--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: BOO007-2.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO007-2 : TPTP v3.7.0. 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 : 0.22 v3.4.0, 0.25 v3.3.0, 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 v3.7.0. 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 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 : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+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 ##;
+ntry (nassumption) ##;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: BOO007-4.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO007-4 : TPTP v3.7.0. 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 : 0.22 v3.4.0, 0.25 v3.3.0, 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 v3.7.0. 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 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 : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+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 ##;
+ntry (nassumption) ##;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: BOO031-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO031-1 : TPTP v3.7.0. 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 : 0.22 v3.4.0, 0.25 v3.3.0, 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 ##;
+ntry (nassumption) ##;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: BOO034-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO034-1 : TPTP v3.7.0. 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 : 0.44 v3.4.0, 0.50 v3.3.0, 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 v3.7.0. 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 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 : 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 ##;
+ntry (nassumption) ##;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: BOO072-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO072-1 : TPTP v3.7.0. 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 : 0.22 v3.4.0, 0.25 v3.3.0, 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 ( 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 ##;
+ntry (nassumption) ##;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: BOO073-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO073-1 : TPTP v3.7.0. 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 : 0.22 v3.4.0, 0.25 v3.3.0, 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 ##;
+ntry (nassumption) ##;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: BOO076-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : BOO076-1 : TPTP v3.7.0. 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 : 0.67 v3.4.0, 0.88 v3.3.0, 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 ##;
+ntry (nassumption) ##;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL003-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL003-1 : TPTP v3.7.0. 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 : 0.67 v3.4.0, 0.75 v3.3.0, 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 ? ? ? ?) ##[
+##2:
+nauto by H0,H1 ##;
+##| ##skip ##]
+ntry (nassumption) ##;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL003-12.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL003-12 : TPTP v3.7.0. Released v2.1.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Strong fixed point for B and W *)
+
+(* 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 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 : J sage [MW87] *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.22 v3.4.0, 0.25 v3.3.0, 0.07 v3.1.0, 0.22 v2.7.0, 0.18 v2.6.0, 0.17 v2.5.0, 0.00 v2.2.1, 0.25 v2.2.0, 0.40 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 : Found by Statman. *)
+
+(* -------------------------------------------------------------------------- *)
+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.
+∀w:Univ.
+∀H0:eq Univ strong_fixed_point (apply (apply b (apply w w)) (apply (apply b w) (apply (apply b b) b))).
+∀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)).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 ##.
+#w ##.
+#H0 ##.
+#H1 ##.
+#H2 ##.
+nauto by H0,H1,H2 ##;
+ntry (nassumption) ##;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL003-20.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL003-20 : TPTP v3.7.0. Released v2.1.0. *)
+
+(* Domain : Combinatory Logic *)
+
+(* Problem : Strong fixed point for B and W *)
+
+(* 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 W alone, where ((Bx)y)z *)
+
+(* = x(yz) and (Wx)y = (xy)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 : *)
+
+(* Status : Unsatisfiable *)
+
+(* Rating : 0.33 v3.4.0, 0.38 v3.3.0, 0.21 v3.1.0, 0.22 v2.7.0, 0.36 v2.6.0, 0.33 v2.5.0, 0.00 v2.2.1, 0.62 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 : 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.
+∀w:Univ.
+∀H0:eq Univ strong_fixed_point (apply (apply b (apply w w)) (apply (apply b (apply b w)) (apply (apply b b) b))).
+∀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)).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 ##.
+#w ##.
+#H0 ##.
+#H1 ##.
+#H2 ##.
