0.04/0.12	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.04/0.13	% Command    : twee %s --tstp --casc --quiet --explain-encoding --conditional-encoding if --smaller --drop-non-horn
0.13/0.34	% Computer   : n013.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 13:11:01 EDT 2019
0.13/0.34	% CPUTime    : 
11.12/11.34	% SZS status Unsatisfiable
11.12/11.34	
11.12/11.34	% SZS output start Proof
11.12/11.34	Take the following subset of the input axioms:
11.40/11.57	  fof(condensed_detachment, axiom, ![X, Y]: true=ifeq(is_a_theorem(implies(X, Y)), true, ifeq(is_a_theorem(X), true, is_a_theorem(Y), true), true)).
11.40/11.57	  fof(ifeq_axiom, axiom, ![B, A, C]: B=ifeq(A, A, B, C)).
11.40/11.57	  fof(mv_1, axiom, ![X, Y]: is_a_theorem(implies(X, implies(Y, X)))=true).
11.40/11.57	  fof(mv_2, axiom, ![X, Y, Z]: is_a_theorem(implies(implies(X, Y), implies(implies(Y, Z), implies(X, Z))))=true).
11.40/11.57	  fof(mv_3, axiom, ![X, Y]: is_a_theorem(implies(implies(implies(X, Y), Y), implies(implies(Y, X), X)))=true).
11.40/11.57	  fof(mv_5, axiom, ![X, Y]: is_a_theorem(implies(implies(not(X), not(Y)), implies(Y, X)))=true).
11.40/11.57	  fof(prove_mv_33, negated_conjecture, is_a_theorem(implies(implies(not(a), b), implies(not(b), a)))!=true).
11.40/11.57	
11.40/11.57	Now clausify the problem and encode Horn clauses using encoding 3 of
11.40/11.57	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
11.40/11.57	We repeatedly replace C & s=t => u=v by the two clauses:
11.40/11.57	  fresh(y, y, x1...xn) = u
11.40/11.57	  C => fresh(s, t, x1...xn) = v
11.40/11.57	where fresh is a fresh function symbol and x1..xn are the free
11.40/11.57	variables of u and v.
11.40/11.57	A predicate p(X) is encoded as p(X)=true (this is sound, because the
11.40/11.57	input problem has no model of domain size 1).
11.40/11.57	
11.40/11.57	The encoding turns the above axioms into the following unit equations and goals:
11.40/11.57	
11.40/11.57	Axiom 1 (ifeq_axiom): X = ifeq(Y, Y, X, Z).
11.40/11.57	Axiom 2 (mv_1): is_a_theorem(implies(X, implies(Y, X))) = true.
11.40/11.57	Axiom 3 (condensed_detachment): true = ifeq(is_a_theorem(implies(X, Y)), true, ifeq(is_a_theorem(X), true, is_a_theorem(Y), true), true).
11.40/11.57	Axiom 4 (mv_3): is_a_theorem(implies(implies(implies(X, Y), Y), implies(implies(Y, X), X))) = true.
11.40/11.57	Axiom 5 (mv_5): is_a_theorem(implies(implies(not(X), not(Y)), implies(Y, X))) = true.
11.40/11.58	Axiom 6 (mv_2): is_a_theorem(implies(implies(X, Y), implies(implies(Y, Z), implies(X, Z)))) = true.
11.40/11.58	
11.40/11.58	Lemma 7: ifeq(is_a_theorem(implies(X, Y)), true, is_a_theorem(implies(implies(Y, Z), implies(X, Z))), true) = true.
11.40/11.58	Proof:
11.40/11.58	  ifeq(is_a_theorem(implies(X, Y)), true, is_a_theorem(implies(implies(Y, Z), implies(X, Z))), true)
11.40/11.58	= { by axiom 1 (ifeq_axiom) }
11.40/11.58	  ifeq(true, true, ifeq(is_a_theorem(implies(X, Y)), true, is_a_theorem(implies(implies(Y, Z), implies(X, Z))), true), true)
11.40/11.58	= { by axiom 6 (mv_2) }
11.40/11.58	  ifeq(is_a_theorem(implies(implies(X, Y), implies(implies(Y, Z), implies(X, Z)))), true, ifeq(is_a_theorem(implies(X, Y)), true, is_a_theorem(implies(implies(Y, Z), implies(X, Z))), true), true)
11.40/11.58	= { by axiom 3 (condensed_detachment) }
11.40/11.58	  true
11.40/11.58	
11.40/11.58	Lemma 8: is_a_theorem(implies(implies(implies(implies(X, Y), implies(Z, Y)), W), implies(implies(Z, X), W))) = true.
11.40/11.58	Proof:
11.40/11.58	  is_a_theorem(implies(implies(implies(implies(X, Y), implies(Z, Y)), W), implies(implies(Z, X), W)))
11.40/11.58	= { by axiom 1 (ifeq_axiom) }
11.40/11.58	  ifeq(true, true, is_a_theorem(implies(implies(implies(implies(X, Y), implies(Z, Y)), W), implies(implies(Z, X), W))), true)
11.40/11.58	= { by axiom 6 (mv_2) }
11.40/11.58	  ifeq(is_a_theorem(implies(implies(Z, X), implies(implies(X, Y), implies(Z, Y)))), true, is_a_theorem(implies(implies(implies(implies(X, Y), implies(Z, Y)), W), implies(implies(Z, X), W))), true)
11.40/11.58	= { by lemma 7 }
11.40/11.58	  true
11.40/11.58	
11.40/11.58	Lemma 9: ifeq(is_a_theorem(implies(implies(X, implies(Y, X)), Z)), true, is_a_theorem(Z), true) = true.
