0.00/0.12	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.00/0.12	% Command    : twee %s --tstp --casc --quiet --explain-encoding --conditional-encoding if --smaller --drop-non-horn
0.12/0.33	% Computer   : n007.cluster.edu
0.12/0.33	% Model      : x86_64 x86_64
0.12/0.33	% CPU        : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
0.12/0.33	% Memory     : 8042.1875MB
0.12/0.33	% OS         : Linux 3.10.0-693.el7.x86_64
0.12/0.33	% CPULimit   : 180
0.12/0.33	% DateTime   : Thu Aug 29 13:33:06 EDT 2019
0.12/0.33	% CPUTime    : 
25.18/25.40	% SZS status Unsatisfiable
25.18/25.40	
25.18/25.40	% SZS output start Proof
25.18/25.40	Take the following subset of the input axioms:
25.28/25.45	  fof(and_defn, axiom, ![P, Q]: and(P, Q)=not(or(not(P), not(Q)))).
25.28/25.45	  fof(axiom_1_2, axiom, ![A]: true=axiom(implies(or(A, A), A))).
25.28/25.45	  fof(axiom_1_3, axiom, ![A, B]: true=axiom(implies(A, or(B, A)))).
25.28/25.45	  fof(axiom_1_4, axiom, ![A, B]: true=axiom(implies(or(A, B), or(B, A)))).
25.28/25.45	  fof(axiom_1_5, axiom, ![A, B, C]: axiom(implies(or(A, or(B, C)), or(B, or(A, C))))=true).
25.28/25.45	  fof(axiom_1_6, axiom, ![A, B, C]: true=axiom(implies(implies(A, B), implies(or(C, A), or(C, B))))).
25.28/25.45	  fof(equivalent_defn, axiom, ![P, Q]: equivalent(P, Q)=and(implies(P, Q), implies(Q, P))).
25.28/25.45	  fof(ifeq_axiom, axiom, ![A, B, C]: B=ifeq(A, A, B, C)).
25.28/25.45	  fof(implies_definition, axiom, ![Y, X]: or(not(X), Y)=implies(X, Y)).
25.28/25.45	  fof(prove_this, negated_conjecture, theorem(equivalent(p, not(not(p))))!=true).
25.28/25.45	  fof(rule_1, axiom, ![X]: ifeq(axiom(X), true, theorem(X), true)=true).
25.28/25.45	  fof(rule_2, axiom, ![Y, X]: ifeq(theorem(implies(Y, X)), true, ifeq(theorem(Y), true, theorem(X), true), true)=true).
25.28/25.45	
25.28/25.45	Now clausify the problem and encode Horn clauses using encoding 3 of
25.28/25.45	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
25.28/25.45	We repeatedly replace C & s=t => u=v by the two clauses:
25.28/25.45	  fresh(y, y, x1...xn) = u
25.28/25.45	  C => fresh(s, t, x1...xn) = v
25.28/25.45	where fresh is a fresh function symbol and x1..xn are the free
25.28/25.45	variables of u and v.
25.28/25.45	A predicate p(X) is encoded as p(X)=true (this is sound, because the
25.28/25.45	input problem has no model of domain size 1).
25.28/25.45	
25.28/25.45	The encoding turns the above axioms into the following unit equations and goals:
25.28/25.45	
25.28/25.45	Axiom 1 (axiom_1_3): true = axiom(implies(X, or(Y, X))).
25.28/25.45	Axiom 2 (axiom_1_5): axiom(implies(or(X, or(Y, Z)), or(Y, or(X, Z)))) = true.
25.28/25.45	Axiom 3 (rule_1): ifeq(axiom(X), true, theorem(X), true) = true.
25.28/25.45	Axiom 4 (axiom_1_4): true = axiom(implies(or(X, Y), or(Y, X))).
25.28/25.45	Axiom 5 (axiom_1_2): true = axiom(implies(or(X, X), X)).
25.28/25.45	Axiom 6 (implies_definition): or(not(X), Y) = implies(X, Y).
25.28/25.45	Axiom 7 (equivalent_defn): equivalent(X, Y) = and(implies(X, Y), implies(Y, X)).
25.28/25.45	Axiom 8 (ifeq_axiom): X = ifeq(Y, Y, X, Z).