+nauto by H0,H1,H2 ##;
+ntry (nassumption) ##;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL006-6.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL006-6 : TPTP v3.7.0. 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 : 0.44 v3.4.0, 0.38 v3.3.0, 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 ##;
+ntry (nassumption) ##;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL011-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL011-1 : TPTP v3.7.0. 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 : 0.22 v3.4.0, 0.25 v3.3.0, 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 ? ? ? ?) ##[
+##2:
+nauto by H0,H1 ##;
+##| ##skip ##]
+ntry (nassumption) ##;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL037-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL037-1 : TPTP v3.7.0. 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 : 0.22 v3.4.0, 0.25 v3.3.0, 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 ? ? ? ?) ##[
+##2:
+nauto by H0,H1,H2 ##;
+##| ##skip ##]
+ntry (nassumption) ##;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL038-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL038-1 : TPTP v3.7.0. 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 : 0.33 v3.4.0, 0.50 v3.3.0, 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 ? ? ? ?) ##[
+##2:
+nauto by H0,H1,H2 ##;
+##| ##skip ##]
+ntry (nassumption) ##;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL043-3.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL043-3 : TPTP v3.7.0. 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 : 0.67 v3.4.0, 0.75 v3.3.0, 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 ##;
+ntry (nassumption) ##;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL044-8.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL044-8 : TPTP v3.7.0. 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 : 0.67 v3.4.0, 0.75 v3.3.0, 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 ##;
+ntry (nassumption) ##;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL046-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL046-1 : TPTP v3.7.0. 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 : 0.33 v3.4.0, 0.50 v3.3.0, 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 ? ? ? ?) ##[
+##2:
+nauto by H0,H1,H2 ##;
+##| ##skip ##]
+ntry (nassumption) ##;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL049-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL049-1 : TPTP v3.7.0. 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 : 0.22 v3.4.0, 0.25 v3.3.0, 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 ? ? ? ?) ##[
+##2:
+nauto by H0,H1,H2 ##;
+##| ##skip ##]
+ntry (nassumption) ##;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL057-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL057-1 : TPTP v3.7.0. 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 : 0.22 v3.4.0, 0.25 v3.3.0, 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 ? ? ? ?) ##[
+##2:
+nauto by H0,H1,H2,H3 ##;
+##| ##skip ##]
+ntry (nassumption) ##;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL060-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL060-1 : TPTP v3.7.0. 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 : 0.33 v3.4.0, 0.38 v3.3.0, 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 ? ? ? ?) ##[
+##2:
+nauto by H0,H1 ##;
+##| ##skip ##]
+ntry (nassumption) ##;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL061-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL061-1 : TPTP v3.7.0. 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 : 0.33 v3.4.0, 0.38 v3.3.0, 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 ? ? ? ?) ##[
+##2:
+nauto by H0,H1 ##;
+##| ##skip ##]
+ntry (nassumption) ##;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL063-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL063-1 : TPTP v3.7.0. 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 : 0.44 v3.4.0, 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 ? ? ? ?) ##[
+##2:
+nauto by H0,H1 ##;
+##| ##skip ##]
+ntry (nassumption) ##;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL064-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL064-1 : TPTP v3.7.0. 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 : 0.56 v3.4.0, 0.50 v3.3.0, 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 ? ? ? ?) ##[
+##2:
+nauto by H0,H1 ##;
+##| ##skip ##]
+ntry (nassumption) ##;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: COL065-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : COL065-1 : TPTP v3.7.0. 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 : 0.56 v3.4.0, 0.62 v3.3.0, 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 ? ? ? ?) ##[
+##2:
+nauto by H0,H1 ##;
+##| ##skip ##]
+ntry (nassumption) ##;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP014-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP014-1 : TPTP v3.7.0. 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 : 0.22 v3.4.0, 0.25 v3.3.0, 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 ##;
+ntry (nassumption) ##;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP024-5.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP024-5 : TPTP v3.7.0. 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 : 0.33 v3.4.0, 0.38 v3.3.0, 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 v3.7.0. 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 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 : [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 ##;
+ntry (nassumption) ##;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP114-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP114-1 : TPTP v3.7.0. 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 : 0.33 v3.4.0, 0.25 v3.3.0, 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 v3.7.0. 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 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 : [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 ##;
+ntry (nassumption) ##;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP164-2.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP164-2 : TPTP v3.7.0. 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 : 0.89 v3.4.0, 0.88 v3.3.0, 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 v3.7.0. 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 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 : [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 v3.7.0. 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 atoms : 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 ##;
+ntry (nassumption) ##;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP167-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP167-1 : TPTP v3.7.0. 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 : 0.22 v3.4.0, 0.25 v3.3.0, 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 v3.7.0. 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 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 : [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 v3.7.0. 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 atoms : 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 ##;
+ntry (nassumption) ##;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP178-2.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP178-2 : TPTP v3.7.0. 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 : 0.22 v3.4.0, 0.25 v3.3.0, 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 v3.7.0. 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 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 : [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 v3.7.0. 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 atoms : 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 ##;
+ntry (nassumption) ##;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP181-4.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP181-4 : TPTP v3.7.0. 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 : 0.33 v3.4.0, 0.25 v3.3.0, 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 v3.7.0. 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 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 : [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 v3.7.0. 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 atoms : 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 ##;
+ntry (nassumption) ##;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP183-4.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP183-4 : TPTP v3.7.0. 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 : 0.33 v3.4.0, 0.38 v3.3.0, 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 v3.7.0. 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 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 : [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 v3.7.0. 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 atoms : 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 ##;
+ntry (nassumption) ##;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP184-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP184-1 : TPTP v3.7.0. 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 : 0.33 v3.4.0, 0.38 v3.3.0, 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 v3.7.0. 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 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 : [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 v3.7.0. 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 atoms : 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 ##;
