1 include "logic/equality.ma".
3 (* Inclusion of: COL075-2.p *)
5 (* -------------------------------------------------------------------------- *)
7 (* File : COL075-2 : TPTP v3.7.0. Released v1.2.0. *)
9 (* Domain : Combinatory Logic *)
11 (* Problem : Lemma 1 for showing the unsatisfiable variant of TRC *)
13 (* Version : [Jec95] (equality) axioms : Reduced > Incomplete. *)
15 (* English : Searching for a diagonal combinator F with the property *)
19 (* Refs : [Jec95] Jech (1995), Otter Experiments in a System of Combinat *)
21 (* Source : [Jec95] *)
23 (* Names : - [Jec95] *)
25 (* Status : Unsatisfiable *)
27 (* Rating : 0.11 v3.4.0, 0.12 v3.3.0, 0.00 v2.0.0 *)
29 (* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 1 RR) *)
31 (* Number of atoms : 3 ( 3 equality) *)
33 (* Maximal clause size : 1 ( 1 average) *)
35 (* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
37 (* Number of functors : 5 ( 2 constant; 0-2 arity) *)
39 (* Number of variables : 6 ( 1 singleton) *)
41 (* Maximal term depth : 4 ( 3 average) *)
45 (* -------------------------------------------------------------------------- *)
47 (* ----Don't include axioms of Type-respecting combinators *)
49 (* include('Axioms/COL001-0.ax'). *)
51 (* -------------------------------------------------------------------------- *)
53 (* ----Replace k function by k combinator *)
55 (* input_clause(k_definition,axiom, *)
57 (* [++equal(apply(k(X),Y),X)]). *)
59 (* ----Replace k function by k combinator *)
61 (* input_clause(abstraction,axiom, *)
63 (* [++equal(apply(apply(apply(abstraction,X),Y),Z),apply(apply(X,k(Z)), *)
67 (* ----Subsitution axioms *)
68 ntheorem prove_diagonal_combinator:
69 (∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
71 ∀apply:∀_:Univ.∀_:Univ.Univ.
75 ∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (apply (apply (apply abstraction X) Y) Z) (apply (apply X (apply k Z)) (apply Y Z)).
76 ∀H1:∀X:Univ.∀Y:Univ.eq Univ (apply (apply k X) Y) X.∃Y:Univ.eq Univ (apply (apply Y (b Y)) (c Y)) (apply (b Y) (b Y)))
89 napply (ex_intro ? ? ? ?) ##[
93 ntry (nassumption) ##;
96 (* -------------------------------------------------------------------------- *)