1 include "logic/equality.ma".
3 (* Inclusion of: RNG033-9.p *)
5 (* -------------------------------------------------------------------------- *)
7 (* File : RNG033-9 : TPTP v3.7.0. Bugfixed v2.3.0. *)
9 (* Domain : Ring Theory (Alternative) *)
11 (* Problem : A fairly complex equation with associators *)
13 (* Version : [Ste87] (equality) axioms : Augmented. *)
15 (* English : assr(X.Y,Z,W)+assr(X,Y,comm(Z,W)) = X.assr(Y,Z,W)+assr(X,Z,W).Y *)
17 (* Refs : [Ste87] Stevens (1987), Some Experiments in Nonassociative Rin *)
23 (* Status : Unknown *)
25 (* Rating : 1.00 v2.3.0 *)
27 (* Syntax : Number of clauses : 24 ( 0 non-Horn; 24 unit; 1 RR) *)
29 (* Number of atoms : 24 ( 24 equality) *)
31 (* Maximal clause size : 1 ( 1 average) *)
33 (* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
35 (* Number of functors : 10 ( 5 constant; 0-3 arity) *)
37 (* Number of variables : 48 ( 2 singleton) *)
39 (* Maximal term depth : 5 ( 3 average) *)
41 (* Comments : This includes the right Moufang identity, which [Ste87] *)
43 (* suggests in a useful lemma for this problem. *)
45 (* Bugfixes : v2.3.0 - Clause right_moufang fixed. *)
47 (* -------------------------------------------------------------------------- *)
49 (* ----Include nonassociative ring axioms *)
51 (* Inclusion of: Axioms/RNG003-0.ax *)
53 (* -------------------------------------------------------------------------- *)
55 (* File : RNG003-0 : TPTP v3.7.0. Released v1.0.0. *)
57 (* Domain : Ring Theory (Alternative) *)
59 (* Axioms : Alternative ring theory (equality) axioms *)
61 (* Version : [Ste87] (equality) axioms. *)
65 (* Refs : [Ste87] Stevens (1987), Some Experiments in Nonassociative Rin *)
67 (* Source : [Ste87] *)
73 (* Syntax : Number of clauses : 15 ( 0 non-Horn; 15 unit; 0 RR) *)
75 (* Number of atoms : 15 ( 15 equality) *)
77 (* Maximal clause size : 1 ( 1 average) *)
79 (* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
81 (* Number of functors : 6 ( 1 constant; 0-3 arity) *)
83 (* Number of variables : 27 ( 2 singleton) *)
85 (* Maximal term depth : 5 ( 2 average) *)
89 (* -------------------------------------------------------------------------- *)
91 (* ----There exists an additive identity element *)
93 (* ----Multiplicative zero *)
95 (* ----Existence of left additive additive_inverse *)
97 (* ----Inverse of additive_inverse of X is X *)
99 (* ----Distributive property of product over sum *)
101 (* ----Commutativity for addition *)
103 (* ----Associativity for addition *)
105 (* ----Right alternative law *)
107 (* ----Left alternative law *)
113 (* -------------------------------------------------------------------------- *)
115 (* -------------------------------------------------------------------------- *)
117 (* ----The next 7 clause are extra lemmas which Stevens found useful *)
119 (* ----Right Moufang *)
120 ntheorem prove_challenge:
121 (∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
122 ∀add:∀_:Univ.∀_:Univ.Univ.
123 ∀additive_identity:Univ.
124 ∀additive_inverse:∀_:Univ.Univ.
125 ∀associator:∀_:Univ.∀_:Univ.∀_:Univ.Univ.
126 ∀commutator:∀_:Univ.∀_:Univ.Univ.
127 ∀multiply:∀_:Univ.∀_:Univ.Univ.
132 ∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply Z (multiply X (multiply Y X))) (multiply (multiply (multiply Z X) Y) X).
133 ∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X Y) (additive_inverse Z)) (add (additive_inverse (multiply X Z)) (additive_inverse (multiply Y Z))).
134 ∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (additive_inverse X) (add Y Z)) (add (additive_inverse (multiply X Y)) (additive_inverse (multiply X Z))).
135 ∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X (additive_inverse Y)) Z) (add (multiply X Z) (additive_inverse (multiply Y Z))).
136 ∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y (additive_inverse Z))) (add (multiply X Y) (additive_inverse (multiply X Z))).
137 ∀H5:∀X:Univ.∀Y:Univ.eq Univ (multiply X (additive_inverse Y)) (additive_inverse (multiply X Y)).
138 ∀H6:∀X:Univ.∀Y:Univ.eq Univ (multiply (additive_inverse X) Y) (additive_inverse (multiply X Y)).
139 ∀H7:∀X:Univ.∀Y:Univ.eq Univ (multiply (additive_inverse X) (additive_inverse Y)) (multiply X Y).
140 ∀H8:∀X:Univ.∀Y:Univ.eq Univ (commutator X Y) (add (multiply Y X) (additive_inverse (multiply X Y))).
141 ∀H9:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (associator X Y Z) (add (multiply (multiply X Y) Z) (additive_inverse (multiply X (multiply Y Z)))).
142 ∀H10:∀X:Univ.∀Y:Univ.eq Univ (multiply (multiply X X) Y) (multiply X (multiply X Y)).
143 ∀H11:∀X:Univ.∀Y:Univ.eq Univ (multiply (multiply X Y) Y) (multiply X (multiply Y Y)).
144 ∀H12:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add X (add Y Z)) (add (add X Y) Z).
145 ∀H13:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).
146 ∀H14:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X Y) Z) (add (multiply X Z) (multiply Y Z)).
147 ∀H15:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y Z)) (add (multiply X Y) (multiply X Z)).
148 ∀H16:∀X:Univ.eq Univ (additive_inverse (additive_inverse X)) X.
149 ∀H17:∀X:Univ.eq Univ (add X (additive_inverse X)) additive_identity.
150 ∀H18:∀X:Univ.eq Univ (add (additive_inverse X) X) additive_identity.
151 ∀H19:∀X:Univ.eq Univ (multiply X additive_identity) additive_identity.
152 ∀H20:∀X:Univ.eq Univ (multiply additive_identity X) additive_identity.
153 ∀H21:∀X:Univ.eq Univ (add X additive_identity) X.
154 ∀H22:∀X:Univ.eq Univ (add additive_identity X) X.eq Univ (add (associator (multiply x y) z w) (associator x y (commutator z w))) (add (multiply x (associator y z w)) (multiply (associator x z w) y)))
161 #additive_identity ##.
162 #additive_inverse ##.
193 nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12,H13,H14,H15,H16,H17,H18,H19,H20,H21,H22 ##;
194 ntry (nassumption) ##;
197 (* -------------------------------------------------------------------------- *)