1 include "logic/equality.ma".
3 (* Inclusion of: COL066-3.p *)
5 (* -------------------------------------------------------------------------- *)
7 (* File : COL066-3 : TPTP v3.7.0. Bugfixed v1.2.0. *)
9 (* Domain : Combinatory Logic *)
11 (* Problem : Find combinator equivalent to P from B, Q and W *)
13 (* Version : [WM88] (equality) axioms. *)
15 (* Theorem formulation : The combinator is provided and checked. *)
17 (* English : Construct from B, Q and W alone a combinator that behaves as *)
19 (* the combinator P does, where ((Bx)y)z = x(yz), ((Qx)y)z = *)
21 (* y(xz), (Wx)y = (xy)y, (((Px)y)y)z = (xy)((xy)z) *)
23 (* Refs : [WM88] Wos & McCune (1988), Challenge Problems Focusing on Eq *)
25 (* : [WW+90] Wos et al. (1990), Automated Reasoning Contributes to *)
31 (* Status : Unsatisfiable *)
33 (* Rating : 0.11 v3.4.0, 0.12 v3.3.0, 0.00 v3.1.0, 0.11 v2.7.0, 0.00 v2.1.0, 0.29 v2.0.0 *)
35 (* Syntax : Number of clauses : 4 ( 0 non-Horn; 4 unit; 1 RR) *)
37 (* Number of atoms : 4 ( 4 equality) *)
39 (* Maximal clause size : 1 ( 1 average) *)
41 (* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
43 (* Number of functors : 7 ( 6 constant; 0-2 arity) *)
45 (* Number of variables : 8 ( 0 singleton) *)
47 (* Maximal term depth : 10 ( 4 average) *)
51 (* Bugfixes : v1.2.0 : Redundant [fgh]_substitution axioms removed. *)
53 (* -------------------------------------------------------------------------- *)
55 (* ----This is the P equivalent *)
56 ntheorem prove_p_combinator:
57 (∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
58 ∀apply:∀_:Univ.∀_:Univ.Univ.
65 ∀H0:∀X:Univ.∀Y:Univ.eq Univ (apply (apply w X) Y) (apply (apply X Y) Y).
66 ∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply q X) Y) Z) (apply Y (apply X Z)).
67 ∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply b X) Y) Z) (apply X (apply Y Z)).eq Univ (apply (apply (apply (apply (apply (apply b (apply w (apply q (apply q q)))) q) x) y) y) z) (apply (apply x y) (apply (apply x y) z)))
84 ntry (nassumption) ##;
87 (* -------------------------------------------------------------------------- *)