (join b d)))) (meet a (join b (meet c (join d (meet a (meet b c))))))
+.
+#Univ.
+#U.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#d.
+#join.
+#meet.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8;
+nqed.
+
+(* ------------------------------------------------------------------------------ *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT168-1.p *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* File : LAT168-1 : TPTP v3.2.0. Released v3.1.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Huntington equation H18_dual implies H58 *)
+
+(* Version : [McC05] (equality) axioms : Especial. *)
+
+(* English : *)
+
+(* Refs : [McC05] McCune (2005), Email to Geoff Sutcliffe *)
+
+(* Source : [McC05] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.21 v3.2.0, 0.14 v3.1.0 *)
+
+(* Syntax : Number of clauses : 10 ( 0 non-Horn; 10 unit; 1 RR) *)
+
+(* Number of atoms : 10 ( 10 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 19 ( 2 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* ----Include Lattice theory (equality) axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ------------------------------------------------------------------------------ *)
+ntheorem prove_H58:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (join X Y) (join X Z)) (join X (meet (join X Y) (meet (join X Z) (join Y (meet X Z))))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H7:∀X:Univ.eq Univ (join X X) X.
+∀H8:∀X:Univ.eq Univ (meet X X) X.eq Univ (meet a (join b c)) (meet a (join b (meet (join a b) (join c (meet a b)))))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#join.
+#meet.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8;
+nqed.
+
+(* ------------------------------------------------------------------------------ *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT169-1.p *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* File : LAT169-1 : TPTP v3.2.0. Released v3.1.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Huntington equation H21_dual implies H58 *)
+
+(* Version : [McC05] (equality) axioms : Especial. *)
+
+(* English : *)
+
+(* Refs : [McC05] McCune (2005), Email to Geoff Sutcliffe *)
+
+(* Source : [McC05] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.86 v3.1.0 *)
+
+(* Syntax : Number of clauses : 10 ( 0 non-Horn; 10 unit; 1 RR) *)
+
+(* Number of atoms : 10 ( 10 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 19 ( 2 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* ----Include Lattice theory (equality) axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ------------------------------------------------------------------------------ *)
+ntheorem prove_H58:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (join X Y) (join X Z)) (join X (meet (join Y (meet X (join Y Z))) (join Z (meet X Y)))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H7:∀X:Univ.eq Univ (join X X) X.
+∀H8:∀X:Univ.eq Univ (meet X X) X.eq Univ (meet a (join b c)) (meet a (join b (meet (join a b) (join c (meet a b)))))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#join.
+#meet.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8;
+nqed.
+
+(* ------------------------------------------------------------------------------ *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT170-1.p *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* File : LAT170-1 : TPTP v3.2.0. Released v3.1.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Huntington equation H49_dual implies H58 *)
+
+(* Version : [McC05] (equality) axioms : Especial. *)
+
+(* English : *)
+
+(* Refs : [McC05] McCune (2005), Email to Geoff Sutcliffe *)
+
+(* Source : [McC05] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.79 v3.1.0 *)
+
+(* Syntax : Number of clauses : 10 ( 0 non-Horn; 10 unit; 1 RR) *)
+
+(* Number of atoms : 10 ( 10 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 20 ( 2 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* ----Include Lattice theory (equality) axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ------------------------------------------------------------------------------ *)
+ntheorem prove_H58:
+ ∀Univ:Type.∀U:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀U:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join X (meet Y (join Z (meet X U)))) (join X (meet Y (meet (join X Z) (join Z (meet Y U))))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H7:∀X:Univ.eq Univ (join X X) X.
+∀H8:∀X:Univ.eq Univ (meet X X) X.eq Univ (meet a (join b c)) (meet a (join b (meet (join a b) (join c (meet a b)))))
+.
+#Univ.
+#U.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#join.
+#meet.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8;
+nqed.
+
+(* ------------------------------------------------------------------------------ *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT171-1.p *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* File : LAT171-1 : TPTP v3.2.0. Released v3.1.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Huntington equation H61_dual implies H6 *)
+
+(* Version : [McC05] (equality) axioms : Especial. *)
+
+(* English : *)
+
+(* Refs : [McC05] McCune (2005), Email to Geoff Sutcliffe *)
+
+(* Source : [McC05] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.57 v3.2.0, 0.50 v3.1.0 *)
+
+(* Syntax : Number of clauses : 10 ( 0 non-Horn; 10 unit; 1 RR) *)
+
+(* Number of atoms : 10 ( 10 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 19 ( 2 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* ----Include Lattice theory (equality) axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ------------------------------------------------------------------------------ *)
+ntheorem prove_H6:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (meet X Y) (meet X Z)) (meet X (join (meet X Y) (meet (join X Y) Z))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H7:∀X:Univ.eq Univ (join X X) X.
+∀H8:∀X:Univ.eq Univ (meet X X) X.eq Univ (meet a (join b (meet a c))) (meet a (join (meet a (join b (meet a c))) (meet c (join a b))))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#join.
+#meet.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8;
+nqed.
+
+(* ------------------------------------------------------------------------------ *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT172-1.p *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* File : LAT172-1 : TPTP v3.2.0. Released v3.1.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Huntington equation H76_dual implies H32 *)
+
+(* Version : [McC05] (equality) axioms : Especial. *)
+
+(* English : *)
+
+(* Refs : [McC05] McCune (2005), Email to Geoff Sutcliffe *)
+
+(* Source : [McC05] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.93 v3.1.0 *)
+
+(* Syntax : Number of clauses : 10 ( 0 non-Horn; 10 unit; 1 RR) *)
+
+(* Number of atoms : 10 ( 10 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 20 ( 2 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* ----Include Lattice theory (equality) axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ------------------------------------------------------------------------------ *)
+ntheorem prove_H32:
+ ∀Univ:Type.∀U:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀d:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀U:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join X (meet Y (join Z (meet Y U)))) (join X (meet Y (join Z (meet U (join X Y))))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H7:∀X:Univ.eq Univ (join X X) X.
+∀H8:∀X:Univ.eq Univ (meet X X) X.eq Univ (meet a (join b (meet a (meet c d)))) (meet a (join b (meet c (join (meet a d) (meet b d)))))
+.
+#Univ.
+#U.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#d.
+#join.
+#meet.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8;
+nqed.
+
+(* ------------------------------------------------------------------------------ *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT173-1.p *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* File : LAT173-1 : TPTP v3.2.0. Released v3.1.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Huntington equation H76_dual implies H40 *)
+
+(* Version : [McC05] (equality) axioms : Especial. *)
+
+(* English : *)
+
+(* Refs : [McC05] McCune (2005), Email to Geoff Sutcliffe *)
+
+(* Source : [McC05] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.93 v3.1.0 *)
+
+(* Syntax : Number of clauses : 10 ( 0 non-Horn; 10 unit; 1 RR) *)
+
+(* Number of atoms : 10 ( 10 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 20 ( 2 singleton) *)
+
+(* Maximal term depth : 7 ( 3 average) *)
+
+(* Comments : *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* ----Include Lattice theory (equality) axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ------------------------------------------------------------------------------ *)
+ntheorem prove_H40:
+ ∀Univ:Type.∀U:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀d:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀U:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join X (meet Y (join Z (meet Y U)))) (join X (meet Y (join Z (meet U (join X Y))))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H7:∀X:Univ.eq Univ (join X X) X.
+∀H8:∀X:Univ.eq Univ (meet X X) X.eq Univ (meet a (join b (meet c (join a d)))) (meet a (join b (meet c (join d (meet c (join a b))))))
+.
+#Univ.
+#U.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#d.
+#join.
+#meet.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8;
+nqed.
+
+(* ------------------------------------------------------------------------------ *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT174-1.p *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* File : LAT174-1 : TPTP v3.2.0. Released v3.1.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Huntington equation H76_dual implies H6 *)
+
+(* Version : [McC05] (equality) axioms : Especial. *)
+
+(* English : *)
+
+(* Refs : [McC05] McCune (2005), Email to Geoff Sutcliffe *)
+
+(* Source : [McC05] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.79 v3.1.0 *)
+
+(* Syntax : Number of clauses : 10 ( 0 non-Horn; 10 unit; 1 RR) *)
+
+(* Number of atoms : 10 ( 10 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 20 ( 2 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* ----Include Lattice theory (equality) axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ------------------------------------------------------------------------------ *)
+ntheorem prove_H6:
+ ∀Univ:Type.∀U:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀U:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join X (meet Y (join Z (meet Y U)))) (join X (meet Y (join Z (meet U (join X Y))))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H7:∀X:Univ.eq Univ (join X X) X.
+∀H8:∀X:Univ.eq Univ (meet X X) X.eq Univ (meet a (join b (meet a c))) (meet a (join (meet a (join b (meet a c))) (meet c (join a b))))
+.
+#Univ.
+#U.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#join.
+#meet.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8;
+nqed.
+
+(* ------------------------------------------------------------------------------ *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT175-1.p *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* File : LAT175-1 : TPTP v3.2.0. Released v3.1.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Huntington equation H79_dual implies H32 *)
+
+(* Version : [McC05] (equality) axioms : Especial. *)
+
+(* English : *)
+
+(* Refs : [McC05] McCune (2005), Email to Geoff Sutcliffe *)
+
+(* Source : [McC05] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.93 v3.1.0 *)
+
+(* Syntax : Number of clauses : 10 ( 0 non-Horn; 10 unit; 1 RR) *)
+
+(* Number of atoms : 10 ( 10 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 20 ( 2 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* ----Include Lattice theory (equality) axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ------------------------------------------------------------------------------ *)
+ntheorem prove_H32:
+ ∀Univ:Type.∀U:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀d:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀U:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join X (meet Y (join Z (meet X U)))) (join X (meet (join X (meet Y (join X Z))) (join Z U))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H7:∀X:Univ.eq Univ (join X X) X.
+∀H8:∀X:Univ.eq Univ (meet X X) X.eq Univ (meet a (join b (meet a (meet c d)))) (meet a (join b (meet c (join (meet a d) (meet b d)))))
+.
+#Univ.
+#U.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#d.
+#join.
+#meet.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8;
+nqed.
+
+(* ------------------------------------------------------------------------------ *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT176-1.p *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* File : LAT176-1 : TPTP v3.2.0. Released v3.1.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Huntington equation H79_dual implies H42 *)
+
+(* Version : [McC05] (equality) axioms : Especial. *)
+
+(* English : *)
+
+(* Refs : [McC05] McCune (2005), Email to Geoff Sutcliffe *)
+
+(* Source : [McC05] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.93 v3.1.0 *)
+
+(* Syntax : Number of clauses : 10 ( 0 non-Horn; 10 unit; 1 RR) *)
+
+(* Number of atoms : 10 ( 10 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 20 ( 2 singleton) *)
+
+(* Maximal term depth : 7 ( 3 average) *)
+
+(* Comments : *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* ----Include Lattice theory (equality) axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ------------------------------------------------------------------------------ *)
+ntheorem prove_H42:
+ ∀Univ:Type.∀U:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀d:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀U:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join X (meet Y (join Z (meet X U)))) (join X (meet (join X (meet Y (join X Z))) (join Z U))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H7:∀X:Univ.eq Univ (join X X) X.
+∀H8:∀X:Univ.eq Univ (meet X X) X.eq Univ (meet a (join b (meet c (join a d)))) (meet a (join b (meet c (join b (join d (meet a c))))))
+.
+#Univ.
+#U.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#d.
+#join.
+#meet.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8;
+nqed.
+
+(* ------------------------------------------------------------------------------ *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT177-1.p *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* File : LAT177-1 : TPTP v3.2.0. Released v3.1.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Problem : Huntington equation H79_dual implies H6 *)
+
+(* Version : [McC05] (equality) axioms : Especial. *)
+
+(* English : *)
+
+(* Refs : [McC05] McCune (2005), Email to Geoff Sutcliffe *)
+
+(* Source : [McC05] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 1.00 v3.1.0 *)
+
+(* Syntax : Number of clauses : 10 ( 0 non-Horn; 10 unit; 1 RR) *)
+
+(* Number of atoms : 10 ( 10 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 20 ( 2 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* ----Include Lattice theory (equality) axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Lattice Theory *)
+
+(* Axioms : Lattice theory (equality) axioms *)
+
+(* Version : [McC88] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
+
+(* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* Source : [McC88] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 2-2 arity) *)
+
+(* Number of variables : 16 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The following 8 clauses characterise lattices *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ------------------------------------------------------------------------------ *)
+ntheorem prove_H6:
+ ∀Univ:Type.∀U:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀join:∀_:Univ.∀_:Univ.Univ.
+∀meet:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀U:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join X (meet Y (join Z (meet X U)))) (join X (meet (join X (meet Y (join X Z))) (join Z U))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
+∀H7:∀X:Univ.eq Univ (join X X) X.
+∀H8:∀X:Univ.eq Univ (meet X X) X.eq Univ (meet a (join b (meet a c))) (meet a (join (meet a (join b (meet a c))) (meet c (join a b))))
+.
+#Univ.
+#U.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#join.
+#meet.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8;
+nqed.
+
+(* ------------------------------------------------------------------------------ *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LCL109-2.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LCL109-2 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Logic Calculi (Many valued sentential) *)
+
+(* Problem : MV-4 depends on the Merideth system *)
+
+(* Version : [Ove90] axioms. *)
+
+(* Theorem formulation : Wajsberg algebra formulation. *)
+
+(* English : An axiomatisation of the many valued sentential calculus *)
+
+(* is {MV-1,MV-2,MV-3,MV-5} by Meredith. Wajsberg provided *)
+
+(* a different axiomatisation. Show that MV-4 depends on the *)
+
+(* Wajsberg system. *)
+
+(* Refs : [Ove90] Overbeek (1990), ATP competition announced at CADE-10 *)
+
+(* : [LM92] Lusk & McCune (1992), Experiments with ROO, a Parallel *)
+
+(* : [LW92] Lusk & Wos (1992), Benchmark Problems in Which Equalit *)
+
+(* : [Ove93] Overbeek (1993), The CADE-11 Competitions: A Personal *)
+
+(* : [LM93] Lusk & McCune (1993), Uniform Strategies: The CADE-11 *)
+
+(* : [Zha93] Zhang (1993), Automated Proofs of Equality Problems in *)
+
+(* Source : [Ove90] *)
+
+(* Names : CADE-11 Competition Eq-5 [Ove90] *)
+
+(* : Luka-5 [LM92] *)
+
+(* : MV4 [LW92] *)
+
+(* : THEOREM EQ-5 [LM93] *)
+
+(* : PROBLEM 5 [Zha93] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.29 v3.1.0, 0.22 v2.7.0, 0.27 v2.6.0, 0.17 v2.5.0, 0.00 v2.4.0, 0.33 v2.2.1, 0.56 v2.2.0, 0.71 v2.1.0, 1.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 1 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 8 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Wajsberg algebra axioms *)
+
+(* Inclusion of: Axioms/LCL001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LCL001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Logic Calculi (Wajsberg Algebras) *)
+
+(* Axioms : Wajsberg algebra axioms *)
+
+(* Version : [Bon91] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [FRT84] Font et al. (1984), Wajsberg Algebras *)
+
+(* : [Bon91] Bonacina (1991), Problems in Lukasiewicz Logic *)
+
+(* : [MW92] McCune & Wos (1992), Experiments in Automated Deductio *)
+
+(* Source : [MW92] *)
+
+(* Names : MV Sentential Calculus [MW92] *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 4 ( 0 non-Horn; 4 unit; 0 RR) *)
+
+(* Number of literals : 4 ( 4 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 8 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_wajsberg_mv_4:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀implies:∀_:Univ.∀_:Univ.Univ.
+∀not:∀_:Univ.Univ.
+∀truth:Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (implies (implies (not X) (not Y)) (implies Y X)) truth.
+∀H1:∀X:Univ.∀Y:Univ.eq Univ (implies (implies X Y) Y) (implies (implies Y X) X).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (implies (implies X Y) (implies (implies Y Z) (implies X Z))) truth.
+∀H3:∀X:Univ.eq Univ (implies truth X) X.eq Univ (implies (implies (implies a b) (implies b a)) (implies b a)) truth
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#implies.
+#not.
+#truth.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LCL109-6.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LCL109-6 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Logic Calculi (Wajsberg Algebra) *)
+
+(* Problem : A ntheorem in the lattice structure of Wajsberg algebras *)
+
+(* Version : [Bon91] (equality) axioms : Augmented. *)
+
+(* Theorem formulation : Alternative Wajsberg algebras lattice *)
+
+(* structure. *)
+
+(* English : *)
+
+(* Refs : [FRT84] Font et al. (1984), Wajsberg Algebras *)
+
+(* : [AB90] Anantharaman & Bonacina (1990), An Application of the *)
+
+(* : [Bon91] Bonacina (1991), Problems in Lukasiewicz Logic *)
+
+(* Source : [Bon91] *)
+
+(* Names : Lattice structure ntheorem 8 [Bon91] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.14 v3.1.0, 0.11 v2.7.0, 0.09 v2.6.0, 0.17 v2.5.0, 0.00 v2.4.0, 0.00 v2.2.1, 0.67 v2.2.0, 0.71 v2.1.0, 0.88 v2.0.0 *)
+
+(* Syntax : Number of clauses : 14 ( 0 non-Horn; 14 unit; 2 RR) *)
+
+(* Number of atoms : 14 ( 14 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 8 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 19 ( 1 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Alternative Wajsberg algebra axioms *)
+
+(* Inclusion of: Axioms/LCL002-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LCL002-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Logic Calculi (Wajsberg Algebras) *)
+
+(* Axioms : Alternative Wajsberg algebra axioms *)
+
+(* Version : [AB90] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [FRT84] Font et al. (1984), Wajsberg Algebras *)
+
+(* : [AB90] Anantharaman & Bonacina (1990), An Application of the *)
+
+(* : [Bon91] Bonacina (1991), Problems in Lukasiewicz Logic *)
+
+(* Source : [Bon91] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 10 ( 1 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : To be used in conjunction with the LAT003 alternative *)
+
+(* Wajsberg algebra definitions. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include some Alternative Wajsberg algebra definitions *)
+
+(* include('Axioms/LCL002-1.ax'). *)
+
+(* ----Definition that and_star is AC and xor is C *)
+
+(* ----Definition of false in terms of true *)
+
+(* ----Include the definition of implies in terms of xor and and_star *)
+ntheorem prove_wajsberg_mv_4:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀and_star:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀falsehood:Univ.
+∀implies:∀_:Univ.∀_:Univ.Univ.
+∀not:∀_:Univ.Univ.
+∀truth:Univ.
+∀xor:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (implies X Y) (xor truth (and_star X (xor truth Y))).
+∀H1:eq Univ (not truth) falsehood.
+∀H2:∀X:Univ.∀Y:Univ.eq Univ (and_star X Y) (and_star Y X).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (and_star (and_star X Y) Z) (and_star X (and_star Y Z)).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (xor X Y) (xor Y X).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (and_star (xor (and_star (xor truth X) Y) truth) Y) (and_star (xor (and_star (xor truth Y) X) truth) X).
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (xor X (xor truth Y)) (xor (xor X truth) Y).
+∀H7:∀X:Univ.eq Univ (and_star (xor truth X) X) falsehood.
+∀H8:∀X:Univ.eq Univ (and_star X falsehood) falsehood.
+∀H9:∀X:Univ.eq Univ (and_star X truth) X.
+∀H10:∀X:Univ.eq Univ (xor X X) falsehood.
+∀H11:∀X:Univ.eq Univ (xor X falsehood) X.
+∀H12:∀X:Univ.eq Univ (not X) (xor X truth).eq Univ (implies (implies (implies a b) (implies b a)) (implies b a)) truth
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#and_star.
+#b.
+#falsehood.
+#implies.
+#not.
+#truth.
+#xor.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LCL110-2.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LCL110-2 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Logic Calculi (Many valued sentential) *)
+
+(* Problem : MV-24 depends on the Meredith system *)
+
+(* Version : [LW92] axioms. *)
+
+(* Theorem formulation : Wajsberg algebra formulation *)
+
+(* English : An axiomatisation of the many valued sentential calculus *)
+
+(* is {MV-1,MV-2,MV-3,MV-5} by Meredith. Wajsberg presented *)
+
+(* an equality axiomatisation. Show that MV-24 depends on the *)
+
+(* Wajsberg axiomatisation. *)
+
+(* Refs : [FRT84] Font et al. (1984), Wajsberg Algebras *)
+
+(* : [MW92] McCune & Wos (1992), Experiments in Automated Deductio *)
+
+(* : [LW92] Lusk & Wos (1992), Benchmark Problems in Which Equalit *)
+
+(* Source : [LW92] *)
+
+(* Names : MV1.1 [LW92] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.2.1, 0.22 v2.2.0, 0.29 v2.1.0, 0.25 v2.0.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 1 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 8 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Wajsberg algebra axioms *)
+
+(* Inclusion of: Axioms/LCL001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LCL001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Logic Calculi (Wajsberg Algebras) *)
+
+(* Axioms : Wajsberg algebra axioms *)
+
+(* Version : [Bon91] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [FRT84] Font et al. (1984), Wajsberg Algebras *)
+
+(* : [Bon91] Bonacina (1991), Problems in Lukasiewicz Logic *)
+
+(* : [MW92] McCune & Wos (1992), Experiments in Automated Deductio *)
+
+(* Source : [MW92] *)
+
+(* Names : MV Sentential Calculus [MW92] *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 4 ( 0 non-Horn; 4 unit; 0 RR) *)
+
+(* Number of literals : 4 ( 4 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 8 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_mv_24:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀implies:∀_:Univ.∀_:Univ.Univ.
+∀not:∀_:Univ.Univ.
+∀truth:Univ.
+∀x:Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (implies (implies (not X) (not Y)) (implies Y X)) truth.
+∀H1:∀X:Univ.∀Y:Univ.eq Univ (implies (implies X Y) Y) (implies (implies Y X) X).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (implies (implies X Y) (implies (implies Y Z) (implies X Z))) truth.
+∀H3:∀X:Univ.eq Univ (implies truth X) X.eq Univ (implies (not (not x)) x) truth
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#implies.
+#not.
+#truth.
+#x.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LCL111-2.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LCL111-2 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Logic Calculi (Many valued sentential) *)
+
+(* Problem : MV-25 depends on the Meredith system *)
+
+(* Version : [Bon91] (equality) axioms. *)
+
+(* Theorem formulation : Wajsberg algebra formulation *)
+
+(* English : An axiomatisation of the many valued sentential calculus *)
+
+(* is {MV-1,MV-2,MV-3,MV-5} by Meredith. Wajsberg presented *)
+
+(* an equality axiomatisation. Show that MV-25 depends on the *)
+
+(* Wajsberg axiomatisation. *)
+
+(* Refs : [FRT84] Font et al. (1984), Wajsberg Algebras *)
+
+(* : [Bon91] Bonacina (1991), Problems in Lukasiewicz Logic *)
+
+(* : [MW92] McCune & Wos (1992), Experiments in Automated Deductio *)
+
+(* : [LW92] Lusk & Wos (1992), Benchmark Problems in Which Equalit *)
+
+(* Source : [Bon91] *)
+
+(* Names : Lemma 6 [Bon91] *)
+
+(* : MV2 [LW92] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.07 v3.1.0, 0.22 v2.7.0, 0.00 v2.4.0, 0.33 v2.2.1, 0.22 v2.2.0, 0.29 v2.1.0, 0.50 v2.0.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 1 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 8 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Wajsberg algebra axioms *)
+
+(* Inclusion of: Axioms/LCL001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LCL001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Logic Calculi (Wajsberg Algebras) *)
+
+(* Axioms : Wajsberg algebra axioms *)
+
+(* Version : [Bon91] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [FRT84] Font et al. (1984), Wajsberg Algebras *)
+
+(* : [Bon91] Bonacina (1991), Problems in Lukasiewicz Logic *)
+
+(* : [MW92] McCune & Wos (1992), Experiments in Automated Deductio *)
+
+(* Source : [MW92] *)
+
+(* Names : MV Sentential Calculus [MW92] *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 4 ( 0 non-Horn; 4 unit; 0 RR) *)
+
+(* Number of literals : 4 ( 4 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 8 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_mv_25:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀implies:∀_:Univ.∀_:Univ.Univ.
+∀not:∀_:Univ.Univ.
+∀truth:Univ.
+∀x:Univ.
+∀y:Univ.
+∀z:Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (implies (implies (not X) (not Y)) (implies Y X)) truth.
+∀H1:∀X:Univ.∀Y:Univ.eq Univ (implies (implies X Y) Y) (implies (implies Y X) X).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (implies (implies X Y) (implies (implies Y Z) (implies X Z))) truth.
+∀H3:∀X:Univ.eq Univ (implies truth X) X.eq Univ (implies (implies x y) (implies (implies z x) (implies z y))) truth
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#implies.
+#not.
+#truth.
+#x.
+#y.
+#z.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LCL112-2.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LCL112-2 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Logic Calculi (Many valued sentential) *)
+
+(* Problem : MV-29 depends on the Meredith system *)
+
+(* Version : [McC92] axioms. *)
+
+(* Theorem formulation : Wajsberg algebra formulation *)
+
+(* English : An axiomatisation of the many valued sentential calculus *)
+
+(* is {MV-1,MV-2,MV-3,MV-5} by Meredith. Wajsberg presented *)
+
+(* an equality axiomatisation. Show that MV-29 depends on the *)
+
+(* Wajsberg axiomatisation. *)
+
+(* Refs : [FRT84] Font et al. (1984), Wajsberg Algebras *)
+
+(* : [MW92] McCune & Wos (1992), Experiments in Automated Deductio *)
+
+(* : [LW92] Lusk & Wos (1992), Benchmark Problems in Which Equalit *)
+
+(* : [McC92] McCune (1992), Email to G. Sutcliffe *)
+
+(* Source : [McC92] *)
+
+(* Names : MV1.2 [LW92] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.2.1, 0.22 v2.2.0, 0.29 v2.1.0, 0.25 v2.0.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 1 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 8 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Wajsberg algebra axioms *)
+
+(* Inclusion of: Axioms/LCL001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LCL001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Logic Calculi (Wajsberg Algebras) *)
+
+(* Axioms : Wajsberg algebra axioms *)
+
+(* Version : [Bon91] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [FRT84] Font et al. (1984), Wajsberg Algebras *)
+
+(* : [Bon91] Bonacina (1991), Problems in Lukasiewicz Logic *)
+
+(* : [MW92] McCune & Wos (1992), Experiments in Automated Deductio *)
+
+(* Source : [MW92] *)
+
+(* Names : MV Sentential Calculus [MW92] *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 4 ( 0 non-Horn; 4 unit; 0 RR) *)
+
+(* Number of literals : 4 ( 4 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 8 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_mv_29:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀implies:∀_:Univ.∀_:Univ.Univ.
+∀not:∀_:Univ.Univ.
+∀truth:Univ.
+∀x:Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (implies (implies (not X) (not Y)) (implies Y X)) truth.
+∀H1:∀X:Univ.∀Y:Univ.eq Univ (implies (implies X Y) Y) (implies (implies Y X) X).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (implies (implies X Y) (implies (implies Y Z) (implies X Z))) truth.
+∀H3:∀X:Univ.eq Univ (implies truth X) X.eq Univ (implies x (not (not x))) truth
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#implies.
+#not.
+#truth.
+#x.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LCL113-2.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LCL113-2 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Logic Calculi (Many valued sentential) *)
+
+(* Problem : MV-33 depends on the Meredith system *)
+
+(* Version : [TPTP] axioms. *)
+
+(* Theorem formulation : Wajsberg algebra formulation *)
+
+(* English : An axiomatisation of the many valued sentential calculus *)
+
+(* is {MV-1,MV-2,MV-3,MV-5} by Meredith. Wajsberg presented *)
+
+(* an equality axiomatisation. Show that MV-33 depends on the *)
+
+(* Wajsberg axiomatisation. *)
+
+(* Refs : [FRT84] Font et al. (1984), Wajsberg Algebras *)
+
+(* : [MW92] McCune & Wos (1992), Experiments in Automated Deductio *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.2.1, 0.22 v2.2.0, 0.29 v2.1.0, 0.25 v2.0.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 1 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 8 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Wajsberg algebra axioms *)
+
+(* Inclusion of: Axioms/LCL001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LCL001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Logic Calculi (Wajsberg Algebras) *)
+
+(* Axioms : Wajsberg algebra axioms *)
+
+(* Version : [Bon91] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [FRT84] Font et al. (1984), Wajsberg Algebras *)
+
+(* : [Bon91] Bonacina (1991), Problems in Lukasiewicz Logic *)
+
+(* : [MW92] McCune & Wos (1992), Experiments in Automated Deductio *)
+
+(* Source : [MW92] *)
+
+(* Names : MV Sentential Calculus [MW92] *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 4 ( 0 non-Horn; 4 unit; 0 RR) *)
+
+(* Number of literals : 4 ( 4 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 8 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_mv_33:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀implies:∀_:Univ.∀_:Univ.Univ.
+∀not:∀_:Univ.Univ.
+∀truth:Univ.
+∀x:Univ.
+∀y:Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (implies (implies (not X) (not Y)) (implies Y X)) truth.
+∀H1:∀X:Univ.∀Y:Univ.eq Univ (implies (implies X Y) Y) (implies (implies Y X) X).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (implies (implies X Y) (implies (implies Y Z) (implies X Z))) truth.
+∀H3:∀X:Univ.eq Univ (implies truth X) X.eq Univ (implies (implies (not x) y) (implies (not y) x)) truth
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#implies.
+#not.
+#truth.
+#x.
+#y.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LCL114-2.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LCL114-2 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Logic Calculi (Many valued sentential) *)
+
+(* Problem : MV-36 depends on the Meredith system *)
+
+(* Version : [LW92] axioms. *)
+
+(* Theorem formulation : Wajsberg algebra formulation *)
+
+(* English : An axiomatisation of the many valued sentential calculus *)
+
+(* is {MV-1,MV-2,MV-3,MV-5} by Meredith. Wajsberg presented *)
+
+(* an equality axiomatisation. Show that MV-36 depends on the *)
+
+(* Wajsberg axiomatisation. *)
+
+(* Refs : [FRT84] Font et al. (1984), Wajsberg Algebras *)
+
+(* : [MW92] McCune & Wos (1992), Experiments in Automated Deductio *)
+
+(* : [LW92] Lusk & Wos (1992), Benchmark Problems in Which Equalit *)
+
+(* Source : [LW92] *)
+
+(* Names : MV3 [LW92] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.2.1, 0.22 v2.2.0, 0.29 v2.1.0, 0.25 v2.0.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 1 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 8 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Wajsberg algebra axioms *)
+
+(* Inclusion of: Axioms/LCL001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LCL001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Logic Calculi (Wajsberg Algebras) *)
+
+(* Axioms : Wajsberg algebra axioms *)
+
+(* Version : [Bon91] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [FRT84] Font et al. (1984), Wajsberg Algebras *)
+
+(* : [Bon91] Bonacina (1991), Problems in Lukasiewicz Logic *)
+
+(* : [MW92] McCune & Wos (1992), Experiments in Automated Deductio *)
+
+(* Source : [MW92] *)
+
+(* Names : MV Sentential Calculus [MW92] *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 4 ( 0 non-Horn; 4 unit; 0 RR) *)
+
+(* Number of literals : 4 ( 4 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 8 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_mv_36:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀implies:∀_:Univ.∀_:Univ.Univ.
+∀not:∀_:Univ.Univ.
+∀truth:Univ.
+∀x:Univ.
+∀y:Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (implies (implies (not X) (not Y)) (implies Y X)) truth.
+∀H1:∀X:Univ.∀Y:Univ.eq Univ (implies (implies X Y) Y) (implies (implies Y X) X).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (implies (implies X Y) (implies (implies Y Z) (implies X Z))) truth.
+∀H3:∀X:Univ.eq Univ (implies truth X) X.eq Univ (implies (implies x y) (implies (not y) (not x))) truth
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#implies.
+#not.
+#truth.
+#x.
+#y.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LCL115-2.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LCL115-2 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Logic Calculi (Many valued sentential) *)
+
+(* Problem : MV-39 depends on the Meredith system *)
+
+(* Version : [TPTP] axioms. *)
+
+(* Theorem formulation : Wajsberg algebra formulation *)
+
+(* English : An axiomatisation of the many valued sentential calculus *)
+
+(* is {MV-1,MV-2,MV-3,MV-5} by Meredith. Wajsberg presented *)
+
+(* an equality axiomatisation. Show that MV-39 depends on the *)
+
+(* Wajsberg axiomatisation. *)
+
+(* Refs : [FRT84] Font et al. (1984), Wajsberg Algebras *)
+
+(* : [MW92] McCune & Wos (1992), Experiments in Automated Deductio *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.2.1, 0.22 v2.2.0, 0.29 v2.1.0, 0.25 v2.0.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 1 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 8 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Wajsberg algebra axioms *)
+
+(* Inclusion of: Axioms/LCL001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LCL001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Logic Calculi (Wajsberg Algebras) *)
+
+(* Axioms : Wajsberg algebra axioms *)
+
+(* Version : [Bon91] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [FRT84] Font et al. (1984), Wajsberg Algebras *)
+
+(* : [Bon91] Bonacina (1991), Problems in Lukasiewicz Logic *)
+
+(* : [MW92] McCune & Wos (1992), Experiments in Automated Deductio *)
+
+(* Source : [MW92] *)
+
+(* Names : MV Sentential Calculus [MW92] *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 4 ( 0 non-Horn; 4 unit; 0 RR) *)
+
+(* Number of literals : 4 ( 4 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 8 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_mv_39:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀implies:∀_:Univ.∀_:Univ.Univ.
+∀not:∀_:Univ.Univ.
+∀truth:Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (implies (implies (not X) (not Y)) (implies Y X)) truth.
+∀H1:∀X:Univ.∀Y:Univ.eq Univ (implies (implies X Y) Y) (implies (implies Y X) X).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (implies (implies X Y) (implies (implies Y Z) (implies X Z))) truth.
+∀H3:∀X:Univ.eq Univ (implies truth X) X.eq Univ (implies (not (implies a b)) (not b)) truth
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#implies.
+#not.
+#truth.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LCL116-2.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LCL116-2 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Logic Calculi (Many valued sentential) *)
+
+(* Problem : MV-50 depends on the Meredith system *)
+
+(* Version : [TPTP] axioms. *)
+
+(* Theorem formulation : Wajsberg algebra formulation *)
+
+(* English : An axiomatisation of the many valued sentential calculus *)
+
+(* is {MV-1,MV-2,MV-3,MV-5} by Meredith. Wajsberg presented *)
+
+(* an equality axiomatisation. Show that MV-50 depends on the *)
+
+(* Wajsberg axiomatisation. *)
+
+(* Refs : [FRT84] Font et al. (1984), Wajsberg Algebras *)
+
+(* : [MW92] McCune & Wos (1992), Experiments in Automated Deductio *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.2.1, 0.22 v2.2.0, 0.29 v2.1.0, 0.25 v2.0.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 1 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 8 ( 0 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Wajsberg algebra axioms *)
+
+(* Inclusion of: Axioms/LCL001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LCL001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Logic Calculi (Wajsberg Algebras) *)
+
+(* Axioms : Wajsberg algebra axioms *)
+
+(* Version : [Bon91] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [FRT84] Font et al. (1984), Wajsberg Algebras *)
+
+(* : [Bon91] Bonacina (1991), Problems in Lukasiewicz Logic *)
+
+(* : [MW92] McCune & Wos (1992), Experiments in Automated Deductio *)
+
+(* Source : [MW92] *)
+
+(* Names : MV Sentential Calculus [MW92] *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 4 ( 0 non-Horn; 4 unit; 0 RR) *)
+
+(* Number of literals : 4 ( 4 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 8 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_mv_50:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀implies:∀_:Univ.∀_:Univ.Univ.
+∀not:∀_:Univ.Univ.
+∀truth:Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (implies (implies (not X) (not Y)) (implies Y X)) truth.
+∀H1:∀X:Univ.∀Y:Univ.eq Univ (implies (implies X Y) Y) (implies (implies Y X) X).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (implies (implies X Y) (implies (implies Y Z) (implies X Z))) truth.
+∀H3:∀X:Univ.eq Univ (implies truth X) X.eq Univ (implies (not a) (implies b (not (implies b a)))) truth
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#implies.
+#not.
+#truth.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LCL132-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LCL132-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Logic Calculi (Wajsberg Algebra) *)
+
+(* Problem : A lemma in Wajsberg algebras *)
+
+(* Version : [Bon91] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [FRT84] Font et al. (1984), Wajsberg Algebras *)
+
+(* : [Bon91] Bonacina (1991), Problems in Lukasiewicz Logic *)
+
+(* Source : [Bon91] *)
+
+(* Names : Lemma 1 [Bon91] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 1 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 8 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Wajsberg algebra axioms *)
+
+(* Inclusion of: Axioms/LCL001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LCL001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Logic Calculi (Wajsberg Algebras) *)
+
+(* Axioms : Wajsberg algebra axioms *)
+
+(* Version : [Bon91] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [FRT84] Font et al. (1984), Wajsberg Algebras *)
+
+(* : [Bon91] Bonacina (1991), Problems in Lukasiewicz Logic *)
+
+(* : [MW92] McCune & Wos (1992), Experiments in Automated Deductio *)
+
+(* Source : [MW92] *)
+
+(* Names : MV Sentential Calculus [MW92] *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 4 ( 0 non-Horn; 4 unit; 0 RR) *)
+
+(* Number of literals : 4 ( 4 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 8 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_wajsberg_lemma:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀implies:∀_:Univ.∀_:Univ.Univ.
+∀not:∀_:Univ.Univ.
+∀truth:Univ.
+∀x:Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (implies (implies (not X) (not Y)) (implies Y X)) truth.
+∀H1:∀X:Univ.∀Y:Univ.eq Univ (implies (implies X Y) Y) (implies (implies Y X) X).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (implies (implies X Y) (implies (implies Y Z) (implies X Z))) truth.
+∀H3:∀X:Univ.eq Univ (implies truth X) X.eq Univ (implies x x) truth
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#implies.
+#not.
+#truth.
+#x.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LCL133-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LCL133-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Logic Calculi (Wajsberg Algebra) *)
+
+(* Problem : A lemma in Wajsberg algebras *)
+
+(* Version : [Bon91] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [FRT84] Font et al. (1984), Wajsberg Algebras *)
+
+(* : [Bon91] Bonacina (1991), Problems in Lukasiewicz Logic *)
+
+(* Source : [Bon91] *)
+
+(* Names : Lemma 2 [Bon91] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.2.1, 0.22 v2.2.0, 0.29 v2.1.0, 0.25 v2.0.0 *)
+
+(* Syntax : Number of clauses : 6 ( 0 non-Horn; 6 unit; 1 RR) *)
+
+(* Number of atoms : 6 ( 6 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 10 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Wajsberg algebra axioms *)
+
+(* Inclusion of: Axioms/LCL001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LCL001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Logic Calculi (Wajsberg Algebras) *)
+
+(* Axioms : Wajsberg algebra axioms *)
+
+(* Version : [Bon91] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [FRT84] Font et al. (1984), Wajsberg Algebras *)
+
+(* : [Bon91] Bonacina (1991), Problems in Lukasiewicz Logic *)
+
+(* : [MW92] McCune & Wos (1992), Experiments in Automated Deductio *)
+
+(* Source : [MW92] *)
+
+(* Names : MV Sentential Calculus [MW92] *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 4 ( 0 non-Horn; 4 unit; 0 RR) *)
+
+(* Number of literals : 4 ( 4 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 8 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_wajsberg_lemma:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀implies:∀_:Univ.∀_:Univ.Univ.
+∀not:∀_:Univ.Univ.
+∀truth:Univ.
+∀x:Univ.
+∀y:Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (implies X Y) (implies Y X).
+∀H1:∀X:Univ.∀Y:Univ.eq Univ (implies (implies (not X) (not Y)) (implies Y X)) truth.
+∀H2:∀X:Univ.∀Y:Univ.eq Univ (implies (implies X Y) Y) (implies (implies Y X) X).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (implies (implies X Y) (implies (implies Y Z) (implies X Z))) truth.
+∀H4:∀X:Univ.eq Univ (implies truth X) X.eq Univ x y
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#implies.
+#not.
+#truth.
+#x.
+#y.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+nauto by H0,H1,H2,H3,H4;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LCL134-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LCL134-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Logic Calculi (Wajsberg Algebra) *)
+
+(* Problem : A lemma in Wajsberg algebras *)
+
+(* Version : [Bon91] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [FRT84] Font et al. (1984), Wajsberg Algebras *)
+
+(* : [Bon91] Bonacina (1991), Problems in Lukasiewicz Logic *)
+
+(* Source : [Bon91] *)
+
+(* Names : Lemma 3 [Bon91] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.2.1, 0.22 v2.2.0, 0.29 v2.1.0, 0.25 v2.0.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 1 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 8 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Wajsberg algebra axioms *)
+
+(* Inclusion of: Axioms/LCL001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LCL001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Logic Calculi (Wajsberg Algebras) *)
+
+(* Axioms : Wajsberg algebra axioms *)
+
+(* Version : [Bon91] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [FRT84] Font et al. (1984), Wajsberg Algebras *)
+
+(* : [Bon91] Bonacina (1991), Problems in Lukasiewicz Logic *)
+
+(* : [MW92] McCune & Wos (1992), Experiments in Automated Deductio *)
+
+(* Source : [MW92] *)
+
+(* Names : MV Sentential Calculus [MW92] *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 4 ( 0 non-Horn; 4 unit; 0 RR) *)
+
+(* Number of literals : 4 ( 4 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 8 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_wajsberg_lemma:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀implies:∀_:Univ.∀_:Univ.Univ.
+∀not:∀_:Univ.Univ.
+∀truth:Univ.
+∀x:Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (implies (implies (not X) (not Y)) (implies Y X)) truth.
+∀H1:∀X:Univ.∀Y:Univ.eq Univ (implies (implies X Y) Y) (implies (implies Y X) X).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (implies (implies X Y) (implies (implies Y Z) (implies X Z))) truth.
+∀H3:∀X:Univ.eq Univ (implies truth X) X.eq Univ (implies x truth) truth
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#implies.
+#not.
+#truth.
+#x.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LCL135-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LCL135-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Logic Calculi (Wajsberg Algebra) *)
+
+(* Problem : A lemma in Wajsberg algebras *)
+
+(* Version : [Bon91] (equality) axioms. *)
+
+(* English : An axiomatisation of the many valued sentential calculus *)
+
+(* is {MV-1,MV-2,MV-3,MV-5} by Meredith. Wajsberg provided *)
+
+(* a different axiomatisation. Show that MV-1 depends on the *)
+
+(* Wajsberg system. *)
+
+(* Refs : [FRT84] Font et al. (1984), Wajsberg Algebras *)
+
+(* : [Bon91] Bonacina (1991), Problems in Lukasiewicz Logic *)
+
+(* : [MW92] McCune & Wos (1992), Experiments in Automated Deductio *)
+
+(* Source : [Bon91] *)
+
+(* Names : Lemma 4 [Bon91] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.2.1, 0.22 v2.2.0, 0.29 v2.1.0, 0.25 v2.0.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 1 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 8 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Wajsberg algebra axioms *)
+
+(* Inclusion of: Axioms/LCL001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LCL001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Logic Calculi (Wajsberg Algebras) *)
+
+(* Axioms : Wajsberg algebra axioms *)
+
+(* Version : [Bon91] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [FRT84] Font et al. (1984), Wajsberg Algebras *)
+
+(* : [Bon91] Bonacina (1991), Problems in Lukasiewicz Logic *)
+
+(* : [MW92] McCune & Wos (1992), Experiments in Automated Deductio *)
+
+(* Source : [MW92] *)
+
+(* Names : MV Sentential Calculus [MW92] *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 4 ( 0 non-Horn; 4 unit; 0 RR) *)
+
+(* Number of literals : 4 ( 4 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 8 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_wajsberg_lemma:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀implies:∀_:Univ.∀_:Univ.Univ.
+∀not:∀_:Univ.Univ.
+∀truth:Univ.
+∀x:Univ.
+∀y:Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (implies (implies (not X) (not Y)) (implies Y X)) truth.
+∀H1:∀X:Univ.∀Y:Univ.eq Univ (implies (implies X Y) Y) (implies (implies Y X) X).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (implies (implies X Y) (implies (implies Y Z) (implies X Z))) truth.
+∀H3:∀X:Univ.eq Univ (implies truth X) X.eq Univ (implies x (implies y x)) truth
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#implies.
+#not.
+#truth.
+#x.
+#y.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LCL136-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LCL136-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Logic Calculi (Wajsberg Algebra) *)
+
+(* Problem : A lemma in Wajsberg algebras *)
+
+(* Version : [Bon91] (equality) axioms. *)
+
+(* English : An axiomatisation of the many valued sentential calculus *)
+
+(* is {MV-1,MV-2,MV-3,MV-5} by Meredith. Wajsberg provided *)
+
+(* a different axiomatisation. Show that a version of MV-2 *)
+
+(* depends on the Wajsberg system. *)
+
+(* Refs : [FRT84] Font et al. (1984), Wajsberg Algebras *)
+
+(* : [Bon91] Bonacina (1991), Problems in Lukasiewicz Logic *)
+
+(* : [MW92] McCune & Wos (1992), Experiments in Automated Deductio *)
+
+(* Source : [Bon91] *)
+
+(* Names : Lemma 5 [Bon91] *)
+
+(* Status : Satisfiable *)
+
+(* Rating : 0.33 v3.2.0, 0.67 v3.1.0, 0.33 v2.4.0, 0.67 v2.2.1, 0.75 v2.2.0, 0.67 v2.1.0, 1.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 6 ( 0 non-Horn; 6 unit; 2 RR) *)
+
+(* Number of atoms : 6 ( 6 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 8 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Wajsberg algebra axioms *)
+
+(* Inclusion of: Axioms/LCL001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LCL001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Logic Calculi (Wajsberg Algebras) *)
+
+(* Axioms : Wajsberg algebra axioms *)
+
+(* Version : [Bon91] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [FRT84] Font et al. (1984), Wajsberg Algebras *)
+
+(* : [Bon91] Bonacina (1991), Problems in Lukasiewicz Logic *)
+
+(* : [MW92] McCune & Wos (1992), Experiments in Automated Deductio *)
+
+(* Source : [MW92] *)
+
+(* Names : MV Sentential Calculus [MW92] *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 4 ( 0 non-Horn; 4 unit; 0 RR) *)
+
+(* Number of literals : 4 ( 4 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 8 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_wajsberg_lemma:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀implies:∀_:Univ.∀_:Univ.Univ.
+∀not:∀_:Univ.Univ.
+∀truth:Univ.
+∀x:Univ.
+∀y:Univ.
+∀z:Univ.
+∀H0:eq Univ (implies x y) (implies y z).
+∀H1:∀X:Univ.∀Y:Univ.eq Univ (implies (implies (not X) (not Y)) (implies Y X)) truth.
+∀H2:∀X:Univ.∀Y:Univ.eq Univ (implies (implies X Y) Y) (implies (implies Y X) X).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (implies (implies X Y) (implies (implies Y Z) (implies X Z))) truth.
+∀H4:∀X:Univ.eq Univ (implies truth X) X.eq Univ (implies x z) truth
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#implies.
+#not.
+#truth.
+#x.
+#y.
+#z.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+nauto by H0,H1,H2,H3,H4;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LCL137-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LCL137-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Logic Calculi (Wajsberg Algebra) *)
+
+(* Problem : A lemma in Wajsberg algebras *)
+
+(* Version : [TPTP] axioms. *)
+
+(* English : An axiomatisation of the many valued sentential calculus *)
+
+(* is {MV-1,MV-2,MV-3,MV-5} by Meredith. Wajsberg provided *)
+
+(* a different axiomatisation. Show that MV-3 depends on the *)
+
+(* Wajsberg system. *)
+
+(* Refs : [FRT84] Font et al. (1984), Wajsberg Algebras *)
+
+(* : [MW92] McCune & Wos (1992), Experiments in Automated Deductio *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Satisfiable *)
+
+(* Rating : 0.33 v3.2.0, 0.67 v3.1.0, 0.33 v2.4.0, 0.67 v2.2.1, 0.75 v2.2.0, 0.67 v2.1.0, 1.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 1 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 8 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Wajsberg algebra axioms *)
+
+(* Inclusion of: Axioms/LCL001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LCL001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Logic Calculi (Wajsberg Algebras) *)
+
+(* Axioms : Wajsberg algebra axioms *)
+
+(* Version : [Bon91] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [FRT84] Font et al. (1984), Wajsberg Algebras *)
+
+(* : [Bon91] Bonacina (1991), Problems in Lukasiewicz Logic *)
+
+(* : [MW92] McCune & Wos (1992), Experiments in Automated Deductio *)
+
+(* Source : [MW92] *)
+
+(* Names : MV Sentential Calculus [MW92] *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 4 ( 0 non-Horn; 4 unit; 0 RR) *)
+
+(* Number of literals : 4 ( 4 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 8 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_wajsberg_lemma:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀implies:∀_:Univ.∀_:Univ.Univ.
+∀not:∀_:Univ.Univ.
+∀truth:Univ.
+∀x:Univ.
+∀y:Univ.
+∀z:Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (implies (implies (not X) (not Y)) (implies Y X)) truth.
+∀H1:∀X:Univ.∀Y:Univ.eq Univ (implies (implies X Y) Y) (implies (implies Y X) X).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (implies (implies X Y) (implies (implies Y Z) (implies X Z))) truth.
+∀H3:∀X:Univ.eq Univ (implies truth X) X.eq Univ (implies (implies (implies x y) y) (implies (implies y z) (implies x z))) truth
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#implies.
+#not.
+#truth.
+#x.
+#y.
+#z.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LCL138-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LCL138-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Logic Calculi (Wajsberg Algebra) *)
+
+(* Problem : A lemma in Wajsberg algebras *)
+
+(* Version : [Bon91] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [FRT84] Font et al. (1984), Wajsberg Algebras *)
+
+(* : [Bon91] Bonacina (1991), Problems in Lukasiewicz Logic *)
+
+(* Source : [Bon91] *)
+
+(* Names : Lemma 7 [Bon91] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.29 v3.2.0, 0.21 v3.1.0, 0.33 v2.7.0, 0.27 v2.6.0, 0.17 v2.5.0, 0.00 v2.4.0, 0.33 v2.3.0, 0.67 v2.2.1, 0.33 v2.2.0, 0.43 v2.1.0, 0.62 v2.0.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 1 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 8 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Wajsberg algebra axioms *)
+
+(* Inclusion of: Axioms/LCL001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LCL001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Logic Calculi (Wajsberg Algebras) *)
+
+(* Axioms : Wajsberg algebra axioms *)
+
+(* Version : [Bon91] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [FRT84] Font et al. (1984), Wajsberg Algebras *)
+
+(* : [Bon91] Bonacina (1991), Problems in Lukasiewicz Logic *)
+
+(* : [MW92] McCune & Wos (1992), Experiments in Automated Deductio *)
+
+(* Source : [MW92] *)
+
+(* Names : MV Sentential Calculus [MW92] *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 4 ( 0 non-Horn; 4 unit; 0 RR) *)
+
+(* Number of literals : 4 ( 4 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 8 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_wajsberg_lemma:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀implies:∀_:Univ.∀_:Univ.Univ.
+∀not:∀_:Univ.Univ.
+∀truth:Univ.
+∀x:Univ.
+∀y:Univ.
+∀z:Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (implies (implies (not X) (not Y)) (implies Y X)) truth.
+∀H1:∀X:Univ.∀Y:Univ.eq Univ (implies (implies X Y) Y) (implies (implies Y X) X).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (implies (implies X Y) (implies (implies Y Z) (implies X Z))) truth.
+∀H3:∀X:Univ.eq Univ (implies truth X) X.eq Univ (implies x (implies y z)) (implies y (implies x z))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#implies.
+#not.
+#truth.
+#x.
+#y.
+#z.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LCL139-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LCL139-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Logic Calculi (Wajsberg Algebra) *)
+
+(* Problem : A lemma in Wajsberg algebras *)
+
+(* Version : [Bon91] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [FRT84] Font et al. (1984), Wajsberg Algebras *)
+
+(* : [Bon91] Bonacina (1991), Problems in Lukasiewicz Logic *)
+
+(* Source : [Bon91] *)
+
+(* Names : Lemma 8 [Bon91] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.2.1, 0.22 v2.2.0, 0.29 v2.1.0, 0.25 v2.0.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 1 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 8 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Wajsberg algebra axioms *)
+
+(* Inclusion of: Axioms/LCL001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LCL001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Logic Calculi (Wajsberg Algebras) *)
+
+(* Axioms : Wajsberg algebra axioms *)
+
+(* Version : [Bon91] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [FRT84] Font et al. (1984), Wajsberg Algebras *)
+
+(* : [Bon91] Bonacina (1991), Problems in Lukasiewicz Logic *)
+
+(* : [MW92] McCune & Wos (1992), Experiments in Automated Deductio *)
+
+(* Source : [MW92] *)
+
+(* Names : MV Sentential Calculus [MW92] *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 4 ( 0 non-Horn; 4 unit; 0 RR) *)
+
+(* Number of literals : 4 ( 4 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 8 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_wajsberg_lemma:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀implies:∀_:Univ.∀_:Univ.Univ.