25.28/25.45	Axiom 9 (rule_2): ifeq(theorem(implies(X, Y)), true, ifeq(theorem(X), true, theorem(Y), true), true) = true.
25.28/25.45	Axiom 10 (and_defn): and(X, Y) = not(or(not(X), not(Y))).
25.28/25.45	Axiom 11 (axiom_1_6): true = axiom(implies(implies(X, Y), implies(or(Z, X), or(Z, Y)))).
25.28/25.45	
25.28/25.45	Lemma 12: theorem(implies(implies(X, or(Y, Z)), or(Y, implies(X, Z)))) = true.
25.28/25.45	Proof:
25.28/25.45	  theorem(implies(implies(X, or(Y, Z)), or(Y, implies(X, Z))))
25.28/25.45	= { by axiom 8 (ifeq_axiom) }
25.28/25.45	  ifeq(true, true, theorem(implies(implies(X, or(Y, Z)), or(Y, implies(X, Z)))), true)
25.28/25.45	= { by axiom 2 (axiom_1_5) }
25.28/25.45	  ifeq(axiom(implies(or(not(X), or(Y, Z)), or(Y, or(not(X), Z)))), true, theorem(implies(implies(X, or(Y, Z)), or(Y, implies(X, Z)))), true)
25.28/25.45	= { by axiom 6 (implies_definition) }
25.28/25.45	  ifeq(axiom(implies(implies(X, or(Y, Z)), or(Y, or(not(X), Z)))), true, theorem(implies(implies(X, or(Y, Z)), or(Y, implies(X, Z)))), true)
25.28/25.45	= { by axiom 6 (implies_definition) }
25.28/25.45	  ifeq(axiom(implies(implies(X, or(Y, Z)), or(Y, implies(X, Z)))), true, theorem(implies(implies(X, or(Y, Z)), or(Y, implies(X, Z)))), true)
25.28/25.45	= { by axiom 3 (rule_1) }
25.28/25.46	  true
25.28/25.46	
25.28/25.46	Lemma 13: theorem(implies(X, X)) = true.
25.28/25.46	Proof:
25.28/25.46	  theorem(implies(X, X))
25.28/25.46	= { by axiom 8 (ifeq_axiom) }
25.28/25.46	  ifeq(true, true, theorem(implies(X, X)), true)
25.28/25.46	= { by axiom 9 (rule_2) }
25.28/25.46	  ifeq(ifeq(theorem(implies(implies(X, or(implies(X, X), X)), or(implies(X, X), implies(X, X)))), true, ifeq(theorem(implies(X, or(implies(X, X), X))), true, theorem(or(implies(X, X), implies(X, X))), true), true), true, theorem(implies(X, X)), true)
25.28/25.46	= { by lemma 12 }
25.28/25.46	  ifeq(ifeq(true, true, ifeq(theorem(implies(X, or(implies(X, X), X))), true, theorem(or(implies(X, X), implies(X, X))), true), true), true, theorem(implies(X, X)), true)
25.28/25.46	= { by axiom 8 (ifeq_axiom) }
25.28/25.46	  ifeq(ifeq(true, true, ifeq(ifeq(true, true, theorem(implies(X, or(implies(X, X), X))), true), true, theorem(or(implies(X, X), implies(X, X))), true), true), true, theorem(implies(X, X)), true)
25.28/25.46	= { by axiom 1 (axiom_1_3) }
25.28/25.46	  ifeq(ifeq(true, true, ifeq(ifeq(axiom(implies(X, or(implies(X, X), X))), true, theorem(implies(X, or(implies(X, X), X))), true), true, theorem(or(implies(X, X), implies(X, X))), true), true), true, theorem(implies(X, X)), true)
25.28/25.46	= { by axiom 3 (rule_1) }
25.28/25.46	  ifeq(ifeq(true, true, ifeq(true, true, theorem(or(implies(X, X), implies(X, X))), true), true), true, theorem(implies(X, X)), true)
25.28/25.46	= { by axiom 8 (ifeq_axiom) }
25.28/25.46	  ifeq(ifeq(true, true, theorem(or(implies(X, X), implies(X, X))), true), true, theorem(implies(X, X)), true)
25.28/25.46	= { by axiom 8 (ifeq_axiom) }
25.28/25.46	  ifeq(theorem(or(implies(X, X), implies(X, X))), true, theorem(implies(X, X)), true)
