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# File     : LCL093-1 : TPTP v2.1.0. Released v1.0.0.
# Domain   : Logic Calculi (Implicational propositional)
# Problem  : IC-4 depends on the 5th Lukasiewicz axiom
# Version  : [TPTP] axioms.
# English  : Axiomatisations of the Implicational propositional calculus 
#            are {IC-2,IC-3,IC-4} by Tarski-Bernays and single Lukasiewicz 
#            axioms.Show that IC-4 depends on the fifth Lukasiewicz axiom.

# Refs     : [Luk48] Lukasiewicz (1948), The Shortest Axiom of the Implicat
#          : [Pfe88] Pfenning (1988), Single Axioms in the Implicational Pr
# Source   : [TPTP]
# Names    : 

# Status   : unsatisfiable
# Rating   : 0.78 v2.1.0, 0.88 v2.0.0
# Syntax   : Number of clauses    :    3 (   0 non-Horn;   2 unit;   2 RR)
#            Number of literals   :    5 (   0 equality)
#            Maximal clause size  :    3 (   1 average)
#            Number of predicates :    1 (   0 propositional; 1-1 arity)
#            Number of functors   :    4 (   3 constant; 0-2 arity)
#            Number of variables  :    7 (   2 singleton)
#            Maximal term depth   :    5 (   2 average)

# Comments : 
#          : tptp2X -f setheo:sign -t rm_equality:rstfp LCL093-1.p 
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# condensed_detachment, axiom.
is_a_theorem(Y) <- 
    is_a_theorem(implies(X, Y)),
    is_a_theorem(X).

# ic_JLukasiewicz_5, axiom.
is_a_theorem(implies(implies(implies(P, Q), implies(R, S)), implies(implies(S, P), implies(T, implies(R, P))))) <- .

# prove_ic_4, conjecture.
 <- is_a_theorem(implies(implies(a, b), implies(implies(b, c), implies(a, c)))).

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