+∀not:∀_:Univ.Univ.
+∀truth:Univ.
+∀x:Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (implies (implies (not X) (not Y)) (implies Y X)) truth.
+∀H1:∀X:Univ.∀Y:Univ.eq Univ (implies (implies X Y) Y) (implies (implies Y X) X).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (implies (implies X Y) (implies (implies Y Z) (implies X Z))) truth.
+∀H3:∀X:Univ.eq Univ (implies truth X) X.eq Univ (implies x (not truth)) (not x)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#implies.
+#not.
+#truth.
+#x.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LCL140-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LCL140-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Logic Calculi (Wajsberg Algebra) *)
+
+(* Problem : A lemma in Wajsberg algebras *)
+
+(* Version : [Bon91] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [FRT84] Font et al. (1984), Wajsberg Algebras *)
+
+(* : [Bon91] Bonacina (1991), Problems in Lukasiewicz Logic *)
+
+(* Source : [Bon91] *)
+
+(* Names : Lemma 9 [Bon91] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.2.1, 0.22 v2.2.0, 0.29 v2.1.0, 0.25 v2.0.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 1 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 8 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Wajsberg algebra axioms *)
+
+(* Inclusion of: Axioms/LCL001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LCL001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Logic Calculi (Wajsberg Algebras) *)
+
+(* Axioms : Wajsberg algebra axioms *)
+
+(* Version : [Bon91] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [FRT84] Font et al. (1984), Wajsberg Algebras *)
+
+(* : [Bon91] Bonacina (1991), Problems in Lukasiewicz Logic *)
+
+(* : [MW92] McCune & Wos (1992), Experiments in Automated Deductio *)
+
+(* Source : [MW92] *)
+
+(* Names : MV Sentential Calculus [MW92] *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 4 ( 0 non-Horn; 4 unit; 0 RR) *)
+
+(* Number of literals : 4 ( 4 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 8 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_wajsberg_lemma:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀implies:∀_:Univ.∀_:Univ.Univ.
+∀not:∀_:Univ.Univ.
+∀truth:Univ.
+∀x:Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (implies (implies (not X) (not Y)) (implies Y X)) truth.
+∀H1:∀X:Univ.∀Y:Univ.eq Univ (implies (implies X Y) Y) (implies (implies Y X) X).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (implies (implies X Y) (implies (implies Y Z) (implies X Z))) truth.
+∀H3:∀X:Univ.eq Univ (implies truth X) X.eq Univ (not (not x)) x
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#implies.
+#not.
+#truth.
+#x.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LCL141-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LCL141-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Logic Calculi (Wajsberg Algebra) *)
+
+(* Problem : A lemma in Wajsberg algebras *)
+
+(* Version : [Bon91] (equality) axioms. *)
+
+(* English : An axiomatisation of the many valued sentential calculus *)
+
+(* is {MV-1,MV-2,MV-3,MV-5} by Meredith. Wajsberg provided *)
+
+(* a different axiomatisation. Show that MV-5 depends on the *)
+
+(* Wajsberg system. *)
+
+(* Refs : [FRT84] Font et al. (1984), Wajsberg Algebras *)
+
+(* : [Bon91] Bonacina (1991), Problems in Lukasiewicz Logic *)
+
+(* : [MW92] McCune & Wos (1992), Experiments in Automated Deductio *)
+
+(* Source : [Bon91] *)
+
+(* Names : Lemma 10 [Bon91] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.2.1, 0.22 v2.2.0, 0.29 v2.1.0, 0.25 v2.0.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 1 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 8 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Wajsberg algebra axioms *)
+
+(* Inclusion of: Axioms/LCL001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LCL001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Logic Calculi (Wajsberg Algebras) *)
+
+(* Axioms : Wajsberg algebra axioms *)
+
+(* Version : [Bon91] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [FRT84] Font et al. (1984), Wajsberg Algebras *)
+
+(* : [Bon91] Bonacina (1991), Problems in Lukasiewicz Logic *)
+
+(* : [MW92] McCune & Wos (1992), Experiments in Automated Deductio *)
+
+(* Source : [MW92] *)
+
+(* Names : MV Sentential Calculus [MW92] *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 4 ( 0 non-Horn; 4 unit; 0 RR) *)
+
+(* Number of literals : 4 ( 4 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 8 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_wajsberg_lemma:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀implies:∀_:Univ.∀_:Univ.Univ.
+∀not:∀_:Univ.Univ.
+∀truth:Univ.
+∀x:Univ.
+∀y:Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (implies (implies (not X) (not Y)) (implies Y X)) truth.
+∀H1:∀X:Univ.∀Y:Univ.eq Univ (implies (implies X Y) Y) (implies (implies Y X) X).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (implies (implies X Y) (implies (implies Y Z) (implies X Z))) truth.
+∀H3:∀X:Univ.eq Univ (implies truth X) X.eq Univ (implies (not x) (not y)) (implies y x)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#implies.
+#not.
+#truth.
+#x.
+#y.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LCL153-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LCL153-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Logic Calculi (Wajsberg Algebra) *)
+
+(* Problem : The 1st alternative Wajsberg algebra axiom *)
+
+(* Version : [Bon91] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [FRT84] Font et al. (1984), Wajsberg Algebras *)
+
+(* : [AB90] Anantharaman & Bonacina (1990), An Application of the *)
+
+(* : [Bon91] Bonacina (1991), Problems in Lukasiewicz Logic *)
+
+(* Source : [Bon91] *)
+
+(* Names : W' axiom 1 [Bon91] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.2.1, 0.22 v2.2.0, 0.29 v2.1.0, 0.38 v2.0.0 *)
+
+(* Syntax : Number of clauses : 17 ( 0 non-Horn; 17 unit; 2 RR) *)
+
+(* Number of atoms : 17 ( 17 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 9 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 33 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Wajsberg algebra axioms *)
+
+(* Inclusion of: Axioms/LCL001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LCL001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Logic Calculi (Wajsberg Algebras) *)
+
+(* Axioms : Wajsberg algebra axioms *)
+
+(* Version : [Bon91] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [FRT84] Font et al. (1984), Wajsberg Algebras *)
+
+(* : [Bon91] Bonacina (1991), Problems in Lukasiewicz Logic *)
+
+(* : [MW92] McCune & Wos (1992), Experiments in Automated Deductio *)
+
+(* Source : [MW92] *)
+
+(* Names : MV Sentential Calculus [MW92] *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 4 ( 0 non-Horn; 4 unit; 0 RR) *)
+
+(* Number of literals : 4 ( 4 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 8 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Wajsberg algebra and and or definitions *)
+
+(* Inclusion of: Axioms/LCL001-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LCL001-2 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Logic Calculi (Wajsberg Algebras) *)
+
+(* Axioms : Wajsberg algebra AND and OR definitions *)
+
+(* Version : [AB90] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [FRT84] Font et al. (1984), Wajsberg Algebras *)
+
+(* : [AB90] Anantharaman & Bonacina (1990), An Application of the *)
+
+(* : [Bon91] Bonacina (1991), Problems in Lukasiewicz Logic *)
+
+(* Source : [Bon91] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 6 ( 0 non-Horn; 6 unit; 0 RR) *)
+
+(* Number of literals : 6 ( 6 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 0 constant; 1-2 arity) *)
+
+(* Number of variables : 14 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 3 average) *)
+
+(* Comments : Requires LCL001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Definitions of or and and, which are AC *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Alternative Wajsberg algebra definitions *)
+
+(* Inclusion of: Axioms/LCL002-1.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LCL002-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Logic Calculi (Wajsberg Algebras) *)
+
+(* Axioms : Alternative Wajsberg algebra definitions *)
+
+(* Version : [AB90] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [FRT84] Font et al. (1984), Wajsberg Algebras *)
+
+(* : [AB90] Anantharaman & Bonacina (1990), An Application of the *)
+
+(* : [Bon91] Bonacina (1991), Problems in Lukasiewicz Logic *)
+
+(* Source : [Bon91] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 6 ( 0 non-Horn; 6 unit; 1 RR) *)
+
+(* Number of literals : 6 ( 6 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 11 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : Requires LCL001-0.ax LCL001-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Definitions of and_star and xor, where and_star is AC and xor is C *)
+
+(* ---I guess the next two can be derived from the AC of and *)
+
+(* ----Definition of false in terms of truth *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_alternative_wajsberg_axiom:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀myand:∀_:Univ.∀_:Univ.Univ.
+∀and_star:∀_:Univ.∀_:Univ.Univ.
+∀falsehood:Univ.
+∀implies:∀_:Univ.∀_:Univ.Univ.
+∀not:∀_:Univ.Univ.
+∀or:∀_:Univ.∀_:Univ.Univ.
+∀truth:Univ.
+∀x:Univ.
+∀xor:∀_:Univ.∀_:Univ.Univ.
+∀H0:eq Univ (not truth) falsehood.
+∀H1:∀X:Univ.∀Y:Univ.eq Univ (and_star X Y) (and_star Y X).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (and_star (and_star X Y) Z) (and_star X (and_star Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (and_star X Y) (not (or (not X) (not Y))).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (xor X Y) (xor Y X).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (xor X Y) (or (myand X (not Y)) (myand (not X) Y)).
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (myand X Y) (myand Y X).
+∀H7:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (myand (myand X Y) Z) (myand X (myand Y Z)).
+∀H8:∀X:Univ.∀Y:Univ.eq Univ (myand X Y) (not (or (not X) (not Y))).
+∀H9:∀X:Univ.∀Y:Univ.eq Univ (or X Y) (or Y X).
+∀H10:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (or (or X Y) Z) (or X (or Y Z)).
+∀H11:∀X:Univ.∀Y:Univ.eq Univ (or X Y) (implies (not X) Y).
+∀H12:∀X:Univ.∀Y:Univ.eq Univ (implies (implies (not X) (not Y)) (implies Y X)) truth.
+∀H13:∀X:Univ.∀Y:Univ.eq Univ (implies (implies X Y) Y) (implies (implies Y X) X).
+∀H14:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (implies (implies X Y) (implies (implies Y Z) (implies X Z))) truth.
+∀H15:∀X:Univ.eq Univ (implies truth X) X.eq Univ (not x) (xor x truth)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#and.
+#and_star.
+#falsehood.
+#implies.
+#not.
+#or.
+#truth.
+#x.
+#xor.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LCL154-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LCL154-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Logic Calculi (Wajsberg Algebra) *)
+
+(* Problem : The 2nd alternative Wajsberg algebra axiom *)
+
+(* Version : [Bon91] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [FRT84] Font et al. (1984), Wajsberg Algebras *)
+
+(* : [AB90] Anantharaman & Bonacina (1990), An Application of the *)
+
+(* : [Bon91] Bonacina (1991), Problems in Lukasiewicz Logic *)
+
+(* Source : [Bon91] *)
+
+(* Names : W' axiom 2 [Bon91] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.2.1, 0.22 v2.2.0, 0.29 v2.1.0, 0.38 v2.0.0 *)
+
+(* Syntax : Number of clauses : 17 ( 0 non-Horn; 17 unit; 2 RR) *)
+
+(* Number of atoms : 17 ( 17 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 9 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 33 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Wajsberg algebra axioms *)
+
+(* Inclusion of: Axioms/LCL001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LCL001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Logic Calculi (Wajsberg Algebras) *)
+
+(* Axioms : Wajsberg algebra axioms *)
+
+(* Version : [Bon91] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [FRT84] Font et al. (1984), Wajsberg Algebras *)
+
+(* : [Bon91] Bonacina (1991), Problems in Lukasiewicz Logic *)
+
+(* : [MW92] McCune & Wos (1992), Experiments in Automated Deductio *)
+
+(* Source : [MW92] *)
+
+(* Names : MV Sentential Calculus [MW92] *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 4 ( 0 non-Horn; 4 unit; 0 RR) *)
+
+(* Number of literals : 4 ( 4 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 8 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Wajsberg algebra and and or definitions *)
+
+(* Inclusion of: Axioms/LCL001-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LCL001-2 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Logic Calculi (Wajsberg Algebras) *)
+
+(* Axioms : Wajsberg algebra AND and OR definitions *)
+
+(* Version : [AB90] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [FRT84] Font et al. (1984), Wajsberg Algebras *)
+
+(* : [AB90] Anantharaman & Bonacina (1990), An Application of the *)
+
+(* : [Bon91] Bonacina (1991), Problems in Lukasiewicz Logic *)
+
+(* Source : [Bon91] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 6 ( 0 non-Horn; 6 unit; 0 RR) *)
+
+(* Number of literals : 6 ( 6 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 0 constant; 1-2 arity) *)
+
+(* Number of variables : 14 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 3 average) *)
+
+(* Comments : Requires LCL001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Definitions of or and and, which are AC *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Alternative Wajsberg algebra definitions *)
+
+(* Inclusion of: Axioms/LCL002-1.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LCL002-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Logic Calculi (Wajsberg Algebras) *)
+
+(* Axioms : Alternative Wajsberg algebra definitions *)
+
+(* Version : [AB90] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [FRT84] Font et al. (1984), Wajsberg Algebras *)
+
+(* : [AB90] Anantharaman & Bonacina (1990), An Application of the *)
+
+(* : [Bon91] Bonacina (1991), Problems in Lukasiewicz Logic *)
+
+(* Source : [Bon91] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 6 ( 0 non-Horn; 6 unit; 1 RR) *)
+
+(* Number of literals : 6 ( 6 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 11 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : Requires LCL001-0.ax LCL001-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Definitions of and_star and xor, where and_star is AC and xor is C *)
+
+(* ---I guess the next two can be derived from the AC of and *)
+
+(* ----Definition of false in terms of truth *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_alternative_wajsberg_axiom:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀myand:∀_:Univ.∀_:Univ.Univ.
+∀and_star:∀_:Univ.∀_:Univ.Univ.
+∀falsehood:Univ.
+∀implies:∀_:Univ.∀_:Univ.Univ.
+∀not:∀_:Univ.Univ.
+∀or:∀_:Univ.∀_:Univ.Univ.
+∀truth:Univ.
+∀x:Univ.
+∀xor:∀_:Univ.∀_:Univ.Univ.
+∀H0:eq Univ (not truth) falsehood.
+∀H1:∀X:Univ.∀Y:Univ.eq Univ (and_star X Y) (and_star Y X).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (and_star (and_star X Y) Z) (and_star X (and_star Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (and_star X Y) (not (or (not X) (not Y))).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (xor X Y) (xor Y X).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (xor X Y) (or (myand X (not Y)) (myand (not X) Y)).
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (myand X Y) (myand Y X).
+∀H7:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (myand (myand X Y) Z) (myand X (myand Y Z)).
+∀H8:∀X:Univ.∀Y:Univ.eq Univ (myand X Y) (not (or (not X) (not Y))).
+∀H9:∀X:Univ.∀Y:Univ.eq Univ (or X Y) (or Y X).
+∀H10:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (or (or X Y) Z) (or X (or Y Z)).
+∀H11:∀X:Univ.∀Y:Univ.eq Univ (or X Y) (implies (not X) Y).
+∀H12:∀X:Univ.∀Y:Univ.eq Univ (implies (implies (not X) (not Y)) (implies Y X)) truth.
+∀H13:∀X:Univ.∀Y:Univ.eq Univ (implies (implies X Y) Y) (implies (implies Y X) X).
+∀H14:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (implies (implies X Y) (implies (implies Y Z) (implies X Z))) truth.
+∀H15:∀X:Univ.eq Univ (implies truth X) X.eq Univ (xor x falsehood) x
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#and.
+#and_star.
+#falsehood.
+#implies.
+#not.
+#or.
+#truth.
+#x.
+#xor.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LCL155-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LCL155-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Logic Calculi (Wajsberg Algebra) *)
+
+(* Problem : The 3rd alternative Wajsberg algebra axiom *)
+
+(* Version : [Bon91] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [FRT84] Font et al. (1984), Wajsberg Algebras *)
+
+(* : [AB90] Anantharaman & Bonacina (1990), An Application of the *)
+
+(* : [Bon91] Bonacina (1991), Problems in Lukasiewicz Logic *)
+
+(* Source : [Bon91] *)
+
+(* Names : W' axiom 3 [Bon91] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.2.1, 0.22 v2.2.0, 0.29 v2.1.0, 0.38 v2.0.0 *)
+
+(* Syntax : Number of clauses : 17 ( 0 non-Horn; 17 unit; 2 RR) *)
+
+(* Number of atoms : 17 ( 17 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 9 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 33 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Wajsberg algebra axioms *)
+
+(* Inclusion of: Axioms/LCL001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LCL001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Logic Calculi (Wajsberg Algebras) *)
+
+(* Axioms : Wajsberg algebra axioms *)
+
+(* Version : [Bon91] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [FRT84] Font et al. (1984), Wajsberg Algebras *)
+
+(* : [Bon91] Bonacina (1991), Problems in Lukasiewicz Logic *)
+
+(* : [MW92] McCune & Wos (1992), Experiments in Automated Deductio *)
+
+(* Source : [MW92] *)
+
+(* Names : MV Sentential Calculus [MW92] *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 4 ( 0 non-Horn; 4 unit; 0 RR) *)
+
+(* Number of literals : 4 ( 4 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 8 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Wajsberg algebra and and or definitions *)
+
+(* Inclusion of: Axioms/LCL001-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LCL001-2 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Logic Calculi (Wajsberg Algebras) *)
+
+(* Axioms : Wajsberg algebra AND and OR definitions *)
+
+(* Version : [AB90] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [FRT84] Font et al. (1984), Wajsberg Algebras *)
+
+(* : [AB90] Anantharaman & Bonacina (1990), An Application of the *)
+
+(* : [Bon91] Bonacina (1991), Problems in Lukasiewicz Logic *)
+
+(* Source : [Bon91] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 6 ( 0 non-Horn; 6 unit; 0 RR) *)
+
+(* Number of literals : 6 ( 6 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 0 constant; 1-2 arity) *)
+
+(* Number of variables : 14 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 3 average) *)
+
+(* Comments : Requires LCL001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Definitions of or and and, which are AC *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Alternative Wajsberg algebra definitions *)
+
+(* Inclusion of: Axioms/LCL002-1.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LCL002-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Logic Calculi (Wajsberg Algebras) *)
+
+(* Axioms : Alternative Wajsberg algebra definitions *)
+
+(* Version : [AB90] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [FRT84] Font et al. (1984), Wajsberg Algebras *)
+
+(* : [AB90] Anantharaman & Bonacina (1990), An Application of the *)
+
+(* : [Bon91] Bonacina (1991), Problems in Lukasiewicz Logic *)
+
+(* Source : [Bon91] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 6 ( 0 non-Horn; 6 unit; 1 RR) *)
+
+(* Number of literals : 6 ( 6 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 11 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : Requires LCL001-0.ax LCL001-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Definitions of and_star and xor, where and_star is AC and xor is C *)
+
+(* ---I guess the next two can be derived from the AC of and *)
+
+(* ----Definition of false in terms of truth *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_alternative_wajsberg_axiom:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀myand:∀_:Univ.∀_:Univ.Univ.
+∀and_star:∀_:Univ.∀_:Univ.Univ.
+∀falsehood:Univ.
+∀implies:∀_:Univ.∀_:Univ.Univ.
+∀not:∀_:Univ.Univ.
+∀or:∀_:Univ.∀_:Univ.Univ.
+∀truth:Univ.
+∀x:Univ.
+∀xor:∀_:Univ.∀_:Univ.Univ.
+∀H0:eq Univ (not truth) falsehood.
+∀H1:∀X:Univ.∀Y:Univ.eq Univ (and_star X Y) (and_star Y X).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (and_star (and_star X Y) Z) (and_star X (and_star Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (and_star X Y) (not (or (not X) (not Y))).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (xor X Y) (xor Y X).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (xor X Y) (or (myand X (not Y)) (myand (not X) Y)).
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (myand X Y) (myand Y X).
+∀H7:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (myand (myand X Y) Z) (myand X (myand Y Z)).
+∀H8:∀X:Univ.∀Y:Univ.eq Univ (myand X Y) (not (or (not X) (not Y))).
+∀H9:∀X:Univ.∀Y:Univ.eq Univ (or X Y) (or Y X).
+∀H10:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (or (or X Y) Z) (or X (or Y Z)).
+∀H11:∀X:Univ.∀Y:Univ.eq Univ (or X Y) (implies (not X) Y).
+∀H12:∀X:Univ.∀Y:Univ.eq Univ (implies (implies (not X) (not Y)) (implies Y X)) truth.
+∀H13:∀X:Univ.∀Y:Univ.eq Univ (implies (implies X Y) Y) (implies (implies Y X) X).
+∀H14:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (implies (implies X Y) (implies (implies Y Z) (implies X Z))) truth.
+∀H15:∀X:Univ.eq Univ (implies truth X) X.eq Univ (xor x x) falsehood
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#and.
+#and_star.
+#falsehood.
+#implies.
+#not.
+#or.
+#truth.
+#x.
+#xor.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LCL156-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LCL156-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Logic Calculi (Wajsberg Algebra) *)
+
+(* Problem : The 4th alternative Wajsberg algebra axiom *)
+
+(* Version : [Bon91] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [FRT84] Font et al. (1984), Wajsberg Algebras *)
+
+(* : [AB90] Anantharaman & Bonacina (1990), An Application of the *)
+
+(* : [Bon91] Bonacina (1991), Problems in Lukasiewicz Logic *)
+
+(* Source : [Bon91] *)
+
+(* Names : W' axiom 4 [Bon91] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.2.1, 0.22 v2.2.0, 0.29 v2.1.0, 0.38 v2.0.0 *)
+
+(* Syntax : Number of clauses : 17 ( 0 non-Horn; 17 unit; 2 RR) *)
+
+(* Number of atoms : 17 ( 17 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 9 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 33 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Wajsberg algebra axioms *)
+
+(* Inclusion of: Axioms/LCL001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LCL001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Logic Calculi (Wajsberg Algebras) *)
+
+(* Axioms : Wajsberg algebra axioms *)
+
+(* Version : [Bon91] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [FRT84] Font et al. (1984), Wajsberg Algebras *)
+
+(* : [Bon91] Bonacina (1991), Problems in Lukasiewicz Logic *)
+
+(* : [MW92] McCune & Wos (1992), Experiments in Automated Deductio *)
+
+(* Source : [MW92] *)
+
+(* Names : MV Sentential Calculus [MW92] *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 4 ( 0 non-Horn; 4 unit; 0 RR) *)
+
+(* Number of literals : 4 ( 4 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 8 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Wajsberg algebra and and or definitions *)
+
+(* Inclusion of: Axioms/LCL001-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LCL001-2 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Logic Calculi (Wajsberg Algebras) *)
+
+(* Axioms : Wajsberg algebra AND and OR definitions *)
+
+(* Version : [AB90] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [FRT84] Font et al. (1984), Wajsberg Algebras *)
+
+(* : [AB90] Anantharaman & Bonacina (1990), An Application of the *)
+
+(* : [Bon91] Bonacina (1991), Problems in Lukasiewicz Logic *)
+
+(* Source : [Bon91] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 6 ( 0 non-Horn; 6 unit; 0 RR) *)
+
+(* Number of literals : 6 ( 6 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 0 constant; 1-2 arity) *)
+
+(* Number of variables : 14 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 3 average) *)
+
+(* Comments : Requires LCL001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Definitions of or and and, which are AC *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Alternative Wajsberg algebra definitions *)
+
+(* Inclusion of: Axioms/LCL002-1.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LCL002-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Logic Calculi (Wajsberg Algebras) *)
+
+(* Axioms : Alternative Wajsberg algebra definitions *)
+
+(* Version : [AB90] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [FRT84] Font et al. (1984), Wajsberg Algebras *)
+
+(* : [AB90] Anantharaman & Bonacina (1990), An Application of the *)
+
+(* : [Bon91] Bonacina (1991), Problems in Lukasiewicz Logic *)
+
+(* Source : [Bon91] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 6 ( 0 non-Horn; 6 unit; 1 RR) *)
+
+(* Number of literals : 6 ( 6 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 11 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : Requires LCL001-0.ax LCL001-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Definitions of and_star and xor, where and_star is AC and xor is C *)
+
+(* ---I guess the next two can be derived from the AC of and *)
+
+(* ----Definition of false in terms of truth *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_alternative_wajsberg_axiom:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀myand:∀_:Univ.∀_:Univ.Univ.
+∀and_star:∀_:Univ.∀_:Univ.Univ.
+∀falsehood:Univ.
+∀implies:∀_:Univ.∀_:Univ.Univ.
+∀not:∀_:Univ.Univ.
+∀or:∀_:Univ.∀_:Univ.Univ.
+∀truth:Univ.
+∀x:Univ.
+∀xor:∀_:Univ.∀_:Univ.Univ.
+∀H0:eq Univ (not truth) falsehood.
+∀H1:∀X:Univ.∀Y:Univ.eq Univ (and_star X Y) (and_star Y X).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (and_star (and_star X Y) Z) (and_star X (and_star Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (and_star X Y) (not (or (not X) (not Y))).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (xor X Y) (xor Y X).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (xor X Y) (or (myand X (not Y)) (myand (not X) Y)).
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (myand X Y) (myand Y X).
+∀H7:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (myand (myand X Y) Z) (myand X (myand Y Z)).
+∀H8:∀X:Univ.∀Y:Univ.eq Univ (myand X Y) (not (or (not X) (not Y))).
+∀H9:∀X:Univ.∀Y:Univ.eq Univ (or X Y) (or Y X).
+∀H10:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (or (or X Y) Z) (or X (or Y Z)).
+∀H11:∀X:Univ.∀Y:Univ.eq Univ (or X Y) (implies (not X) Y).
+∀H12:∀X:Univ.∀Y:Univ.eq Univ (implies (implies (not X) (not Y)) (implies Y X)) truth.
+∀H13:∀X:Univ.∀Y:Univ.eq Univ (implies (implies X Y) Y) (implies (implies Y X) X).
+∀H14:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (implies (implies X Y) (implies (implies Y Z) (implies X Z))) truth.
+∀H15:∀X:Univ.eq Univ (implies truth X) X.eq Univ (and_star x truth) x
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#and.
+#and_star.
+#falsehood.
+#implies.
+#not.
+#or.
+#truth.
+#x.
+#xor.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LCL157-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LCL157-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Logic Calculi (Wajsberg Algebra) *)
+
+(* Problem : The 5th alternative Wajsberg algebra axiom *)
+
+(* Version : [Bon91] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [FRT84] Font et al. (1984), Wajsberg Algebras *)
+
+(* : [AB90] Anantharaman & Bonacina (1990), An Application of the *)
+
+(* : [Bon91] Bonacina (1991), Problems in Lukasiewicz Logic *)
+
+(* Source : [Bon91] *)
+
+(* Names : W' axiom 5 [Bon91] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.2.1, 0.22 v2.2.0, 0.29 v2.1.0, 0.38 v2.0.0 *)
+
+(* Syntax : Number of clauses : 17 ( 0 non-Horn; 17 unit; 2 RR) *)
+
+(* Number of atoms : 17 ( 17 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 9 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 33 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Wajsberg algebra axioms *)
+
+(* Inclusion of: Axioms/LCL001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LCL001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Logic Calculi (Wajsberg Algebras) *)
+
+(* Axioms : Wajsberg algebra axioms *)
+
+(* Version : [Bon91] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [FRT84] Font et al. (1984), Wajsberg Algebras *)
+
+(* : [Bon91] Bonacina (1991), Problems in Lukasiewicz Logic *)
+
+(* : [MW92] McCune & Wos (1992), Experiments in Automated Deductio *)
+
+(* Source : [MW92] *)
+
+(* Names : MV Sentential Calculus [MW92] *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 4 ( 0 non-Horn; 4 unit; 0 RR) *)
+
+(* Number of literals : 4 ( 4 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 8 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Wajsberg algebra and and or definitions *)
+
+(* Inclusion of: Axioms/LCL001-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LCL001-2 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Logic Calculi (Wajsberg Algebras) *)
+
+(* Axioms : Wajsberg algebra AND and OR definitions *)
+
+(* Version : [AB90] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [FRT84] Font et al. (1984), Wajsberg Algebras *)
+
+(* : [AB90] Anantharaman & Bonacina (1990), An Application of the *)
+
+(* : [Bon91] Bonacina (1991), Problems in Lukasiewicz Logic *)
+
+(* Source : [Bon91] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 6 ( 0 non-Horn; 6 unit; 0 RR) *)
+
+(* Number of literals : 6 ( 6 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 0 constant; 1-2 arity) *)
+
+(* Number of variables : 14 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 3 average) *)
+
+(* Comments : Requires LCL001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Definitions of or and and, which are AC *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Alternative Wajsberg algebra definitions *)
+
+(* Inclusion of: Axioms/LCL002-1.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LCL002-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Logic Calculi (Wajsberg Algebras) *)
+
+(* Axioms : Alternative Wajsberg algebra definitions *)
+
+(* Version : [AB90] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [FRT84] Font et al. (1984), Wajsberg Algebras *)
+
+(* : [AB90] Anantharaman & Bonacina (1990), An Application of the *)
+
+(* : [Bon91] Bonacina (1991), Problems in Lukasiewicz Logic *)
+
+(* Source : [Bon91] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 6 ( 0 non-Horn; 6 unit; 1 RR) *)
+
+(* Number of literals : 6 ( 6 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 11 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : Requires LCL001-0.ax LCL001-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Definitions of and_star and xor, where and_star is AC and xor is C *)
+
+(* ---I guess the next two can be derived from the AC of and *)
+
+(* ----Definition of false in terms of truth *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_alternative_wajsberg_axiom:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀myand:∀_:Univ.∀_:Univ.Univ.
+∀and_star:∀_:Univ.∀_:Univ.Univ.
+∀falsehood:Univ.
+∀implies:∀_:Univ.∀_:Univ.Univ.
+∀not:∀_:Univ.Univ.
+∀or:∀_:Univ.∀_:Univ.Univ.
+∀truth:Univ.
+∀x:Univ.
+∀xor:∀_:Univ.∀_:Univ.Univ.
+∀H0:eq Univ (not truth) falsehood.
+∀H1:∀X:Univ.∀Y:Univ.eq Univ (and_star X Y) (and_star Y X).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (and_star (and_star X Y) Z) (and_star X (and_star Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (and_star X Y) (not (or (not X) (not Y))).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (xor X Y) (xor Y X).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (xor X Y) (or (myand X (not Y)) (myand (not X) Y)).
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (myand X Y) (myand Y X).
+∀H7:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (myand (myand X Y) Z) (myand X (myand Y Z)).
+∀H8:∀X:Univ.∀Y:Univ.eq Univ (myand X Y) (not (or (not X) (not Y))).
+∀H9:∀X:Univ.∀Y:Univ.eq Univ (or X Y) (or Y X).
+∀H10:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (or (or X Y) Z) (or X (or Y Z)).
+∀H11:∀X:Univ.∀Y:Univ.eq Univ (or X Y) (implies (not X) Y).
+∀H12:∀X:Univ.∀Y:Univ.eq Univ (implies (implies (not X) (not Y)) (implies Y X)) truth.
+∀H13:∀X:Univ.∀Y:Univ.eq Univ (implies (implies X Y) Y) (implies (implies Y X) X).
+∀H14:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (implies (implies X Y) (implies (implies Y Z) (implies X Z))) truth.
+∀H15:∀X:Univ.eq Univ (implies truth X) X.eq Univ (and_star x falsehood) falsehood
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#and.
+#and_star.
+#falsehood.
+#implies.
+#not.
+#or.
+#truth.
+#x.
+#xor.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LCL158-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LCL158-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Logic Calculi (Wajsberg Algebra) *)
+
+(* Problem : The 6th alternative Wajsberg algebra axiom *)
+
+(* Version : [Bon91] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [FRT84] Font et al. (1984), Wajsberg Algebras *)
+
+(* : [AB90] Anantharaman & Bonacina (1990), An Application of the *)
+
+(* : [Bon91] Bonacina (1991), Problems in Lukasiewicz Logic *)
+
+(* Source : [Bon91] *)
+
+(* Names : W' axiom 6 [Bon91] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.2.1, 0.22 v2.2.0, 0.29 v2.1.0, 0.38 v2.0.0 *)
+
+(* Syntax : Number of clauses : 17 ( 0 non-Horn; 17 unit; 2 RR) *)
+
+(* Number of atoms : 17 ( 17 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 9 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 33 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Wajsberg algebra axioms *)
+
+(* Inclusion of: Axioms/LCL001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LCL001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Logic Calculi (Wajsberg Algebras) *)
+
+(* Axioms : Wajsberg algebra axioms *)
+
+(* Version : [Bon91] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [FRT84] Font et al. (1984), Wajsberg Algebras *)
+
+(* : [Bon91] Bonacina (1991), Problems in Lukasiewicz Logic *)
+
+(* : [MW92] McCune & Wos (1992), Experiments in Automated Deductio *)
+
+(* Source : [MW92] *)
+
+(* Names : MV Sentential Calculus [MW92] *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 4 ( 0 non-Horn; 4 unit; 0 RR) *)
+
+(* Number of literals : 4 ( 4 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 8 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Wajsberg algebra and and or definitions *)
+
+(* Inclusion of: Axioms/LCL001-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LCL001-2 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Logic Calculi (Wajsberg Algebras) *)
+
+(* Axioms : Wajsberg algebra AND and OR definitions *)
+
+(* Version : [AB90] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [FRT84] Font et al. (1984), Wajsberg Algebras *)
+
+(* : [AB90] Anantharaman & Bonacina (1990), An Application of the *)
+
+(* : [Bon91] Bonacina (1991), Problems in Lukasiewicz Logic *)
+
+(* Source : [Bon91] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 6 ( 0 non-Horn; 6 unit; 0 RR) *)
+
+(* Number of literals : 6 ( 6 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 0 constant; 1-2 arity) *)
+
+(* Number of variables : 14 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 3 average) *)
+
+(* Comments : Requires LCL001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Definitions of or and and, which are AC *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Alternative Wajsberg algebra definitions *)
+
+(* Inclusion of: Axioms/LCL002-1.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LCL002-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Logic Calculi (Wajsberg Algebras) *)
+
+(* Axioms : Alternative Wajsberg algebra definitions *)
+
+(* Version : [AB90] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [FRT84] Font et al. (1984), Wajsberg Algebras *)
+
+(* : [AB90] Anantharaman & Bonacina (1990), An Application of the *)
+
+(* : [Bon91] Bonacina (1991), Problems in Lukasiewicz Logic *)
+
+(* Source : [Bon91] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 6 ( 0 non-Horn; 6 unit; 1 RR) *)
+
+(* Number of literals : 6 ( 6 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 11 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : Requires LCL001-0.ax LCL001-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Definitions of and_star and xor, where and_star is AC and xor is C *)
+
+(* ---I guess the next two can be derived from the AC of and *)
+
+(* ----Definition of false in terms of truth *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_alternative_wajsberg_axiom:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀myand:∀_:Univ.∀_:Univ.Univ.
+∀and_star:∀_:Univ.∀_:Univ.Univ.
+∀falsehood:Univ.
+∀implies:∀_:Univ.∀_:Univ.Univ.
+∀not:∀_:Univ.Univ.
+∀or:∀_:Univ.∀_:Univ.Univ.
+∀truth:Univ.
+∀x:Univ.
+∀xor:∀_:Univ.∀_:Univ.Univ.
+∀H0:eq Univ (not truth) falsehood.
+∀H1:∀X:Univ.∀Y:Univ.eq Univ (and_star X Y) (and_star Y X).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (and_star (and_star X Y) Z) (and_star X (and_star Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (and_star X Y) (not (or (not X) (not Y))).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (xor X Y) (xor Y X).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (xor X Y) (or (myand X (not Y)) (myand (not X) Y)).
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (myand X Y) (myand Y X).
+∀H7:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (myand (myand X Y) Z) (myand X (myand Y Z)).
+∀H8:∀X:Univ.∀Y:Univ.eq Univ (myand X Y) (not (or (not X) (not Y))).
+∀H9:∀X:Univ.∀Y:Univ.eq Univ (or X Y) (or Y X).
+∀H10:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (or (or X Y) Z) (or X (or Y Z)).
+∀H11:∀X:Univ.∀Y:Univ.eq Univ (or X Y) (implies (not X) Y).
+∀H12:∀X:Univ.∀Y:Univ.eq Univ (implies (implies (not X) (not Y)) (implies Y X)) truth.
+∀H13:∀X:Univ.∀Y:Univ.eq Univ (implies (implies X Y) Y) (implies (implies Y X) X).
+∀H14:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (implies (implies X Y) (implies (implies Y Z) (implies X Z))) truth.
+∀H15:∀X:Univ.eq Univ (implies truth X) X.eq Univ (and_star (xor truth x) x) falsehood
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#and.
+#and_star.
+#falsehood.
+#implies.
+#not.
+#or.
+#truth.
+#x.
+#xor.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LCL159-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LCL159-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Logic Calculi (Wajsberg Algebra) *)
+
+(* Problem : The 7th alternative Wajsberg algebra axiom *)
+
+(* Version : [Bon91] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [FRT84] Font et al. (1984), Wajsberg Algebras *)
+
+(* : [AB90] Anantharaman & Bonacina (1990), An Application of the *)
+
+(* : [Bon91] Bonacina (1991), Problems in Lukasiewicz Logic *)
+
+(* Source : [Bon91] *)
+
+(* Names : W' axiom 7 [Bon91] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.2.1, 0.22 v2.2.0, 0.29 v2.1.0, 0.38 v2.0.0 *)
+
+(* Syntax : Number of clauses : 17 ( 0 non-Horn; 17 unit; 2 RR) *)
+
+(* Number of atoms : 17 ( 17 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 10 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 33 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Wajsberg algebra axioms *)
+
+(* Inclusion of: Axioms/LCL001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LCL001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Logic Calculi (Wajsberg Algebras) *)
+
+(* Axioms : Wajsberg algebra axioms *)
+
+(* Version : [Bon91] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [FRT84] Font et al. (1984), Wajsberg Algebras *)
+
+(* : [Bon91] Bonacina (1991), Problems in Lukasiewicz Logic *)
+
+(* : [MW92] McCune & Wos (1992), Experiments in Automated Deductio *)
+
+(* Source : [MW92] *)
+
+(* Names : MV Sentential Calculus [MW92] *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 4 ( 0 non-Horn; 4 unit; 0 RR) *)
+
+(* Number of literals : 4 ( 4 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 8 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Wajsberg algebra and and or definitions *)
+
+(* Inclusion of: Axioms/LCL001-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LCL001-2 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Logic Calculi (Wajsberg Algebras) *)
+
+(* Axioms : Wajsberg algebra AND and OR definitions *)
+
+(* Version : [AB90] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [FRT84] Font et al. (1984), Wajsberg Algebras *)
+
+(* : [AB90] Anantharaman & Bonacina (1990), An Application of the *)
+
+(* : [Bon91] Bonacina (1991), Problems in Lukasiewicz Logic *)
+
+(* Source : [Bon91] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 6 ( 0 non-Horn; 6 unit; 0 RR) *)
+
+(* Number of literals : 6 ( 6 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 0 constant; 1-2 arity) *)
+
+(* Number of variables : 14 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 3 average) *)
+
+(* Comments : Requires LCL001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Definitions of or and and, which are AC *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Alternative Wajsberg algebra definitions *)
+
+(* Inclusion of: Axioms/LCL002-1.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LCL002-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Logic Calculi (Wajsberg Algebras) *)
+
+(* Axioms : Alternative Wajsberg algebra definitions *)
+
+(* Version : [AB90] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [FRT84] Font et al. (1984), Wajsberg Algebras *)
+
+(* : [AB90] Anantharaman & Bonacina (1990), An Application of the *)
+
+(* : [Bon91] Bonacina (1991), Problems in Lukasiewicz Logic *)
+
+(* Source : [Bon91] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 6 ( 0 non-Horn; 6 unit; 1 RR) *)
+
+(* Number of literals : 6 ( 6 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 11 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : Requires LCL001-0.ax LCL001-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Definitions of and_star and xor, where and_star is AC and xor is C *)
+
+(* ---I guess the next two can be derived from the AC of and *)
+
+(* ----Definition of false in terms of truth *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_alternative_wajsberg_axiom:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀myand:∀_:Univ.∀_:Univ.Univ.
+∀and_star:∀_:Univ.∀_:Univ.Univ.
+∀falsehood:Univ.
+∀implies:∀_:Univ.∀_:Univ.Univ.
+∀not:∀_:Univ.Univ.
+∀or:∀_:Univ.∀_:Univ.Univ.
+∀truth:Univ.
+∀x:Univ.
+∀xor:∀_:Univ.∀_:Univ.Univ.
+∀y:Univ.
+∀H0:eq Univ (not truth) falsehood.
+∀H1:∀X:Univ.∀Y:Univ.eq Univ (and_star X Y) (and_star Y X).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (and_star (and_star X Y) Z) (and_star X (and_star Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (and_star X Y) (not (or (not X) (not Y))).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (xor X Y) (xor Y X).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (xor X Y) (or (myand X (not Y)) (myand (not X) Y)).
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (myand X Y) (myand Y X).
+∀H7:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (myand (myand X Y) Z) (myand X (myand Y Z)).
+∀H8:∀X:Univ.∀Y:Univ.eq Univ (myand X Y) (not (or (not X) (not Y))).
+∀H9:∀X:Univ.∀Y:Univ.eq Univ (or X Y) (or Y X).
+∀H10:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (or (or X Y) Z) (or X (or Y Z)).
+∀H11:∀X:Univ.∀Y:Univ.eq Univ (or X Y) (implies (not X) Y).
+∀H12:∀X:Univ.∀Y:Univ.eq Univ (implies (implies (not X) (not Y)) (implies Y X)) truth.
+∀H13:∀X:Univ.∀Y:Univ.eq Univ (implies (implies X Y) Y) (implies (implies Y X) X).
+∀H14:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (implies (implies X Y) (implies (implies Y Z) (implies X Z))) truth.
+∀H15:∀X:Univ.eq Univ (implies truth X) X.eq Univ (xor x (xor truth y)) (xor (xor x truth) y)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#and.
+#and_star.
+#falsehood.
+#implies.
+#not.
+#or.
+#truth.
+#x.
+#xor.
+#y.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LCL160-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LCL160-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Logic Calculi (Wajsberg Algebra) *)
+
+(* Problem : The 8th alternative Wajsberg algebra axiom *)
+
+(* Version : [Bon91] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [FRT84] Font et al. (1984), Wajsberg Algebras *)
+
+(* : [AB90] Anantharaman & Bonacina (1990), An Application of the *)
+
+(* : [Bon91] Bonacina (1991), Problems in Lukasiewicz Logic *)
+
+(* Source : [Bon91] *)
+
+(* Names : W' axiom 8 [Bon91] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.14 v3.1.0, 0.00 v2.7.0, 0.27 v2.6.0, 0.33 v2.5.0, 0.00 v2.2.1, 0.33 v2.2.0, 0.57 v2.1.0, 0.88 v2.0.0 *)
+
+(* Syntax : Number of clauses : 17 ( 0 non-Horn; 17 unit; 2 RR) *)
+
+(* Number of atoms : 17 ( 17 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 10 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 33 ( 0 singleton) *)
+
+(* Maximal term depth : 5 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Wajsberg algebra axioms *)
+
+(* Inclusion of: Axioms/LCL001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LCL001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Logic Calculi (Wajsberg Algebras) *)
+
+(* Axioms : Wajsberg algebra axioms *)
+
+(* Version : [Bon91] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [FRT84] Font et al. (1984), Wajsberg Algebras *)
+
+(* : [Bon91] Bonacina (1991), Problems in Lukasiewicz Logic *)
+
+(* : [MW92] McCune & Wos (1992), Experiments in Automated Deductio *)
+
+(* Source : [MW92] *)
+
+(* Names : MV Sentential Calculus [MW92] *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 4 ( 0 non-Horn; 4 unit; 0 RR) *)
+
+(* Number of literals : 4 ( 4 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 8 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Wajsberg algebra and and or definitions *)
+
+(* Inclusion of: Axioms/LCL001-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LCL001-2 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Logic Calculi (Wajsberg Algebras) *)
+
+(* Axioms : Wajsberg algebra AND and OR definitions *)
+
+(* Version : [AB90] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [FRT84] Font et al. (1984), Wajsberg Algebras *)
+
+(* : [AB90] Anantharaman & Bonacina (1990), An Application of the *)
+
+(* : [Bon91] Bonacina (1991), Problems in Lukasiewicz Logic *)
+
+(* Source : [Bon91] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 6 ( 0 non-Horn; 6 unit; 0 RR) *)
+
+(* Number of literals : 6 ( 6 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 0 constant; 1-2 arity) *)
+
+(* Number of variables : 14 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 3 average) *)
+
+(* Comments : Requires LCL001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Definitions of or and and, which are AC *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Alternative Wajsberg algebra definitions *)
+
+(* Inclusion of: Axioms/LCL002-1.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LCL002-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Logic Calculi (Wajsberg Algebras) *)
+
+(* Axioms : Alternative Wajsberg algebra definitions *)
+
+(* Version : [AB90] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [FRT84] Font et al. (1984), Wajsberg Algebras *)
+
+(* : [AB90] Anantharaman & Bonacina (1990), An Application of the *)
+
+(* : [Bon91] Bonacina (1991), Problems in Lukasiewicz Logic *)
+
+(* Source : [Bon91] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 6 ( 0 non-Horn; 6 unit; 1 RR) *)
+
+(* Number of literals : 6 ( 6 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 11 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : Requires LCL001-0.ax LCL001-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Definitions of and_star and xor, where and_star is AC and xor is C *)
+
+(* ---I guess the next two can be derived from the AC of and *)
+
+(* ----Definition of false in terms of truth *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_alternative_wajsberg_axiom:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀myand:∀_:Univ.∀_:Univ.Univ.
+∀and_star:∀_:Univ.∀_:Univ.Univ.
+∀falsehood:Univ.
+∀implies:∀_:Univ.∀_:Univ.Univ.
+∀not:∀_:Univ.Univ.
+∀or:∀_:Univ.∀_:Univ.Univ.
+∀truth:Univ.
+∀x:Univ.
+∀xor:∀_:Univ.∀_:Univ.Univ.
+∀y:Univ.
+∀H0:eq Univ (not truth) falsehood.
+∀H1:∀X:Univ.∀Y:Univ.eq Univ (and_star X Y) (and_star Y X).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (and_star (and_star X Y) Z) (and_star X (and_star Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (and_star X Y) (not (or (not X) (not Y))).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (xor X Y) (xor Y X).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (xor X Y) (or (myand X (not Y)) (myand (not X) Y)).
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (myand X Y) (myand Y X).
+∀H7:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (myand (myand X Y) Z) (myand X (myand Y Z)).
+∀H8:∀X:Univ.∀Y:Univ.eq Univ (myand X Y) (not (or (not X) (not Y))).
+∀H9:∀X:Univ.∀Y:Univ.eq Univ (or X Y) (or Y X).
+∀H10:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (or (or X Y) Z) (or X (or Y Z)).
+∀H11:∀X:Univ.∀Y:Univ.eq Univ (or X Y) (implies (not X) Y).
+∀H12:∀X:Univ.∀Y:Univ.eq Univ (implies (implies (not X) (not Y)) (implies Y X)) truth.
+∀H13:∀X:Univ.∀Y:Univ.eq Univ (implies (implies X Y) Y) (implies (implies Y X) X).
+∀H14:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (implies (implies X Y) (implies (implies Y Z) (implies X Z))) truth.
+∀H15:∀X:Univ.eq Univ (implies truth X) X.eq Univ (and_star (xor (and_star (xor truth x) y) truth) y) (and_star (xor (and_star (xor truth y) x) truth) x)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#and.
+#and_star.
+#falsehood.
+#implies.
+#not.
+#or.
+#truth.
+#x.
+#xor.
+#y.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LCL161-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LCL161-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Logic Calculi (Wajsberg Algebra) *)
+
+(* Problem : The 1st Wajsberg algebra axiom, from the alternative axioms *)
+
+(* Version : [Bon91] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [FRT84] Font et al. (1984), Wajsberg Algebras *)
+
+(* : [AB90] Anantharaman & Bonacina (1990), An Application of the *)
+
+(* : [Bon91] Bonacina (1991), Problems in Lukasiewicz Logic *)
+
+(* Source : [Bon91] *)
+
+(* Names : W axiom 1 [Bon91] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.2.1, 0.11 v2.2.0, 0.14 v2.1.0, 0.13 v2.0.0 *)
+
+(* Syntax : Number of clauses : 14 ( 0 non-Horn; 14 unit; 2 RR) *)
+
+(* Number of atoms : 14 ( 14 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 19 ( 1 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Alternative Wajsberg algebra axioms *)
+
+(* Inclusion of: Axioms/LCL002-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LCL002-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Logic Calculi (Wajsberg Algebras) *)
+
+(* Axioms : Alternative Wajsberg algebra axioms *)
+
+(* Version : [AB90] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [FRT84] Font et al. (1984), Wajsberg Algebras *)
+
+(* : [AB90] Anantharaman & Bonacina (1990), An Application of the *)
+
+(* : [Bon91] Bonacina (1991), Problems in Lukasiewicz Logic *)
+
+(* Source : [Bon91] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 10 ( 1 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : To be used in conjunction with the LAT003 alternative *)
+
+(* Wajsberg algebra definitions. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include some Alternative Wajsberg algebra definitions *)
+
+(* include('Axioms/LCL002-1.ax'). *)
+
+(* ----Definition that and_star is AC and xor is C *)
+
+(* ----Definition of false in terms of true *)
+
+(* ----Include the definition of implies in terms of xor and and_star *)
+ntheorem prove_wajsberg_axiom:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀and_star:∀_:Univ.∀_:Univ.Univ.
+∀falsehood:Univ.
+∀implies:∀_:Univ.∀_:Univ.Univ.
+∀not:∀_:Univ.Univ.
+∀truth:Univ.
+∀x:Univ.
+∀xor:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (implies X Y) (xor truth (and_star X (xor truth Y))).
+∀H1:eq Univ (not truth) falsehood.
+∀H2:∀X:Univ.∀Y:Univ.eq Univ (and_star X Y) (and_star Y X).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (and_star (and_star X Y) Z) (and_star X (and_star Y Z)).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (xor X Y) (xor Y X).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (and_star (xor (and_star (xor truth X) Y) truth) Y) (and_star (xor (and_star (xor truth Y) X) truth) X).
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (xor X (xor truth Y)) (xor (xor X truth) Y).
+∀H7:∀X:Univ.eq Univ (and_star (xor truth X) X) falsehood.
+∀H8:∀X:Univ.eq Univ (and_star X falsehood) falsehood.
+∀H9:∀X:Univ.eq Univ (and_star X truth) X.
+∀H10:∀X:Univ.eq Univ (xor X X) falsehood.
+∀H11:∀X:Univ.eq Univ (xor X falsehood) X.
+∀H12:∀X:Univ.eq Univ (not X) (xor X truth).eq Univ (implies truth x) x
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#and_star.
+#falsehood.
+#implies.
+#not.
+#truth.
+#x.
+#xor.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LCL162-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LCL162-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Logic Calculi (Wajsberg Algebra) *)
+
+(* Problem : The 2nd Wajsberg algebra axiom, from the alternative axioms *)
+
+(* Version : [Bon91] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [FRT84] Font et al. (1984), Wajsberg Algebras *)
+
+(* : [AB90] Anantharaman & Bonacina (1990), An Application of the *)
+
+(* : [Bon91] Bonacina (1991), Problems in Lukasiewicz Logic *)
+
+(* Source : [Bon91] *)
+
+(* Names : W axiom 2 [Bon91] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.07 v3.2.0, 0.00 v3.1.0, 0.11 v2.7.0, 0.09 v2.6.0, 0.00 v2.4.0, 0.00 v2.2.1, 0.56 v2.2.0, 0.57 v2.1.0, 0.75 v2.0.0 *)
+
+(* Syntax : Number of clauses : 14 ( 0 non-Horn; 14 unit; 2 RR) *)
+
+(* Number of atoms : 14 ( 14 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 9 ( 5 constant; 0-2 arity) *)
+
+(* Number of variables : 19 ( 1 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Alternative Wajsberg algebra axioms *)
+
+(* Inclusion of: Axioms/LCL002-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LCL002-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Logic Calculi (Wajsberg Algebras) *)
+
+(* Axioms : Alternative Wajsberg algebra axioms *)
+
+(* Version : [AB90] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [FRT84] Font et al. (1984), Wajsberg Algebras *)
+
+(* : [AB90] Anantharaman & Bonacina (1990), An Application of the *)
+
+(* : [Bon91] Bonacina (1991), Problems in Lukasiewicz Logic *)
+
+(* Source : [Bon91] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 10 ( 1 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : To be used in conjunction with the LAT003 alternative *)
+
+(* Wajsberg algebra definitions. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include some Alternative Wajsberg algebra definitions *)
+
+(* include('Axioms/LCL002-1.ax'). *)
+
+(* ----Definition that and_star is AC and xor is C *)
+
+(* ----Definition of false in terms of true *)
+
+(* ----Include the definition of implies in terms of xor and and_star *)
+ntheorem prove_wajsberg_axiom:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀and_star:∀_:Univ.∀_:Univ.Univ.