25.28/25.46	= { by axiom 8 (ifeq_axiom) }
25.28/25.46	  ifeq(true, true, ifeq(theorem(or(implies(X, X), implies(X, X))), true, theorem(implies(X, X)), true), true)
25.28/25.46	= { by axiom 3 (rule_1) }
25.28/25.46	  ifeq(ifeq(axiom(implies(or(implies(X, X), implies(X, X)), implies(X, X))), true, theorem(implies(or(implies(X, X), implies(X, X)), implies(X, X))), true), true, ifeq(theorem(or(implies(X, X), implies(X, X))), true, theorem(implies(X, X)), true), true)
25.28/25.46	= { by axiom 5 (axiom_1_2) }
25.28/25.46	  ifeq(ifeq(true, true, theorem(implies(or(implies(X, X), implies(X, X)), implies(X, X))), true), true, ifeq(theorem(or(implies(X, X), implies(X, X))), true, theorem(implies(X, X)), true), true)
25.28/25.46	= { by axiom 8 (ifeq_axiom) }
25.28/25.46	  ifeq(theorem(implies(or(implies(X, X), implies(X, X)), implies(X, X))), true, ifeq(theorem(or(implies(X, X), implies(X, X))), true, theorem(implies(X, X)), true), true)
25.28/25.46	= { by axiom 9 (rule_2) }
25.28/25.46	  true
25.28/25.46	
25.28/25.46	Lemma 14: ifeq(theorem(implies(X, Y)), true, theorem(or(Y, not(X))), true) = true.
25.28/25.46	Proof:
25.28/25.46	  ifeq(theorem(implies(X, Y)), true, theorem(or(Y, not(X))), true)
25.28/25.46	= { by axiom 8 (ifeq_axiom) }
25.28/25.46	  ifeq(true, true, ifeq(theorem(implies(X, Y)), true, theorem(or(Y, not(X))), true), true)
25.28/25.46	= { by axiom 3 (rule_1) }
25.28/25.46	  ifeq(ifeq(axiom(implies(implies(X, Y), or(Y, not(X)))), true, theorem(implies(implies(X, Y), or(Y, not(X)))), true), true, ifeq(theorem(implies(X, Y)), true, theorem(or(Y, not(X))), true), true)
25.28/25.46	= { by axiom 6 (implies_definition) }
25.28/25.46	  ifeq(ifeq(axiom(implies(or(not(X), Y), or(Y, not(X)))), true, theorem(implies(implies(X, Y), or(Y, not(X)))), true), true, ifeq(theorem(implies(X, Y)), true, theorem(or(Y, not(X))), true), true)
25.28/25.46	= { by axiom 4 (axiom_1_4) }
25.28/25.46	  ifeq(ifeq(true, true, theorem(implies(implies(X, Y), or(Y, not(X)))), true), true, ifeq(theorem(implies(X, Y)), true, theorem(or(Y, not(X))), true), true)
25.28/25.46	= { by axiom 8 (ifeq_axiom) }
25.28/25.46	  ifeq(theorem(implies(implies(X, Y), or(Y, not(X)))), true, ifeq(theorem(implies(X, Y)), true, theorem(or(Y, not(X))), true), true)
25.28/25.46	= { by axiom 9 (rule_2) }
25.28/25.46	  true
25.28/25.46	
25.28/25.46	Lemma 15: theorem(or(X, not(X))) = true.
25.28/25.46	Proof:
25.28/25.46	  theorem(or(X, not(X)))
25.28/25.46	= { by axiom 8 (ifeq_axiom) }
25.28/25.46	  ifeq(true, true, theorem(or(X, not(X))), true)
25.28/25.46	= { by lemma 13 }
25.28/25.46	  ifeq(theorem(implies(X, X)), true, theorem(or(X, not(X))), true)
25.28/25.46	= { by lemma 14 }
25.28/25.46	  true
25.28/25.46	
25.28/25.46	Lemma 16: theorem(implies(X, not(not(X)))) = true.
25.28/25.46	Proof:
25.28/25.46	  theorem(implies(X, not(not(X))))
25.28/25.46	= { by axiom 6 (implies_definition) }
25.28/25.46	  theorem(or(not(X), not(not(X))))
25.28/25.46	= { by lemma 15 }
25.28/25.46	  true
25.28/25.46	
25.28/25.46	Lemma 17: not(implies(X, not(Y))) = and(X, Y).
