0.03/0.12	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.03/0.13	% Command    : twee %s --tstp --casc --quiet --explain-encoding --conditional-encoding if --smaller --drop-non-horn
0.13/0.34	% Computer   : n012.cluster.edu
0.13/0.34	% Model      : x86_64 x86_64
0.13/0.34	% CPU        : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
0.13/0.34	% Memory     : 8042.1875MB
0.13/0.34	% OS         : Linux 3.10.0-693.el7.x86_64
0.13/0.34	% CPULimit   : 180
0.13/0.34	% DateTime   : Thu Aug 29 11:30:11 EDT 2019
0.13/0.34	% CPUTime    : 
9.10/9.33	% SZS status Unsatisfiable
9.10/9.33	
9.10/9.34	% SZS output start Proof
9.10/9.34	Take the following subset of the input axioms:
9.34/9.53	  fof(axiom_1_2, axiom, ![A]: axiom(implies(or(A, A), A))=true).
9.34/9.53	  fof(axiom_1_3, axiom, ![A, B]: axiom(implies(A, or(B, A)))=true).
9.34/9.53	  fof(axiom_1_4, axiom, ![A, B]: true=axiom(implies(or(A, B), or(B, A)))).
9.34/9.53	  fof(axiom_1_5, axiom, ![A, B, C]: axiom(implies(or(A, or(B, C)), or(B, or(A, C))))=true).
9.34/9.53	  fof(axiom_1_6, axiom, ![A, B, C]: true=axiom(implies(implies(A, B), implies(or(C, A), or(C, B))))).
9.34/9.53	  fof(ifeq_axiom, axiom, ![A, B, C]: ifeq(A, A, B, C)=B).
9.34/9.53	  fof(implies_definition, axiom, ![Y, X]: implies(X, Y)=or(not(X), Y)).
9.34/9.53	  fof(prove_this, negated_conjecture, theorem(implies(not(or(p, q)), or(not(p), not(q))))!=true).
9.34/9.53	  fof(rule_1, axiom, ![X]: ifeq(axiom(X), true, theorem(X), true)=true).
9.34/9.53	  fof(rule_2, axiom, ![Y, X]: ifeq(theorem(implies(Y, X)), true, ifeq(theorem(Y), true, theorem(X), true), true)=true).
9.34/9.53	
9.34/9.53	Now clausify the problem and encode Horn clauses using encoding 3 of
9.34/9.53	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
9.34/9.53	We repeatedly replace C & s=t => u=v by the two clauses:
9.34/9.53	  fresh(y, y, x1...xn) = u
9.34/9.53	  C => fresh(s, t, x1...xn) = v
9.34/9.53	where fresh is a fresh function symbol and x1..xn are the free
9.34/9.53	variables of u and v.
9.34/9.53	A predicate p(X) is encoded as p(X)=true (this is sound, because the
9.34/9.53	input problem has no model of domain size 1).
9.34/9.53	
9.34/9.53	The encoding turns the above axioms into the following unit equations and goals:
9.34/9.53	
9.34/9.53	Axiom 1 (rule_1): ifeq(axiom(X), true, theorem(X), true) = true.
9.34/9.53	Axiom 2 (axiom_1_6): true = axiom(implies(implies(X, Y), implies(or(Z, X), or(Z, Y)))).
9.34/9.53	Axiom 3 (axiom_1_2): axiom(implies(or(X, X), X)) = true.
9.34/9.53	Axiom 4 (axiom_1_3): axiom(implies(X, or(Y, X))) = true.
9.34/9.53	Axiom 5 (axiom_1_5): axiom(implies(or(X, or(Y, Z)), or(Y, or(X, Z)))) = true.
9.34/9.53	Axiom 6 (implies_definition): implies(X, Y) = or(not(X), Y).
9.34/9.53	Axiom 7 (rule_2): ifeq(theorem(implies(X, Y)), true, ifeq(theorem(X), true, theorem(Y), true), true) = true.
9.34/9.53	Axiom 8 (ifeq_axiom): ifeq(X, X, Y, Z) = Y.
9.34/9.53	Axiom 9 (axiom_1_4): true = axiom(implies(or(X, Y), or(Y, X))).
9.34/9.53	
9.34/9.53	Lemma 10: theorem(implies(implies(X, Y), implies(implies(Z, X), implies(Z, Y)))) = true.
9.34/9.53	Proof:
9.34/9.53	  theorem(implies(implies(X, Y), implies(implies(Z, X), implies(Z, Y))))
9.34/9.53	= { by axiom 8 (ifeq_axiom) }
9.34/9.53	  ifeq(true, true, theorem(implies(implies(X, Y), implies(implies(Z, X), implies(Z, Y)))), true)
9.34/9.53	= { by axiom 2 (axiom_1_6) }
9.34/9.53	  ifeq(axiom(implies(implies(X, Y), implies(or(not(Z), X), or(not(Z), Y)))), true, theorem(implies(implies(X, Y), implies(implies(Z, X), implies(Z, Y)))), true)
9.34/9.53	= { by axiom 6 (implies_definition) }
9.34/9.53	  ifeq(axiom(implies(implies(X, Y), implies(implies(Z, X), or(not(Z), Y)))), true, theorem(implies(implies(X, Y), implies(implies(Z, X), implies(Z, Y)))), true)
9.34/9.53	= { by axiom 6 (implies_definition) }
9.34/9.53	  ifeq(axiom(implies(implies(X, Y), implies(implies(Z, X), implies(Z, Y)))), true, theorem(implies(implies(X, Y), implies(implies(Z, X), implies(Z, Y)))), true)
9.34/9.53	= { by axiom 1 (rule_1) }
9.34/9.53	  true
9.34/9.53	
9.34/9.53	Lemma 11: theorem(implies(or(X, Y), or(Y, X))) = true.
