1 include "logic/equality.ma".
3 (* Inclusion of: RNG035-7.p *)
5 (* -------------------------------------------------------------------------- *)
7 (* File : RNG035-7 : TPTP v3.7.0. Released v1.0.0. *)
9 (* Domain : Ring Theory *)
11 (* Problem : If X*X*X*X = X then the ring is commutative *)
13 (* Version : [LW91] (equality) axioms. *)
15 (* English : Given a ring in which for all x, x * x * x * x = x, prove *)
17 (* that for all x and y, x * y = y * x. *)
19 (* Refs : [LW91] Lusk & Wos (1991), Benchmark Problems in Which Equalit *)
23 (* Names : RT3 [LW91] *)
25 (* Status : Unsatisfiable *)
27 (* Rating : 0.89 v3.4.0, 0.88 v3.3.0, 0.79 v3.2.0, 0.86 v3.1.0, 0.67 v2.7.0, 0.73 v2.6.0, 0.50 v2.5.0, 0.25 v2.4.0, 0.33 v2.2.1, 1.00 v2.0.0 *)
29 (* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 2 RR) *)
31 (* Number of atoms : 12 ( 12 equality) *)
33 (* Maximal clause size : 1 ( 1 average) *)
35 (* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
37 (* Number of functors : 7 ( 4 constant; 0-2 arity) *)
39 (* Number of variables : 19 ( 0 singleton) *)
41 (* Maximal term depth : 4 ( 2 average) *)
45 (* -------------------------------------------------------------------------- *)
47 (* ----Include ring theory axioms *)
49 (* Inclusion of: Axioms/RNG005-0.ax *)
51 (* -------------------------------------------------------------------------- *)
53 (* File : RNG005-0 : TPTP v3.7.0. Released v1.0.0. *)
55 (* Domain : Ring Theory *)
57 (* Axioms : Ring theory (equality) axioms *)
59 (* Version : [LW92] (equality) axioms. *)
63 (* Refs : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
65 (* : [LW92] Lusk & Wos (1992), Benchmark Problems in Which Equalit *)
73 (* Syntax : Number of clauses : 9 ( 0 non-Horn; 9 unit; 0 RR) *)
75 (* Number of atoms : 9 ( 9 equality) *)
77 (* Maximal clause size : 1 ( 1 average) *)
79 (* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
81 (* Number of functors : 4 ( 1 constant; 0-2 arity) *)
83 (* Number of variables : 18 ( 0 singleton) *)
85 (* Maximal term depth : 3 ( 2 average) *)
87 (* Comments : These axioms are used in [Wos88] p.203. *)
89 (* -------------------------------------------------------------------------- *)
91 (* ----There exists an additive identity element *)
93 (* ----Existence of left additive additive_inverse *)
95 (* ----Associativity for addition *)
97 (* ----Commutativity for addition *)
99 (* ----Associativity for multiplication *)
101 (* ----Distributive property of product over sum *)
103 (* -------------------------------------------------------------------------- *)
105 (* -------------------------------------------------------------------------- *)
106 ntheorem prove_commutativity:
107 (∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
109 ∀add:∀_:Univ.∀_:Univ.Univ.
110 ∀additive_identity:Univ.
111 ∀additive_inverse:∀_:Univ.Univ.
114 ∀multiply:∀_:Univ.∀_:Univ.Univ.
115 ∀H0:eq Univ (multiply a b) c.
116 ∀H1:∀X:Univ.eq Univ (multiply X (multiply X (multiply X X))) X.
117 ∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X Y) Z) (add (multiply X Z) (multiply Y Z)).
118 ∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y Z)) (add (multiply X Y) (multiply X Z)).
119 ∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (multiply Y Z)) (multiply (multiply X Y) Z).
120 ∀H5:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).
121 ∀H6:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add X (add Y Z)) (add (add X Y) Z).
122 ∀H7:∀X:Univ.eq Univ (add X (additive_inverse X)) additive_identity.
123 ∀H8:∀X:Univ.eq Univ (add (additive_inverse X) X) additive_identity.
124 ∀H9:∀X:Univ.eq Univ (add X additive_identity) X.
125 ∀H10:∀X:Univ.eq Univ (add additive_identity X) X.eq Univ (multiply b a) c)
133 #additive_identity ##.
134 #additive_inverse ##.
149 nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10 ##;
150 ntry (nassumption) ##;
153 (* -------------------------------------------------------------------------- *)