-ng implemented

[helm.git] / helm / software / matita / contribs / ng_TPTP / RNG008-3.ma

diff --git a/helm/software/matita/contribs/ng_TPTP/RNG008-3.ma b/helm/software/matita/contribs/ng_TPTP/RNG008-3.ma
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+include "logic/equality.ma".
+
+(* Inclusion of: RNG008-3.p *)
+
+(* -------------------------------------------------------------------------- *)
+
+(*  File     : RNG008-3 : TPTP v3.2.0. Released v1.0.0. *)
+
+(*  Domain   : Ring Theory *)
+
+(*  Problem  : Boolean rings are commutative *)
+
+(*  Version  : [PS81] (equality) axioms : Augmented. *)
+
+(*             Theorem formulation : Equality. *)
+
+(*  English  : Given a ring in which for all x, x * x = x, prove that for  *)
+
+(*             all x and y, x * y = y * x. *)
+
+(*  Refs     : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
+
+(*           : [PS81]  Peterson & Stickel (1981), Complete Sets of Reductions *)
+
+(*  Source   : [ANL] *)
+
+(*  Names    : commute.ver2.in [ANL] *)
+
+(*  Status   : Unsatisfiable *)
+
+(*  Rating   : 0.07 v3.1.0, 0.11 v2.7.0, 0.00 v2.2.1, 0.22 v2.2.0, 0.29 v2.1.0, 0.25 v2.0.0 *)
+
+(*  Syntax   : Number of clauses     :   19 (   0 non-Horn;  19 unit;   3 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 (   4 constant; 0-2 arity) *)
+
+(*             Number of variables   :   28 (   2 singleton) *)
+
+(*             Maximal term depth    :    3 (   2 average) *)
+
+(*  Comments :  *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Include ring theory axioms  *)
+
+(* Inclusion of: Axioms/RNG002-0.ax *)
+
+(* -------------------------------------------------------------------------- *)
+
+(*  File     : RNG002-0 : TPTP v3.2.0. Released v1.0.0. *)
+
+(*  Domain   : Ring Theory *)
+
+(*  Axioms   : Ring theory (equality) axioms *)
+
+(*  Version  : [PS81] (equality) axioms : *)
+
+(*             Reduced & Augmented > Complete. *)
+
+(*  English  :  *)
+
+(*  Refs     : [PS81]  Peterson & Stickel (1981), Complete Sets of Reductions *)
+
+(*  Source   : [ANL] *)
+
+(*  Names    :  *)
+
+(*  Status   :  *)
+
+(*  Syntax   : Number of clauses    :   14 (   0 non-Horn;  14 unit;   1 RR) *)
+
+(*             Number of literals   :   14 (  14 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  :   25 (   2 singleton) *)
+
+(*             Maximal term depth   :    3 (   2 average) *)
+
+(*  Comments : Not sure if these are complete. I don't know if the reductions *)
+
+(*             given in [PS81] are suitable for ATP. *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Existence of left identity for addition  *)
+
+(* ----Existence of left additive additive_inverse  *)
+
+(* ----Distributive property of product over sum  *)
+
+(* ----Inverse of identity is identity, stupid  *)
+
+(* ----Inverse of additive_inverse of X is X  *)
+
+(* ----Behavior of 0 and the multiplication operation  *)
+
+(* ----Inverse of (x + y) is additive_inverse(x) + additive_inverse(y)  *)
+
+(* ----x * additive_inverse(y) = additive_inverse (x * y)  *)
+
+(* ----Associativity of addition  *)
+
+(* ----Commutativity of addition  *)
+
+(* ----Associativity of product  *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* -------------------------------------------------------------------------- *)
+
+(* ----Right identity and inverse are dependent lemmas  *)
+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 X) X.
+∀H2:∀X:Univ.eq Univ (add X (additive_inverse X)) additive_identity.
+∀H3:∀X:Univ.eq Univ (add X additive_identity) X.
+∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply 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 (add X Y) Z) (add X (add Y Z)).
+∀H7:∀X:Univ.∀Y:Univ.eq Univ (multiply (additive_inverse X) Y) (additive_inverse (multiply X Y)).
+∀H8:∀X:Univ.∀Y:Univ.eq Univ (multiply X (additive_inverse Y)) (additive_inverse (multiply X Y)).
+∀H9:∀X:Univ.∀Y:Univ.eq Univ (additive_inverse (add X Y)) (add (additive_inverse X) (additive_inverse Y)).
+∀H10:∀X:Univ.eq Univ (multiply additive_identity X) additive_identity.
+∀H11:∀X:Univ.eq Univ (multiply X additive_identity) additive_identity.
+∀H12:∀X:Univ.eq Univ (additive_inverse (additive_inverse X)) X.
+∀H13:eq Univ (additive_inverse additive_identity) additive_identity.
+∀H14:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X Y) Z) (add (multiply X Z) (multiply Y Z)).
+∀H15:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y Z)) (add (multiply X Y) (multiply X Z)).
+∀H16:∀X:Univ.eq Univ (add (additive_inverse X) X) additive_identity.
+∀H17:∀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.
+#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;
+nqed.
+
+(* -------------------------------------------------------------------------- *)