0.11/0.12	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.11/0.12	% Command    : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
0.12/0.33	% Computer   : n006.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   : 1200
0.12/0.33	% WCLimit    : 120
0.12/0.33	% DateTime   : Tue Jul 13 16:12:11 EDT 2021
0.12/0.34	% CPUTime    : 
336.87/42.93	% SZS status Theorem
336.87/42.93	
336.87/42.98	% SZS output start Proof
336.87/42.98	Take the following subset of the input axioms:
336.87/42.98	  fof(cn1, axiom, ![P, Q, R]: is_a_theorem(implies(implies(P, Q), implies(implies(Q, R), implies(P, R)))) <=> cn1).
336.87/42.98	  fof(kn3, axiom, kn3 <=> ![P, Q, R]: is_a_theorem(implies(implies(P, Q), implies(not(and(Q, R)), not(and(R, P)))))).
336.87/42.98	  fof(modus_ponens, axiom, ![X, Y]: (is_a_theorem(Y) <= (is_a_theorem(implies(X, Y)) & is_a_theorem(X))) <=> modus_ponens).
336.87/42.98	  fof(op_and, axiom, op_and => ![X, Y]: not(or(not(X), not(Y)))=and(X, Y)).
336.87/42.98	  fof(op_equiv, axiom, op_equiv => ![X, Y]: and(implies(X, Y), implies(Y, X))=equiv(X, Y)).
336.87/42.98	  fof(op_implies_and, axiom, op_implies_and => ![X, Y]: implies(X, Y)=not(and(X, not(Y)))).
336.87/42.98	  fof(op_implies_or, axiom, ![X, Y]: implies(X, Y)=or(not(X), Y) <= op_implies_or).
336.87/42.98	  fof(op_or, axiom, op_or => ![X, Y]: or(X, Y)=not(and(not(X), not(Y)))).
336.87/42.98	  fof(principia_modus_ponens, axiom, modus_ponens).
336.87/42.98	  fof(principia_op_and, axiom, op_and).
336.87/42.98	  fof(principia_op_equiv, axiom, op_equiv).
336.87/42.98	  fof(principia_op_implies_or, axiom, op_implies_or).
336.87/42.98	  fof(principia_r1, axiom, r1).
336.87/42.98	  fof(principia_r2, axiom, r2).
336.87/42.98	  fof(principia_r3, axiom, r3).
336.87/42.98	  fof(principia_r4, axiom, r4).
336.87/42.98	  fof(principia_r5, axiom, r5).
336.87/42.98	  fof(r1, axiom, ![P]: is_a_theorem(implies(or(P, P), P)) <=> r1).
336.87/42.98	  fof(r2, axiom, r2 <=> ![P, Q]: is_a_theorem(implies(Q, or(P, Q)))).
336.87/42.98	  fof(r3, axiom, r3 <=> ![P, Q]: is_a_theorem(implies(or(P, Q), or(Q, P)))).
336.87/42.98	  fof(r4, axiom, r4 <=> ![P, Q, R]: is_a_theorem(implies(or(P, or(Q, R)), or(Q, or(P, R))))).
336.87/42.98	  fof(r5, axiom, ![P, Q, R]: is_a_theorem(implies(implies(Q, R), implies(or(P, Q), or(P, R)))) <=> r5).
336.87/42.98	  fof(rosser_kn3, conjecture, kn3).
336.87/42.98	  fof(rosser_op_implies_and, axiom, op_implies_and).
336.87/42.98	  fof(rosser_op_or, axiom, op_or).
336.87/42.98	  fof(substitution_of_equivalents, axiom, ![X, Y]: (is_a_theorem(equiv(X, Y)) => X=Y) <=> substitution_of_equivalents).
336.87/42.98	  fof(substitution_of_equivalents, axiom, substitution_of_equivalents).
336.87/42.98	
336.87/42.98	Now clausify the problem and encode Horn clauses using encoding 3 of
336.87/42.98	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
336.87/42.98	We repeatedly replace C & s=t => u=v by the two clauses:
336.87/42.98	  fresh(y, y, x1...xn) = u
336.87/42.98	  C => fresh(s, t, x1...xn) = v
336.87/42.98	where fresh is a fresh function symbol and x1..xn are the free
336.87/42.98	variables of u and v.
336.87/42.98	A predicate p(X) is encoded as p(X)=true (this is sound, because the
336.87/42.98	input problem has no model of domain size 1).
336.87/42.98	
336.87/42.98	The encoding turns the above axioms into the following unit equations and goals:
336.87/42.98	
336.87/42.98	Axiom 1 (principia_modus_ponens): modus_ponens = true.
336.87/42.98	Axiom 2 (principia_r1): r1 = true.
336.87/42.98	Axiom 3 (principia_r3): r3 = true.
336.87/42.98	Axiom 4 (principia_r5): r5 = true.