+∀falsehood:Univ.
+∀implies:∀_:Univ.∀_:Univ.Univ.
+∀not:∀_:Univ.Univ.
+∀truth:Univ.
+∀x:Univ.
+∀xor:∀_:Univ.∀_:Univ.Univ.
+∀y:Univ.
+∀z:Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (implies X Y) (xor truth (and_star X (xor truth Y))).
+∀H1:eq Univ (not truth) falsehood.
+∀H2:∀X:Univ.∀Y:Univ.eq Univ (and_star X Y) (and_star Y X).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (and_star (and_star X Y) Z) (and_star X (and_star Y Z)).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (xor X Y) (xor Y X).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (and_star (xor (and_star (xor truth X) Y) truth) Y) (and_star (xor (and_star (xor truth Y) X) truth) X).
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (xor X (xor truth Y)) (xor (xor X truth) Y).
+∀H7:∀X:Univ.eq Univ (and_star (xor truth X) X) falsehood.
+∀H8:∀X:Univ.eq Univ (and_star X falsehood) falsehood.
+∀H9:∀X:Univ.eq Univ (and_star X truth) X.
+∀H10:∀X:Univ.eq Univ (xor X X) falsehood.
+∀H11:∀X:Univ.eq Univ (xor X falsehood) X.
+∀H12:∀X:Univ.eq Univ (not X) (xor X truth).eq Univ (implies (implies x y) (implies (implies y z) (implies x z))) truth
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#and_star.
+#falsehood.
+#implies.
+#not.
+#truth.
+#x.
+#xor.
+#y.
+#z.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LCL163-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LCL163-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Logic Calculi (Wajsberg Algebra) *)
+
+(* Problem : The 3rd Wajsberg algebra axiom, from the alternative axioms *)
+
+(* Version : [Bon91] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [FRT84] Font et al. (1984), Wajsberg Algebras *)
+
+(* : [AB90] Anantharaman & Bonacina (1990), An Application of the *)
+
+(* : [Bon91] Bonacina (1991), Problems in Lukasiewicz Logic *)
+
+(* Source : [Bon91] *)
+
+(* Names : W axiom 3 [Bon91] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.14 v3.1.0, 0.11 v2.7.0, 0.09 v2.6.0, 0.00 v2.4.0, 0.00 v2.2.1, 0.44 v2.2.0, 0.43 v2.1.0, 0.75 v2.0.0 *)
+
+(* Syntax : Number of clauses : 14 ( 0 non-Horn; 14 unit; 2 RR) *)
+
+(* Number of atoms : 14 ( 14 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 8 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 19 ( 1 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Alternative Wajsberg algebra axioms *)
+
+(* Inclusion of: Axioms/LCL002-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LCL002-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Logic Calculi (Wajsberg Algebras) *)
+
+(* Axioms : Alternative Wajsberg algebra axioms *)
+
+(* Version : [AB90] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [FRT84] Font et al. (1984), Wajsberg Algebras *)
+
+(* : [AB90] Anantharaman & Bonacina (1990), An Application of the *)
+
+(* : [Bon91] Bonacina (1991), Problems in Lukasiewicz Logic *)
+
+(* Source : [Bon91] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 10 ( 1 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : To be used in conjunction with the LAT003 alternative *)
+
+(* Wajsberg algebra definitions. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include some Alternative Wajsberg algebra definitions *)
+
+(* include('Axioms/LCL002-1.ax'). *)
+
+(* ----Definition that and_star is AC and xor is C *)
+
+(* ----Definition of false in terms of true *)
+
+(* ----Include the definition of implies in terms of xor and and_star *)
+ntheorem prove_wajsberg_axiom:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀and_star:∀_:Univ.∀_:Univ.Univ.
+∀falsehood:Univ.
+∀implies:∀_:Univ.∀_:Univ.Univ.
+∀not:∀_:Univ.Univ.
+∀truth:Univ.
+∀x:Univ.
+∀xor:∀_:Univ.∀_:Univ.Univ.
+∀y:Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (implies X Y) (xor truth (and_star X (xor truth Y))).
+∀H1:eq Univ (not truth) falsehood.
+∀H2:∀X:Univ.∀Y:Univ.eq Univ (and_star X Y) (and_star Y X).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (and_star (and_star X Y) Z) (and_star X (and_star Y Z)).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (xor X Y) (xor Y X).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (and_star (xor (and_star (xor truth X) Y) truth) Y) (and_star (xor (and_star (xor truth Y) X) truth) X).
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (xor X (xor truth Y)) (xor (xor X truth) Y).
+∀H7:∀X:Univ.eq Univ (and_star (xor truth X) X) falsehood.
+∀H8:∀X:Univ.eq Univ (and_star X falsehood) falsehood.
+∀H9:∀X:Univ.eq Univ (and_star X truth) X.
+∀H10:∀X:Univ.eq Univ (xor X X) falsehood.
+∀H11:∀X:Univ.eq Univ (xor X falsehood) X.
+∀H12:∀X:Univ.eq Univ (not X) (xor X truth).eq Univ (implies (implies x y) y) (implies (implies y x) x)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#and_star.
+#falsehood.
+#implies.
+#not.
+#truth.
+#x.
+#xor.
+#y.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LCL164-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LCL164-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Logic Calculi (Wajsberg Algebra) *)
+
+(* Problem : The 4th Wajsberg algebra axiom, from the alternative axioms *)
+
+(* Version : [Bon91] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [FRT84] Font et al. (1984), Wajsberg Algebras *)
+
+(* : [AB90] Anantharaman & Bonacina (1990), An Application of the *)
+
+(* : [Bon91] Bonacina (1991), Problems in Lukasiewicz Logic *)
+
+(* Source : [Bon91] *)
+
+(* Names : W axiom 4 [Bon91] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.2.1, 0.22 v2.2.0, 0.29 v2.1.0, 0.38 v2.0.0 *)
+
+(* Syntax : Number of clauses : 14 ( 0 non-Horn; 14 unit; 2 RR) *)
+
+(* Number of atoms : 14 ( 14 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 8 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 19 ( 1 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Alternative Wajsberg algebra axioms *)
+
+(* Inclusion of: Axioms/LCL002-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LCL002-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Logic Calculi (Wajsberg Algebras) *)
+
+(* Axioms : Alternative Wajsberg algebra axioms *)
+
+(* Version : [AB90] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [FRT84] Font et al. (1984), Wajsberg Algebras *)
+
+(* : [AB90] Anantharaman & Bonacina (1990), An Application of the *)
+
+(* : [Bon91] Bonacina (1991), Problems in Lukasiewicz Logic *)
+
+(* Source : [Bon91] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 10 ( 1 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : To be used in conjunction with the LAT003 alternative *)
+
+(* Wajsberg algebra definitions. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include some Alternative Wajsberg algebra definitions *)
+
+(* include('Axioms/LCL002-1.ax'). *)
+
+(* ----Definition that and_star is AC and xor is C *)
+
+(* ----Definition of false in terms of true *)
+
+(* ----Include the definition of implies in terms of xor and and_star *)
+ntheorem prove_wajsberg_axiom:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀and_star:∀_:Univ.∀_:Univ.Univ.
+∀falsehood:Univ.
+∀implies:∀_:Univ.∀_:Univ.Univ.
+∀not:∀_:Univ.Univ.
+∀truth:Univ.
+∀x:Univ.
+∀xor:∀_:Univ.∀_:Univ.Univ.
+∀y:Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (implies X Y) (xor truth (and_star X (xor truth Y))).
+∀H1:eq Univ (not truth) falsehood.
+∀H2:∀X:Univ.∀Y:Univ.eq Univ (and_star X Y) (and_star Y X).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (and_star (and_star X Y) Z) (and_star X (and_star Y Z)).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (xor X Y) (xor Y X).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (and_star (xor (and_star (xor truth X) Y) truth) Y) (and_star (xor (and_star (xor truth Y) X) truth) X).
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (xor X (xor truth Y)) (xor (xor X truth) Y).
+∀H7:∀X:Univ.eq Univ (and_star (xor truth X) X) falsehood.
+∀H8:∀X:Univ.eq Univ (and_star X falsehood) falsehood.
+∀H9:∀X:Univ.eq Univ (and_star X truth) X.
+∀H10:∀X:Univ.eq Univ (xor X X) falsehood.
+∀H11:∀X:Univ.eq Univ (xor X falsehood) X.
+∀H12:∀X:Univ.eq Univ (not X) (xor X truth).eq Univ (implies (implies (not x) (not y)) (implies y x)) truth
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#and_star.
+#falsehood.
+#implies.
+#not.
+#truth.
+#x.
+#xor.
+#y.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LCL165-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LCL165-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Logic Calculi (Wajsberg Algebra) *)
+
+(* Problem : A ntheorem in Wajsberg algebras *)
+
+(* Version : [Bon91] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [FRT84] Font et al. (1984), Wajsberg Algebras *)
+
+(* : [AB90] Anantharaman & Bonacina (1990), An Application of the *)
+
+(* : [Bon91] Bonacina (1991), Problems in Lukasiewicz Logic *)
+
+(* Source : [Bon91] *)
+
+(* Names : Third problem [Bon91] *)
+
+(* Status : Satisfiable *)
+
+(* Rating : 0.33 v3.2.0, 0.67 v3.1.0, 0.33 v2.5.0, 0.67 v2.4.0, 1.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 11 ( 0 non-Horn; 11 unit; 1 RR) *)
+
+(* Number of atoms : 11 ( 11 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 22 ( 0 singleton) *)
+
+(* Maximal term depth : 5 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Wajsberg algebra axioms *)
+
+(* Inclusion of: Axioms/LCL001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LCL001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Logic Calculi (Wajsberg Algebras) *)
+
+(* Axioms : Wajsberg algebra axioms *)
+
+(* Version : [Bon91] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [FRT84] Font et al. (1984), Wajsberg Algebras *)
+
+(* : [Bon91] Bonacina (1991), Problems in Lukasiewicz Logic *)
+
+(* : [MW92] McCune & Wos (1992), Experiments in Automated Deductio *)
+
+(* Source : [MW92] *)
+
+(* Names : MV Sentential Calculus [MW92] *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 4 ( 0 non-Horn; 4 unit; 0 RR) *)
+
+(* Number of literals : 4 ( 4 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 8 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Wajsberg algebra and and or definitions *)
+
+(* Inclusion of: Axioms/LCL001-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LCL001-2 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Logic Calculi (Wajsberg Algebras) *)
+
+(* Axioms : Wajsberg algebra AND and OR definitions *)
+
+(* Version : [AB90] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [FRT84] Font et al. (1984), Wajsberg Algebras *)
+
+(* : [AB90] Anantharaman & Bonacina (1990), An Application of the *)
+
+(* : [Bon91] Bonacina (1991), Problems in Lukasiewicz Logic *)
+
+(* Source : [Bon91] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 6 ( 0 non-Horn; 6 unit; 0 RR) *)
+
+(* Number of literals : 6 ( 6 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 0 constant; 1-2 arity) *)
+
+(* Number of variables : 14 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 3 average) *)
+
+(* Comments : Requires LCL001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Definitions of or and and, which are AC *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_wajsberg_ntheorem:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀myand:∀_:Univ.∀_:Univ.Univ.
+∀implies:∀_:Univ.∀_:Univ.Univ.
+∀not:∀_:Univ.Univ.
+∀or:∀_:Univ.∀_:Univ.Univ.
+∀truth:Univ.
+∀x:Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (myand X Y) (myand Y X).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (myand (myand X Y) Z) (myand X (myand Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.eq Univ (myand X Y) (not (or (not X) (not Y))).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (or X Y) (or Y X).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (or (or X Y) Z) (or X (or Y Z)).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (or X Y) (implies (not X) Y).
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (implies (implies (not X) (not Y)) (implies Y X)) truth.
+∀H7:∀X:Univ.∀Y:Univ.eq Univ (implies (implies X Y) Y) (implies (implies Y X) X).
+∀H8:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (implies (implies X Y) (implies (implies Y Z) (implies X Z))) truth.
+∀H9:∀X:Univ.eq Univ (implies truth X) X.eq Univ (not (or (myand x (or x x)) (myand x x))) (myand (not x) (or (or (not x) (not x)) (myand (not x) (not x))))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#and.
+#implies.
+#not.
+#or.
+#truth.
+#x.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LCL407-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LCL407-1 : TPTP v3.2.0. Released v2.5.0. *)
+
+(* Domain : Logic Calculi (Wajsberg Algebras) *)
+
+(* Problem : Wajsberg algebra axioms *)
+
+(* Version : [Bon91] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [FRT84] Font et al. (1984), Wajsberg Algebras *)
+
+(* : [Bon91] Bonacina (1991), Problems in Lukasiewicz Logic *)
+
+(* : [MW92] McCune & Wos (1992), Experiments in Automated Deductio *)
+
+(* Source : [MW92] *)
+
+(* Names : MV Sentential Calculus [MW92] *)
+
+(* Status : Satisfiable *)
+
+(* Rating : 0.00 v3.2.0, 0.33 v3.1.0, 0.00 v2.5.0 *)
+
+(* Syntax : Number of clauses : 4 ( 0 non-Horn; 4 unit; 0 RR) *)
+
+(* Number of atoms : 4 ( 4 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 8 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Wajsberg algebra axioms *)
+
+(* Inclusion of: Axioms/LCL001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LCL001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Logic Calculi (Wajsberg Algebras) *)
+
+(* Axioms : Wajsberg algebra axioms *)
+
+(* Version : [Bon91] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [FRT84] Font et al. (1984), Wajsberg Algebras *)
+
+(* : [Bon91] Bonacina (1991), Problems in Lukasiewicz Logic *)
+
+(* : [MW92] McCune & Wos (1992), Experiments in Automated Deductio *)
+
+(* Source : [MW92] *)
+
+(* Names : MV Sentential Calculus [MW92] *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 4 ( 0 non-Horn; 4 unit; 0 RR) *)
+
+(* Number of literals : 4 ( 4 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 8 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LCL407-2.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LCL407-2 : TPTP v3.2.0. Released v2.5.0. *)
+
+(* Domain : Logic Calculi (Wajsberg Algebras) *)
+
+(* Problem : Alternative Wajsberg algebra axioms *)
+
+(* Version : [AB90] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [FRT84] Font et al. (1984), Wajsberg Algebras *)
+
+(* : [AB90] Anantharaman & Bonacina (1990), An Application of the *)
+
+(* : [Bon91] Bonacina (1991), Problems in Lukasiewicz Logic *)
+
+(* Source : [Bon91] *)
+
+(* Names : *)
+
+(* Status : Satisfiable *)
+
+(* Rating : 0.00 v3.2.0, 0.33 v3.1.0, 0.00 v2.5.0 *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of atoms : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 10 ( 1 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Alternative Wajsberg algebra axioms *)
+
+(* Inclusion of: Axioms/LCL002-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LCL002-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Logic Calculi (Wajsberg Algebras) *)
+
+(* Axioms : Alternative Wajsberg algebra axioms *)
+
+(* Version : [AB90] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [FRT84] Font et al. (1984), Wajsberg Algebras *)
+
+(* : [AB90] Anantharaman & Bonacina (1990), An Application of the *)
+
+(* : [Bon91] Bonacina (1991), Problems in Lukasiewicz Logic *)
+
+(* Source : [Bon91] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 10 ( 1 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : To be used in conjunction with the LAT003 alternative *)
+
+(* Wajsberg algebra definitions. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LCL409-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LCL409-1 : TPTP v3.2.0. Released v2.5.0. *)
+
+(* Domain : Logic Calculi (Wajsberg Algebras) *)
+
+(* Problem : Wajsberg algebra AND and OR definitions *)
+
+(* Version : [AB90] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [FRT84] Font et al. (1984), Wajsberg Algebras *)
+
+(* : [AB90] Anantharaman & Bonacina (1990), An Application of the *)
+
+(* : [Bon91] Bonacina (1991), Problems in Lukasiewicz Logic *)
+
+(* Source : [Bon91] *)
+
+(* Names : *)
+
+(* Status : Satisfiable *)
+
+(* Rating : 0.00 v3.2.0, 0.33 v3.1.0, 0.00 v2.5.0 *)
+
+(* Syntax : Number of clauses : 10 ( 0 non-Horn; 10 unit; 0 RR) *)
+
+(* Number of atoms : 10 ( 10 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 22 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Wajsberg algebra axioms *)
+
+(* Inclusion of: Axioms/LCL001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LCL001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Logic Calculi (Wajsberg Algebras) *)
+
+(* Axioms : Wajsberg algebra axioms *)
+
+(* Version : [Bon91] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [FRT84] Font et al. (1984), Wajsberg Algebras *)
+
+(* : [Bon91] Bonacina (1991), Problems in Lukasiewicz Logic *)
+
+(* : [MW92] McCune & Wos (1992), Experiments in Automated Deductio *)
+
+(* Source : [MW92] *)
+
+(* Names : MV Sentential Calculus [MW92] *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 4 ( 0 non-Horn; 4 unit; 0 RR) *)
+
+(* Number of literals : 4 ( 4 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 8 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Wajsberg algebra AND and OR definitions *)
+
+(* Inclusion of: Axioms/LCL001-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LCL001-2 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Logic Calculi (Wajsberg Algebras) *)
+
+(* Axioms : Wajsberg algebra AND and OR definitions *)
+
+(* Version : [AB90] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [FRT84] Font et al. (1984), Wajsberg Algebras *)
+
+(* : [AB90] Anantharaman & Bonacina (1990), An Application of the *)
+
+(* : [Bon91] Bonacina (1991), Problems in Lukasiewicz Logic *)
+
+(* Source : [Bon91] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 6 ( 0 non-Horn; 6 unit; 0 RR) *)
+
+(* Number of literals : 6 ( 6 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 0 constant; 1-2 arity) *)
+
+(* Number of variables : 14 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 3 average) *)
+
+(* Comments : Requires LCL001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Definitions of or and and, which are AC *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LCL410-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LCL410-1 : TPTP v3.2.0. Released v2.5.0. *)
+
+(* Domain : Logic Calculi (Wajsberg Algebras) *)
+
+(* Problem : Alternative Wajsberg algebra definitions *)
+
+(* Version : [AB90] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [FRT84] Font et al. (1984), Wajsberg Algebras *)
+
+(* : [AB90] Anantharaman & Bonacina (1990), An Application of the *)
+
+(* : [Bon91] Bonacina (1991), Problems in Lukasiewicz Logic *)
+
+(* Source : [Bon91] *)
+
+(* Names : *)
+
+(* Status : Satisfiable *)
+
+(* Rating : 0.00 v3.2.0, 0.33 v3.1.0, 0.00 v2.5.0 *)
+
+(* Syntax : Number of clauses : 20 ( 0 non-Horn; 20 unit; 1 RR) *)
+
+(* Number of atoms : 20 ( 20 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 8 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 35 ( 1 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Alternative Wajsberg algebra axioms *)
+
+(* Inclusion of: Axioms/LCL002-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LCL002-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Logic Calculi (Wajsberg Algebras) *)
+
+(* Axioms : Alternative Wajsberg algebra axioms *)
+
+(* Version : [AB90] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [FRT84] Font et al. (1984), Wajsberg Algebras *)
+
+(* : [AB90] Anantharaman & Bonacina (1990), An Application of the *)
+
+(* : [Bon91] Bonacina (1991), Problems in Lukasiewicz Logic *)
+
+(* Source : [Bon91] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
+
+(* Number of literals : 8 ( 8 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 10 ( 1 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : To be used in conjunction with the LAT003 alternative *)
+
+(* Wajsberg algebra definitions. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Wajsberg algebra AND and OR definitions *)
+
+(* Inclusion of: Axioms/LCL001-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LCL001-2 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Logic Calculi (Wajsberg Algebras) *)
+
+(* Axioms : Wajsberg algebra AND and OR definitions *)
+
+(* Version : [AB90] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [FRT84] Font et al. (1984), Wajsberg Algebras *)
+
+(* : [AB90] Anantharaman & Bonacina (1990), An Application of the *)
+
+(* : [Bon91] Bonacina (1991), Problems in Lukasiewicz Logic *)
+
+(* Source : [Bon91] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 6 ( 0 non-Horn; 6 unit; 0 RR) *)
+
+(* Number of literals : 6 ( 6 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 0 constant; 1-2 arity) *)
+
+(* Number of variables : 14 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 3 average) *)
+
+(* Comments : Requires LCL001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Definitions of or and and, which are AC *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Alternative Wajsberg algebra definitions *)
+
+(* Inclusion of: Axioms/LCL002-1.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LCL002-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Logic Calculi (Wajsberg Algebras) *)
+
+(* Axioms : Alternative Wajsberg algebra definitions *)
+
+(* Version : [AB90] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [FRT84] Font et al. (1984), Wajsberg Algebras *)
+
+(* : [AB90] Anantharaman & Bonacina (1990), An Application of the *)
+
+(* : [Bon91] Bonacina (1991), Problems in Lukasiewicz Logic *)
+
+(* Source : [Bon91] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 6 ( 0 non-Horn; 6 unit; 1 RR) *)
+
+(* Number of literals : 6 ( 6 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 11 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : Requires LCL001-0.ax LCL001-2.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Definitions of and_star and xor, where and_star is AC and xor is C *)
+
+(* ---I guess the next two can be derived from the AC of and *)
+
+(* ----Definition of false in terms of truth *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LDA001-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LDA001-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : LD-Algebras *)
+
+(* Problem : Verify 3*2*U = UUU, where U = 2*2 *)
+
+(* Version : [Jec93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Jec93] Jech (1993), LD-Algebras *)
+
+(* Source : [Jec93] *)
+
+(* Names : Problem 1 [Jec93] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.2.1, 0.22 v2.2.0, 0.29 v2.1.0, 0.25 v2.0.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 4 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 3 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----A1: x(yz)=xy(xz) *)
+
+(* ----3*2*U = U*U*U *)
+ntheorem prove_equation:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀f:∀_:Univ.∀_:Univ.Univ.
+∀n1:Univ.
+∀n2:Univ.
+∀n3:Univ.
+∀u:Univ.
+∀H0:eq Univ u (f n2 n2).
+∀H1:eq Univ n3 (f n2 n1).
+∀H2:eq Univ n2 (f n1 n1).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (f X (f Y Z)) (f (f X Y) (f X Z)).eq Univ (f (f n3 n2) u) (f (f u u) u)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#f.
+#n1.
+#n2.
+#n3.
+#u.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LDA002-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LDA002-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : LD-Algebras *)
+
+(* Problem : Verify 3*2(U2)(UU(UU)) = U1(U3)(UU(UU)) *)
+
+(* Version : [Jec93] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Jec93] Jech (1993), LD-Algebras *)
+
+(* Source : [Jec93] *)
+
+(* Names : Problem 2 [Jec93] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.2.1, 0.22 v2.2.0, 0.29 v2.1.0, 0.38 v2.0.0 *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 11 RR) *)
+
+(* Number of atoms : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 12 ( 11 constant; 0-2 arity) *)
+
+(* Number of variables : 3 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----A1: x(yz)=xy(xz) *)
+
+(* ----3*2*U2*(UU*UU) = U1*U3*(uU*UU) *)
+ntheorem prove_equation:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀f:∀_:Univ.∀_:Univ.Univ.
+∀n1:Univ.
+∀n2:Univ.
+∀n3:Univ.
+∀u:Univ.
+∀u1:Univ.
+∀u2:Univ.
+∀u3:Univ.
+∀uu:Univ.
+∀v:Univ.
+∀H0:eq Univ v (f uu uu).
+∀H1:eq Univ b (f u1 u3).
+∀H2:eq Univ a (f (f n3 n2) u2).
+∀H3:eq Univ uu (f u u).
+∀H4:eq Univ u3 (f u n3).
+∀H5:eq Univ u2 (f u n2).
+∀H6:eq Univ u1 (f u n1).
+∀H7:eq Univ u (f n2 n2).
+∀H8:eq Univ n3 (f n2 n1).
+∀H9:eq Univ n2 (f n1 n1).
+∀H10:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (f X (f Y Z)) (f (f X Y) (f X Z)).eq Univ (f a v) (f b v)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#f.
+#n1.
+#n2.
+#n3.
+#u.
+#u1.
+#u2.
+#u3.
+#uu.
+#v.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LDA007-3.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LDA007-3 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : LD-Algebras (Embedding algebras) *)
+
+(* Problem : Let g = cr(t). Show that t(tsg) = tt(ts)(tg) *)
+
+(* Version : [Jec93] axioms : Incomplete > Reduced & Augmented > Incomplete. *)
+
+(* English : *)
+
+(* Refs : [Jec93] Jech (1993), LD-Algebras *)
+
+(* Source : [Jec93] *)
+
+(* Names : Problem 8 [Jec93] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.2.1, 0.11 v2.2.0, 0.14 v2.1.0, 0.13 v2.0.0 *)
+
+(* Syntax : Number of clauses : 7 ( 0 non-Horn; 7 unit; 6 RR) *)
+
+(* Number of atoms : 7 ( 7 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 9 ( 8 constant; 0-2 arity) *)
+
+(* Number of variables : 3 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Embedding algebra axioms *)
+
+(* include('Axioms/LDA001-0.ax'). *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----t(tsk) = tt(ts)(tk), where k=crit(t) *)
+ntheorem prove_equation:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀f:∀_:Univ.∀_:Univ.Univ.
+∀k:Univ.
+∀s:Univ.
+∀t:Univ.
+∀tk:Univ.
+∀ts:Univ.
+∀tsk:Univ.
+∀tt:Univ.
+∀tt_ts:Univ.
+∀H0:eq Univ tsk (f ts k).
+∀H1:eq Univ tk (f t k).
+∀H2:eq Univ tt_ts (f tt ts).
+∀H3:eq Univ ts (f t s).
+∀H4:eq Univ tt (f t t).
+∀H5:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (f X (f Y Z)) (f (f X Y) (f X Z)).eq Univ (f t tsk) (f tt_ts tk)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#f.
+#k.
+#s.
+#t.
+#tk.
+#ts.
+#tsk.
+#tt.
+#tt_ts.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+nauto by H0,H1,H2,H3,H4,H5;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: RNG007-4.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG007-4 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Ring Theory *)
+
+(* Problem : In Boolean rings, X is its own inverse *)
+
+(* Version : [Peterson & Stickel, 1981] (equality) axioms. *)
+
+(* Theorem formulation : Equality. *)
+
+(* English : Given a ring in which for all x, x * x = x, prove that for *)
+
+(* all x, x + x = additive_identity *)
+
+(* Refs : [PS81] Peterson & Stickel (1981), Complete Sets of Reductions *)
+
+(* Source : [ANL] *)
+
+(* Names : lemma.ver2.in [ANL] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.2.1, 0.11 v2.2.0, 0.14 v2.1.0, 0.13 v2.0.0 *)
+
+(* Syntax : Number of clauses : 16 ( 0 non-Horn; 16 unit; 2 RR) *)
+
+(* Number of atoms : 16 ( 16 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 26 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include ring theory axioms *)
+
+(* Inclusion of: Axioms/RNG002-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG002-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Ring Theory *)
+
+(* Axioms : Ring theory (equality) axioms *)
+
+(* Version : [PS81] (equality) axioms : *)
+
+(* Reduced & Augmented > Complete. *)
+
+(* English : *)
+
+(* Refs : [PS81] Peterson & Stickel (1981), Complete Sets of Reductions *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 14 ( 0 non-Horn; 14 unit; 1 RR) *)
+
+(* Number of literals : 14 ( 14 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 25 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Not sure if these are complete. I don't know if the reductions *)
+
+(* given in [PS81] are suitable for ATP. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Existence of left identity for addition *)
+
+(* ----Existence of left additive additive_inverse *)
+
+(* ----Distributive property of product over sum *)
+
+(* ----Inverse of identity is identity, stupid *)
+
+(* ----Inverse of additive_inverse of X is X *)
+
+(* ----Behavior of 0 and the multiplication operation *)
+
+(* ----Inverse of (x + y) is additive_inverse(x) + additive_inverse(y) *)
+
+(* ----x * additive_inverse(y) = additive_inverse (x * y) *)
+
+(* ----Associativity of addition *)
+
+(* ----Commutativity of addition *)
+
+(* ----Associativity of product *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_inverse:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀additive_identity:Univ.
+∀additive_inverse:∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.eq Univ (multiply X X) X.
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add (add X Y) Z) (add X (add Y Z)).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (multiply (additive_inverse X) Y) (additive_inverse (multiply X Y)).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (multiply X (additive_inverse Y)) (additive_inverse (multiply X Y)).
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (additive_inverse (add X Y)) (add (additive_inverse X) (additive_inverse Y)).
+∀H7:∀X:Univ.eq Univ (multiply additive_identity X) additive_identity.
+∀H8:∀X:Univ.eq Univ (multiply X additive_identity) additive_identity.
+∀H9:∀X:Univ.eq Univ (additive_inverse (additive_inverse X)) X.
+∀H10:eq Univ (additive_inverse additive_identity) additive_identity.
+∀H11:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X Y) Z) (add (multiply X Z) (multiply Y Z)).
+∀H12:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y Z)) (add (multiply X Y) (multiply X Z)).
+∀H13:∀X:Univ.eq Univ (add (additive_inverse X) X) additive_identity.
+∀H14:∀X:Univ.eq Univ (add additive_identity X) X.eq Univ (add a a) additive_identity
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#add.
+#additive_identity.
+#additive_inverse.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: RNG008-3.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG008-3 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Ring Theory *)
+
+(* Problem : Boolean rings are commutative *)
+
+(* Version : [PS81] (equality) axioms : Augmented. *)
+
+(* Theorem formulation : Equality. *)
+
+(* English : Given a ring in which for all x, x * x = x, prove that for *)
+
+(* all x and y, x * y = y * x. *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [PS81] Peterson & Stickel (1981), Complete Sets of Reductions *)
+
+(* Source : [ANL] *)
+
+(* Names : commute.ver2.in [ANL] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.07 v3.1.0, 0.11 v2.7.0, 0.00 v2.2.1, 0.22 v2.2.0, 0.29 v2.1.0, 0.25 v2.0.0 *)
+
+(* Syntax : Number of clauses : 19 ( 0 non-Horn; 19 unit; 3 RR) *)
+
+(* Number of atoms : 19 ( 19 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 28 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include ring theory axioms *)
+
+(* Inclusion of: Axioms/RNG002-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG002-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Ring Theory *)
+
+(* Axioms : Ring theory (equality) axioms *)
+
+(* Version : [PS81] (equality) axioms : *)
+
+(* Reduced & Augmented > Complete. *)
+
+(* English : *)
+
+(* Refs : [PS81] Peterson & Stickel (1981), Complete Sets of Reductions *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 14 ( 0 non-Horn; 14 unit; 1 RR) *)
+
+(* Number of literals : 14 ( 14 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 25 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Not sure if these are complete. I don't know if the reductions *)
+
+(* given in [PS81] are suitable for ATP. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Existence of left identity for addition *)
+
+(* ----Existence of left additive additive_inverse *)
+
+(* ----Distributive property of product over sum *)
+
+(* ----Inverse of identity is identity, stupid *)
+
+(* ----Inverse of additive_inverse of X is X *)
+
+(* ----Behavior of 0 and the multiplication operation *)
+
+(* ----Inverse of (x + y) is additive_inverse(x) + additive_inverse(y) *)
+
+(* ----x * additive_inverse(y) = additive_inverse (x * y) *)
+
+(* ----Associativity of addition *)
+
+(* ----Commutativity of addition *)
+
+(* ----Associativity of product *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Right identity and inverse are dependent lemmas *)
+ntheorem prove_commutativity:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀additive_identity:Univ.
+∀additive_inverse:∀_:Univ.Univ.
+∀b:Univ.
+∀c:Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:eq Univ (multiply a b) c.
+∀H1:∀X:Univ.eq Univ (multiply X X) X.
+∀H2:∀X:Univ.eq Univ (add X (additive_inverse X)) additive_identity.
+∀H3:∀X:Univ.eq Univ (add X additive_identity) X.
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).
+∀H6:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add (add X Y) Z) (add X (add Y Z)).
+∀H7:∀X:Univ.∀Y:Univ.eq Univ (multiply (additive_inverse X) Y) (additive_inverse (multiply X Y)).
+∀H8:∀X:Univ.∀Y:Univ.eq Univ (multiply X (additive_inverse Y)) (additive_inverse (multiply X Y)).
+∀H9:∀X:Univ.∀Y:Univ.eq Univ (additive_inverse (add X Y)) (add (additive_inverse X) (additive_inverse Y)).
+∀H10:∀X:Univ.eq Univ (multiply additive_identity X) additive_identity.
+∀H11:∀X:Univ.eq Univ (multiply X additive_identity) additive_identity.
+∀H12:∀X:Univ.eq Univ (additive_inverse (additive_inverse X)) X.
+∀H13:eq Univ (additive_inverse additive_identity) additive_identity.
+∀H14:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X Y) Z) (add (multiply X Z) (multiply Y Z)).
+∀H15:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y Z)) (add (multiply X Y) (multiply X Z)).
+∀H16:∀X:Univ.eq Univ (add (additive_inverse X) X) additive_identity.
+∀H17:∀X:Univ.eq Univ (add additive_identity X) X.eq Univ (multiply b a) c
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#add.
+#additive_identity.
+#additive_inverse.
+#b.
+#c.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+#H16.
+#H17.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15,H16,H17;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: RNG008-4.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG008-4 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Ring Theory *)
+
+(* Problem : Boolean rings are commutative *)
+
+(* Version : [PS81] (equality) axioms. *)
+
+(* Theorem formulation : Equality. *)
+
+(* English : Given a ring in which for all x, x * x = x, prove that for *)
+
+(* all x and y, x * y = y * x. *)
+
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(* : [PS81] Peterson & Stickel (1981), Complete Sets of Reductions *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.07 v3.1.0, 0.11 v2.7.0, 0.00 v2.2.1, 0.22 v2.2.0, 0.29 v2.1.0, 0.25 v2.0.0 *)
+
+(* Syntax : Number of clauses : 17 ( 0 non-Horn; 17 unit; 3 RR) *)
+
+(* Number of atoms : 17 ( 17 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 26 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include ring theory axioms *)
+
+(* Inclusion of: Axioms/RNG002-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG002-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Ring Theory *)
+
+(* Axioms : Ring theory (equality) axioms *)
+
+(* Version : [PS81] (equality) axioms : *)
+
+(* Reduced & Augmented > Complete. *)
+
+(* English : *)
+
+(* Refs : [PS81] Peterson & Stickel (1981), Complete Sets of Reductions *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 14 ( 0 non-Horn; 14 unit; 1 RR) *)
+
+(* Number of literals : 14 ( 14 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 25 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Not sure if these are complete. I don't know if the reductions *)
+
+(* given in [PS81] are suitable for ATP. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Existence of left identity for addition *)
+
+(* ----Existence of left additive additive_inverse *)
+
+(* ----Distributive property of product over sum *)
+
+(* ----Inverse of identity is identity, stupid *)
+
+(* ----Inverse of additive_inverse of X is X *)
+
+(* ----Behavior of 0 and the multiplication operation *)
+
+(* ----Inverse of (x + y) is additive_inverse(x) + additive_inverse(y) *)
+
+(* ----x * additive_inverse(y) = additive_inverse (x * y) *)
+
+(* ----Associativity of addition *)
+
+(* ----Commutativity of addition *)
+
+(* ----Associativity of product *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_commutativity:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀additive_identity:Univ.
+∀additive_inverse:∀_:Univ.Univ.
+∀b:Univ.
+∀c:Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:eq Univ (multiply a b) c.
+∀H1:∀X:Univ.eq Univ (multiply X X) X.
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add (add X Y) Z) (add X (add Y Z)).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (multiply (additive_inverse X) Y) (additive_inverse (multiply X Y)).
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (multiply X (additive_inverse Y)) (additive_inverse (multiply X Y)).
+∀H7:∀X:Univ.∀Y:Univ.eq Univ (additive_inverse (add X Y)) (add (additive_inverse X) (additive_inverse Y)).
+∀H8:∀X:Univ.eq Univ (multiply additive_identity X) additive_identity.
+∀H9:∀X:Univ.eq Univ (multiply X additive_identity) additive_identity.
+∀H10:∀X:Univ.eq Univ (additive_inverse (additive_inverse X)) X.
+∀H11:eq Univ (additive_inverse additive_identity) additive_identity.
+∀H12:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X Y) Z) (add (multiply X Z) (multiply Y Z)).
+∀H13:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y Z)) (add (multiply X Y) (multiply X Z)).
+∀H14:∀X:Univ.eq Univ (add (additive_inverse X) X) additive_identity.
+∀H15:∀X:Univ.eq Univ (add additive_identity X) X.eq Univ (multiply b a) c
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#add.
+#additive_identity.
+#additive_inverse.
+#b.
+#c.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: RNG008-7.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG008-7 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Ring Theory *)
+
+(* Problem : Boolean rings are commutative *)
+
+(* Version : [LW91] (equality) axioms. *)
+
+(* English : Given a ring in which for all x, x * x = x, prove that for *)
+
+(* all x and y, x * y = y * x. *)
+
+(* Refs : [LO85] Lusk & Overbeek (1985), Reasoning about Equality *)
+
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* : [LW91] Lusk & Wos (1991), Benchmark Problems in Which Equalit *)
+
+(* Source : [LW91] *)
+
+(* Names : Problem 3 [LO85] *)
+
+(* : Test Problem 8 [Wos88] *)
+
+(* : Boolean Rings [Wos88] *)
+
+(* : RT1 [LW91] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.07 v3.1.0, 0.11 v2.7.0, 0.00 v2.2.1, 0.22 v2.2.0, 0.29 v2.1.0, 0.25 v2.0.0 *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 2 RR) *)
+
+(* Number of atoms : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 19 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : This is very similar to ring_x2.in [OTTER]. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include ring theory axioms *)
+
+(* Inclusion of: Axioms/RNG005-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG005-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Ring Theory *)
+
+(* Axioms : Ring theory (equality) axioms *)
+
+(* Version : [LW92] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* : [LW92] Lusk & Wos (1992), Benchmark Problems in Which Equalit *)
+
+(* Source : [LW92] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 9 ( 0 non-Horn; 9 unit; 0 RR) *)
+
+(* Number of literals : 9 ( 9 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 18 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : These axioms are used in [Wos88] p.203. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----There exists an additive identity element *)
+
+(* ----Existence of left additive additive_inverse *)
+
+(* ----Associativity for addition *)
+
+(* ----Commutativity for addition *)
+
+(* ----Associativity for multiplication *)
+
+(* ----Distributive property of product over sum *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_commutativity:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀additive_identity:Univ.
+∀additive_inverse:∀_:Univ.Univ.
+∀b:Univ.
+∀c:Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:eq Univ (multiply a b) c.
+∀H1:∀X:Univ.eq Univ (multiply X X) X.
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X Y) Z) (add (multiply X Z) (multiply Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y Z)) (add (multiply X Y) (multiply X Z)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (multiply Y Z)) (multiply (multiply X Y) Z).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).
+∀H6:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add X (add Y Z)) (add (add X Y) Z).
+∀H7:∀X:Univ.eq Univ (add X (additive_inverse X)) additive_identity.
+∀H8:∀X:Univ.eq Univ (add (additive_inverse X) X) additive_identity.
+∀H9:∀X:Univ.eq Univ (add X additive_identity) X.
+∀H10:∀X:Univ.eq Univ (add additive_identity X) X.eq Univ (multiply b a) c
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#add.
+#additive_identity.
+#additive_inverse.
+#b.
+#c.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: RNG009-5.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG009-5 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Ring Theory *)
+
+(* Problem : If X*X*X = X then the ring is commutative *)
+
+(* Version : [Peterson & Stickel, 1981] (equality) axioms : *)
+
+(* Reduced > Incomplete. *)
+
+(* English : Given a ring in which for all x, x * x * x = x, prove that *)
+
+(* for all x and y, x * y = y * x. *)
+
+(* Refs : [PS81] Peterson & Stickel (1981), Complete Sets of Reductions *)
+
+(* : [Ove90] Overbeek (1990), ATP competition announced at CADE-10 *)
+
+(* : [Ove93] Overbeek (1993), The CADE-11 Competitions: A Personal *)
+
+(* : [LM93] Lusk & McCune (1993), Uniform Strategies: The CADE-11 *)
+
+(* : [Zha93] Zhang (1993), Automated Proofs of Equality Problems in *)
+
+(* Source : [Ove90] *)
+
+(* Names : CADE-11 Competition Eq-7 [Ove90] *)
+
+(* : THEOREM EQ-7 [LM93] *)
+
+(* : PROBLEM 7 [Zha93] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.50 v3.1.0, 0.44 v2.7.0, 0.36 v2.6.0, 0.17 v2.5.0, 0.25 v2.4.0, 0.00 v2.2.1, 0.67 v2.2.0, 0.71 v2.1.0, 1.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 9 ( 0 non-Horn; 9 unit; 1 RR) *)
+
+(* Number of atoms : 9 ( 9 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 17 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Right identity and inverse *)
+
+(* ----Distributive property of product over sum *)
+
+(* ----Associativity of addition *)
+
+(* ----Commutativity of addition *)
+
+(* ----Associativity of product *)
+ntheorem prove_commutativity:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀additive_identity:Univ.
+∀additive_inverse:∀_:Univ.Univ.
+∀b:Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.eq Univ (multiply X (multiply X X)) X.
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add (add X Y) Z) (add X (add Y Z)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X Y) Z) (add (multiply X Z) (multiply Y Z)).
+∀H5:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y Z)) (add (multiply X Y) (multiply X Z)).
+∀H6:∀X:Univ.eq Univ (add X (additive_inverse X)) additive_identity.
+∀H7:∀X:Univ.eq Univ (add X additive_identity) X.eq Univ (multiply a b) (multiply b a)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#add.
+#additive_identity.
+#additive_inverse.
+#b.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: RNG009-7.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG009-7 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Ring Theory *)
+
+(* Problem : If X*X*X = X then the ring is commutative *)
+
+(* Version : [LW91] (equality) axioms. *)
+
+(* English : Given a ring in which for all x, x * x * x = x, prove that *)
+
+(* for all x and y, x * y = y * x. *)
+
+(* Refs : [LO85] Lusk & Overbeek (1985), Reasoning about Equality *)
+
+(* : [LW91] Lusk & Wos (1991), Benchmark Problems in Which Equalit *)
+
+(* Source : [LW91] *)
+
+(* Names : Problem 6 [LO85] *)
+
+(* : RT2 [LW91] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.50 v3.1.0, 0.33 v2.7.0, 0.36 v2.6.0, 0.17 v2.5.0, 0.25 v2.4.0, 0.00 v2.2.1, 0.56 v2.2.0, 0.71 v2.1.0, 1.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 2 RR) *)
+
+(* Number of atoms : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 19 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include ring theory axioms *)
+
+(* Inclusion of: Axioms/RNG005-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG005-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Ring Theory *)
+
+(* Axioms : Ring theory (equality) axioms *)
+
+(* Version : [LW92] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* : [LW92] Lusk & Wos (1992), Benchmark Problems in Which Equalit *)
+
+(* Source : [LW92] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 9 ( 0 non-Horn; 9 unit; 0 RR) *)
+
+(* Number of literals : 9 ( 9 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 18 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : These axioms are used in [Wos88] p.203. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----There exists an additive identity element *)
+
+(* ----Existence of left additive additive_inverse *)
+
+(* ----Associativity for addition *)
+
+(* ----Commutativity for addition *)
+
+(* ----Associativity for multiplication *)
+
+(* ----Distributive property of product over sum *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_commutativity:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀additive_identity:Univ.
+∀additive_inverse:∀_:Univ.Univ.
+∀b:Univ.
+∀c:Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:eq Univ (multiply a b) c.
+∀H1:∀X:Univ.eq Univ (multiply X (multiply X X)) X.
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X Y) Z) (add (multiply X Z) (multiply Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y Z)) (add (multiply X Y) (multiply X Z)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (multiply Y Z)) (multiply (multiply X Y) Z).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).
+∀H6:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add X (add Y Z)) (add (add X Y) Z).
+∀H7:∀X:Univ.eq Univ (add X (additive_inverse X)) additive_identity.
+∀H8:∀X:Univ.eq Univ (add (additive_inverse X) X) additive_identity.
+∀H9:∀X:Univ.eq Univ (add X additive_identity) X.
+∀H10:∀X:Univ.eq Univ (add additive_identity X) X.eq Univ (multiply b a) c
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#add.
+#additive_identity.
+#additive_inverse.
+#b.
+#c.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: RNG010-5.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG010-5 : TPTP v3.2.0. Bugfixed v2.3.0. *)
+
+(* Domain : Ring Theory (Right alternative) *)
+
+(* Problem : Skew symmetry of the auxilliary function *)
+
+(* Version : [Ove90] (equality) axioms : *)
+
+(* Incomplete > Augmented > Incomplete. *)
+
+(* English : The three Moufang identities imply the skew symmetry *)
+
+(* of s(W,X,Y,Z) = (W*X,Y,Z) - X*(W,Y,Z) - (X,Y,Z)*W. *)
+
+(* Recall that skew symmetry means that the function sign *)
+
+(* changes when any two arguments are swapped. This problem *)
+
+(* proves the case for swapping the first two arguments. *)
+
+(* Refs : [Ove90] Overbeek (1990), ATP competition announced at CADE-10 *)
+
+(* : [Ove93] Overbeek (1993), The CADE-11 Competitions: A Personal *)
+
+(* : [LM93] Lusk & McCune (1993), Uniform Strategies: The CADE-11 *)
+
+(* : [Zha93] Zhang (1993), Automated Proofs of Equality Problems in *)
+
+(* Source : [Ove90] *)
+
+(* Names : CADE-11 Competition Eq-9 [Ove90] *)
+
+(* : THEOREM EQ-9 [LM93] *)
+
+(* : PROBLEM 9 [Zha93] *)
+
+(* Status : Unknown *)
+
+(* Rating : 1.00 v2.3.0 *)
+
+(* Syntax : Number of clauses : 27 ( 0 non-Horn; 27 unit; 2 RR) *)
+
+(* Number of atoms : 27 ( 27 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 11 ( 5 constant; 0-4 arity) *)
+
+(* Number of variables : 52 ( 2 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : I copied this directly. I think the Moufang identities may *)
+
+(* be wrong. At least they're in another form. *)
+
+(* : Yes, now they known to be wrong, and bugfixed in v2.3.0. *)
+
+(* Bugfixes : v2.3.0 - Clauses right_moufang and left_moufang fixed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Commutativity of addition *)
+
+(* ----Associativity of addition *)
+
+(* ----Additive identity *)
+
+(* ----Additive inverse *)
+
+(* ----Inverse of identity is identity, stupid *)
+
+(* ----Axiom of Overbeek *)
+
+(* ----Inverse of (x + y) is additive_inverse(x) + additive_inverse(y), *)
+
+(* ----Inverse of additive_inverse of X is X *)
+
+(* ----Behavior of 0 and the multiplication operation *)
+
+(* ----Axiom of Overbeek *)
+
+(* ----x * additive_inverse(y) = additive_inverse (x * y), *)
+
+(* ----Distributive property of product over sum *)
+
+(* ----Right alternative law *)
+
+(* ----Associator *)
+
+(* ----Commutator *)
+
+(* ----Middle associator identity *)
+
+(* ----Left alternative law *)
+
+(* ----Definition of s *)
+
+(* ----Right Moufang *)
+
+(* ----Left Moufang *)
+
+(* ----Middle Moufang *)
+ntheorem prove_skew_symmetry:
+ ∀Univ:Type.∀W:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀additive_identity:Univ.
+∀additive_inverse:∀_:Univ.Univ.
+∀associator:∀_:Univ.∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀c:Univ.
+∀commutator:∀_:Univ.∀_:Univ.Univ.
+∀d:Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀s:∀_:Univ.∀_:Univ.∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) (multiply Z X)) (multiply (multiply X (multiply Y Z)) X).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X (multiply Y X)) Z) (multiply X (multiply Y (multiply X Z))).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply Z (multiply X (multiply Y X))) (multiply (multiply (multiply Z X) Y) X).
+∀H3:∀W:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (s W X Y Z) (add (add (associator (multiply W X) Y Z) (additive_inverse (multiply X (associator W Y Z)))) (additive_inverse (multiply (associator X Y Z) W))).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (multiply (multiply X X) Y) (multiply X (multiply X Y)).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (multiply (multiply (associator X X Y) X) (associator X X Y)) additive_identity.
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (commutator X Y) (add (multiply Y X) (additive_inverse (multiply X Y))).
+∀H7:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (associator X Y Z) (add (multiply (multiply X Y) Z) (additive_inverse (multiply X (multiply Y Z)))).
+∀H8:∀X:Univ.∀Y:Univ.eq Univ (multiply (multiply X Y) Y) (multiply X (multiply Y Y)).
+∀H9:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X Y) Z) (add (multiply X Z) (multiply Y Z)).
+∀H10:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y Z)) (add (multiply X Y) (multiply X Z)).
+∀H11:∀X:Univ.∀Y:Univ.eq Univ (multiply (additive_inverse X) Y) (additive_inverse (multiply X Y)).
+∀H12:∀X:Univ.∀Y:Univ.eq Univ (multiply X (additive_inverse Y)) (additive_inverse (multiply X Y)).
+∀H13:∀X:Univ.∀Y:Univ.eq Univ (multiply (additive_inverse X) (additive_inverse Y)) (multiply X Y).
+∀H14:∀X:Univ.eq Univ (multiply additive_identity X) additive_identity.
+∀H15:∀X:Univ.eq Univ (multiply X additive_identity) additive_identity.
+∀H16:∀X:Univ.eq Univ (additive_inverse (additive_inverse X)) X.
+∀H17:∀X:Univ.∀Y:Univ.eq Univ (additive_inverse (add X Y)) (add (additive_inverse X) (additive_inverse Y)).
+∀H18:∀X:Univ.∀Y:Univ.eq Univ (add X (add (additive_inverse X) Y)) Y.
+∀H19:eq Univ (additive_inverse additive_identity) additive_identity.
+∀H20:∀X:Univ.eq Univ (add (additive_inverse X) X) additive_identity.
+∀H21:∀X:Univ.eq Univ (add X (additive_inverse X)) additive_identity.
+∀H22:∀X:Univ.eq Univ (add additive_identity X) X.
+∀H23:∀X:Univ.eq Univ (add X additive_identity) X.
+∀H24:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add (add X Y) Z) (add X (add Y Z)).
+∀H25:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).eq Univ (s a b c d) (additive_inverse (s b a c d))
+.
+#Univ.
+#W.
+#X.
+#Y.
+#Z.
+#a.
+#add.
+#additive_identity.
+#additive_inverse.
+#associator.
+#b.
+#c.
+#commutator.
+#d.
+#multiply.
+#s.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+#H16.
+#H17.
+#H18.
+#H19.
+#H20.
+#H21.
+#H22.
+#H23.
+#H24.
+#H25.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15,H16,H17,H18,H19,H20,H21,H22,H23,H24,H25;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: RNG010-6.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG010-6 : TPTP v3.2.0. Bugfixed v2.3.0. *)
+
+(* Domain : Ring Theory (Right alternative) *)
+
+(* Problem : Skew symmetry of the auxilliary function *)
+
+(* Version : [Ste87] (equality) axioms. *)
+
+(* English : The three Moufang identities imply the skew symmetry *)
+
+(* of s(W,X,Y,Z) = (W*X,Y,Z) - X*(W,Y,Z) - (X,Y,Z)*W. *)
+
+(* Recall that skew symmetry means that the function sign *)
+
+(* changes when any two arguments are swapped. This problem *)
+
+(* proves the case for swapping the first two arguments. *)
+
+(* Refs : [Ste87] Stevens (1987), Some Experiments in Nonassociative Rin *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unknown *)
+
+(* Rating : 1.00 v2.3.0 *)
+
+(* Syntax : Number of clauses : 20 ( 0 non-Horn; 20 unit; 1 RR) *)
+
+(* Number of atoms : 20 ( 20 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 11 ( 5 constant; 0-4 arity) *)
+
+(* Number of variables : 40 ( 2 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : *)
+
+(* Bugfixes : v2.3.0 - Clause prove_skew_symmetry fixed. *)
+
+(* : v2.3.0 - Left alternative law added in. *)
+
+(* : v2.3.0 - Clauses right_moufang and left_moufang fixed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include nonassociative ring axioms. *)
+
+(* Inclusion of: Axioms/RNG003-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG003-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Ring Theory (Alternative) *)
+
+(* Axioms : Alternative ring theory (equality) axioms *)
+
+(* Version : [Ste87] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Ste87] Stevens (1987), Some Experiments in Nonassociative Rin *)
+
+(* Source : [Ste87] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 15 ( 0 non-Horn; 15 unit; 0 RR) *)
+
+(* Number of literals : 15 ( 15 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 1 constant; 0-3 arity) *)
+
+(* Number of variables : 27 ( 2 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----There exists an additive identity element *)
+
+(* ----Multiplicative zero *)
+
+(* ----Existence of left additive additive_inverse *)
+
+(* ----Inverse of additive_inverse of X is X *)
+
+(* ----Distributive property of product over sum *)
+
+(* ----Commutativity for addition *)
+
+(* ----Associativity for addition *)
+
+(* ----Right alternative law *)
+
+(* ----Left alternative law *)
+
+(* ----Associator *)
+
+(* ----Commutator *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Definition of s *)
+
+(* ----Right Moufang *)
+
+(* ----Left Moufang *)
+ntheorem prove_skew_symmetry:
+ ∀Univ:Type.∀W:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀additive_identity:Univ.