9.34/9.53	Proof:
9.34/9.53	  theorem(implies(or(X, Y), or(Y, X)))
9.34/9.53	= { by axiom 8 (ifeq_axiom) }
9.34/9.53	  ifeq(true, true, theorem(implies(or(X, Y), or(Y, X))), true)
9.34/9.53	= { by axiom 9 (axiom_1_4) }
9.34/9.53	  ifeq(axiom(implies(or(X, Y), or(Y, X))), true, theorem(implies(or(X, Y), or(Y, X))), true)
9.34/9.53	= { by axiom 1 (rule_1) }
9.34/9.53	  true
9.34/9.53	
9.34/9.53	Lemma 12: ifeq(theorem(implies(implies(X, or(Y, X)), Z)), true, theorem(Z), true) = true.
9.34/9.53	Proof:
9.34/9.53	  ifeq(theorem(implies(implies(X, or(Y, X)), Z)), true, theorem(Z), true)
9.34/9.53	= { by axiom 8 (ifeq_axiom) }
9.34/9.53	  ifeq(theorem(implies(implies(X, or(Y, X)), Z)), true, ifeq(true, true, theorem(Z), true), true)
9.34/9.53	= { by axiom 1 (rule_1) }
9.34/9.53	  ifeq(theorem(implies(implies(X, or(Y, X)), Z)), true, ifeq(ifeq(axiom(implies(X, or(Y, X))), true, theorem(implies(X, or(Y, X))), true), true, theorem(Z), true), true)
9.34/9.53	= { by axiom 4 (axiom_1_3) }
9.34/9.53	  ifeq(theorem(implies(implies(X, or(Y, X)), Z)), true, ifeq(ifeq(true, true, theorem(implies(X, or(Y, X))), true), true, theorem(Z), true), true)
9.34/9.53	= { by axiom 8 (ifeq_axiom) }
9.34/9.53	  ifeq(theorem(implies(implies(X, or(Y, X)), Z)), true, ifeq(theorem(implies(X, or(Y, X))), true, theorem(Z), true), true)
9.34/9.53	= { by axiom 7 (rule_2) }
9.34/9.53	  true
9.34/9.53	
9.34/9.53	Lemma 13: theorem(implies(X, or(X, Y))) = true.
9.34/9.53	Proof:
9.34/9.53	  theorem(implies(X, or(X, Y)))
9.34/9.53	= { by axiom 8 (ifeq_axiom) }
9.34/9.53	  ifeq(true, true, theorem(implies(X, or(X, Y))), true)
9.34/9.53	= { by axiom 7 (rule_2) }
9.34/9.53	  ifeq(ifeq(theorem(implies(implies(or(Y, X), or(X, Y)), implies(implies(X, or(Y, X)), implies(X, or(X, Y))))), true, ifeq(theorem(implies(or(Y, X), or(X, Y))), true, theorem(implies(implies(X, or(Y, X)), implies(X, or(X, Y)))), true), true), true, theorem(implies(X, or(X, Y))), true)
9.34/9.53	= { by lemma 10 }
9.34/9.53	  ifeq(ifeq(true, true, ifeq(theorem(implies(or(Y, X), or(X, Y))), true, theorem(implies(implies(X, or(Y, X)), implies(X, or(X, Y)))), true), true), true, theorem(implies(X, or(X, Y))), true)
9.34/9.53	= { by lemma 11 }
9.34/9.54	  ifeq(ifeq(true, true, ifeq(true, true, theorem(implies(implies(X, or(Y, X)), implies(X, or(X, Y)))), true), true), true, theorem(implies(X, or(X, Y))), true)
9.34/9.54	= { by axiom 8 (ifeq_axiom) }
9.34/9.54	  ifeq(ifeq(true, true, theorem(implies(implies(X, or(Y, X)), implies(X, or(X, Y)))), true), true, theorem(implies(X, or(X, Y))), true)
9.34/9.54	= { by axiom 8 (ifeq_axiom) }
9.34/9.54	  ifeq(theorem(implies(implies(X, or(Y, X)), implies(X, or(X, Y)))), true, theorem(implies(X, or(X, Y))), true)
9.34/9.54	= { by lemma 12 }
9.55/9.74	  true
9.55/9.74	
9.55/9.74	Goal 1 (prove_this): theorem(implies(not(or(p, q)), or(not(p), not(q)))) = true.