11.40/11.58	Proof:
11.40/11.58	  ifeq(is_a_theorem(implies(implies(X, implies(Y, X)), Z)), true, is_a_theorem(Z), true)
11.40/11.58	= { by axiom 1 (ifeq_axiom) }
11.40/11.58	  ifeq(is_a_theorem(implies(implies(X, implies(Y, X)), Z)), true, ifeq(true, true, is_a_theorem(Z), true), true)
11.40/11.58	= { by axiom 2 (mv_1) }
11.40/11.58	  ifeq(is_a_theorem(implies(implies(X, implies(Y, X)), Z)), true, ifeq(is_a_theorem(implies(X, implies(Y, X))), true, is_a_theorem(Z), true), true)
11.40/11.58	= { by axiom 3 (condensed_detachment) }
11.40/11.58	  true
11.40/11.58	
11.40/11.58	Lemma 10: ifeq(is_a_theorem(implies(implies(X, Y), Z)), true, is_a_theorem(implies(Y, Z)), true) = true.
11.40/11.58	Proof:
11.40/11.58	  ifeq(is_a_theorem(implies(implies(X, Y), Z)), true, is_a_theorem(implies(Y, Z)), true)
11.40/11.58	= { by axiom 1 (ifeq_axiom) }
11.40/11.58	  ifeq(true, true, ifeq(is_a_theorem(implies(implies(X, Y), Z)), true, is_a_theorem(implies(Y, Z)), true), true)
11.40/11.58	= { by lemma 9 }
11.40/11.58	  ifeq(ifeq(is_a_theorem(implies(implies(Y, implies(X, Y)), implies(implies(implies(X, Y), Z), implies(Y, Z)))), true, is_a_theorem(implies(implies(implies(X, Y), Z), implies(Y, Z))), true), true, ifeq(is_a_theorem(implies(implies(X, Y), Z)), true, is_a_theorem(implies(Y, Z)), true), true)
11.40/11.58	= { by axiom 6 (mv_2) }
11.40/11.58	  ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(X, Y), Z), implies(Y, Z))), true), true, ifeq(is_a_theorem(implies(implies(X, Y), Z)), true, is_a_theorem(implies(Y, Z)), true), true)
11.40/11.58	= { by axiom 1 (ifeq_axiom) }
11.40/11.58	  ifeq(is_a_theorem(implies(implies(implies(X, Y), Z), implies(Y, Z))), true, ifeq(is_a_theorem(implies(implies(X, Y), Z)), true, is_a_theorem(implies(Y, Z)), true), true)
11.40/11.58	= { by axiom 3 (condensed_detachment) }
11.40/11.58	  true
11.40/11.58	
11.40/11.58	Lemma 11: is_a_theorem(implies(X, implies(implies(X, Y), Y))) = true.
11.40/11.58	Proof:
11.40/11.58	  is_a_theorem(implies(X, implies(implies(X, Y), Y)))
11.40/11.58	= { by axiom 1 (ifeq_axiom) }
11.40/11.58	  ifeq(true, true, is_a_theorem(implies(X, implies(implies(X, Y), Y))), true)
11.40/11.58	= { by axiom 4 (mv_3) }
11.40/11.58	  ifeq(is_a_theorem(implies(implies(implies(Y, X), X), implies(implies(X, Y), Y))), true, is_a_theorem(implies(X, implies(implies(X, Y), Y))), true)
11.40/11.58	= { by lemma 10 }
11.40/11.58	  true
11.40/11.58	
11.40/11.58	Lemma 12: is_a_theorem(implies(implies(implies(implies(X, Y), Y), Z), implies(X, Z))) = true.
11.40/11.58	Proof:
11.40/11.58	  is_a_theorem(implies(implies(implies(implies(X, Y), Y), Z), implies(X, Z)))
11.40/11.58	= { by axiom 1 (ifeq_axiom) }
11.40/11.58	  ifeq(true, true, is_a_theorem(implies(implies(implies(implies(X, Y), Y), Z), implies(X, Z))), true)
11.40/11.58	= { by lemma 11 }
11.40/11.58	  ifeq(is_a_theorem(implies(X, implies(implies(X, Y), Y))), true, is_a_theorem(implies(implies(implies(implies(X, Y), Y), Z), implies(X, Z))), true)
11.40/11.58	= { by lemma 7 }
11.40/11.58	  true
11.40/11.58	
11.40/11.58	Lemma 13: ifeq(is_a_theorem(implies(implies(implies(X, Y), implies(Z, Y)), W)), true, is_a_theorem(implies(implies(Z, X), W)), true) = true.