+∀additive_inverse:∀_:Univ.Univ.
+∀associator:∀_:Univ.∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀c:Univ.
+∀commutator:∀_:Univ.∀_:Univ.Univ.
+∀d:Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀s:∀_:Univ.∀_:Univ.∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) (multiply Z X)) (multiply (multiply X (multiply Y Z)) X).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X (multiply Y X)) Z) (multiply X (multiply Y (multiply X Z))).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply Z (multiply X (multiply Y X))) (multiply (multiply (multiply Z X) Y) X).
+∀H3:∀W:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (s W X Y Z) (add (add (associator (multiply W X) Y Z) (additive_inverse (multiply X (associator W Y Z)))) (additive_inverse (multiply (associator X Y Z) W))).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (commutator X Y) (add (multiply Y X) (additive_inverse (multiply X Y))).
+∀H5:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (associator X Y Z) (add (multiply (multiply X Y) Z) (additive_inverse (multiply X (multiply Y Z)))).
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (multiply (multiply X X) Y) (multiply X (multiply X Y)).
+∀H7:∀X:Univ.∀Y:Univ.eq Univ (multiply (multiply X Y) Y) (multiply X (multiply Y Y)).
+∀H8:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add X (add Y Z)) (add (add X Y) Z).
+∀H9:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).
+∀H10:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X Y) Z) (add (multiply X Z) (multiply Y Z)).
+∀H11:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y Z)) (add (multiply X Y) (multiply X Z)).
+∀H12:∀X:Univ.eq Univ (additive_inverse (additive_inverse X)) X.
+∀H13:∀X:Univ.eq Univ (add X (additive_inverse X)) additive_identity.
+∀H14:∀X:Univ.eq Univ (add (additive_inverse X) X) additive_identity.
+∀H15:∀X:Univ.eq Univ (multiply X additive_identity) additive_identity.
+∀H16:∀X:Univ.eq Univ (multiply additive_identity X) additive_identity.
+∀H17:∀X:Univ.eq Univ (add X additive_identity) X.
+∀H18:∀X:Univ.eq Univ (add additive_identity X) X.eq Univ (s a b c d) (additive_inverse (s b a c d))
+.
+#Univ.
+#W.
+#X.
+#Y.
+#Z.
+#a.
+#add.
+#additive_identity.
+#additive_inverse.
+#associator.
+#b.
+#c.
+#commutator.
+#d.
+#multiply.
+#s.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+#H16.
+#H17.
+#H18.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15,H16,H17,H18;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: RNG010-7.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG010-7 : TPTP v3.2.0. Bugfixed v2.3.0. *)
+
+(* Domain : Ring Theory (Right alternative) *)
+
+(* Problem : Skew symmetry of the auxilliary function *)
+
+(* Version : [Ste87] (equality) axioms : Augmented. *)
+
+(* English : The three Moufang identities imply the skew symmetry *)
+
+(* of s(W,X,Y,Z) = (W*X,Y,Z) - X*(W,Y,Z) - (X,Y,Z)*W. *)
+
+(* Recall that skew symmetry means that the function sign *)
+
+(* changes when any two arguments are swapped. This problem *)
+
+(* proves the case for swapping the first two arguments. *)
+
+(* Refs : [Ste87] Stevens (1987), Some Experiments in Nonassociative Rin *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unknown *)
+
+(* Rating : 1.00 v2.3.0 *)
+
+(* Syntax : Number of clauses : 27 ( 0 non-Horn; 27 unit; 1 RR) *)
+
+(* Number of atoms : 27 ( 27 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 11 ( 5 constant; 0-4 arity) *)
+
+(* Number of variables : 58 ( 2 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : Extra lemmas added to help the ITP prover. *)
+
+(* Bugfixes : v2.3.0 - Clause prove_skew_symmetry fixed. *)
+
+(* : v2.3.0 - Left alternative law added in. *)
+
+(* : v2.3.0 - Clauses right_moufang and left_moufang fixed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include nonassociative ring axioms. *)
+
+(* Inclusion of: Axioms/RNG003-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG003-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Ring Theory (Alternative) *)
+
+(* Axioms : Alternative ring theory (equality) axioms *)
+
+(* Version : [Ste87] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Ste87] Stevens (1987), Some Experiments in Nonassociative Rin *)
+
+(* Source : [Ste87] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 15 ( 0 non-Horn; 15 unit; 0 RR) *)
+
+(* Number of literals : 15 ( 15 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 1 constant; 0-3 arity) *)
+
+(* Number of variables : 27 ( 2 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----There exists an additive identity element *)
+
+(* ----Multiplicative zero *)
+
+(* ----Existence of left additive additive_inverse *)
+
+(* ----Inverse of additive_inverse of X is X *)
+
+(* ----Distributive property of product over sum *)
+
+(* ----Commutativity for addition *)
+
+(* ----Associativity for addition *)
+
+(* ----Right alternative law *)
+
+(* ----Left alternative law *)
+
+(* ----Associator *)
+
+(* ----Commutator *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The next 7 clauses are extra lemmas which Stevens found useful *)
+
+(* ----Definition of s *)
+
+(* ----Right Moufang *)
+
+(* ----Left Moufang *)
+ntheorem prove_skew_symmetry:
+ ∀Univ:Type.∀W:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀additive_identity:Univ.
+∀additive_inverse:∀_:Univ.Univ.
+∀associator:∀_:Univ.∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀c:Univ.
+∀commutator:∀_:Univ.∀_:Univ.Univ.
+∀d:Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀s:∀_:Univ.∀_:Univ.∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) (multiply Z X)) (multiply (multiply X (multiply Y Z)) X).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X (multiply Y X)) Z) (multiply X (multiply Y (multiply X Z))).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply Z (multiply X (multiply Y X))) (multiply (multiply (multiply Z X) Y) X).
+∀H3:∀W:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (s W X Y Z) (add (add (associator (multiply W X) Y Z) (additive_inverse (multiply X (associator W Y Z)))) (additive_inverse (multiply (associator X Y Z) W))).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X Y) (additive_inverse Z)) (add (additive_inverse (multiply X Z)) (additive_inverse (multiply Y Z))).
+∀H5:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (additive_inverse X) (add Y Z)) (add (additive_inverse (multiply X Y)) (additive_inverse (multiply X Z))).
+∀H6:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X (additive_inverse Y)) Z) (add (multiply X Z) (additive_inverse (multiply Y Z))).
+∀H7:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y (additive_inverse Z))) (add (multiply X Y) (additive_inverse (multiply X Z))).
+∀H8:∀X:Univ.∀Y:Univ.eq Univ (multiply X (additive_inverse Y)) (additive_inverse (multiply X Y)).
+∀H9:∀X:Univ.∀Y:Univ.eq Univ (multiply (additive_inverse X) Y) (additive_inverse (multiply X Y)).
+∀H10:∀X:Univ.∀Y:Univ.eq Univ (multiply (additive_inverse X) (additive_inverse Y)) (multiply X Y).
+∀H11:∀X:Univ.∀Y:Univ.eq Univ (commutator X Y) (add (multiply Y X) (additive_inverse (multiply X Y))).
+∀H12:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (associator X Y Z) (add (multiply (multiply X Y) Z) (additive_inverse (multiply X (multiply Y Z)))).
+∀H13:∀X:Univ.∀Y:Univ.eq Univ (multiply (multiply X X) Y) (multiply X (multiply X Y)).
+∀H14:∀X:Univ.∀Y:Univ.eq Univ (multiply (multiply X Y) Y) (multiply X (multiply Y Y)).
+∀H15:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add X (add Y Z)) (add (add X Y) Z).
+∀H16:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).
+∀H17:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X Y) Z) (add (multiply X Z) (multiply Y Z)).
+∀H18:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y Z)) (add (multiply X Y) (multiply X Z)).
+∀H19:∀X:Univ.eq Univ (additive_inverse (additive_inverse X)) X.
+∀H20:∀X:Univ.eq Univ (add X (additive_inverse X)) additive_identity.
+∀H21:∀X:Univ.eq Univ (add (additive_inverse X) X) additive_identity.
+∀H22:∀X:Univ.eq Univ (multiply X additive_identity) additive_identity.
+∀H23:∀X:Univ.eq Univ (multiply additive_identity X) additive_identity.
+∀H24:∀X:Univ.eq Univ (add X additive_identity) X.
+∀H25:∀X:Univ.eq Univ (add additive_identity X) X.eq Univ (s a b c d) (additive_inverse (s b a c d))
+.
+#Univ.
+#W.
+#X.
+#Y.
+#Z.
+#a.
+#add.
+#additive_identity.
+#additive_inverse.
+#associator.
+#b.
+#c.
+#commutator.
+#d.
+#multiply.
+#s.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+#H16.
+#H17.
+#H18.
+#H19.
+#H20.
+#H21.
+#H22.
+#H23.
+#H24.
+#H25.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15,H16,H17,H18,H19,H20,H21,H22,H23,H24,H25;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: RNG011-5.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG011-5 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Ring Theory *)
+
+(* Problem : In a right alternative ring (((X,X,Y)*X)*(X,X,Y)) = Add Id *)
+
+(* Version : [Ove90] (equality) axioms : *)
+
+(* Incomplete > Augmented > Incomplete. *)
+
+(* English : *)
+
+(* Refs : [Ove90] Overbeek (1990), ATP competition announced at CADE-10 *)
+
+(* : [Ove93] Overbeek (1993), The CADE-11 Competitions: A Personal *)
+
+(* : [LM93] Lusk & McCune (1993), Uniform Strategies: The CADE-11 *)
+
+(* : [Zha93] Zhang (1993), Automated Proofs of Equality Problems in *)
+
+(* Source : [Ove90] *)
+
+(* Names : CADE-11 Competition Eq-10 [Ove90] *)
+
+(* : THEOREM EQ-10 [LM93] *)
+
+(* : PROBLEM 10 [Zha93] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 22 ( 0 non-Horn; 22 unit; 2 RR) *)
+
+(* Number of atoms : 22 ( 22 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 8 ( 3 constant; 0-3 arity) *)
+
+(* Number of variables : 37 ( 2 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Commutativity of addition *)
+
+(* ----Associativity of addition *)
+
+(* ----Additive identity *)
+
+(* ----Additive inverse *)
+
+(* ----Inverse of identity is identity, stupid *)
+
+(* ----Axiom of Overbeek *)
+
+(* ----Inverse of (x + y) is additive_inverse(x) + additive_inverse(y), *)
+
+(* ----Inverse of additive_inverse of X is X *)
+
+(* ----Behavior of 0 and the multiplication operation *)
+
+(* ----Axiom of Overbeek *)
+
+(* ----x * additive_inverse(y) = additive_inverse (x * y), *)
+
+(* ----Distributive property of product over sum *)
+
+(* ----Right alternative law *)
+
+(* ----Associator *)
+
+(* ----Commutator *)
+
+(* ----Middle associator identity *)
+ntheorem prove_equality:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀additive_identity:Univ.
+∀additive_inverse:∀_:Univ.Univ.
+∀associator:∀_:Univ.∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀commutator:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (multiply (multiply (associator X X Y) X) (associator X X Y)) additive_identity.
+∀H1:∀X:Univ.∀Y:Univ.eq Univ (commutator X Y) (add (multiply Y X) (additive_inverse (multiply X Y))).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (associator X Y Z) (add (multiply (multiply X Y) Z) (additive_inverse (multiply X (multiply Y Z)))).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (multiply (multiply X Y) Y) (multiply X (multiply Y Y)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X Y) Z) (add (multiply X Z) (multiply Y Z)).
+∀H5:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y Z)) (add (multiply X Y) (multiply X Z)).
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (multiply (additive_inverse X) Y) (additive_inverse (multiply X Y)).
+∀H7:∀X:Univ.∀Y:Univ.eq Univ (multiply X (additive_inverse Y)) (additive_inverse (multiply X Y)).
+∀H8:∀X:Univ.∀Y:Univ.eq Univ (multiply (additive_inverse X) (additive_inverse Y)) (multiply X Y).
+∀H9:∀X:Univ.eq Univ (multiply additive_identity X) additive_identity.
+∀H10:∀X:Univ.eq Univ (multiply X additive_identity) additive_identity.
+∀H11:∀X:Univ.eq Univ (additive_inverse (additive_inverse X)) X.
+∀H12:∀X:Univ.∀Y:Univ.eq Univ (additive_inverse (add X Y)) (add (additive_inverse X) (additive_inverse Y)).
+∀H13:∀X:Univ.∀Y:Univ.eq Univ (add X (add (additive_inverse X) Y)) Y.
+∀H14:eq Univ (additive_inverse additive_identity) additive_identity.
+∀H15:∀X:Univ.eq Univ (add (additive_inverse X) X) additive_identity.
+∀H16:∀X:Univ.eq Univ (add X (additive_inverse X)) additive_identity.
+∀H17:∀X:Univ.eq Univ (add additive_identity X) X.
+∀H18:∀X:Univ.eq Univ (add X additive_identity) X.
+∀H19:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add (add X Y) Z) (add X (add Y Z)).
+∀H20:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).eq Univ (multiply (multiply (associator a a b) a) (associator a a b)) additive_identity
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#add.
+#additive_identity.
+#additive_inverse.
+#associator.
+#b.
+#commutator.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+#H16.
+#H17.
+#H18.
+#H19.
+#H20.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15,H16,H17,H18,H19,H20;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: RNG012-6.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG012-6 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Ring Theory (Alternative) *)
+
+(* Problem : Product of inverses equal product *)
+
+(* Version : [Ste87] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Ste87] Stevens (1987), Some Experiments in Nonassociative Rin *)
+
+(* Source : [Ste87] *)
+
+(* Names : c15 [Ste87] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.2.1, 0.22 v2.2.0, 0.29 v2.1.0, 0.25 v2.0.0 *)
+
+(* Syntax : Number of clauses : 16 ( 0 non-Horn; 16 unit; 1 RR) *)
+
+(* Number of atoms : 16 ( 16 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 8 ( 3 constant; 0-3 arity) *)
+
+(* Number of variables : 27 ( 2 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include nonassociative ring axioms *)
+
+(* Inclusion of: Axioms/RNG003-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG003-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Ring Theory (Alternative) *)
+
+(* Axioms : Alternative ring theory (equality) axioms *)
+
+(* Version : [Ste87] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Ste87] Stevens (1987), Some Experiments in Nonassociative Rin *)
+
+(* Source : [Ste87] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 15 ( 0 non-Horn; 15 unit; 0 RR) *)
+
+(* Number of literals : 15 ( 15 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 1 constant; 0-3 arity) *)
+
+(* Number of variables : 27 ( 2 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----There exists an additive identity element *)
+
+(* ----Multiplicative zero *)
+
+(* ----Existence of left additive additive_inverse *)
+
+(* ----Inverse of additive_inverse of X is X *)
+
+(* ----Distributive property of product over sum *)
+
+(* ----Commutativity for addition *)
+
+(* ----Associativity for addition *)
+
+(* ----Right alternative law *)
+
+(* ----Left alternative law *)
+
+(* ----Associator *)
+
+(* ----Commutator *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_equation:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀additive_identity:Univ.
+∀additive_inverse:∀_:Univ.Univ.
+∀associator:∀_:Univ.∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀commutator:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (commutator X Y) (add (multiply Y X) (additive_inverse (multiply X Y))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (associator X Y Z) (add (multiply (multiply X Y) Z) (additive_inverse (multiply X (multiply Y Z)))).
+∀H2:∀X:Univ.∀Y:Univ.eq Univ (multiply (multiply X X) Y) (multiply X (multiply X Y)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (multiply (multiply X Y) Y) (multiply X (multiply Y Y)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add X (add Y Z)) (add (add X Y) Z).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).
+∀H6:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X Y) Z) (add (multiply X Z) (multiply Y Z)).
+∀H7:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y Z)) (add (multiply X Y) (multiply X Z)).
+∀H8:∀X:Univ.eq Univ (additive_inverse (additive_inverse X)) X.
+∀H9:∀X:Univ.eq Univ (add X (additive_inverse X)) additive_identity.
+∀H10:∀X:Univ.eq Univ (add (additive_inverse X) X) additive_identity.
+∀H11:∀X:Univ.eq Univ (multiply X additive_identity) additive_identity.
+∀H12:∀X:Univ.eq Univ (multiply additive_identity X) additive_identity.
+∀H13:∀X:Univ.eq Univ (add X additive_identity) X.
+∀H14:∀X:Univ.eq Univ (add additive_identity X) X.eq Univ (multiply (additive_inverse a) (additive_inverse b)) (multiply a b)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#add.
+#additive_identity.
+#additive_inverse.
+#associator.
+#b.
+#commutator.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: RNG013-6.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG013-6 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Ring Theory (Alternative) *)
+
+(* Problem : -X*Y = -(X*Y) *)
+
+(* Version : [Ste87] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Ste87] Stevens (1987), Some Experiments in Nonassociative Rin *)
+
+(* Source : [Ste87] *)
+
+(* Names : c16 [Ste87] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.2.1, 0.11 v2.2.0, 0.14 v2.1.0, 0.25 v2.0.0 *)
+
+(* Syntax : Number of clauses : 16 ( 0 non-Horn; 16 unit; 1 RR) *)
+
+(* Number of atoms : 16 ( 16 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 8 ( 3 constant; 0-3 arity) *)
+
+(* Number of variables : 27 ( 2 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include nonassociative ring axioms *)
+
+(* Inclusion of: Axioms/RNG003-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG003-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Ring Theory (Alternative) *)
+
+(* Axioms : Alternative ring theory (equality) axioms *)
+
+(* Version : [Ste87] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Ste87] Stevens (1987), Some Experiments in Nonassociative Rin *)
+
+(* Source : [Ste87] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 15 ( 0 non-Horn; 15 unit; 0 RR) *)
+
+(* Number of literals : 15 ( 15 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 1 constant; 0-3 arity) *)
+
+(* Number of variables : 27 ( 2 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----There exists an additive identity element *)
+
+(* ----Multiplicative zero *)
+
+(* ----Existence of left additive additive_inverse *)
+
+(* ----Inverse of additive_inverse of X is X *)
+
+(* ----Distributive property of product over sum *)
+
+(* ----Commutativity for addition *)
+
+(* ----Associativity for addition *)
+
+(* ----Right alternative law *)
+
+(* ----Left alternative law *)
+
+(* ----Associator *)
+
+(* ----Commutator *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_equation:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀additive_identity:Univ.
+∀additive_inverse:∀_:Univ.Univ.
+∀associator:∀_:Univ.∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀commutator:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (commutator X Y) (add (multiply Y X) (additive_inverse (multiply X Y))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (associator X Y Z) (add (multiply (multiply X Y) Z) (additive_inverse (multiply X (multiply Y Z)))).
+∀H2:∀X:Univ.∀Y:Univ.eq Univ (multiply (multiply X X) Y) (multiply X (multiply X Y)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (multiply (multiply X Y) Y) (multiply X (multiply Y Y)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add X (add Y Z)) (add (add X Y) Z).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).
+∀H6:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X Y) Z) (add (multiply X Z) (multiply Y Z)).
+∀H7:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y Z)) (add (multiply X Y) (multiply X Z)).
+∀H8:∀X:Univ.eq Univ (additive_inverse (additive_inverse X)) X.
+∀H9:∀X:Univ.eq Univ (add X (additive_inverse X)) additive_identity.
+∀H10:∀X:Univ.eq Univ (add (additive_inverse X) X) additive_identity.
+∀H11:∀X:Univ.eq Univ (multiply X additive_identity) additive_identity.
+∀H12:∀X:Univ.eq Univ (multiply additive_identity X) additive_identity.
+∀H13:∀X:Univ.eq Univ (add X additive_identity) X.
+∀H14:∀X:Univ.eq Univ (add additive_identity X) X.eq Univ (multiply (additive_inverse a) b) (additive_inverse (multiply a b))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#add.
+#additive_identity.
+#additive_inverse.
+#associator.
+#b.
+#commutator.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: RNG014-6.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG014-6 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Ring Theory (Alternative) *)
+
+(* Problem : -X*Y = -(X*Y) *)
+
+(* Version : [Ste87] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Ste87] Stevens (1987), Some Experiments in Nonassociative Rin *)
+
+(* Source : [Ste87] *)
+
+(* Names : c17 [Ste87] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.2.1, 0.11 v2.2.0, 0.14 v2.1.0, 0.25 v2.0.0 *)
+
+(* Syntax : Number of clauses : 16 ( 0 non-Horn; 16 unit; 1 RR) *)
+
+(* Number of atoms : 16 ( 16 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 8 ( 3 constant; 0-3 arity) *)
+
+(* Number of variables : 27 ( 2 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include nonassociative ring axioms *)
+
+(* Inclusion of: Axioms/RNG003-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG003-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Ring Theory (Alternative) *)
+
+(* Axioms : Alternative ring theory (equality) axioms *)
+
+(* Version : [Ste87] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Ste87] Stevens (1987), Some Experiments in Nonassociative Rin *)
+
+(* Source : [Ste87] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 15 ( 0 non-Horn; 15 unit; 0 RR) *)
+
+(* Number of literals : 15 ( 15 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 1 constant; 0-3 arity) *)
+
+(* Number of variables : 27 ( 2 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----There exists an additive identity element *)
+
+(* ----Multiplicative zero *)
+
+(* ----Existence of left additive additive_inverse *)
+
+(* ----Inverse of additive_inverse of X is X *)
+
+(* ----Distributive property of product over sum *)
+
+(* ----Commutativity for addition *)
+
+(* ----Associativity for addition *)
+
+(* ----Right alternative law *)
+
+(* ----Left alternative law *)
+
+(* ----Associator *)
+
+(* ----Commutator *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_equation:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀additive_identity:Univ.
+∀additive_inverse:∀_:Univ.Univ.
+∀associator:∀_:Univ.∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀commutator:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (commutator X Y) (add (multiply Y X) (additive_inverse (multiply X Y))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (associator X Y Z) (add (multiply (multiply X Y) Z) (additive_inverse (multiply X (multiply Y Z)))).
+∀H2:∀X:Univ.∀Y:Univ.eq Univ (multiply (multiply X X) Y) (multiply X (multiply X Y)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (multiply (multiply X Y) Y) (multiply X (multiply Y Y)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add X (add Y Z)) (add (add X Y) Z).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).
+∀H6:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X Y) Z) (add (multiply X Z) (multiply Y Z)).
+∀H7:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y Z)) (add (multiply X Y) (multiply X Z)).
+∀H8:∀X:Univ.eq Univ (additive_inverse (additive_inverse X)) X.
+∀H9:∀X:Univ.eq Univ (add X (additive_inverse X)) additive_identity.
+∀H10:∀X:Univ.eq Univ (add (additive_inverse X) X) additive_identity.
+∀H11:∀X:Univ.eq Univ (multiply X additive_identity) additive_identity.
+∀H12:∀X:Univ.eq Univ (multiply additive_identity X) additive_identity.
+∀H13:∀X:Univ.eq Univ (add X additive_identity) X.
+∀H14:∀X:Univ.eq Univ (add additive_identity X) X.eq Univ (multiply a (additive_inverse b)) (additive_inverse (multiply a b))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#add.
+#additive_identity.
+#additive_inverse.
+#associator.
+#b.
+#commutator.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: RNG015-6.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG015-6 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Ring Theory (Alternative) *)
+
+(* Problem : X*(Y+ -Z) = (X*Y) + -(X*Z) *)
+
+(* Version : [Ste87] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Ste87] Stevens (1987), Some Experiments in Nonassociative Rin *)
+
+(* Source : [Ste87] *)
+
+(* Names : c18 [Ste87] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.2.1, 0.22 v2.2.0, 0.14 v2.1.0, 0.25 v2.0.0 *)
+
+(* Syntax : Number of clauses : 16 ( 0 non-Horn; 16 unit; 1 RR) *)
+
+(* Number of atoms : 16 ( 16 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 9 ( 4 constant; 0-3 arity) *)
+
+(* Number of variables : 27 ( 2 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include nonassociative ring axioms *)
+
+(* Inclusion of: Axioms/RNG003-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG003-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Ring Theory (Alternative) *)
+
+(* Axioms : Alternative ring theory (equality) axioms *)
+
+(* Version : [Ste87] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Ste87] Stevens (1987), Some Experiments in Nonassociative Rin *)
+
+(* Source : [Ste87] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 15 ( 0 non-Horn; 15 unit; 0 RR) *)
+
+(* Number of literals : 15 ( 15 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 1 constant; 0-3 arity) *)
+
+(* Number of variables : 27 ( 2 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----There exists an additive identity element *)
+
+(* ----Multiplicative zero *)
+
+(* ----Existence of left additive additive_inverse *)
+
+(* ----Inverse of additive_inverse of X is X *)
+
+(* ----Distributive property of product over sum *)
+
+(* ----Commutativity for addition *)
+
+(* ----Associativity for addition *)
+
+(* ----Right alternative law *)
+
+(* ----Left alternative law *)
+
+(* ----Associator *)
+
+(* ----Commutator *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_distributivity:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀additive_identity:Univ.
+∀additive_inverse:∀_:Univ.Univ.
+∀associator:∀_:Univ.∀_:Univ.∀_:Univ.Univ.
+∀commutator:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀x:Univ.
+∀y:Univ.
+∀z:Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (commutator X Y) (add (multiply Y X) (additive_inverse (multiply X Y))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (associator X Y Z) (add (multiply (multiply X Y) Z) (additive_inverse (multiply X (multiply Y Z)))).
+∀H2:∀X:Univ.∀Y:Univ.eq Univ (multiply (multiply X X) Y) (multiply X (multiply X Y)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (multiply (multiply X Y) Y) (multiply X (multiply Y Y)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add X (add Y Z)) (add (add X Y) Z).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).
+∀H6:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X Y) Z) (add (multiply X Z) (multiply Y Z)).
+∀H7:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y Z)) (add (multiply X Y) (multiply X Z)).
+∀H8:∀X:Univ.eq Univ (additive_inverse (additive_inverse X)) X.
+∀H9:∀X:Univ.eq Univ (add X (additive_inverse X)) additive_identity.
+∀H10:∀X:Univ.eq Univ (add (additive_inverse X) X) additive_identity.
+∀H11:∀X:Univ.eq Univ (multiply X additive_identity) additive_identity.
+∀H12:∀X:Univ.eq Univ (multiply additive_identity X) additive_identity.
+∀H13:∀X:Univ.eq Univ (add X additive_identity) X.
+∀H14:∀X:Univ.eq Univ (add additive_identity X) X.eq Univ (multiply x (add y (additive_inverse z))) (add (multiply x y) (additive_inverse (multiply x z)))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#add.
+#additive_identity.
+#additive_inverse.
+#associator.
+#commutator.
+#multiply.
+#x.
+#y.
+#z.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: RNG016-6.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG016-6 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Ring Theory (Alternative) *)
+
+(* Problem : (X+ -Y)*Z = (X*Z) + -(Y*Z) *)
+
+(* Version : [Ste87] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Ste87] Stevens (1987), Some Experiments in Nonassociative Rin *)
+
+(* Source : [Ste87] *)
+
+(* Names : c19 [Ste87] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.2.1, 0.11 v2.2.0, 0.14 v2.1.0, 0.38 v2.0.0 *)
+
+(* Syntax : Number of clauses : 16 ( 0 non-Horn; 16 unit; 1 RR) *)
+
+(* Number of atoms : 16 ( 16 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 9 ( 4 constant; 0-3 arity) *)
+
+(* Number of variables : 27 ( 2 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include nonassociative ring axioms *)
+
+(* Inclusion of: Axioms/RNG003-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG003-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Ring Theory (Alternative) *)
+
+(* Axioms : Alternative ring theory (equality) axioms *)
+
+(* Version : [Ste87] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Ste87] Stevens (1987), Some Experiments in Nonassociative Rin *)
+
+(* Source : [Ste87] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 15 ( 0 non-Horn; 15 unit; 0 RR) *)
+
+(* Number of literals : 15 ( 15 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 1 constant; 0-3 arity) *)
+
+(* Number of variables : 27 ( 2 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----There exists an additive identity element *)
+
+(* ----Multiplicative zero *)
+
+(* ----Existence of left additive additive_inverse *)
+
+(* ----Inverse of additive_inverse of X is X *)
+
+(* ----Distributive property of product over sum *)
+
+(* ----Commutativity for addition *)
+
+(* ----Associativity for addition *)
+
+(* ----Right alternative law *)
+
+(* ----Left alternative law *)
+
+(* ----Associator *)
+
+(* ----Commutator *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_distributivity:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀additive_identity:Univ.
+∀additive_inverse:∀_:Univ.Univ.
+∀associator:∀_:Univ.∀_:Univ.∀_:Univ.Univ.
+∀commutator:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀x:Univ.
+∀y:Univ.
+∀z:Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (commutator X Y) (add (multiply Y X) (additive_inverse (multiply X Y))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (associator X Y Z) (add (multiply (multiply X Y) Z) (additive_inverse (multiply X (multiply Y Z)))).
+∀H2:∀X:Univ.∀Y:Univ.eq Univ (multiply (multiply X X) Y) (multiply X (multiply X Y)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (multiply (multiply X Y) Y) (multiply X (multiply Y Y)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add X (add Y Z)) (add (add X Y) Z).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).
+∀H6:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X Y) Z) (add (multiply X Z) (multiply Y Z)).
+∀H7:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y Z)) (add (multiply X Y) (multiply X Z)).
+∀H8:∀X:Univ.eq Univ (additive_inverse (additive_inverse X)) X.
+∀H9:∀X:Univ.eq Univ (add X (additive_inverse X)) additive_identity.
+∀H10:∀X:Univ.eq Univ (add (additive_inverse X) X) additive_identity.
+∀H11:∀X:Univ.eq Univ (multiply X additive_identity) additive_identity.
+∀H12:∀X:Univ.eq Univ (multiply additive_identity X) additive_identity.
+∀H13:∀X:Univ.eq Univ (add X additive_identity) X.
+∀H14:∀X:Univ.eq Univ (add additive_identity X) X.eq Univ (multiply (add x (additive_inverse y)) z) (add (multiply x z) (additive_inverse (multiply y z)))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#add.
+#additive_identity.
+#additive_inverse.
+#associator.
+#commutator.
+#multiply.
+#x.
+#y.
+#z.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: RNG017-6.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG017-6 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Ring Theory (Alternative) *)
+
+(* Problem : -X*(Y+Z) = -(X*Y) + -(X*Z) *)
+
+(* Version : [Ste87] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Ste87] Stevens (1987), Some Experiments in Nonassociative Rin *)
+
+(* Source : [Ste87] *)
+
+(* Names : c20 [Ste87] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.2.1, 0.22 v2.2.0, 0.29 v2.1.0, 0.38 v2.0.0 *)
+
+(* Syntax : Number of clauses : 16 ( 0 non-Horn; 16 unit; 1 RR) *)
+
+(* Number of atoms : 16 ( 16 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 9 ( 4 constant; 0-3 arity) *)
+
+(* Number of variables : 27 ( 2 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include nonassociative ring axioms *)
+
+(* Inclusion of: Axioms/RNG003-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG003-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Ring Theory (Alternative) *)
+
+(* Axioms : Alternative ring theory (equality) axioms *)
+
+(* Version : [Ste87] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Ste87] Stevens (1987), Some Experiments in Nonassociative Rin *)
+
+(* Source : [Ste87] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 15 ( 0 non-Horn; 15 unit; 0 RR) *)
+
+(* Number of literals : 15 ( 15 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 1 constant; 0-3 arity) *)
+
+(* Number of variables : 27 ( 2 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----There exists an additive identity element *)
+
+(* ----Multiplicative zero *)
+
+(* ----Existence of left additive additive_inverse *)
+
+(* ----Inverse of additive_inverse of X is X *)
+
+(* ----Distributive property of product over sum *)
+
+(* ----Commutativity for addition *)
+
+(* ----Associativity for addition *)
+
+(* ----Right alternative law *)
+
+(* ----Left alternative law *)
+
+(* ----Associator *)
+
+(* ----Commutator *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_distributivity:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀additive_identity:Univ.
+∀additive_inverse:∀_:Univ.Univ.
+∀associator:∀_:Univ.∀_:Univ.∀_:Univ.Univ.
+∀commutator:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀x:Univ.
+∀y:Univ.
+∀z:Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (commutator X Y) (add (multiply Y X) (additive_inverse (multiply X Y))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (associator X Y Z) (add (multiply (multiply X Y) Z) (additive_inverse (multiply X (multiply Y Z)))).
+∀H2:∀X:Univ.∀Y:Univ.eq Univ (multiply (multiply X X) Y) (multiply X (multiply X Y)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (multiply (multiply X Y) Y) (multiply X (multiply Y Y)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add X (add Y Z)) (add (add X Y) Z).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).
+∀H6:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X Y) Z) (add (multiply X Z) (multiply Y Z)).
+∀H7:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y Z)) (add (multiply X Y) (multiply X Z)).
+∀H8:∀X:Univ.eq Univ (additive_inverse (additive_inverse X)) X.
+∀H9:∀X:Univ.eq Univ (add X (additive_inverse X)) additive_identity.
+∀H10:∀X:Univ.eq Univ (add (additive_inverse X) X) additive_identity.
+∀H11:∀X:Univ.eq Univ (multiply X additive_identity) additive_identity.
+∀H12:∀X:Univ.eq Univ (multiply additive_identity X) additive_identity.
+∀H13:∀X:Univ.eq Univ (add X additive_identity) X.
+∀H14:∀X:Univ.eq Univ (add additive_identity X) X.eq Univ (multiply (additive_inverse x) (add y z)) (add (additive_inverse (multiply x y)) (additive_inverse (multiply x z)))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#add.
+#additive_identity.
+#additive_inverse.
+#associator.
+#commutator.
+#multiply.
+#x.
+#y.
+#z.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: RNG018-6.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG018-6 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Ring Theory (Alternative) *)
+
+(* Problem : (X+Y)* -Z = -(X*Z) + -(Y*Z) *)
+
+(* Version : [Ste87] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Ste87] Stevens (1987), Some Experiments in Nonassociative Rin *)
+
+(* Source : [Ste87] *)
+
+(* Names : c21 [Ste87] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.2.1, 0.22 v2.2.0, 0.29 v2.1.0, 0.25 v2.0.0 *)
+
+(* Syntax : Number of clauses : 16 ( 0 non-Horn; 16 unit; 1 RR) *)
+
+(* Number of atoms : 16 ( 16 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 9 ( 4 constant; 0-3 arity) *)
+
+(* Number of variables : 27 ( 2 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include nonassociative ring axioms *)
+
+(* Inclusion of: Axioms/RNG003-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG003-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Ring Theory (Alternative) *)
+
+(* Axioms : Alternative ring theory (equality) axioms *)
+
+(* Version : [Ste87] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Ste87] Stevens (1987), Some Experiments in Nonassociative Rin *)
+
+(* Source : [Ste87] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 15 ( 0 non-Horn; 15 unit; 0 RR) *)
+
+(* Number of literals : 15 ( 15 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 1 constant; 0-3 arity) *)
+
+(* Number of variables : 27 ( 2 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----There exists an additive identity element *)
+
+(* ----Multiplicative zero *)
+
+(* ----Existence of left additive additive_inverse *)
+
+(* ----Inverse of additive_inverse of X is X *)
+
+(* ----Distributive property of product over sum *)
+
+(* ----Commutativity for addition *)
+
+(* ----Associativity for addition *)
+
+(* ----Right alternative law *)
+
+(* ----Left alternative law *)
+
+(* ----Associator *)
+
+(* ----Commutator *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_distributivity:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀additive_identity:Univ.
+∀additive_inverse:∀_:Univ.Univ.
+∀associator:∀_:Univ.∀_:Univ.∀_:Univ.Univ.
+∀commutator:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀x:Univ.
+∀y:Univ.
+∀z:Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (commutator X Y) (add (multiply Y X) (additive_inverse (multiply X Y))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (associator X Y Z) (add (multiply (multiply X Y) Z) (additive_inverse (multiply X (multiply Y Z)))).
+∀H2:∀X:Univ.∀Y:Univ.eq Univ (multiply (multiply X X) Y) (multiply X (multiply X Y)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (multiply (multiply X Y) Y) (multiply X (multiply Y Y)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add X (add Y Z)) (add (add X Y) Z).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).
+∀H6:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X Y) Z) (add (multiply X Z) (multiply Y Z)).
+∀H7:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y Z)) (add (multiply X Y) (multiply X Z)).
+∀H8:∀X:Univ.eq Univ (additive_inverse (additive_inverse X)) X.
+∀H9:∀X:Univ.eq Univ (add X (additive_inverse X)) additive_identity.
+∀H10:∀X:Univ.eq Univ (add (additive_inverse X) X) additive_identity.
+∀H11:∀X:Univ.eq Univ (multiply X additive_identity) additive_identity.
+∀H12:∀X:Univ.eq Univ (multiply additive_identity X) additive_identity.
+∀H13:∀X:Univ.eq Univ (add X additive_identity) X.
+∀H14:∀X:Univ.eq Univ (add additive_identity X) X.eq Univ (multiply (add x y) (additive_inverse z)) (add (additive_inverse (multiply x z)) (additive_inverse (multiply y z)))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#add.
+#additive_identity.
+#additive_inverse.
+#associator.
+#commutator.
+#multiply.
+#x.
+#y.
+#z.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: RNG019-6.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG019-6 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Ring Theory (Alternative) *)
+
+(* Problem : First part of the linearised form of the associator *)
+
+(* Version : [Ste87] (equality) axioms. *)
+
+(* English : The associator can be expressed in another form called *)
+
+(* a linearised form. There are three clauses to be proved *)
+
+(* to establish the equivalence of the two forms. *)
+
+(* Refs : [Ste87] Stevens (1987), Some Experiments in Nonassociative Rin *)
+
+(* Source : [Ste87] *)
+
+(* Names : c24 [Ste87] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.14 v3.2.0, 0.21 v3.1.0, 0.11 v2.7.0, 0.18 v2.6.0, 0.33 v2.5.0, 0.25 v2.4.0, 0.67 v2.2.1, 0.78 v2.2.0, 0.71 v2.1.0, 0.50 v2.0.0 *)
+
+(* Syntax : Number of clauses : 16 ( 0 non-Horn; 16 unit; 1 RR) *)
+
+(* Number of atoms : 16 ( 16 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 10 ( 5 constant; 0-3 arity) *)
+
+(* Number of variables : 27 ( 2 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include nonassociative ring axioms *)
+
+(* Inclusion of: Axioms/RNG003-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG003-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Ring Theory (Alternative) *)
+
+(* Axioms : Alternative ring theory (equality) axioms *)
+
+(* Version : [Ste87] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Ste87] Stevens (1987), Some Experiments in Nonassociative Rin *)
+
+(* Source : [Ste87] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 15 ( 0 non-Horn; 15 unit; 0 RR) *)
+
+(* Number of literals : 15 ( 15 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 1 constant; 0-3 arity) *)
+
+(* Number of variables : 27 ( 2 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----There exists an additive identity element *)
+
+(* ----Multiplicative zero *)
+
+(* ----Existence of left additive additive_inverse *)
+
+(* ----Inverse of additive_inverse of X is X *)
+
+(* ----Distributive property of product over sum *)
+
+(* ----Commutativity for addition *)
+
+(* ----Associativity for addition *)
+
+(* ----Right alternative law *)
+
+(* ----Left alternative law *)
+
+(* ----Associator *)
+
+(* ----Commutator *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_linearised_form1:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀additive_identity:Univ.
+∀additive_inverse:∀_:Univ.Univ.
+∀associator:∀_:Univ.∀_:Univ.∀_:Univ.Univ.
+∀commutator:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀u:Univ.
+∀v:Univ.
+∀x:Univ.
+∀y:Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (commutator X Y) (add (multiply Y X) (additive_inverse (multiply X Y))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (associator X Y Z) (add (multiply (multiply X Y) Z) (additive_inverse (multiply X (multiply Y Z)))).
+∀H2:∀X:Univ.∀Y:Univ.eq Univ (multiply (multiply X X) Y) (multiply X (multiply X Y)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (multiply (multiply X Y) Y) (multiply X (multiply Y Y)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add X (add Y Z)) (add (add X Y) Z).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).
+∀H6:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X Y) Z) (add (multiply X Z) (multiply Y Z)).
+∀H7:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y Z)) (add (multiply X Y) (multiply X Z)).
+∀H8:∀X:Univ.eq Univ (additive_inverse (additive_inverse X)) X.
+∀H9:∀X:Univ.eq Univ (add X (additive_inverse X)) additive_identity.
+∀H10:∀X:Univ.eq Univ (add (additive_inverse X) X) additive_identity.
+∀H11:∀X:Univ.eq Univ (multiply X additive_identity) additive_identity.
+∀H12:∀X:Univ.eq Univ (multiply additive_identity X) additive_identity.
+∀H13:∀X:Univ.eq Univ (add X additive_identity) X.
+∀H14:∀X:Univ.eq Univ (add additive_identity X) X.eq Univ (associator x y (add u v)) (add (associator x y u) (associator x y v))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#add.
+#additive_identity.
+#additive_inverse.
+#associator.
+#commutator.
+#multiply.
+#u.
+#v.
+#x.
+#y.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: RNG019-7.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG019-7 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Ring Theory (Alternative) *)
+
+(* Problem : First part of the linearised form of the associator *)
+
+(* Version : [Ste87] (equality) axioms : Augmented. *)
+
+(* English : The associator can be expressed in another form called *)
+
+(* a linearised form. There are three clauses to be proved *)
+
+(* to establish the equivalence of the two forms. *)
+
+(* Refs : [Ste87] Stevens (1987), Some Experiments in Nonassociative Rin *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.14 v3.2.0, 0.21 v3.1.0, 0.22 v2.7.0, 0.27 v2.6.0, 0.33 v2.5.0, 0.25 v2.4.0, 0.67 v2.2.1, 0.78 v2.2.0, 0.71 v2.1.0, 0.50 v2.0.0 *)
+
+(* Syntax : Number of clauses : 23 ( 0 non-Horn; 23 unit; 1 RR) *)
+
+(* Number of atoms : 23 ( 23 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 10 ( 5 constant; 0-3 arity) *)
+
+(* Number of variables : 45 ( 2 singleton) *)
+
+(* Maximal term depth : 5 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include nonassociative ring axioms *)
+
+(* Inclusion of: Axioms/RNG003-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG003-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Ring Theory (Alternative) *)
+
+(* Axioms : Alternative ring theory (equality) axioms *)
+
+(* Version : [Ste87] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Ste87] Stevens (1987), Some Experiments in Nonassociative Rin *)
+
+(* Source : [Ste87] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 15 ( 0 non-Horn; 15 unit; 0 RR) *)
+
+(* Number of literals : 15 ( 15 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 1 constant; 0-3 arity) *)
+
+(* Number of variables : 27 ( 2 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----There exists an additive identity element *)
+
+(* ----Multiplicative zero *)
+
+(* ----Existence of left additive additive_inverse *)
+
+(* ----Inverse of additive_inverse of X is X *)
+
+(* ----Distributive property of product over sum *)
+
+(* ----Commutativity for addition *)
+
+(* ----Associativity for addition *)
+
+(* ----Right alternative law *)
+
+(* ----Left alternative law *)
+
+(* ----Associator *)
+
+(* ----Commutator *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The next 7 clause are extra lemmas which Stevens found useful *)
+ntheorem prove_linearised_form1:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀additive_identity:Univ.
+∀additive_inverse:∀_:Univ.Univ.
+∀associator:∀_:Univ.∀_:Univ.∀_:Univ.Univ.
+∀commutator:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀u:Univ.
+∀v:Univ.
+∀x:Univ.
+∀y:Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X Y) (additive_inverse Z)) (add (additive_inverse (multiply X Z)) (additive_inverse (multiply Y Z))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (additive_inverse X) (add Y Z)) (add (additive_inverse (multiply X Y)) (additive_inverse (multiply X Z))).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X (additive_inverse Y)) Z) (add (multiply X Z) (additive_inverse (multiply Y Z))).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y (additive_inverse Z))) (add (multiply X Y) (additive_inverse (multiply X Z))).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (multiply X (additive_inverse Y)) (additive_inverse (multiply X Y)).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (multiply (additive_inverse X) Y) (additive_inverse (multiply X Y)).
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (multiply (additive_inverse X) (additive_inverse Y)) (multiply X Y).
+∀H7:∀X:Univ.∀Y:Univ.eq Univ (commutator X Y) (add (multiply Y X) (additive_inverse (multiply X Y))).
+∀H8:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (associator X Y Z) (add (multiply (multiply X Y) Z) (additive_inverse (multiply X (multiply Y Z)))).
+∀H9:∀X:Univ.∀Y:Univ.eq Univ (multiply (multiply X X) Y) (multiply X (multiply X Y)).
+∀H10:∀X:Univ.∀Y:Univ.eq Univ (multiply (multiply X Y) Y) (multiply X (multiply Y Y)).
+∀H11:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add X (add Y Z)) (add (add X Y) Z).
+∀H12:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).
+∀H13:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X Y) Z) (add (multiply X Z) (multiply Y Z)).
+∀H14:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y Z)) (add (multiply X Y) (multiply X Z)).
+∀H15:∀X:Univ.eq Univ (additive_inverse (additive_inverse X)) X.
+∀H16:∀X:Univ.eq Univ (add X (additive_inverse X)) additive_identity.
+∀H17:∀X:Univ.eq Univ (add (additive_inverse X) X) additive_identity.
+∀H18:∀X:Univ.eq Univ (multiply X additive_identity) additive_identity.
+∀H19:∀X:Univ.eq Univ (multiply additive_identity X) additive_identity.
+∀H20:∀X:Univ.eq Univ (add X additive_identity) X.
+∀H21:∀X:Univ.eq Univ (add additive_identity X) X.eq Univ (associator x y (add u v)) (add (associator x y u) (associator x y v))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#add.
+#additive_identity.
+#additive_inverse.
+#associator.
+#commutator.
+#multiply.
+#u.
+#v.
+#x.
+#y.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+#H16.
+#H17.
+#H18.
+#H19.
+#H20.
+#H21.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15,H16,H17,H18,H19,H20,H21;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: RNG020-6.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG020-6 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Ring Theory (Alternative) *)
+
+(* Problem : Second part of the linearised form of the associator *)
+
+(* Version : [Ste87] (equality) axioms. *)
+
+(* English : The associator can be expressed in another form called *)
+
+(* a linearised form. There are three clauses to be proved *)
+
+(* to establish the equivalence of the two forms. *)
+
+(* Refs : [Ste87] Stevens (1987), Some Experiments in Nonassociative Rin *)
+
+(* Source : [Ste87] *)
+
+(* Names : c25 [Ste87] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.21 v3.2.0, 0.29 v3.1.0, 0.33 v2.7.0, 0.36 v2.6.0, 0.33 v2.5.0, 0.25 v2.4.0, 0.67 v2.2.1, 0.78 v2.2.0, 0.71 v2.1.0, 0.88 v2.0.0 *)
+
+(* Syntax : Number of clauses : 16 ( 0 non-Horn; 16 unit; 1 RR) *)
+
+(* Number of atoms : 16 ( 16 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 10 ( 5 constant; 0-3 arity) *)
+
+(* Number of variables : 27 ( 2 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include nonassociative ring axioms *)
+
+(* Inclusion of: Axioms/RNG003-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG003-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Ring Theory (Alternative) *)
+
+(* Axioms : Alternative ring theory (equality) axioms *)
+
+(* Version : [Ste87] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Ste87] Stevens (1987), Some Experiments in Nonassociative Rin *)
+
+(* Source : [Ste87] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 15 ( 0 non-Horn; 15 unit; 0 RR) *)
+
+(* Number of literals : 15 ( 15 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 1 constant; 0-3 arity) *)
+
+(* Number of variables : 27 ( 2 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----There exists an additive identity element *)
+
+(* ----Multiplicative zero *)
+
+(* ----Existence of left additive additive_inverse *)
+
+(* ----Inverse of additive_inverse of X is X *)
+
+(* ----Distributive property of product over sum *)
+
+(* ----Commutativity for addition *)
+
+(* ----Associativity for addition *)
+
+(* ----Right alternative law *)
+
+(* ----Left alternative law *)
+
+(* ----Associator *)
+
+(* ----Commutator *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_linearised_form2:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀additive_identity:Univ.
+∀additive_inverse:∀_:Univ.Univ.
+∀associator:∀_:Univ.∀_:Univ.∀_:Univ.Univ.
+∀commutator:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀u:Univ.
+∀v:Univ.
+∀x:Univ.
+∀y:Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (commutator X Y) (add (multiply Y X) (additive_inverse (multiply X Y))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (associator X Y Z) (add (multiply (multiply X Y) Z) (additive_inverse (multiply X (multiply Y Z)))).
+∀H2:∀X:Univ.∀Y:Univ.eq Univ (multiply (multiply X X) Y) (multiply X (multiply X Y)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (multiply (multiply X Y) Y) (multiply X (multiply Y Y)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add X (add Y Z)) (add (add X Y) Z).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).
+∀H6:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X Y) Z) (add (multiply X Z) (multiply Y Z)).
+∀H7:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y Z)) (add (multiply X Y) (multiply X Z)).
+∀H8:∀X:Univ.eq Univ (additive_inverse (additive_inverse X)) X.
+∀H9:∀X:Univ.eq Univ (add X (additive_inverse X)) additive_identity.
+∀H10:∀X:Univ.eq Univ (add (additive_inverse X) X) additive_identity.
+∀H11:∀X:Univ.eq Univ (multiply X additive_identity) additive_identity.
+∀H12:∀X:Univ.eq Univ (multiply additive_identity X) additive_identity.
+∀H13:∀X:Univ.eq Univ (add X additive_identity) X.
+∀H14:∀X:Univ.eq Univ (add additive_identity X) X.eq Univ (associator x (add u v) y) (add (associator x u y) (associator x v y))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#add.
+#additive_identity.
+#additive_inverse.
+#associator.
+#commutator.
+#multiply.
+#u.
+#v.
+#x.
+#y.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: RNG020-7.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG020-7 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Ring Theory (Alternative) *)
+
+(* Problem : Second part of the linearised form of the associator *)
+
+(* Version : [Ste87] (equality) axioms : Augmented. *)
+
+(* English : The associator can be expressed in another form called *)
+
+(* a linearised form. There are three clauses to be proved *)
+
+(* to establish the equivalence of the two forms. *)
+
+(* Refs : [Ste87] Stevens (1987), Some Experiments in Nonassociative Rin *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.14 v3.2.0, 0.21 v3.1.0, 0.33 v2.7.0, 0.36 v2.6.0, 0.33 v2.5.0, 0.25 v2.4.0, 0.67 v2.2.1, 0.78 v2.2.0, 0.71 v2.1.0, 0.88 v2.0.0 *)
+
+(* Syntax : Number of clauses : 23 ( 0 non-Horn; 23 unit; 1 RR) *)
+
+(* Number of atoms : 23 ( 23 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 10 ( 5 constant; 0-3 arity) *)
+
+(* Number of variables : 45 ( 2 singleton) *)
+
+(* Maximal term depth : 5 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include nonassociative ring axioms *)
+
+(* Inclusion of: Axioms/RNG003-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG003-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Ring Theory (Alternative) *)
+
+(* Axioms : Alternative ring theory (equality) axioms *)
+
+(* Version : [Ste87] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Ste87] Stevens (1987), Some Experiments in Nonassociative Rin *)
+
+(* Source : [Ste87] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 15 ( 0 non-Horn; 15 unit; 0 RR) *)
+
+(* Number of literals : 15 ( 15 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 1 constant; 0-3 arity) *)
+
+(* Number of variables : 27 ( 2 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----There exists an additive identity element *)
+
+(* ----Multiplicative zero *)
+
+(* ----Existence of left additive additive_inverse *)
+
+(* ----Inverse of additive_inverse of X is X *)
+
+(* ----Distributive property of product over sum *)
+
+(* ----Commutativity for addition *)
+
+(* ----Associativity for addition *)
+
+(* ----Right alternative law *)
+
+(* ----Left alternative law *)
+
+(* ----Associator *)
+
+(* ----Commutator *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The next 7 clause are extra lemmas which Stevens found useful *)
+ntheorem prove_linearised_form2:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀additive_identity:Univ.
+∀additive_inverse:∀_:Univ.Univ.
+∀associator:∀_:Univ.∀_:Univ.∀_:Univ.Univ.
+∀commutator:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀u:Univ.
+∀v:Univ.
+∀x:Univ.
+∀y:Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X Y) (additive_inverse Z)) (add (additive_inverse (multiply X Z)) (additive_inverse (multiply Y Z))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (additive_inverse X) (add Y Z)) (add (additive_inverse (multiply X Y)) (additive_inverse (multiply X Z))).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X (additive_inverse Y)) Z) (add (multiply X Z) (additive_inverse (multiply Y Z))).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y (additive_inverse Z))) (add (multiply X Y) (additive_inverse (multiply X Z))).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (multiply X (additive_inverse Y)) (additive_inverse (multiply X Y)).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (multiply (additive_inverse X) Y) (additive_inverse (multiply X Y)).
