0.00/0.03	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.00/0.04	% Command    : twee %s --tstp --casc --quiet --conditional-encoding if --smaller --drop-non-horn
0.02/0.23	% Computer   : n178.star.cs.uiowa.edu
0.02/0.23	% Model      : x86_64 x86_64
0.02/0.23	% CPU        : Intel(R) Xeon(R) CPU E5-2609 0 @ 2.40GHz
0.02/0.23	% Memory     : 32218.625MB
0.02/0.23	% OS         : Linux 3.10.0-693.2.2.el7.x86_64
0.02/0.23	% CPULimit   : 300
0.02/0.23	% DateTime   : Sat Jul 14 04:57:55 CDT 2018
0.02/0.23	% CPUTime    : 
0.07/0.49	% SZS status Theorem
0.07/0.49	
0.07/0.49	% SZS output start Proof
0.07/0.49	Take the following subset of the input axioms:
0.40/0.57	  fof(equivalence_2, axiom,
0.40/0.57	      ![X, Y]: is_a_theorem(implies(equiv(X, Y), implies(Y, X)))
0.40/0.57	        <=> equivalence_2).
0.40/0.57	  fof(hilbert_equivalence_2, conjecture, equivalence_2).
0.40/0.57	  fof(hilbert_op_implies_and, axiom, op_implies_and).
0.40/0.57	  fof(hilbert_op_or, axiom, op_or).
0.40/0.57	  fof(modus_ponens, axiom,
0.40/0.57	      modus_ponens
0.40/0.57	        <=> ![X, Y]:
0.40/0.57	              ((is_a_theorem(X) & is_a_theorem(implies(X, Y)))
0.40/0.57	                 => is_a_theorem(Y))).
0.40/0.57	  fof(op_and, axiom,
0.40/0.57	      ![X, Y]: and(X, Y)=not(or(not(X), not(Y))) <= op_and).
0.40/0.57	  fof(op_equiv, axiom,
0.40/0.57	      op_equiv
0.40/0.57	        => ![X, Y]: equiv(X, Y)=and(implies(X, Y), implies(Y, X))).
0.40/0.57	  fof(op_implies_and, axiom,
0.40/0.57	      ![X, Y]: not(and(X, not(Y)))=implies(X, Y) <= op_implies_and).
0.40/0.57	  fof(op_implies_or, axiom,
0.40/0.57	      ![X, Y]: or(not(X), Y)=implies(X, Y) <= op_implies_or).
0.40/0.57	  fof(op_or, axiom,
0.40/0.57	      ![X, Y]: or(X, Y)=not(and(not(X), not(Y))) <= op_or).
0.40/0.57	  fof(or_2, axiom,
0.40/0.57	      ![X, Y]: is_a_theorem(implies(Y, or(X, Y))) <=> or_2).
0.40/0.57	  fof(principia_modus_ponens, axiom, modus_ponens).
0.40/0.57	  fof(principia_op_and, axiom, op_and).
0.40/0.57	  fof(principia_op_equiv, axiom, op_equiv).
0.40/0.57	  fof(principia_op_implies_or, axiom, op_implies_or).
0.40/0.57	  fof(principia_r2, axiom, r2).
0.40/0.57	  fof(principia_r3, axiom, r3).
0.40/0.57	  fof(r2, axiom, r2 <=> ![P, Q]: is_a_theorem(implies(Q, or(P, Q)))).
0.40/0.57	  fof(r3, axiom,
0.40/0.57	      r3 <=> ![P, Q]: is_a_theorem(implies(or(P, Q), or(Q, P)))).
0.40/0.57	
0.40/0.57	Now clausify the problem and encode Horn clauses using encoding 3 of
0.40/0.57	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
0.40/0.57	We repeatedly replace C & s=t => u=v by the two clauses:
0.40/0.57	  $$fresh(y, y, x1...xn) = u
0.40/0.57	  C => $$fresh(s, t, x1...xn) = v
0.40/0.57	where $$fresh is a fresh function symbol and x1..xn are the free
0.40/0.57	variables of u and v.