11.40/11.58	Proof:
11.40/11.58	  ifeq(is_a_theorem(implies(implies(implies(X, Y), implies(Z, Y)), W)), true, is_a_theorem(implies(implies(Z, X), W)), true)
11.40/11.58	= { by axiom 1 (ifeq_axiom) }
11.40/11.58	  ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(X, Y), implies(Z, Y)), W)), true, is_a_theorem(implies(implies(Z, X), W)), true), true)
11.40/11.58	= { by lemma 8 }
11.40/11.58	  ifeq(is_a_theorem(implies(implies(implies(implies(X, Y), implies(Z, Y)), W), implies(implies(Z, X), W))), true, ifeq(is_a_theorem(implies(implies(implies(X, Y), implies(Z, Y)), W)), true, is_a_theorem(implies(implies(Z, X), W)), true), true)
11.40/11.58	= { by axiom 3 (condensed_detachment) }
11.65/11.82	  true
11.65/11.82	
11.65/11.82	Goal 1 (prove_mv_33): is_a_theorem(implies(implies(not(a), b), implies(not(b), a))) = true.
11.65/11.82	Proof:
11.65/11.82	  is_a_theorem(implies(implies(not(a), b), implies(not(b), a)))
11.65/11.82	= { by axiom 1 (ifeq_axiom) }
11.65/11.82	  ifeq(true, true, is_a_theorem(implies(implies(not(a), b), implies(not(b), a))), true)
11.65/11.82	= { by axiom 3 (condensed_detachment) }
11.65/11.82	  ifeq(ifeq(is_a_theorem(implies(implies(implies(not(a), b), implies(implies(b, not(not(b))), implies(not(b), a))), implies(implies(b, not(not(b))), implies(implies(not(a), b), implies(not(b), a))))), true, ifeq(is_a_theorem(implies(implies(not(a), b), implies(implies(b, not(not(b))), implies(not(b), a)))), true, is_a_theorem(implies(implies(b, not(not(b))), implies(implies(not(a), b), implies(not(b), a)))), true), true), true, is_a_theorem(implies(implies(not(a), b), implies(not(b), a))), true)
11.65/11.82	= { by axiom 1 (ifeq_axiom) }
11.65/11.82	  ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(not(a), b), implies(implies(b, not(not(b))), implies(not(b), a))), implies(implies(b, not(not(b))), implies(implies(not(a), b), implies(not(b), a))))), true), true, ifeq(is_a_theorem(implies(implies(not(a), b), implies(implies(b, not(not(b))), implies(not(b), a)))), true, is_a_theorem(implies(implies(b, not(not(b))), implies(implies(not(a), b), implies(not(b), a)))), true), true), true, is_a_theorem(implies(implies(not(a), b), implies(not(b), a))), true)
11.65/11.82	= { by lemma 12 }
11.65/11.82	  ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(implies(b, not(not(b))), implies(not(b), a)), implies(not(b), a)), implies(implies(not(a), b), implies(not(b), a))), implies(implies(b, not(not(b))), implies(implies(not(a), b), implies(not(b), a))))), true, is_a_theorem(implies(implies(implies(not(a), b), implies(implies(b, not(not(b))), implies(not(b), a))), implies(implies(b, not(not(b))), implies(implies(not(a), b), implies(not(b), a))))), true), true, ifeq(is_a_theorem(implies(implies(not(a), b), implies(implies(b, not(not(b))), implies(not(b), a)))), true, is_a_theorem(implies(implies(b, not(not(b))), implies(implies(not(a), b), implies(not(b), a)))), true), true), true, is_a_theorem(implies(implies(not(a), b), implies(not(b), a))), true)
11.65/11.82	= { by lemma 13 }
11.65/11.82	  ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(not(a), b), implies(implies(b, not(not(b))), implies(not(b), a)))), true, is_a_theorem(implies(implies(b, not(not(b))), implies(implies(not(a), b), implies(not(b), a)))), true), true), true, is_a_theorem(implies(implies(not(a), b), implies(not(b), a))), true)
11.65/11.82	= { by axiom 1 (ifeq_axiom) }
11.65/11.82	  ifeq(ifeq(is_a_theorem(implies(implies(not(a), b), implies(implies(b, not(not(b))), implies(not(b), a)))), true, is_a_theorem(implies(implies(b, not(not(b))), implies(implies(not(a), b), implies(not(b), a)))), true), true, is_a_theorem(implies(implies(not(a), b), implies(not(b), a))), true)
11.65/11.82	= { by axiom 1 (ifeq_axiom) }
11.65/11.82	  ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(not(a), b), implies(implies(b, not(not(b))), implies(not(b), a)))), true), true, is_a_theorem(implies(implies(b, not(not(b))), implies(implies(not(a), b), implies(not(b), a)))), true), true, is_a_theorem(implies(implies(not(a), b), implies(not(b), a))), true)
11.65/11.82	= { by axiom 3 (condensed_detachment) }
11.65/11.82	  ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(a), not(not(b))), implies(not(b), a)), implies(implies(implies(b, not(not(b))), implies(not(a), not(not(b)))), implies(implies(b, not(not(b))), implies(not(b), a))))), true, ifeq(is_a_theorem(implies(implies(not(a), not(not(b))), implies(not(b), a))), true, is_a_theorem(implies(implies(implies(b, not(not(b))), implies(not(a), not(not(b)))), implies(implies(b, not(not(b))), implies(not(b), a)))), true), true), true, is_a_theorem(implies(implies(not(a), b), implies(implies(b, not(not(b))), implies(not(b), a)))), true), true, is_a_theorem(implies(implies(b, not(not(b))), implies(implies(not(a), b), implies(not(b), a)))), true), true, is_a_theorem(implies(implies(not(a), b), implies(not(b), a))), true)