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (multiply (additive_inverse X) (additive_inverse Y)) (multiply X Y).
+∀H7:∀X:Univ.∀Y:Univ.eq Univ (commutator X Y) (add (multiply Y X) (additive_inverse (multiply X Y))).
+∀H8:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (associator X Y Z) (add (multiply (multiply X Y) Z) (additive_inverse (multiply X (multiply Y Z)))).
+∀H9:∀X:Univ.∀Y:Univ.eq Univ (multiply (multiply X X) Y) (multiply X (multiply X Y)).
+∀H10:∀X:Univ.∀Y:Univ.eq Univ (multiply (multiply X Y) Y) (multiply X (multiply Y Y)).
+∀H11:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add X (add Y Z)) (add (add X Y) Z).
+∀H12:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).
+∀H13:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X Y) Z) (add (multiply X Z) (multiply Y Z)).
+∀H14:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y Z)) (add (multiply X Y) (multiply X Z)).
+∀H15:∀X:Univ.eq Univ (additive_inverse (additive_inverse X)) X.
+∀H16:∀X:Univ.eq Univ (add X (additive_inverse X)) additive_identity.
+∀H17:∀X:Univ.eq Univ (add (additive_inverse X) X) additive_identity.
+∀H18:∀X:Univ.eq Univ (multiply X additive_identity) additive_identity.
+∀H19:∀X:Univ.eq Univ (multiply additive_identity X) additive_identity.
+∀H20:∀X:Univ.eq Univ (add X additive_identity) X.
+∀H21:∀X:Univ.eq Univ (add additive_identity X) X.eq Univ (associator x (add u v) y) (add (associator x u y) (associator x v y))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#add.
+#additive_identity.
+#additive_inverse.
+#associator.
+#commutator.
+#multiply.
+#u.
+#v.
+#x.
+#y.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+#H16.
+#H17.
+#H18.
+#H19.
+#H20.
+#H21.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15,H16,H17,H18,H19,H20,H21;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: RNG021-6.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG021-6 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Ring Theory (Alternative) *)
+
+(* Problem : Third part of the linearised form of the associator *)
+
+(* Version : [Ste87] (equality) axioms. *)
+
+(* English : The associator can be expressed in another form called *)
+
+(* a linearised form. There are three clauses to be proved *)
+
+(* to establish the equivalence of the two forms. *)
+
+(* Refs : [Ste87] Stevens (1987), Some Experiments in Nonassociative Rin *)
+
+(* Source : [Ste87] *)
+
+(* Names : c26 [Ste87] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.07 v3.2.0, 0.14 v3.1.0, 0.22 v2.7.0, 0.27 v2.6.0, 0.33 v2.5.0, 0.25 v2.4.0, 0.67 v2.2.1, 0.78 v2.2.0, 0.71 v2.1.0, 0.88 v2.0.0 *)
+
+(* Syntax : Number of clauses : 16 ( 0 non-Horn; 16 unit; 1 RR) *)
+
+(* Number of atoms : 16 ( 16 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 10 ( 5 constant; 0-3 arity) *)
+
+(* Number of variables : 27 ( 2 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include nonassociative ring axioms *)
+
+(* Inclusion of: Axioms/RNG003-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG003-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Ring Theory (Alternative) *)
+
+(* Axioms : Alternative ring theory (equality) axioms *)
+
+(* Version : [Ste87] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Ste87] Stevens (1987), Some Experiments in Nonassociative Rin *)
+
+(* Source : [Ste87] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 15 ( 0 non-Horn; 15 unit; 0 RR) *)
+
+(* Number of literals : 15 ( 15 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 1 constant; 0-3 arity) *)
+
+(* Number of variables : 27 ( 2 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----There exists an additive identity element *)
+
+(* ----Multiplicative zero *)
+
+(* ----Existence of left additive additive_inverse *)
+
+(* ----Inverse of additive_inverse of X is X *)
+
+(* ----Distributive property of product over sum *)
+
+(* ----Commutativity for addition *)
+
+(* ----Associativity for addition *)
+
+(* ----Right alternative law *)
+
+(* ----Left alternative law *)
+
+(* ----Associator *)
+
+(* ----Commutator *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_linearised_form3:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀additive_identity:Univ.
+∀additive_inverse:∀_:Univ.Univ.
+∀associator:∀_:Univ.∀_:Univ.∀_:Univ.Univ.
+∀commutator:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀u:Univ.
+∀v:Univ.
+∀x:Univ.
+∀y:Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (commutator X Y) (add (multiply Y X) (additive_inverse (multiply X Y))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (associator X Y Z) (add (multiply (multiply X Y) Z) (additive_inverse (multiply X (multiply Y Z)))).
+∀H2:∀X:Univ.∀Y:Univ.eq Univ (multiply (multiply X X) Y) (multiply X (multiply X Y)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (multiply (multiply X Y) Y) (multiply X (multiply Y Y)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add X (add Y Z)) (add (add X Y) Z).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).
+∀H6:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X Y) Z) (add (multiply X Z) (multiply Y Z)).
+∀H7:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y Z)) (add (multiply X Y) (multiply X Z)).
+∀H8:∀X:Univ.eq Univ (additive_inverse (additive_inverse X)) X.
+∀H9:∀X:Univ.eq Univ (add X (additive_inverse X)) additive_identity.
+∀H10:∀X:Univ.eq Univ (add (additive_inverse X) X) additive_identity.
+∀H11:∀X:Univ.eq Univ (multiply X additive_identity) additive_identity.
+∀H12:∀X:Univ.eq Univ (multiply additive_identity X) additive_identity.
+∀H13:∀X:Univ.eq Univ (add X additive_identity) X.
+∀H14:∀X:Univ.eq Univ (add additive_identity X) X.eq Univ (associator (add u v) x y) (add (associator u x y) (associator v x y))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#add.
+#additive_identity.
+#additive_inverse.
+#associator.
+#commutator.
+#multiply.
+#u.
+#v.
+#x.
+#y.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: RNG021-7.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG021-7 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Ring Theory (Alternative) *)
+
+(* Problem : Third part of the linearised form of the associator *)
+
+(* Version : [Ste87] (equality) axioms : Augmented. *)
+
+(* English : The associator can be expressed in another form called *)
+
+(* a linearised form. There are three clauses to be proved *)
+
+(* to establish the equivalence of the two forms. *)
+
+(* Refs : [Ste87] Stevens (1987), Some Experiments in Nonassociative Rin *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.14 v3.2.0, 0.21 v3.1.0, 0.33 v2.7.0, 0.45 v2.6.0, 0.33 v2.5.0, 0.25 v2.4.0, 0.67 v2.2.1, 0.78 v2.2.0, 0.71 v2.1.0, 0.88 v2.0.0 *)
+
+(* Syntax : Number of clauses : 23 ( 0 non-Horn; 23 unit; 1 RR) *)
+
+(* Number of atoms : 23 ( 23 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 10 ( 5 constant; 0-3 arity) *)
+
+(* Number of variables : 45 ( 2 singleton) *)
+
+(* Maximal term depth : 5 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include nonassociative ring axioms *)
+
+(* Inclusion of: Axioms/RNG003-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG003-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Ring Theory (Alternative) *)
+
+(* Axioms : Alternative ring theory (equality) axioms *)
+
+(* Version : [Ste87] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Ste87] Stevens (1987), Some Experiments in Nonassociative Rin *)
+
+(* Source : [Ste87] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 15 ( 0 non-Horn; 15 unit; 0 RR) *)
+
+(* Number of literals : 15 ( 15 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 1 constant; 0-3 arity) *)
+
+(* Number of variables : 27 ( 2 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----There exists an additive identity element *)
+
+(* ----Multiplicative zero *)
+
+(* ----Existence of left additive additive_inverse *)
+
+(* ----Inverse of additive_inverse of X is X *)
+
+(* ----Distributive property of product over sum *)
+
+(* ----Commutativity for addition *)
+
+(* ----Associativity for addition *)
+
+(* ----Right alternative law *)
+
+(* ----Left alternative law *)
+
+(* ----Associator *)
+
+(* ----Commutator *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The next 7 clause are extra lemmas which Stevens found useful *)
+ntheorem prove_linearised_form3:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀additive_identity:Univ.
+∀additive_inverse:∀_:Univ.Univ.
+∀associator:∀_:Univ.∀_:Univ.∀_:Univ.Univ.
+∀commutator:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀u:Univ.
+∀v:Univ.
+∀x:Univ.
+∀y:Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X Y) (additive_inverse Z)) (add (additive_inverse (multiply X Z)) (additive_inverse (multiply Y Z))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (additive_inverse X) (add Y Z)) (add (additive_inverse (multiply X Y)) (additive_inverse (multiply X Z))).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X (additive_inverse Y)) Z) (add (multiply X Z) (additive_inverse (multiply Y Z))).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y (additive_inverse Z))) (add (multiply X Y) (additive_inverse (multiply X Z))).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (multiply X (additive_inverse Y)) (additive_inverse (multiply X Y)).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (multiply (additive_inverse X) Y) (additive_inverse (multiply X Y)).
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (multiply (additive_inverse X) (additive_inverse Y)) (multiply X Y).
+∀H7:∀X:Univ.∀Y:Univ.eq Univ (commutator X Y) (add (multiply Y X) (additive_inverse (multiply X Y))).
+∀H8:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (associator X Y Z) (add (multiply (multiply X Y) Z) (additive_inverse (multiply X (multiply Y Z)))).
+∀H9:∀X:Univ.∀Y:Univ.eq Univ (multiply (multiply X X) Y) (multiply X (multiply X Y)).
+∀H10:∀X:Univ.∀Y:Univ.eq Univ (multiply (multiply X Y) Y) (multiply X (multiply Y Y)).
+∀H11:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add X (add Y Z)) (add (add X Y) Z).
+∀H12:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).
+∀H13:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X Y) Z) (add (multiply X Z) (multiply Y Z)).
+∀H14:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y Z)) (add (multiply X Y) (multiply X Z)).
+∀H15:∀X:Univ.eq Univ (additive_inverse (additive_inverse X)) X.
+∀H16:∀X:Univ.eq Univ (add X (additive_inverse X)) additive_identity.
+∀H17:∀X:Univ.eq Univ (add (additive_inverse X) X) additive_identity.
+∀H18:∀X:Univ.eq Univ (multiply X additive_identity) additive_identity.
+∀H19:∀X:Univ.eq Univ (multiply additive_identity X) additive_identity.
+∀H20:∀X:Univ.eq Univ (add X additive_identity) X.
+∀H21:∀X:Univ.eq Univ (add additive_identity X) X.eq Univ (associator (add u v) x y) (add (associator u x y) (associator v x y))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#add.
+#additive_identity.
+#additive_inverse.
+#associator.
+#commutator.
+#multiply.
+#u.
+#v.
+#x.
+#y.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+#H16.
+#H17.
+#H18.
+#H19.
+#H20.
+#H21.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15,H16,H17,H18,H19,H20,H21;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: RNG023-6.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG023-6 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Ring Theory (Alternative) *)
+
+(* Problem : Left alternative *)
+
+(* Version : [Ste87] (equality) axioms. *)
+
+(* Theorem formulation : In terms of associators *)
+
+(* English : *)
+
+(* Refs : [Ste87] Stevens (1987), Some Experiments in Nonassociative Rin *)
+
+(* : [Ste92] Stevens (1992), Unpublished Note *)
+
+(* Source : [Ste92] *)
+
+(* Names : - [Ste87] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.1.0, 0.13 v2.0.0 *)
+
+(* Syntax : Number of clauses : 16 ( 0 non-Horn; 16 unit; 1 RR) *)
+
+(* Number of atoms : 16 ( 16 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 8 ( 3 constant; 0-3 arity) *)
+
+(* Number of variables : 27 ( 2 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include nonassociative ring axioms *)
+
+(* Inclusion of: Axioms/RNG003-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG003-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Ring Theory (Alternative) *)
+
+(* Axioms : Alternative ring theory (equality) axioms *)
+
+(* Version : [Ste87] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Ste87] Stevens (1987), Some Experiments in Nonassociative Rin *)
+
+(* Source : [Ste87] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 15 ( 0 non-Horn; 15 unit; 0 RR) *)
+
+(* Number of literals : 15 ( 15 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 1 constant; 0-3 arity) *)
+
+(* Number of variables : 27 ( 2 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----There exists an additive identity element *)
+
+(* ----Multiplicative zero *)
+
+(* ----Existence of left additive additive_inverse *)
+
+(* ----Inverse of additive_inverse of X is X *)
+
+(* ----Distributive property of product over sum *)
+
+(* ----Commutativity for addition *)
+
+(* ----Associativity for addition *)
+
+(* ----Right alternative law *)
+
+(* ----Left alternative law *)
+
+(* ----Associator *)
+
+(* ----Commutator *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_left_alternative:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀additive_identity:Univ.
+∀additive_inverse:∀_:Univ.Univ.
+∀associator:∀_:Univ.∀_:Univ.∀_:Univ.Univ.
+∀commutator:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀x:Univ.
+∀y:Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (commutator X Y) (add (multiply Y X) (additive_inverse (multiply X Y))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (associator X Y Z) (add (multiply (multiply X Y) Z) (additive_inverse (multiply X (multiply Y Z)))).
+∀H2:∀X:Univ.∀Y:Univ.eq Univ (multiply (multiply X X) Y) (multiply X (multiply X Y)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (multiply (multiply X Y) Y) (multiply X (multiply Y Y)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add X (add Y Z)) (add (add X Y) Z).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).
+∀H6:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X Y) Z) (add (multiply X Z) (multiply Y Z)).
+∀H7:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y Z)) (add (multiply X Y) (multiply X Z)).
+∀H8:∀X:Univ.eq Univ (additive_inverse (additive_inverse X)) X.
+∀H9:∀X:Univ.eq Univ (add X (additive_inverse X)) additive_identity.
+∀H10:∀X:Univ.eq Univ (add (additive_inverse X) X) additive_identity.
+∀H11:∀X:Univ.eq Univ (multiply X additive_identity) additive_identity.
+∀H12:∀X:Univ.eq Univ (multiply additive_identity X) additive_identity.
+∀H13:∀X:Univ.eq Univ (add X additive_identity) X.
+∀H14:∀X:Univ.eq Univ (add additive_identity X) X.eq Univ (associator x x y) additive_identity
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#add.
+#additive_identity.
+#additive_inverse.
+#associator.
+#commutator.
+#multiply.
+#x.
+#y.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: RNG023-7.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG023-7 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Ring Theory (Alternative) *)
+
+(* Problem : Left alternative *)
+
+(* Version : [Ste87] (equality) axioms : Augmented. *)
+
+(* Theorem formulation : In terms of associators *)
+
+(* English : *)
+
+(* Refs : [Ste87] Stevens (1987), Some Experiments in Nonassociative Rin *)
+
+(* : [Ste92] Stevens (1992), Unpublished Note *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.2.1, 0.11 v2.2.0, 0.00 v2.1.0, 0.13 v2.0.0 *)
+
+(* Syntax : Number of clauses : 23 ( 0 non-Horn; 23 unit; 1 RR) *)
+
+(* Number of atoms : 23 ( 23 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 8 ( 3 constant; 0-3 arity) *)
+
+(* Number of variables : 45 ( 2 singleton) *)
+
+(* Maximal term depth : 5 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include nonassociative ring axioms *)
+
+(* Inclusion of: Axioms/RNG003-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG003-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Ring Theory (Alternative) *)
+
+(* Axioms : Alternative ring theory (equality) axioms *)
+
+(* Version : [Ste87] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Ste87] Stevens (1987), Some Experiments in Nonassociative Rin *)
+
+(* Source : [Ste87] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 15 ( 0 non-Horn; 15 unit; 0 RR) *)
+
+(* Number of literals : 15 ( 15 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 1 constant; 0-3 arity) *)
+
+(* Number of variables : 27 ( 2 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----There exists an additive identity element *)
+
+(* ----Multiplicative zero *)
+
+(* ----Existence of left additive additive_inverse *)
+
+(* ----Inverse of additive_inverse of X is X *)
+
+(* ----Distributive property of product over sum *)
+
+(* ----Commutativity for addition *)
+
+(* ----Associativity for addition *)
+
+(* ----Right alternative law *)
+
+(* ----Left alternative law *)
+
+(* ----Associator *)
+
+(* ----Commutator *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The next 7 clause are extra lemmas which Stevens found useful *)
+ntheorem prove_left_alternative:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀additive_identity:Univ.
+∀additive_inverse:∀_:Univ.Univ.
+∀associator:∀_:Univ.∀_:Univ.∀_:Univ.Univ.
+∀commutator:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀x:Univ.
+∀y:Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X Y) (additive_inverse Z)) (add (additive_inverse (multiply X Z)) (additive_inverse (multiply Y Z))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (additive_inverse X) (add Y Z)) (add (additive_inverse (multiply X Y)) (additive_inverse (multiply X Z))).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X (additive_inverse Y)) Z) (add (multiply X Z) (additive_inverse (multiply Y Z))).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y (additive_inverse Z))) (add (multiply X Y) (additive_inverse (multiply X Z))).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (multiply X (additive_inverse Y)) (additive_inverse (multiply X Y)).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (multiply (additive_inverse X) Y) (additive_inverse (multiply X Y)).
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (multiply (additive_inverse X) (additive_inverse Y)) (multiply X Y).
+∀H7:∀X:Univ.∀Y:Univ.eq Univ (commutator X Y) (add (multiply Y X) (additive_inverse (multiply X Y))).
+∀H8:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (associator X Y Z) (add (multiply (multiply X Y) Z) (additive_inverse (multiply X (multiply Y Z)))).
+∀H9:∀X:Univ.∀Y:Univ.eq Univ (multiply (multiply X X) Y) (multiply X (multiply X Y)).
+∀H10:∀X:Univ.∀Y:Univ.eq Univ (multiply (multiply X Y) Y) (multiply X (multiply Y Y)).
+∀H11:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add X (add Y Z)) (add (add X Y) Z).
+∀H12:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).
+∀H13:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X Y) Z) (add (multiply X Z) (multiply Y Z)).
+∀H14:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y Z)) (add (multiply X Y) (multiply X Z)).
+∀H15:∀X:Univ.eq Univ (additive_inverse (additive_inverse X)) X.
+∀H16:∀X:Univ.eq Univ (add X (additive_inverse X)) additive_identity.
+∀H17:∀X:Univ.eq Univ (add (additive_inverse X) X) additive_identity.
+∀H18:∀X:Univ.eq Univ (multiply X additive_identity) additive_identity.
+∀H19:∀X:Univ.eq Univ (multiply additive_identity X) additive_identity.
+∀H20:∀X:Univ.eq Univ (add X additive_identity) X.
+∀H21:∀X:Univ.eq Univ (add additive_identity X) X.eq Univ (associator x x y) additive_identity
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#add.
+#additive_identity.
+#additive_inverse.
+#associator.
+#commutator.
+#multiply.
+#x.
+#y.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+#H16.
+#H17.
+#H18.
+#H19.
+#H20.
+#H21.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15,H16,H17,H18,H19,H20,H21;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: RNG024-6.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG024-6 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Ring Theory (Alternative) *)
+
+(* Problem : Right alternative *)
+
+(* Version : [Ste87] (equality) axioms. *)
+
+(* Theorem formulation : In terms of associators *)
+
+(* English : *)
+
+(* Refs : [Ste87] Stevens (1987), Some Experiments in Nonassociative Rin *)
+
+(* : [Ste92] Stevens (1992), Unpublished Note *)
+
+(* Source : [Ste92] *)
+
+(* Names : - [Ste87] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.1.0, 0.13 v2.0.0 *)
+
+(* Syntax : Number of clauses : 16 ( 0 non-Horn; 16 unit; 1 RR) *)
+
+(* Number of atoms : 16 ( 16 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 8 ( 3 constant; 0-3 arity) *)
+
+(* Number of variables : 27 ( 2 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include nonassociative ring axioms *)
+
+(* Inclusion of: Axioms/RNG003-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG003-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Ring Theory (Alternative) *)
+
+(* Axioms : Alternative ring theory (equality) axioms *)
+
+(* Version : [Ste87] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Ste87] Stevens (1987), Some Experiments in Nonassociative Rin *)
+
+(* Source : [Ste87] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 15 ( 0 non-Horn; 15 unit; 0 RR) *)
+
+(* Number of literals : 15 ( 15 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 1 constant; 0-3 arity) *)
+
+(* Number of variables : 27 ( 2 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----There exists an additive identity element *)
+
+(* ----Multiplicative zero *)
+
+(* ----Existence of left additive additive_inverse *)
+
+(* ----Inverse of additive_inverse of X is X *)
+
+(* ----Distributive property of product over sum *)
+
+(* ----Commutativity for addition *)
+
+(* ----Associativity for addition *)
+
+(* ----Right alternative law *)
+
+(* ----Left alternative law *)
+
+(* ----Associator *)
+
+(* ----Commutator *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_right_alternative:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀additive_identity:Univ.
+∀additive_inverse:∀_:Univ.Univ.
+∀associator:∀_:Univ.∀_:Univ.∀_:Univ.Univ.
+∀commutator:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀x:Univ.
+∀y:Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (commutator X Y) (add (multiply Y X) (additive_inverse (multiply X Y))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (associator X Y Z) (add (multiply (multiply X Y) Z) (additive_inverse (multiply X (multiply Y Z)))).
+∀H2:∀X:Univ.∀Y:Univ.eq Univ (multiply (multiply X X) Y) (multiply X (multiply X Y)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (multiply (multiply X Y) Y) (multiply X (multiply Y Y)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add X (add Y Z)) (add (add X Y) Z).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).
+∀H6:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X Y) Z) (add (multiply X Z) (multiply Y Z)).
+∀H7:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y Z)) (add (multiply X Y) (multiply X Z)).
+∀H8:∀X:Univ.eq Univ (additive_inverse (additive_inverse X)) X.
+∀H9:∀X:Univ.eq Univ (add X (additive_inverse X)) additive_identity.
+∀H10:∀X:Univ.eq Univ (add (additive_inverse X) X) additive_identity.
+∀H11:∀X:Univ.eq Univ (multiply X additive_identity) additive_identity.
+∀H12:∀X:Univ.eq Univ (multiply additive_identity X) additive_identity.
+∀H13:∀X:Univ.eq Univ (add X additive_identity) X.
+∀H14:∀X:Univ.eq Univ (add additive_identity X) X.eq Univ (associator x y y) additive_identity
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#add.
+#additive_identity.
+#additive_inverse.
+#associator.
+#commutator.
+#multiply.
+#x.
+#y.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: RNG024-7.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG024-7 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Ring Theory (Alternative) *)
+
+(* Problem : Right alternative *)
+
+(* Version : [Ste87] (equality) axioms : Augmented. *)
+
+(* Theorem formulation : In terms of associators *)
+
+(* English : *)
+
+(* Refs : [Ste87] Stevens (1987), Some Experiments in Nonassociative Rin *)
+
+(* : [Ste92] Stevens (1992), Unpublished Note *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.2.1, 0.11 v2.2.0, 0.00 v2.1.0, 0.13 v2.0.0 *)
+
+(* Syntax : Number of clauses : 23 ( 0 non-Horn; 23 unit; 1 RR) *)
+
+(* Number of atoms : 23 ( 23 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 8 ( 3 constant; 0-3 arity) *)
+
+(* Number of variables : 45 ( 2 singleton) *)
+
+(* Maximal term depth : 5 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include nonassociative ring axioms *)
+
+(* Inclusion of: Axioms/RNG003-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG003-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Ring Theory (Alternative) *)
+
+(* Axioms : Alternative ring theory (equality) axioms *)
+
+(* Version : [Ste87] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Ste87] Stevens (1987), Some Experiments in Nonassociative Rin *)
+
+(* Source : [Ste87] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 15 ( 0 non-Horn; 15 unit; 0 RR) *)
+
+(* Number of literals : 15 ( 15 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 1 constant; 0-3 arity) *)
+
+(* Number of variables : 27 ( 2 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----There exists an additive identity element *)
+
+(* ----Multiplicative zero *)
+
+(* ----Existence of left additive additive_inverse *)
+
+(* ----Inverse of additive_inverse of X is X *)
+
+(* ----Distributive property of product over sum *)
+
+(* ----Commutativity for addition *)
+
+(* ----Associativity for addition *)
+
+(* ----Right alternative law *)
+
+(* ----Left alternative law *)
+
+(* ----Associator *)
+
+(* ----Commutator *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The next 7 clause are extra lemmas which Stevens found useful *)
+ntheorem prove_right_alternative:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀additive_identity:Univ.
+∀additive_inverse:∀_:Univ.Univ.
+∀associator:∀_:Univ.∀_:Univ.∀_:Univ.Univ.
+∀commutator:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀x:Univ.
+∀y:Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X Y) (additive_inverse Z)) (add (additive_inverse (multiply X Z)) (additive_inverse (multiply Y Z))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (additive_inverse X) (add Y Z)) (add (additive_inverse (multiply X Y)) (additive_inverse (multiply X Z))).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X (additive_inverse Y)) Z) (add (multiply X Z) (additive_inverse (multiply Y Z))).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y (additive_inverse Z))) (add (multiply X Y) (additive_inverse (multiply X Z))).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (multiply X (additive_inverse Y)) (additive_inverse (multiply X Y)).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (multiply (additive_inverse X) Y) (additive_inverse (multiply X Y)).
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (multiply (additive_inverse X) (additive_inverse Y)) (multiply X Y).
+∀H7:∀X:Univ.∀Y:Univ.eq Univ (commutator X Y) (add (multiply Y X) (additive_inverse (multiply X Y))).
+∀H8:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (associator X Y Z) (add (multiply (multiply X Y) Z) (additive_inverse (multiply X (multiply Y Z)))).
+∀H9:∀X:Univ.∀Y:Univ.eq Univ (multiply (multiply X X) Y) (multiply X (multiply X Y)).
+∀H10:∀X:Univ.∀Y:Univ.eq Univ (multiply (multiply X Y) Y) (multiply X (multiply Y Y)).
+∀H11:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add X (add Y Z)) (add (add X Y) Z).
+∀H12:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).
+∀H13:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X Y) Z) (add (multiply X Z) (multiply Y Z)).
+∀H14:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y Z)) (add (multiply X Y) (multiply X Z)).
+∀H15:∀X:Univ.eq Univ (additive_inverse (additive_inverse X)) X.
+∀H16:∀X:Univ.eq Univ (add X (additive_inverse X)) additive_identity.
+∀H17:∀X:Univ.eq Univ (add (additive_inverse X) X) additive_identity.
+∀H18:∀X:Univ.eq Univ (multiply X additive_identity) additive_identity.
+∀H19:∀X:Univ.eq Univ (multiply additive_identity X) additive_identity.
+∀H20:∀X:Univ.eq Univ (add X additive_identity) X.
+∀H21:∀X:Univ.eq Univ (add additive_identity X) X.eq Univ (associator x y y) additive_identity
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#add.
+#additive_identity.
+#additive_inverse.
+#associator.
+#commutator.
+#multiply.
+#x.
+#y.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+#H16.
+#H17.
+#H18.
+#H19.
+#H20.
+#H21.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15,H16,H17,H18,H19,H20,H21;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: RNG025-4.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG025-4 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Ring Theory (Alternative) *)
+
+(* Problem : Middle or Flexible Law *)
+
+(* Version : [Ste87] (equality) axioms. *)
+
+(* Theorem formulation : Linearized. *)
+
+(* English : *)
+
+(* Refs : [Ste87] Stevens (1987), Some Experiments in Nonassociative Rin *)
+
+(* : [Ste92] Stevens (1992), Unpublished Note *)
+
+(* Source : [Ste92] *)
+
+(* Names : - [Ste87] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.36 v3.2.0, 0.29 v3.1.0, 0.44 v2.7.0, 0.45 v2.6.0, 0.67 v2.5.0, 0.75 v2.4.0, 0.67 v2.2.1, 0.89 v2.2.0, 0.86 v2.1.0, 0.88 v2.0.0 *)
+
+(* Syntax : Number of clauses : 16 ( 0 non-Horn; 16 unit; 1 RR) *)
+
+(* Number of atoms : 16 ( 16 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 9 ( 4 constant; 0-3 arity) *)
+
+(* Number of variables : 27 ( 2 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include nonassociative ring axioms *)
+
+(* Inclusion of: Axioms/RNG003-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG003-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Ring Theory (Alternative) *)
+
+(* Axioms : Alternative ring theory (equality) axioms *)
+
+(* Version : [Ste87] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Ste87] Stevens (1987), Some Experiments in Nonassociative Rin *)
+
+(* Source : [Ste87] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 15 ( 0 non-Horn; 15 unit; 0 RR) *)
+
+(* Number of literals : 15 ( 15 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 1 constant; 0-3 arity) *)
+
+(* Number of variables : 27 ( 2 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----There exists an additive identity element *)
+
+(* ----Multiplicative zero *)
+
+(* ----Existence of left additive additive_inverse *)
+
+(* ----Inverse of additive_inverse of X is X *)
+
+(* ----Distributive property of product over sum *)
+
+(* ----Commutativity for addition *)
+
+(* ----Associativity for addition *)
+
+(* ----Right alternative law *)
+
+(* ----Left alternative law *)
+
+(* ----Associator *)
+
+(* ----Commutator *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_equation:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀additive_identity:Univ.
+∀additive_inverse:∀_:Univ.Univ.
+∀associator:∀_:Univ.∀_:Univ.∀_:Univ.Univ.
+∀commutator:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀x:Univ.
+∀y:Univ.
+∀z:Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (commutator X Y) (add (multiply Y X) (additive_inverse (multiply X Y))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (associator X Y Z) (add (multiply (multiply X Y) Z) (additive_inverse (multiply X (multiply Y Z)))).
+∀H2:∀X:Univ.∀Y:Univ.eq Univ (multiply (multiply X X) Y) (multiply X (multiply X Y)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (multiply (multiply X Y) Y) (multiply X (multiply Y Y)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add X (add Y Z)) (add (add X Y) Z).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).
+∀H6:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X Y) Z) (add (multiply X Z) (multiply Y Z)).
+∀H7:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y Z)) (add (multiply X Y) (multiply X Z)).
+∀H8:∀X:Univ.eq Univ (additive_inverse (additive_inverse X)) X.
+∀H9:∀X:Univ.eq Univ (add X (additive_inverse X)) additive_identity.
+∀H10:∀X:Univ.eq Univ (add (additive_inverse X) X) additive_identity.
+∀H11:∀X:Univ.eq Univ (multiply X additive_identity) additive_identity.
+∀H12:∀X:Univ.eq Univ (multiply additive_identity X) additive_identity.
+∀H13:∀X:Univ.eq Univ (add X additive_identity) X.
+∀H14:∀X:Univ.eq Univ (add additive_identity X) X.eq Univ (add (associator x y z) (associator x z y)) additive_identity
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#add.
+#additive_identity.
+#additive_inverse.
+#associator.
+#commutator.
+#multiply.
+#x.
+#y.
+#z.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: RNG025-5.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG025-5 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Ring Theory (Alternative) *)
+
+(* Problem : Middle or Flexible Law *)
+
+(* Version : [Ste87] (equality) axioms : Augmented. *)
+
+(* Theorem formulation : Linearized. *)
+
+(* English : *)
+
+(* Refs : [Ste87] Stevens (1987), Some Experiments in Nonassociative Rin *)
+
+(* : [Ste92] Stevens (1992), Unpublished Note *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.43 v3.2.0, 0.36 v3.1.0, 0.67 v2.7.0, 0.55 v2.6.0, 0.67 v2.5.0, 0.75 v2.4.0, 0.67 v2.2.1, 0.89 v2.2.0, 0.86 v2.1.0, 0.88 v2.0.0 *)
+
+(* Syntax : Number of clauses : 23 ( 0 non-Horn; 23 unit; 1 RR) *)
+
+(* Number of atoms : 23 ( 23 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 9 ( 4 constant; 0-3 arity) *)
+
+(* Number of variables : 45 ( 2 singleton) *)
+
+(* Maximal term depth : 5 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include nonassociative ring axioms *)
+
+(* Inclusion of: Axioms/RNG003-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG003-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Ring Theory (Alternative) *)
+
+(* Axioms : Alternative ring theory (equality) axioms *)
+
+(* Version : [Ste87] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Ste87] Stevens (1987), Some Experiments in Nonassociative Rin *)
+
+(* Source : [Ste87] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 15 ( 0 non-Horn; 15 unit; 0 RR) *)
+
+(* Number of literals : 15 ( 15 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 1 constant; 0-3 arity) *)
+
+(* Number of variables : 27 ( 2 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----There exists an additive identity element *)
+
+(* ----Multiplicative zero *)
+
+(* ----Existence of left additive additive_inverse *)
+
+(* ----Inverse of additive_inverse of X is X *)
+
+(* ----Distributive property of product over sum *)
+
+(* ----Commutativity for addition *)
+
+(* ----Associativity for addition *)
+
+(* ----Right alternative law *)
+
+(* ----Left alternative law *)
+
+(* ----Associator *)
+
+(* ----Commutator *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The next 7 clause are extra lemmas which Stevens found useful *)
+ntheorem prove_equation:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀additive_identity:Univ.
+∀additive_inverse:∀_:Univ.Univ.
+∀associator:∀_:Univ.∀_:Univ.∀_:Univ.Univ.
+∀commutator:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀x:Univ.
+∀y:Univ.
+∀z:Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X Y) (additive_inverse Z)) (add (additive_inverse (multiply X Z)) (additive_inverse (multiply Y Z))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (additive_inverse X) (add Y Z)) (add (additive_inverse (multiply X Y)) (additive_inverse (multiply X Z))).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X (additive_inverse Y)) Z) (add (multiply X Z) (additive_inverse (multiply Y Z))).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y (additive_inverse Z))) (add (multiply X Y) (additive_inverse (multiply X Z))).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (multiply X (additive_inverse Y)) (additive_inverse (multiply X Y)).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (multiply (additive_inverse X) Y) (additive_inverse (multiply X Y)).
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (multiply (additive_inverse X) (additive_inverse Y)) (multiply X Y).
+∀H7:∀X:Univ.∀Y:Univ.eq Univ (commutator X Y) (add (multiply Y X) (additive_inverse (multiply X Y))).
+∀H8:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (associator X Y Z) (add (multiply (multiply X Y) Z) (additive_inverse (multiply X (multiply Y Z)))).
+∀H9:∀X:Univ.∀Y:Univ.eq Univ (multiply (multiply X X) Y) (multiply X (multiply X Y)).
+∀H10:∀X:Univ.∀Y:Univ.eq Univ (multiply (multiply X Y) Y) (multiply X (multiply Y Y)).
+∀H11:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add X (add Y Z)) (add (add X Y) Z).
+∀H12:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).
+∀H13:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X Y) Z) (add (multiply X Z) (multiply Y Z)).
+∀H14:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y Z)) (add (multiply X Y) (multiply X Z)).
+∀H15:∀X:Univ.eq Univ (additive_inverse (additive_inverse X)) X.
+∀H16:∀X:Univ.eq Univ (add X (additive_inverse X)) additive_identity.
+∀H17:∀X:Univ.eq Univ (add (additive_inverse X) X) additive_identity.
+∀H18:∀X:Univ.eq Univ (multiply X additive_identity) additive_identity.
+∀H19:∀X:Univ.eq Univ (multiply additive_identity X) additive_identity.
+∀H20:∀X:Univ.eq Univ (add X additive_identity) X.
+∀H21:∀X:Univ.eq Univ (add additive_identity X) X.eq Univ (add (associator x y z) (associator x z y)) additive_identity
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#add.
+#additive_identity.
+#additive_inverse.
+#associator.
+#commutator.
+#multiply.
+#x.
+#y.
+#z.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+#H16.
+#H17.
+#H18.
+#H19.
+#H20.
+#H21.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15,H16,H17,H18,H19,H20,H21;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: RNG025-6.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG025-6 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Ring Theory (Alternative) *)
+
+(* Problem : Middle or Flexible Law *)
+
+(* Version : [Ste87] (equality) axioms. *)
+
+(* Theorem formulation : In terms of associators *)
+
+(* English : *)
+
+(* Refs : [Ste87] Stevens (1987), Some Experiments in Nonassociative Rin *)
+
+(* : [Ste92] Stevens (1992), Unpublished Note *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.29 v3.1.0, 0.44 v2.7.0, 0.36 v2.6.0, 0.67 v2.5.0, 0.75 v2.4.0, 0.33 v2.2.1, 0.89 v2.2.0, 0.86 v2.1.0, 0.88 v2.0.0 *)
+
+(* Syntax : Number of clauses : 16 ( 0 non-Horn; 16 unit; 1 RR) *)
+
+(* Number of atoms : 16 ( 16 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 8 ( 3 constant; 0-3 arity) *)
+
+(* Number of variables : 27 ( 2 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include nonassociative ring axioms *)
+
+(* Inclusion of: Axioms/RNG003-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG003-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Ring Theory (Alternative) *)
+
+(* Axioms : Alternative ring theory (equality) axioms *)
+
+(* Version : [Ste87] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Ste87] Stevens (1987), Some Experiments in Nonassociative Rin *)
+
+(* Source : [Ste87] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 15 ( 0 non-Horn; 15 unit; 0 RR) *)
+
+(* Number of literals : 15 ( 15 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 1 constant; 0-3 arity) *)
+
+(* Number of variables : 27 ( 2 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----There exists an additive identity element *)
+
+(* ----Multiplicative zero *)
+
+(* ----Existence of left additive additive_inverse *)
+
+(* ----Inverse of additive_inverse of X is X *)
+
+(* ----Distributive property of product over sum *)
+
+(* ----Commutativity for addition *)
+
+(* ----Associativity for addition *)
+
+(* ----Right alternative law *)
+
+(* ----Left alternative law *)
+
+(* ----Associator *)
+
+(* ----Commutator *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_flexible_law:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀additive_identity:Univ.
+∀additive_inverse:∀_:Univ.Univ.
+∀associator:∀_:Univ.∀_:Univ.∀_:Univ.Univ.
+∀commutator:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀x:Univ.
+∀y:Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (commutator X Y) (add (multiply Y X) (additive_inverse (multiply X Y))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (associator X Y Z) (add (multiply (multiply X Y) Z) (additive_inverse (multiply X (multiply Y Z)))).
+∀H2:∀X:Univ.∀Y:Univ.eq Univ (multiply (multiply X X) Y) (multiply X (multiply X Y)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (multiply (multiply X Y) Y) (multiply X (multiply Y Y)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add X (add Y Z)) (add (add X Y) Z).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).
+∀H6:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X Y) Z) (add (multiply X Z) (multiply Y Z)).
+∀H7:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y Z)) (add (multiply X Y) (multiply X Z)).
+∀H8:∀X:Univ.eq Univ (additive_inverse (additive_inverse X)) X.
+∀H9:∀X:Univ.eq Univ (add X (additive_inverse X)) additive_identity.
+∀H10:∀X:Univ.eq Univ (add (additive_inverse X) X) additive_identity.
+∀H11:∀X:Univ.eq Univ (multiply X additive_identity) additive_identity.
+∀H12:∀X:Univ.eq Univ (multiply additive_identity X) additive_identity.
+∀H13:∀X:Univ.eq Univ (add X additive_identity) X.
+∀H14:∀X:Univ.eq Univ (add additive_identity X) X.eq Univ (associator x y x) additive_identity
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#add.
+#additive_identity.
+#additive_inverse.
+#associator.
+#commutator.
+#multiply.
+#x.
+#y.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: RNG025-7.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG025-7 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Ring Theory (Alternative) *)
+
+(* Problem : Middle or Flexible Law *)
+
+(* Version : [Ste87] (equality) axioms : Augmented. *)
+
+(* Theorem formulation : In terms of associators *)
+
+(* English : *)
+
+(* Refs : [Ste87] Stevens (1987), Some Experiments in Nonassociative Rin *)
+
+(* : [Ste92] Stevens (1992), Unpublished Note *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.36 v3.2.0, 0.43 v3.1.0, 0.56 v2.7.0, 0.55 v2.6.0, 0.67 v2.5.0, 0.75 v2.4.0, 0.67 v2.2.1, 0.89 v2.2.0, 0.86 v2.1.0, 0.88 v2.0.0 *)
+
+(* Syntax : Number of clauses : 23 ( 0 non-Horn; 23 unit; 1 RR) *)
+
+(* Number of atoms : 23 ( 23 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 8 ( 3 constant; 0-3 arity) *)
+
+(* Number of variables : 45 ( 2 singleton) *)
+
+(* Maximal term depth : 5 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include nonassociative ring axioms *)
+
+(* Inclusion of: Axioms/RNG003-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG003-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Ring Theory (Alternative) *)
+
+(* Axioms : Alternative ring theory (equality) axioms *)
+
+(* Version : [Ste87] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Ste87] Stevens (1987), Some Experiments in Nonassociative Rin *)
+
+(* Source : [Ste87] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 15 ( 0 non-Horn; 15 unit; 0 RR) *)
+
+(* Number of literals : 15 ( 15 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 1 constant; 0-3 arity) *)
+
+(* Number of variables : 27 ( 2 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----There exists an additive identity element *)
+
+(* ----Multiplicative zero *)
+
+(* ----Existence of left additive additive_inverse *)
+
+(* ----Inverse of additive_inverse of X is X *)
+
+(* ----Distributive property of product over sum *)
+
+(* ----Commutativity for addition *)
+
+(* ----Associativity for addition *)
+
+(* ----Right alternative law *)
+
+(* ----Left alternative law *)
+
+(* ----Associator *)
+
+(* ----Commutator *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The next 7 clause are extra lemmas which Stevens found useful *)
+ntheorem prove_flexible_law:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀additive_identity:Univ.
+∀additive_inverse:∀_:Univ.Univ.
+∀associator:∀_:Univ.∀_:Univ.∀_:Univ.Univ.
+∀commutator:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀x:Univ.
+∀y:Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X Y) (additive_inverse Z)) (add (additive_inverse (multiply X Z)) (additive_inverse (multiply Y Z))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (additive_inverse X) (add Y Z)) (add (additive_inverse (multiply X Y)) (additive_inverse (multiply X Z))).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X (additive_inverse Y)) Z) (add (multiply X Z) (additive_inverse (multiply Y Z))).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y (additive_inverse Z))) (add (multiply X Y) (additive_inverse (multiply X Z))).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (multiply X (additive_inverse Y)) (additive_inverse (multiply X Y)).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (multiply (additive_inverse X) Y) (additive_inverse (multiply X Y)).
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (multiply (additive_inverse X) (additive_inverse Y)) (multiply X Y).
+∀H7:∀X:Univ.∀Y:Univ.eq Univ (commutator X Y) (add (multiply Y X) (additive_inverse (multiply X Y))).
+∀H8:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (associator X Y Z) (add (multiply (multiply X Y) Z) (additive_inverse (multiply X (multiply Y Z)))).
+∀H9:∀X:Univ.∀Y:Univ.eq Univ (multiply (multiply X X) Y) (multiply X (multiply X Y)).
+∀H10:∀X:Univ.∀Y:Univ.eq Univ (multiply (multiply X Y) Y) (multiply X (multiply Y Y)).
+∀H11:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add X (add Y Z)) (add (add X Y) Z).
+∀H12:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).
+∀H13:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X Y) Z) (add (multiply X Z) (multiply Y Z)).
+∀H14:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y Z)) (add (multiply X Y) (multiply X Z)).
+∀H15:∀X:Univ.eq Univ (additive_inverse (additive_inverse X)) X.
+∀H16:∀X:Univ.eq Univ (add X (additive_inverse X)) additive_identity.
+∀H17:∀X:Univ.eq Univ (add (additive_inverse X) X) additive_identity.
+∀H18:∀X:Univ.eq Univ (multiply X additive_identity) additive_identity.
+∀H19:∀X:Univ.eq Univ (multiply additive_identity X) additive_identity.
+∀H20:∀X:Univ.eq Univ (add X additive_identity) X.
+∀H21:∀X:Univ.eq Univ (add additive_identity X) X.eq Univ (associator x y x) additive_identity
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#add.
+#additive_identity.
+#additive_inverse.
+#associator.
+#commutator.
+#multiply.
+#x.
+#y.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+#H16.
+#H17.
+#H18.
+#H19.
+#H20.
+#H21.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15,H16,H17,H18,H19,H20,H21;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: RNG025-8.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG025-8 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Ring Theory (Alternative) *)
+
+(* Problem : Middle or Flexible Law *)
+
+(* Version : [Ste87] (equality) axioms : Reduced & Augmented > Complete. *)
+
+(* Theorem formulation : Linearized. *)
+
+(* English : *)
+
+(* Refs : [Ste87] Stevens (1987), Some Experiments in Nonassociative Rin *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Satisfiable *)
+
+(* Rating : 0.33 v3.2.0, 0.67 v3.1.0, 0.33 v2.4.0, 0.67 v2.2.1, 0.75 v2.2.0, 0.67 v2.1.0, 1.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 18 ( 0 non-Horn; 18 unit; 1 RR) *)
+
+(* Number of atoms : 18 ( 18 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 9 ( 4 constant; 0-3 arity) *)
+
+(* Number of variables : 36 ( 2 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Don't Include nonassociative ring axioms. *)
+
+(* ----The associator has to be replaced by its linearised form. *)
+
+(* include('axioms/RNG003-0.ax'). *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Commutativity for addition *)
+
+(* ----Associativity for addition *)
+
+(* ----There exists an additive identity element *)
+
+(* ----Multiplicative zero *)
+
+(* ----Existence of left additive additive_inverse *)
+
+(* ----Distributive property of product over sum *)
+
+(* ----Inverse of additive_inverse of X is X *)
+
+(* ----Right alternative law *)
+
+(* ----Left alternative law *)
+
+(* ----Associator *)
+
+(* input_clause(associator,axiom, *)
+
+(* [++equal(associator(X,Y,Z),add(multiply(multiply(X,Y),Z), *)
+
+(* additive_inverse(multiply(X,multiply(Y,Z)))))]). *)
+
+(* ----Linearised for of the associator *)
+
+(* ----Commutator *)
+ntheorem prove_flexible_law:
+ ∀Univ:Type.∀U:Univ.∀V:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀additive_identity:Univ.
+∀additive_inverse:∀_:Univ.Univ.
+∀associator:∀_:Univ.∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀c:Univ.
+∀commutator:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (commutator X Y) (add (multiply Y X) (additive_inverse (multiply X Y))).
+∀H1:∀U:Univ.∀V:Univ.∀X:Univ.∀Y:Univ.eq Univ (associator (add U V) X Y) (add (associator U X Y) (associator V X Y)).
+∀H2:∀U:Univ.∀V:Univ.∀X:Univ.∀Y:Univ.eq Univ (associator X (add U V) Y) (add (associator X U Y) (associator X V Y)).
+∀H3:∀U:Univ.∀V:Univ.∀X:Univ.∀Y:Univ.eq Univ (associator X Y (add U V)) (add (associator X Y U) (associator X Y V)).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (multiply (multiply X X) Y) (multiply X (multiply X Y)).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (multiply (multiply X Y) Y) (multiply X (multiply Y Y)).
+∀H6:∀X:Univ.eq Univ (additive_inverse (additive_inverse X)) X.
+∀H7:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X Y) Z) (add (multiply X Z) (multiply Y Z)).
+∀H8:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y Z)) (add (multiply X Y) (multiply X Z)).
+∀H9:∀X:Univ.eq Univ (add X (additive_inverse X)) additive_identity.
+∀H10:∀X:Univ.eq Univ (add (additive_inverse X) X) additive_identity.
+∀H11:∀X:Univ.eq Univ (multiply X additive_identity) additive_identity.
+∀H12:∀X:Univ.eq Univ (multiply additive_identity X) additive_identity.
+∀H13:∀X:Univ.eq Univ (add X additive_identity) X.
+∀H14:∀X:Univ.eq Univ (add additive_identity X) X.
+∀H15:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add X (add Y Z)) (add (add X Y) Z).
+∀H16:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).eq Univ (add (associator a b c) (associator a c b)) additive_identity
+.
+#Univ.
+#U.
+#V.
+#X.
+#Y.
+#Z.
+#a.
+#add.
+#additive_identity.
+#additive_inverse.
+#associator.
+#b.
+#c.
+#commutator.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+#H16.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15,H16;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: RNG025-9.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG025-9 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Ring Theory (Alternative) *)
+
+(* Problem : Middle or Flexible Law *)
+
+(* Version : [Ste87] (equality) axioms : Biased. *)
+
+(* Theorem formulation : Linearized. *)
+
+(* English : *)
+
+(* Refs : [Ste87] Stevens (1987), Some Experiments in Nonassociative Rin *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Satisfiable *)
+
+(* Rating : 0.33 v3.2.0, 0.67 v3.1.0, 0.33 v2.5.0, 0.67 v2.4.0, 0.67 v2.2.1, 0.75 v2.2.0, 0.67 v2.1.0, 1.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 25 ( 0 non-Horn; 25 unit; 1 RR) *)
+
+(* Number of atoms : 25 ( 25 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 9 ( 4 constant; 0-3 arity) *)
+
+(* Number of variables : 54 ( 2 singleton) *)
+
+(* Maximal term depth : 4 ( 3 average) *)
+
+(* Comments : Biased towards Otter. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Don't Include nonassociative ring axioms. *)
+
+(* ----The associator has to be replaced by its linearised form. *)
+
+(* include('axioms/RNG003-0.ax'). *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The next 7 clause are extra lemmas which Stevens found useful *)
+
+(* ----Commutativity for addition *)
+
+(* ----Associativity for addition *)
+
+(* ----There exists an additive identity element *)
+
+(* ----Multiplicative zero *)
+
+(* ----Existence of left additive additive_inverse *)
+
+(* ----Distributive property of product over sum *)
+
+(* ----Inverse of additive_inverse of X is X *)
+
+(* ----Right alternative law *)
+
+(* ----Left alternative law *)
+
+(* ----Associator *)
+
+(* input_clause(associator,axiom, *)
+
+(* [++equal(associator(X,Y,Z),add(multiply(multiply(X,Y),Z), *)
+
+(* additive_inverse(multiply(X,multiply(Y,Z)))))]). *)
+
+(* ----Linearised for of the associator *)
+
+(* ----Commutator *)
+ntheorem prove_flexible_law:
+ ∀Univ:Type.∀U:Univ.∀V:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀additive_identity:Univ.
+∀additive_inverse:∀_:Univ.Univ.
+∀associator:∀_:Univ.∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀c:Univ.
+∀commutator:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (commutator X Y) (add (multiply Y X) (additive_inverse (multiply X Y))).
+∀H1:∀U:Univ.∀V:Univ.∀X:Univ.∀Y:Univ.eq Univ (associator (add U V) X Y) (add (associator U X Y) (associator V X Y)).
+∀H2:∀U:Univ.∀V:Univ.∀X:Univ.∀Y:Univ.eq Univ (associator X (add U V) Y) (add (associator X U Y) (associator X V Y)).
+∀H3:∀U:Univ.∀V:Univ.∀X:Univ.∀Y:Univ.eq Univ (associator X Y (add U V)) (add (associator X Y U) (associator X Y V)).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (multiply (multiply X X) Y) (multiply X (multiply X Y)).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (multiply (multiply X Y) Y) (multiply X (multiply Y Y)).
+∀H6:∀X:Univ.eq Univ (additive_inverse (additive_inverse X)) X.
+∀H7:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X Y) Z) (add (multiply X Z) (multiply Y Z)).
+∀H8:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y Z)) (add (multiply X Y) (multiply X Z)).
+∀H9:∀X:Univ.eq Univ (add X (additive_inverse X)) additive_identity.
+∀H10:∀X:Univ.eq Univ (add (additive_inverse X) X) additive_identity.
+∀H11:∀X:Univ.eq Univ (multiply X additive_identity) additive_identity.
+∀H12:∀X:Univ.eq Univ (multiply additive_identity X) additive_identity.
+∀H13:∀X:Univ.eq Univ (add X additive_identity) X.
+∀H14:∀X:Univ.eq Univ (add additive_identity X) X.
+∀H15:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add X (add Y Z)) (add (add X Y) Z).
+∀H16:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).
+∀H17:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X Y) (additive_inverse Z)) (add (additive_inverse (multiply X Z)) (additive_inverse (multiply Y Z))).
+∀H18:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (additive_inverse X) (add Y Z)) (add (additive_inverse (multiply X Y)) (additive_inverse (multiply X Z))).
+∀H19:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X (additive_inverse Y)) Z) (add (multiply X Z) (additive_inverse (multiply Y Z))).
+∀H20:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y (additive_inverse Z))) (add (multiply X Y) (additive_inverse (multiply X Z))).
+∀H21:∀X:Univ.∀Y:Univ.eq Univ (multiply X (additive_inverse Y)) (additive_inverse (multiply X Y)).
+∀H22:∀X:Univ.∀Y:Univ.eq Univ (multiply (additive_inverse X) Y) (additive_inverse (multiply X Y)).
+∀H23:∀X:Univ.∀Y:Univ.eq Univ (multiply (additive_inverse X) (additive_inverse Y)) (multiply X Y).eq Univ (add (associator a b c) (associator a c b)) additive_identity
+.