0.40/0.57	A predicate p(X) is encoded as p(X)=$$true (this is sound, because the
0.40/0.57	input problem has no model of domain size 1).
0.40/0.57	
0.40/0.57	The encoding turns the above axioms into the following unit equations and goals:
0.40/0.57	
0.40/0.57	Axiom 15 (equivalence_2): $$fresh44(X, X) = $$true.
0.40/0.57	Axiom 31 (modus_ponens_2): $$fresh28(X, X, Y, Z) = is_a_theorem(Z).
0.40/0.57	Axiom 32 (modus_ponens_2): $$fresh59(X, X, Y) = $$true.
0.40/0.57	Axiom 33 (modus_ponens_2): $$fresh60(X, X, Y, Z) = $$fresh59(is_a_theorem(Y), $$true, Z).
0.40/0.57	Axiom 37 (op_and): $$fresh24(X, X, Y, Z) = and(Y, Z).
0.40/0.57	Axiom 38 (op_equiv): $$fresh23(X, X, Y, Z) = equiv(Y, Z).
0.40/0.57	Axiom 39 (op_implies_and): $$fresh22(X, X, Y, Z) = implies(Y, Z).
0.40/0.57	Axiom 40 (op_implies_or): $$fresh21(X, X, Y, Z) = implies(Y, Z).
0.40/0.57	Axiom 41 (op_or): $$fresh20(X, X, Y, Z) = or(Y, Z).
0.40/0.57	Axiom 51 (r2_1): $$fresh10(X, X, Y, Z) = $$true.
0.40/0.57	Axiom 53 (r3_1): $$fresh8(X, X, Y, Z) = $$true.
0.40/0.57	Axiom 63 (equivalence_2_1): $$fresh43(equivalence_2, $$true, X, Y) = is_a_theorem(implies(equiv(X, Y), implies(Y, X))).
0.40/0.57	Axiom 64 (equivalence_2): $$fresh44(is_a_theorem(implies(equiv(sK52_equivalence_2_X, sK51_equivalence_2_Y), implies(sK51_equivalence_2_Y, sK52_equivalence_2_X))), $$true) = equivalence_2.
0.40/0.57	Axiom 81 (or_2_1): $$fresh16(or_2, $$true, X, Y) = is_a_theorem(implies(Y, or(X, Y))).
0.40/0.57	Axiom 84 (modus_ponens_2): $$fresh60(modus_ponens, $$true, X, Y) = $$fresh28(is_a_theorem(implies(X, Y)), $$true, X, Y).
0.40/0.57	Axiom 91 (r2_1): $$fresh10(r2, $$true, X, Y) = is_a_theorem(implies(Y, or(X, Y))).
0.40/0.57	Axiom 97 (r3_1): $$fresh8(r3, $$true, X, Y) = is_a_theorem(implies(or(X, Y), or(Y, X))).
0.40/0.57	Axiom 113 (op_or): $$fresh20(op_or, $$true, X, Y) = not(and(not(X), not(Y))).
0.40/0.57	Axiom 114 (op_and): $$fresh24(op_and, $$true, X, Y) = not(or(not(X), not(Y))).
0.40/0.57	Axiom 115 (op_implies_and): $$fresh22(op_implies_and, $$true, X, Y) = not(and(X, not(Y))).
0.40/0.57	Axiom 116 (op_implies_or): $$fresh21(op_implies_or, $$true, X, Y) = or(not(X), Y).
0.40/0.57	Axiom 117 (op_equiv): $$fresh23(op_equiv, $$true, X, Y) = and(implies(X, Y), implies(Y, X)).
0.40/0.57	Axiom 119 (principia_op_equiv): op_equiv = $$true.
0.40/0.57	Axiom 120 (principia_r2): r2 = $$true.
0.40/0.57	Axiom 122 (principia_r3): r3 = $$true.