9.55/9.74	Proof:
9.55/9.74	  theorem(implies(not(or(p, q)), or(not(p), not(q))))
9.55/9.74	= { by axiom 6 (implies_definition) }
9.55/9.74	  theorem(implies(not(or(p, q)), implies(p, not(q))))
9.55/9.74	= { by axiom 8 (ifeq_axiom) }
9.55/9.74	  ifeq(true, true, theorem(implies(not(or(p, q)), implies(p, not(q)))), true)
9.55/9.74	= { by axiom 7 (rule_2) }
9.55/9.74	  ifeq(ifeq(theorem(implies(implies(p, or(p, q)), implies(p, implies(not(or(p, q)), not(q))))), true, ifeq(theorem(implies(p, or(p, q))), true, theorem(implies(p, implies(not(or(p, q)), not(q)))), true), true), true, theorem(implies(not(or(p, q)), implies(p, not(q)))), true)
9.55/9.74	= { by axiom 8 (ifeq_axiom) }
9.55/9.74	  ifeq(ifeq(ifeq(true, true, theorem(implies(implies(p, or(p, q)), implies(p, implies(not(or(p, q)), not(q))))), true), true, ifeq(theorem(implies(p, or(p, q))), true, theorem(implies(p, implies(not(or(p, q)), not(q)))), true), true), true, theorem(implies(not(or(p, q)), implies(p, not(q)))), true)
9.55/9.74	= { by axiom 7 (rule_2) }
9.55/9.74	  ifeq(ifeq(ifeq(ifeq(theorem(implies(or(not(or(p, q)), not(not(or(p, q)))), or(not(or(p, q)), or(not(not(or(p, q))), not(q))))), true, ifeq(theorem(or(not(or(p, q)), not(not(or(p, q))))), true, theorem(or(not(or(p, q)), or(not(not(or(p, q))), not(q)))), true), true), true, theorem(implies(implies(p, or(p, q)), implies(p, implies(not(or(p, q)), not(q))))), true), true, ifeq(theorem(implies(p, or(p, q))), true, theorem(implies(p, implies(not(or(p, q)), not(q)))), true), true), true, theorem(implies(not(or(p, q)), implies(p, not(q)))), true)
9.55/9.74	= { by axiom 8 (ifeq_axiom) }
9.55/9.74	  ifeq(ifeq(ifeq(ifeq(ifeq(true, true, theorem(implies(or(not(or(p, q)), not(not(or(p, q)))), or(not(or(p, q)), or(not(not(or(p, q))), not(q))))), true), true, ifeq(theorem(or(not(or(p, q)), not(not(or(p, q))))), true, theorem(or(not(or(p, q)), or(not(not(or(p, q))), not(q)))), true), true), true, theorem(implies(implies(p, or(p, q)), implies(p, implies(not(or(p, q)), not(q))))), true), true, ifeq(theorem(implies(p, or(p, q))), true, theorem(implies(p, implies(not(or(p, q)), not(q)))), true), true), true, theorem(implies(not(or(p, q)), implies(p, not(q)))), true)
9.55/9.74	= { by lemma 13 }
9.55/9.74	  ifeq(ifeq(ifeq(ifeq(ifeq(theorem(implies(not(not(or(p, q))), or(not(not(or(p, q))), not(q)))), true, theorem(implies(or(not(or(p, q)), not(not(or(p, q)))), or(not(or(p, q)), or(not(not(or(p, q))), not(q))))), true), true, ifeq(theorem(or(not(or(p, q)), not(not(or(p, q))))), true, theorem(or(not(or(p, q)), or(not(not(or(p, q))), not(q)))), true), true), true, theorem(implies(implies(p, or(p, q)), implies(p, implies(not(or(p, q)), not(q))))), true), true, ifeq(theorem(implies(p, or(p, q))), true, theorem(implies(p, implies(not(or(p, q)), not(q)))), true), true), true, theorem(implies(not(or(p, q)), implies(p, not(q)))), true)
9.55/9.74	= { by axiom 8 (ifeq_axiom) }
9.55/9.74	  ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(theorem(implies(not(not(or(p, q))), or(not(not(or(p, q))), not(q)))), true, theorem(implies(or(not(or(p, q)), not(not(or(p, q)))), or(not(or(p, q)), or(not(not(or(p, q))), not(q))))), true), true), true, ifeq(theorem(or(not(or(p, q)), not(not(or(p, q))))), true, theorem(or(not(or(p, q)), or(not(not(or(p, q))), not(q)))), true), true), true, theorem(implies(implies(p, or(p, q)), implies(p, implies(not(or(p, q)), not(q))))), true), true, ifeq(theorem(implies(p, or(p, q))), true, theorem(implies(p, implies(not(or(p, q)), not(q)))), true), true), true, theorem(implies(not(or(p, q)), implies(p, not(q)))), true)
9.55/9.74	= { by axiom 1 (rule_1) }