11.65/11.82	= { by axiom 1 (ifeq_axiom) }
11.65/11.82	  ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(not(a), not(not(b))), implies(not(b), a)), implies(implies(implies(b, not(not(b))), implies(not(a), not(not(b)))), implies(implies(b, not(not(b))), implies(not(b), a))))), true), true, ifeq(is_a_theorem(implies(implies(not(a), not(not(b))), implies(not(b), a))), true, is_a_theorem(implies(implies(implies(b, not(not(b))), implies(not(a), not(not(b)))), implies(implies(b, not(not(b))), implies(not(b), a)))), true), true), true, is_a_theorem(implies(implies(not(a), b), implies(implies(b, not(not(b))), implies(not(b), a)))), true), true, is_a_theorem(implies(implies(b, not(not(b))), implies(implies(not(a), b), implies(not(b), a)))), true), true, is_a_theorem(implies(implies(not(a), b), implies(not(b), a))), true)
11.65/11.82	= { by lemma 8 }
11.65/11.82	  ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(implies(not(a), not(not(b))), implies(not(b), a)), implies(implies(b, not(not(b))), implies(not(b), a))), implies(implies(b, not(not(b))), implies(not(b), a))), implies(implies(implies(b, not(not(b))), implies(not(a), not(not(b)))), implies(implies(b, not(not(b))), implies(not(b), a))))), true, is_a_theorem(implies(implies(implies(not(a), not(not(b))), implies(not(b), a)), implies(implies(implies(b, not(not(b))), implies(not(a), not(not(b)))), implies(implies(b, not(not(b))), implies(not(b), a))))), true), true, ifeq(is_a_theorem(implies(implies(not(a), not(not(b))), implies(not(b), a))), true, is_a_theorem(implies(implies(implies(b, not(not(b))), implies(not(a), not(not(b)))), implies(implies(b, not(not(b))), implies(not(b), a)))), true), true), true, is_a_theorem(implies(implies(not(a), b), implies(implies(b, not(not(b))), implies(not(b), a)))), true), true, is_a_theorem(implies(implies(b, not(not(b))), implies(implies(not(a), b), implies(not(b), a)))), true), true, is_a_theorem(implies(implies(not(a), b), implies(not(b), a))), true)
11.65/11.82	= { by axiom 1 (ifeq_axiom) }
11.65/11.82	  ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(not(a), not(not(b))), implies(not(b), a)), implies(implies(b, not(not(b))), implies(not(b), a))), implies(implies(b, not(not(b))), implies(not(b), a))), implies(implies(implies(b, not(not(b))), implies(not(a), not(not(b)))), implies(implies(b, not(not(b))), implies(not(b), a))))), true, is_a_theorem(implies(implies(implies(not(a), not(not(b))), implies(not(b), a)), implies(implies(implies(b, not(not(b))), implies(not(a), not(not(b)))), implies(implies(b, not(not(b))), implies(not(b), a))))), true), true), true, ifeq(is_a_theorem(implies(implies(not(a), not(not(b))), implies(not(b), a))), true, is_a_theorem(implies(implies(implies(b, not(not(b))), implies(not(a), not(not(b)))), implies(implies(b, not(not(b))), implies(not(b), a)))), true), true), true, is_a_theorem(implies(implies(not(a), b), implies(implies(b, not(not(b))), implies(not(b), a)))), true), true, is_a_theorem(implies(implies(b, not(not(b))), implies(implies(not(a), b), implies(not(b), a)))), true), true, is_a_theorem(implies(implies(not(a), b), implies(not(b), a))), true)
11.65/11.82	= { by lemma 12 }
11.65/11.82	  ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(implies(implies(not(a), not(not(b))), implies(not(b), a)), implies(implies(b, not(not(b))), implies(not(b), a))), implies(implies(b, not(not(b))), implies(not(b), a))), implies(implies(implies(b, not(not(b))), implies(not(a), not(not(b)))), implies(implies(b, not(not(b))), implies(not(b), a)))), implies(implies(implies(not(a), not(not(b))), implies(not(b), a)), implies(implies(implies(b, not(not(b))), implies(not(a), not(not(b)))), implies(implies(b, not(not(b))), implies(not(b), a)))))), true, ifeq(is_a_theorem(implies(implies(implies(implies(implies(not(a), not(not(b))), implies(not(b), a)), implies(implies(b, not(not(b))), implies(not(b), a))), implies(implies(b, not(not(b))), implies(not(b), a))), implies(implies(implies(b, not(not(b))), implies(not(a), not(not(b)))), implies(implies(b, not(not(b))), implies(not(b), a))))), true, is_a_theorem(implies(implies(implies(not(a), not(not(b))), implies(not(b), a)), implies(implies(implies(b, not(not(b))), implies(not(a), not(not(b)))), implies(implies(b, not(not(b))), implies(not(b), a))))), true), true), true, ifeq(is_a_theorem(implies(implies(not(a), not(not(b))), implies(not(b), a))), true, is_a_theorem(implies(implies(implies(b, not(not(b))), implies(not(a), not(not(b)))), implies(implies(b, not(not(b))), implies(not(b), a)))), true), true), true, is_a_theorem(implies(implies(not(a), b), implies(implies(b, not(not(b))), implies(not(b), a)))), true), true, is_a_theorem(implies(implies(b, not(not(b))), implies(implies(not(a), b), implies(not(b), a)))), true), true, is_a_theorem(implies(implies(not(a), b), implies(not(b), a))), true)