25.28/25.46	Proof:
25.28/25.46	  not(implies(X, not(Y)))
25.28/25.46	= { by axiom 6 (implies_definition) }
25.28/25.46	  not(or(not(X), not(Y)))
25.28/25.46	= { by axiom 10 (and_defn) }
25.28/25.46	  and(X, Y)
25.28/25.46	
25.28/25.46	Lemma 18: implies(implies(X, not(Y)), Z) = or(and(X, Y), Z).
25.28/25.46	Proof:
25.28/25.46	  implies(implies(X, not(Y)), Z)
25.28/25.46	= { by axiom 6 (implies_definition) }
25.28/25.46	  or(not(implies(X, not(Y))), Z)
25.28/25.46	= { by lemma 17 }
25.34/25.51	  or(and(X, Y), Z)
25.34/25.51	
25.34/25.51	Goal 1 (prove_this): theorem(equivalent(p, not(not(p)))) = true.
25.34/25.51	Proof:
25.34/25.51	  theorem(equivalent(p, not(not(p))))
25.34/25.51	= { by axiom 8 (ifeq_axiom) }
25.34/25.51	  ifeq(true, true, theorem(equivalent(p, not(not(p)))), true)
25.34/25.51	= { by axiom 9 (rule_2) }
25.34/25.51	  ifeq(ifeq(theorem(implies(or(p, not(not(not(p)))), or(not(not(not(p))), p))), true, ifeq(theorem(or(p, not(not(not(p))))), true, theorem(or(not(not(not(p))), p)), true), true), true, theorem(equivalent(p, not(not(p)))), true)
25.34/25.51	= { by axiom 8 (ifeq_axiom) }
25.34/25.51	  ifeq(ifeq(ifeq(true, true, theorem(implies(or(p, not(not(not(p)))), or(not(not(not(p))), p))), true), true, ifeq(theorem(or(p, not(not(not(p))))), true, theorem(or(not(not(not(p))), p)), true), true), true, theorem(equivalent(p, not(not(p)))), true)
25.34/25.51	= { by axiom 4 (axiom_1_4) }
25.34/25.51	  ifeq(ifeq(ifeq(axiom(implies(or(p, not(not(not(p)))), or(not(not(not(p))), p))), true, theorem(implies(or(p, not(not(not(p)))), or(not(not(not(p))), p))), true), true, ifeq(theorem(or(p, not(not(not(p))))), true, theorem(or(not(not(not(p))), p)), true), true), true, theorem(equivalent(p, not(not(p)))), true)
25.34/25.51	= { by axiom 3 (rule_1) }
25.34/25.51	  ifeq(ifeq(true, true, ifeq(theorem(or(p, not(not(not(p))))), true, theorem(or(not(not(not(p))), p)), true), true), true, theorem(equivalent(p, not(not(p)))), true)
25.34/25.51	= { by axiom 8 (ifeq_axiom) }
25.34/25.51	  ifeq(ifeq(theorem(or(p, not(not(not(p))))), true, theorem(or(not(not(not(p))), p)), true), true, theorem(equivalent(p, not(not(p)))), true)
25.34/25.51	= { by axiom 8 (ifeq_axiom) }
25.34/25.51	  ifeq(ifeq(ifeq(true, true, theorem(or(p, not(not(not(p))))), true), true, theorem(or(not(not(not(p))), p)), true), true, theorem(equivalent(p, not(not(p)))), true)
25.34/25.51	= { by axiom 9 (rule_2) }
25.34/25.51	  ifeq(ifeq(ifeq(ifeq(theorem(implies(implies(not(p), not(not(not(p)))), implies(or(p, not(p)), or(p, not(not(not(p))))))), true, ifeq(theorem(implies(not(p), not(not(not(p))))), true, theorem(implies(or(p, not(p)), or(p, not(not(not(p)))))), true), true), true, theorem(or(p, not(not(not(p))))), true), true, theorem(or(not(not(not(p))), p)), true), true, theorem(equivalent(p, not(not(p)))), true)
25.34/25.51	= { by axiom 8 (ifeq_axiom) }
25.34/25.51	  ifeq(ifeq(ifeq(ifeq(ifeq(true, true, theorem(implies(implies(not(p), not(not(not(p)))), implies(or(p, not(p)), or(p, not(not(not(p))))))), true), true, ifeq(theorem(implies(not(p), not(not(not(p))))), true, theorem(implies(or(p, not(p)), or(p, not(not(not(p)))))), true), true), true, theorem(or(p, not(not(not(p))))), true), true, theorem(or(not(not(not(p))), p)), true), true, theorem(equivalent(p, not(not(p)))), true)
