1 include "logic/equality.ma".
3 (* Inclusion of: ALG005-1.p *)
5 (* -------------------------------------------------------------------------- *)
7 (* File : ALG005-1 : TPTP v3.7.0. Released v2.2.0. *)
9 (* Domain : General Algebra *)
11 (* Problem : Associativity of intersection in terms of set difference. *)
13 (* Version : [MP96] (equality) axioms : Especial. *)
15 (* English : Starting with Kalman's basis for families of sets closed under *)
17 (* set difference, we define intersection and show it to be *)
21 (* Refs : [McC98] McCune (1998), Email to G. Sutcliffe *)
23 (* : [MP96] McCune & Padmanabhan (1996), Automated Deduction in Eq *)
25 (* Source : [McC98] *)
27 (* Names : SD-2-a [MP96] *)
29 (* Status : Unsatisfiable *)
31 (* Rating : 0.11 v3.4.0, 0.12 v3.3.0, 0.07 v3.1.0, 0.22 v2.7.0, 0.00 v2.2.1 *)
33 (* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 1 RR) *)
35 (* Number of atoms : 5 ( 5 equality) *)
37 (* Maximal clause size : 1 ( 1 average) *)
39 (* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
41 (* Number of functors : 5 ( 3 constant; 0-2 arity) *)
43 (* Number of variables : 9 ( 1 singleton) *)
45 (* Maximal term depth : 3 ( 3 average) *)
49 (* -------------------------------------------------------------------------- *)
51 (* ----Kalman's axioms for set difference: *)
53 (* ----Definition of intersection: *)
55 (* ----Denial of associativity: *)
56 ntheorem prove_associativity_of_multiply:
57 (∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
61 ∀difference:∀_:Univ.∀_:Univ.Univ.
62 ∀multiply:∀_:Univ.∀_:Univ.Univ.
63 ∀H0:∀X:Univ.∀Y:Univ.eq Univ (multiply X Y) (difference X (difference X Y)).
64 ∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (difference (difference X Y) Z) (difference (difference X Z) (difference Y Z)).
65 ∀H2:∀X:Univ.∀Y:Univ.eq Univ (difference X (difference X Y)) (difference Y (difference Y X)).
66 ∀H3:∀X:Univ.∀Y:Univ.eq Univ (difference X (difference Y X)) X.eq Univ (multiply (multiply a b) c) (multiply a (multiply b c)))
81 nauto by H0,H1,H2,H3 ##;
82 ntry (nassumption) ##;
85 (* -------------------------------------------------------------------------- *)