9.55/9.74	  ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(axiom(implies(implies(not(not(or(p, q))), or(not(not(or(p, q))), not(q))), implies(or(not(or(p, q)), not(not(or(p, q)))), or(not(or(p, q)), or(not(not(or(p, q))), not(q)))))), true, theorem(implies(implies(not(not(or(p, q))), or(not(not(or(p, q))), not(q))), implies(or(not(or(p, q)), not(not(or(p, q)))), or(not(or(p, q)), or(not(not(or(p, q))), not(q)))))), true), true, ifeq(theorem(implies(not(not(or(p, q))), or(not(not(or(p, q))), not(q)))), true, theorem(implies(or(not(or(p, q)), not(not(or(p, q)))), or(not(or(p, q)), or(not(not(or(p, q))), not(q))))), true), true), true, ifeq(theorem(or(not(or(p, q)), not(not(or(p, q))))), true, theorem(or(not(or(p, q)), or(not(not(or(p, q))), not(q)))), true), true), true, theorem(implies(implies(p, or(p, q)), implies(p, implies(not(or(p, q)), not(q))))), true), true, ifeq(theorem(implies(p, or(p, q))), true, theorem(implies(p, implies(not(or(p, q)), not(q)))), true), true), true, theorem(implies(not(or(p, q)), implies(p, not(q)))), true)
9.55/9.74	= { by axiom 2 (axiom_1_6) }
9.55/9.74	  ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, theorem(implies(implies(not(not(or(p, q))), or(not(not(or(p, q))), not(q))), implies(or(not(or(p, q)), not(not(or(p, q)))), or(not(or(p, q)), or(not(not(or(p, q))), not(q)))))), true), true, ifeq(theorem(implies(not(not(or(p, q))), or(not(not(or(p, q))), not(q)))), true, theorem(implies(or(not(or(p, q)), not(not(or(p, q)))), or(not(or(p, q)), or(not(not(or(p, q))), not(q))))), true), true), true, ifeq(theorem(or(not(or(p, q)), not(not(or(p, q))))), true, theorem(or(not(or(p, q)), or(not(not(or(p, q))), not(q)))), true), true), true, theorem(implies(implies(p, or(p, q)), implies(p, implies(not(or(p, q)), not(q))))), true), true, ifeq(theorem(implies(p, or(p, q))), true, theorem(implies(p, implies(not(or(p, q)), not(q)))), true), true), true, theorem(implies(not(or(p, q)), implies(p, not(q)))), true)
9.55/9.74	= { by axiom 8 (ifeq_axiom) }
9.55/9.74	  ifeq(ifeq(ifeq(ifeq(ifeq(theorem(implies(implies(not(not(or(p, q))), or(not(not(or(p, q))), not(q))), implies(or(not(or(p, q)), not(not(or(p, q)))), or(not(or(p, q)), or(not(not(or(p, q))), not(q)))))), true, ifeq(theorem(implies(not(not(or(p, q))), or(not(not(or(p, q))), not(q)))), true, theorem(implies(or(not(or(p, q)), not(not(or(p, q)))), or(not(or(p, q)), or(not(not(or(p, q))), not(q))))), true), true), true, ifeq(theorem(or(not(or(p, q)), not(not(or(p, q))))), true, theorem(or(not(or(p, q)), or(not(not(or(p, q))), not(q)))), true), true), true, theorem(implies(implies(p, or(p, q)), implies(p, implies(not(or(p, q)), not(q))))), true), true, ifeq(theorem(implies(p, or(p, q))), true, theorem(implies(p, implies(not(or(p, q)), not(q)))), true), true), true, theorem(implies(not(or(p, q)), implies(p, not(q)))), true)
9.55/9.74	= { by axiom 7 (rule_2) }
9.55/9.74	  ifeq(ifeq(ifeq(ifeq(true, true, ifeq(theorem(or(not(or(p, q)), not(not(or(p, q))))), true, theorem(or(not(or(p, q)), or(not(not(or(p, q))), not(q)))), true), true), true, theorem(implies(implies(p, or(p, q)), implies(p, implies(not(or(p, q)), not(q))))), true), true, ifeq(theorem(implies(p, or(p, q))), true, theorem(implies(p, implies(not(or(p, q)), not(q)))), true), true), true, theorem(implies(not(or(p, q)), implies(p, not(q)))), true)
9.55/9.74	= { by axiom 8 (ifeq_axiom) }
9.55/9.74	  ifeq(ifeq(ifeq(ifeq(true, true, ifeq(ifeq(true, true, theorem(or(not(or(p, q)), not(not(or(p, q))))), true), true, theorem(or(not(or(p, q)), or(not(not(or(p, q))), not(q)))), true), true), true, theorem(implies(implies(p, or(p, q)), implies(p, implies(not(or(p, q)), not(q))))), true), true, ifeq(theorem(implies(p, or(p, q))), true, theorem(implies(p, implies(not(or(p, q)), not(q)))), true), true), true, theorem(implies(not(or(p, q)), implies(p, not(q)))), true)
9.55/9.74	= { by lemma 12 }
9.58/9.74	  ifeq(ifeq(ifeq(ifeq(true, true, ifeq(ifeq(ifeq(theorem(implies(implies(not(or(p, q)), or(not(or(p, q)), not(or(p, q)))), implies(not(or(p, q)), not(or(p, q))))), true, theorem(implies(not(or(p, q)), not(or(p, q)))), true), true, theorem(or(not(or(p, q)), not(not(or(p, q))))), true), true, theorem(or(not(or(p, q)), or(not(not(or(p, q))), not(q)))), true), true), true, theorem(implies(implies(p, or(p, q)), implies(p, implies(not(or(p, q)), not(q))))), true), true, ifeq(theorem(implies(p, or(p, q))), true, theorem(implies(p, implies(not(or(p, q)), not(q)))), true), true), true, theorem(implies(not(or(p, q)), implies(p, not(q)))), true)
