0.10/0.11	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.10/0.12	% Command    : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
0.13/0.33	% Computer   : n020.cluster.edu
0.13/0.33	% Model      : x86_64 x86_64
0.13/0.33	% CPU        : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
0.13/0.33	% Memory     : 8042.1875MB
0.13/0.33	% OS         : Linux 3.10.0-693.el7.x86_64
0.13/0.33	% CPULimit   : 1200
0.13/0.33	% WCLimit    : 120
0.13/0.33	% DateTime   : Tue Jul 13 13:43:51 EDT 2021
0.13/0.33	% CPUTime    : 
0.13/0.37	% SZS status Theorem
0.13/0.37	
0.13/0.37	% SZS output start Proof
0.13/0.37	Take the following subset of the input axioms:
0.13/0.38	  fof(thm147, conjecture, ~(![P, Q]: (~a_truth(implies(P, Q)) | (a_truth(Q) | ~a_truth(P))) & (![P, Q]: a_truth(implies(P, implies(Q, P))) & (![P, Q, R]: a_truth(implies(implies(P, implies(Q, R)), implies(implies(P, Q), implies(P, R)))) & (![P, Q]: a_truth(implies(implies(not(P), not(Q)), implies(Q, P))) & ?[A]: ~a_truth(implies(A, A))))))).
0.13/0.38	
0.13/0.38	Now clausify the problem and encode Horn clauses using encoding 3 of
0.13/0.38	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
0.13/0.38	We repeatedly replace C & s=t => u=v by the two clauses:
0.13/0.38	  fresh(y, y, x1...xn) = u
0.13/0.38	  C => fresh(s, t, x1...xn) = v
0.13/0.38	where fresh is a fresh function symbol and x1..xn are the free
0.13/0.38	variables of u and v.
0.13/0.38	A predicate p(X) is encoded as p(X)=true (this is sound, because the
0.13/0.38	input problem has no model of domain size 1).
0.13/0.38	
0.13/0.38	The encoding turns the above axioms into the following unit equations and goals:
0.13/0.38	
0.13/0.38	Axiom 1 (thm147_3): fresh2(X, X, Y) = true.
0.13/0.38	Axiom 2 (thm147_3): fresh(X, X, Y, Z) = a_truth(Z).
0.13/0.38	Axiom 3 (thm147): a_truth(implies(X, implies(Y, X))) = true.
0.13/0.38	Axiom 4 (thm147_3): fresh(a_truth(implies(X, Y)), true, X, Y) = fresh2(a_truth(X), true, Y).
0.13/0.38	Axiom 5 (thm147_1): a_truth(implies(implies(X, implies(Y, Z)), implies(implies(X, Y), implies(X, Z)))) = true.
0.13/0.38	
0.13/0.38	Goal 1 (thm147_4): a_truth(implies(a, a)) = true.
0.13/0.38	Proof:
0.13/0.38	  a_truth(implies(a, a))
0.13/0.38	= { by axiom 2 (thm147_3) R->L }
0.13/0.38	  fresh(true, true, implies(a, implies(X, a)), implies(a, a))
0.13/0.38	= { by axiom 1 (thm147_3) R->L }
0.13/0.38	  fresh(fresh2(true, true, implies(implies(a, implies(X, a)), implies(a, a))), true, implies(a, implies(X, a)), implies(a, a))
0.13/0.38	= { by axiom 3 (thm147) R->L }
0.13/0.38	  fresh(fresh2(a_truth(implies(a, implies(implies(X, a), a))), true, implies(implies(a, implies(X, a)), implies(a, a))), true, implies(a, implies(X, a)), implies(a, a))
0.13/0.38	= { by axiom 4 (thm147_3) R->L }
0.13/0.38	  fresh(fresh(a_truth(implies(implies(a, implies(implies(X, a), a)), implies(implies(a, implies(X, a)), implies(a, a)))), true, implies(a, implies(implies(X, a), a)), implies(implies(a, implies(X, a)), implies(a, a))), true, implies(a, implies(X, a)), implies(a, a))
0.13/0.38	= { by axiom 5 (thm147_1) }
0.13/0.38	  fresh(fresh(true, true, implies(a, implies(implies(X, a), a)), implies(implies(a, implies(X, a)), implies(a, a))), true, implies(a, implies(X, a)), implies(a, a))
0.13/0.38	= { by axiom 2 (thm147_3) }
0.13/0.38	  fresh(a_truth(implies(implies(a, implies(X, a)), implies(a, a))), true, implies(a, implies(X, a)), implies(a, a))
0.13/0.38	= { by axiom 4 (thm147_3) }
0.13/0.38	  fresh2(a_truth(implies(a, implies(X, a))), true, implies(a, a))
0.13/0.38	= { by axiom 3 (thm147) }
0.13/0.38	  fresh2(true, true, implies(a, a))
0.13/0.38	= { by axiom 1 (thm147_3) }
0.13/0.38	  true
0.13/0.38	% SZS output end Proof
0.13/0.38	
0.13/0.38	RESULT: Theorem (the conjecture is true).
0.13/0.38	EOF