+#Univ.
+#U.
+#V.
+#X.
+#Y.
+#Z.
+#a.
+#add.
+#additive_identity.
+#additive_inverse.
+#associator.
+#b.
+#c.
+#commutator.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+#H16.
+#H17.
+#H18.
+#H19.
+#H20.
+#H21.
+#H22.
+#H23.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15,H16,H17,H18,H19,H20,H21,H22,H23;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: RNG026-6.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG026-6 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Ring Theory (Alternative) *)
+
+(* Problem : Teichmuller Identity *)
+
+(* Version : [Ste87] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Ste87] Stevens (1987), Some Experiments in Nonassociative Rin *)
+
+(* Source : [Ste87] *)
+
+(* Names : Teichmuller Identity [Ste87] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.50 v3.2.0, 0.57 v3.1.0, 0.33 v2.7.0, 0.64 v2.6.0, 0.50 v2.5.0, 0.25 v2.4.0, 0.67 v2.2.1, 0.89 v2.2.0, 0.86 v2.1.0, 0.88 v2.0.0 *)
+
+(* Syntax : Number of clauses : 16 ( 0 non-Horn; 16 unit; 1 RR) *)
+
+(* Number of atoms : 16 ( 16 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 10 ( 5 constant; 0-3 arity) *)
+
+(* Number of variables : 27 ( 2 singleton) *)
+
+(* Maximal term depth : 7 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include nonassociative ring axioms *)
+
+(* Inclusion of: Axioms/RNG003-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG003-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Ring Theory (Alternative) *)
+
+(* Axioms : Alternative ring theory (equality) axioms *)
+
+(* Version : [Ste87] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Ste87] Stevens (1987), Some Experiments in Nonassociative Rin *)
+
+(* Source : [Ste87] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 15 ( 0 non-Horn; 15 unit; 0 RR) *)
+
+(* Number of literals : 15 ( 15 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 1 constant; 0-3 arity) *)
+
+(* Number of variables : 27 ( 2 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----There exists an additive identity element *)
+
+(* ----Multiplicative zero *)
+
+(* ----Existence of left additive additive_inverse *)
+
+(* ----Inverse of additive_inverse of X is X *)
+
+(* ----Distributive property of product over sum *)
+
+(* ----Commutativity for addition *)
+
+(* ----Associativity for addition *)
+
+(* ----Right alternative law *)
+
+(* ----Left alternative law *)
+
+(* ----Associator *)
+
+(* ----Commutator *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_teichmuller_identity:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀additive_identity:Univ.
+∀additive_inverse:∀_:Univ.Univ.
+∀associator:∀_:Univ.∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀c:Univ.
+∀commutator:∀_:Univ.∀_:Univ.Univ.
+∀d:Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (commutator X Y) (add (multiply Y X) (additive_inverse (multiply X Y))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (associator X Y Z) (add (multiply (multiply X Y) Z) (additive_inverse (multiply X (multiply Y Z)))).
+∀H2:∀X:Univ.∀Y:Univ.eq Univ (multiply (multiply X X) Y) (multiply X (multiply X Y)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (multiply (multiply X Y) Y) (multiply X (multiply Y Y)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add X (add Y Z)) (add (add X Y) Z).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).
+∀H6:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X Y) Z) (add (multiply X Z) (multiply Y Z)).
+∀H7:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y Z)) (add (multiply X Y) (multiply X Z)).
+∀H8:∀X:Univ.eq Univ (additive_inverse (additive_inverse X)) X.
+∀H9:∀X:Univ.eq Univ (add X (additive_inverse X)) additive_identity.
+∀H10:∀X:Univ.eq Univ (add (additive_inverse X) X) additive_identity.
+∀H11:∀X:Univ.eq Univ (multiply X additive_identity) additive_identity.
+∀H12:∀X:Univ.eq Univ (multiply additive_identity X) additive_identity.
+∀H13:∀X:Univ.eq Univ (add X additive_identity) X.
+∀H14:∀X:Univ.eq Univ (add additive_identity X) X.eq Univ (add (add (associator (multiply a b) c d) (associator a b (multiply c d))) (additive_inverse (add (add (associator a (multiply b c) d) (multiply a (associator b c d))) (multiply (associator a b c) d)))) additive_identity
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#add.
+#additive_identity.
+#additive_inverse.
+#associator.
+#b.
+#c.
+#commutator.
+#d.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: RNG026-7.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG026-7 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Ring Theory (Alternative) *)
+
+(* Problem : Teichmuller Identity *)
+
+(* Version : [Ste87] (equality) axioms : Augmented. *)
+
+(* English : *)
+
+(* Refs : [Ste87] Stevens (1987), Some Experiments in Nonassociative Rin *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.43 v3.2.0, 0.50 v3.1.0, 0.33 v2.7.0, 0.64 v2.6.0, 0.50 v2.5.0, 0.25 v2.4.0, 0.67 v2.2.1, 0.89 v2.2.0, 0.86 v2.1.0, 1.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 23 ( 0 non-Horn; 23 unit; 1 RR) *)
+
+(* Number of atoms : 23 ( 23 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 10 ( 5 constant; 0-3 arity) *)
+
+(* Number of variables : 45 ( 2 singleton) *)
+
+(* Maximal term depth : 7 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include nonassociative ring axioms *)
+
+(* Inclusion of: Axioms/RNG003-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG003-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Ring Theory (Alternative) *)
+
+(* Axioms : Alternative ring theory (equality) axioms *)
+
+(* Version : [Ste87] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Ste87] Stevens (1987), Some Experiments in Nonassociative Rin *)
+
+(* Source : [Ste87] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 15 ( 0 non-Horn; 15 unit; 0 RR) *)
+
+(* Number of literals : 15 ( 15 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 1 constant; 0-3 arity) *)
+
+(* Number of variables : 27 ( 2 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----There exists an additive identity element *)
+
+(* ----Multiplicative zero *)
+
+(* ----Existence of left additive additive_inverse *)
+
+(* ----Inverse of additive_inverse of X is X *)
+
+(* ----Distributive property of product over sum *)
+
+(* ----Commutativity for addition *)
+
+(* ----Associativity for addition *)
+
+(* ----Right alternative law *)
+
+(* ----Left alternative law *)
+
+(* ----Associator *)
+
+(* ----Commutator *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The next 7 clause are extra lemmas which Stevens found useful *)
+ntheorem prove_teichmuller_identity:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀additive_identity:Univ.
+∀additive_inverse:∀_:Univ.Univ.
+∀associator:∀_:Univ.∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀c:Univ.
+∀commutator:∀_:Univ.∀_:Univ.Univ.
+∀d:Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X Y) (additive_inverse Z)) (add (additive_inverse (multiply X Z)) (additive_inverse (multiply Y Z))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (additive_inverse X) (add Y Z)) (add (additive_inverse (multiply X Y)) (additive_inverse (multiply X Z))).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X (additive_inverse Y)) Z) (add (multiply X Z) (additive_inverse (multiply Y Z))).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y (additive_inverse Z))) (add (multiply X Y) (additive_inverse (multiply X Z))).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (multiply X (additive_inverse Y)) (additive_inverse (multiply X Y)).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (multiply (additive_inverse X) Y) (additive_inverse (multiply X Y)).
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (multiply (additive_inverse X) (additive_inverse Y)) (multiply X Y).
+∀H7:∀X:Univ.∀Y:Univ.eq Univ (commutator X Y) (add (multiply Y X) (additive_inverse (multiply X Y))).
+∀H8:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (associator X Y Z) (add (multiply (multiply X Y) Z) (additive_inverse (multiply X (multiply Y Z)))).
+∀H9:∀X:Univ.∀Y:Univ.eq Univ (multiply (multiply X X) Y) (multiply X (multiply X Y)).
+∀H10:∀X:Univ.∀Y:Univ.eq Univ (multiply (multiply X Y) Y) (multiply X (multiply Y Y)).
+∀H11:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add X (add Y Z)) (add (add X Y) Z).
+∀H12:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).
+∀H13:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X Y) Z) (add (multiply X Z) (multiply Y Z)).
+∀H14:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y Z)) (add (multiply X Y) (multiply X Z)).
+∀H15:∀X:Univ.eq Univ (additive_inverse (additive_inverse X)) X.
+∀H16:∀X:Univ.eq Univ (add X (additive_inverse X)) additive_identity.
+∀H17:∀X:Univ.eq Univ (add (additive_inverse X) X) additive_identity.
+∀H18:∀X:Univ.eq Univ (multiply X additive_identity) additive_identity.
+∀H19:∀X:Univ.eq Univ (multiply additive_identity X) additive_identity.
+∀H20:∀X:Univ.eq Univ (add X additive_identity) X.
+∀H21:∀X:Univ.eq Univ (add additive_identity X) X.eq Univ (add (add (associator (multiply a b) c d) (associator a b (multiply c d))) (additive_inverse (add (add (associator a (multiply b c) d) (multiply a (associator b c d))) (multiply (associator a b c) d)))) additive_identity
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#add.
+#additive_identity.
+#additive_inverse.
+#associator.
+#b.
+#c.
+#commutator.
+#d.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+#H16.
+#H17.
+#H18.
+#H19.
+#H20.
+#H21.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15,H16,H17,H18,H19,H20,H21;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: RNG027-5.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG027-5 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Ring Theory (Alternative) *)
+
+(* Problem : Right Moufang identity *)
+
+(* Version : [Ste87] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Ste87] Stevens (1987), Some Experiments in Nonassociative Rin *)
+
+(* : [Ste88] Stevens (1988), Challenge Problems from Nonassociative *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.86 v3.1.0, 0.89 v2.7.0, 0.91 v2.6.0, 0.83 v2.5.0, 0.75 v2.4.0, 0.67 v2.2.1, 1.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 16 ( 0 non-Horn; 16 unit; 1 RR) *)
+
+(* Number of atoms : 16 ( 16 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 9 ( 4 constant; 0-3 arity) *)
+
+(* Number of variables : 27 ( 2 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : [Ste88] presents a slightly different set of axioms for proving *)
+
+(* this ntheorem. The axioms are so similar to those in RNG004.ax *)
+
+(* that a separate problems has not been created. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include nonassociative ring axioms *)
+
+(* Inclusion of: Axioms/RNG003-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG003-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Ring Theory (Alternative) *)
+
+(* Axioms : Alternative ring theory (equality) axioms *)
+
+(* Version : [Ste87] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Ste87] Stevens (1987), Some Experiments in Nonassociative Rin *)
+
+(* Source : [Ste87] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 15 ( 0 non-Horn; 15 unit; 0 RR) *)
+
+(* Number of literals : 15 ( 15 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 1 constant; 0-3 arity) *)
+
+(* Number of variables : 27 ( 2 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----There exists an additive identity element *)
+
+(* ----Multiplicative zero *)
+
+(* ----Existence of left additive additive_inverse *)
+
+(* ----Inverse of additive_inverse of X is X *)
+
+(* ----Distributive property of product over sum *)
+
+(* ----Commutativity for addition *)
+
+(* ----Associativity for addition *)
+
+(* ----Right alternative law *)
+
+(* ----Left alternative law *)
+
+(* ----Associator *)
+
+(* ----Commutator *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_right_moufang:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀additive_identity:Univ.
+∀additive_inverse:∀_:Univ.Univ.
+∀associator:∀_:Univ.∀_:Univ.∀_:Univ.Univ.
+∀commutator:∀_:Univ.∀_:Univ.Univ.
+∀cx:Univ.
+∀cy:Univ.
+∀cz:Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (commutator X Y) (add (multiply Y X) (additive_inverse (multiply X Y))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (associator X Y Z) (add (multiply (multiply X Y) Z) (additive_inverse (multiply X (multiply Y Z)))).
+∀H2:∀X:Univ.∀Y:Univ.eq Univ (multiply (multiply X X) Y) (multiply X (multiply X Y)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (multiply (multiply X Y) Y) (multiply X (multiply Y Y)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add X (add Y Z)) (add (add X Y) Z).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).
+∀H6:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X Y) Z) (add (multiply X Z) (multiply Y Z)).
+∀H7:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y Z)) (add (multiply X Y) (multiply X Z)).
+∀H8:∀X:Univ.eq Univ (additive_inverse (additive_inverse X)) X.
+∀H9:∀X:Univ.eq Univ (add X (additive_inverse X)) additive_identity.
+∀H10:∀X:Univ.eq Univ (add (additive_inverse X) X) additive_identity.
+∀H11:∀X:Univ.eq Univ (multiply X additive_identity) additive_identity.
+∀H12:∀X:Univ.eq Univ (multiply additive_identity X) additive_identity.
+∀H13:∀X:Univ.eq Univ (add X additive_identity) X.
+∀H14:∀X:Univ.eq Univ (add additive_identity X) X.eq Univ (multiply cz (multiply cx (multiply cy cx))) (multiply (multiply (multiply cz cx) cy) cx)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#add.
+#additive_identity.
+#additive_inverse.
+#associator.
+#commutator.
+#cx.
+#cy.
+#cz.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: RNG027-7.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG027-7 : TPTP v3.2.0. Bugfixed v2.3.0. *)
+
+(* Domain : Ring Theory (Alternative) *)
+
+(* Problem : Right Moufang identity *)
+
+(* Version : [Ste87] (equality) axioms : Augmented. *)
+
+(* Theorem formulation : In terms of associators *)
+
+(* English : *)
+
+(* Refs : [Ste87] Stevens (1987), Some Experiments in Nonassociative Rin *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.86 v3.1.0, 0.89 v2.7.0, 0.91 v2.6.0, 0.83 v2.5.0, 0.75 v2.4.0, 0.67 v2.3.0 *)
+
+(* Syntax : Number of clauses : 23 ( 0 non-Horn; 23 unit; 1 RR) *)
+
+(* Number of atoms : 23 ( 23 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 9 ( 4 constant; 0-3 arity) *)
+
+(* Number of variables : 45 ( 2 singleton) *)
+
+(* Maximal term depth : 5 ( 3 average) *)
+
+(* Comments : *)
+
+(* Bugfixes : v2.3.0 - Clause prove_right_moufang fixed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include nonassociative ring axioms *)
+
+(* Inclusion of: Axioms/RNG003-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG003-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Ring Theory (Alternative) *)
+
+(* Axioms : Alternative ring theory (equality) axioms *)
+
+(* Version : [Ste87] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Ste87] Stevens (1987), Some Experiments in Nonassociative Rin *)
+
+(* Source : [Ste87] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 15 ( 0 non-Horn; 15 unit; 0 RR) *)
+
+(* Number of literals : 15 ( 15 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 1 constant; 0-3 arity) *)
+
+(* Number of variables : 27 ( 2 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----There exists an additive identity element *)
+
+(* ----Multiplicative zero *)
+
+(* ----Existence of left additive additive_inverse *)
+
+(* ----Inverse of additive_inverse of X is X *)
+
+(* ----Distributive property of product over sum *)
+
+(* ----Commutativity for addition *)
+
+(* ----Associativity for addition *)
+
+(* ----Right alternative law *)
+
+(* ----Left alternative law *)
+
+(* ----Associator *)
+
+(* ----Commutator *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The next 7 clause are extra lemmas which Stevens found useful *)
+ntheorem prove_right_moufang:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀additive_identity:Univ.
+∀additive_inverse:∀_:Univ.Univ.
+∀associator:∀_:Univ.∀_:Univ.∀_:Univ.Univ.
+∀commutator:∀_:Univ.∀_:Univ.Univ.
+∀cx:Univ.
+∀cy:Univ.
+∀cz:Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X Y) (additive_inverse Z)) (add (additive_inverse (multiply X Z)) (additive_inverse (multiply Y Z))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (additive_inverse X) (add Y Z)) (add (additive_inverse (multiply X Y)) (additive_inverse (multiply X Z))).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X (additive_inverse Y)) Z) (add (multiply X Z) (additive_inverse (multiply Y Z))).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y (additive_inverse Z))) (add (multiply X Y) (additive_inverse (multiply X Z))).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (multiply X (additive_inverse Y)) (additive_inverse (multiply X Y)).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (multiply (additive_inverse X) Y) (additive_inverse (multiply X Y)).
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (multiply (additive_inverse X) (additive_inverse Y)) (multiply X Y).
+∀H7:∀X:Univ.∀Y:Univ.eq Univ (commutator X Y) (add (multiply Y X) (additive_inverse (multiply X Y))).
+∀H8:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (associator X Y Z) (add (multiply (multiply X Y) Z) (additive_inverse (multiply X (multiply Y Z)))).
+∀H9:∀X:Univ.∀Y:Univ.eq Univ (multiply (multiply X X) Y) (multiply X (multiply X Y)).
+∀H10:∀X:Univ.∀Y:Univ.eq Univ (multiply (multiply X Y) Y) (multiply X (multiply Y Y)).
+∀H11:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add X (add Y Z)) (add (add X Y) Z).
+∀H12:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).
+∀H13:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X Y) Z) (add (multiply X Z) (multiply Y Z)).
+∀H14:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y Z)) (add (multiply X Y) (multiply X Z)).
+∀H15:∀X:Univ.eq Univ (additive_inverse (additive_inverse X)) X.
+∀H16:∀X:Univ.eq Univ (add X (additive_inverse X)) additive_identity.
+∀H17:∀X:Univ.eq Univ (add (additive_inverse X) X) additive_identity.
+∀H18:∀X:Univ.eq Univ (multiply X additive_identity) additive_identity.
+∀H19:∀X:Univ.eq Univ (multiply additive_identity X) additive_identity.
+∀H20:∀X:Univ.eq Univ (add X additive_identity) X.
+∀H21:∀X:Univ.eq Univ (add additive_identity X) X.eq Univ (multiply cz (multiply cx (multiply cy cx))) (multiply (multiply (multiply cz cx) cy) cx)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#add.
+#additive_identity.
+#additive_inverse.
+#associator.
+#commutator.
+#cx.
+#cy.
+#cz.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+#H16.
+#H17.
+#H18.
+#H19.
+#H20.
+#H21.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15,H16,H17,H18,H19,H20,H21;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: RNG027-8.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG027-8 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Ring Theory (Alternative) *)
+
+(* Problem : Right Moufang identity *)
+
+(* Version : [Ste87] (equality) axioms. *)
+
+(* Theorem formulation : Associators. *)
+
+(* English : *)
+
+(* Refs : [Ste87] Stevens (1987), Some Experiments in Nonassociative Rin *)
+
+(* : [Ste88] Stevens (1988), Challenge Problems from Nonassociative *)
+
+(* Source : [Ste87] *)
+
+(* Names : m1' [Ste87] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.86 v3.1.0, 0.89 v2.7.0, 0.91 v2.6.0, 0.83 v2.5.0, 0.75 v2.4.0, 0.67 v2.2.1, 1.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 16 ( 0 non-Horn; 16 unit; 1 RR) *)
+
+(* Number of atoms : 16 ( 16 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 9 ( 4 constant; 0-3 arity) *)
+
+(* Number of variables : 27 ( 2 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : [Ste88] presents a slightly different set of axioms for proving *)
+
+(* this ntheorem. The axioms are so similar to those in RNG004.ax *)
+
+(* that a separate problems has not been created. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include nonassociative ring axioms *)
+
+(* Inclusion of: Axioms/RNG003-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG003-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Ring Theory (Alternative) *)
+
+(* Axioms : Alternative ring theory (equality) axioms *)
+
+(* Version : [Ste87] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Ste87] Stevens (1987), Some Experiments in Nonassociative Rin *)
+
+(* Source : [Ste87] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 15 ( 0 non-Horn; 15 unit; 0 RR) *)
+
+(* Number of literals : 15 ( 15 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 1 constant; 0-3 arity) *)
+
+(* Number of variables : 27 ( 2 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----There exists an additive identity element *)
+
+(* ----Multiplicative zero *)
+
+(* ----Existence of left additive additive_inverse *)
+
+(* ----Inverse of additive_inverse of X is X *)
+
+(* ----Distributive property of product over sum *)
+
+(* ----Commutativity for addition *)
+
+(* ----Associativity for addition *)
+
+(* ----Right alternative law *)
+
+(* ----Left alternative law *)
+
+(* ----Associator *)
+
+(* ----Commutator *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_right_moufang:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀additive_identity:Univ.
+∀additive_inverse:∀_:Univ.Univ.
+∀associator:∀_:Univ.∀_:Univ.∀_:Univ.Univ.
+∀commutator:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀x:Univ.
+∀y:Univ.
+∀z:Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (commutator X Y) (add (multiply Y X) (additive_inverse (multiply X Y))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (associator X Y Z) (add (multiply (multiply X Y) Z) (additive_inverse (multiply X (multiply Y Z)))).
+∀H2:∀X:Univ.∀Y:Univ.eq Univ (multiply (multiply X X) Y) (multiply X (multiply X Y)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (multiply (multiply X Y) Y) (multiply X (multiply Y Y)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add X (add Y Z)) (add (add X Y) Z).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).
+∀H6:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X Y) Z) (add (multiply X Z) (multiply Y Z)).
+∀H7:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y Z)) (add (multiply X Y) (multiply X Z)).
+∀H8:∀X:Univ.eq Univ (additive_inverse (additive_inverse X)) X.
+∀H9:∀X:Univ.eq Univ (add X (additive_inverse X)) additive_identity.
+∀H10:∀X:Univ.eq Univ (add (additive_inverse X) X) additive_identity.
+∀H11:∀X:Univ.eq Univ (multiply X additive_identity) additive_identity.
+∀H12:∀X:Univ.eq Univ (multiply additive_identity X) additive_identity.
+∀H13:∀X:Univ.eq Univ (add X additive_identity) X.
+∀H14:∀X:Univ.eq Univ (add additive_identity X) X.eq Univ (associator x (multiply x y) z) (multiply (associator x y z) x)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#add.
+#additive_identity.
+#additive_inverse.
+#associator.
+#commutator.
+#multiply.
+#x.
+#y.
+#z.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: RNG027-9.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG027-9 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Ring Theory (Alternative) *)
+
+(* Problem : Right Moufang identity *)
+
+(* Version : [Ste87] (equality) axioms : Augmented. *)
+
+(* Theorem formulation : Associators. *)
+
+(* English : *)
+
+(* Refs : [Ste87] Stevens (1987), Some Experiments in Nonassociative Rin *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.86 v3.1.0, 0.89 v2.7.0, 0.91 v2.6.0, 0.83 v2.5.0, 0.75 v2.4.0, 0.67 v2.2.1, 1.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 23 ( 0 non-Horn; 23 unit; 1 RR) *)
+
+(* Number of atoms : 23 ( 23 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 9 ( 4 constant; 0-3 arity) *)
+
+(* Number of variables : 45 ( 2 singleton) *)
+
+(* Maximal term depth : 5 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include nonassociative ring axioms *)
+
+(* Inclusion of: Axioms/RNG003-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG003-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Ring Theory (Alternative) *)
+
+(* Axioms : Alternative ring theory (equality) axioms *)
+
+(* Version : [Ste87] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Ste87] Stevens (1987), Some Experiments in Nonassociative Rin *)
+
+(* Source : [Ste87] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 15 ( 0 non-Horn; 15 unit; 0 RR) *)
+
+(* Number of literals : 15 ( 15 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 1 constant; 0-3 arity) *)
+
+(* Number of variables : 27 ( 2 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----There exists an additive identity element *)
+
+(* ----Multiplicative zero *)
+
+(* ----Existence of left additive additive_inverse *)
+
+(* ----Inverse of additive_inverse of X is X *)
+
+(* ----Distributive property of product over sum *)
+
+(* ----Commutativity for addition *)
+
+(* ----Associativity for addition *)
+
+(* ----Right alternative law *)
+
+(* ----Left alternative law *)
+
+(* ----Associator *)
+
+(* ----Commutator *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The next 7 clause are extra lemmas which Stevens found useful *)
+ntheorem prove_right_moufang:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀additive_identity:Univ.
+∀additive_inverse:∀_:Univ.Univ.
+∀associator:∀_:Univ.∀_:Univ.∀_:Univ.Univ.
+∀commutator:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀x:Univ.
+∀y:Univ.
+∀z:Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X Y) (additive_inverse Z)) (add (additive_inverse (multiply X Z)) (additive_inverse (multiply Y Z))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (additive_inverse X) (add Y Z)) (add (additive_inverse (multiply X Y)) (additive_inverse (multiply X Z))).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X (additive_inverse Y)) Z) (add (multiply X Z) (additive_inverse (multiply Y Z))).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y (additive_inverse Z))) (add (multiply X Y) (additive_inverse (multiply X Z))).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (multiply X (additive_inverse Y)) (additive_inverse (multiply X Y)).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (multiply (additive_inverse X) Y) (additive_inverse (multiply X Y)).
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (multiply (additive_inverse X) (additive_inverse Y)) (multiply X Y).
+∀H7:∀X:Univ.∀Y:Univ.eq Univ (commutator X Y) (add (multiply Y X) (additive_inverse (multiply X Y))).
+∀H8:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (associator X Y Z) (add (multiply (multiply X Y) Z) (additive_inverse (multiply X (multiply Y Z)))).
+∀H9:∀X:Univ.∀Y:Univ.eq Univ (multiply (multiply X X) Y) (multiply X (multiply X Y)).
+∀H10:∀X:Univ.∀Y:Univ.eq Univ (multiply (multiply X Y) Y) (multiply X (multiply Y Y)).
+∀H11:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add X (add Y Z)) (add (add X Y) Z).
+∀H12:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).
+∀H13:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X Y) Z) (add (multiply X Z) (multiply Y Z)).
+∀H14:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y Z)) (add (multiply X Y) (multiply X Z)).
+∀H15:∀X:Univ.eq Univ (additive_inverse (additive_inverse X)) X.
+∀H16:∀X:Univ.eq Univ (add X (additive_inverse X)) additive_identity.
+∀H17:∀X:Univ.eq Univ (add (additive_inverse X) X) additive_identity.
+∀H18:∀X:Univ.eq Univ (multiply X additive_identity) additive_identity.
+∀H19:∀X:Univ.eq Univ (multiply additive_identity X) additive_identity.
+∀H20:∀X:Univ.eq Univ (add X additive_identity) X.
+∀H21:∀X:Univ.eq Univ (add additive_identity X) X.eq Univ (associator x (multiply x y) z) (multiply (associator x y z) x)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#add.
+#additive_identity.
+#additive_inverse.
+#associator.
+#commutator.
+#multiply.
+#x.
+#y.
+#z.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+#H16.
+#H17.
+#H18.
+#H19.
+#H20.
+#H21.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15,H16,H17,H18,H19,H20,H21;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: RNG028-5.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG028-5 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Ring Theory (Alternative) *)
+
+(* Problem : Left Moufang identity *)
+
+(* Version : [Ste87] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Ste87] Stevens (1987), Some Experiments in Nonassociative Rin *)
+
+(* : [Ste88] Stevens (1988), Challenge Problems from Nonassociative *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.86 v3.1.0, 0.89 v2.7.0, 0.91 v2.6.0, 0.83 v2.5.0, 0.75 v2.4.0, 0.67 v2.2.1, 1.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 16 ( 0 non-Horn; 16 unit; 1 RR) *)
+
+(* Number of atoms : 16 ( 16 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 9 ( 4 constant; 0-3 arity) *)
+
+(* Number of variables : 27 ( 2 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : [Ste88] presents a slightly different set of axioms for proving *)
+
+(* this ntheorem. The axioms are so similar to those in RNG004.ax *)
+
+(* that a separate problems has not been created. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include nonassociative ring axioms *)
+
+(* Inclusion of: Axioms/RNG003-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG003-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Ring Theory (Alternative) *)
+
+(* Axioms : Alternative ring theory (equality) axioms *)
+
+(* Version : [Ste87] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Ste87] Stevens (1987), Some Experiments in Nonassociative Rin *)
+
+(* Source : [Ste87] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 15 ( 0 non-Horn; 15 unit; 0 RR) *)
+
+(* Number of literals : 15 ( 15 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 1 constant; 0-3 arity) *)
+
+(* Number of variables : 27 ( 2 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----There exists an additive identity element *)
+
+(* ----Multiplicative zero *)
+
+(* ----Existence of left additive additive_inverse *)
+
+(* ----Inverse of additive_inverse of X is X *)
+
+(* ----Distributive property of product over sum *)
+
+(* ----Commutativity for addition *)
+
+(* ----Associativity for addition *)
+
+(* ----Right alternative law *)
+
+(* ----Left alternative law *)
+
+(* ----Associator *)
+
+(* ----Commutator *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_left_moufang:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀additive_identity:Univ.
+∀additive_inverse:∀_:Univ.Univ.
+∀associator:∀_:Univ.∀_:Univ.∀_:Univ.Univ.
+∀commutator:∀_:Univ.∀_:Univ.Univ.
+∀cx:Univ.
+∀cy:Univ.
+∀cz:Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (commutator X Y) (add (multiply Y X) (additive_inverse (multiply X Y))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (associator X Y Z) (add (multiply (multiply X Y) Z) (additive_inverse (multiply X (multiply Y Z)))).
+∀H2:∀X:Univ.∀Y:Univ.eq Univ (multiply (multiply X X) Y) (multiply X (multiply X Y)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (multiply (multiply X Y) Y) (multiply X (multiply Y Y)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add X (add Y Z)) (add (add X Y) Z).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).
+∀H6:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X Y) Z) (add (multiply X Z) (multiply Y Z)).
+∀H7:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y Z)) (add (multiply X Y) (multiply X Z)).
+∀H8:∀X:Univ.eq Univ (additive_inverse (additive_inverse X)) X.
+∀H9:∀X:Univ.eq Univ (add X (additive_inverse X)) additive_identity.
+∀H10:∀X:Univ.eq Univ (add (additive_inverse X) X) additive_identity.
+∀H11:∀X:Univ.eq Univ (multiply X additive_identity) additive_identity.
+∀H12:∀X:Univ.eq Univ (multiply additive_identity X) additive_identity.
+∀H13:∀X:Univ.eq Univ (add X additive_identity) X.
+∀H14:∀X:Univ.eq Univ (add additive_identity X) X.eq Univ (multiply (multiply cx (multiply cy cx)) cz) (multiply cx (multiply cy (multiply cx cz)))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#add.
+#additive_identity.
+#additive_inverse.
+#associator.
+#commutator.
+#cx.
+#cy.
+#cz.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: RNG028-7.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG028-7 : TPTP v3.2.0. Bugfixed v2.3.0. *)
+
+(* Domain : Ring Theory (Alternative) *)
+
+(* Problem : Left Moufang identity *)
+
+(* Version : [Ste87] (equality) axioms : Augmented. *)
+
+(* Theorem formulation : In terms of associators *)
+
+(* English : *)
+
+(* Refs : [Ste87] Stevens (1987), Some Experiments in Nonassociative Rin *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.86 v3.1.0, 0.89 v2.7.0, 0.91 v2.6.0, 0.83 v2.5.0, 0.75 v2.4.0, 0.67 v2.3.0 *)
+
+(* Syntax : Number of clauses : 23 ( 0 non-Horn; 23 unit; 1 RR) *)
+
+(* Number of atoms : 23 ( 23 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 9 ( 4 constant; 0-3 arity) *)
+
+(* Number of variables : 45 ( 2 singleton) *)
+
+(* Maximal term depth : 5 ( 3 average) *)
+
+(* Comments : *)
+
+(* Bugfixes : v2.3.0 - Clause prove_left_moufang fixed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include nonassociative ring axioms *)
+
+(* Inclusion of: Axioms/RNG003-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG003-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Ring Theory (Alternative) *)
+
+(* Axioms : Alternative ring theory (equality) axioms *)
+
+(* Version : [Ste87] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Ste87] Stevens (1987), Some Experiments in Nonassociative Rin *)
+
+(* Source : [Ste87] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 15 ( 0 non-Horn; 15 unit; 0 RR) *)
+
+(* Number of literals : 15 ( 15 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 1 constant; 0-3 arity) *)
+
+(* Number of variables : 27 ( 2 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----There exists an additive identity element *)
+
+(* ----Multiplicative zero *)
+
+(* ----Existence of left additive additive_inverse *)
+
+(* ----Inverse of additive_inverse of X is X *)
+
+(* ----Distributive property of product over sum *)
+
+(* ----Commutativity for addition *)
+
+(* ----Associativity for addition *)
+
+(* ----Right alternative law *)
+
+(* ----Left alternative law *)
+
+(* ----Associator *)
+
+(* ----Commutator *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The next 7 clause are extra lemmas which Stevens found useful *)
+ntheorem prove_left_moufang:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀additive_identity:Univ.
+∀additive_inverse:∀_:Univ.Univ.
+∀associator:∀_:Univ.∀_:Univ.∀_:Univ.Univ.
+∀commutator:∀_:Univ.∀_:Univ.Univ.
+∀cx:Univ.
+∀cy:Univ.
+∀cz:Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X Y) (additive_inverse Z)) (add (additive_inverse (multiply X Z)) (additive_inverse (multiply Y Z))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (additive_inverse X) (add Y Z)) (add (additive_inverse (multiply X Y)) (additive_inverse (multiply X Z))).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X (additive_inverse Y)) Z) (add (multiply X Z) (additive_inverse (multiply Y Z))).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y (additive_inverse Z))) (add (multiply X Y) (additive_inverse (multiply X Z))).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (multiply X (additive_inverse Y)) (additive_inverse (multiply X Y)).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (multiply (additive_inverse X) Y) (additive_inverse (multiply X Y)).
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (multiply (additive_inverse X) (additive_inverse Y)) (multiply X Y).
+∀H7:∀X:Univ.∀Y:Univ.eq Univ (commutator X Y) (add (multiply Y X) (additive_inverse (multiply X Y))).
+∀H8:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (associator X Y Z) (add (multiply (multiply X Y) Z) (additive_inverse (multiply X (multiply Y Z)))).
+∀H9:∀X:Univ.∀Y:Univ.eq Univ (multiply (multiply X X) Y) (multiply X (multiply X Y)).
+∀H10:∀X:Univ.∀Y:Univ.eq Univ (multiply (multiply X Y) Y) (multiply X (multiply Y Y)).
+∀H11:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add X (add Y Z)) (add (add X Y) Z).
+∀H12:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).
+∀H13:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X Y) Z) (add (multiply X Z) (multiply Y Z)).
+∀H14:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y Z)) (add (multiply X Y) (multiply X Z)).
+∀H15:∀X:Univ.eq Univ (additive_inverse (additive_inverse X)) X.
+∀H16:∀X:Univ.eq Univ (add X (additive_inverse X)) additive_identity.
+∀H17:∀X:Univ.eq Univ (add (additive_inverse X) X) additive_identity.
+∀H18:∀X:Univ.eq Univ (multiply X additive_identity) additive_identity.
+∀H19:∀X:Univ.eq Univ (multiply additive_identity X) additive_identity.
+∀H20:∀X:Univ.eq Univ (add X additive_identity) X.
+∀H21:∀X:Univ.eq Univ (add additive_identity X) X.eq Univ (multiply (multiply cx (multiply cy cx)) cz) (multiply cx (multiply cy (multiply cx cz)))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#add.
+#additive_identity.
+#additive_inverse.
+#associator.
+#commutator.
+#cx.
+#cy.
+#cz.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+#H16.
+#H17.
+#H18.
+#H19.
+#H20.
+#H21.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15,H16,H17,H18,H19,H20,H21;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: RNG028-8.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG028-8 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Ring Theory (Alternative) *)
+
+(* Problem : Left Moufang identity *)
+
+(* Version : [Ste87] (equality) axioms. *)
+
+(* Theorem formulation : Associators. *)
+
+(* English : *)
+
+(* Refs : [Ste87] Stevens (1987), Some Experiments in Nonassociative Rin *)
+
+(* : [Ste88] Stevens (1988), Challenge Problems from Nonassociative *)
+
+(* Source : [Ste87] *)
+
+(* Names : m2' [Ste87] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.86 v3.1.0, 0.89 v2.7.0, 0.91 v2.6.0, 0.83 v2.5.0, 0.75 v2.4.0, 0.67 v2.2.1, 1.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 16 ( 0 non-Horn; 16 unit; 1 RR) *)
+
+(* Number of atoms : 16 ( 16 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 9 ( 4 constant; 0-3 arity) *)
+
+(* Number of variables : 27 ( 2 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : [Ste88] presents a slightly different set of axioms for proving *)
+
+(* this ntheorem. The axioms are so similar to those in RNG004.ax *)
+
+(* that a separate problems has not been created. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include nonassociative ring axioms *)
+
+(* Inclusion of: Axioms/RNG003-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG003-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Ring Theory (Alternative) *)
+
+(* Axioms : Alternative ring theory (equality) axioms *)
+
+(* Version : [Ste87] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Ste87] Stevens (1987), Some Experiments in Nonassociative Rin *)
+
+(* Source : [Ste87] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 15 ( 0 non-Horn; 15 unit; 0 RR) *)
+
+(* Number of literals : 15 ( 15 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 1 constant; 0-3 arity) *)
+
+(* Number of variables : 27 ( 2 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----There exists an additive identity element *)
+
+(* ----Multiplicative zero *)
+
+(* ----Existence of left additive additive_inverse *)
+
+(* ----Inverse of additive_inverse of X is X *)
+
+(* ----Distributive property of product over sum *)
+
+(* ----Commutativity for addition *)
+
+(* ----Associativity for addition *)
+
+(* ----Right alternative law *)
+
+(* ----Left alternative law *)
+
+(* ----Associator *)
+
+(* ----Commutator *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_left_moufang:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀additive_identity:Univ.
+∀additive_inverse:∀_:Univ.Univ.
+∀associator:∀_:Univ.∀_:Univ.∀_:Univ.Univ.
+∀commutator:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀x:Univ.
+∀y:Univ.
+∀z:Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (commutator X Y) (add (multiply Y X) (additive_inverse (multiply X Y))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (associator X Y Z) (add (multiply (multiply X Y) Z) (additive_inverse (multiply X (multiply Y Z)))).
+∀H2:∀X:Univ.∀Y:Univ.eq Univ (multiply (multiply X X) Y) (multiply X (multiply X Y)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (multiply (multiply X Y) Y) (multiply X (multiply Y Y)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add X (add Y Z)) (add (add X Y) Z).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).
+∀H6:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X Y) Z) (add (multiply X Z) (multiply Y Z)).
+∀H7:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y Z)) (add (multiply X Y) (multiply X Z)).
+∀H8:∀X:Univ.eq Univ (additive_inverse (additive_inverse X)) X.
+∀H9:∀X:Univ.eq Univ (add X (additive_inverse X)) additive_identity.
+∀H10:∀X:Univ.eq Univ (add (additive_inverse X) X) additive_identity.
+∀H11:∀X:Univ.eq Univ (multiply X additive_identity) additive_identity.
+∀H12:∀X:Univ.eq Univ (multiply additive_identity X) additive_identity.
+∀H13:∀X:Univ.eq Univ (add X additive_identity) X.
+∀H14:∀X:Univ.eq Univ (add additive_identity X) X.eq Univ (associator x (multiply y x) z) (multiply x (associator x y z))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#add.
+#additive_identity.
+#additive_inverse.
+#associator.
+#commutator.
+#multiply.
+#x.
+#y.
+#z.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: RNG028-9.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG028-9 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Ring Theory (Alternative) *)
+
+(* Problem : Left Moufang identity *)
+
+(* Version : [Ste87] (equality) axioms : Augmented. *)
+
+(* Theorem formulation : Associators. *)
+
+(* English : *)
+
+(* Refs : [Ste87] Stevens (1987), Some Experiments in Nonassociative Rin *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.86 v3.1.0, 0.89 v2.7.0, 0.91 v2.6.0, 0.83 v2.5.0, 0.75 v2.4.0, 0.67 v2.2.1, 1.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 23 ( 0 non-Horn; 23 unit; 1 RR) *)
+
+(* Number of atoms : 23 ( 23 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 9 ( 4 constant; 0-3 arity) *)
+
+(* Number of variables : 45 ( 2 singleton) *)
+
+(* Maximal term depth : 5 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include nonassociative ring axioms *)
+
+(* Inclusion of: Axioms/RNG003-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG003-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Ring Theory (Alternative) *)
+
+(* Axioms : Alternative ring theory (equality) axioms *)
+
+(* Version : [Ste87] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Ste87] Stevens (1987), Some Experiments in Nonassociative Rin *)
+
+(* Source : [Ste87] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 15 ( 0 non-Horn; 15 unit; 0 RR) *)
+
+(* Number of literals : 15 ( 15 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 1 constant; 0-3 arity) *)
+
+(* Number of variables : 27 ( 2 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----There exists an additive identity element *)
+
+(* ----Multiplicative zero *)
+
+(* ----Existence of left additive additive_inverse *)
+
+(* ----Inverse of additive_inverse of X is X *)
+
+(* ----Distributive property of product over sum *)
+
+(* ----Commutativity for addition *)
+
+(* ----Associativity for addition *)
+
+(* ----Right alternative law *)
+
+(* ----Left alternative law *)
+
+(* ----Associator *)
+
+(* ----Commutator *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The next 7 clause are extra lemmas which Stevens found useful *)
+ntheorem prove_left_moufang:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀additive_identity:Univ.
+∀additive_inverse:∀_:Univ.Univ.
+∀associator:∀_:Univ.∀_:Univ.∀_:Univ.Univ.
+∀commutator:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀x:Univ.
+∀y:Univ.
+∀z:Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X Y) (additive_inverse Z)) (add (additive_inverse (multiply X Z)) (additive_inverse (multiply Y Z))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (additive_inverse X) (add Y Z)) (add (additive_inverse (multiply X Y)) (additive_inverse (multiply X Z))).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X (additive_inverse Y)) Z) (add (multiply X Z) (additive_inverse (multiply Y Z))).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y (additive_inverse Z))) (add (multiply X Y) (additive_inverse (multiply X Z))).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (multiply X (additive_inverse Y)) (additive_inverse (multiply X Y)).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (multiply (additive_inverse X) Y) (additive_inverse (multiply X Y)).
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (multiply (additive_inverse X) (additive_inverse Y)) (multiply X Y).
+∀H7:∀X:Univ.∀Y:Univ.eq Univ (commutator X Y) (add (multiply Y X) (additive_inverse (multiply X Y))).
+∀H8:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (associator X Y Z) (add (multiply (multiply X Y) Z) (additive_inverse (multiply X (multiply Y Z)))).
+∀H9:∀X:Univ.∀Y:Univ.eq Univ (multiply (multiply X X) Y) (multiply X (multiply X Y)).
+∀H10:∀X:Univ.∀Y:Univ.eq Univ (multiply (multiply X Y) Y) (multiply X (multiply Y Y)).
+∀H11:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add X (add Y Z)) (add (add X Y) Z).
+∀H12:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).
+∀H13:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X Y) Z) (add (multiply X Z) (multiply Y Z)).
+∀H14:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y Z)) (add (multiply X Y) (multiply X Z)).
+∀H15:∀X:Univ.eq Univ (additive_inverse (additive_inverse X)) X.
+∀H16:∀X:Univ.eq Univ (add X (additive_inverse X)) additive_identity.
+∀H17:∀X:Univ.eq Univ (add (additive_inverse X) X) additive_identity.
+∀H18:∀X:Univ.eq Univ (multiply X additive_identity) additive_identity.
+∀H19:∀X:Univ.eq Univ (multiply additive_identity X) additive_identity.
+∀H20:∀X:Univ.eq Univ (add X additive_identity) X.
+∀H21:∀X:Univ.eq Univ (add additive_identity X) X.eq Univ (associator x (multiply y x) z) (multiply x (associator x y z))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#add.
+#additive_identity.
+#additive_inverse.
+#associator.
+#commutator.
+#multiply.
+#x.
+#y.
+#z.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+#H16.
+#H17.
+#H18.
+#H19.
+#H20.
+#H21.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15,H16,H17,H18,H19,H20,H21;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: RNG029-5.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG029-5 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Ring Theory (Alternative) *)
+
+(* Problem : Middle Moufang identity *)
+
+(* Version : [Ste87] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Ste87] Stevens (1987), Some Experiments in Nonassociative Rin *)
+
+(* : [Ste88] Stevens (1988), Challenge Problems from Nonassociative *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.86 v3.1.0, 0.89 v2.7.0, 0.91 v2.6.0, 0.83 v2.5.0, 0.75 v2.4.0, 0.67 v2.2.1, 1.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 16 ( 0 non-Horn; 16 unit; 1 RR) *)
+
+(* Number of atoms : 16 ( 16 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 9 ( 4 constant; 0-3 arity) *)
+
+(* Number of variables : 27 ( 2 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : [Ste88] presents a slightly different set of axioms for proving *)
+
+(* this ntheorem. The axioms are so similar to those in RNG004.ax *)
+
+(* that a separate problems has not been created. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include nonassociative ring axioms *)
+
+(* Inclusion of: Axioms/RNG003-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG003-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Ring Theory (Alternative) *)
+
+(* Axioms : Alternative ring theory (equality) axioms *)
+
+(* Version : [Ste87] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Ste87] Stevens (1987), Some Experiments in Nonassociative Rin *)
+
+(* Source : [Ste87] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 15 ( 0 non-Horn; 15 unit; 0 RR) *)
+
+(* Number of literals : 15 ( 15 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 1 constant; 0-3 arity) *)
+
+(* Number of variables : 27 ( 2 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----There exists an additive identity element *)
+
+(* ----Multiplicative zero *)
+
+(* ----Existence of left additive additive_inverse *)
+
+(* ----Inverse of additive_inverse of X is X *)
+
+(* ----Distributive property of product over sum *)
+
+(* ----Commutativity for addition *)
+
+(* ----Associativity for addition *)
+
+(* ----Right alternative law *)
+
+(* ----Left alternative law *)
+
+(* ----Associator *)
+
+(* ----Commutator *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_middle_law:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀additive_identity:Univ.
+∀additive_inverse:∀_:Univ.Univ.
+∀associator:∀_:Univ.∀_:Univ.∀_:Univ.Univ.
+∀commutator:∀_:Univ.∀_:Univ.Univ.
+∀cx:Univ.
+∀cy:Univ.
+∀cz:Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (commutator X Y) (add (multiply Y X) (additive_inverse (multiply X Y))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (associator X Y Z) (add (multiply (multiply X Y) Z) (additive_inverse (multiply X (multiply Y Z)))).
+∀H2:∀X:Univ.∀Y:Univ.eq Univ (multiply (multiply X X) Y) (multiply X (multiply X Y)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (multiply (multiply X Y) Y) (multiply X (multiply Y Y)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add X (add Y Z)) (add (add X Y) Z).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).
+∀H6:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X Y) Z) (add (multiply X Z) (multiply Y Z)).
+∀H7:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y Z)) (add (multiply X Y) (multiply X Z)).
+∀H8:∀X:Univ.eq Univ (additive_inverse (additive_inverse X)) X.
+∀H9:∀X:Univ.eq Univ (add X (additive_inverse X)) additive_identity.
+∀H10:∀X:Univ.eq Univ (add (additive_inverse X) X) additive_identity.
+∀H11:∀X:Univ.eq Univ (multiply X additive_identity) additive_identity.
+∀H12:∀X:Univ.eq Univ (multiply additive_identity X) additive_identity.
+∀H13:∀X:Univ.eq Univ (add X additive_identity) X.
+∀H14:∀X:Univ.eq Univ (add additive_identity X) X.eq Univ (multiply (multiply cx cy) (multiply cz cx)) (multiply cx (multiply (multiply cy cz) cx))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#add.
+#additive_identity.
+#additive_inverse.
+#associator.
+#commutator.
+#cx.
+#cy.
+#cz.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: RNG029-6.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG029-6 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Ring Theory (Alternative) *)
+
+(* Problem : Middle Moufang identity *)
+
+(* Version : [Ste87] (equality) axioms. *)
+
+(* Theorem formulation : In terms of associators *)
+
+(* English : *)
+
+(* Refs : [Ste87] Stevens (1987), Some Experiments in Nonassociative Rin *)
+
+(* Source : [Ste87] *)
+
+(* Names : m3 [Ste87] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.86 v3.1.0, 0.89 v2.7.0, 0.91 v2.6.0, 0.83 v2.5.0, 0.75 v2.4.0, 0.67 v2.2.1, 1.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 16 ( 0 non-Horn; 16 unit; 1 RR) *)
+
+(* Number of atoms : 16 ( 16 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 9 ( 4 constant; 0-3 arity) *)
+
+(* Number of variables : 27 ( 2 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include nonassociative ring axioms *)
+
+(* Inclusion of: Axioms/RNG003-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG003-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Ring Theory (Alternative) *)
+
+(* Axioms : Alternative ring theory (equality) axioms *)
+
+(* Version : [Ste87] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Ste87] Stevens (1987), Some Experiments in Nonassociative Rin *)
+
+(* Source : [Ste87] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 15 ( 0 non-Horn; 15 unit; 0 RR) *)
+
+(* Number of literals : 15 ( 15 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 1 constant; 0-3 arity) *)
+
+(* Number of variables : 27 ( 2 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----There exists an additive identity element *)
+
+(* ----Multiplicative zero *)
+
+(* ----Existence of left additive additive_inverse *)
+
+(* ----Inverse of additive_inverse of X is X *)
+
+(* ----Distributive property of product over sum *)
+
+(* ----Commutativity for addition *)
+
+(* ----Associativity for addition *)
+
+(* ----Right alternative law *)
+
+(* ----Left alternative law *)
+
+(* ----Associator *)
+
+(* ----Commutator *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_middle_moufang:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀additive_identity:Univ.
+∀additive_inverse:∀_:Univ.Univ.
+∀associator:∀_:Univ.∀_:Univ.∀_:Univ.Univ.
+∀commutator:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀x:Univ.
+∀y:Univ.
+∀z:Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (commutator X Y) (add (multiply Y X) (additive_inverse (multiply X Y))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (associator X Y Z) (add (multiply (multiply X Y) Z) (additive_inverse (multiply X (multiply Y Z)))).
+∀H2:∀X:Univ.∀Y:Univ.eq Univ (multiply (multiply X X) Y) (multiply X (multiply X Y)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (multiply (multiply X Y) Y) (multiply X (multiply Y Y)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add X (add Y Z)) (add (add X Y) Z).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).
+∀H6:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X Y) Z) (add (multiply X Z) (multiply Y Z)).
+∀H7:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y Z)) (add (multiply X Y) (multiply X Z)).
+∀H8:∀X:Univ.eq Univ (additive_inverse (additive_inverse X)) X.
+∀H9:∀X:Univ.eq Univ (add X (additive_inverse X)) additive_identity.
+∀H10:∀X:Univ.eq Univ (add (additive_inverse X) X) additive_identity.
+∀H11:∀X:Univ.eq Univ (multiply X additive_identity) additive_identity.
+∀H12:∀X:Univ.eq Univ (multiply additive_identity X) additive_identity.
+∀H13:∀X:Univ.eq Univ (add X additive_identity) X.
+∀H14:∀X:Univ.eq Univ (add additive_identity X) X.eq Univ (multiply (multiply x y) (multiply z x)) (multiply (multiply x (multiply y z)) x)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#add.
+#additive_identity.
+#additive_inverse.
+#associator.
+#commutator.
+#multiply.
+#x.
+#y.
+#z.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: RNG029-7.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG029-7 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Ring Theory (Alternative) *)
+
+(* Problem : Middle Moufang identity *)
+
+(* Version : [Ste87] (equality) axioms : Augmented. *)
+
+(* Theorem formulation : In terms of associators *)
+
+(* English : *)
+
+(* Refs : [Ste87] Stevens (1987), Some Experiments in Nonassociative Rin *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.86 v3.1.0, 0.89 v2.7.0, 0.91 v2.6.0, 0.83 v2.5.0, 0.75 v2.4.0, 0.67 v2.2.1, 1.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 23 ( 0 non-Horn; 23 unit; 1 RR) *)
+
+(* Number of atoms : 23 ( 23 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 9 ( 4 constant; 0-3 arity) *)
+
+(* Number of variables : 45 ( 2 singleton) *)
+
+(* Maximal term depth : 5 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include nonassociative ring axioms *)
+
+(* Inclusion of: Axioms/RNG003-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG003-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Ring Theory (Alternative) *)
+
+(* Axioms : Alternative ring theory (equality) axioms *)
+
+(* Version : [Ste87] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Ste87] Stevens (1987), Some Experiments in Nonassociative Rin *)
+
+(* Source : [Ste87] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 15 ( 0 non-Horn; 15 unit; 0 RR) *)
+
+(* Number of literals : 15 ( 15 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 1 constant; 0-3 arity) *)
+
+(* Number of variables : 27 ( 2 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----There exists an additive identity element *)
+
+(* ----Multiplicative zero *)
+
+(* ----Existence of left additive additive_inverse *)
+
+(* ----Inverse of additive_inverse of X is X *)
+
+(* ----Distributive property of product over sum *)
+
+(* ----Commutativity for addition *)
+
+(* ----Associativity for addition *)
+
+(* ----Right alternative law *)
+
+(* ----Left alternative law *)
+
+(* ----Associator *)
+
+(* ----Commutator *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The next 7 clause are extra lemmas which Stevens found useful *)
+ntheorem prove_middle_moufang:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀additive_identity:Univ.
+∀additive_inverse:∀_:Univ.Univ.
+∀associator:∀_:Univ.∀_:Univ.∀_:Univ.Univ.
