0.03/0.13	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.03/0.14	% Command    : twee %s --tstp --casc --quiet --explain-encoding --conditional-encoding if --smaller --drop-non-horn
0.13/0.36	% Computer   : n026.cluster.edu
0.13/0.36	% Model      : x86_64 x86_64
0.13/0.36	% CPU        : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
0.13/0.36	% Memory     : 8042.1875MB
0.13/0.36	% OS         : Linux 3.10.0-693.el7.x86_64
0.13/0.36	% CPULimit   : 180
0.13/0.36	% DateTime   : Thu Aug 29 12:34:05 EDT 2019
0.13/0.36	% CPUTime    : 
1.62/1.79	% SZS status Unsatisfiable
1.62/1.79	
1.62/1.79	% SZS output start Proof
1.62/1.79	Take the following subset of the input axioms:
1.62/1.79	  fof(axiom_1_3, axiom, ![A, B]: axiom(or(not(A), or(B, A)))=true).
1.62/1.79	  fof(axiom_1_4, axiom, ![A, B]: axiom(or(not(or(A, B)), or(B, A)))=true).
1.62/1.79	  fof(axiom_1_5, axiom, ![A, B, C]: true=axiom(or(not(or(A, or(B, C))), or(B, or(A, C))))).
1.62/1.79	  fof(ifeq_axiom, axiom, ![A, B, C]: ifeq(A, A, B, C)=B).
1.62/1.79	  fof(prove_this, negated_conjecture, theorem(or(not(or(not(or(p, q)), q)), or(not(p), q)))!=true).
1.62/1.79	  fof(rule_1, axiom, ![X]: true=ifeq(axiom(X), true, theorem(X), true)).
1.62/1.79	  fof(rule_2, axiom, ![Y, X]: ifeq(theorem(Y), true, ifeq(axiom(or(not(Y), X)), true, theorem(X), true), true)=true).
1.62/1.79	  fof(rule_3, axiom, ![Y, Z, X]: true=ifeq(theorem(or(not(Y), Z)), true, ifeq(axiom(or(not(X), Y)), true, theorem(or(not(X), Z)), true), true)).
1.62/1.79	
1.62/1.79	Now clausify the problem and encode Horn clauses using encoding 3 of
1.62/1.79	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
1.62/1.79	We repeatedly replace C & s=t => u=v by the two clauses:
1.62/1.79	  fresh(y, y, x1...xn) = u
1.62/1.79	  C => fresh(s, t, x1...xn) = v
1.62/1.79	where fresh is a fresh function symbol and x1..xn are the free
1.62/1.79	variables of u and v.
1.62/1.79	A predicate p(X) is encoded as p(X)=true (this is sound, because the
1.62/1.79	input problem has no model of domain size 1).
1.62/1.79	
1.62/1.79	The encoding turns the above axioms into the following unit equations and goals:
1.62/1.79	
1.62/1.79	Axiom 1 (rule_1): true = ifeq(axiom(X), true, theorem(X), true).
1.62/1.79	Axiom 2 (axiom_1_3): axiom(or(not(X), or(Y, X))) = true.
1.62/1.79	Axiom 3 (rule_2): ifeq(theorem(X), true, ifeq(axiom(or(not(X), Y)), true, theorem(Y), true), true) = true.
1.62/1.79	Axiom 4 (rule_3): true = ifeq(theorem(or(not(X), Y)), true, ifeq(axiom(or(not(Z), X)), true, theorem(or(not(Z), Y)), true), true).
1.62/1.79	Axiom 5 (axiom_1_4): axiom(or(not(or(X, Y)), or(Y, X))) = true.
1.62/1.79	Axiom 6 (ifeq_axiom): ifeq(X, X, Y, Z) = Y.
1.62/1.79	Axiom 7 (axiom_1_5): true = axiom(or(not(or(X, or(Y, Z))), or(Y, or(X, Z)))).
1.62/1.79	
1.62/1.79	Lemma 8: ifeq(theorem(or(X, or(Y, Z))), true, theorem(or(Y, or(X, Z))), true) = true.