336.87/42.98	Axiom 5 (principia_r4): r4 = true.
336.87/42.98	Axiom 6 (principia_r2): r2 = true.
336.87/42.98	Axiom 7 (substitution_of_equivalents): substitution_of_equivalents = true.
336.87/42.98	Axiom 8 (principia_op_equiv): op_equiv = true.
336.87/42.98	Axiom 9 (principia_op_implies_or): op_implies_or = true.
336.87/42.98	Axiom 10 (rosser_op_implies_and): op_implies_and = true.
336.87/42.98	Axiom 11 (rosser_op_or): op_or = true.
336.87/42.98	Axiom 12 (principia_op_and): op_and = true.
336.87/42.98	Axiom 13 (kn3): fresh30(X, X) = true.
336.87/42.98	Axiom 14 (modus_ponens_2): fresh60(X, X, Y) = true.
336.87/42.98	Axiom 15 (r1_1): fresh12(X, X, Y) = true.
336.87/42.98	Axiom 16 (substitution_of_equivalents_2): fresh(X, X, Y, Z) = Z.
336.87/42.98	Axiom 17 (modus_ponens_2): fresh59(X, X, Y, Z) = fresh60(is_a_theorem(Y), true, Z).
336.87/42.98	Axiom 18 (modus_ponens_2): fresh28(X, X, Y, Z) = is_a_theorem(Z).
336.87/42.98	Axiom 19 (op_and): fresh24(X, X, Y, Z) = and(Y, Z).
336.87/42.98	Axiom 20 (op_equiv): fresh23(X, X, Y, Z) = equiv(Y, Z).
336.87/42.98	Axiom 21 (op_implies_and): fresh22(X, X, Y, Z) = implies(Y, Z).
336.87/42.98	Axiom 22 (op_implies_and): fresh22(op_implies_and, true, X, Y) = not(and(X, not(Y))).
336.87/42.98	Axiom 23 (op_implies_or): fresh21(X, X, Y, Z) = implies(Y, Z).
336.87/42.98	Axiom 24 (op_implies_or): fresh21(op_implies_or, true, X, Y) = or(not(X), Y).
336.87/42.98	Axiom 25 (op_or): fresh20(X, X, Y, Z) = or(Y, Z).
336.87/42.98	Axiom 26 (r2_1): fresh10(X, X, Y, Z) = true.
336.87/42.98	Axiom 27 (r3_1): fresh8(X, X, Y, Z) = true.
336.87/42.98	Axiom 28 (substitution_of_equivalents_2): fresh2(X, X, Y, Z) = Y.
336.87/42.98	Axiom 29 (r2_1): fresh10(r2, true, X, Y) = is_a_theorem(implies(Y, or(X, Y))).
336.87/42.98	Axiom 30 (r1_1): fresh12(r1, true, X) = is_a_theorem(implies(or(X, X), X)).
336.87/42.98	Axiom 31 (op_and): fresh24(op_and, true, X, Y) = not(or(not(X), not(Y))).
336.87/42.98	Axiom 32 (op_or): fresh20(op_or, true, X, Y) = not(and(not(X), not(Y))).
336.87/42.98	Axiom 33 (r4_1): fresh6(X, X, Y, Z, W) = true.
336.87/42.98	Axiom 34 (r5_1): fresh4(X, X, Y, Z, W) = true.
336.87/42.98	Axiom 35 (op_equiv): fresh23(op_equiv, true, X, Y) = and(implies(X, Y), implies(Y, X)).
336.87/42.98	Axiom 36 (r3_1): fresh8(r3, true, X, Y) = is_a_theorem(implies(or(X, Y), or(Y, X))).
336.87/42.98	Axiom 37 (substitution_of_equivalents_2): fresh2(substitution_of_equivalents, true, X, Y) = fresh(is_a_theorem(equiv(X, Y)), true, X, Y).
336.87/42.98	Axiom 38 (modus_ponens_2): fresh59(modus_ponens, true, X, Y) = fresh28(is_a_theorem(implies(X, Y)), true, X, Y).
336.87/42.98	Axiom 39 (cn1_1): fresh51(cn1, true, X, Y, Z) = is_a_theorem(implies(implies(X, Y), implies(implies(Y, Z), implies(X, Z)))).
336.87/42.98	Axiom 40 (r5_1): fresh4(r5, true, X, Y, Z) = is_a_theorem(implies(implies(Y, Z), implies(or(X, Y), or(X, Z)))).
336.87/42.98	Axiom 41 (r4_1): fresh6(r4, true, X, Y, Z) = is_a_theorem(implies(or(X, or(Y, Z)), or(Y, or(X, Z)))).
336.87/42.98	Axiom 42 (kn3_1): fresh29(kn3, true, X, Y, Z) = is_a_theorem(implies(implies(X, Y), implies(not(and(Y, Z)), not(and(Z, X))))).
