--- /dev/null
+
+include "logic/equality.ma".
+(* Inclusion of: GRP012-4.p *)
+(* -------------------------------------------------------------------------- *)
+(* File : GRP012-4 : TPTP v3.1.1. Released v1.0.0. *)
+(* Domain : Group Theory *)
+(* Problem : Inverse of products = Product of inverses *)
+(* Version : [MOW76] (equality) axioms : Augmented. *)
+(* English : The inverse of products equals the product of the inverse, *)
+(* in opposite order *)
+(* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+(* Source : [ANL] *)
+(* Names : - [ANL] *)
+(* Status : Unsatisfiable *)
+(* Rating : 0.00 v2.2.1, 0.22 v2.2.0, 0.29 v2.1.0, 0.25 v2.0.0 *)
+(* Syntax : Number of clauses : 6 ( 0 non-Horn; 6 unit; 1 RR) *)
+(* Number of atoms : 6 ( 6 equality) *)
+(* Maximal clause size : 1 ( 1 average) *)
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+(* Number of functors : 5 ( 3 constant; 0-2 arity) *)
+(* Number of variables : 7 ( 0 singleton) *)
+(* Maximal term depth : 3 ( 2 average) *)
+(* Comments : In Lemmas.eq.clauses of [ANL] *)
+(* -------------------------------------------------------------------------- *)
+(* ----Include equality group theory axioms *)
+(* Inclusion of: Axioms/GRP004-0.ax *)
+(* -------------------------------------------------------------------------- *)
+(* File : GRP004-0 : TPTP v3.1.1. 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 *)
+(* -------------------------------------------------------------------------- *)
+(* -------------------------------------------------------------------------- *)
+(* ----Redundant two axioms *)
+theorem prove_inverse_of_product_is_product_of_inverses:
+ \forall Univ:Set.
+\forall a:Univ.
+\forall b:Univ.
+\forall identity:Univ.
+\forall inverse:\forall _:Univ.Univ.
+\forall multiply:\forall _:Univ.\forall _:Univ.Univ.
+\forall H0:\forall X:Univ.eq Univ (multiply X (inverse X)) identity.
+\forall H1:\forall X:Univ.eq Univ (multiply X identity) X.
+\forall H2:\forall X:Univ.\forall Y:Univ.\forall Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+\forall H3:\forall X:Univ.eq Univ (multiply (inverse X) X) identity.
+\forall H4:\forall X:Univ.eq Univ (multiply identity X) X.eq Univ (inverse (multiply a b)) (multiply (inverse b) (inverse a))
+.
+intros.
+autobatch paramodulation timeout=100;
+try assumption.
+print proofterm.
+qed.
+(* -------------------------------------------------------------------------- *)