11.65/11.82	= { by axiom 3 (condensed_detachment) }
11.65/11.82	  ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(not(a), not(not(b))), implies(not(b), a))), true, is_a_theorem(implies(implies(implies(b, not(not(b))), implies(not(a), not(not(b)))), implies(implies(b, not(not(b))), implies(not(b), a)))), true), true), true, is_a_theorem(implies(implies(not(a), b), implies(implies(b, not(not(b))), implies(not(b), a)))), true), true, is_a_theorem(implies(implies(b, not(not(b))), implies(implies(not(a), b), implies(not(b), a)))), true), true, is_a_theorem(implies(implies(not(a), b), implies(not(b), a))), true)
11.65/11.82	= { by axiom 5 (mv_5) }
11.65/11.82	  ifeq(ifeq(ifeq(ifeq(true, true, ifeq(true, true, is_a_theorem(implies(implies(implies(b, not(not(b))), implies(not(a), not(not(b)))), implies(implies(b, not(not(b))), implies(not(b), a)))), true), true), true, is_a_theorem(implies(implies(not(a), b), implies(implies(b, not(not(b))), implies(not(b), a)))), true), true, is_a_theorem(implies(implies(b, not(not(b))), implies(implies(not(a), b), implies(not(b), a)))), true), true, is_a_theorem(implies(implies(not(a), b), implies(not(b), a))), true)
11.65/11.82	= { by axiom 1 (ifeq_axiom) }
11.65/11.82	  ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(b, not(not(b))), implies(not(a), not(not(b)))), implies(implies(b, not(not(b))), implies(not(b), a)))), true), true, is_a_theorem(implies(implies(not(a), b), implies(implies(b, not(not(b))), implies(not(b), a)))), true), true, is_a_theorem(implies(implies(b, not(not(b))), implies(implies(not(a), b), implies(not(b), a)))), true), true, is_a_theorem(implies(implies(not(a), b), implies(not(b), a))), true)
11.65/11.82	= { by axiom 1 (ifeq_axiom) }
11.65/11.82	  ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(b, not(not(b))), implies(not(a), not(not(b)))), implies(implies(b, not(not(b))), implies(not(b), a)))), true, is_a_theorem(implies(implies(not(a), b), implies(implies(b, not(not(b))), implies(not(b), a)))), true), true, is_a_theorem(implies(implies(b, not(not(b))), implies(implies(not(a), b), implies(not(b), a)))), true), true, is_a_theorem(implies(implies(not(a), b), implies(not(b), a))), true)
11.65/11.82	= { by lemma 13 }
11.65/11.82	  ifeq(ifeq(true, true, is_a_theorem(implies(implies(b, not(not(b))), implies(implies(not(a), b), implies(not(b), a)))), true), true, is_a_theorem(implies(implies(not(a), b), implies(not(b), a))), true)
11.65/11.82	= { by axiom 1 (ifeq_axiom) }
11.65/11.82	  ifeq(is_a_theorem(implies(implies(b, not(not(b))), implies(implies(not(a), b), implies(not(b), a)))), true, is_a_theorem(implies(implies(not(a), b), implies(not(b), a))), true)
11.65/11.82	= { by axiom 1 (ifeq_axiom) }
11.65/11.82	  ifeq(is_a_theorem(implies(implies(b, not(not(b))), implies(implies(not(a), b), implies(not(b), a)))), true, ifeq(true, true, is_a_theorem(implies(implies(not(a), b), implies(not(b), a))), true), true)
11.65/11.82	= { by axiom 3 (condensed_detachment) }
11.65/11.82	  ifeq(is_a_theorem(implies(implies(b, not(not(b))), implies(implies(not(a), b), implies(not(b), a)))), true, ifeq(ifeq(is_a_theorem(implies(implies(not(not(not(b))), not(b)), implies(b, not(not(b))))), true, ifeq(is_a_theorem(implies(not(not(not(b))), not(b))), true, is_a_theorem(implies(b, not(not(b)))), true), true), true, is_a_theorem(implies(implies(not(a), b), implies(not(b), a))), true), true)
11.65/11.82	= { by axiom 5 (mv_5) }
11.65/11.82	  ifeq(is_a_theorem(implies(implies(b, not(not(b))), implies(implies(not(a), b), implies(not(b), a)))), true, ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(not(not(not(b))), not(b))), true, is_a_theorem(implies(b, not(not(b)))), true), true), true, is_a_theorem(implies(implies(not(a), b), implies(not(b), a))), true), true)
11.65/11.82	= { by axiom 1 (ifeq_axiom) }
11.65/11.82	  ifeq(is_a_theorem(implies(implies(b, not(not(b))), implies(implies(not(a), b), implies(not(b), a)))), true, ifeq(ifeq(is_a_theorem(implies(not(not(not(b))), not(b))), true, is_a_theorem(implies(b, not(not(b)))), true), true, is_a_theorem(implies(implies(not(a), b), implies(not(b), a))), true), true)
11.65/11.82	= { by axiom 1 (ifeq_axiom) }