+ntry (nassumption) ##;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP184-3.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP184-3 : TPTP v3.7.0. 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 : 0.33 v3.4.0, 0.25 v3.3.0, 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 v3.7.0. 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 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 : [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 v3.7.0. 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 atoms : 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 ##;
+ntry (nassumption) ##;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP185-2.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP185-2 : TPTP v3.7.0. 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 : 0.22 v3.4.0, 0.12 v3.3.0, 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 v3.7.0. 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 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 : [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 v3.7.0. 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 atoms : 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 ##;
+ntry (nassumption) ##;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP185-3.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP185-3 : TPTP v3.7.0. 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 : 0.33 v3.4.0, 0.25 v3.3.0, 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 v3.7.0. 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 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 : [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 v3.7.0. 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 atoms : 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 ##;
+ntry (nassumption) ##;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP186-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP186-1 : TPTP v3.7.0. 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 : 0.44 v3.4.0, 0.62 v3.3.0, 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 v3.7.0. 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 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 : [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 v3.7.0. 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 atoms : 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 ##;
+ntry (nassumption) ##;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP186-2.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP186-2 : TPTP v3.7.0. 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 : 0.33 v3.4.0, 0.38 v3.3.0, 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 v3.7.0. 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 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 : [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 v3.7.0. 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 atoms : 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 ##;
+ntry (nassumption) ##;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP187-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP187-1 : TPTP v3.7.0. 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 : 0.56 v3.4.0, 0.62 v3.3.0, 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 v3.7.0. 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 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 : [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 v3.7.0. 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 atoms : 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 ##;
+ntry (nassumption) ##;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP200-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP200-1 : TPTP v3.7.0. 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 : 0.22 v3.4.0, 0.25 v3.3.0, 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 ##;
+ntry (nassumption) ##;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP202-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP202-1 : TPTP v3.7.0. 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 : 0.22 v3.4.0, 0.38 v3.3.0, 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 ##;
+ntry (nassumption) ##;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP404-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP404-1 : TPTP v3.7.0. 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 : 0.22 v3.4.0, 0.25 v3.3.0, 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 ##;
+ntry (nassumption) ##;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP405-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP405-1 : TPTP v3.7.0. 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 : 0.22 v3.4.0, 0.25 v3.3.0, 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 ##;
+ntry (nassumption) ##;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP422-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP422-1 : TPTP v3.7.0. 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 : 0.22 v3.4.0, 0.25 v3.3.0, 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 ##;
+ntry (nassumption) ##;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP423-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP423-1 : TPTP v3.7.0. 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 : 0.22 v3.4.0, 0.25 v3.3.0, 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 ##;
+ntry (nassumption) ##;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP444-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP444-1 : TPTP v3.7.0. 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 : 0.22 v3.4.0, 0.25 v3.3.0, 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 ##;
+ntry (nassumption) ##;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP452-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP452-1 : TPTP v3.7.0. 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 : 0.22 v3.4.0, 0.25 v3.3.0, 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 ##;
+ntry (nassumption) ##;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP453-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP453-1 : TPTP v3.7.0. 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 : 0.33 v3.4.0, 0.38 v3.3.0, 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 ##;
+ntry (nassumption) ##;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP471-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP471-1 : TPTP v3.7.0. 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 : 0.22 v3.4.0, 0.25 v3.3.0, 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 ##;
+ntry (nassumption) ##;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP477-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP477-1 : TPTP v3.7.0. 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 : 0.22 v3.4.0, 0.25 v3.3.0, 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 ##;
+ntry (nassumption) ##;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP506-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP506-1 : TPTP v3.7.0. 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 : 0.67 v3.4.0, 0.75 v3.3.0, 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 ##;
+ntry (nassumption) ##;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: GRP508-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : GRP508-1 : TPTP v3.7.0. 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 : 0.56 v3.4.0, 0.62 v3.3.0, 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 ##;
+ntry (nassumption) ##;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT080-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT080-1 : TPTP v3.7.0. 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 : 0.56 v3.4.0, 0.50 v3.3.0, 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 ##;
+ntry (nassumption) ##;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT087-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT087-1 : TPTP v3.7.0. 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 : 0.67 v3.4.0, 0.62 v3.3.0, 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 ##;
+ntry (nassumption) ##;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT093-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT093-1 : TPTP v3.7.0. 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 : 0.56 v3.4.0, 0.50 v3.3.0, 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 ##;
+ntry (nassumption) ##;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT138-1.p *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* File : LAT138-1 : TPTP v3.7.0. 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 : 0.89 v3.4.0, 0.88 v3.3.0, 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 : 7 ( 3 average) *)
+
+(* Comments : *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* ----Include Lattice theory (equality) axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.7.0. 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 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 : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----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 ##;
+ntry (nassumption) ##;
+nqed.