336.87/42.98	Axiom 43 (kn3): fresh30(is_a_theorem(implies(implies(p7, q5), implies(not(and(q5, r7)), not(and(r7, p7))))), true) = kn3.
336.87/42.98	
336.87/42.98	Lemma 44: not(and(X, not(Y))) = implies(X, Y).
336.87/42.98	Proof:
336.87/42.98	  not(and(X, not(Y)))
336.87/42.98	= { by axiom 22 (op_implies_and) R->L }
336.87/42.98	  fresh22(op_implies_and, true, X, Y)
336.87/42.98	= { by axiom 10 (rosser_op_implies_and) }
336.87/42.98	  fresh22(true, true, X, Y)
336.87/42.98	= { by axiom 21 (op_implies_and) }
336.87/42.98	  implies(X, Y)
336.87/42.98	
336.87/42.98	Lemma 45: implies(not(X), Y) = or(X, Y).
336.87/42.98	Proof:
336.87/42.98	  implies(not(X), Y)
336.87/42.98	= { by lemma 44 R->L }
336.87/42.98	  not(and(not(X), not(Y)))
336.87/42.98	= { by axiom 32 (op_or) R->L }
336.87/42.98	  fresh20(op_or, true, X, Y)
336.87/42.98	= { by axiom 11 (rosser_op_or) }
336.87/42.98	  fresh20(true, true, X, Y)
336.87/42.98	= { by axiom 25 (op_or) }
336.87/42.98	  or(X, Y)
336.87/42.98	
336.87/42.98	Lemma 46: or(not(X), Y) = implies(X, Y).
336.87/42.98	Proof:
336.87/42.98	  or(not(X), Y)
336.87/42.98	= { by axiom 24 (op_implies_or) R->L }
336.87/42.98	  fresh21(op_implies_or, true, X, Y)
336.87/42.98	= { by axiom 9 (principia_op_implies_or) }
336.87/42.98	  fresh21(true, true, X, Y)
336.87/42.98	= { by axiom 23 (op_implies_or) }
336.87/42.98	  implies(X, Y)
336.87/42.98	
336.87/42.98	Lemma 47: not(implies(X, not(Y))) = and(X, Y).
336.87/42.98	Proof:
336.87/42.98	  not(implies(X, not(Y)))
336.87/42.98	= { by lemma 46 R->L }
336.87/42.98	  not(or(not(X), not(Y)))
336.87/42.98	= { by axiom 31 (op_and) R->L }
336.87/42.98	  fresh24(op_and, true, X, Y)
336.87/42.98	= { by axiom 12 (principia_op_and) }
336.87/42.98	  fresh24(true, true, X, Y)
336.87/42.98	= { by axiom 19 (op_and) }
336.87/42.98	  and(X, Y)
336.87/42.98	
336.87/42.98	Lemma 48: and(X, implies(Y, not(Z))) = not(implies(X, and(Y, Z))).
336.87/42.98	Proof:
336.87/42.98	  and(X, implies(Y, not(Z)))
336.87/42.98	= { by lemma 47 R->L }
336.87/42.98	  not(implies(X, not(implies(Y, not(Z)))))
336.87/42.98	= { by lemma 47 }
336.87/42.98	  not(implies(X, and(Y, Z)))
336.87/42.98	
336.87/42.98	Lemma 49: implies(implies(X, not(Y)), Z) = or(and(X, Y), Z).
336.87/42.98	Proof:
336.87/42.98	  implies(implies(X, not(Y)), Z)
336.87/42.98	= { by lemma 46 R->L }
336.87/42.98	  or(not(implies(X, not(Y))), Z)
336.87/42.98	= { by lemma 47 }
336.87/42.98	  or(and(X, Y), Z)
336.87/42.98	
336.87/42.98	Lemma 50: is_a_theorem(implies(or(X, X), X)) = true.
336.87/42.98	Proof:
336.87/42.98	  is_a_theorem(implies(or(X, X), X))
336.87/42.98	= { by axiom 30 (r1_1) R->L }
336.87/42.98	  fresh12(r1, true, X)
336.87/42.98	= { by axiom 2 (principia_r1) }
336.87/42.98	  fresh12(true, true, X)
336.87/42.98	= { by axiom 15 (r1_1) }
336.87/42.98	  true
336.87/42.98	
336.87/42.98	Lemma 51: fresh28(is_a_theorem(implies(X, Y)), true, X, Y) = fresh60(is_a_theorem(X), true, Y).
336.87/42.98	Proof:
336.87/42.98	  fresh28(is_a_theorem(implies(X, Y)), true, X, Y)
336.87/42.98	= { by axiom 38 (modus_ponens_2) R->L }
336.87/42.98	  fresh59(modus_ponens, true, X, Y)
336.87/42.98	= { by axiom 1 (principia_modus_ponens) }
336.87/42.98	  fresh59(true, true, X, Y)
336.87/42.98	= { by axiom 17 (modus_ponens_2) }
336.87/42.98	  fresh60(is_a_theorem(X), true, Y)
336.87/42.98	
336.87/42.98	Lemma 52: is_a_theorem(implies(or(X, Y), or(Y, X))) = true.