11.65/11.82	  ifeq(is_a_theorem(implies(implies(b, not(not(b))), implies(implies(not(a), b), implies(not(b), a)))), true, ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(not(not(not(b))), not(b))), true), true, is_a_theorem(implies(b, not(not(b)))), true), true, is_a_theorem(implies(implies(not(a), b), implies(not(b), a))), true), true)
11.65/11.82	= { by axiom 3 (condensed_detachment) }
11.65/11.82	  ifeq(is_a_theorem(implies(implies(b, not(not(b))), implies(implies(not(a), b), implies(not(b), a)))), true, ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(?, ?), not(b)), not(b)), implies(implies(not(not(b)), not(implies(?, ?))), not(b)))), true, ifeq(is_a_theorem(implies(implies(implies(?, ?), not(b)), not(b))), true, is_a_theorem(implies(implies(not(not(b)), not(implies(?, ?))), not(b))), true), true), true, is_a_theorem(implies(not(not(not(b))), not(b))), true), true, is_a_theorem(implies(b, not(not(b)))), true), true, is_a_theorem(implies(implies(not(a), b), implies(not(b), a))), true), true)
11.65/11.82	= { by axiom 1 (ifeq_axiom) }
11.65/11.82	  ifeq(is_a_theorem(implies(implies(b, not(not(b))), implies(implies(not(a), b), implies(not(b), a)))), true, ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(?, ?), not(b)), not(b)), implies(implies(not(not(b)), not(implies(?, ?))), not(b)))), true), true, ifeq(is_a_theorem(implies(implies(implies(?, ?), not(b)), not(b))), true, is_a_theorem(implies(implies(not(not(b)), not(implies(?, ?))), not(b))), true), true), true, is_a_theorem(implies(not(not(not(b))), not(b))), true), true, is_a_theorem(implies(b, not(not(b)))), true), true, is_a_theorem(implies(implies(not(a), b), implies(not(b), a))), true), true)
11.65/11.82	= { by axiom 5 (mv_5) }
11.65/11.82	  ifeq(is_a_theorem(implies(implies(b, not(not(b))), implies(implies(not(a), b), implies(not(b), a)))), true, ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(not(b)), not(implies(?, ?))), implies(implies(?, ?), not(b)))), true, is_a_theorem(implies(implies(implies(implies(?, ?), not(b)), not(b)), implies(implies(not(not(b)), not(implies(?, ?))), not(b)))), true), true, ifeq(is_a_theorem(implies(implies(implies(?, ?), not(b)), not(b))), true, is_a_theorem(implies(implies(not(not(b)), not(implies(?, ?))), not(b))), true), true), true, is_a_theorem(implies(not(not(not(b))), not(b))), true), true, is_a_theorem(implies(b, not(not(b)))), true), true, is_a_theorem(implies(implies(not(a), b), implies(not(b), a))), true), true)
11.65/11.82	= { by lemma 7 }
11.65/11.82	  ifeq(is_a_theorem(implies(implies(b, not(not(b))), implies(implies(not(a), b), implies(not(b), a)))), true, ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(?, ?), not(b)), not(b))), true, is_a_theorem(implies(implies(not(not(b)), not(implies(?, ?))), not(b))), true), true), true, is_a_theorem(implies(not(not(not(b))), not(b))), true), true, is_a_theorem(implies(b, not(not(b)))), true), true, is_a_theorem(implies(implies(not(a), b), implies(not(b), a))), true), true)
11.65/11.82	= { by axiom 1 (ifeq_axiom) }
11.65/11.82	  ifeq(is_a_theorem(implies(implies(b, not(not(b))), implies(implies(not(a), b), implies(not(b), a)))), true, ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(?, ?), not(b)), not(b))), true, is_a_theorem(implies(implies(not(not(b)), not(implies(?, ?))), not(b))), true), true, is_a_theorem(implies(not(not(not(b))), not(b))), true), true, is_a_theorem(implies(b, not(not(b)))), true), true, is_a_theorem(implies(implies(not(a), b), implies(not(b), a))), true), true)
11.65/11.82	= { by axiom 1 (ifeq_axiom) }
11.65/11.82	  ifeq(is_a_theorem(implies(implies(b, not(not(b))), implies(implies(not(a), b), implies(not(b), a)))), true, ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(?, ?), not(b)), not(b))), true), true, is_a_theorem(implies(implies(not(not(b)), not(implies(?, ?))), not(b))), true), true, is_a_theorem(implies(not(not(not(b))), not(b))), true), true, is_a_theorem(implies(b, not(not(b)))), true), true, is_a_theorem(implies(implies(not(a), b), implies(not(b), a))), true), true)
11.65/11.82	= { by lemma 10 }
11.65/11.82	  ifeq(is_a_theorem(implies(implies(b, not(not(b))), implies(implies(not(a), b), implies(not(b), a)))), true, ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(?, implies(?, ?)), ?), ?)), true, is_a_theorem(implies(?, ?)), true), true, is_a_theorem(implies(implies(implies(?, ?), not(b)), not(b))), true), true, is_a_theorem(implies(implies(not(not(b)), not(implies(?, ?))), not(b))), true), true, is_a_theorem(implies(not(not(not(b))), not(b))), true), true, is_a_theorem(implies(b, not(not(b)))), true), true, is_a_theorem(implies(implies(not(a), b), implies(not(b), a))), true), true)
11.65/11.82	= { by axiom 1 (ifeq_axiom) }
