%--------------------------------------------------------------------------
% File     : COL006-5 : TPTP v2.1.0. Released v2.1.0.
% Domain   : Combinatory Logic
% Problem  : Strong fixed point for S and K
% Version  : [WM88] (equality) axioms.
%            Theorem formulation : The fixed point is provided and checked.
% English  : The strong fixed point property holds for the set 
%            P consisting of the combinators S and K alone, where
%            ((Sx)y)z = (xz)(yz), (Kx)y = x.

% Refs     : [WM88]  Wos & McCune (1988), Challenge Problems Focusing on Eq
% Source   : [TPTP]
% Names    : 

% Status   : unknown
% Rating   : ? v2.1.0
% Syntax   : Number of clauses    :    4 (   0 non-Horn;   4 unit;   2 RR)
%            Number of literals   :    4 (   4 equality)
%            Maximal clause size  :    1 (   1 average)
%            Number of predicates :    1 (   0 propositional; 2-2 arity)
%            Number of functors   :    5 (   4 constant; 0-2 arity)
%            Number of variables  :    5 (   1 singleton)
%            Maximal term depth   :    8 (   3 average)

% Comments : 
%          : tptp2X -f tptp -t rm_equality:rstfp COL006-5.p 
%--------------------------------------------------------------------------
input_clause(s_definition,axiom,
    [++ equal(apply(apply(apply(s, X), Y), Z), apply(apply(X, Z), apply(Y, Z)))]).

input_clause(k_definition,axiom,
    [++ equal(apply(apply(k, X), Y), X)]).

input_clause(strong_fixed_point,axiom,
    [++ equal(strong_fixed_point, apply(apply(s, apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k)))), apply(apply(s, apply(k, apply(apply(s, s), apply(s, k)))), apply(apply(s, apply(k, s)), k))))]).

input_clause(prove_strong_fixed_point,conjecture,
    [-- equal(apply(strong_fixed_point, fixed_pt), apply(fixed_pt, apply(strong_fixed_point, fixed_pt)))]).
%--------------------------------------------------------------------------