+∀commutator:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀x:Univ.
+∀y:Univ.
+∀z:Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X Y) (additive_inverse Z)) (add (additive_inverse (multiply X Z)) (additive_inverse (multiply Y Z))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (additive_inverse X) (add Y Z)) (add (additive_inverse (multiply X Y)) (additive_inverse (multiply X Z))).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X (additive_inverse Y)) Z) (add (multiply X Z) (additive_inverse (multiply Y Z))).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y (additive_inverse Z))) (add (multiply X Y) (additive_inverse (multiply X Z))).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (multiply X (additive_inverse Y)) (additive_inverse (multiply X Y)).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (multiply (additive_inverse X) Y) (additive_inverse (multiply X Y)).
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (multiply (additive_inverse X) (additive_inverse Y)) (multiply X Y).
+∀H7:∀X:Univ.∀Y:Univ.eq Univ (commutator X Y) (add (multiply Y X) (additive_inverse (multiply X Y))).
+∀H8:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (associator X Y Z) (add (multiply (multiply X Y) Z) (additive_inverse (multiply X (multiply Y Z)))).
+∀H9:∀X:Univ.∀Y:Univ.eq Univ (multiply (multiply X X) Y) (multiply X (multiply X Y)).
+∀H10:∀X:Univ.∀Y:Univ.eq Univ (multiply (multiply X Y) Y) (multiply X (multiply Y Y)).
+∀H11:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add X (add Y Z)) (add (add X Y) Z).
+∀H12:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).
+∀H13:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X Y) Z) (add (multiply X Z) (multiply Y Z)).
+∀H14:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y Z)) (add (multiply X Y) (multiply X Z)).
+∀H15:∀X:Univ.eq Univ (additive_inverse (additive_inverse X)) X.
+∀H16:∀X:Univ.eq Univ (add X (additive_inverse X)) additive_identity.
+∀H17:∀X:Univ.eq Univ (add (additive_inverse X) X) additive_identity.
+∀H18:∀X:Univ.eq Univ (multiply X additive_identity) additive_identity.
+∀H19:∀X:Univ.eq Univ (multiply additive_identity X) additive_identity.
+∀H20:∀X:Univ.eq Univ (add X additive_identity) X.
+∀H21:∀X:Univ.eq Univ (add additive_identity X) X.eq Univ (multiply (multiply x y) (multiply z x)) (multiply (multiply x (multiply y z)) x)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#add.
+#additive_identity.
+#additive_inverse.
+#associator.
+#commutator.
+#multiply.
+#x.
+#y.
+#z.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+#H16.
+#H17.
+#H18.
+#H19.
+#H20.
+#H21.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15,H16,H17,H18,H19,H20,H21;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: RNG030-6.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG030-6 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Ring Theory (Right alternative) *)
+
+(* Problem : 2*assr(X,X,Y)^3 = additive identity *)
+
+(* Version : [Ste87] (equality) axioms : Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [Ste87] Stevens (1987), Some Experiments in Nonassociative Rin *)
+
+(* Source : [Ste87] *)
+
+(* Names : Conjecture 1 [Ste87] *)
+
+(* Status : Open *)
+
+(* Rating : 1.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 15 ( 0 non-Horn; 15 unit; 1 RR) *)
+
+(* Number of atoms : 15 ( 15 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 8 ( 3 constant; 0-3 arity) *)
+
+(* Number of variables : 25 ( 2 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Don't Include nonassociative ring axioms. *)
+
+(* ----The left alternative law has to be omitted. *)
+
+(* include('axioms/RNG003-0.ax'). *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Commutativity for addition *)
+
+(* ----Associativity for addition *)
+
+(* ----There exists an additive identity element *)
+
+(* ----Multiplicative zero *)
+
+(* ----Existence of left additive additive_inverse *)
+
+(* ----Distributive property of product over sum *)
+
+(* ----Inverse of additive_inverse of X is X *)
+
+(* ----Right alternative law *)
+
+(* ----Left alternative law *)
+
+(* input_clause(left_alternative,axiom, *)
+
+(* [++equal(multiply(multiply(X,X),Y),multiply(X,multiply(X,Y)))]). *)
+
+(* ----Associator *)
+
+(* ----Commutator *)
+ntheorem prove_conjecture_1:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀additive_identity:Univ.
+∀additive_inverse:∀_:Univ.Univ.
+∀associator:∀_:Univ.∀_:Univ.∀_:Univ.Univ.
+∀commutator:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀x:Univ.
+∀y:Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (commutator X Y) (add (multiply Y X) (additive_inverse (multiply X Y))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (associator X Y Z) (add (multiply (multiply X Y) Z) (additive_inverse (multiply X (multiply Y Z)))).
+∀H2:∀X:Univ.∀Y:Univ.eq Univ (multiply (multiply X Y) Y) (multiply X (multiply Y Y)).
+∀H3:∀X:Univ.eq Univ (additive_inverse (additive_inverse X)) X.
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X Y) Z) (add (multiply X Z) (multiply Y Z)).
+∀H5:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y Z)) (add (multiply X Y) (multiply X Z)).
+∀H6:∀X:Univ.eq Univ (add X (additive_inverse X)) additive_identity.
+∀H7:∀X:Univ.eq Univ (add (additive_inverse X) X) additive_identity.
+∀H8:∀X:Univ.eq Univ (multiply X additive_identity) additive_identity.
+∀H9:∀X:Univ.eq Univ (multiply additive_identity X) additive_identity.
+∀H10:∀X:Univ.eq Univ (add X additive_identity) X.
+∀H11:∀X:Univ.eq Univ (add additive_identity X) X.
+∀H12:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add X (add Y Z)) (add (add X Y) Z).
+∀H13:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).eq Univ (add (multiply (associator x x y) (multiply (associator x x y) (associator x x y))) (multiply (associator x x y) (multiply (associator x x y) (associator x x y)))) additive_identity
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#add.
+#additive_identity.
+#additive_inverse.
+#associator.
+#commutator.
+#multiply.
+#x.
+#y.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: RNG030-7.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG030-7 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Ring Theory (Right alternative) *)
+
+(* Problem : 2*assr(X,X,Y)^3 = additive identity *)
+
+(* Version : [Ste87] (equality) axioms : Augmented. *)
+
+(* English : *)
+
+(* Refs : [Ste87] Stevens (1987), Some Experiments in Nonassociative Rin *)
+
+(* Source : [Ste87] *)
+
+(* Names : Conjecture 1 [Ste87] *)
+
+(* Status : Open *)
+
+(* Rating : 1.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 22 ( 0 non-Horn; 22 unit; 1 RR) *)
+
+(* Number of atoms : 22 ( 22 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 8 ( 3 constant; 0-3 arity) *)
+
+(* Number of variables : 43 ( 2 singleton) *)
+
+(* Maximal term depth : 5 ( 3 average) *)
+
+(* Comments : Extra lemmas added to help the ITP prover. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Don't Include nonassociative ring axioms. *)
+
+(* ----The left alternative law has to be omitted. *)
+
+(* include('axioms/RNG003-0.ax'). *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The next 7 clause are extra lemmas which Stevens found useful *)
+
+(* ----Commutativity for addition *)
+
+(* ----Associativity for addition *)
+
+(* ----There exists an additive identity element *)
+
+(* ----Multiplicative zero *)
+
+(* ----Existence of left additive additive_inverse *)
+
+(* ----Distributive property of product over sum *)
+
+(* ----Inverse of additive_inverse of X is X *)
+
+(* ----Right alternative law *)
+
+(* ----Left alternative law *)
+
+(* input_clause(left_alternative,axiom, *)
+
+(* [++equal(multiply(multiply(X,X),Y),multiply(X,multiply(X,Y)))]). *)
+
+(* ----Associator *)
+
+(* ----Commutator *)
+ntheorem prove_conjecture_1:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀additive_identity:Univ.
+∀additive_inverse:∀_:Univ.Univ.
+∀associator:∀_:Univ.∀_:Univ.∀_:Univ.Univ.
+∀commutator:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀x:Univ.
+∀y:Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (commutator X Y) (add (multiply Y X) (additive_inverse (multiply X Y))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (associator X Y Z) (add (multiply (multiply X Y) Z) (additive_inverse (multiply X (multiply Y Z)))).
+∀H2:∀X:Univ.∀Y:Univ.eq Univ (multiply (multiply X Y) Y) (multiply X (multiply Y Y)).
+∀H3:∀X:Univ.eq Univ (additive_inverse (additive_inverse X)) X.
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X Y) Z) (add (multiply X Z) (multiply Y Z)).
+∀H5:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y Z)) (add (multiply X Y) (multiply X Z)).
+∀H6:∀X:Univ.eq Univ (add X (additive_inverse X)) additive_identity.
+∀H7:∀X:Univ.eq Univ (add (additive_inverse X) X) additive_identity.
+∀H8:∀X:Univ.eq Univ (multiply X additive_identity) additive_identity.
+∀H9:∀X:Univ.eq Univ (multiply additive_identity X) additive_identity.
+∀H10:∀X:Univ.eq Univ (add X additive_identity) X.
+∀H11:∀X:Univ.eq Univ (add additive_identity X) X.
+∀H12:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add X (add Y Z)) (add (add X Y) Z).
+∀H13:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).
+∀H14:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X Y) (additive_inverse Z)) (add (additive_inverse (multiply X Z)) (additive_inverse (multiply Y Z))).
+∀H15:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (additive_inverse X) (add Y Z)) (add (additive_inverse (multiply X Y)) (additive_inverse (multiply X Z))).
+∀H16:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X (additive_inverse Y)) Z) (add (multiply X Z) (additive_inverse (multiply Y Z))).
+∀H17:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y (additive_inverse Z))) (add (multiply X Y) (additive_inverse (multiply X Z))).
+∀H18:∀X:Univ.∀Y:Univ.eq Univ (multiply X (additive_inverse Y)) (additive_inverse (multiply X Y)).
+∀H19:∀X:Univ.∀Y:Univ.eq Univ (multiply (additive_inverse X) Y) (additive_inverse (multiply X Y)).
+∀H20:∀X:Univ.∀Y:Univ.eq Univ (multiply (additive_inverse X) (additive_inverse Y)) (multiply X Y).eq Univ (add (multiply (associator x x y) (multiply (associator x x y) (associator x x y))) (multiply (associator x x y) (multiply (associator x x y) (associator x x y)))) additive_identity
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#add.
+#additive_identity.
+#additive_inverse.
+#associator.
+#commutator.
+#multiply.
+#x.
+#y.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+#H16.
+#H17.
+#H18.
+#H19.
+#H20.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15,H16,H17,H18,H19,H20;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: RNG031-6.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG031-6 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Ring Theory (Right alternative) *)
+
+(* Problem : (W*W)*X*(W*W) = additive identity *)
+
+(* Version : [Ste87] (equality) axioms : Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [Ste87] Stevens (1987), Some Experiments in Nonassociative Rin *)
+
+(* : [Ste92] Stevens (1992), Unpublished Note *)
+
+(* Source : [Ste87] *)
+
+(* Names : Conjecture 2 [Ste87] *)
+
+(* Status : Satisfiable *)
+
+(* Rating : 0.33 v3.2.0, 0.67 v3.1.0, 0.33 v2.6.0, 0.67 v2.5.0, 1.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 15 ( 0 non-Horn; 15 unit; 1 RR) *)
+
+(* Number of atoms : 15 ( 15 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 8 ( 3 constant; 0-3 arity) *)
+
+(* Number of variables : 25 ( 2 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : This conjecture has been shown true. See [Ste92]. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Don't Include nonassociative ring axioms. *)
+
+(* ----The left alternative law has to be omitted. *)
+
+(* include('axioms/RNG003-0.ax'). *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Commutativity for addition *)
+
+(* ----Associativity for addition *)
+
+(* ----There exists an additive identity element *)
+
+(* ----Multiplicative zero *)
+
+(* ----Existence of left additive additive_inverse *)
+
+(* ----Distributive property of product over sum *)
+
+(* ----Inverse of additive_inverse of X is X *)
+
+(* ----Right alternative law *)
+
+(* ----Left alternative law *)
+
+(* input_clause(left_alternative,axiom, *)
+
+(* [++equal(multiply(multiply(X,X),Y),multiply(X,multiply(X,Y)))]). *)
+
+(* ----Associator *)
+
+(* ----Commutator *)
+ntheorem prove_conjecture_2:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀additive_identity:Univ.
+∀additive_inverse:∀_:Univ.Univ.
+∀associator:∀_:Univ.∀_:Univ.∀_:Univ.Univ.
+∀commutator:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀x:Univ.
+∀y:Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (commutator X Y) (add (multiply Y X) (additive_inverse (multiply X Y))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (associator X Y Z) (add (multiply (multiply X Y) Z) (additive_inverse (multiply X (multiply Y Z)))).
+∀H2:∀X:Univ.∀Y:Univ.eq Univ (multiply (multiply X Y) Y) (multiply X (multiply Y Y)).
+∀H3:∀X:Univ.eq Univ (additive_inverse (additive_inverse X)) X.
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X Y) Z) (add (multiply X Z) (multiply Y Z)).
+∀H5:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y Z)) (add (multiply X Y) (multiply X Z)).
+∀H6:∀X:Univ.eq Univ (add X (additive_inverse X)) additive_identity.
+∀H7:∀X:Univ.eq Univ (add (additive_inverse X) X) additive_identity.
+∀H8:∀X:Univ.eq Univ (multiply X additive_identity) additive_identity.
+∀H9:∀X:Univ.eq Univ (multiply additive_identity X) additive_identity.
+∀H10:∀X:Univ.eq Univ (add X additive_identity) X.
+∀H11:∀X:Univ.eq Univ (add additive_identity X) X.
+∀H12:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add X (add Y Z)) (add (add X Y) Z).
+∀H13:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).eq Univ (multiply (multiply (multiply (associator x x y) (associator x x y)) x) (multiply (associator x x y) (associator x x y))) additive_identity
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#add.
+#additive_identity.
+#additive_inverse.
+#associator.
+#commutator.
+#multiply.
+#x.
+#y.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: RNG031-7.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG031-7 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Ring Theory (Right alternative) *)
+
+(* Problem : (W*W)*X*(W*W) = additive identity *)
+
+(* Version : [Ste87] (equality) axioms : Augmented. *)
+
+(* English : *)
+
+(* Refs : [Ste87] Stevens (1987), Some Experiments in Nonassociative Rin *)
+
+(* : [Ste92] Stevens (1992), Unpublished Note *)
+
+(* Source : [Ste87] *)
+
+(* Names : Conjecture 2 [Ste87] *)
+
+(* Status : Satisfiable *)
+
+(* Rating : 0.33 v3.2.0, 0.67 v3.1.0, 0.33 v2.6.0, 0.67 v2.5.0, 1.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 22 ( 0 non-Horn; 22 unit; 1 RR) *)
+
+(* Number of atoms : 22 ( 22 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 8 ( 3 constant; 0-3 arity) *)
+
+(* Number of variables : 43 ( 2 singleton) *)
+
+(* Maximal term depth : 5 ( 3 average) *)
+
+(* Comments : This conjecture has been shown true. See [Ste92]. *)
+
+(* : Extra lemmas added to help the ITP prover. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Don't Include nonassociative ring axioms. *)
+
+(* ----The left alternative law has to be omitted. *)
+
+(* include('axioms/RNG003-0.ax'). *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The next 7 clause are extra lemmas which Stevens found useful *)
+
+(* ----Commutativity for addition *)
+
+(* ----Associativity for addition *)
+
+(* ----There exists an additive identity element *)
+
+(* ----Multiplicative zero *)
+
+(* ----Existence of left additive additive_inverse *)
+
+(* ----Distributive property of product over sum *)
+
+(* ----Inverse of additive_inverse of X is X *)
+
+(* ----Right alternative law *)
+
+(* ----Left alternative law *)
+
+(* input_clause(left_alternative,axiom, *)
+
+(* [++equal(multiply(multiply(X,X),Y),multiply(X,multiply(X,Y)))]). *)
+
+(* ----Associator *)
+
+(* ----Commutator *)
+ntheorem prove_conjecture_2:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀additive_identity:Univ.
+∀additive_inverse:∀_:Univ.Univ.
+∀associator:∀_:Univ.∀_:Univ.∀_:Univ.Univ.
+∀commutator:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀x:Univ.
+∀y:Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (commutator X Y) (add (multiply Y X) (additive_inverse (multiply X Y))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (associator X Y Z) (add (multiply (multiply X Y) Z) (additive_inverse (multiply X (multiply Y Z)))).
+∀H2:∀X:Univ.∀Y:Univ.eq Univ (multiply (multiply X Y) Y) (multiply X (multiply Y Y)).
+∀H3:∀X:Univ.eq Univ (additive_inverse (additive_inverse X)) X.
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X Y) Z) (add (multiply X Z) (multiply Y Z)).
+∀H5:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y Z)) (add (multiply X Y) (multiply X Z)).
+∀H6:∀X:Univ.eq Univ (add X (additive_inverse X)) additive_identity.
+∀H7:∀X:Univ.eq Univ (add (additive_inverse X) X) additive_identity.
+∀H8:∀X:Univ.eq Univ (multiply X additive_identity) additive_identity.
+∀H9:∀X:Univ.eq Univ (multiply additive_identity X) additive_identity.
+∀H10:∀X:Univ.eq Univ (add X additive_identity) X.
+∀H11:∀X:Univ.eq Univ (add additive_identity X) X.
+∀H12:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add X (add Y Z)) (add (add X Y) Z).
+∀H13:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).
+∀H14:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X Y) (additive_inverse Z)) (add (additive_inverse (multiply X Z)) (additive_inverse (multiply Y Z))).
+∀H15:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (additive_inverse X) (add Y Z)) (add (additive_inverse (multiply X Y)) (additive_inverse (multiply X Z))).
+∀H16:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X (additive_inverse Y)) Z) (add (multiply X Z) (additive_inverse (multiply Y Z))).
+∀H17:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y (additive_inverse Z))) (add (multiply X Y) (additive_inverse (multiply X Z))).
+∀H18:∀X:Univ.∀Y:Univ.eq Univ (multiply X (additive_inverse Y)) (additive_inverse (multiply X Y)).
+∀H19:∀X:Univ.∀Y:Univ.eq Univ (multiply (additive_inverse X) Y) (additive_inverse (multiply X Y)).
+∀H20:∀X:Univ.∀Y:Univ.eq Univ (multiply (additive_inverse X) (additive_inverse Y)) (multiply X Y).eq Univ (multiply (multiply (multiply (associator x x y) (associator x x y)) x) (multiply (associator x x y) (associator x x y))) additive_identity
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#add.
+#additive_identity.
+#additive_inverse.
+#associator.
+#commutator.
+#multiply.
+#x.
+#y.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+#H16.
+#H17.
+#H18.
+#H19.
+#H20.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15,H16,H17,H18,H19,H20;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: RNG032-6.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG032-6 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Ring Theory (Right alternative) *)
+
+(* Problem : 6*assr(X,X,Y)^6 = additive identity *)
+
+(* Version : [Ste87] (equality) axioms : Reduced > Complete. *)
+
+(* English : *)
+
+(* Refs : [Ste87] Stevens (1987), Some Experiments in Nonassociative Rin *)
+
+(* Source : [Ste87] *)
+
+(* Names : Conjecture 3 [Ste87] *)
+
+(* Status : Open *)
+
+(* Rating : 1.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 15 ( 0 non-Horn; 15 unit; 1 RR) *)
+
+(* Number of atoms : 15 ( 15 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 8 ( 3 constant; 0-3 arity) *)
+
+(* Number of variables : 25 ( 2 singleton) *)
+
+(* Maximal term depth : 9 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Don't Include nonassociative ring axioms. *)
+
+(* ----The left alternative law has to be omitted. *)
+
+(* include('axioms/RNG003-0.ax'). *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Commutativity for addition *)
+
+(* ----Associativity for addition *)
+
+(* ----There exists an additive identity element *)
+
+(* ----Multiplicative zero *)
+
+(* ----Existence of left additive additive_inverse *)
+
+(* ----Distributive property of product over sum *)
+
+(* ----Inverse of additive_inverse of X is X *)
+
+(* ----Right alternative law *)
+
+(* ----Left alternative law *)
+
+(* input_clause(left_alternative,axiom, *)
+
+(* [++equal(multiply(multiply(X,X),Y),multiply(X,multiply(X,Y)))]). *)
+
+(* ----Associator *)
+
+(* ----Commutator *)
+ntheorem prove_conjecture_3:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀additive_identity:Univ.
+∀additive_inverse:∀_:Univ.Univ.
+∀associator:∀_:Univ.∀_:Univ.∀_:Univ.Univ.
+∀commutator:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀x:Univ.
+∀y:Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (commutator X Y) (add (multiply Y X) (additive_inverse (multiply X Y))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (associator X Y Z) (add (multiply (multiply X Y) Z) (additive_inverse (multiply X (multiply Y Z)))).
+∀H2:∀X:Univ.∀Y:Univ.eq Univ (multiply (multiply X Y) Y) (multiply X (multiply Y Y)).
+∀H3:∀X:Univ.eq Univ (additive_inverse (additive_inverse X)) X.
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X Y) Z) (add (multiply X Z) (multiply Y Z)).
+∀H5:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y Z)) (add (multiply X Y) (multiply X Z)).
+∀H6:∀X:Univ.eq Univ (add X (additive_inverse X)) additive_identity.
+∀H7:∀X:Univ.eq Univ (add (additive_inverse X) X) additive_identity.
+∀H8:∀X:Univ.eq Univ (multiply X additive_identity) additive_identity.
+∀H9:∀X:Univ.eq Univ (multiply additive_identity X) additive_identity.
+∀H10:∀X:Univ.eq Univ (add X additive_identity) X.
+∀H11:∀X:Univ.eq Univ (add additive_identity X) X.
+∀H12:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add X (add Y Z)) (add (add X Y) Z).
+∀H13:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).eq Univ (add (add (add (add (add (multiply (associator x x y) (multiply (associator x x y) (associator x x y))) (multiply (associator x x y) (multiply (associator x x y) (associator x x y)))) (multiply (associator x x y) (multiply (associator x x y) (associator x x y)))) (multiply (associator x x y) (multiply (associator x x y) (associator x x y)))) (multiply (associator x x y) (multiply (associator x x y) (associator x x y)))) (multiply (associator x x y) (multiply (associator x x y) (associator x x y)))) additive_identity
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#add.
+#additive_identity.
+#additive_inverse.
+#associator.
+#commutator.
+#multiply.
+#x.
+#y.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: RNG032-7.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG032-7 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Ring Theory (Right alternative) *)
+
+(* Problem : 6*assr(X,X,Y)^6 = additive identity *)
+
+(* Version : [Ste87] (equality) axioms : Augmented. *)
+
+(* English : *)
+
+(* Refs : [Ste87] Stevens (1987), Some Experiments in Nonassociative Rin *)
+
+(* Source : [Ste87] *)
+
+(* Names : Conjecture 3 [Ste87] *)
+
+(* Status : Open *)
+
+(* Rating : 1.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 22 ( 0 non-Horn; 22 unit; 1 RR) *)
+
+(* Number of atoms : 22 ( 22 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 8 ( 3 constant; 0-3 arity) *)
+
+(* Number of variables : 43 ( 2 singleton) *)
+
+(* Maximal term depth : 9 ( 3 average) *)
+
+(* Comments : Extra lemmas added to help the ITP prover. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Don't Include nonassociative ring axioms. *)
+
+(* ----The left alternative law has to be omitted. *)
+
+(* include('axioms/RNG003-0.ax'). *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The next 7 clause are extra lemmas which Stevens found useful *)
+
+(* ----Commutativity for addition *)
+
+(* ----Associativity for addition *)
+
+(* ----There exists an additive identity element *)
+
+(* ----Multiplicative zero *)
+
+(* ----Existence of left additive additive_inverse *)
+
+(* ----Distributive property of product over sum *)
+
+(* ----Inverse of additive_inverse of X is X *)
+
+(* ----Right alternative law *)
+
+(* ----Left alternative law *)
+
+(* input_clause(left_alternative,axiom, *)
+
+(* [++equal(multiply(multiply(X,X),Y),multiply(X,multiply(X,Y)))]). *)
+
+(* ----Associator *)
+
+(* ----Commutator *)
+ntheorem prove_conjecture_3:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀additive_identity:Univ.
+∀additive_inverse:∀_:Univ.Univ.
+∀associator:∀_:Univ.∀_:Univ.∀_:Univ.Univ.
+∀commutator:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀x:Univ.
+∀y:Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (commutator X Y) (add (multiply Y X) (additive_inverse (multiply X Y))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (associator X Y Z) (add (multiply (multiply X Y) Z) (additive_inverse (multiply X (multiply Y Z)))).
+∀H2:∀X:Univ.∀Y:Univ.eq Univ (multiply (multiply X Y) Y) (multiply X (multiply Y Y)).
+∀H3:∀X:Univ.eq Univ (additive_inverse (additive_inverse X)) X.
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X Y) Z) (add (multiply X Z) (multiply Y Z)).
+∀H5:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y Z)) (add (multiply X Y) (multiply X Z)).
+∀H6:∀X:Univ.eq Univ (add X (additive_inverse X)) additive_identity.
+∀H7:∀X:Univ.eq Univ (add (additive_inverse X) X) additive_identity.
+∀H8:∀X:Univ.eq Univ (multiply X additive_identity) additive_identity.
+∀H9:∀X:Univ.eq Univ (multiply additive_identity X) additive_identity.
+∀H10:∀X:Univ.eq Univ (add X additive_identity) X.
+∀H11:∀X:Univ.eq Univ (add additive_identity X) X.
+∀H12:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add X (add Y Z)) (add (add X Y) Z).
+∀H13:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).
+∀H14:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X Y) (additive_inverse Z)) (add (additive_inverse (multiply X Z)) (additive_inverse (multiply Y Z))).
+∀H15:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (additive_inverse X) (add Y Z)) (add (additive_inverse (multiply X Y)) (additive_inverse (multiply X Z))).
+∀H16:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X (additive_inverse Y)) Z) (add (multiply X Z) (additive_inverse (multiply Y Z))).
+∀H17:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y (additive_inverse Z))) (add (multiply X Y) (additive_inverse (multiply X Z))).
+∀H18:∀X:Univ.∀Y:Univ.eq Univ (multiply X (additive_inverse Y)) (additive_inverse (multiply X Y)).
+∀H19:∀X:Univ.∀Y:Univ.eq Univ (multiply (additive_inverse X) Y) (additive_inverse (multiply X Y)).
+∀H20:∀X:Univ.∀Y:Univ.eq Univ (multiply (additive_inverse X) (additive_inverse Y)) (multiply X Y).eq Univ (add (add (add (add (add (multiply (associator x x y) (multiply (associator x x y) (associator x x y))) (multiply (associator x x y) (multiply (associator x x y) (associator x x y)))) (multiply (associator x x y) (multiply (associator x x y) (associator x x y)))) (multiply (associator x x y) (multiply (associator x x y) (associator x x y)))) (multiply (associator x x y) (multiply (associator x x y) (associator x x y)))) (multiply (associator x x y) (multiply (associator x x y) (associator x x y)))) additive_identity
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#add.
+#additive_identity.
+#additive_inverse.
+#associator.
+#commutator.
+#multiply.
+#x.
+#y.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+#H16.
+#H17.
+#H18.
+#H19.
+#H20.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15,H16,H17,H18,H19,H20;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: RNG033-6.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG033-6 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Ring Theory (Alternative) *)
+
+(* Problem : A fairly complex equation with associators *)
+
+(* Version : [Ste87] (equality) axioms. *)
+
+(* English : assr(X.Y,Z,W)+assr(X,Y,comm(Z,W)) = X.assr(Y,Z,W)+assr(X,Z,W).Y *)
+
+(* Refs : [Ste87] Stevens (1987), Some Experiments in Nonassociative Rin *)
+
+(* Source : [Ste87] *)
+
+(* Names : ch [Ste87] *)
+
+(* Status : Unknown *)
+
+(* Rating : 1.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 16 ( 0 non-Horn; 16 unit; 1 RR) *)
+
+(* Number of atoms : 16 ( 16 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 10 ( 5 constant; 0-3 arity) *)
+
+(* Number of variables : 27 ( 2 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include nonassociative ring axioms *)
+
+(* Inclusion of: Axioms/RNG003-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG003-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Ring Theory (Alternative) *)
+
+(* Axioms : Alternative ring theory (equality) axioms *)
+
+(* Version : [Ste87] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Ste87] Stevens (1987), Some Experiments in Nonassociative Rin *)
+
+(* Source : [Ste87] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 15 ( 0 non-Horn; 15 unit; 0 RR) *)
+
+(* Number of literals : 15 ( 15 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 1 constant; 0-3 arity) *)
+
+(* Number of variables : 27 ( 2 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----There exists an additive identity element *)
+
+(* ----Multiplicative zero *)
+
+(* ----Existence of left additive additive_inverse *)
+
+(* ----Inverse of additive_inverse of X is X *)
+
+(* ----Distributive property of product over sum *)
+
+(* ----Commutativity for addition *)
+
+(* ----Associativity for addition *)
+
+(* ----Right alternative law *)
+
+(* ----Left alternative law *)
+
+(* ----Associator *)
+
+(* ----Commutator *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_challenge:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀additive_identity:Univ.
+∀additive_inverse:∀_:Univ.Univ.
+∀associator:∀_:Univ.∀_:Univ.∀_:Univ.Univ.
+∀commutator:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀w:Univ.
+∀x:Univ.
+∀y:Univ.
+∀z:Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (commutator X Y) (add (multiply Y X) (additive_inverse (multiply X Y))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (associator X Y Z) (add (multiply (multiply X Y) Z) (additive_inverse (multiply X (multiply Y Z)))).
+∀H2:∀X:Univ.∀Y:Univ.eq Univ (multiply (multiply X X) Y) (multiply X (multiply X Y)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (multiply (multiply X Y) Y) (multiply X (multiply Y Y)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add X (add Y Z)) (add (add X Y) Z).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).
+∀H6:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X Y) Z) (add (multiply X Z) (multiply Y Z)).
+∀H7:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y Z)) (add (multiply X Y) (multiply X Z)).
+∀H8:∀X:Univ.eq Univ (additive_inverse (additive_inverse X)) X.
+∀H9:∀X:Univ.eq Univ (add X (additive_inverse X)) additive_identity.
+∀H10:∀X:Univ.eq Univ (add (additive_inverse X) X) additive_identity.
+∀H11:∀X:Univ.eq Univ (multiply X additive_identity) additive_identity.
+∀H12:∀X:Univ.eq Univ (multiply additive_identity X) additive_identity.
+∀H13:∀X:Univ.eq Univ (add X additive_identity) X.
+∀H14:∀X:Univ.eq Univ (add additive_identity X) X.eq Univ (add (associator (multiply x y) z w) (associator x y (commutator z w))) (add (multiply x (associator y z w)) (multiply (associator x z w) y))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#add.
+#additive_identity.
+#additive_inverse.
+#associator.
+#commutator.
+#multiply.
+#w.
+#x.
+#y.
+#z.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: RNG033-7.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG033-7 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Ring Theory (Alternative) *)
+
+(* Problem : A fairly complex equation with associators *)
+
+(* Version : [Ste87] (equality) axioms : Augmented. *)
+
+(* English : assr(X.Y,Z,W)+assr(X,Y,comm(Z,W)) = X.assr(Y,Z,W)+assr(X,Z,W).Y *)
+
+(* Refs : [Ste87] Stevens (1987), Some Experiments in Nonassociative Rin *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unknown *)
+
+(* Rating : 1.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 23 ( 0 non-Horn; 23 unit; 1 RR) *)
+
+(* Number of atoms : 23 ( 23 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 10 ( 5 constant; 0-3 arity) *)
+
+(* Number of variables : 45 ( 2 singleton) *)
+
+(* Maximal term depth : 5 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include nonassociative ring axioms *)
+
+(* Inclusion of: Axioms/RNG003-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG003-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Ring Theory (Alternative) *)
+
+(* Axioms : Alternative ring theory (equality) axioms *)
+
+(* Version : [Ste87] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Ste87] Stevens (1987), Some Experiments in Nonassociative Rin *)
+
+(* Source : [Ste87] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 15 ( 0 non-Horn; 15 unit; 0 RR) *)
+
+(* Number of literals : 15 ( 15 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 1 constant; 0-3 arity) *)
+
+(* Number of variables : 27 ( 2 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----There exists an additive identity element *)
+
+(* ----Multiplicative zero *)
+
+(* ----Existence of left additive additive_inverse *)
+
+(* ----Inverse of additive_inverse of X is X *)
+
+(* ----Distributive property of product over sum *)
+
+(* ----Commutativity for addition *)
+
+(* ----Associativity for addition *)
+
+(* ----Right alternative law *)
+
+(* ----Left alternative law *)
+
+(* ----Associator *)
+
+(* ----Commutator *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The next 7 clause are extra lemmas which Stevens found useful *)
+ntheorem prove_challenge:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀additive_identity:Univ.
+∀additive_inverse:∀_:Univ.Univ.
+∀associator:∀_:Univ.∀_:Univ.∀_:Univ.Univ.
+∀commutator:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀w:Univ.
+∀x:Univ.
+∀y:Univ.
+∀z:Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X Y) (additive_inverse Z)) (add (additive_inverse (multiply X Z)) (additive_inverse (multiply Y Z))).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (additive_inverse X) (add Y Z)) (add (additive_inverse (multiply X Y)) (additive_inverse (multiply X Z))).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X (additive_inverse Y)) Z) (add (multiply X Z) (additive_inverse (multiply Y Z))).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y (additive_inverse Z))) (add (multiply X Y) (additive_inverse (multiply X Z))).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (multiply X (additive_inverse Y)) (additive_inverse (multiply X Y)).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (multiply (additive_inverse X) Y) (additive_inverse (multiply X Y)).
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (multiply (additive_inverse X) (additive_inverse Y)) (multiply X Y).
+∀H7:∀X:Univ.∀Y:Univ.eq Univ (commutator X Y) (add (multiply Y X) (additive_inverse (multiply X Y))).
+∀H8:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (associator X Y Z) (add (multiply (multiply X Y) Z) (additive_inverse (multiply X (multiply Y Z)))).
+∀H9:∀X:Univ.∀Y:Univ.eq Univ (multiply (multiply X X) Y) (multiply X (multiply X Y)).
+∀H10:∀X:Univ.∀Y:Univ.eq Univ (multiply (multiply X Y) Y) (multiply X (multiply Y Y)).
+∀H11:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add X (add Y Z)) (add (add X Y) Z).
+∀H12:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).
+∀H13:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X Y) Z) (add (multiply X Z) (multiply Y Z)).
+∀H14:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y Z)) (add (multiply X Y) (multiply X Z)).
+∀H15:∀X:Univ.eq Univ (additive_inverse (additive_inverse X)) X.
+∀H16:∀X:Univ.eq Univ (add X (additive_inverse X)) additive_identity.
+∀H17:∀X:Univ.eq Univ (add (additive_inverse X) X) additive_identity.
+∀H18:∀X:Univ.eq Univ (multiply X additive_identity) additive_identity.
+∀H19:∀X:Univ.eq Univ (multiply additive_identity X) additive_identity.
+∀H20:∀X:Univ.eq Univ (add X additive_identity) X.
+∀H21:∀X:Univ.eq Univ (add additive_identity X) X.eq Univ (add (associator (multiply x y) z w) (associator x y (commutator z w))) (add (multiply x (associator y z w)) (multiply (associator x z w) y))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#add.
+#additive_identity.
+#additive_inverse.
+#associator.
+#commutator.
+#multiply.
+#w.
+#x.
+#y.
+#z.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+#H16.
+#H17.
+#H18.
+#H19.
+#H20.
+#H21.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15,H16,H17,H18,H19,H20,H21;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: RNG033-8.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG033-8 : TPTP v3.2.0. Bugfixed v2.3.0. *)
+
+(* Domain : Ring Theory (Alternative) *)
+
+(* Problem : A fairly complex equation with associators *)
+
+(* Version : [Ste87] (equality) axioms : Augmented. *)
+
+(* English : assr(X.Y,Z,W)+assr(X,Y,comm(Z,W)) = X.assr(Y,Z,W)+assr(X,Z,W).Y *)
+
+(* Refs : [Ste87] Stevens (1987), Some Experiments in Nonassociative Rin *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unknown *)
+
+(* Rating : 1.00 v2.3.0 *)
+
+(* Syntax : Number of clauses : 17 ( 0 non-Horn; 17 unit; 1 RR) *)
+
+(* Number of atoms : 17 ( 17 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 10 ( 5 constant; 0-3 arity) *)
+
+(* Number of variables : 30 ( 2 singleton) *)
+
+(* Maximal term depth : 5 ( 3 average) *)
+
+(* Comments : This includes the right Moufang identity, which [Ste87] *)
+
+(* suggests in a useful lemma for this problem. *)
+
+(* Bugfixes : v2.3.0 - Clause right_moufang fixed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include nonassociative ring axioms *)
+
+(* Inclusion of: Axioms/RNG003-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG003-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Ring Theory (Alternative) *)
+
+(* Axioms : Alternative ring theory (equality) axioms *)
+
+(* Version : [Ste87] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Ste87] Stevens (1987), Some Experiments in Nonassociative Rin *)
+
+(* Source : [Ste87] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 15 ( 0 non-Horn; 15 unit; 0 RR) *)
+
+(* Number of literals : 15 ( 15 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 1 constant; 0-3 arity) *)
+
+(* Number of variables : 27 ( 2 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----There exists an additive identity element *)
+
+(* ----Multiplicative zero *)
+
+(* ----Existence of left additive additive_inverse *)
+
+(* ----Inverse of additive_inverse of X is X *)
+
+(* ----Distributive property of product over sum *)
+
+(* ----Commutativity for addition *)
+
+(* ----Associativity for addition *)
+
+(* ----Right alternative law *)
+
+(* ----Left alternative law *)
+
+(* ----Associator *)
+
+(* ----Commutator *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Right Moufang *)
+ntheorem prove_challenge:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀additive_identity:Univ.
+∀additive_inverse:∀_:Univ.Univ.
+∀associator:∀_:Univ.∀_:Univ.∀_:Univ.Univ.
+∀commutator:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀w:Univ.
+∀x:Univ.
+∀y:Univ.
+∀z:Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply Z (multiply X (multiply Y X))) (multiply (multiply (multiply Z X) Y) X).
+∀H1:∀X:Univ.∀Y:Univ.eq Univ (commutator X Y) (add (multiply Y X) (additive_inverse (multiply X Y))).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (associator X Y Z) (add (multiply (multiply X Y) Z) (additive_inverse (multiply X (multiply Y Z)))).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (multiply (multiply X X) Y) (multiply X (multiply X Y)).
+∀H4:∀X:Univ.∀Y:Univ.eq Univ (multiply (multiply X Y) Y) (multiply X (multiply Y Y)).
+∀H5:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add X (add Y Z)) (add (add X Y) Z).
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).
+∀H7:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X Y) Z) (add (multiply X Z) (multiply Y Z)).
+∀H8:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y Z)) (add (multiply X Y) (multiply X Z)).
+∀H9:∀X:Univ.eq Univ (additive_inverse (additive_inverse X)) X.
+∀H10:∀X:Univ.eq Univ (add X (additive_inverse X)) additive_identity.
+∀H11:∀X:Univ.eq Univ (add (additive_inverse X) X) additive_identity.
+∀H12:∀X:Univ.eq Univ (multiply X additive_identity) additive_identity.
+∀H13:∀X:Univ.eq Univ (multiply additive_identity X) additive_identity.
+∀H14:∀X:Univ.eq Univ (add X additive_identity) X.
+∀H15:∀X:Univ.eq Univ (add additive_identity X) X.eq Univ (add (associator (multiply x y) z w) (associator x y (commutator z w))) (add (multiply x (associator y z w)) (multiply (associator x z w) y))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#add.
+#additive_identity.
+#additive_inverse.
+#associator.
+#commutator.
+#multiply.
+#w.
+#x.
+#y.
+#z.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: RNG033-9.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG033-9 : TPTP v3.2.0. Bugfixed v2.3.0. *)
+
+(* Domain : Ring Theory (Alternative) *)
+
+(* Problem : A fairly complex equation with associators *)
+
+(* Version : [Ste87] (equality) axioms : Augmented. *)
+
+(* English : assr(X.Y,Z,W)+assr(X,Y,comm(Z,W)) = X.assr(Y,Z,W)+assr(X,Z,W).Y *)
+
+(* Refs : [Ste87] Stevens (1987), Some Experiments in Nonassociative Rin *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unknown *)
+
+(* Rating : 1.00 v2.3.0 *)
+
+(* Syntax : Number of clauses : 24 ( 0 non-Horn; 24 unit; 1 RR) *)
+
+(* Number of atoms : 24 ( 24 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 10 ( 5 constant; 0-3 arity) *)
+
+(* Number of variables : 48 ( 2 singleton) *)
+
+(* Maximal term depth : 5 ( 3 average) *)
+
+(* Comments : This includes the right Moufang identity, which [Ste87] *)
+
+(* suggests in a useful lemma for this problem. *)
+
+(* Bugfixes : v2.3.0 - Clause right_moufang fixed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include nonassociative ring axioms *)
+
+(* Inclusion of: Axioms/RNG003-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG003-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Ring Theory (Alternative) *)
+
+(* Axioms : Alternative ring theory (equality) axioms *)
+
+(* Version : [Ste87] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Ste87] Stevens (1987), Some Experiments in Nonassociative Rin *)
+
+(* Source : [Ste87] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 15 ( 0 non-Horn; 15 unit; 0 RR) *)
+
+(* Number of literals : 15 ( 15 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 1 constant; 0-3 arity) *)
+
+(* Number of variables : 27 ( 2 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----There exists an additive identity element *)
+
+(* ----Multiplicative zero *)
+
+(* ----Existence of left additive additive_inverse *)
+
+(* ----Inverse of additive_inverse of X is X *)
+
+(* ----Distributive property of product over sum *)
+
+(* ----Commutativity for addition *)
+
+(* ----Associativity for addition *)
+
+(* ----Right alternative law *)
+
+(* ----Left alternative law *)
+
+(* ----Associator *)
+
+(* ----Commutator *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----The next 7 clause are extra lemmas which Stevens found useful *)
+
+(* ----Right Moufang *)
+ntheorem prove_challenge:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀additive_identity:Univ.
+∀additive_inverse:∀_:Univ.Univ.
+∀associator:∀_:Univ.∀_:Univ.∀_:Univ.Univ.
+∀commutator:∀_:Univ.∀_:Univ.Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀w:Univ.
+∀x:Univ.
+∀y:Univ.
+∀z:Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply Z (multiply X (multiply Y X))) (multiply (multiply (multiply Z X) Y) X).
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X Y) (additive_inverse Z)) (add (additive_inverse (multiply X Z)) (additive_inverse (multiply Y Z))).
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (additive_inverse X) (add Y Z)) (add (additive_inverse (multiply X Y)) (additive_inverse (multiply X Z))).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X (additive_inverse Y)) Z) (add (multiply X Z) (additive_inverse (multiply Y Z))).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y (additive_inverse Z))) (add (multiply X Y) (additive_inverse (multiply X Z))).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (multiply X (additive_inverse Y)) (additive_inverse (multiply X Y)).
+∀H6:∀X:Univ.∀Y:Univ.eq Univ (multiply (additive_inverse X) Y) (additive_inverse (multiply X Y)).
+∀H7:∀X:Univ.∀Y:Univ.eq Univ (multiply (additive_inverse X) (additive_inverse Y)) (multiply X Y).
+∀H8:∀X:Univ.∀Y:Univ.eq Univ (commutator X Y) (add (multiply Y X) (additive_inverse (multiply X Y))).
+∀H9:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (associator X Y Z) (add (multiply (multiply X Y) Z) (additive_inverse (multiply X (multiply Y Z)))).
+∀H10:∀X:Univ.∀Y:Univ.eq Univ (multiply (multiply X X) Y) (multiply X (multiply X Y)).
+∀H11:∀X:Univ.∀Y:Univ.eq Univ (multiply (multiply X Y) Y) (multiply X (multiply Y Y)).
+∀H12:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add X (add Y Z)) (add (add X Y) Z).
+∀H13:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).
+∀H14:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X Y) Z) (add (multiply X Z) (multiply Y Z)).
+∀H15:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y Z)) (add (multiply X Y) (multiply X Z)).
+∀H16:∀X:Univ.eq Univ (additive_inverse (additive_inverse X)) X.
+∀H17:∀X:Univ.eq Univ (add X (additive_inverse X)) additive_identity.
+∀H18:∀X:Univ.eq Univ (add (additive_inverse X) X) additive_identity.
+∀H19:∀X:Univ.eq Univ (multiply X additive_identity) additive_identity.
+∀H20:∀X:Univ.eq Univ (multiply additive_identity X) additive_identity.
+∀H21:∀X:Univ.eq Univ (add X additive_identity) X.
+∀H22:∀X:Univ.eq Univ (add additive_identity X) X.eq Univ (add (associator (multiply x y) z w) (associator x y (commutator z w))) (add (multiply x (associator y z w)) (multiply (associator x z w) y))
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#add.
+#additive_identity.
+#additive_inverse.
+#associator.
+#commutator.
+#multiply.
+#w.
+#x.
+#y.
+#z.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+#H11.
+#H12.
+#H13.
+#H14.
+#H15.
+#H16.
+#H17.
+#H18.
+#H19.
+#H20.
+#H21.
+#H22.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15,H16,H17,H18,H19,H20,H21,H22;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: RNG035-7.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG035-7 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Ring Theory *)
+
+(* Problem : If X*X*X*X = X then the ring is commutative *)
+
+(* Version : [LW91] (equality) axioms. *)
+
+(* English : Given a ring in which for all x, x * x * x * x = x, prove *)
+
+(* that for all x and y, x * y = y * x. *)
+
+(* Refs : [LW91] Lusk & Wos (1991), Benchmark Problems in Which Equalit *)
+
+(* Source : [LW91] *)
+
+(* Names : RT3 [LW91] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.79 v3.2.0, 0.86 v3.1.0, 0.67 v2.7.0, 0.73 v2.6.0, 0.50 v2.5.0, 0.25 v2.4.0, 0.33 v2.2.1, 1.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 2 RR) *)
+
+(* Number of atoms : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 19 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include ring theory axioms *)
+
+(* Inclusion of: Axioms/RNG005-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG005-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Ring Theory *)
+
+(* Axioms : Ring theory (equality) axioms *)
+
+(* Version : [LW92] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* : [LW92] Lusk & Wos (1992), Benchmark Problems in Which Equalit *)
+
+(* Source : [LW92] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 9 ( 0 non-Horn; 9 unit; 0 RR) *)
+
+(* Number of literals : 9 ( 9 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 18 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : These axioms are used in [Wos88] p.203. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----There exists an additive identity element *)
+
+(* ----Existence of left additive additive_inverse *)
+
+(* ----Associativity for addition *)
+
+(* ----Commutativity for addition *)
+
+(* ----Associativity for multiplication *)
+
+(* ----Distributive property of product over sum *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_commutativity:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀additive_identity:Univ.
+∀additive_inverse:∀_:Univ.Univ.
+∀b:Univ.
+∀c:Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:eq Univ (multiply a b) c.
+∀H1:∀X:Univ.eq Univ (multiply X (multiply X (multiply X X))) X.
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X Y) Z) (add (multiply X Z) (multiply Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y Z)) (add (multiply X Y) (multiply X Z)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (multiply Y Z)) (multiply (multiply X Y) Z).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).
+∀H6:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add X (add Y Z)) (add (add X Y) Z).
+∀H7:∀X:Univ.eq Univ (add X (additive_inverse X)) additive_identity.
+∀H8:∀X:Univ.eq Univ (add (additive_inverse X) X) additive_identity.
+∀H9:∀X:Univ.eq Univ (add X additive_identity) X.
+∀H10:∀X:Univ.eq Univ (add additive_identity X) X.eq Univ (multiply b a) c
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#add.
+#additive_identity.
+#additive_inverse.
+#b.
+#c.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: RNG036-7.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG036-7 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Ring Theory *)
+
+(* Problem : If X*X*X*X*X = X then the ring is commutative *)
+
+(* Version : [LW91] (equality) axioms. *)
+
+(* English : Given a ring in which for all x, x * x * x * x * x = x, prove *)
+
+(* that for all x and y, x * y = y * x. *)
+
+(* Refs : [LW91] Lusk & Wos (1991), Benchmark Problems in Which Equalit *)
+
+(* Source : [LW91] *)
+
+(* Names : RT4 [LW91] *)
+
+(* Status : Unknown *)
+
+(* Rating : 1.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 2 RR) *)
+
+(* Number of atoms : 12 ( 12 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 7 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 19 ( 0 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include ring theory axioms *)
+
+(* Inclusion of: Axioms/RNG005-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG005-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Ring Theory *)
+
+(* Axioms : Ring theory (equality) axioms *)
+
+(* Version : [LW92] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* : [LW92] Lusk & Wos (1992), Benchmark Problems in Which Equalit *)
+
+(* Source : [LW92] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 9 ( 0 non-Horn; 9 unit; 0 RR) *)
+
+(* Number of literals : 9 ( 9 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 18 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : These axioms are used in [Wos88] p.203. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----There exists an additive identity element *)
+
+(* ----Existence of left additive additive_inverse *)
+
+(* ----Associativity for addition *)
+
+(* ----Commutativity for addition *)
+
+(* ----Associativity for multiplication *)
+
+(* ----Distributive property of product over sum *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_commutativity:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀additive_identity:Univ.
+∀additive_inverse:∀_:Univ.Univ.
+∀b:Univ.
+∀c:Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:eq Univ (multiply a b) c.
+∀H1:∀X:Univ.eq Univ (multiply X (multiply X (multiply X (multiply X X)))) X.
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X Y) Z) (add (multiply X Z) (multiply Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y Z)) (add (multiply X Y) (multiply X Z)).
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (multiply Y Z)) (multiply (multiply X Y) Z).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).
+∀H6:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add X (add Y Z)) (add (add X Y) Z).
+∀H7:∀X:Univ.eq Univ (add X (additive_inverse X)) additive_identity.
+∀H8:∀X:Univ.eq Univ (add (additive_inverse X) X) additive_identity.
+∀H9:∀X:Univ.eq Univ (add X additive_identity) X.
+∀H10:∀X:Univ.eq Univ (add additive_identity X) X.eq Univ (multiply b a) c
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#add.
+#additive_identity.
+#additive_inverse.
+#b.
+#c.
+#multiply.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+#H6.
+#H7.
+#H8.
+#H9.