1.62/1.79	Proof:
1.62/1.79	  ifeq(theorem(or(X, or(Y, Z))), true, theorem(or(Y, or(X, Z))), true)
1.62/1.79	= { by axiom 6 (ifeq_axiom) }
1.62/1.79	  ifeq(theorem(or(X, or(Y, Z))), true, ifeq(true, true, theorem(or(Y, or(X, Z))), true), true)
1.62/1.79	= { by axiom 7 (axiom_1_5) }
1.62/1.79	  ifeq(theorem(or(X, or(Y, Z))), true, ifeq(axiom(or(not(or(X, or(Y, Z))), or(Y, or(X, Z)))), true, theorem(or(Y, or(X, Z))), true), true)
1.62/1.79	= { by axiom 3 (rule_2) }
1.62/1.79	  true
1.62/1.79	
1.62/1.79	Lemma 9: ifeq(theorem(or(not(or(X, Y)), Z)), true, theorem(or(not(or(Y, X)), Z)), true) = true.
1.62/1.79	Proof:
1.62/1.79	  ifeq(theorem(or(not(or(X, Y)), Z)), true, theorem(or(not(or(Y, X)), Z)), true)
1.62/1.79	= { by axiom 6 (ifeq_axiom) }
1.62/1.79	  ifeq(theorem(or(not(or(X, Y)), Z)), true, ifeq(true, true, theorem(or(not(or(Y, X)), Z)), true), true)
1.62/1.79	= { by axiom 5 (axiom_1_4) }
1.62/1.79	  ifeq(theorem(or(not(or(X, Y)), Z)), true, ifeq(axiom(or(not(or(Y, X)), or(X, Y))), true, theorem(or(not(or(Y, X)), Z)), true), true)
1.62/1.79	= { by axiom 4 (rule_3) }
1.62/1.80	  true
1.62/1.80	
1.62/1.80	Goal 1 (prove_this): theorem(or(not(or(not(or(p, q)), q)), or(not(p), q))) = true.
1.62/1.80	Proof:
1.62/1.80	  theorem(or(not(or(not(or(p, q)), q)), or(not(p), q)))
1.62/1.80	= { by axiom 6 (ifeq_axiom) }
1.62/1.80	  ifeq(true, true, theorem(or(not(or(not(or(p, q)), q)), or(not(p), q))), true)
1.62/1.80	= { by lemma 8 }
1.62/1.80	  ifeq(ifeq(theorem(or(not(p), or(not(or(q, not(or(p, q)))), q))), true, theorem(or(not(or(q, not(or(p, q)))), or(not(p), q))), true), true, theorem(or(not(or(not(or(p, q)), q)), or(not(p), q))), true)
1.62/1.80	= { by axiom 6 (ifeq_axiom) }
1.62/1.80	  ifeq(ifeq(ifeq(true, true, theorem(or(not(p), or(not(or(q, not(or(p, q)))), q))), true), true, theorem(or(not(or(q, not(or(p, q)))), or(not(p), q))), true), true, theorem(or(not(or(not(or(p, q)), q)), or(not(p), q))), true)
1.62/1.80	= { by lemma 9 }
1.62/1.80	  ifeq(ifeq(ifeq(ifeq(theorem(or(not(or(p, q)), or(not(or(q, not(or(p, q)))), q))), true, theorem(or(not(or(q, p)), or(not(or(q, not(or(p, q)))), q))), true), true, theorem(or(not(p), or(not(or(q, not(or(p, q)))), q))), true), true, theorem(or(not(or(q, not(or(p, q)))), or(not(p), q))), true), true, theorem(or(not(or(not(or(p, q)), q)), or(not(p), q))), true)
1.62/1.80	= { by axiom 6 (ifeq_axiom) }
1.62/1.80	  ifeq(ifeq(ifeq(ifeq(ifeq(true, true, theorem(or(not(or(p, q)), or(not(or(q, not(or(p, q)))), q))), true), true, theorem(or(not(or(q, p)), or(not(or(q, not(or(p, q)))), q))), true), true, theorem(or(not(p), or(not(or(q, not(or(p, q)))), q))), true), true, theorem(or(not(or(q, not(or(p, q)))), or(not(p), q))), true), true, theorem(or(not(or(not(or(p, q)), q)), or(not(p), q))), true)
1.62/1.80	= { by axiom 1 (rule_1) }
1.62/1.80	  ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(axiom(or(not(or(q, not(or(p, q)))), or(not(or(p, q)), q))), true, theorem(or(not(or(q, not(or(p, q)))), or(not(or(p, q)), q))), true), true, theorem(or(not(or(p, q)), or(not(or(q, not(or(p, q)))), q))), true), true, theorem(or(not(or(q, p)), or(not(or(q, not(or(p, q)))), q))), true), true, theorem(or(not(p), or(not(or(q, not(or(p, q)))), q))), true), true, theorem(or(not(or(q, not(or(p, q)))), or(not(p), q))), true), true, theorem(or(not(or(not(or(p, q)), q)), or(not(p), q))), true)
