Stuff moved from old Matita.

[helm.git] / matita / matita / contribs / ng_TPTP / GRP002-2.ma

diff --git a/matita/matita/contribs/ng_TPTP/GRP002-2.ma b/matita/matita/contribs/ng_TPTP/GRP002-2.ma
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+include "logic/equality.ma".
+
+(* Inclusion of: GRP002-2.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(*  File     : GRP002-2 : TPTP v3.7.0. Bugfixed v1.2.1. *)
+
+(*  Domain   : Group Theory *)
+
+(*  Problem  : Commutator equals identity in groups of order 3 *)
+
+(*  Version  : [MOW76] (equality) axioms. *)
+
+(*  English  : In a group, if (for all x) the cube of x is the identity  *)
+
+(*             (i.e. a group of order 3), then the equation [[x,y],y]=  *)
+
+(*             identity holds, where [x,y] is the product of x, y, the  *)
+
+(*             inverse of x and the inverse of y (i.e. the commutator  *)
+
+(*             of x and y). *)
+
+(*  Refs     : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(*  Source   : [ANL] *)
+
+(*  Names    : commutator.ver2.in [ANL] *)
+
+(*  Status   : Unsatisfiable *)
+
+(*  Rating   : 0.11 v3.4.0, 0.12 v3.3.0, 0.00 v2.2.1, 0.22 v2.2.0, 0.29 v2.1.0, 0.29 v2.0.0 *)
+
+(*  Syntax   : Number of clauses     :   12 (   0 non-Horn;  12 unit;   6 RR) *)
+
+(*             Number of atoms       :   12 (  12 equality) *)
+
+(*             Maximal clause size   :    1 (   1 average) *)
+
+(*             Number of predicates  :    1 (   0 propositional; 2-2 arity) *)
+
+(*             Number of functors    :   10 (   8 constant; 0-2 arity) *)
+
+(*             Number of variables   :    8 (   0 singleton) *)
+
+(*             Maximal term depth    :    3 (   2 average) *)
+
+(*  Comments :  *)
+
+(*  Bugfixes : v1.2.1 - Clause x_cubed_is_identity fixed. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* Inclusion of: Axioms/GRP004-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(*  File     : GRP004-0 : TPTP v3.7.0. 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  *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Redundant two axioms, but established in standard axiomatizations. *)
+
+(* ----This hypothesis is omitted in the ANL source version  *)
+ntheorem prove_k_times_inverse_b_is_e:
+ (∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
+∀a:Univ.
+∀b:Univ.
+∀c:Univ.
+∀d:Univ.
+∀h:Univ.
+∀identity:Univ.
+∀inverse:∀_:Univ.Univ.
+∀j:Univ.
+∀k:Univ.
+∀multiply:∀_:Univ.∀_:Univ.Univ.
+∀H0:eq Univ (multiply j (inverse h)) k.
+∀H1:eq Univ (multiply h b) j.
+∀H2:eq Univ (multiply d (inverse b)) h.
+∀H3:eq Univ (multiply c (inverse a)) d.
+∀H4:eq Univ (multiply a b) c.
+∀H5:∀X:Univ.eq Univ (multiply X (multiply X X)) identity.
+∀H6:∀X:Univ.eq Univ (multiply X (inverse X)) identity.
+∀H7:∀X:Univ.eq Univ (multiply X identity) X.
+∀H8:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
+∀H9:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
+∀H10:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (multiply k (inverse b)) identity)
+.
+#Univ ##.
+#X ##.
+#Y ##.
+#Z ##.
+#a ##.
+#b ##.
+#c ##.
+#d ##.
+#h ##.
+#identity ##.
+#inverse ##.
+#j ##.
+#k ##.
+#multiply ##.
+#H0 ##.
+#H1 ##.
+#H2 ##.
+#H3 ##.
+#H4 ##.
+#H5 ##.
+#H6 ##.
+#H7 ##.
+#H8 ##.
+#H9 ##.
+#H10 ##.
+nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10 ##;
+ntry (nassumption) ##;
+nqed.
+
+(* -------------------------------------------------------------------------- *)