9.58/9.74	= { by axiom 8 (ifeq_axiom) }
9.58/9.74	  ifeq(ifeq(ifeq(ifeq(true, true, ifeq(ifeq(ifeq(ifeq(true, true, theorem(implies(implies(not(or(p, q)), or(not(or(p, q)), not(or(p, q)))), implies(not(or(p, q)), not(or(p, q))))), true), true, theorem(implies(not(or(p, q)), not(or(p, q)))), true), true, theorem(or(not(or(p, q)), not(not(or(p, q))))), true), true, theorem(or(not(or(p, q)), or(not(not(or(p, q))), not(q)))), true), true), true, theorem(implies(implies(p, or(p, q)), implies(p, implies(not(or(p, q)), not(q))))), true), true, ifeq(theorem(implies(p, or(p, q))), true, theorem(implies(p, implies(not(or(p, q)), not(q)))), true), true), true, theorem(implies(not(or(p, q)), implies(p, not(q)))), true)
9.58/9.74	= { by lemma 10 }
9.58/9.74	  ifeq(ifeq(ifeq(ifeq(true, true, ifeq(ifeq(ifeq(ifeq(theorem(implies(implies(or(not(or(p, q)), not(or(p, q))), not(or(p, q))), implies(implies(not(or(p, q)), or(not(or(p, q)), not(or(p, q)))), implies(not(or(p, q)), not(or(p, q)))))), true, theorem(implies(implies(not(or(p, q)), or(not(or(p, q)), not(or(p, q)))), implies(not(or(p, q)), not(or(p, q))))), true), true, theorem(implies(not(or(p, q)), not(or(p, q)))), true), true, theorem(or(not(or(p, q)), not(not(or(p, q))))), true), true, theorem(or(not(or(p, q)), or(not(not(or(p, q))), not(q)))), true), true), true, theorem(implies(implies(p, or(p, q)), implies(p, implies(not(or(p, q)), not(q))))), true), true, ifeq(theorem(implies(p, or(p, q))), true, theorem(implies(p, implies(not(or(p, q)), not(q)))), true), true), true, theorem(implies(not(or(p, q)), implies(p, not(q)))), true)
9.58/9.74	= { by axiom 8 (ifeq_axiom) }
9.58/9.74	  ifeq(ifeq(ifeq(ifeq(true, true, ifeq(ifeq(ifeq(ifeq(theorem(implies(implies(or(not(or(p, q)), not(or(p, q))), not(or(p, q))), implies(implies(not(or(p, q)), or(not(or(p, q)), not(or(p, q)))), implies(not(or(p, q)), not(or(p, q)))))), true, ifeq(true, true, theorem(implies(implies(not(or(p, q)), or(not(or(p, q)), not(or(p, q)))), implies(not(or(p, q)), not(or(p, q))))), true), true), true, theorem(implies(not(or(p, q)), not(or(p, q)))), true), true, theorem(or(not(or(p, q)), not(not(or(p, q))))), true), true, theorem(or(not(or(p, q)), or(not(not(or(p, q))), not(q)))), true), true), true, theorem(implies(implies(p, or(p, q)), implies(p, implies(not(or(p, q)), not(q))))), true), true, ifeq(theorem(implies(p, or(p, q))), true, theorem(implies(p, implies(not(or(p, q)), not(q)))), true), true), true, theorem(implies(not(or(p, q)), implies(p, not(q)))), true)
9.58/9.74	= { by axiom 1 (rule_1) }
9.58/9.74	  ifeq(ifeq(ifeq(ifeq(true, true, ifeq(ifeq(ifeq(ifeq(theorem(implies(implies(or(not(or(p, q)), not(or(p, q))), not(or(p, q))), implies(implies(not(or(p, q)), or(not(or(p, q)), not(or(p, q)))), implies(not(or(p, q)), not(or(p, q)))))), true, ifeq(ifeq(axiom(implies(or(not(or(p, q)), not(or(p, q))), not(or(p, q)))), true, theorem(implies(or(not(or(p, q)), not(or(p, q))), not(or(p, q)))), true), true, theorem(implies(implies(not(or(p, q)), or(not(or(p, q)), not(or(p, q)))), implies(not(or(p, q)), not(or(p, q))))), true), true), true, theorem(implies(not(or(p, q)), not(or(p, q)))), true), true, theorem(or(not(or(p, q)), not(not(or(p, q))))), true), true, theorem(or(not(or(p, q)), or(not(not(or(p, q))), not(q)))), true), true), true, theorem(implies(implies(p, or(p, q)), implies(p, implies(not(or(p, q)), not(q))))), true), true, ifeq(theorem(implies(p, or(p, q))), true, theorem(implies(p, implies(not(or(p, q)), not(q)))), true), true), true, theorem(implies(not(or(p, q)), implies(p, not(q)))), true)
9.58/9.74	= { by axiom 3 (axiom_1_2) }