0.40/0.57	Axiom 123 (principia_modus_ponens): modus_ponens = $$true.
0.40/0.57	Axiom 125 (principia_op_implies_or): op_implies_or = $$true.
0.40/0.57	Axiom 126 (principia_op_and): op_and = $$true.
0.40/0.57	Axiom 128 (hilbert_op_or): op_or = $$true.
0.40/0.57	Axiom 129 (hilbert_op_implies_and): op_implies_and = $$true.
0.40/0.57	
0.40/0.57	Lemma 130: or(not(X), Y) = implies(X, Y).
0.40/0.57	Proof:
0.40/0.57	  or(not(X), Y)
0.40/0.57	= { by axiom 116 (op_implies_or) }
0.40/0.57	  $$fresh21(op_implies_or, $$true, X, Y)
0.40/0.57	= { by axiom 125 (principia_op_implies_or) }
0.40/0.57	  $$fresh21($$true, $$true, X, Y)
0.40/0.57	= { by axiom 40 (op_implies_or) }
0.40/0.57	  implies(X, Y)
0.40/0.57	
0.40/0.57	Lemma 131: not(and(X, not(Y))) = implies(X, Y).
0.40/0.57	Proof:
0.40/0.57	  not(and(X, not(Y)))
0.40/0.57	= { by axiom 115 (op_implies_and) }
0.40/0.57	  $$fresh22(op_implies_and, $$true, X, Y)
0.40/0.57	= { by axiom 129 (hilbert_op_implies_and) }
0.40/0.57	  $$fresh22($$true, $$true, X, Y)
0.40/0.57	= { by axiom 39 (op_implies_and) }
0.40/0.57	  implies(X, Y)
0.40/0.57	
0.40/0.57	Lemma 132: implies(not(X), Y) = or(X, Y).
0.40/0.57	Proof:
0.40/0.57	  implies(not(X), Y)
0.40/0.57	= { by lemma 131 }
0.40/0.57	  not(and(not(X), not(Y)))
0.40/0.57	= { by axiom 113 (op_or) }
0.40/0.57	  $$fresh20(op_or, $$true, X, Y)
0.40/0.57	= { by axiom 128 (hilbert_op_or) }
0.40/0.57	  $$fresh20($$true, $$true, X, Y)
0.40/0.57	= { by axiom 41 (op_or) }
0.40/0.57	  or(X, Y)
0.40/0.57	
0.40/0.57	Lemma 133: not(implies(X, not(Y))) = and(X, Y).
0.40/0.57	Proof:
0.40/0.57	  not(implies(X, not(Y)))
0.40/0.57	= { by lemma 130 }
0.40/0.57	  not(or(not(X), not(Y)))
0.40/0.57	= { by axiom 114 (op_and) }
0.40/0.57	  $$fresh24(op_and, $$true, X, Y)
0.40/0.57	= { by axiom 126 (principia_op_and) }
0.40/0.57	  $$fresh24($$true, $$true, X, Y)
0.40/0.57	= { by axiom 37 (op_and) }
0.40/0.58	  and(X, Y)
0.40/0.58	
0.40/0.58	Lemma 134: implies(implies(X, not(Y)), Z) = or(and(X, Y), Z).
0.40/0.58	Proof:
0.40/0.58	  implies(implies(X, not(Y)), Z)
0.40/0.58	= { by lemma 130 }
0.40/0.58	  implies(or(not(X), not(Y)), Z)
0.40/0.58	= { by lemma 130 }
0.40/0.58	  or(not(or(not(X), not(Y))), Z)
0.40/0.58	= { by axiom 114 (op_and) }
0.40/0.58	  or($$fresh24(op_and, $$true, X, Y), Z)
0.40/0.58	= { by axiom 126 (principia_op_and) }
0.40/0.58	  or($$fresh24($$true, $$true, X, Y), Z)
0.40/0.58	= { by axiom 37 (op_and) }
0.40/0.58	  or(and(X, Y), Z)
0.40/0.58	
0.40/0.58	Lemma 135: is_a_theorem(implies(equiv(X, not(Y)), or(Y, X))) = $$fresh43(equivalence_2, $$true, X, not(Y)).