11.65/11.82	  ifeq(is_a_theorem(implies(implies(b, not(not(b))), implies(implies(not(a), b), implies(not(b), a)))), true, ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(?, implies(?, ?)), ?), ?)), true), true, is_a_theorem(implies(?, ?)), true), true, is_a_theorem(implies(implies(implies(?, ?), not(b)), not(b))), true), true, is_a_theorem(implies(implies(not(not(b)), not(implies(?, ?))), not(b))), true), true, is_a_theorem(implies(not(not(not(b))), not(b))), true), true, is_a_theorem(implies(b, not(not(b)))), true), true, is_a_theorem(implies(implies(not(a), b), implies(not(b), a))), true), true)
11.65/11.82	= { by lemma 11 }
11.65/11.82	  ifeq(is_a_theorem(implies(implies(b, not(not(b))), implies(implies(not(a), b), implies(not(b), a)))), true, ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(?, implies(?, ?)), implies(implies(implies(?, implies(?, ?)), ?), ?))), true, is_a_theorem(implies(implies(implies(?, implies(?, ?)), ?), ?)), true), true, is_a_theorem(implies(?, ?)), true), true, is_a_theorem(implies(implies(implies(?, ?), not(b)), not(b))), true), true, is_a_theorem(implies(implies(not(not(b)), not(implies(?, ?))), not(b))), true), true, is_a_theorem(implies(not(not(not(b))), not(b))), true), true, is_a_theorem(implies(b, not(not(b)))), true), true, is_a_theorem(implies(implies(not(a), b), implies(not(b), a))), true), true)
11.65/11.82	= { by lemma 9 }
11.65/11.82	  ifeq(is_a_theorem(implies(implies(b, not(not(b))), implies(implies(not(a), b), implies(not(b), a)))), true, ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(?, ?)), true), true, is_a_theorem(implies(implies(implies(?, ?), not(b)), not(b))), true), true, is_a_theorem(implies(implies(not(not(b)), not(implies(?, ?))), not(b))), true), true, is_a_theorem(implies(not(not(not(b))), not(b))), true), true, is_a_theorem(implies(b, not(not(b)))), true), true, is_a_theorem(implies(implies(not(a), b), implies(not(b), a))), true), true)
11.65/11.82	= { by axiom 1 (ifeq_axiom) }
11.65/11.82	  ifeq(is_a_theorem(implies(implies(b, not(not(b))), implies(implies(not(a), b), implies(not(b), a)))), true, ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(?, ?)), true, is_a_theorem(implies(implies(implies(?, ?), not(b)), not(b))), true), true, is_a_theorem(implies(implies(not(not(b)), not(implies(?, ?))), not(b))), true), true, is_a_theorem(implies(not(not(not(b))), not(b))), true), true, is_a_theorem(implies(b, not(not(b)))), true), true, is_a_theorem(implies(implies(not(a), b), implies(not(b), a))), true), true)
11.65/11.82	= { by axiom 1 (ifeq_axiom) }
11.65/11.83	  ifeq(is_a_theorem(implies(implies(b, not(not(b))), implies(implies(not(a), b), implies(not(b), a)))), true, ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(?, ?)), true, is_a_theorem(implies(implies(implies(?, ?), not(b)), not(b))), true), true), true, is_a_theorem(implies(implies(not(not(b)), not(implies(?, ?))), not(b))), true), true, is_a_theorem(implies(not(not(not(b))), not(b))), true), true, is_a_theorem(implies(b, not(not(b)))), true), true, is_a_theorem(implies(implies(not(a), b), implies(not(b), a))), true), true)
11.65/11.83	= { by lemma 11 }
11.65/11.83	  ifeq(is_a_theorem(implies(implies(b, not(not(b))), implies(implies(not(a), b), implies(not(b), a)))), true, ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(?, ?), implies(implies(implies(?, ?), not(b)), not(b)))), true, ifeq(is_a_theorem(implies(?, ?)), true, is_a_theorem(implies(implies(implies(?, ?), not(b)), not(b))), true), true), true, is_a_theorem(implies(implies(not(not(b)), not(implies(?, ?))), not(b))), true), true, is_a_theorem(implies(not(not(not(b))), not(b))), true), true, is_a_theorem(implies(b, not(not(b)))), true), true, is_a_theorem(implies(implies(not(a), b), implies(not(b), a))), true), true)
11.65/11.83	= { by axiom 3 (condensed_detachment) }
11.65/11.83	  ifeq(is_a_theorem(implies(implies(b, not(not(b))), implies(implies(not(a), b), implies(not(b), a)))), true, ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(not(not(b)), not(implies(?, ?))), not(b))), true), true, is_a_theorem(implies(not(not(not(b))), not(b))), true), true, is_a_theorem(implies(b, not(not(b)))), true), true, is_a_theorem(implies(implies(not(a), b), implies(not(b), a))), true), true)
11.65/11.83	= { by axiom 1 (ifeq_axiom) }
11.65/11.83	  ifeq(is_a_theorem(implies(implies(b, not(not(b))), implies(implies(not(a), b), implies(not(b), a)))), true, ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(not(b)), not(implies(?, ?))), not(b))), true, is_a_theorem(implies(not(not(not(b))), not(b))), true), true, is_a_theorem(implies(b, not(not(b)))), true), true, is_a_theorem(implies(implies(not(a), b), implies(not(b), a))), true), true)