336.87/42.98	Proof:
336.87/42.98	  is_a_theorem(implies(or(X, Y), or(Y, X)))
336.87/42.98	= { by axiom 36 (r3_1) R->L }
336.87/42.98	  fresh8(r3, true, X, Y)
336.87/42.98	= { by axiom 3 (principia_r3) }
336.87/42.98	  fresh8(true, true, X, Y)
336.87/42.98	= { by axiom 27 (r3_1) }
336.87/42.98	  true
336.87/42.98	
336.87/42.98	Lemma 53: is_a_theorem(implies(or(X, or(Y, Z)), or(Y, or(X, Z)))) = true.
336.87/42.98	Proof:
336.87/42.98	  is_a_theorem(implies(or(X, or(Y, Z)), or(Y, or(X, Z))))
336.87/42.98	= { by axiom 41 (r4_1) R->L }
336.87/42.98	  fresh6(r4, true, X, Y, Z)
336.87/42.98	= { by axiom 5 (principia_r4) }
336.87/42.98	  fresh6(true, true, X, Y, Z)
336.87/42.98	= { by axiom 33 (r4_1) }
336.87/42.98	  true
336.87/42.98	
336.87/42.98	Lemma 54: not(not(X)) = X.
336.87/42.98	Proof:
336.87/42.98	  not(not(X))
336.87/42.98	= { by axiom 28 (substitution_of_equivalents_2) R->L }
336.87/42.98	  fresh2(true, true, not(not(X)), X)
336.87/42.98	= { by axiom 7 (substitution_of_equivalents) R->L }
336.87/42.98	  fresh2(substitution_of_equivalents, true, not(not(X)), X)
336.87/42.98	= { by axiom 37 (substitution_of_equivalents_2) }
336.87/42.98	  fresh(is_a_theorem(equiv(not(not(X)), X)), true, not(not(X)), X)
336.87/42.98	= { by axiom 20 (op_equiv) R->L }
336.87/42.98	  fresh(is_a_theorem(fresh23(true, true, not(not(X)), X)), true, not(not(X)), X)
336.87/42.98	= { by axiom 8 (principia_op_equiv) R->L }
336.87/42.99	  fresh(is_a_theorem(fresh23(op_equiv, true, not(not(X)), X)), true, not(not(X)), X)
336.87/42.99	= { by axiom 35 (op_equiv) }
336.87/42.99	  fresh(is_a_theorem(and(implies(not(not(X)), X), implies(X, not(not(X))))), true, not(not(X)), X)
336.87/42.99	= { by lemma 45 }
336.87/42.99	  fresh(is_a_theorem(and(or(not(X), X), implies(X, not(not(X))))), true, not(not(X)), X)
336.87/42.99	= { by lemma 48 }
336.87/42.99	  fresh(is_a_theorem(not(implies(or(not(X), X), and(X, not(X))))), true, not(not(X)), X)
336.87/42.99	= { by lemma 46 }
336.87/42.99	  fresh(is_a_theorem(not(implies(implies(X, X), and(X, not(X))))), true, not(not(X)), X)
336.87/42.99	= { by lemma 48 R->L }
336.87/42.99	  fresh(is_a_theorem(and(implies(X, X), implies(X, not(not(X))))), true, not(not(X)), X)
336.87/42.99	= { by lemma 44 R->L }
336.87/42.99	  fresh(is_a_theorem(and(not(and(X, not(X))), implies(X, not(not(X))))), true, not(not(X)), X)
336.87/42.99	= { by lemma 47 R->L }
336.87/42.99	  fresh(is_a_theorem(not(implies(not(and(X, not(X))), not(implies(X, not(not(X))))))), true, not(not(X)), X)
336.87/42.99	= { by lemma 45 }
336.87/42.99	  fresh(is_a_theorem(not(or(and(X, not(X)), not(implies(X, not(not(X))))))), true, not(not(X)), X)
336.87/42.99	= { by lemma 49 R->L }
336.87/42.99	  fresh(is_a_theorem(not(implies(implies(X, not(not(X))), not(implies(X, not(not(X))))))), true, not(not(X)), X)
337.50/42.99	= { by lemma 47 }
337.50/42.99	  fresh(is_a_theorem(and(implies(X, not(not(X))), implies(X, not(not(X))))), true, not(not(X)), X)
337.50/42.99	= { by axiom 18 (modus_ponens_2) R->L }
337.50/42.99	  fresh(fresh28(true, true, implies(X, not(not(X))), and(implies(X, not(not(X))), implies(X, not(not(X))))), true, not(not(X)), X)
337.50/42.99	= { by axiom 14 (modus_ponens_2) R->L }
337.50/42.99	  fresh(fresh28(fresh60(true, true, or(not(implies(X, not(not(X)))), and(implies(X, not(not(X))), implies(X, not(not(X)))))), true, implies(X, not(not(X))), and(implies(X, not(not(X))), implies(X, not(not(X))))), true, not(not(X)), X)
337.50/42.99	= { by lemma 50 R->L }