0.40/0.58	Proof:
0.40/0.58	  is_a_theorem(implies(equiv(X, not(Y)), or(Y, X)))
0.40/0.58	= { by lemma 132 }
0.40/0.58	  is_a_theorem(implies(equiv(X, not(Y)), implies(not(Y), X)))
0.40/0.58	= { by axiom 63 (equivalence_2_1) }
0.46/0.65	  $$fresh43(equivalence_2, $$true, X, not(Y))
0.46/0.65	
0.46/0.65	Goal 1 (hilbert_equivalence_2): equivalence_2 = $$true.
0.46/0.65	Proof:
0.46/0.65	  equivalence_2
0.46/0.65	= { by axiom 64 (equivalence_2) }
0.46/0.65	  $$fresh44(is_a_theorem(implies(equiv(sK52_equivalence_2_X, sK51_equivalence_2_Y), implies(sK51_equivalence_2_Y, sK52_equivalence_2_X))), $$true)
0.46/0.65	= { by axiom 38 (op_equiv) }
0.46/0.65	  $$fresh44(is_a_theorem(implies($$fresh23($$true, $$true, sK52_equivalence_2_X, sK51_equivalence_2_Y), implies(sK51_equivalence_2_Y, sK52_equivalence_2_X))), $$true)
0.46/0.65	= { by axiom 119 (principia_op_equiv) }
0.46/0.65	  $$fresh44(is_a_theorem(implies($$fresh23(op_equiv, $$true, sK52_equivalence_2_X, sK51_equivalence_2_Y), implies(sK51_equivalence_2_Y, sK52_equivalence_2_X))), $$true)
0.46/0.65	= { by axiom 117 (op_equiv) }
0.46/0.65	  $$fresh44(is_a_theorem(implies(and(implies(sK52_equivalence_2_X, sK51_equivalence_2_Y), implies(sK51_equivalence_2_Y, sK52_equivalence_2_X)), implies(sK51_equivalence_2_Y, sK52_equivalence_2_X))), $$true)
0.46/0.65	= { by lemma 131 }
0.46/0.65	  $$fresh44(is_a_theorem(implies(and(not(and(sK52_equivalence_2_X, not(sK51_equivalence_2_Y))), implies(sK51_equivalence_2_Y, sK52_equivalence_2_X)), implies(sK51_equivalence_2_Y, sK52_equivalence_2_X))), $$true)
0.46/0.65	= { by lemma 130 }
0.46/0.65	  $$fresh44(is_a_theorem(implies(and(not(and(sK52_equivalence_2_X, not(sK51_equivalence_2_Y))), or(not(sK51_equivalence_2_Y), sK52_equivalence_2_X)), implies(sK51_equivalence_2_Y, sK52_equivalence_2_X))), $$true)
0.46/0.65	= { by lemma 132 }
0.46/0.65	  $$fresh44(is_a_theorem(implies(and(not(and(sK52_equivalence_2_X, not(sK51_equivalence_2_Y))), implies(not(not(sK51_equivalence_2_Y)), sK52_equivalence_2_X)), implies(sK51_equivalence_2_Y, sK52_equivalence_2_X))), $$true)
0.46/0.65	= { by lemma 133 }
0.46/0.65	  $$fresh44(is_a_theorem(implies(not(implies(not(and(sK52_equivalence_2_X, not(sK51_equivalence_2_Y))), not(implies(not(not(sK51_equivalence_2_Y)), sK52_equivalence_2_X)))), implies(sK51_equivalence_2_Y, sK52_equivalence_2_X))), $$true)
0.46/0.65	= { by lemma 132 }
0.46/0.65	  $$fresh44(is_a_theorem(implies(not(or(and(sK52_equivalence_2_X, not(sK51_equivalence_2_Y)), not(implies(not(not(sK51_equivalence_2_Y)), sK52_equivalence_2_X)))), implies(sK51_equivalence_2_Y, sK52_equivalence_2_X))), $$true)
