1 include "logic/equality.ma".
3 (* Inclusion of: COL003-2.p *)
5 (* -------------------------------------------------------------------------- *)
7 (* File : COL003-2 : TPTP v3.2.0. Released v1.0.0. *)
9 (* Domain : Combinatory Logic *)
11 (* Problem : Strong fixed point for B and W *)
13 (* Version : [WM88] (equality) axioms : Augmented > Especial. *)
15 (* Theorem formulation : The fixed point is provided and checked. *)
17 (* English : The strong fixed point property holds for the set *)
19 (* P consisting of the combinators B and W alone, where ((Bx)y)z *)
21 (* = x(yz) and (Wx)y = (xy)y. *)
23 (* Refs : [Smu85] Smullyan (1978), To Mock a Mocking Bird and Other Logi *)
25 (* : [MW87] McCune & Wos (1987), A Case Study in Automated Theorem *)
27 (* : [WM88] Wos & McCune (1988), Challenge Problems Focusing on Eq *)
29 (* : [Wos93] Wos (1993), The Kernel Strategy and Its Use for the St *)
35 (* Status : Unsatisfiable *)
37 (* Rating : 0.29 v3.1.0, 0.33 v2.7.0, 0.17 v2.6.0, 0.29 v2.5.0, 0.20 v2.4.0, 0.33 v2.2.1, 0.67 v2.2.0, 0.71 v2.1.0, 1.00 v2.0.0 *)
39 (* Syntax : Number of clauses : 4 ( 0 non-Horn; 3 unit; 2 RR) *)
41 (* Number of atoms : 5 ( 3 equality) *)
43 (* Maximal clause size : 2 ( 1 average) *)
45 (* Number of predicates : 2 ( 0 propositional; 1-2 arity) *)
47 (* Number of functors : 4 ( 3 constant; 0-2 arity) *)
49 (* Number of variables : 6 ( 0 singleton) *)
51 (* Maximal term depth : 5 ( 3 average) *)
53 (* Comments : This the J sage of [McCune & Wos, 1987], found by Statman. *)
55 (* -------------------------------------------------------------------------- *)
56 ntheorem prove_strong_fixed_point:
57 (∀Univ:Type.∀Strong_fixed_point:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.
58 ∀apply:∀_:Univ.∀_:Univ.Univ.
60 ∀fixed_point:∀_:Univ.Prop.
63 ∀H0:∀Strong_fixed_point:Univ.∀_:eq Univ (apply Strong_fixed_point fixed_pt) (apply fixed_pt (apply Strong_fixed_point fixed_pt)).fixed_point Strong_fixed_point.
64 ∀H1:∀X:Univ.∀Y:Univ.eq Univ (apply (apply w X) Y) (apply (apply X Y) Y).
65 ∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply b X) Y) Z) (apply X (apply Y Z)).fixed_point (apply (apply b (apply w w)) (apply (apply b w) (apply (apply b b) b))))
68 #Strong_fixed_point ##.
83 (* -------------------------------------------------------------------------- *)