337.50/42.99	  fresh(fresh28(fresh60(is_a_theorem(implies(or(not(implies(X, not(not(X)))), not(implies(X, not(not(X))))), not(implies(X, not(not(X)))))), true, or(not(implies(X, not(not(X)))), and(implies(X, not(not(X))), implies(X, not(not(X)))))), true, implies(X, not(not(X))), and(implies(X, not(not(X))), implies(X, not(not(X))))), true, not(not(X)), X)
337.50/42.99	= { by lemma 46 }
337.50/42.99	  fresh(fresh28(fresh60(is_a_theorem(implies(implies(implies(X, not(not(X))), not(implies(X, not(not(X))))), not(implies(X, not(not(X)))))), true, or(not(implies(X, not(not(X)))), and(implies(X, not(not(X))), implies(X, not(not(X)))))), true, implies(X, not(not(X))), and(implies(X, not(not(X))), implies(X, not(not(X))))), true, not(not(X)), X)
337.50/42.99	= { by lemma 49 }
337.50/42.99	  fresh(fresh28(fresh60(is_a_theorem(or(and(implies(X, not(not(X))), implies(X, not(not(X)))), not(implies(X, not(not(X)))))), true, or(not(implies(X, not(not(X)))), and(implies(X, not(not(X))), implies(X, not(not(X)))))), true, implies(X, not(not(X))), and(implies(X, not(not(X))), implies(X, not(not(X))))), true, not(not(X)), X)
337.50/42.99	= { by lemma 51 R->L }
337.50/42.99	  fresh(fresh28(fresh28(is_a_theorem(implies(or(and(implies(X, not(not(X))), implies(X, not(not(X)))), not(implies(X, not(not(X))))), or(not(implies(X, not(not(X)))), and(implies(X, not(not(X))), implies(X, not(not(X))))))), true, or(and(implies(X, not(not(X))), implies(X, not(not(X)))), not(implies(X, not(not(X))))), or(not(implies(X, not(not(X)))), and(implies(X, not(not(X))), implies(X, not(not(X)))))), true, implies(X, not(not(X))), and(implies(X, not(not(X))), implies(X, not(not(X))))), true, not(not(X)), X)
337.50/42.99	= { by lemma 52 }
337.50/42.99	  fresh(fresh28(fresh28(true, true, or(and(implies(X, not(not(X))), implies(X, not(not(X)))), not(implies(X, not(not(X))))), or(not(implies(X, not(not(X)))), and(implies(X, not(not(X))), implies(X, not(not(X)))))), true, implies(X, not(not(X))), and(implies(X, not(not(X))), implies(X, not(not(X))))), true, not(not(X)), X)
337.50/42.99	= { by axiom 18 (modus_ponens_2) }
337.50/42.99	  fresh(fresh28(is_a_theorem(or(not(implies(X, not(not(X)))), and(implies(X, not(not(X))), implies(X, not(not(X)))))), true, implies(X, not(not(X))), and(implies(X, not(not(X))), implies(X, not(not(X))))), true, not(not(X)), X)
337.50/42.99	= { by lemma 46 }
337.50/42.99	  fresh(fresh28(is_a_theorem(implies(implies(X, not(not(X))), and(implies(X, not(not(X))), implies(X, not(not(X)))))), true, implies(X, not(not(X))), and(implies(X, not(not(X))), implies(X, not(not(X))))), true, not(not(X)), X)
337.50/42.99	= { by lemma 51 }
337.50/42.99	  fresh(fresh60(is_a_theorem(implies(X, not(not(X)))), true, and(implies(X, not(not(X))), implies(X, not(not(X))))), true, not(not(X)), X)
337.50/42.99	= { by lemma 46 R->L }
337.50/42.99	  fresh(fresh60(is_a_theorem(or(not(X), not(not(X)))), true, and(implies(X, not(not(X))), implies(X, not(not(X))))), true, not(not(X)), X)
337.50/42.99	= { by axiom 18 (modus_ponens_2) R->L }
337.50/42.99	  fresh(fresh60(fresh28(true, true, or(or(not(X), not(not(X))), or(not(X), not(not(X)))), or(not(X), not(not(X)))), true, and(implies(X, not(not(X))), implies(X, not(not(X))))), true, not(not(X)), X)
337.50/42.99	= { by lemma 50 R->L }
337.50/42.99	  fresh(fresh60(fresh28(is_a_theorem(implies(or(or(not(X), not(not(X))), or(not(X), not(not(X)))), or(not(X), not(not(X))))), true, or(or(not(X), not(not(X))), or(not(X), not(not(X)))), or(not(X), not(not(X)))), true, and(implies(X, not(not(X))), implies(X, not(not(X))))), true, not(not(X)), X)
