Stuff moved from old Matita.

[helm.git] / matita / matita / contribs / ng_TPTP / RNG036-7.ma

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+include "logic/equality.ma".
+
+(* Inclusion of: RNG036-7.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(*  File     : RNG036-7 : TPTP v3.7.0. Released v1.0.0. *)
+
+(*  Domain   : Ring Theory *)
+
+(*  Problem  : If X*X*X*X*X = X then the ring is commutative *)
+
+(*  Version  : [LW91] (equality) axioms. *)
+
+(*  English  : Given a ring in which for all x, x * x * x * x * x = x, prove  *)
+
+(*             that for all x and y, x * y = y * x. *)
+
+(*  Refs     : [LW91]  Lusk & Wos (1991), Benchmark Problems in Which Equalit *)
+
+(*  Source   : [LW91] *)
+
+(*  Names    : RT4 [LW91] *)
+
+(*  Status   : Unknown *)
+
+(*  Rating   : 1.00 v2.0.0 *)
+
+(*  Syntax   : Number of clauses     :   12 (   0 non-Horn;  12 unit;   2 RR) *)
+
+(*             Number of atoms       :   12 (  12 equality) *)
+
+(*             Maximal clause size   :    1 (   1 average) *)
+
+(*             Number of predicates  :    1 (   0 propositional; 2-2 arity) *)
+
+(*             Number of functors    :    7 (   4 constant; 0-2 arity) *)
+
+(*             Number of variables   :   19 (   0 singleton) *)
+
+(*             Maximal term depth    :    5 (   2 average) *)
+
+(*  Comments :  *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include ring theory axioms  *)
+
+(* Inclusion of: Axioms/RNG005-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(*  File     : RNG005-0 : TPTP v3.7.0. 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 (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.
+
+(* -------------------------------------------------------------------------- *)