+
+(* ------------------------------------------------------------------------------ *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT140-1.p *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* File : LAT140-1 : TPTP v3.7.0. 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 : 0.89 v3.4.0, 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 : 7 ( 3 average) *)
+
+(* Comments : *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* ----Include Lattice theory (equality) axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.7.0. 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 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 : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----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 ##;
+ntry (nassumption) ##;
+nqed.
+
+(* ------------------------------------------------------------------------------ *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT146-1.p *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* File : LAT146-1 : TPTP v3.7.0. 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 : 0.56 v3.4.0, 0.62 v3.3.0, 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 : 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 v3.7.0. 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 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 : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----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 ##;
+ntry (nassumption) ##;
+nqed.
+
+(* ------------------------------------------------------------------------------ *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT148-1.p *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* File : LAT148-1 : TPTP v3.7.0. 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 : 0.56 v3.4.0, 0.50 v3.3.0, 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 v3.7.0. 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 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 : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----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 ##;
+ntry (nassumption) ##;
+nqed.
+
+(* ------------------------------------------------------------------------------ *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT152-1.p *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* File : LAT152-1 : TPTP v3.7.0. 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 : 0.78 v3.4.0, 0.62 v3.3.0, 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 v3.7.0. 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 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 : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----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 ##;
+ntry (nassumption) ##;
+nqed.
+
+(* ------------------------------------------------------------------------------ *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT156-1.p *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* File : LAT156-1 : TPTP v3.7.0. 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 : 0.56 v3.4.0, 0.38 v3.3.0, 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 v3.7.0. 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 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 : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----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 ##;
+ntry (nassumption) ##;
+nqed.
+
+(* ------------------------------------------------------------------------------ *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT159-1.p *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* File : LAT159-1 : TPTP v3.7.0. 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 : 0.78 v3.4.0, 0.62 v3.3.0, 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 v3.7.0. 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 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 : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----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 ##;
+ntry (nassumption) ##;
+nqed.
+
+(* ------------------------------------------------------------------------------ *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT164-1.p *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* File : LAT164-1 : TPTP v3.7.0. 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 : 0.78 v3.4.0, 0.75 v3.3.0, 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 : 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 v3.7.0. 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 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 : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----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 ##;
+ntry (nassumption) ##;
+nqed.
+
+(* ------------------------------------------------------------------------------ *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT165-1.p *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* File : LAT165-1 : TPTP v3.7.0. 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 : 0.89 v3.4.0, 0.75 v3.3.0, 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 v3.7.0. 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 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 : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----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 ##;
+ntry (nassumption) ##;
+nqed.
+
+(* ------------------------------------------------------------------------------ *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT166-1.p *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* File : LAT166-1 : TPTP v3.7.0. 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 : 0.89 v3.4.0, 0.88 v3.3.0, 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 v3.7.0. 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 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 : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----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 ##;
+ntry (nassumption) ##;
+nqed.
+
+(* ------------------------------------------------------------------------------ *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT169-1.p *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* File : LAT169-1 : TPTP v3.7.0. 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 : 0.78 v3.4.0, 0.75 v3.3.0, 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 v3.7.0. 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 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 : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----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 ##;
+ntry (nassumption) ##;
+nqed.
+
+(* ------------------------------------------------------------------------------ *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT170-1.p *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* File : LAT170-1 : TPTP v3.7.0. 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 : 0.67 v3.4.0, 0.62 v3.3.0, 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 : 6 ( 3 average) *)
+
+(* Comments : *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* ----Include Lattice theory (equality) axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.7.0. 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 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 : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----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 ##;
+ntry (nassumption) ##;
+nqed.
