--- /dev/null
+
+include "logic/equality.ma".
+(* Inclusion of: GRP153-1.p *)
+(* -------------------------------------------------------------------------- *)
+(* File : GRP153-1 : TPTP v3.1.1. Bugfixed v1.2.1. *)
+(* Domain : Group Theory (Lattice Ordered) *)
+(* Problem : Prove least upper-bound axiom using the GLB transformation *)
+(* Version : [Fuc94] (equality) axioms. *)
+(* English : This problem proves the original least upper-bound axiom from *)
+(* the equational axiomatization. *)
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+(* Source : [Sch95] *)
+(* Names : ax_lub3b [Sch95] *)
+(* Status : Unsatisfiable *)
+(* Rating : 0.00 v2.0.0 *)
+(* Syntax : Number of clauses : 16 ( 0 non-Horn; 16 unit; 1 RR) *)
+(* Number of atoms : 16 ( 16 equality) *)
+(* Maximal clause size : 1 ( 1 average) *)
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+(* Number of functors : 7 ( 3 constant; 0-2 arity) *)
+(* Number of variables : 33 ( 2 singleton) *)
+(* Maximal term depth : 3 ( 2 average) *)
+(* Comments : ORDERING LPO inverse > product > greatest_lower_bound > *)
+(* least_upper_bound > identity > a > b *)
+(* Bugfixes : v1.2.1 - Duplicate axioms in GRP004-2.ax removed. *)
+(* -------------------------------------------------------------------------- *)
+(* ----Include equality group theory axioms *)
+(* Inclusion of: Axioms/GRP004-0.ax *)
+(* -------------------------------------------------------------------------- *)
+(* File : GRP004-0 : TPTP v3.1.1. Released v1.0.0. *)
+(* Domain : Group Theory *)
+(* Axioms : Group theory (equality) axioms *)
+(* Version : [MOW76] (equality) axioms : *)
+(* Reduced > Complete. *)
+(* English : *)
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+(* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
+(* Source : [ANL] *)
+(* Names : *)
+(* Status : *)
+(* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
+(* Number of literals : 3 ( 3 equality) *)
+(* Maximal clause size : 1 ( 1 average) *)
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+(* Number of variables : 5 ( 0 singleton) *)
+(* Maximal term depth : 3 ( 2 average) *)
+(* Comments : [MOW76] also contains redundant right_identity and *)
+(* right_inverse axioms. *)
+(* : These axioms are also used in [Wos88] p.186, also with *)
+(* right_identity and right_inverse. *)
+(* -------------------------------------------------------------------------- *)
+(* ----For any x and y in the group x*y is also in the group. No clause *)
+(* ----is needed here since this is an instance of reflexivity *)
+(* ----There exists an identity element *)
+(* ----For any x in the group, there exists an element y such that x*y = y*x *)
+(* ----= identity. *)
+(* ----The operation '*' is associative *)
+(* -------------------------------------------------------------------------- *)
+(* ----Include Lattice ordered group (equality) axioms *)
+(* Inclusion of: Axioms/GRP004-2.ax *)
+(* -------------------------------------------------------------------------- *)
+(* File : GRP004-2 : TPTP v3.1.1. Bugfixed v1.2.0. *)
+(* Domain : Group Theory (Lattice Ordered) *)
+(* Axioms : Lattice ordered group (equality) axioms *)
+(* Version : [Fuc94] (equality) axioms. *)
+(* English : *)
+(* Refs : [Fuc94] Fuchs (1994), The Application of Goal-Orientated Heuri *)
+(* : [Sch95] Schulz (1995), Explanation Based Learning for Distribu *)
+(* Source : [Sch95] *)
+(* Names : *)
+(* Status : *)
+(* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 0 RR) *)
+(* Number of literals : 12 ( 12 equality) *)
+(* Maximal clause size : 1 ( 1 average) *)
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+(* Number of functors : 3 ( 0 constant; 2-2 arity) *)
+(* Number of variables : 28 ( 2 singleton) *)
+(* Maximal term depth : 3 ( 2 average) *)
+(* Comments : Requires GRP004-0.ax *)
+(* -------------------------------------------------------------------------- *)
+(* ----Specification of the least upper bound and greatest lower bound *)
+(* ----Monotony of multiply *)
+(* -------------------------------------------------------------------------- *)
+(* -------------------------------------------------------------------------- *)
+theorem prove_ax_lub3b:
+ \forall Univ:Set.
+\forall a:Univ.
+\forall b:Univ.
+\forall greatest_lower_bound:\forall _:Univ.\forall _:Univ.Univ.
+\forall identity:Univ.
+\forall inverse:\forall _:Univ.Univ.
+\forall least_upper_bound:\forall _:Univ.\forall _:Univ.Univ.
+\forall multiply:\forall _:Univ.\forall _:Univ.Univ.
+\forall H0:\forall X:Univ.\forall Y:Univ.\forall Z:Univ.eq Univ (multiply (greatest_lower_bound Y Z) X) (greatest_lower_bound (multiply Y X) (multiply Z X)).
+\forall H1:\forall X:Univ.\forall Y:Univ.\forall Z:Univ.eq Univ (multiply (least_upper_bound Y Z) X) (least_upper_bound (multiply Y X) (multiply Z X)).
+\forall H2:\forall X:Univ.\forall Y:Univ.\forall Z:Univ.eq Univ (multiply X (greatest_lower_bound Y Z)) (greatest_lower_bound (multiply X Y) (multiply X Z)).
+\forall H3:\forall X:Univ.\forall Y:Univ.\forall Z:Univ.eq Univ (multiply X (least_upper_bound Y Z)) (least_upper_bound (multiply X Y) (multiply X Z)).
+\forall H4:\forall X:Univ.\forall Y:Univ.eq Univ (greatest_lower_bound X (least_upper_bound X Y)) X.
+\forall H5:\forall X:Univ.\forall Y:Univ.eq Univ (least_upper_bound X (greatest_lower_bound X Y)) X.
+\forall H6:\forall X:Univ.eq Univ (greatest_lower_bound X X) X.
+\forall H7:\forall X:Univ.eq Univ (least_upper_bound X X) X.
+\forall H8:\forall X:Univ.\forall Y:Univ.\forall Z:Univ.eq Univ (least_upper_bound X (least_upper_bound Y Z)) (least_upper_bound (least_upper_bound X Y) Z).
+\forall H9:\forall X:Univ.\forall Y:Univ.\forall Z:Univ.eq Univ (greatest_lower_bound X (greatest_lower_bound Y Z)) (greatest_lower_bound (greatest_lower_bound X Y) Z).
+\forall H10:\forall X:Univ.\forall Y:Univ.eq Univ (least_upper_bound X Y) (least_upper_bound Y X).
+\forall H11:\forall X:Univ.\forall Y:Univ.eq Univ (greatest_lower_bound X Y) (greatest_lower_bound Y X).
+\forall H12:\forall X:Univ.\forall Y:Univ.\forall Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+\forall H13:\forall X:Univ.eq Univ (multiply (inverse X) X) identity.
+\forall H14:\forall X:Univ.eq Univ (multiply identity X) X.eq Univ (greatest_lower_bound b (least_upper_bound a b)) b
+.
+intros.
+autobatch paramodulation timeout=100;
+try assumption.
+print proofterm.
+qed.
+(* -------------------------------------------------------------------------- *)
projects / helm.git / blobdiff