1 include "logic/equality.ma".
3 (* Inclusion of: RNG008-7.p *)
5 (* -------------------------------------------------------------------------- *)
7 (* File : RNG008-7 : TPTP v3.7.0. Released v1.0.0. *)
9 (* Domain : Ring Theory *)
11 (* Problem : Boolean rings are commutative *)
13 (* Version : [LW91] (equality) axioms. *)
15 (* English : Given a ring in which for all x, x * x = x, prove that for *)
17 (* all x and y, x * y = y * x. *)
19 (* Refs : [LO85] Lusk & Overbeek (1985), Reasoning about Equality *)
21 (* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
23 (* : [LW91] Lusk & Wos (1991), Benchmark Problems in Which Equalit *)
27 (* Names : Problem 3 [LO85] *)
29 (* : Test Problem 8 [Wos88] *)
31 (* : Boolean Rings [Wos88] *)
35 (* Status : Unsatisfiable *)
37 (* Rating : 0.00 v3.3.0, 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 *)
39 (* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 2 RR) *)
41 (* Number of atoms : 12 ( 12 equality) *)
43 (* Maximal clause size : 1 ( 1 average) *)
45 (* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
47 (* Number of functors : 7 ( 4 constant; 0-2 arity) *)
49 (* Number of variables : 19 ( 0 singleton) *)
51 (* Maximal term depth : 3 ( 2 average) *)
53 (* Comments : This is very similar to ring_x2.in [OTTER]. *)
55 (* -------------------------------------------------------------------------- *)
57 (* ----Include ring theory axioms *)
59 (* Inclusion of: Axioms/RNG005-0.ax *)
61 (* -------------------------------------------------------------------------- *)
63 (* File : RNG005-0 : TPTP v3.7.0. Released v1.0.0. *)
65 (* Domain : Ring Theory *)
67 (* Axioms : Ring theory (equality) axioms *)
69 (* Version : [LW92] (equality) axioms. *)
73 (* Refs : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
75 (* : [LW92] Lusk & Wos (1992), Benchmark Problems in Which Equalit *)
83 (* Syntax : Number of clauses : 9 ( 0 non-Horn; 9 unit; 0 RR) *)
85 (* Number of atoms : 9 ( 9 equality) *)
87 (* Maximal clause size : 1 ( 1 average) *)
89 (* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
91 (* Number of functors : 4 ( 1 constant; 0-2 arity) *)
93 (* Number of variables : 18 ( 0 singleton) *)
95 (* Maximal term depth : 3 ( 2 average) *)
97 (* Comments : These axioms are used in [Wos88] p.203. *)
99 (* -------------------------------------------------------------------------- *)
101 (* ----There exists an additive identity element *)
103 (* ----Existence of left additive additive_inverse *)
105 (* ----Associativity for addition *)
107 (* ----Commutativity for addition *)
109 (* ----Associativity for multiplication *)
111 (* ----Distributive property of product over sum *)
113 (* -------------------------------------------------------------------------- *)
115 (* -------------------------------------------------------------------------- *)
116 ntheorem prove_commutativity:
117 (∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
119 ∀add:∀_:Univ.∀_:Univ.Univ.
120 ∀additive_identity:Univ.
121 ∀additive_inverse:∀_:Univ.Univ.
124 ∀multiply:∀_:Univ.∀_:Univ.Univ.
125 ∀H0:eq Univ (multiply a b) c.
126 ∀H1:∀X:Univ.eq Univ (multiply X X) X.
127 ∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (add X Y) Z) (add (multiply X Z) (multiply Y Z)).
128 ∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y Z)) (add (multiply X Y) (multiply X Z)).
129 ∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (multiply Y Z)) (multiply (multiply X Y) Z).
130 ∀H5:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).
131 ∀H6:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add X (add Y Z)) (add (add X Y) Z).
132 ∀H7:∀X:Univ.eq Univ (add X (additive_inverse X)) additive_identity.
133 ∀H8:∀X:Univ.eq Univ (add (additive_inverse X) X) additive_identity.
134 ∀H9:∀X:Univ.eq Univ (add X additive_identity) X.
135 ∀H10:∀X:Univ.eq Univ (add additive_identity X) X.eq Univ (multiply b a) c)
143 #additive_identity ##.
144 #additive_inverse ##.
159 nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10 ##;
160 ntry (nassumption) ##;
163 (* -------------------------------------------------------------------------- *)