+
+(* ------------------------------------------------------------------------------ *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT173-1.p *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* File : LAT173-1 : TPTP v3.7.0. 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 : 0.89 v3.4.0, 0.88 v3.3.0, 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 v3.7.0. 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 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 : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----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 ##;
+ntry (nassumption) ##;
+nqed.
+
+(* ------------------------------------------------------------------------------ *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: LAT175-1.p *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* File : LAT175-1 : TPTP v3.7.0. 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 : 0.89 v3.4.0, 0.88 v3.3.0, 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 : 6 ( 3 average) *)
+
+(* Comments : *)
+
+(* ------------------------------------------------------------------------------ *)
+
+(* ----Include Lattice theory (equality) axioms *)
+
+(* Inclusion of: Axioms/LAT001-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : LAT001-0 : TPTP v3.7.0. 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 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 : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----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 ##;
+ntry (nassumption) ##;
+nqed.
+
+(* ------------------------------------------------------------------------------ *)
--- /dev/null
+BIN=../../../
+DIR=$(shell basename $$PWD)
+
+$(DIR) all:
+ $(BIN)matitac
+$(DIR).opt opt all.opt:
+ $(BIN)matitac.opt
+clean:
+ $(BIN)matitaclean
+clean.opt:
+ $(BIN)matitaclean.opt
+depend:
+ $(BIN)matitadep
+depend.opt:
+ $(BIN)matitadep.opt
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: RNG009-7.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG009-7 : TPTP v3.7.0. 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 : 0.56 v3.4.0, 0.62 v3.3.0, 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 v3.7.0. 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 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 : 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 ##;
+ntry (nassumption) ##;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: RNG019-6.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG019-6 : TPTP v3.7.0. 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 : 0.22 v3.4.0, 0.38 v3.3.0, 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 v3.7.0. 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 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 : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----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 ##;
+ntry (nassumption) ##;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: RNG019-7.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG019-7 : TPTP v3.7.0. 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 : 0.33 v3.4.0, 0.38 v3.3.0, 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 v3.7.0. 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 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 : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----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 ##;
+ntry (nassumption) ##;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: RNG020-6.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG020-6 : TPTP v3.7.0. 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 : 0.33 v3.4.0, 0.38 v3.3.0, 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 v3.7.0. 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 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 : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----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 ##;
+ntry (nassumption) ##;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: RNG026-6.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG026-6 : TPTP v3.7.0. 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 : 0.56 v3.4.0, 0.62 v3.3.0, 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 v3.7.0. 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 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 : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----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 ##;
+ntry (nassumption) ##;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: RNG027-7.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG027-7 : TPTP v3.7.0. 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 : 0.78 v3.4.0, 0.75 v3.3.0, 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 v3.7.0. 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 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 : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----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 ##;
+ntry (nassumption) ##;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: RNG028-9.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG028-9 : TPTP v3.7.0. 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 : 0.78 v3.4.0, 0.75 v3.3.0, 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 v3.7.0. 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 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 : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----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 ##;
+ntry (nassumption) ##;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: RNG029-7.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG029-7 : TPTP v3.7.0. 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 : 0.78 v3.4.0, 0.75 v3.3.0, 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 v3.7.0. 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 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 : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----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 ##;
+ntry (nassumption) ##;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: RNG035-7.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : RNG035-7 : TPTP v3.7.0. 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 : 0.89 v3.4.0, 0.88 v3.3.0, 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 v3.7.0. 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 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 : 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 ##;
+ntry (nassumption) ##;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: ROB006-1.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : ROB006-1 : TPTP v3.7.0. 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 : 0.78 v3.4.0, 0.88 v3.3.0, 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 v3.7.0. 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 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 : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+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 ##;
+ntry (nassumption) ##;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
+include "logic/equality.ma".
+
+(* Inclusion of: ROB006-2.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* File : ROB006-2 : TPTP v3.7.0. 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 : 0.78 v3.4.0, 0.88 v3.3.0, 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 v3.7.0. 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 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 : *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+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 ? ? ? ?) ##[
+##2:
+nauto by H0,H1,H2,H3 ##;
+##| ##skip ##]
+ntry (nassumption) ##;
+nqed.
+
+(* -------------------------------------------------------------------------- *)
--- /dev/null
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+logic/equality.ma
--- /dev/null
+baseuri=cic:/matita/ngtptp
projects / helm.git / commitdiff