2 include "logic/equality.ma".
3 (* Inclusion of: RNG007-4.p *)
4 (* -------------------------------------------------------------------------- *)
5 (* File : RNG007-4 : TPTP v3.1.1. Released v1.0.0. *)
6 (* Domain : Ring Theory *)
7 (* Problem : In Boolean rings, X is its own inverse *)
8 (* Version : [Peterson & Stickel, 1981] (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, x + x = additive_identity *)
12 (* Refs : [PS81] Peterson & Stickel (1981), Complete Sets of Reductions *)
14 (* Names : lemma.ver2.in [ANL] *)
15 (* Status : Unsatisfiable *)
16 (* Rating : 0.00 v2.2.1, 0.11 v2.2.0, 0.14 v2.1.0, 0.13 v2.0.0 *)
17 (* Syntax : Number of clauses : 16 ( 0 non-Horn; 16 unit; 2 RR) *)
18 (* Number of atoms : 16 ( 16 equality) *)
19 (* Maximal clause size : 1 ( 1 average) *)
20 (* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
21 (* Number of functors : 5 ( 2 constant; 0-2 arity) *)
22 (* Number of variables : 26 ( 2 singleton) *)
23 (* Maximal term depth : 3 ( 2 average) *)
25 (* -------------------------------------------------------------------------- *)
26 (* ----Include ring theory axioms *)
27 (* Inclusion of: Axioms/RNG002-0.ax *)
28 (* -------------------------------------------------------------------------- *)
29 (* File : RNG002-0 : TPTP v3.1.1. Released v1.0.0. *)
30 (* Domain : Ring Theory *)
31 (* Axioms : Ring theory (equality) axioms *)
32 (* Version : [PS81] (equality) axioms : *)
33 (* Reduced & Augmented > Complete. *)
35 (* Refs : [PS81] Peterson & Stickel (1981), Complete Sets of Reductions *)
39 (* Syntax : Number of clauses : 14 ( 0 non-Horn; 14 unit; 1 RR) *)
40 (* Number of literals : 14 ( 14 equality) *)
41 (* Maximal clause size : 1 ( 1 average) *)
42 (* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
43 (* Number of functors : 4 ( 1 constant; 0-2 arity) *)
44 (* Number of variables : 25 ( 2 singleton) *)
45 (* Maximal term depth : 3 ( 2 average) *)
46 (* Comments : Not sure if these are complete. I don't know if the reductions *)
47 (* given in [PS81] are suitable for ATP. *)
48 (* -------------------------------------------------------------------------- *)
49 (* ----Existence of left identity for addition *)
50 (* ----Existence of left additive additive_inverse *)
51 (* ----Distributive property of product over sum *)
52 (* ----Inverse of identity is identity, stupid *)
53 (* ----Inverse of additive_inverse of X is X *)
54 (* ----Behavior of 0 and the multiplication operation *)
55 (* ----Inverse of (x + y) is additive_inverse(x) + additive_inverse(y) *)
56 (* ----x * additive_inverse(y) = additive_inverse (x * y) *)
57 (* ----Associativity of addition *)
58 (* ----Commutativity of addition *)
59 (* ----Associativity of product *)
60 (* -------------------------------------------------------------------------- *)
61 (* -------------------------------------------------------------------------- *)
62 theorem prove_inverse:
65 \forall add:\forall _:Univ.\forall _:Univ.Univ.
66 \forall additive_identity:Univ.
67 \forall additive_inverse:\forall _:Univ.Univ.
68 \forall multiply:\forall _:Univ.\forall _:Univ.Univ.
69 \forall H0:\forall X:Univ.eq Univ (multiply X X) X.
70 \forall H1:\forall X:Univ.\forall Y:Univ.\forall Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
71 \forall H2:\forall X:Univ.\forall Y:Univ.eq Univ (add X Y) (add Y X).
72 \forall H3:\forall X:Univ.\forall Y:Univ.\forall Z:Univ.eq Univ (add (add X Y) Z) (add X (add Y Z)).
73 \forall H4:\forall X:Univ.\forall Y:Univ.eq Univ (multiply (additive_inverse X) Y) (additive_inverse (multiply X Y)).
74 \forall H5:\forall X:Univ.\forall Y:Univ.eq Univ (multiply X (additive_inverse Y)) (additive_inverse (multiply X Y)).
75 \forall H6:\forall X:Univ.\forall Y:Univ.eq Univ (additive_inverse (add X Y)) (add (additive_inverse X) (additive_inverse Y)).
76 \forall H7:\forall X:Univ.eq Univ (multiply additive_identity X) additive_identity.
77 \forall H8:\forall X:Univ.eq Univ (multiply X additive_identity) additive_identity.
78 \forall H9:\forall X:Univ.eq Univ (additive_inverse (additive_inverse X)) X.
79 \forall H10:eq Univ (additive_inverse additive_identity) additive_identity.
80 \forall H11:\forall X:Univ.\forall Y:Univ.\forall Z:Univ.eq Univ (multiply (add X Y) Z) (add (multiply X Z) (multiply Y Z)).
81 \forall H12:\forall X:Univ.\forall Y:Univ.\forall Z:Univ.eq Univ (multiply X (add Y Z)) (add (multiply X Y) (multiply X Z)).
82 \forall H13:\forall X:Univ.eq Univ (add (additive_inverse X) X) additive_identity.
83 \forall H14:\forall X:Univ.eq Univ (add additive_identity X) X.eq Univ (add a a) additive_identity
86 autobatch paramodulation timeout=100;
90 (* -------------------------------------------------------------------------- *)