25.34/25.51	= { by axiom 11 (axiom_1_6) }
25.34/25.51	  ifeq(ifeq(ifeq(ifeq(ifeq(axiom(implies(implies(not(p), not(not(not(p)))), implies(or(p, not(p)), or(p, not(not(not(p))))))), true, theorem(implies(implies(not(p), not(not(not(p)))), implies(or(p, not(p)), or(p, not(not(not(p))))))), true), true, ifeq(theorem(implies(not(p), not(not(not(p))))), true, theorem(implies(or(p, not(p)), or(p, not(not(not(p)))))), true), true), true, theorem(or(p, not(not(not(p))))), true), true, theorem(or(not(not(not(p))), p)), true), true, theorem(equivalent(p, not(not(p)))), true)
25.34/25.51	= { by axiom 3 (rule_1) }
25.34/25.51	  ifeq(ifeq(ifeq(ifeq(true, true, ifeq(theorem(implies(not(p), not(not(not(p))))), true, theorem(implies(or(p, not(p)), or(p, not(not(not(p)))))), true), true), true, theorem(or(p, not(not(not(p))))), true), true, theorem(or(not(not(not(p))), p)), true), true, theorem(equivalent(p, not(not(p)))), true)
25.34/25.51	= { by axiom 8 (ifeq_axiom) }
25.34/25.51	  ifeq(ifeq(ifeq(ifeq(theorem(implies(not(p), not(not(not(p))))), true, theorem(implies(or(p, not(p)), or(p, not(not(not(p)))))), true), true, theorem(or(p, not(not(not(p))))), true), true, theorem(or(not(not(not(p))), p)), true), true, theorem(equivalent(p, not(not(p)))), true)
25.34/25.51	= { by lemma 16 }
25.34/25.51	  ifeq(ifeq(ifeq(ifeq(true, true, theorem(implies(or(p, not(p)), or(p, not(not(not(p)))))), true), true, theorem(or(p, not(not(not(p))))), true), true, theorem(or(not(not(not(p))), p)), true), true, theorem(equivalent(p, not(not(p)))), true)
25.34/25.51	= { by axiom 8 (ifeq_axiom) }
25.34/25.51	  ifeq(ifeq(ifeq(theorem(implies(or(p, not(p)), or(p, not(not(not(p)))))), true, theorem(or(p, not(not(not(p))))), true), true, theorem(or(not(not(not(p))), p)), true), true, theorem(equivalent(p, not(not(p)))), true)
25.34/25.51	= { by axiom 8 (ifeq_axiom) }
25.34/25.51	  ifeq(ifeq(ifeq(theorem(implies(or(p, not(p)), or(p, not(not(not(p)))))), true, ifeq(true, true, theorem(or(p, not(not(not(p))))), true), true), true, theorem(or(not(not(not(p))), p)), true), true, theorem(equivalent(p, not(not(p)))), true)
25.34/25.51	= { by lemma 15 }
25.34/25.51	  ifeq(ifeq(ifeq(theorem(implies(or(p, not(p)), or(p, not(not(not(p)))))), true, ifeq(theorem(or(p, not(p))), true, theorem(or(p, not(not(not(p))))), true), true), true, theorem(or(not(not(not(p))), p)), true), true, theorem(equivalent(p, not(not(p)))), true)
25.34/25.51	= { by axiom 9 (rule_2) }
25.34/25.51	  ifeq(ifeq(true, true, theorem(or(not(not(not(p))), p)), true), true, theorem(equivalent(p, not(not(p)))), true)
25.34/25.51	= { by axiom 8 (ifeq_axiom) }
25.34/25.51	  ifeq(theorem(or(not(not(not(p))), p)), true, theorem(equivalent(p, not(not(p)))), true)
25.34/25.51	= { by axiom 6 (implies_definition) }
25.34/25.51	  ifeq(theorem(implies(not(not(p)), p)), true, theorem(equivalent(p, not(not(p)))), true)
25.34/25.51	= { by axiom 8 (ifeq_axiom) }
25.34/25.51	  ifeq(true, true, ifeq(theorem(implies(not(not(p)), p)), true, theorem(equivalent(p, not(not(p)))), true), true)
25.34/25.51	= { by lemma 14 }
