#--------------------------------------------------------------------------
# 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 setheo:sign -t rm_equality:rstfp COL006-5.p 
#--------------------------------------------------------------------------
# s_definition, axiom.
equal(apply(apply(apply(s, X), Y), Z), apply(apply(X, Z), apply(Y, Z))) <- .

# k_definition, axiom.
equal(apply(apply(k, X), Y), X) <- .

# 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)))) <- .

# prove_strong_fixed_point, conjecture.
 <- equal(apply(strong_fixed_point, fixed_pt), apply(fixed_pt, apply(strong_fixed_point, fixed_pt))).

#--------------------------------------------------------------------------