#--------------------------------------------------------------------------
# File     : COL003-5 : TPTP v2.1.0. Released v1.0.0.
# Domain   : Combinatory Logic
# Problem  : Strong fixed point for B and W
# Version  : [WM88] (equality) axioms : Augmented > Special.
#            Theorem formulation : The fixed point is provided and checked.
# English  : The strong fixed point property holds for the set 
#            P consisting of the combinators B and W alone, where ((Bx)y)z 
#            = x(yz) and (Wx)y = (xy)y.

# Refs     : [Smu85] Smullyan (1978), To Mock a Mocking Bird and Other Logi
#          : [MW87]  McCune & Wos (1987), A Case Study in Automated Theorem
#          : [WM88]  Wos & McCune (1988), Challenge Problems Focusing on Eq
#          : [Wos93] Wos (1993), The Kernel Strategy and Its Use for the St
# Source   : [TPTP]
# Names    : 

# Status   : unsatisfiable
# Rating   : 0.83 v2.1.0, 0.80 v2.0.0
# Syntax   : Number of clauses    :    4 (   0 non-Horn;   3 unit;   2 RR)
#            Number of literals   :    5 (   3 equality)
#            Maximal clause size  :    2 (   1 average)
#            Number of predicates :    2 (   0 propositional; 1-2 arity)
#            Number of functors   :    4 (   3 constant; 0-2 arity)
#            Number of variables  :    6 (   0 singleton)
#            Maximal term depth   :    6 (   3 average)

# Comments : This the M sage of [McCune & Wos, 1987].
#          : fixed_point/1 substitution axioms are not included as it is
#            simply a way of introducing the required copies of the strong
#            fixed point.
#          : tptp2X -f setheo:sign -t rm_equality:rstfp COL003-5.p 
#--------------------------------------------------------------------------
# b_definition, axiom.
equal(apply(apply(apply(b, X), Y), Z), apply(X, apply(Y, Z))) <- .

# w_definition, axiom.
equal(apply(apply(w, X), Y), apply(apply(X, Y), Y)) <- .

# strong_fixed_point, axiom.
fixed_point(Strong_fixed_point) <- 
    equal(apply(Strong_fixed_point, fixed_pt), apply(fixed_pt, apply(Strong_fixed_point, fixed_pt))).

# prove_strong_fixed_point, conjecture.
 <- fixed_point(apply(apply(b, apply(apply(b, apply(w, w)), w)), apply(apply(b, b), b))).

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