337.50/42.99	= { by lemma 51 }
337.50/42.99	  fresh(fresh60(fresh60(is_a_theorem(or(or(not(X), not(not(X))), or(not(X), not(not(X))))), true, or(not(X), not(not(X)))), true, and(implies(X, not(not(X))), implies(X, not(not(X))))), true, not(not(X)), X)
337.50/42.99	= { by axiom 18 (modus_ponens_2) R->L }
337.50/42.99	  fresh(fresh60(fresh60(fresh28(true, true, or(not(X), or(or(not(X), not(not(X))), not(not(X)))), or(or(not(X), not(not(X))), or(not(X), not(not(X))))), true, or(not(X), not(not(X)))), true, and(implies(X, not(not(X))), implies(X, not(not(X))))), true, not(not(X)), X)
337.50/42.99	= { by lemma 53 R->L }
337.50/42.99	  fresh(fresh60(fresh60(fresh28(is_a_theorem(implies(or(not(X), or(or(not(X), not(not(X))), not(not(X)))), or(or(not(X), not(not(X))), or(not(X), not(not(X)))))), true, or(not(X), or(or(not(X), not(not(X))), not(not(X)))), or(or(not(X), not(not(X))), or(not(X), not(not(X))))), true, or(not(X), not(not(X)))), true, and(implies(X, not(not(X))), implies(X, not(not(X))))), true, not(not(X)), X)
337.50/42.99	= { by lemma 51 }
337.50/42.99	  fresh(fresh60(fresh60(fresh60(is_a_theorem(or(not(X), or(or(not(X), not(not(X))), not(not(X))))), true, or(or(not(X), not(not(X))), or(not(X), not(not(X))))), true, or(not(X), not(not(X)))), true, and(implies(X, not(not(X))), implies(X, not(not(X))))), true, not(not(X)), X)
337.50/42.99	= { by lemma 45 R->L }
337.50/42.99	  fresh(fresh60(fresh60(fresh60(is_a_theorem(implies(not(not(X)), or(or(not(X), not(not(X))), not(not(X))))), true, or(or(not(X), not(not(X))), or(not(X), not(not(X))))), true, or(not(X), not(not(X)))), true, and(implies(X, not(not(X))), implies(X, not(not(X))))), true, not(not(X)), X)
337.50/42.99	= { by axiom 29 (r2_1) R->L }
337.50/42.99	  fresh(fresh60(fresh60(fresh60(fresh10(r2, true, or(not(X), not(not(X))), not(not(X))), true, or(or(not(X), not(not(X))), or(not(X), not(not(X))))), true, or(not(X), not(not(X)))), true, and(implies(X, not(not(X))), implies(X, not(not(X))))), true, not(not(X)), X)
337.50/42.99	= { by axiom 6 (principia_r2) }
337.50/42.99	  fresh(fresh60(fresh60(fresh60(fresh10(true, true, or(not(X), not(not(X))), not(not(X))), true, or(or(not(X), not(not(X))), or(not(X), not(not(X))))), true, or(not(X), not(not(X)))), true, and(implies(X, not(not(X))), implies(X, not(not(X))))), true, not(not(X)), X)
337.50/42.99	= { by axiom 26 (r2_1) }
337.50/42.99	  fresh(fresh60(fresh60(fresh60(true, true, or(or(not(X), not(not(X))), or(not(X), not(not(X))))), true, or(not(X), not(not(X)))), true, and(implies(X, not(not(X))), implies(X, not(not(X))))), true, not(not(X)), X)
337.50/42.99	= { by axiom 14 (modus_ponens_2) }
337.50/42.99	  fresh(fresh60(fresh60(true, true, or(not(X), not(not(X)))), true, and(implies(X, not(not(X))), implies(X, not(not(X))))), true, not(not(X)), X)
337.50/42.99	= { by axiom 14 (modus_ponens_2) }
337.50/42.99	  fresh(fresh60(true, true, and(implies(X, not(not(X))), implies(X, not(not(X))))), true, not(not(X)), X)
337.50/42.99	= { by axiom 14 (modus_ponens_2) }
337.50/42.99	  fresh(true, true, not(not(X)), X)
337.50/42.99	= { by axiom 16 (substitution_of_equivalents_2) }
337.50/42.99	  X
337.50/42.99	
337.50/42.99	Lemma 55: or(and(X, not(Y)), Z) = implies(implies(X, Y), Z).
337.50/42.99	Proof:
337.50/42.99	  or(and(X, not(Y)), Z)
337.50/42.99	= { by lemma 45 R->L }
337.50/42.99	  implies(not(and(X, not(Y))), Z)
337.50/42.99	= { by lemma 44 }
337.50/42.99	  implies(implies(X, Y), Z)
337.50/42.99	
337.50/42.99	Lemma 56: fresh51(cn1, true, X, Y, Z) = true.