0.46/0.65	= { by lemma 134 }
0.46/0.65	  $$fresh44(is_a_theorem(implies(not(implies(implies(sK52_equivalence_2_X, not(not(sK51_equivalence_2_Y))), not(implies(not(not(sK51_equivalence_2_Y)), sK52_equivalence_2_X)))), implies(sK51_equivalence_2_Y, sK52_equivalence_2_X))), $$true)
0.46/0.65	= { by lemma 133 }
0.46/0.65	  $$fresh44(is_a_theorem(implies(and(implies(sK52_equivalence_2_X, not(not(sK51_equivalence_2_Y))), implies(not(not(sK51_equivalence_2_Y)), sK52_equivalence_2_X)), implies(sK51_equivalence_2_Y, sK52_equivalence_2_X))), $$true)
0.46/0.65	= { by axiom 117 (op_equiv) }
0.46/0.65	  $$fresh44(is_a_theorem(implies($$fresh23(op_equiv, $$true, sK52_equivalence_2_X, not(not(sK51_equivalence_2_Y))), implies(sK51_equivalence_2_Y, sK52_equivalence_2_X))), $$true)
0.46/0.65	= { by axiom 119 (principia_op_equiv) }
0.46/0.65	  $$fresh44(is_a_theorem(implies($$fresh23($$true, $$true, sK52_equivalence_2_X, not(not(sK51_equivalence_2_Y))), implies(sK51_equivalence_2_Y, sK52_equivalence_2_X))), $$true)
0.46/0.65	= { by axiom 38 (op_equiv) }
0.46/0.65	  $$fresh44(is_a_theorem(implies(equiv(sK52_equivalence_2_X, not(not(sK51_equivalence_2_Y))), implies(sK51_equivalence_2_Y, sK52_equivalence_2_X))), $$true)
0.46/0.65	= { by lemma 130 }
0.46/0.65	  $$fresh44(is_a_theorem(implies(equiv(sK52_equivalence_2_X, not(not(sK51_equivalence_2_Y))), or(not(sK51_equivalence_2_Y), sK52_equivalence_2_X))), $$true)
0.46/0.65	= { by axiom 41 (op_or) }
0.46/0.65	  $$fresh44(is_a_theorem(implies(equiv(sK52_equivalence_2_X, not(not(sK51_equivalence_2_Y))), $$fresh20($$true, $$true, not(sK51_equivalence_2_Y), sK52_equivalence_2_X))), $$true)
0.46/0.65	= { by axiom 128 (hilbert_op_or) }
0.46/0.65	  $$fresh44(is_a_theorem(implies(equiv(sK52_equivalence_2_X, not(not(sK51_equivalence_2_Y))), $$fresh20(op_or, $$true, not(sK51_equivalence_2_Y), sK52_equivalence_2_X))), $$true)
0.46/0.65	= { by axiom 113 (op_or) }
0.46/0.65	  $$fresh44(is_a_theorem(implies(equiv(sK52_equivalence_2_X, not(not(sK51_equivalence_2_Y))), not(and(not(not(sK51_equivalence_2_Y)), not(sK52_equivalence_2_X))))), $$true)
0.46/0.65	= { by lemma 130 }
0.46/0.65	  $$fresh44(is_a_theorem(or(not(equiv(sK52_equivalence_2_X, not(not(sK51_equivalence_2_Y)))), not(and(not(not(sK51_equivalence_2_Y)), not(sK52_equivalence_2_X))))), $$true)
0.46/0.65	= { by axiom 38 (op_equiv) }
0.46/0.65	  $$fresh44(is_a_theorem(or(not($$fresh23($$true, $$true, sK52_equivalence_2_X, not(not(sK51_equivalence_2_Y)))), not(and(not(not(sK51_equivalence_2_Y)), not(sK52_equivalence_2_X))))), $$true)
0.46/0.65	= { by axiom 119 (principia_op_equiv) }