9.58/9.74	  ifeq(ifeq(ifeq(ifeq(true, true, ifeq(ifeq(ifeq(ifeq(theorem(implies(implies(or(not(or(p, q)), not(or(p, q))), not(or(p, q))), implies(implies(not(or(p, q)), or(not(or(p, q)), not(or(p, q)))), implies(not(or(p, q)), not(or(p, q)))))), true, ifeq(ifeq(true, true, theorem(implies(or(not(or(p, q)), not(or(p, q))), not(or(p, q)))), true), true, theorem(implies(implies(not(or(p, q)), or(not(or(p, q)), not(or(p, q)))), implies(not(or(p, q)), not(or(p, q))))), true), true), true, theorem(implies(not(or(p, q)), not(or(p, q)))), true), true, theorem(or(not(or(p, q)), not(not(or(p, q))))), true), true, theorem(or(not(or(p, q)), or(not(not(or(p, q))), not(q)))), true), true), true, theorem(implies(implies(p, or(p, q)), implies(p, implies(not(or(p, q)), not(q))))), true), true, ifeq(theorem(implies(p, or(p, q))), true, theorem(implies(p, implies(not(or(p, q)), not(q)))), true), true), true, theorem(implies(not(or(p, q)), implies(p, not(q)))), true)
9.58/9.74	= { by axiom 8 (ifeq_axiom) }
9.58/9.74	  ifeq(ifeq(ifeq(ifeq(true, true, ifeq(ifeq(ifeq(ifeq(theorem(implies(implies(or(not(or(p, q)), not(or(p, q))), not(or(p, q))), implies(implies(not(or(p, q)), or(not(or(p, q)), not(or(p, q)))), implies(not(or(p, q)), not(or(p, q)))))), true, ifeq(theorem(implies(or(not(or(p, q)), not(or(p, q))), not(or(p, q)))), true, theorem(implies(implies(not(or(p, q)), or(not(or(p, q)), not(or(p, q)))), implies(not(or(p, q)), not(or(p, q))))), true), true), true, theorem(implies(not(or(p, q)), not(or(p, q)))), true), true, theorem(or(not(or(p, q)), not(not(or(p, q))))), true), true, theorem(or(not(or(p, q)), or(not(not(or(p, q))), not(q)))), true), true), true, theorem(implies(implies(p, or(p, q)), implies(p, implies(not(or(p, q)), not(q))))), true), true, ifeq(theorem(implies(p, or(p, q))), true, theorem(implies(p, implies(not(or(p, q)), not(q)))), true), true), true, theorem(implies(not(or(p, q)), implies(p, not(q)))), true)
9.58/9.74	= { by axiom 7 (rule_2) }
9.58/9.75	  ifeq(ifeq(ifeq(ifeq(true, true, ifeq(ifeq(ifeq(true, true, theorem(implies(not(or(p, q)), not(or(p, q)))), true), true, theorem(or(not(or(p, q)), not(not(or(p, q))))), true), true, theorem(or(not(or(p, q)), or(not(not(or(p, q))), not(q)))), true), true), true, theorem(implies(implies(p, or(p, q)), implies(p, implies(not(or(p, q)), not(q))))), true), true, ifeq(theorem(implies(p, or(p, q))), true, theorem(implies(p, implies(not(or(p, q)), not(q)))), true), true), true, theorem(implies(not(or(p, q)), implies(p, not(q)))), true)
9.58/9.75	= { by axiom 8 (ifeq_axiom) }
9.58/9.75	  ifeq(ifeq(ifeq(ifeq(true, true, ifeq(ifeq(theorem(implies(not(or(p, q)), not(or(p, q)))), true, theorem(or(not(or(p, q)), not(not(or(p, q))))), true), true, theorem(or(not(or(p, q)), or(not(not(or(p, q))), not(q)))), true), true), true, theorem(implies(implies(p, or(p, q)), implies(p, implies(not(or(p, q)), not(q))))), true), true, ifeq(theorem(implies(p, or(p, q))), true, theorem(implies(p, implies(not(or(p, q)), not(q)))), true), true), true, theorem(implies(not(or(p, q)), implies(p, not(q)))), true)
9.58/9.75	= { by axiom 8 (ifeq_axiom) }
9.58/9.75	  ifeq(ifeq(ifeq(ifeq(true, true, ifeq(ifeq(true, true, ifeq(theorem(implies(not(or(p, q)), not(or(p, q)))), true, theorem(or(not(or(p, q)), not(not(or(p, q))))), true), true), true, theorem(or(not(or(p, q)), or(not(not(or(p, q))), not(q)))), true), true), true, theorem(implies(implies(p, or(p, q)), implies(p, implies(not(or(p, q)), not(q))))), true), true, ifeq(theorem(implies(p, or(p, q))), true, theorem(implies(p, implies(not(or(p, q)), not(q)))), true), true), true, theorem(implies(not(or(p, q)), implies(p, not(q)))), true)
9.58/9.75	= { by lemma 11 }