11.65/11.83	= { by axiom 1 (ifeq_axiom) }
11.65/11.83	  ifeq(is_a_theorem(implies(implies(b, not(not(b))), implies(implies(not(a), b), implies(not(b), a)))), true, ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(not(not(b)), not(implies(?, ?))), not(b))), true, is_a_theorem(implies(not(not(not(b))), not(b))), true), true), true, is_a_theorem(implies(b, not(not(b)))), true), true, is_a_theorem(implies(implies(not(a), b), implies(not(b), a))), true), true)
11.65/11.83	= { by lemma 7 }
11.65/11.83	  ifeq(is_a_theorem(implies(implies(b, not(not(b))), implies(implies(not(a), b), implies(not(b), a)))), true, ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(not(not(not(b))), implies(not(not(b)), not(implies(?, ?))))), true, is_a_theorem(implies(implies(implies(not(not(b)), not(implies(?, ?))), not(b)), implies(not(not(not(b))), not(b)))), true), true, ifeq(is_a_theorem(implies(implies(not(not(b)), not(implies(?, ?))), not(b))), true, is_a_theorem(implies(not(not(not(b))), not(b))), true), true), true, is_a_theorem(implies(b, not(not(b)))), true), true, is_a_theorem(implies(implies(not(a), b), implies(not(b), a))), true), true)
11.65/11.83	= { by axiom 1 (ifeq_axiom) }
11.65/11.83	  ifeq(is_a_theorem(implies(implies(b, not(not(b))), implies(implies(not(a), b), implies(not(b), a)))), true, ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(not(not(not(b))), implies(not(not(b)), not(implies(?, ?))))), true), true, is_a_theorem(implies(implies(implies(not(not(b)), not(implies(?, ?))), not(b)), implies(not(not(not(b))), not(b)))), true), true, ifeq(is_a_theorem(implies(implies(not(not(b)), not(implies(?, ?))), not(b))), true, is_a_theorem(implies(not(not(not(b))), not(b))), true), true), true, is_a_theorem(implies(b, not(not(b)))), true), true, is_a_theorem(implies(implies(not(a), b), implies(not(b), a))), true), true)
11.65/11.83	= { by axiom 5 (mv_5) }
11.65/11.83	  ifeq(is_a_theorem(implies(implies(b, not(not(b))), implies(implies(not(a), b), implies(not(b), a)))), true, ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(not(implies(?, ?))), not(not(not(b)))), implies(not(not(b)), not(implies(?, ?))))), true, is_a_theorem(implies(not(not(not(b))), implies(not(not(b)), not(implies(?, ?))))), true), true, is_a_theorem(implies(implies(implies(not(not(b)), not(implies(?, ?))), not(b)), implies(not(not(not(b))), not(b)))), true), true, ifeq(is_a_theorem(implies(implies(not(not(b)), not(implies(?, ?))), not(b))), true, is_a_theorem(implies(not(not(not(b))), not(b))), true), true), true, is_a_theorem(implies(b, not(not(b)))), true), true, is_a_theorem(implies(implies(not(a), b), implies(not(b), a))), true), true)
11.65/11.83	= { by lemma 10 }
11.65/11.83	  ifeq(is_a_theorem(implies(implies(b, not(not(b))), implies(implies(not(a), b), implies(not(b), a)))), true, ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(not(not(b)), not(implies(?, ?))), not(b)), implies(not(not(not(b))), not(b)))), true), true, ifeq(is_a_theorem(implies(implies(not(not(b)), not(implies(?, ?))), not(b))), true, is_a_theorem(implies(not(not(not(b))), not(b))), true), true), true, is_a_theorem(implies(b, not(not(b)))), true), true, is_a_theorem(implies(implies(not(a), b), implies(not(b), a))), true), true)
11.65/11.83	= { by axiom 1 (ifeq_axiom) }
11.65/11.83	  ifeq(is_a_theorem(implies(implies(b, not(not(b))), implies(implies(not(a), b), implies(not(b), a)))), true, ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(not(b)), not(implies(?, ?))), not(b)), implies(not(not(not(b))), not(b)))), true, ifeq(is_a_theorem(implies(implies(not(not(b)), not(implies(?, ?))), not(b))), true, is_a_theorem(implies(not(not(not(b))), not(b))), true), true), true, is_a_theorem(implies(b, not(not(b)))), true), true, is_a_theorem(implies(implies(not(a), b), implies(not(b), a))), true), true)
11.65/11.83	= { by axiom 3 (condensed_detachment) }
11.65/11.83	  ifeq(is_a_theorem(implies(implies(b, not(not(b))), implies(implies(not(a), b), implies(not(b), a)))), true, ifeq(ifeq(true, true, is_a_theorem(implies(b, not(not(b)))), true), true, is_a_theorem(implies(implies(not(a), b), implies(not(b), a))), true), true)
11.65/11.83	= { by axiom 1 (ifeq_axiom) }
11.65/11.83	  ifeq(is_a_theorem(implies(implies(b, not(not(b))), implies(implies(not(a), b), implies(not(b), a)))), true, ifeq(is_a_theorem(implies(b, not(not(b)))), true, is_a_theorem(implies(implies(not(a), b), implies(not(b), a))), true), true)
11.65/11.83	= { by axiom 3 (condensed_detachment) }
11.65/11.83	  true
11.65/11.83	% SZS output end Proof
11.65/11.83	
11.65/11.83	RESULT: Unsatisfiable (the axioms are contradictory).
11.65/11.83	EOF