1.62/1.80	= { by axiom 5 (axiom_1_4) }
1.62/1.80	  ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, theorem(or(not(or(q, not(or(p, q)))), or(not(or(p, q)), q))), true), true, theorem(or(not(or(p, q)), or(not(or(q, not(or(p, q)))), q))), true), true, theorem(or(not(or(q, p)), or(not(or(q, not(or(p, q)))), q))), true), true, theorem(or(not(p), or(not(or(q, not(or(p, q)))), q))), true), true, theorem(or(not(or(q, not(or(p, q)))), or(not(p), q))), true), true, theorem(or(not(or(not(or(p, q)), q)), or(not(p), q))), true)
1.62/1.80	= { by axiom 6 (ifeq_axiom) }
1.62/1.80	  ifeq(ifeq(ifeq(ifeq(ifeq(theorem(or(not(or(q, not(or(p, q)))), or(not(or(p, q)), q))), true, theorem(or(not(or(p, q)), or(not(or(q, not(or(p, q)))), q))), true), true, theorem(or(not(or(q, p)), or(not(or(q, not(or(p, q)))), q))), true), true, theorem(or(not(p), or(not(or(q, not(or(p, q)))), q))), true), true, theorem(or(not(or(q, not(or(p, q)))), or(not(p), q))), true), true, theorem(or(not(or(not(or(p, q)), q)), or(not(p), q))), true)
1.62/1.80	= { by lemma 8 }
1.62/1.80	  ifeq(ifeq(ifeq(ifeq(true, true, theorem(or(not(or(q, p)), or(not(or(q, not(or(p, q)))), q))), true), true, theorem(or(not(p), or(not(or(q, not(or(p, q)))), q))), true), true, theorem(or(not(or(q, not(or(p, q)))), or(not(p), q))), true), true, theorem(or(not(or(not(or(p, q)), q)), or(not(p), q))), true)
1.62/1.80	= { by axiom 6 (ifeq_axiom) }
1.62/1.80	  ifeq(ifeq(ifeq(theorem(or(not(or(q, p)), or(not(or(q, not(or(p, q)))), q))), true, theorem(or(not(p), or(not(or(q, not(or(p, q)))), q))), true), true, theorem(or(not(or(q, not(or(p, q)))), or(not(p), q))), true), true, theorem(or(not(or(not(or(p, q)), q)), or(not(p), q))), true)
1.62/1.80	= { by axiom 6 (ifeq_axiom) }
1.62/1.80	  ifeq(ifeq(ifeq(theorem(or(not(or(q, p)), or(not(or(q, not(or(p, q)))), q))), true, ifeq(true, true, theorem(or(not(p), or(not(or(q, not(or(p, q)))), q))), true), true), true, theorem(or(not(or(q, not(or(p, q)))), or(not(p), q))), true), true, theorem(or(not(or(not(or(p, q)), q)), or(not(p), q))), true)
1.62/1.80	= { by axiom 2 (axiom_1_3) }
1.62/1.80	  ifeq(ifeq(ifeq(theorem(or(not(or(q, p)), or(not(or(q, not(or(p, q)))), q))), true, ifeq(axiom(or(not(p), or(q, p))), true, theorem(or(not(p), or(not(or(q, not(or(p, q)))), q))), true), true), true, theorem(or(not(or(q, not(or(p, q)))), or(not(p), q))), true), true, theorem(or(not(or(not(or(p, q)), q)), or(not(p), q))), true)
1.62/1.80	= { by axiom 4 (rule_3) }
1.62/1.80	  ifeq(ifeq(true, true, theorem(or(not(or(q, not(or(p, q)))), or(not(p), q))), true), true, theorem(or(not(or(not(or(p, q)), q)), or(not(p), q))), true)
1.62/1.80	= { by axiom 6 (ifeq_axiom) }
1.62/1.80	  ifeq(theorem(or(not(or(q, not(or(p, q)))), or(not(p), q))), true, theorem(or(not(or(not(or(p, q)), q)), or(not(p), q))), true)
1.62/1.80	= { by lemma 9 }
1.62/1.80	  true
1.62/1.80	% SZS output end Proof
1.62/1.80	
1.62/1.80	RESULT: Unsatisfiable (the axioms are contradictory).
1.62/1.81	EOF