9.58/9.75	  ifeq(ifeq(ifeq(ifeq(true, true, ifeq(ifeq(theorem(implies(or(not(not(or(p, q))), not(or(p, q))), or(not(or(p, q)), not(not(or(p, q)))))), true, ifeq(theorem(implies(not(or(p, q)), not(or(p, q)))), true, theorem(or(not(or(p, q)), not(not(or(p, q))))), true), true), true, theorem(or(not(or(p, q)), or(not(not(or(p, q))), not(q)))), true), true), true, theorem(implies(implies(p, or(p, q)), implies(p, implies(not(or(p, q)), not(q))))), true), true, ifeq(theorem(implies(p, or(p, q))), true, theorem(implies(p, implies(not(or(p, q)), not(q)))), true), true), true, theorem(implies(not(or(p, q)), implies(p, not(q)))), true)
9.58/9.75	= { by axiom 6 (implies_definition) }
9.58/9.75	  ifeq(ifeq(ifeq(ifeq(true, true, ifeq(ifeq(theorem(implies(implies(not(or(p, q)), not(or(p, q))), or(not(or(p, q)), not(not(or(p, q)))))), true, ifeq(theorem(implies(not(or(p, q)), not(or(p, q)))), true, theorem(or(not(or(p, q)), not(not(or(p, q))))), true), true), true, theorem(or(not(or(p, q)), or(not(not(or(p, q))), not(q)))), true), true), true, theorem(implies(implies(p, or(p, q)), implies(p, implies(not(or(p, q)), not(q))))), true), true, ifeq(theorem(implies(p, or(p, q))), true, theorem(implies(p, implies(not(or(p, q)), not(q)))), true), true), true, theorem(implies(not(or(p, q)), implies(p, not(q)))), true)
9.58/9.75	= { by axiom 7 (rule_2) }
9.58/9.75	  ifeq(ifeq(ifeq(ifeq(true, true, ifeq(true, true, theorem(or(not(or(p, q)), or(not(not(or(p, q))), not(q)))), true), true), true, theorem(implies(implies(p, or(p, q)), implies(p, implies(not(or(p, q)), not(q))))), true), true, ifeq(theorem(implies(p, or(p, q))), true, theorem(implies(p, implies(not(or(p, q)), not(q)))), true), true), true, theorem(implies(not(or(p, q)), implies(p, not(q)))), true)
9.58/9.75	= { by axiom 8 (ifeq_axiom) }
9.58/9.75	  ifeq(ifeq(ifeq(ifeq(true, true, theorem(or(not(or(p, q)), or(not(not(or(p, q))), not(q)))), true), true, theorem(implies(implies(p, or(p, q)), implies(p, implies(not(or(p, q)), not(q))))), true), true, ifeq(theorem(implies(p, or(p, q))), true, theorem(implies(p, implies(not(or(p, q)), not(q)))), true), true), true, theorem(implies(not(or(p, q)), implies(p, not(q)))), true)
9.58/9.75	= { by axiom 8 (ifeq_axiom) }
9.58/9.75	  ifeq(ifeq(ifeq(theorem(or(not(or(p, q)), or(not(not(or(p, q))), not(q)))), true, theorem(implies(implies(p, or(p, q)), implies(p, implies(not(or(p, q)), not(q))))), true), true, ifeq(theorem(implies(p, or(p, q))), true, theorem(implies(p, implies(not(or(p, q)), not(q)))), true), true), true, theorem(implies(not(or(p, q)), implies(p, not(q)))), true)
9.58/9.75	= { by axiom 6 (implies_definition) }
9.58/9.75	  ifeq(ifeq(ifeq(theorem(or(not(or(p, q)), implies(not(or(p, q)), not(q)))), true, theorem(implies(implies(p, or(p, q)), implies(p, implies(not(or(p, q)), not(q))))), true), true, ifeq(theorem(implies(p, or(p, q))), true, theorem(implies(p, implies(not(or(p, q)), not(q)))), true), true), true, theorem(implies(not(or(p, q)), implies(p, not(q)))), true)
9.58/9.75	= { by axiom 6 (implies_definition) }
9.58/9.75	  ifeq(ifeq(ifeq(theorem(implies(or(p, q), implies(not(or(p, q)), not(q)))), true, theorem(implies(implies(p, or(p, q)), implies(p, implies(not(or(p, q)), not(q))))), true), true, ifeq(theorem(implies(p, or(p, q))), true, theorem(implies(p, implies(not(or(p, q)), not(q)))), true), true), true, theorem(implies(not(or(p, q)), implies(p, not(q)))), true)
9.58/9.75	= { by axiom 8 (ifeq_axiom) }
9.58/9.75	  ifeq(ifeq(ifeq(true, true, ifeq(theorem(implies(or(p, q), implies(not(or(p, q)), not(q)))), true, theorem(implies(implies(p, or(p, q)), implies(p, implies(not(or(p, q)), not(q))))), true), true), true, ifeq(theorem(implies(p, or(p, q))), true, theorem(implies(p, implies(not(or(p, q)), not(q)))), true), true), true, theorem(implies(not(or(p, q)), implies(p, not(q)))), true)
9.58/9.75	= { by lemma 10 }
9.58/9.75	  ifeq(ifeq(ifeq(theorem(implies(implies(or(p, q), implies(not(or(p, q)), not(q))), implies(implies(p, or(p, q)), implies(p, implies(not(or(p, q)), not(q)))))), true, ifeq(theorem(implies(or(p, q), implies(not(or(p, q)), not(q)))), true, theorem(implies(implies(p, or(p, q)), implies(p, implies(not(or(p, q)), not(q))))), true), true), true, ifeq(theorem(implies(p, or(p, q))), true, theorem(implies(p, implies(not(or(p, q)), not(q)))), true), true), true, theorem(implies(not(or(p, q)), implies(p, not(q)))), true)