25.34/25.51	  ifeq(ifeq(theorem(implies(or(and(p, not(p)), not(implies(not(not(p)), p))), not(implies(not(not(p)), p)))), true, theorem(or(not(implies(not(not(p)), p)), not(or(and(p, not(p)), not(implies(not(not(p)), p)))))), true), true, ifeq(theorem(implies(not(not(p)), p)), true, theorem(equivalent(p, not(not(p)))), true), true)
25.34/25.51	= { by lemma 18 }
25.34/25.51	  ifeq(ifeq(theorem(implies(implies(implies(p, not(not(p))), not(implies(not(not(p)), p))), not(implies(not(not(p)), p)))), true, theorem(or(not(implies(not(not(p)), p)), not(or(and(p, not(p)), not(implies(not(not(p)), p)))))), true), true, ifeq(theorem(implies(not(not(p)), p)), true, theorem(equivalent(p, not(not(p)))), true), true)
25.34/25.51	= { by axiom 8 (ifeq_axiom) }
25.34/25.51	  ifeq(ifeq(ifeq(true, true, theorem(implies(implies(implies(p, not(not(p))), not(implies(not(not(p)), p))), not(implies(not(not(p)), p)))), true), true, theorem(or(not(implies(not(not(p)), p)), not(or(and(p, not(p)), not(implies(not(not(p)), p)))))), true), true, ifeq(theorem(implies(not(not(p)), p)), true, theorem(equivalent(p, not(not(p)))), true), true)
25.34/25.51	= { by lemma 16 }
25.34/25.51	  ifeq(ifeq(ifeq(theorem(implies(p, not(not(p)))), true, theorem(implies(implies(implies(p, not(not(p))), not(implies(not(not(p)), p))), not(implies(not(not(p)), p)))), true), true, theorem(or(not(implies(not(not(p)), p)), not(or(and(p, not(p)), not(implies(not(not(p)), p)))))), true), true, ifeq(theorem(implies(not(not(p)), p)), true, theorem(equivalent(p, not(not(p)))), true), true)
25.34/25.51	= { by axiom 8 (ifeq_axiom) }
25.34/25.51	  ifeq(ifeq(ifeq(true, true, ifeq(theorem(implies(p, not(not(p)))), true, theorem(implies(implies(implies(p, not(not(p))), not(implies(not(not(p)), p))), not(implies(not(not(p)), p)))), true), true), true, theorem(or(not(implies(not(not(p)), p)), not(or(and(p, not(p)), not(implies(not(not(p)), p)))))), true), true, ifeq(theorem(implies(not(not(p)), p)), true, theorem(equivalent(p, not(not(p)))), true), true)
25.34/25.51	= { by axiom 9 (rule_2) }
25.34/25.51	  ifeq(ifeq(ifeq(ifeq(theorem(implies(implies(implies(implies(p, not(not(p))), not(implies(not(not(p)), p))), implies(implies(p, not(not(p))), not(implies(not(not(p)), p)))), implies(implies(p, not(not(p))), implies(implies(implies(p, not(not(p))), not(implies(not(not(p)), p))), not(implies(not(not(p)), p)))))), true, ifeq(theorem(implies(implies(implies(p, not(not(p))), not(implies(not(not(p)), p))), implies(implies(p, not(not(p))), not(implies(not(not(p)), p))))), true, theorem(implies(implies(p, not(not(p))), implies(implies(implies(p, not(not(p))), not(implies(not(not(p)), p))), not(implies(not(not(p)), p))))), true), true), true, ifeq(theorem(implies(p, not(not(p)))), true, theorem(implies(implies(implies(p, not(not(p))), not(implies(not(not(p)), p))), not(implies(not(not(p)), p)))), true), true), true, theorem(or(not(implies(not(not(p)), p)), not(or(and(p, not(p)), not(implies(not(not(p)), p)))))), true), true, ifeq(theorem(implies(not(not(p)), p)), true, theorem(equivalent(p, not(not(p)))), true), true)
25.34/25.51	= { by axiom 6 (implies_definition) }