0.46/0.65	  $$fresh44(is_a_theorem(or(not($$fresh23(op_equiv, $$true, sK52_equivalence_2_X, not(not(sK51_equivalence_2_Y)))), not(and(not(not(sK51_equivalence_2_Y)), not(sK52_equivalence_2_X))))), $$true)
0.46/0.65	= { by axiom 117 (op_equiv) }
0.46/0.65	  $$fresh44(is_a_theorem(or(not(and(implies(sK52_equivalence_2_X, not(not(sK51_equivalence_2_Y))), implies(not(not(sK51_equivalence_2_Y)), sK52_equivalence_2_X))), not(and(not(not(sK51_equivalence_2_Y)), not(sK52_equivalence_2_X))))), $$true)
0.46/0.65	= { by lemma 131 }
0.46/0.65	  $$fresh44(is_a_theorem(or(not(and(implies(sK52_equivalence_2_X, not(not(sK51_equivalence_2_Y))), not(and(not(not(sK51_equivalence_2_Y)), not(sK52_equivalence_2_X))))), not(and(not(not(sK51_equivalence_2_Y)), not(sK52_equivalence_2_X))))), $$true)
0.46/0.65	= { by lemma 131 }
0.46/0.65	  $$fresh44(is_a_theorem(or(implies(implies(sK52_equivalence_2_X, not(not(sK51_equivalence_2_Y))), and(not(not(sK51_equivalence_2_Y)), not(sK52_equivalence_2_X))), not(and(not(not(sK51_equivalence_2_Y)), not(sK52_equivalence_2_X))))), $$true)
0.46/0.65	= { by lemma 134 }
0.46/0.65	  $$fresh44(is_a_theorem(or(or(and(sK52_equivalence_2_X, not(sK51_equivalence_2_Y)), and(not(not(sK51_equivalence_2_Y)), not(sK52_equivalence_2_X))), not(and(not(not(sK51_equivalence_2_Y)), not(sK52_equivalence_2_X))))), $$true)
0.46/0.65	= { by axiom 31 (modus_ponens_2) }
0.46/0.65	  $$fresh44($$fresh28($$true, $$true, implies(and(not(not(sK51_equivalence_2_Y)), not(sK52_equivalence_2_X)), or(and(sK52_equivalence_2_X, not(sK51_equivalence_2_Y)), and(not(not(sK51_equivalence_2_Y)), not(sK52_equivalence_2_X)))), or(or(and(sK52_equivalence_2_X, not(sK51_equivalence_2_Y)), and(not(not(sK51_equivalence_2_Y)), not(sK52_equivalence_2_X))), not(and(not(not(sK51_equivalence_2_Y)), not(sK52_equivalence_2_X))))), $$true)
0.46/0.65	= { by axiom 53 (r3_1) }
0.46/0.65	  $$fresh44($$fresh28($$fresh8($$true, $$true, not(and(not(not(sK51_equivalence_2_Y)), not(sK52_equivalence_2_X))), or(and(sK52_equivalence_2_X, not(sK51_equivalence_2_Y)), and(not(not(sK51_equivalence_2_Y)), not(sK52_equivalence_2_X)))), $$true, implies(and(not(not(sK51_equivalence_2_Y)), not(sK52_equivalence_2_X)), or(and(sK52_equivalence_2_X, not(sK51_equivalence_2_Y)), and(not(not(sK51_equivalence_2_Y)), not(sK52_equivalence_2_X)))), or(or(and(sK52_equivalence_2_X, not(sK51_equivalence_2_Y)), and(not(not(sK51_equivalence_2_Y)), not(sK52_equivalence_2_X))), not(and(not(not(sK51_equivalence_2_Y)), not(sK52_equivalence_2_X))))), $$true)
0.46/0.65	= { by axiom 122 (principia_r3) }