337.50/42.99	Proof:
337.50/42.99	  fresh51(cn1, true, X, Y, Z)
337.50/42.99	= { by axiom 39 (cn1_1) }
337.50/42.99	  is_a_theorem(implies(implies(X, Y), implies(implies(Y, Z), implies(X, Z))))
337.50/42.99	= { by lemma 54 R->L }
337.50/42.99	  is_a_theorem(implies(implies(X, Y), implies(implies(Y, Z), implies(X, not(not(Z))))))
337.50/42.99	= { by lemma 55 R->L }
337.50/42.99	  is_a_theorem(implies(implies(X, Y), or(and(Y, not(Z)), implies(X, not(not(Z))))))
337.50/42.99	= { by axiom 18 (modus_ponens_2) R->L }
337.50/42.99	  fresh28(true, true, or(and(Y, not(Z)), implies(implies(X, Y), implies(X, not(not(Z))))), implies(implies(X, Y), or(and(Y, not(Z)), implies(X, not(not(Z))))))
337.50/42.99	= { by lemma 53 R->L }
337.50/42.99	  fresh28(is_a_theorem(implies(or(and(Y, not(Z)), or(not(implies(X, Y)), implies(X, not(not(Z))))), or(not(implies(X, Y)), or(and(Y, not(Z)), implies(X, not(not(Z))))))), true, or(and(Y, not(Z)), implies(implies(X, Y), implies(X, not(not(Z))))), implies(implies(X, Y), or(and(Y, not(Z)), implies(X, not(not(Z))))))
337.50/42.99	= { by lemma 46 }
337.50/42.99	  fresh28(is_a_theorem(implies(or(and(Y, not(Z)), implies(implies(X, Y), implies(X, not(not(Z))))), or(not(implies(X, Y)), or(and(Y, not(Z)), implies(X, not(not(Z))))))), true, or(and(Y, not(Z)), implies(implies(X, Y), implies(X, not(not(Z))))), implies(implies(X, Y), or(and(Y, not(Z)), implies(X, not(not(Z))))))
337.50/42.99	= { by lemma 46 }
337.50/42.99	  fresh28(is_a_theorem(implies(or(and(Y, not(Z)), implies(implies(X, Y), implies(X, not(not(Z))))), implies(implies(X, Y), or(and(Y, not(Z)), implies(X, not(not(Z))))))), true, or(and(Y, not(Z)), implies(implies(X, Y), implies(X, not(not(Z))))), implies(implies(X, Y), or(and(Y, not(Z)), implies(X, not(not(Z))))))
337.50/42.99	= { by lemma 51 }
337.50/42.99	  fresh60(is_a_theorem(or(and(Y, not(Z)), implies(implies(X, Y), implies(X, not(not(Z)))))), true, implies(implies(X, Y), or(and(Y, not(Z)), implies(X, not(not(Z))))))
337.50/42.99	= { by lemma 49 R->L }
337.50/42.99	  fresh60(is_a_theorem(implies(implies(Y, not(not(Z))), implies(implies(X, Y), implies(X, not(not(Z)))))), true, implies(implies(X, Y), or(and(Y, not(Z)), implies(X, not(not(Z))))))
337.50/42.99	= { by lemma 46 R->L }
337.50/42.99	  fresh60(is_a_theorem(implies(implies(Y, not(not(Z))), implies(implies(X, Y), or(not(X), not(not(Z)))))), true, implies(implies(X, Y), or(and(Y, not(Z)), implies(X, not(not(Z))))))
337.50/42.99	= { by lemma 46 R->L }
337.50/42.99	  fresh60(is_a_theorem(implies(implies(Y, not(not(Z))), implies(or(not(X), Y), or(not(X), not(not(Z)))))), true, implies(implies(X, Y), or(and(Y, not(Z)), implies(X, not(not(Z))))))
337.50/42.99	= { by axiom 40 (r5_1) R->L }
337.50/42.99	  fresh60(fresh4(r5, true, not(X), Y, not(not(Z))), true, implies(implies(X, Y), or(and(Y, not(Z)), implies(X, not(not(Z))))))
337.50/42.99	= { by axiom 4 (principia_r5) }
337.50/42.99	  fresh60(fresh4(true, true, not(X), Y, not(not(Z))), true, implies(implies(X, Y), or(and(Y, not(Z)), implies(X, not(not(Z))))))
337.50/42.99	= { by axiom 34 (r5_1) }
337.50/42.99	  fresh60(true, true, implies(implies(X, Y), or(and(Y, not(Z)), implies(X, not(not(Z))))))
337.50/42.99	= { by axiom 14 (modus_ponens_2) }
337.50/42.99	  true
337.50/42.99	
337.50/42.99	Lemma 57: is_a_theorem(implies(implies(X, Y), or(and(Y, Z), not(and(Z, X))))) = fresh29(kn3, true, X, Y, Z).