9.58/9.75	= { by axiom 7 (rule_2) }
9.58/9.75	  ifeq(ifeq(true, true, ifeq(theorem(implies(p, or(p, q))), true, theorem(implies(p, implies(not(or(p, q)), not(q)))), true), true), true, theorem(implies(not(or(p, q)), implies(p, not(q)))), true)
9.58/9.75	= { by lemma 13 }
9.58/9.75	  ifeq(ifeq(true, true, ifeq(true, true, theorem(implies(p, implies(not(or(p, q)), not(q)))), true), true), true, theorem(implies(not(or(p, q)), implies(p, not(q)))), true)
9.58/9.75	= { by axiom 8 (ifeq_axiom) }
9.58/9.75	  ifeq(ifeq(true, true, theorem(implies(p, implies(not(or(p, q)), not(q)))), true), true, theorem(implies(not(or(p, q)), implies(p, not(q)))), true)
9.58/9.75	= { by axiom 8 (ifeq_axiom) }
9.58/9.75	  ifeq(theorem(implies(p, implies(not(or(p, q)), not(q)))), true, theorem(implies(not(or(p, q)), implies(p, not(q)))), true)
9.58/9.75	= { by axiom 8 (ifeq_axiom) }
9.58/9.75	  ifeq(true, true, ifeq(theorem(implies(p, implies(not(or(p, q)), not(q)))), true, theorem(implies(not(or(p, q)), implies(p, not(q)))), true), true)
9.58/9.75	= { by axiom 1 (rule_1) }
9.58/9.75	  ifeq(ifeq(axiom(implies(implies(p, or(not(not(or(p, q))), not(q))), or(not(not(or(p, q))), implies(p, not(q))))), true, theorem(implies(implies(p, or(not(not(or(p, q))), not(q))), or(not(not(or(p, q))), implies(p, not(q))))), true), true, ifeq(theorem(implies(p, implies(not(or(p, q)), not(q)))), true, theorem(implies(not(or(p, q)), implies(p, not(q)))), true), true)
9.58/9.75	= { by axiom 6 (implies_definition) }
9.58/9.75	  ifeq(ifeq(axiom(implies(or(not(p), or(not(not(or(p, q))), not(q))), or(not(not(or(p, q))), implies(p, not(q))))), true, theorem(implies(implies(p, or(not(not(or(p, q))), not(q))), or(not(not(or(p, q))), implies(p, not(q))))), true), true, ifeq(theorem(implies(p, implies(not(or(p, q)), not(q)))), true, theorem(implies(not(or(p, q)), implies(p, not(q)))), true), true)
9.58/9.75	= { by axiom 6 (implies_definition) }
9.58/9.75	  ifeq(ifeq(axiom(implies(or(not(p), or(not(not(or(p, q))), not(q))), or(not(not(or(p, q))), or(not(p), not(q))))), true, theorem(implies(implies(p, or(not(not(or(p, q))), not(q))), or(not(not(or(p, q))), implies(p, not(q))))), true), true, ifeq(theorem(implies(p, implies(not(or(p, q)), not(q)))), true, theorem(implies(not(or(p, q)), implies(p, not(q)))), true), true)
9.58/9.75	= { by axiom 5 (axiom_1_5) }
9.58/9.75	  ifeq(ifeq(true, true, theorem(implies(implies(p, or(not(not(or(p, q))), not(q))), or(not(not(or(p, q))), implies(p, not(q))))), true), true, ifeq(theorem(implies(p, implies(not(or(p, q)), not(q)))), true, theorem(implies(not(or(p, q)), implies(p, not(q)))), true), true)
9.58/9.75	= { by axiom 8 (ifeq_axiom) }
9.58/9.75	  ifeq(theorem(implies(implies(p, or(not(not(or(p, q))), not(q))), or(not(not(or(p, q))), implies(p, not(q))))), true, ifeq(theorem(implies(p, implies(not(or(p, q)), not(q)))), true, theorem(implies(not(or(p, q)), implies(p, not(q)))), true), true)
9.58/9.75	= { by axiom 6 (implies_definition) }
9.58/9.75	  ifeq(theorem(implies(implies(p, implies(not(or(p, q)), not(q))), or(not(not(or(p, q))), implies(p, not(q))))), true, ifeq(theorem(implies(p, implies(not(or(p, q)), not(q)))), true, theorem(implies(not(or(p, q)), implies(p, not(q)))), true), true)
9.58/9.75	= { by axiom 6 (implies_definition) }
9.58/9.75	  ifeq(theorem(implies(implies(p, implies(not(or(p, q)), not(q))), implies(not(or(p, q)), implies(p, not(q))))), true, ifeq(theorem(implies(p, implies(not(or(p, q)), not(q)))), true, theorem(implies(not(or(p, q)), implies(p, not(q)))), true), true)
9.58/9.75	= { by axiom 7 (rule_2) }
9.58/9.75	  true
9.58/9.75	% SZS output end Proof
9.58/9.75	
9.58/9.75	RESULT: Unsatisfiable (the axioms are contradictory).
9.58/9.76	EOF