0.46/0.65	  $$fresh44($$fresh28($$fresh8(r3, $$true, not(and(not(not(sK51_equivalence_2_Y)), not(sK52_equivalence_2_X))), or(and(sK52_equivalence_2_X, not(sK51_equivalence_2_Y)), and(not(not(sK51_equivalence_2_Y)), not(sK52_equivalence_2_X)))), $$true, implies(and(not(not(sK51_equivalence_2_Y)), not(sK52_equivalence_2_X)), or(and(sK52_equivalence_2_X, not(sK51_equivalence_2_Y)), and(not(not(sK51_equivalence_2_Y)), not(sK52_equivalence_2_X)))), or(or(and(sK52_equivalence_2_X, not(sK51_equivalence_2_Y)), and(not(not(sK51_equivalence_2_Y)), not(sK52_equivalence_2_X))), not(and(not(not(sK51_equivalence_2_Y)), not(sK52_equivalence_2_X))))), $$true)
0.46/0.65	= { by axiom 97 (r3_1) }
0.46/0.65	  $$fresh44($$fresh28(is_a_theorem(implies(or(not(and(not(not(sK51_equivalence_2_Y)), not(sK52_equivalence_2_X))), or(and(sK52_equivalence_2_X, not(sK51_equivalence_2_Y)), and(not(not(sK51_equivalence_2_Y)), not(sK52_equivalence_2_X)))), or(or(and(sK52_equivalence_2_X, not(sK51_equivalence_2_Y)), and(not(not(sK51_equivalence_2_Y)), not(sK52_equivalence_2_X))), not(and(not(not(sK51_equivalence_2_Y)), not(sK52_equivalence_2_X)))))), $$true, implies(and(not(not(sK51_equivalence_2_Y)), not(sK52_equivalence_2_X)), or(and(sK52_equivalence_2_X, not(sK51_equivalence_2_Y)), and(not(not(sK51_equivalence_2_Y)), not(sK52_equivalence_2_X)))), or(or(and(sK52_equivalence_2_X, not(sK51_equivalence_2_Y)), and(not(not(sK51_equivalence_2_Y)), not(sK52_equivalence_2_X))), not(and(not(not(sK51_equivalence_2_Y)), not(sK52_equivalence_2_X))))), $$true)
0.46/0.65	= { by lemma 130 }
0.46/0.65	  $$fresh44($$fresh28(is_a_theorem(implies(implies(and(not(not(sK51_equivalence_2_Y)), not(sK52_equivalence_2_X)), or(and(sK52_equivalence_2_X, not(sK51_equivalence_2_Y)), and(not(not(sK51_equivalence_2_Y)), not(sK52_equivalence_2_X)))), or(or(and(sK52_equivalence_2_X, not(sK51_equivalence_2_Y)), and(not(not(sK51_equivalence_2_Y)), not(sK52_equivalence_2_X))), not(and(not(not(sK51_equivalence_2_Y)), not(sK52_equivalence_2_X)))))), $$true, implies(and(not(not(sK51_equivalence_2_Y)), not(sK52_equivalence_2_X)), or(and(sK52_equivalence_2_X, not(sK51_equivalence_2_Y)), and(not(not(sK51_equivalence_2_Y)), not(sK52_equivalence_2_X)))), or(or(and(sK52_equivalence_2_X, not(sK51_equivalence_2_Y)), and(not(not(sK51_equivalence_2_Y)), not(sK52_equivalence_2_X))), not(and(not(not(sK51_equivalence_2_Y)), not(sK52_equivalence_2_X))))), $$true)
0.46/0.65	= { by axiom 84 (modus_ponens_2) }
0.46/0.65	  $$fresh44($$fresh60(modus_ponens, $$true, implies(and(not(not(sK51_equivalence_2_Y)), not(sK52_equivalence_2_X)), or(and(sK52_equivalence_2_X, not(sK51_equivalence_2_Y)), and(not(not(sK51_equivalence_2_Y)), not(sK52_equivalence_2_X)))), or(or(and(sK52_equivalence_2_X, not(sK51_equivalence_2_Y)), and(not(not(sK51_equivalence_2_Y)), not(sK52_equivalence_2_X))), not(and(not(not(sK51_equivalence_2_Y)), not(sK52_equivalence_2_X))))), $$true)