+#H10.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: RNG042-2.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG042-2 : TPTP v3.2.0. Released v2.5.0. *)
+
+(* Domain : Ring Theory *)
+
+(* Problem : Ring theory (equality) axioms *)
+
+(* Version : [PS81] (equality) axioms : *)
+
+(* Reduced & Augmented > Complete. *)
+
+(* English : *)
+
+(* Refs : [PS81] Peterson & Stickel (1981), Complete Sets of Reductions *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : Satisfiable *)
+
+(* Rating : 0.00 v3.2.0, 0.33 v3.1.0, 0.00 v2.5.0 *)
+
+(* Syntax : Number of clauses : 14 ( 0 non-Horn; 14 unit; 1 RR) *)
+
+(* Number of atoms : 14 ( 14 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 25 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Ring theory (equality) axioms *)
+
+(* Inclusion of: Axioms/RNG002-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG002-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Ring Theory *)
+
+(* Axioms : Ring theory (equality) axioms *)
+
+(* Version : [PS81] (equality) axioms : *)
+
+(* Reduced & Augmented > Complete. *)
+
+(* English : *)
+
+(* Refs : [PS81] Peterson & Stickel (1981), Complete Sets of Reductions *)
+
+(* Source : [ANL] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 14 ( 0 non-Horn; 14 unit; 1 RR) *)
+
+(* Number of literals : 14 ( 14 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 25 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : Not sure if these are complete. I don't know if the reductions *)
+
+(* given in [PS81] are suitable for ATP. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Existence of left identity for addition *)
+
+(* ----Existence of left additive additive_inverse *)
+
+(* ----Distributive property of product over sum *)
+
+(* ----Inverse of identity is identity, stupid *)
+
+(* ----Inverse of additive_inverse of X is X *)
+
+(* ----Behavior of 0 and the multiplication operation *)
+
+(* ----Inverse of (x + y) is additive_inverse(x) + additive_inverse(y) *)
+
+(* ----x * additive_inverse(y) = additive_inverse (x * y) *)
+
+(* ----Associativity of addition *)
+
+(* ----Commutativity of addition *)
+
+(* ----Associativity of product *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: RNG042-3.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG042-3 : TPTP v3.2.0. Released v2.5.0. *)
+
+(* Domain : Ring Theory *)
+
+(* Problem : Ring theory (equality) axioms *)
+
+(* Version : [LW92] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* : [LW92] Lusk & Wos (1992), Benchmark Problems in Which Equalit *)
+
+(* Source : [LW92] *)
+
+(* Names : *)
+
+(* Status : Satisfiable *)
+
+(* Rating : 0.00 v3.2.0, 0.33 v3.1.0, 0.00 v2.5.0 *)
+
+(* Syntax : Number of clauses : 9 ( 0 non-Horn; 9 unit; 0 RR) *)
+
+(* Number of atoms : 9 ( 9 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 18 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Ring theory (equality) axioms *)
+
+(* Inclusion of: Axioms/RNG005-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG005-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Ring Theory *)
+
+(* Axioms : Ring theory (equality) axioms *)
+
+(* Version : [LW92] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+
+(* : [LW92] Lusk & Wos (1992), Benchmark Problems in Which Equalit *)
+
+(* Source : [LW92] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 9 ( 0 non-Horn; 9 unit; 0 RR) *)
+
+(* Number of literals : 9 ( 9 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 1 constant; 0-2 arity) *)
+
+(* Number of variables : 18 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : These axioms are used in [Wos88] p.203. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----There exists an additive identity element *)
+
+(* ----Existence of left additive additive_inverse *)
+
+(* ----Associativity for addition *)
+
+(* ----Commutativity for addition *)
+
+(* ----Associativity for multiplication *)
+
+(* ----Distributive property of product over sum *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: RNG043-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG043-1 : TPTP v3.2.0. Released v2.5.0. *)
+
+(* Domain : Ring Theory (Alternative) *)
+
+(* Problem : Alternative ring theory (equality) axioms *)
+
+(* Version : [Ste87] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Ste87] Stevens (1987), Some Experiments in Nonassociative Rin *)
+
+(* Source : [Ste87] *)
+
+(* Names : *)
+
+(* Status : Satisfiable *)
+
+(* Rating : 0.00 v3.2.0, 0.33 v3.1.0, 0.00 v2.5.0 *)
+
+(* Syntax : Number of clauses : 15 ( 0 non-Horn; 15 unit; 0 RR) *)
+
+(* Number of atoms : 15 ( 15 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 1 constant; 0-3 arity) *)
+
+(* Number of variables : 27 ( 2 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Alternative ring theory (equality) axioms *)
+
+(* Inclusion of: Axioms/RNG003-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG003-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Ring Theory (Alternative) *)
+
+(* Axioms : Alternative ring theory (equality) axioms *)
+
+(* Version : [Ste87] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Ste87] Stevens (1987), Some Experiments in Nonassociative Rin *)
+
+(* Source : [Ste87] *)
+
+(* Names : *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 15 ( 0 non-Horn; 15 unit; 0 RR) *)
+
+(* Number of literals : 15 ( 15 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 1 constant; 0-3 arity) *)
+
+(* Number of variables : 27 ( 2 singleton) *)
+
+(* Maximal term depth : 5 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----There exists an additive identity element *)
+
+(* ----Multiplicative zero *)
+
+(* ----Existence of left additive additive_inverse *)
+
+(* ----Inverse of additive_inverse of X is X *)
+
+(* ----Distributive property of product over sum *)
+
+(* ----Commutativity for addition *)
+
+(* ----Associativity for addition *)
+
+(* ----Right alternative law *)
+
+(* ----Left alternative law *)
+
+(* ----Associator *)
+
+(* ----Commutator *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: ROB001-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : ROB001-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Robbins Algebra *)
+
+(* Problem : Is every Robbins algebra Boolean? *)
+
+(* Version : [Win90] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [HMT71] Henkin et al. (1971), Cylindrical Algebras *)
+
+(* : [Win90] Winker (1990), Robbins Algebra: Conditions that make a *)
+
+(* Source : [TPTP] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 1.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 4 ( 0 non-Horn; 4 unit; 1 RR) *)
+
+(* Number of atoms : 4 ( 4 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : Commutativity, associativity, and Huntington's axiom *)
+
+(* axiomatize Boolean algebra. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include axioms for Robbins algebra *)
+
+(* Inclusion of: Axioms/ROB001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : ROB001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Robbins algebra *)
+
+(* Axioms : Robbins algebra axioms *)
+
+(* Version : [Win90] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [HMT71] Henkin et al. (1971), Cylindrical Algebras *)
+
+(* : [Win90] Winker (1990), Robbins Algebra: Conditions that make a *)
+
+(* Source : [OTTER] *)
+
+(* Names : Lemma 2.2 [Win90] *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 1-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_huntingtons_axiom:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀negate:∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (negate (add (negate (add X Y)) (negate (add X (negate Y))))) X.
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add (add X Y) Z) (add X (add Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).eq Univ (add (negate (add a (negate b))) (negate (add (negate a) (negate b)))) b
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#add.
+#b.
+#negate.
+#H0.
+#H1.
+#H2.
+nauto by H0,H1,H2;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: ROB002-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : ROB002-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Robbins Algebra *)
+
+(* Problem : --X = X => Boolean *)
+
+(* Version : [Win90] (equality) axioms. *)
+
+(* English : If --X = X then the algebra is Boolean. *)
+
+(* Refs : [HMT71] Henkin et al. (1971), Cylindrical Algebras *)
+
+(* : [Win90] Winker (1990), Robbins Algebra: Conditions that make a *)
+
+(* Source : [Win90] *)
+
+(* Names : Lemma 2.1 [Win90] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.2.1, 0.11 v2.2.0, 0.14 v2.1.0, 0.13 v2.0.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 1 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 8 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : Commutativity, associativity, and Huntington's axiom *)
+
+(* axiomatize Boolean algebra. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include axioms for Robbins algebra *)
+
+(* Inclusion of: Axioms/ROB001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : ROB001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Robbins algebra *)
+
+(* Axioms : Robbins algebra axioms *)
+
+(* Version : [Win90] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [HMT71] Henkin et al. (1971), Cylindrical Algebras *)
+
+(* : [Win90] Winker (1990), Robbins Algebra: Conditions that make a *)
+
+(* Source : [OTTER] *)
+
+(* Names : Lemma 2.2 [Win90] *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 1-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_huntingtons_axiom:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀negate:∀_:Univ.Univ.
+∀H0:∀X:Univ.eq Univ (negate (negate X)) X.
+∀H1:∀X:Univ.∀Y:Univ.eq Univ (negate (add (negate (add X Y)) (negate (add X (negate Y))))) X.
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add (add X Y) Z) (add X (add Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).eq Univ (add (negate (add a (negate b))) (negate (add (negate a) (negate b)))) b
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#add.
+#b.
+#negate.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: ROB003-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : ROB003-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Robbins Algebra *)
+
+(* Problem : X + c=c => Boolean *)
+
+(* Version : [Win90] (equality) axioms. *)
+
+(* English : If there exists c such that X+c=c, then the algebra *)
+
+(* is Boolean. *)
+
+(* Refs : [HMT71] Henkin et al. (1971), Cylindrical Algebras *)
+
+(* : [Win90] Winker (1990), Robbins Algebra: Conditions that make a *)
+
+(* : [LW92] Lusk & Wos (1992), Benchmark Problems in Which Equalit *)
+
+(* Source : [OTTER] *)
+
+(* Names : Lemma 2.2 [Win90] *)
+
+(* : RA1 [LW92] *)
+
+(* : robbins.in [OTTER] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.2.1, 0.22 v2.2.0, 0.29 v2.1.0, 0.25 v2.0.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 1 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 8 ( 1 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : Commutativity, associativity, and Huntington's axiom *)
+
+(* axiomatize Boolean algebra. *)
+
+(* : In Overbeek's version, the hypothesis is slightly different : *)
+
+(* ...an element c such that c+c=c, then... Mail from McCune says *)
+
+(* that this is a simpler problem. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include axioms for Robbins algebra *)
+
+(* Inclusion of: Axioms/ROB001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : ROB001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Robbins algebra *)
+
+(* Axioms : Robbins algebra axioms *)
+
+(* Version : [Win90] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [HMT71] Henkin et al. (1971), Cylindrical Algebras *)
+
+(* : [Win90] Winker (1990), Robbins Algebra: Conditions that make a *)
+
+(* Source : [OTTER] *)
+
+(* Names : Lemma 2.2 [Win90] *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 1-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_huntingtons_axiom:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀c:Univ.
+∀negate:∀_:Univ.Univ.
+∀H0:∀X:Univ.eq Univ (add X c) c.
+∀H1:∀X:Univ.∀Y:Univ.eq Univ (negate (add (negate (add X Y)) (negate (add X (negate Y))))) X.
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add (add X Y) Z) (add X (add Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).eq Univ (add (negate (add a (negate b))) (negate (add (negate a) (negate b)))) b
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#add.
+#b.
+#c.
+#negate.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: ROB004-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : ROB004-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Robbins Algebra *)
+
+(* Problem : c = -d, c + d=d, and c + c=c => Boolean *)
+
+(* Version : [Win90] (equality) axioms. *)
+
+(* English : If there exist c, d such that c = -d, c+d=d, and c+c=c, then *)
+
+(* the algebra is Boolean. *)
+
+(* Refs : [HMT71] Henkin et al. (1971), Cylindrical Algebras *)
+
+(* : [Win90] Winker (1990), Robbins Algebra: Conditions that make a *)
+
+(* Source : [Win90] *)
+
+(* Names : Lemma 2.3 [Win90] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.2.1, 0.22 v2.2.0, 0.29 v2.1.0, 0.38 v2.0.0 *)
+
+(* Syntax : Number of clauses : 7 ( 0 non-Horn; 7 unit; 4 RR) *)
+
+(* Number of atoms : 7 ( 7 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 2 average) *)
+
+(* Comments : Commutativity, associativity, and Huntington's axiom *)
+
+(* axiomatize Boolean algebra. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include axioms for Robbins algebra *)
+
+(* Inclusion of: Axioms/ROB001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : ROB001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Robbins algebra *)
+
+(* Axioms : Robbins algebra axioms *)
+
+(* Version : [Win90] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [HMT71] Henkin et al. (1971), Cylindrical Algebras *)
+
+(* : [Win90] Winker (1990), Robbins Algebra: Conditions that make a *)
+
+(* Source : [OTTER] *)
+
+(* Names : Lemma 2.2 [Win90] *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 1-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_huntingtons_axiom:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀c:Univ.
+∀d:Univ.
+∀negate:∀_:Univ.Univ.
+∀H0:eq Univ (add c c) c.
+∀H1:eq Univ (add c d) d.
+∀H2:eq Univ (negate d) c.
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (negate (add (negate (add X Y)) (negate (add X (negate Y))))) X.
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add (add X Y) Z) (add X (add Y Z)).
+∀H5:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).eq Univ (add (negate (add a (negate b))) (negate (add (negate a) (negate b)))) b
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#add.
+#b.
+#c.
+#d.
+#negate.
+#H0.
+#H1.
+#H2.
+#H3.
+#H4.
+#H5.
+nauto by H0,H1,H2,H3,H4,H5;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: ROB005-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : ROB005-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Robbins Algebra *)
+
+(* Problem : Exists an idempotent element => Boolean *)
+
+(* Version : [Win90] (equality) axioms. *)
+
+(* English : If there is an element c such that c+c=c, then the algebra *)
+
+(* is Boolean. *)
+
+(* Refs : [HMT71] Henkin et al. (1971), Cylindrical Algebras *)
+
+(* : [Win90] Winker (1990), Robbins Algebra: Conditions that make a *)
+
+(* : [Ove90] Overbeek (1990), ATP competition announced at CADE-10 *)
+
+(* : [LW92] Lusk & Wos (1992), Benchmark Problems in Which Equalit *)
+
+(* : [Ove93] Overbeek (1993), The CADE-11 Competitions: A Personal *)
+
+(* : [LM93] Lusk & McCune (1993), Uniform Strategies: The CADE-11 *)
+
+(* : [Zha93] Zhang (1993), Automated Proofs of Equality Problems in *)
+
+(* Source : [Ove90] *)
+
+(* Names : CADE-11 Competition Eq-2 [Ove90] *)
+
+(* : Lemma 2.4 [Win90] *)
+
+(* : RA3 [LW92] *)
+
+(* : THEOREM EQ-2 [LM93] *)
+
+(* : PROBLEM 2 [Zha93] *)
+
+(* : robbins.occ.in [OTTER] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.14 v3.2.0, 0.07 v3.1.0, 0.11 v2.7.0, 0.09 v2.6.0, 0.17 v2.5.0, 0.00 v2.4.0, 0.33 v2.3.0, 0.67 v2.2.1, 0.67 v2.2.0, 0.71 v2.1.0, 0.88 v2.0.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 2 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : Commutativity, associativity, and Huntington's axiom *)
+
+(* axiomatize Boolean algebra. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include axioms for Robbins algebra *)
+
+(* Inclusion of: Axioms/ROB001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : ROB001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Robbins algebra *)
+
+(* Axioms : Robbins algebra axioms *)
+
+(* Version : [Win90] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [HMT71] Henkin et al. (1971), Cylindrical Algebras *)
+
+(* : [Win90] Winker (1990), Robbins Algebra: Conditions that make a *)
+
+(* Source : [OTTER] *)
+
+(* Names : Lemma 2.2 [Win90] *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 1-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_huntingtons_axiom:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀c:Univ.
+∀negate:∀_:Univ.Univ.
+∀H0:eq Univ (add c c) c.
+∀H1:∀X:Univ.∀Y:Univ.eq Univ (negate (add (negate (add X Y)) (negate (add X (negate Y))))) X.
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add (add X Y) Z) (add X (add Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).eq Univ (add (negate (add a (negate b))) (negate (add (negate a) (negate b)))) b
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#add.
+#b.
+#c.
+#negate.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: ROB006-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : ROB006-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Robbins Algebra *)
+
+(* Problem : Exists absorbed element => Boolean *)
+
+(* Version : [Win90] (equality) axioms. *)
+
+(* Theorem formulation : Denies Huntington's axiom. *)
+
+(* English : If there are elements c and d such that c+d=d, then the *)
+
+(* algebra is Boolean. *)
+
+(* Refs : [HMT71] Henkin et al. (1971), Cylindrical Algebras *)
+
+(* : [Win90] Winker (1990), Robbins Algebra: Conditions that make a *)
+
+(* : [Wos92] Wos (1992), An Opportunity to Test Your Skills, and th *)
+
+(* : [LW92] Lusk & Wos (1992), Benchmark Problems in Which Equalit *)
+
+(* Source : [Wos92] *)
+
+(* Names : Theorem 1.1 [Win90] *)
+
+(* : RA4 [LW92] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.86 v3.1.0, 0.89 v2.7.0, 0.91 v2.6.0, 0.83 v2.5.0, 0.75 v2.4.0, 0.67 v2.3.0, 1.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 2 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : Commutativity, associativity, and Huntington's axiom *)
+
+(* axiomatize Boolean algebra. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include axioms for Robbins algebra *)
+
+(* Inclusion of: Axioms/ROB001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : ROB001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Robbins algebra *)
+
+(* Axioms : Robbins algebra axioms *)
+
+(* Version : [Win90] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [HMT71] Henkin et al. (1971), Cylindrical Algebras *)
+
+(* : [Win90] Winker (1990), Robbins Algebra: Conditions that make a *)
+
+(* Source : [OTTER] *)
+
+(* Names : Lemma 2.2 [Win90] *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 1-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_huntingtons_axiom:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀c:Univ.
+∀d:Univ.
+∀negate:∀_:Univ.Univ.
+∀H0:eq Univ (add c d) d.
+∀H1:∀X:Univ.∀Y:Univ.eq Univ (negate (add (negate (add X Y)) (negate (add X (negate Y))))) X.
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add (add X Y) Z) (add X (add Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).eq Univ (add (negate (add a (negate b))) (negate (add (negate a) (negate b)))) b
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#add.
+#b.
+#c.
+#d.
+#negate.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: ROB006-2.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : ROB006-2 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Robbins Algebra *)
+
+(* Problem : Exists absorbed element => Exists idempotent element *)
+
+(* Version : [Win90] (equality) axioms. *)
+
+(* Theorem formulation : Denies idempotence. *)
+
+(* English : If there are elements c and d such that c+d=d, then the *)
+
+(* algebra is Boolean. *)
+
+(* Refs : [HMT71] Henkin et al. (1971), Cylindrical Algebras *)
+
+(* : [Win90] Winker (1990), Robbins Algebra: Conditions that make a *)
+
+(* : [Wos92] Wos (1992), An Opportunity to Test Your Skills, and th *)
+
+(* Source : [Wos92] *)
+
+(* Names : Theorem 1.1 [Win90] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.86 v3.1.0, 0.89 v2.7.0, 0.91 v2.6.0, 0.83 v2.5.0, 0.75 v2.4.0, 0.67 v2.3.0, 1.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 2 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 8 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 2 average) *)
+
+(* Comments : Commutativity, associativity, and Huntington's axiom *)
+
+(* axiomatize Boolean algebra. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include axioms for Robbins algebra *)
+
+(* Inclusion of: Axioms/ROB001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : ROB001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Robbins algebra *)
+
+(* Axioms : Robbins algebra axioms *)
+
+(* Version : [Win90] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [HMT71] Henkin et al. (1971), Cylindrical Algebras *)
+
+(* : [Win90] Winker (1990), Robbins Algebra: Conditions that make a *)
+
+(* Source : [OTTER] *)
+
+(* Names : Lemma 2.2 [Win90] *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 1-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_idempotence:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀c:Univ.
+∀d:Univ.
+∀negate:∀_:Univ.Univ.
+∀H0:eq Univ (add c d) d.
+∀H1:∀X:Univ.∀Y:Univ.eq Univ (negate (add (negate (add X Y)) (negate (add X (negate Y))))) X.
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add (add X Y) Z) (add X (add Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).∃X:Univ.eq Univ (add X X) X
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#add.
+#c.
+#d.
+#negate.
+#H0.
+#H1.
+#H2.
+#H3.
+napply ex_intro[
+nid2:
+nauto by H0,H1,H2,H3;
+nid|
+skip]
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: ROB007-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : ROB007-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Robbins Algebra *)
+
+(* Problem : Absorbed within negation element => Boolean *)
+
+(* Version : [Win90] (equality) axioms. *)
+
+(* English : If there exist a, b such that -(a+b) = -b, then the algebra *)
+
+(* is Boolean. *)
+
+(* Refs : [HMT71] Henkin et al. (1971), Cylindrical Algebras *)
+
+(* : [Win90] Winker (1990), Robbins Algebra: Conditions that make a *)
+
+(* : [LW92] Lusk & Wos (1992), Benchmark Problems in Which Equalit *)
+
+(* Source : [Win90] *)
+
+(* Names : Theorem 1.2 [Win90] *)
+
+(* : RA5 [LW92] *)
+
+(* Status : Unknown *)
+
+(* Rating : 1.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 2 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : Commutativity, associativity, and Huntington's axiom *)
+
+(* axiomatize Boolean algebra. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include axioms for Robbins algebra *)
+
+(* Inclusion of: Axioms/ROB001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : ROB001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Robbins algebra *)
+
+(* Axioms : Robbins algebra axioms *)
+
+(* Version : [Win90] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [HMT71] Henkin et al. (1971), Cylindrical Algebras *)
+
+(* : [Win90] Winker (1990), Robbins Algebra: Conditions that make a *)
+
+(* Source : [OTTER] *)
+
+(* Names : Lemma 2.2 [Win90] *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 1-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_huntingtons_axiom:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀negate:∀_:Univ.Univ.
+∀H0:eq Univ (negate (add a b)) (negate b).
+∀H1:∀X:Univ.∀Y:Univ.eq Univ (negate (add (negate (add X Y)) (negate (add X (negate Y))))) X.
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add (add X Y) Z) (add X (add Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).eq Univ (add (negate (add a (negate b))) (negate (add (negate a) (negate b)))) b
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#add.
+#b.
+#negate.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: ROB007-2.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : ROB007-2 : TPTP v3.2.0. Bugfixed v3.1.0. *)
+
+(* Domain : Robbins Algebra *)
+
+(* Problem : Absorbed within negation element => Exists idempotent element *)
+
+(* Version : [Win90] (equality) axioms. *)
+
+(* English : If there exist a, b such that -(a+b) = -b, then the algebra *)
+
+(* is Boolean. *)
+
+(* Refs : [HMT71] Henkin et al. (1971), Cylindrical Algebras *)
+
+(* : [Win90] Winker (1990), Robbins Algebra: Conditions that make a *)
+
+(* Source : [Win90] *)
+
+(* Names : Theorem 1.2 [Win90] *)
+
+(* Status : Unknown *)
+
+(* Rating : 1.00 v3.1.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 2 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 8 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 2 average) *)
+
+(* Comments : Commutativity, associativity, and Huntington's axiom *)
+
+(* axiomatize Boolean algebra. *)
+
+(* Bugfixes : v3.1.0 - Removed extra negated_conjecture clauses. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include axioms for Robbins algebra *)
+
+(* Inclusion of: Axioms/ROB001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : ROB001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Robbins algebra *)
+
+(* Axioms : Robbins algebra axioms *)
+
+(* Version : [Win90] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [HMT71] Henkin et al. (1971), Cylindrical Algebras *)
+
+(* : [Win90] Winker (1990), Robbins Algebra: Conditions that make a *)
+
+(* Source : [OTTER] *)
+
+(* Names : Lemma 2.2 [Win90] *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 1-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_idempotence:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀negate:∀_:Univ.Univ.
+∀H0:eq Univ (negate (add a b)) (negate b).
+∀H1:∀X:Univ.∀Y:Univ.eq Univ (negate (add (negate (add X Y)) (negate (add X (negate Y))))) X.
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add (add X Y) Z) (add X (add Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).∃X:Univ.eq Univ (add X X) X
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#add.
+#b.
+#negate.
+#H0.
+#H1.
+#H2.
+#H3.
+napply ex_intro[
+nid2:
+nauto by H0,H1,H2,H3;
+nid|
+skip]
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: ROB008-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : ROB008-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Robbins Algebra *)
+
+(* Problem : If -(a + -(b + c)) = -(a + b + -c) then a+b=a *)
+
+(* Version : [Win90] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Win90] Winker (1990), Robbins Algebra: Conditions that make a *)
+
+(* Source : [Win90] *)
+
+(* Names : Lemma 3.1 [Win90] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.2.1, 0.44 v2.2.0, 0.43 v2.1.0, 0.62 v2.0.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 2 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include axioms for Robbins algebra *)
+
+(* Inclusion of: Axioms/ROB001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : ROB001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Robbins algebra *)
+
+(* Axioms : Robbins algebra axioms *)
+
+(* Version : [Win90] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [HMT71] Henkin et al. (1971), Cylindrical Algebras *)
+
+(* : [Win90] Winker (1990), Robbins Algebra: Conditions that make a *)
+
+(* Source : [OTTER] *)
+
+(* Names : Lemma 2.2 [Win90] *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 1-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_result:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀c:Univ.
+∀negate:∀_:Univ.Univ.
+∀H0:eq Univ (negate (add a (negate (add b c)))) (negate (add a (add b (negate c)))).
+∀H1:∀X:Univ.∀Y:Univ.eq Univ (negate (add (negate (add X Y)) (negate (add X (negate Y))))) X.
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add (add X Y) Z) (add X (add Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).eq Univ (add a b) a
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#add.
+#b.
+#c.
+#negate.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: ROB009-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : ROB009-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Robbins Algebra *)
+
+(* Problem : If -(a + -(b + c)) = -(b + -(a + c)) then a = b *)
+
+(* Version : [Win90] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Win90] Winker (1990), Robbins Algebra: Conditions that make a *)
+
+(* Source : [Win90] *)
+
+(* Names : Lemma 3.2 [Win90] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.2.1, 0.22 v2.2.0, 0.14 v2.1.0, 0.50 v2.0.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 2 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include axioms for Robbins algebra *)
+
+(* Inclusion of: Axioms/ROB001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : ROB001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Robbins algebra *)
+
+(* Axioms : Robbins algebra axioms *)
+
+(* Version : [Win90] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [HMT71] Henkin et al. (1971), Cylindrical Algebras *)
+
+(* : [Win90] Winker (1990), Robbins Algebra: Conditions that make a *)
+
+(* Source : [OTTER] *)
+
+(* Names : Lemma 2.2 [Win90] *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 1-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_result:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀c:Univ.
+∀negate:∀_:Univ.Univ.
+∀H0:eq Univ (negate (add a (negate (add b c)))) (negate (add b (negate (add a c)))).
+∀H1:∀X:Univ.∀Y:Univ.eq Univ (negate (add (negate (add X Y)) (negate (add X (negate Y))))) X.
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add (add X Y) Z) (add X (add Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).eq Univ a b
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#add.
+#b.
+#c.
+#negate.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: ROB010-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : ROB010-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Robbins Algebra *)
+
+(* Problem : If -(a + -b) = c then -(c + -(b + a)) = a *)
+
+(* Version : [Win90] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Win90] Winker (1990), Robbins Algebra: Conditions that make a *)
+
+(* : [LW92] Lusk & Wos (1992), Benchmark Problems in Which Equalit *)
+
+(* Source : [Win90] *)
+
+(* Names : Lemma 3.3 [Win90] *)
+
+(* : RA2 [LW92] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 2 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include axioms for Robbins algebra *)
+
+(* Inclusion of: Axioms/ROB001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : ROB001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Robbins algebra *)
+
+(* Axioms : Robbins algebra axioms *)
+
+(* Version : [Win90] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [HMT71] Henkin et al. (1971), Cylindrical Algebras *)
+
+(* : [Win90] Winker (1990), Robbins Algebra: Conditions that make a *)
+
+(* Source : [OTTER] *)
+
+(* Names : Lemma 2.2 [Win90] *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 1-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_result:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀c:Univ.
+∀negate:∀_:Univ.Univ.
+∀H0:eq Univ (negate (add a (negate b))) c.
+∀H1:∀X:Univ.∀Y:Univ.eq Univ (negate (add (negate (add X Y)) (negate (add X (negate Y))))) X.
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add (add X Y) Z) (add X (add Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).eq Univ (negate (add c (negate (add b a)))) a
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#add.
+#b.
+#c.
+#negate.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: ROB013-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : ROB013-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Robbins Algebra *)
+
+(* Problem : If -(a + b) = c then -(c + -(-b + a)) = a *)
+
+(* Version : [Win90] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [Win90] Winker (1990), Robbins Algebra: Conditions that make a *)
+
+(* Source : [Win90] *)
+
+(* Names : Lemma 3.5 [Win90] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 2 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include axioms for Robbins algebra *)
+
+(* Inclusion of: Axioms/ROB001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : ROB001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Robbins algebra *)
+
+(* Axioms : Robbins algebra axioms *)
+
+(* Version : [Win90] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [HMT71] Henkin et al. (1971), Cylindrical Algebras *)
+
+(* : [Win90] Winker (1990), Robbins Algebra: Conditions that make a *)
+
+(* Source : [OTTER] *)
+
+(* Names : Lemma 2.2 [Win90] *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 1-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_result:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀c:Univ.
+∀negate:∀_:Univ.Univ.
+∀H0:eq Univ (negate (add a b)) c.
+∀H1:∀X:Univ.∀Y:Univ.eq Univ (negate (add (negate (add X Y)) (negate (add X (negate Y))))) X.
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add (add X Y) Z) (add X (add Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).eq Univ (negate (add c (negate (add (negate b) a)))) a
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#add.
+#b.
+#c.
+#negate.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: ROB020-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : ROB020-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Robbins Algebra *)
+
+(* Problem : -(a + -b)=b => Boolean *)
+
+(* Version : [Win90] (equality) axioms. *)
+
+(* Theorem formulation : Denies Huntington's axiom. *)
+
+(* English : If there exist a, b such that -(a + -b) = b, the algebra *)
+
+(* is Boolean. *)
+
+(* Refs : [HMT71] Henkin et al. (1971), Cylindrical Algebras *)
+
+(* : [Win90] Winker (1990), Robbins Algebra: Conditions that make a *)
+
+(* Source : [Win90] *)
+
+(* Names : Corollary 3.10 [Win90] *)
+
+(* Status : Unknown *)
+
+(* Rating : 1.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 2 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : Commutativity, associativity, and Huntington's axiom *)
+
+(* axiomatize Boolean algebra. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include axioms for Robbins algebra *)
+
+(* Inclusion of: Axioms/ROB001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : ROB001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Robbins algebra *)
+
+(* Axioms : Robbins algebra axioms *)
+
+(* Version : [Win90] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [HMT71] Henkin et al. (1971), Cylindrical Algebras *)
+
+(* : [Win90] Winker (1990), Robbins Algebra: Conditions that make a *)
+
+(* Source : [OTTER] *)
+
+(* Names : Lemma 2.2 [Win90] *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 1-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_huntingtons_axiom:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀negate:∀_:Univ.Univ.
+∀H0:eq Univ (negate (add a (negate b))) b.
+∀H1:∀X:Univ.∀Y:Univ.eq Univ (negate (add (negate (add X Y)) (negate (add X (negate Y))))) X.
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add (add X Y) Z) (add X (add Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).eq Univ (add (negate (add a (negate b))) (negate (add (negate a) (negate b)))) b
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#add.
+#b.
+#negate.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: ROB020-2.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : ROB020-2 : TPTP v3.2.0. Bugfixed v3.1.0. *)
+
+(* Domain : Robbins Algebra *)
+
+(* Problem : -(a + -b)=b => Exists idempotent element *)
+
+(* Version : [Win90] (equality) axioms. *)
+
+(* Theorem formulation : Denies Huntington's axiom, double *)
+
+(* negation, and idempotence. *)
+
+(* English : If there exist a, b such that -(a + -b) = b, the algebra *)
+
+(* is Boolean. *)
+
+(* Refs : [HMT71] Henkin et al. (1971), Cylindrical Algebras *)
+
+(* : [Win90] Winker (1990), Robbins Algebra: Conditions that make a *)
+
+(* Source : [Win90] *)
+
+(* Names : Corollary 3.10 [Win90] *)
+
+(* Status : Unknown *)
+
+(* Rating : 1.00 v3.1.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 2 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 8 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 2 average) *)
+
+(* Comments : Commutativity, associativity, and Huntington's axiom *)
+
+(* axiomatize Boolean algebra. *)
+
+(* Bugfixes : v3.1.0 - Removed extra negated_conjecture clauses. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include axioms for Robbins algebra *)
+
+(* Inclusion of: Axioms/ROB001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : ROB001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Robbins algebra *)
+
+(* Axioms : Robbins algebra axioms *)
+
+(* Version : [Win90] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [HMT71] Henkin et al. (1971), Cylindrical Algebras *)
+
+(* : [Win90] Winker (1990), Robbins Algebra: Conditions that make a *)
+
+(* Source : [OTTER] *)
+
+(* Names : Lemma 2.2 [Win90] *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 1-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_idempotence:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀negate:∀_:Univ.Univ.
+∀H0:eq Univ (negate (add a (negate b))) b.
+∀H1:∀X:Univ.∀Y:Univ.eq Univ (negate (add (negate (add X Y)) (negate (add X (negate Y))))) X.
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add (add X Y) Z) (add X (add Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).∃X:Univ.eq Univ (add X X) X
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#add.
+#b.
+#negate.
+#H0.
+#H1.
+#H2.
+#H3.
+napply ex_intro[
+nid2:
+nauto by H0,H1,H2,H3;
+nid|
+skip]
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: ROB022-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : ROB022-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Robbins Algebra *)
+
+(* Problem : c + -c=c => Boolean *)
+
+(* Version : [Win90] (equality) axioms. *)
+
+(* English : If there is an element c such that c + -c = c then the *)
+
+(* algebra is Boolean. *)
+
+(* Refs : [HMT71] Henkin et al. (1971), Cylindrical Algebras *)
+
+(* : [Win90] Winker (1990), Robbins Algebra: Conditions that make a *)
+
+(* : [McC92] McCune (1992), Email to G. Sutcliffe *)
+
+(* Source : [McC92] *)
+
+(* Names : - [McC92] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.2.1, 0.33 v2.2.0, 0.29 v2.1.0, 0.75 v2.0.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 2 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : Commutativity, associativity, and Huntington's axiom *)
+
+(* axiomatize Boolean algebra. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include axioms for Robbins algebra *)
+
+(* Inclusion of: Axioms/ROB001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : ROB001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Robbins algebra *)
+
+(* Axioms : Robbins algebra axioms *)
+
+(* Version : [Win90] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [HMT71] Henkin et al. (1971), Cylindrical Algebras *)
+
+(* : [Win90] Winker (1990), Robbins Algebra: Conditions that make a *)
+
+(* Source : [OTTER] *)
+
+(* Names : Lemma 2.2 [Win90] *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 1-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_huntingtons_axiom:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀c:Univ.
+∀negate:∀_:Univ.Univ.
+∀H0:eq Univ (add c (negate c)) c.
+∀H1:∀X:Univ.∀Y:Univ.eq Univ (negate (add (negate (add X Y)) (negate (add X (negate Y))))) X.
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add (add X Y) Z) (add X (add Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).eq Univ (add (negate (add a (negate b))) (negate (add (negate a) (negate b)))) b
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#add.
+#b.
+#c.
+#negate.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: ROB023-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : ROB023-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Robbins Algebra *)
+
+(* Problem : X + X=X => Boolean *)
+
+(* Version : [Win90] (equality) axioms. *)
+
+(* English : If for all X X + X = X then the algebra is Boolean. *)
+
+(* Refs : [HMT71] Henkin et al. (1971), Cylindrical Algebras *)
+
+(* : [Win90] Winker (1990), Robbins Algebra: Conditions that make a *)
+
+(* : [LM92] Lusk & McCune (1992), Experiments with ROO, a Parallel *)
+
+(* : [McC92] McCune (1992), Email to G. Sutcliffe *)
+
+(* Source : [McC92] *)
+
+(* Names : Robbins [LM92] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.2.1, 0.44 v2.2.0, 0.43 v2.1.0, 0.38 v2.0.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 1 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 8 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : Commutativity, associativity, and Huntington's axiom *)
+
+(* axiomatize Boolean algebra. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include axioms for Robbins algebra *)
+
+(* Inclusion of: Axioms/ROB001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : ROB001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Robbins algebra *)
+
+(* Axioms : Robbins algebra axioms *)
+
+(* Version : [Win90] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [HMT71] Henkin et al. (1971), Cylindrical Algebras *)
+
+(* : [Win90] Winker (1990), Robbins Algebra: Conditions that make a *)
+
+(* Source : [OTTER] *)
+
+(* Names : Lemma 2.2 [Win90] *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 1-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_huntingtons_axiom:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀negate:∀_:Univ.Univ.
+∀H0:∀X:Univ.eq Univ (add X X) X.
+∀H1:∀X:Univ.∀Y:Univ.eq Univ (negate (add (negate (add X Y)) (negate (add X (negate Y))))) X.
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add (add X Y) Z) (add X (add Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).eq Univ (add (negate (add a (negate b))) (negate (add (negate a) (negate b)))) b
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#add.
+#b.
+#negate.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: ROB024-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : ROB024-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Robbins Algebra *)
+
+(* Problem : -(a + (a + b)) + -(a + -b) = a => Boolean *)
+
+(* Version : [Win90] (equality) axioms. *)
+
+(* English : If there exist a and b so that -(a + (a + b)) + -(a + -b) *)
+
+(* = a then the algebra is Boolean. *)
+
+(* Refs : [Win90] Winker (1990), Robbins Algebra: Conditions that make a *)
+
+(* : [WW+90] Wos et al. (1990), Automated Reasoning Contributes to *)
+
+(* Source : [WW+90] *)
+
+(* Names : RA-1 [WW+90] *)
+
+(* Status : Unknown *)
+
+(* Rating : 1.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 2 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : Commutativity, associativity, and Huntington's axiom *)
+
+(* axiomatize Boolean algebra. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include axioms for Robbins algebra *)
+
+(* Inclusion of: Axioms/ROB001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : ROB001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Robbins algebra *)
+
+(* Axioms : Robbins algebra axioms *)
+
+(* Version : [Win90] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [HMT71] Henkin et al. (1971), Cylindrical Algebras *)
+
+(* : [Win90] Winker (1990), Robbins Algebra: Conditions that make a *)
+
+(* Source : [OTTER] *)
+
+(* Names : Lemma 2.2 [Win90] *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 1-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_huntingtons_axiom:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀negate:∀_:Univ.Univ.
+∀H0:eq Univ (negate (add (negate (add a (add a b))) (negate (add a (negate b))))) a.
+∀H1:∀X:Univ.∀Y:Univ.eq Univ (negate (add (negate (add X Y)) (negate (add X (negate Y))))) X.
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add (add X Y) Z) (add X (add Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).eq Univ (add (negate (add a (negate b))) (negate (add (negate a) (negate b)))) b
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#add.
+#b.
+#negate.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: ROB026-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : ROB026-1 : TPTP v3.2.0. Released v1.2.0. *)
+
+(* Domain : Robbins Algebra *)
+
+(* Problem : c + d = c => Boolean *)
+
+(* Version : [Win90] (equality) axioms. *)
+
+(* Theorem formulation : Denies Huntington's axiom. *)
+
+(* English : If there are elements c and d such that c+d=d, then the *)
+
+(* algebra is Boolean. *)
+
+(* Refs : [HMT71] Henkin et al. (1971), Cylindrical Algebras *)
+
+(* : [Win90] Winker (1990), Robbins Algebra: Conditions that make a *)
+
+(* : [Wos94] Wos (1994), Two Challenge Problems *)
+
+(* Source : [Wos94] *)
+
+(* Names : - [Wos94] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.86 v3.1.0, 0.89 v2.7.0, 0.91 v2.6.0, 0.83 v2.5.0, 0.75 v2.4.0, 0.67 v2.3.0, 1.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 2 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 6 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : Commutativity, associativity, and Huntington's axiom *)
+
+(* axiomatize Boolean algebra. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include axioms for Robbins algebra *)
+
+(* Inclusion of: Axioms/ROB001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : ROB001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Robbins algebra *)
+
+(* Axioms : Robbins algebra axioms *)
+
+(* Version : [Win90] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [HMT71] Henkin et al. (1971), Cylindrical Algebras *)
+
+(* : [Win90] Winker (1990), Robbins Algebra: Conditions that make a *)
+
+(* Source : [OTTER] *)
+
+(* Names : Lemma 2.2 [Win90] *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 1-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_huntingtons_axiom:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀c:Univ.
+∀d:Univ.
+∀negate:∀_:Univ.Univ.
+∀H0:eq Univ (add c d) c.
+∀H1:∀X:Univ.∀Y:Univ.eq Univ (negate (add (negate (add X Y)) (negate (add X (negate Y))))) X.
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add (add X Y) Z) (add X (add Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).eq Univ (add (negate (add a (negate b))) (negate (add (negate a) (negate b)))) b
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#add.
+#b.
+#c.
+#d.
+#negate.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: ROB027-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : ROB027-1 : TPTP v3.2.0. Released v1.2.0. *)
+
+(* Domain : Robbins Algebra *)
+
+(* Problem : -(-c) = c => Boolean *)
+
+(* Version : [Win90] (equality) axioms. *)
+
+(* Theorem formulation : Denies Huntington's axiom. *)
+
+(* English : If there are elements c and d such that c+d=d, then the *)
+
+(* algebra is Boolean. *)
+
+(* Refs : [HMT71] Henkin et al. (1971), Cylindrical Algebras *)
+
+(* : [Win90] Winker (1990), Robbins Algebra: Conditions that make a *)
+
+(* : [Wos94] Wos (1994), Two Challenge Problems *)
+
+(* Source : [Wos94] *)
+
+(* Names : - [Wos94] *)
+
+(* Status : Open *)
+
+(* Rating : 1.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 2 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : Commutativity, associativity, and Huntington's axiom *)
+
+(* axiomatize Boolean algebra. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include axioms for Robbins algebra *)
+
+(* Inclusion of: Axioms/ROB001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : ROB001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Robbins algebra *)
+
+(* Axioms : Robbins algebra axioms *)
+
+(* Version : [Win90] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [HMT71] Henkin et al. (1971), Cylindrical Algebras *)
+
+(* : [Win90] Winker (1990), Robbins Algebra: Conditions that make a *)
+
+(* Source : [OTTER] *)
+
+(* Names : Lemma 2.2 [Win90] *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 1-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_huntingtons_axiom:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀b:Univ.
+∀c:Univ.
+∀negate:∀_:Univ.Univ.
+∀H0:eq Univ (negate (negate c)) c.
+∀H1:∀X:Univ.∀Y:Univ.eq Univ (negate (add (negate (add X Y)) (negate (add X (negate Y))))) X.
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add (add X Y) Z) (add X (add Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).eq Univ (add (negate (add a (negate b))) (negate (add (negate a) (negate b)))) b
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#add.
+#b.
+#c.
+#negate.
+#H0.
+#H1.
+#H2.
+#H3.
+nauto by H0,H1,H2,H3;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: ROB028-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : ROB028-1 : TPTP v3.2.0. Released v2.5.0. *)
+
+(* Domain : Robbins algebra *)
+
+(* Problem : Robbins algebra axioms *)
+
+(* Version : [Win90] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [HMT71] Henkin et al. (1971), Cylindrical Algebras *)
+
+(* : [Win90] Winker (1990), Robbins Algebra: Conditions that make a *)
+
+(* Source : [OTTER] *)
+
+(* Names : Lemma 2.2 [Win90] *)
+
+(* Status : Satisfiable *)
+
+(* Rating : 0.00 v3.2.0, 0.33 v3.1.0, 0.00 v2.5.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 1-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include Robbins algebra axioms *)
+
+(* Inclusion of: Axioms/ROB001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : ROB001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Robbins algebra *)
+
+(* Axioms : Robbins algebra axioms *)
+
+(* Version : [Win90] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [HMT71] Henkin et al. (1971), Cylindrical Algebras *)
+
+(* : [Win90] Winker (1990), Robbins Algebra: Conditions that make a *)
+
+(* Source : [OTTER] *)
+
+(* Names : Lemma 2.2 [Win90] *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 1-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: ROB030-1.p *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* File : ROB030-1 : TPTP v3.2.0. Released v3.1.0. *)
+
+(* Domain : Robbins Algebra *)
+
+(* Problem : Exists absorbed element => Exists absorbed within negation element *)
+
+(* Version : [Win90] (equality) axioms. *)
+
+(* Theorem formulation : Denies Huntington's axiom. *)
+
+(* English : If there are elements c and d such that c+d=d, then the *)
+
+(* algebra is Boolean. *)
+
+(* Refs : [Win90] Winker (1990), Robbins Algebra: Conditions that make a *)
+
+(* : [Loe04] Loechner (2004), Email to Geoff Sutcliffe *)
+
+(* Source : [Loe04] *)
+
+(* Names : (1) [Loe04] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v3.1.0 *)
+
+(* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 2 RR) *)
+
+(* Number of atoms : 5 ( 5 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 9 ( 1 singleton) *)
+
+(* Maximal term depth : 6 ( 2 average) *)
+
+(* Comments : *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* ----Include axioms for Robbins algebra *)
+
+(* Inclusion of: Axioms/ROB001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : ROB001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Robbins algebra *)
+
+(* Axioms : Robbins algebra axioms *)
+
+(* Version : [Win90] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [HMT71] Henkin et al. (1971), Cylindrical Algebras *)
+
+(* : [Win90] Winker (1990), Robbins Algebra: Conditions that make a *)
+
+(* Source : [OTTER] *)
+
+(* Names : Lemma 2.2 [Win90] *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 1-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ------------------------------------------------------------------------------ *)
+ntheorem prove_absorption_within_negation:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀c:Univ.
+∀d:Univ.
+∀negate:∀_:Univ.Univ.
+∀H0:eq Univ (add c d) d.
+∀H1:∀X:Univ.∀Y:Univ.eq Univ (negate (add (negate (add X Y)) (negate (add X (negate Y))))) X.
+∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add (add X Y) Z) (add X (add Y Z)).
+∀H3:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).∃A:Univ.∃B:Univ.eq Univ (negate (add A B)) (negate B)
+.
+#Univ.
+#A.
+#B.
+#X.
+#Y.
+#Z.
+#add.
+#c.
+#d.
+#negate.
+#H0.
+#H1.
+#H2.
+#H3.
+napply ex_intro[
+nid2:
+napply ex_intro[
+nid2:
+nauto by H0,H1,H2,H3;
+nid|
+skip]
+nid|
+skip]
+nqed.
+
+(* ------------------------------------------------------------------------------ *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: ROB031-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : ROB031-1 : TPTP v3.2.0. Released v3.1.0. *)
+
+(* Domain : Robbins Algebra *)
+
+(* Problem : Robbins => Exists absorbed within negation element *)
+
+(* Version : [Win90] (equality) axioms. *)
+
+(* Theorem formulation : Denies Huntington's axiom. *)
+
+(* English : If there are elements c and d such that c+d=d, then the *)
+
+(* algebra is Boolean. *)
+
+(* Refs : [Win90] Winker (1990), Robbins Algebra: Conditions that make a *)
+
+(* : [Loe04] Loechner (2004), Email to Geoff Sutcliffe *)
+
+(* Source : [Loe04] *)
+
+(* Names : (3) [Loe04] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 1.00 v3.1.0 *)
+
+(* Syntax : Number of clauses : 4 ( 0 non-Horn; 4 unit; 1 RR) *)
+
+(* Number of atoms : 4 ( 4 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 1-2 arity) *)
+
+(* Number of variables : 9 ( 1 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include axioms for Robbins algebra *)
+
+(* Inclusion of: Axioms/ROB001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : ROB001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Robbins algebra *)
+
+(* Axioms : Robbins algebra axioms *)
+
+(* Version : [Win90] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [HMT71] Henkin et al. (1971), Cylindrical Algebras *)
+
+(* : [Win90] Winker (1990), Robbins Algebra: Conditions that make a *)
+
+(* Source : [OTTER] *)
+
+(* Names : Lemma 2.2 [Win90] *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 1-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_absorption_within_negation:
+ ∀Univ:Type.∀A:Univ.∀B:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀negate:∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (negate (add (negate (add X Y)) (negate (add X (negate Y))))) X.
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add (add X Y) Z) (add X (add Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).∃A:Univ.∃B:Univ.eq Univ (negate (add A B)) (negate B)
+.
+#Univ.
+#A.
+#B.
+#X.
+#Y.
+#Z.
+#add.
+#negate.
+#H0.
+#H1.
+#H2.
+napply ex_intro[
+nid2:
+napply ex_intro[
+nid2:
+nauto by H0,H1,H2;
+nid|
+skip]
+nid|
+skip]
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: ROB032-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : ROB032-1 : TPTP v3.2.0. Released v3.1.0. *)
+
+(* Domain : Robbins Algebra *)
+
+(* Problem : Robbins => Exists absorbed element *)
+
+(* Version : [Win90] (equality) axioms. *)
+
+(* Theorem formulation : Denies Huntington's axiom. *)
+
+(* English : If there are elements c and d such that c+d=d, then the *)
+
+(* algebra is Boolean. *)
+
+(* Refs : [Win90] Winker (1990), Robbins Algebra: Conditions that make a *)
+
+(* : [Wos92] Wos (1992), An Opportunity to Test Your Skills, and th *)
+
+(* : [Loe04] Loechner (2004), Email to Geoff Sutcliffe *)
+
+(* Source : [Loe04] *)
+
+(* Names : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 1.00 v3.1.0 *)
+
+(* Syntax : Number of clauses : 4 ( 0 non-Horn; 4 unit; 1 RR) *)
+
+(* Number of atoms : 4 ( 4 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 1-2 arity) *)
+
+(* Number of variables : 9 ( 1 singleton) *)
+
+(* Maximal term depth : 6 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include axioms for Robbins algebra *)
+
+(* Inclusion of: Axioms/ROB001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : ROB001-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Robbins algebra *)
+
+(* Axioms : Robbins algebra axioms *)
+
+(* Version : [Win90] (equality) axioms. *)
+
+(* English : *)
+
+(* Refs : [HMT71] Henkin et al. (1971), Cylindrical Algebras *)
+
+(* : [Win90] Winker (1990), Robbins Algebra: Conditions that make a *)
+
+(* Source : [OTTER] *)
+
+(* Names : Lemma 2.2 [Win90] *)
+
+(* Status : *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+
+(* Number of literals : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 2 ( 0 constant; 1-2 arity) *)
+
+(* Number of variables : 7 ( 0 singleton) *)
+
+(* Maximal term depth : 6 ( 3 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_absorbtion:
+ ∀Univ:Type.∀C:Univ.∀D:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀add:∀_:Univ.∀_:Univ.Univ.
+∀negate:∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (negate (add (negate (add X Y)) (negate (add X (negate Y))))) X.
+∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add (add X Y) Z) (add X (add Y Z)).
+∀H2:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).∃C:Univ.∃D:Univ.eq Univ (add C D) D
+.
+#Univ.
+#C.
+#D.
+#X.
+#Y.
+#Z.
+#add.
+#negate.
+#H0.
+#H1.
+#H2.
+napply ex_intro[
+nid2:
+napply ex_intro[
+nid2:
+nauto by H0,H1,H2;
+nid|
+skip]
+nid|
+skip]
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: SYN080-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : SYN080-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Syntactic *)
+
+(* Problem : Pelletier Problem 58 *)
+
+(* Version : Especial. *)
+
+(* English : *)
+
+(* Refs : [Pel86] Pelletier (1986), Seventy-five Problems for Testing Au *)
+
+(* Source : [Pel86] *)
+
+(* Names : Pelletier 58 [Pel86] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 2 constant; 0-1 arity) *)
+
+(* Number of variables : 2 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_this:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.
+∀a:Univ.
+∀b:Univ.
+∀f:∀_:Univ.Univ.
+∀g:∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.eq Univ (f X) (g Y).eq Univ (f (f a)) (f (g b))
+.
+#Univ.
+#X.
+#Y.
+#a.
+#b.
+#f.
+#g.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: SYN083-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : SYN083-1 : TPTP v3.2.0. Released v1.0.0. *)
+
+(* Domain : Syntactic *)
+
+(* Problem : Pelletier Problem 61 *)
+
+(* Version : Especial. *)
+
+(* English : *)
+
+(* Refs : [Pel86] Pelletier (1986), Seventy-five Problems for Testing Au *)
+
+(* Source : [Pel86] *)
+
+(* Names : Pelletier 61 [Pel86] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 5 ( 4 constant; 0-2 arity) *)
+
+(* Number of variables : 3 ( 0 singleton) *)
+
+(* Maximal term depth : 4 ( 4 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem prove_this:
+ ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀d:Univ.
+∀f:∀_:Univ.∀_:Univ.Univ.
+∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (f X (f Y Z)) (f (f X Y) Z).eq Univ (f a (f b (f c d))) (f (f (f a b) c) d)
+.
+#Univ.
+#X.
+#Y.
+#Z.
+#a.
+#b.
+#c.
+#d.
+#f.
+#H0.
+nauto by H0;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: SYN305-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : SYN305-1 : TPTP v3.2.0. Released v1.2.0. *)
+
+(* Domain : Syntactic *)
+
+(* Problem : Problem for testing satisfiability *)
+
+(* Version : Especial. *)
+
+(* English : *)
+
+(* Refs : [BCP94] Bourely et al. (1994), A Method for Building Models Au *)
+
+(* Source : [BCP94] *)
+
+(* Names : Example 3 [BCP94] *)
+
+(* Status : Satisfiable *)
+
+(* Rating : 0.67 v2.4.0, 0.33 v2.2.1, 0.25 v2.2.0, 0.67 v2.1.0, 0.00 v2.0.0 *)
+
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
+
+(* Number of atoms : 3 ( 3 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 3 ( 0 constant; 1-1 arity) *)
+
+(* Number of variables : 3 ( 0 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : *)
+
+(* -------------------------------------------------------------------------- *)
+ntheorem clause3:
+ ∀Univ:Type.∀X:Univ.
+∀f:∀_:Univ.Univ.
+∀g1:∀_:Univ.Univ.
+∀g2:∀_:Univ.Univ.
+∀H0:∀X:Univ.eq Univ (f (g2 X)) X.
+∀H1:∀X:Univ.eq Univ (f (g1 X)) X.∃X:Univ.eq Univ (g1 X) (g2 X)
+.
+#Univ.
+#X.
+#f.
+#g1.
+#g2.
+#H0.
+#H1.
+napply ex_intro[
+nid2:
+nauto by H0,H1;
+nid|
+skip]
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: SYN552-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : SYN552-1 : TPTP v3.2.0. Bugfixed v2.5.0. *)
+
+(* Domain : Syntactic *)
+
+(* Problem : The E Killer *)
+
+(* Version : Biased. *)
+
+(* English : *)
+
+(* Refs : [Sch99] Schulz (1999), Email to G. Sutcliffe *)
+
+(* Source : [Sch99] *)
+
+(* Names : *)
+
+(* Status : Satisfiable *)
+
+(* Rating : 0.00 v2.5.0 *)
+
+(* Syntax : Number of clauses : 2 ( 0 non-Horn; 2 unit; 1 RR) *)
+
+(* Number of atoms : 2 ( 2 equality) *)
+
+(* Maximal clause size : 1 ( 1 average) *)
+
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+
+(* Number of functors : 4 ( 2 constant; 0-2 arity) *)
+
+(* Number of variables : 2 ( 2 singleton) *)
+
+(* Maximal term depth : 3 ( 2 average) *)
+
+(* Comments : This is an example that brings out the bug in E 0.5 and *)
+
+(* earlier, in the simplify-reflect code. *)
+
+(* : Biased against E 0.5 *)
+
+(* Bugfixes : v2.5.0 - Added missing equality axioms. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
projects / helm.git / commitdiff