25.34/25.51	  ifeq(ifeq(ifeq(ifeq(theorem(implies(implies(implies(implies(p, not(not(p))), not(implies(not(not(p)), p))), or(not(implies(p, not(not(p)))), not(implies(not(not(p)), p)))), implies(implies(p, not(not(p))), implies(implies(implies(p, not(not(p))), not(implies(not(not(p)), p))), not(implies(not(not(p)), p)))))), true, ifeq(theorem(implies(implies(implies(p, not(not(p))), not(implies(not(not(p)), p))), implies(implies(p, not(not(p))), not(implies(not(not(p)), p))))), true, theorem(implies(implies(p, not(not(p))), implies(implies(implies(p, not(not(p))), not(implies(not(not(p)), p))), not(implies(not(not(p)), p))))), true), true), true, ifeq(theorem(implies(p, not(not(p)))), true, theorem(implies(implies(implies(p, not(not(p))), not(implies(not(not(p)), p))), not(implies(not(not(p)), p)))), true), true), true, theorem(or(not(implies(not(not(p)), p)), not(or(and(p, not(p)), not(implies(not(not(p)), p)))))), true), true, ifeq(theorem(implies(not(not(p)), p)), true, theorem(equivalent(p, not(not(p)))), true), true)
25.34/25.51	= { by axiom 6 (implies_definition) }
25.34/25.51	  ifeq(ifeq(ifeq(ifeq(theorem(implies(implies(implies(implies(p, not(not(p))), not(implies(not(not(p)), p))), or(not(implies(p, not(not(p)))), not(implies(not(not(p)), p)))), or(not(implies(p, not(not(p)))), implies(implies(implies(p, not(not(p))), not(implies(not(not(p)), p))), not(implies(not(not(p)), p)))))), true, ifeq(theorem(implies(implies(implies(p, not(not(p))), not(implies(not(not(p)), p))), implies(implies(p, not(not(p))), not(implies(not(not(p)), p))))), true, theorem(implies(implies(p, not(not(p))), implies(implies(implies(p, not(not(p))), not(implies(not(not(p)), p))), not(implies(not(not(p)), p))))), true), true), true, ifeq(theorem(implies(p, not(not(p)))), true, theorem(implies(implies(implies(p, not(not(p))), not(implies(not(not(p)), p))), not(implies(not(not(p)), p)))), true), true), true, theorem(or(not(implies(not(not(p)), p)), not(or(and(p, not(p)), not(implies(not(not(p)), p)))))), true), true, ifeq(theorem(implies(not(not(p)), p)), true, theorem(equivalent(p, not(not(p)))), true), true)
25.34/25.51	= { by lemma 12 }
25.34/25.51	  ifeq(ifeq(ifeq(ifeq(true, true, ifeq(theorem(implies(implies(implies(p, not(not(p))), not(implies(not(not(p)), p))), implies(implies(p, not(not(p))), not(implies(not(not(p)), p))))), true, theorem(implies(implies(p, not(not(p))), implies(implies(implies(p, not(not(p))), not(implies(not(not(p)), p))), not(implies(not(not(p)), p))))), true), true), true, ifeq(theorem(implies(p, not(not(p)))), true, theorem(implies(implies(implies(p, not(not(p))), not(implies(not(not(p)), p))), not(implies(not(not(p)), p)))), true), true), true, theorem(or(not(implies(not(not(p)), p)), not(or(and(p, not(p)), not(implies(not(not(p)), p)))))), true), true, ifeq(theorem(implies(not(not(p)), p)), true, theorem(equivalent(p, not(not(p)))), true), true)
25.34/25.51	= { by lemma 13 }
25.34/25.51	  ifeq(ifeq(ifeq(ifeq(true, true, ifeq(true, true, theorem(implies(implies(p, not(not(p))), implies(implies(implies(p, not(not(p))), not(implies(not(not(p)), p))), not(implies(not(not(p)), p))))), true), true), true, ifeq(theorem(implies(p, not(not(p)))), true, theorem(implies(implies(implies(p, not(not(p))), not(implies(not(not(p)), p))), not(implies(not(not(p)), p)))), true), true), true, theorem(or(not(implies(not(not(p)), p)), not(or(and(p, not(p)), not(implies(not(not(p)), p)))))), true), true, ifeq(theorem(implies(not(not(p)), p)), true, theorem(equivalent(p, not(not(p)))), true), true)