0.46/0.65	= { by axiom 123 (principia_modus_ponens) }
0.46/0.65	  $$fresh44($$fresh60($$true, $$true, implies(and(not(not(sK51_equivalence_2_Y)), not(sK52_equivalence_2_X)), or(and(sK52_equivalence_2_X, not(sK51_equivalence_2_Y)), and(not(not(sK51_equivalence_2_Y)), not(sK52_equivalence_2_X)))), or(or(and(sK52_equivalence_2_X, not(sK51_equivalence_2_Y)), and(not(not(sK51_equivalence_2_Y)), not(sK52_equivalence_2_X))), not(and(not(not(sK51_equivalence_2_Y)), not(sK52_equivalence_2_X))))), $$true)
0.46/0.65	= { by axiom 33 (modus_ponens_2) }
0.46/0.65	  $$fresh44($$fresh59(is_a_theorem(implies(and(not(not(sK51_equivalence_2_Y)), not(sK52_equivalence_2_X)), or(and(sK52_equivalence_2_X, not(sK51_equivalence_2_Y)), and(not(not(sK51_equivalence_2_Y)), not(sK52_equivalence_2_X))))), $$true, or(or(and(sK52_equivalence_2_X, not(sK51_equivalence_2_Y)), and(not(not(sK51_equivalence_2_Y)), not(sK52_equivalence_2_X))), not(and(not(not(sK51_equivalence_2_Y)), not(sK52_equivalence_2_X))))), $$true)
0.46/0.65	= { by axiom 91 (r2_1) }
0.46/0.66	  $$fresh44($$fresh59($$fresh10(r2, $$true, and(sK52_equivalence_2_X, not(sK51_equivalence_2_Y)), and(not(not(sK51_equivalence_2_Y)), not(sK52_equivalence_2_X))), $$true, or(or(and(sK52_equivalence_2_X, not(sK51_equivalence_2_Y)), and(not(not(sK51_equivalence_2_Y)), not(sK52_equivalence_2_X))), not(and(not(not(sK51_equivalence_2_Y)), not(sK52_equivalence_2_X))))), $$true)
0.46/0.66	= { by axiom 120 (principia_r2) }
0.46/0.66	  $$fresh44($$fresh59($$fresh10($$true, $$true, and(sK52_equivalence_2_X, not(sK51_equivalence_2_Y)), and(not(not(sK51_equivalence_2_Y)), not(sK52_equivalence_2_X))), $$true, or(or(and(sK52_equivalence_2_X, not(sK51_equivalence_2_Y)), and(not(not(sK51_equivalence_2_Y)), not(sK52_equivalence_2_X))), not(and(not(not(sK51_equivalence_2_Y)), not(sK52_equivalence_2_X))))), $$true)
0.46/0.66	= { by axiom 51 (r2_1) }
0.46/0.66	  $$fresh44($$fresh59($$true, $$true, or(or(and(sK52_equivalence_2_X, not(sK51_equivalence_2_Y)), and(not(not(sK51_equivalence_2_Y)), not(sK52_equivalence_2_X))), not(and(not(not(sK51_equivalence_2_Y)), not(sK52_equivalence_2_X))))), $$true)
0.46/0.66	= { by axiom 32 (modus_ponens_2) }
0.46/0.66	  $$fresh44($$true, $$true)
0.46/0.66	= { by axiom 15 (equivalence_2) }
0.46/0.66	  $$true
0.46/0.66	% SZS output end Proof
0.46/0.66	
0.46/0.66	RESULT: Theorem (the conjecture is true).
0.46/0.66	EOF