2 include "logic/equality.ma".
3 (* Inclusion of: RNG008-4.p *)
4 (* -------------------------------------------------------------------------- *)
5 (* File : RNG008-4 : TPTP v3.1.1. Released v1.0.0. *)
6 (* Domain : Ring Theory *)
7 (* Problem : Boolean rings are commutative *)
8 (* Version : [PS81] (equality) axioms. *)
9 (* Theorem formulation : Equality. *)
10 (* English : Given a ring in which for all x, x * x = x, prove that for *)
11 (* all x and y, x * y = y * x. *)
12 (* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
13 (* : [PS81] Peterson & Stickel (1981), Complete Sets of Reductions *)
16 (* Status : Unsatisfiable *)
17 (* 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 *)
18 (* Syntax : Number of clauses : 17 ( 0 non-Horn; 17 unit; 3 RR) *)
19 (* Number of atoms : 17 ( 17 equality) *)
20 (* Maximal clause size : 1 ( 1 average) *)
21 (* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
22 (* Number of functors : 7 ( 4 constant; 0-2 arity) *)
23 (* Number of variables : 26 ( 2 singleton) *)
24 (* Maximal term depth : 3 ( 2 average) *)
26 (* -------------------------------------------------------------------------- *)
27 (* ----Include ring theory axioms *)
28 (* Inclusion of: Axioms/RNG002-0.ax *)
29 (* -------------------------------------------------------------------------- *)
30 (* File : RNG002-0 : TPTP v3.1.1. Released v1.0.0. *)
31 (* Domain : Ring Theory *)
32 (* Axioms : Ring theory (equality) axioms *)
33 (* Version : [PS81] (equality) axioms : *)
34 (* Reduced & Augmented > Complete. *)
36 (* Refs : [PS81] Peterson & Stickel (1981), Complete Sets of Reductions *)
40 (* Syntax : Number of clauses : 14 ( 0 non-Horn; 14 unit; 1 RR) *)
41 (* Number of literals : 14 ( 14 equality) *)
42 (* Maximal clause size : 1 ( 1 average) *)
43 (* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
44 (* Number of functors : 4 ( 1 constant; 0-2 arity) *)
45 (* Number of variables : 25 ( 2 singleton) *)
46 (* Maximal term depth : 3 ( 2 average) *)
47 (* Comments : Not sure if these are complete. I don't know if the reductions *)
48 (* given in [PS81] are suitable for ATP. *)
49 (* -------------------------------------------------------------------------- *)
50 (* ----Existence of left identity for addition *)
51 (* ----Existence of left additive additive_inverse *)
52 (* ----Distributive property of product over sum *)
53 (* ----Inverse of identity is identity, stupid *)
54 (* ----Inverse of additive_inverse of X is X *)
55 (* ----Behavior of 0 and the multiplication operation *)
56 (* ----Inverse of (x + y) is additive_inverse(x) + additive_inverse(y) *)
57 (* ----x * additive_inverse(y) = additive_inverse (x * y) *)
58 (* ----Associativity of addition *)
59 (* ----Commutativity of addition *)
60 (* ----Associativity of product *)
61 (* -------------------------------------------------------------------------- *)
62 (* -------------------------------------------------------------------------- *)
63 theorem prove_commutativity:
66 \forall add:\forall _:Univ.\forall _:Univ.Univ.
67 \forall additive_identity:Univ.
68 \forall additive_inverse:\forall _:Univ.Univ.
71 \forall multiply:\forall _:Univ.\forall _:Univ.Univ.
72 \forall H0:eq Univ (multiply a b) c.
73 \forall H1:\forall X:Univ.eq Univ (multiply X X) X.
74 \forall H2:\forall X:Univ.\forall Y:Univ.\forall Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
75 \forall H3:\forall X:Univ.\forall Y:Univ.eq Univ (add X Y) (add Y X).
76 \forall H4:\forall X:Univ.\forall Y:Univ.\forall Z:Univ.eq Univ (add (add X Y) Z) (add X (add Y Z)).
77 \forall H5:\forall X:Univ.\forall Y:Univ.eq Univ (multiply (additive_inverse X) Y) (additive_inverse (multiply X Y)).
78 \forall H6:\forall X:Univ.\forall Y:Univ.eq Univ (multiply X (additive_inverse Y)) (additive_inverse (multiply X Y)).
79 \forall H7:\forall X:Univ.\forall Y:Univ.eq Univ (additive_inverse (add X Y)) (add (additive_inverse X) (additive_inverse Y)).
80 \forall H8:\forall X:Univ.eq Univ (multiply additive_identity X) additive_identity.
81 \forall H9:\forall X:Univ.eq Univ (multiply X additive_identity) additive_identity.
82 \forall H10:\forall X:Univ.eq Univ (additive_inverse (additive_inverse X)) X.
83 \forall H11:eq Univ (additive_inverse additive_identity) additive_identity.
84 \forall H12:\forall X:Univ.\forall Y:Univ.\forall Z:Univ.eq Univ (multiply (add X Y) Z) (add (multiply X Z) (multiply Y Z)).
85 \forall H13:\forall X:Univ.\forall Y:Univ.\forall Z:Univ.eq Univ (multiply X (add Y Z)) (add (multiply X Y) (multiply X Z)).
86 \forall H14:\forall X:Univ.eq Univ (add (additive_inverse X) X) additive_identity.
87 \forall H15:\forall X:Univ.eq Univ (add additive_identity X) X.eq Univ (multiply b a) c
90 autobatch paramodulation timeout=100;
94 (* -------------------------------------------------------------------------- *)