25.34/25.51	= { by axiom 8 (ifeq_axiom) }
25.34/25.51	  ifeq(ifeq(ifeq(ifeq(true, true, theorem(implies(implies(p, not(not(p))), implies(implies(implies(p, not(not(p))), not(implies(not(not(p)), p))), not(implies(not(not(p)), p))))), true), true, ifeq(theorem(implies(p, not(not(p)))), true, theorem(implies(implies(implies(p, not(not(p))), not(implies(not(not(p)), p))), not(implies(not(not(p)), p)))), true), true), true, theorem(or(not(implies(not(not(p)), p)), not(or(and(p, not(p)), not(implies(not(not(p)), p)))))), true), true, ifeq(theorem(implies(not(not(p)), p)), true, theorem(equivalent(p, not(not(p)))), true), true)
25.34/25.51	= { by axiom 8 (ifeq_axiom) }
25.34/25.51	  ifeq(ifeq(ifeq(theorem(implies(implies(p, not(not(p))), implies(implies(implies(p, not(not(p))), not(implies(not(not(p)), p))), not(implies(not(not(p)), p))))), true, ifeq(theorem(implies(p, not(not(p)))), true, theorem(implies(implies(implies(p, not(not(p))), not(implies(not(not(p)), p))), not(implies(not(not(p)), p)))), true), true), true, theorem(or(not(implies(not(not(p)), p)), not(or(and(p, not(p)), not(implies(not(not(p)), p)))))), true), true, ifeq(theorem(implies(not(not(p)), p)), true, theorem(equivalent(p, not(not(p)))), true), true)
25.34/25.51	= { by axiom 9 (rule_2) }
25.34/25.51	  ifeq(ifeq(true, true, theorem(or(not(implies(not(not(p)), p)), not(or(and(p, not(p)), not(implies(not(not(p)), p)))))), true), true, ifeq(theorem(implies(not(not(p)), p)), true, theorem(equivalent(p, not(not(p)))), true), true)
25.34/25.51	= { by axiom 6 (implies_definition) }
25.34/25.51	  ifeq(ifeq(true, true, theorem(implies(implies(not(not(p)), p), not(or(and(p, not(p)), not(implies(not(not(p)), p)))))), true), true, ifeq(theorem(implies(not(not(p)), p)), true, theorem(equivalent(p, not(not(p)))), true), true)
25.34/25.51	= { by axiom 8 (ifeq_axiom) }
25.34/25.51	  ifeq(theorem(implies(implies(not(not(p)), p), not(or(and(p, not(p)), not(implies(not(not(p)), p)))))), true, ifeq(theorem(implies(not(not(p)), p)), true, theorem(equivalent(p, not(not(p)))), true), true)
25.34/25.51	= { by lemma 18 }
25.34/25.51	  ifeq(theorem(implies(implies(not(not(p)), p), not(implies(implies(p, not(not(p))), not(implies(not(not(p)), p)))))), true, ifeq(theorem(implies(not(not(p)), p)), true, theorem(equivalent(p, not(not(p)))), true), true)
25.34/25.51	= { by lemma 17 }
25.34/25.51	  ifeq(theorem(implies(implies(not(not(p)), p), and(implies(p, not(not(p))), implies(not(not(p)), p)))), true, ifeq(theorem(implies(not(not(p)), p)), true, theorem(equivalent(p, not(not(p)))), true), true)
25.34/25.51	= { by axiom 7 (equivalent_defn) }
25.34/25.51	  ifeq(theorem(implies(implies(not(not(p)), p), equivalent(p, not(not(p))))), true, ifeq(theorem(implies(not(not(p)), p)), true, theorem(equivalent(p, not(not(p)))), true), true)
25.34/25.51	= { by axiom 9 (rule_2) }
25.34/25.51	  true
25.34/25.51	% SZS output end Proof
25.34/25.51	
25.34/25.51	RESULT: Unsatisfiable (the axioms are contradictory).
25.34/25.52	EOF