337.50/42.99	Proof:
337.50/42.99	  is_a_theorem(implies(implies(X, Y), or(and(Y, Z), not(and(Z, X)))))
337.50/42.99	= { by lemma 45 R->L }
337.50/42.99	  is_a_theorem(implies(implies(X, Y), implies(not(and(Y, Z)), not(and(Z, X)))))
337.50/42.99	= { by axiom 42 (kn3_1) R->L }
337.50/42.99	  fresh29(kn3, true, X, Y, Z)
337.50/42.99	
337.50/42.99	Lemma 58: fresh60(is_a_theorem(implies(or(X, Y), Z)), true, implies(or(Y, X), Z)) = is_a_theorem(implies(or(Y, X), Z)).
337.50/42.99	Proof:
337.50/42.99	  fresh60(is_a_theorem(implies(or(X, Y), Z)), true, implies(or(Y, X), Z))
337.50/42.99	= { by lemma 51 R->L }
337.50/42.99	  fresh28(is_a_theorem(implies(implies(or(X, Y), Z), implies(or(Y, X), Z))), true, implies(or(X, Y), Z), implies(or(Y, X), Z))
337.50/42.99	= { by axiom 18 (modus_ponens_2) R->L }
337.50/42.99	  fresh28(fresh28(true, true, implies(or(Y, X), or(X, Y)), implies(implies(or(X, Y), Z), implies(or(Y, X), Z))), true, implies(or(X, Y), Z), implies(or(Y, X), Z))
337.50/42.99	= { by lemma 56 R->L }
337.50/42.99	  fresh28(fresh28(fresh51(cn1, true, or(Y, X), or(X, Y), Z), true, implies(or(Y, X), or(X, Y)), implies(implies(or(X, Y), Z), implies(or(Y, X), Z))), true, implies(or(X, Y), Z), implies(or(Y, X), Z))
337.50/42.99	= { by axiom 39 (cn1_1) }
337.50/43.00	  fresh28(fresh28(is_a_theorem(implies(implies(or(Y, X), or(X, Y)), implies(implies(or(X, Y), Z), implies(or(Y, X), Z)))), true, implies(or(Y, X), or(X, Y)), implies(implies(or(X, Y), Z), implies(or(Y, X), Z))), true, implies(or(X, Y), Z), implies(or(Y, X), Z))
337.50/43.00	= { by lemma 51 }
337.50/43.00	  fresh28(fresh60(is_a_theorem(implies(or(Y, X), or(X, Y))), true, implies(implies(or(X, Y), Z), implies(or(Y, X), Z))), true, implies(or(X, Y), Z), implies(or(Y, X), Z))
337.50/43.00	= { by lemma 52 }
337.50/43.00	  fresh28(fresh60(true, true, implies(implies(or(X, Y), Z), implies(or(Y, X), Z))), true, implies(or(X, Y), Z), implies(or(Y, X), Z))
337.50/43.00	= { by axiom 14 (modus_ponens_2) }
337.50/43.00	  fresh28(true, true, implies(or(X, Y), Z), implies(or(Y, X), Z))
337.50/43.00	= { by axiom 18 (modus_ponens_2) }
337.50/43.00	  is_a_theorem(implies(or(Y, X), Z))
337.50/43.00	
337.50/43.00	Goal 1 (rosser_kn3): kn3 = true.
337.50/43.00	Proof:
337.50/43.00	  kn3
337.50/43.00	= { by axiom 43 (kn3) R->L }
337.50/43.00	  fresh30(is_a_theorem(implies(implies(p7, q5), implies(not(and(q5, r7)), not(and(r7, p7))))), true)
337.50/43.00	= { by lemma 45 }
337.50/43.00	  fresh30(is_a_theorem(implies(implies(p7, q5), or(and(q5, r7), not(and(r7, p7))))), true)
337.50/43.00	= { by lemma 57 }
337.50/43.00	  fresh30(fresh29(kn3, true, p7, q5, r7), true)
337.50/43.00	= { by lemma 54 R->L }
337.50/43.00	  fresh30(fresh29(kn3, true, not(not(p7)), q5, r7), true)
337.50/43.00	= { by lemma 54 R->L }
337.50/43.00	  fresh30(fresh29(kn3, true, not(not(p7)), q5, not(not(r7))), true)
337.50/43.00	= { by lemma 57 R->L }
337.50/43.00	  fresh30(is_a_theorem(implies(implies(not(not(p7)), q5), or(and(q5, not(not(r7))), not(and(not(not(r7)), not(not(p7))))))), true)
337.50/43.00	= { by lemma 44 }
337.50/43.00	  fresh30(is_a_theorem(implies(implies(not(not(p7)), q5), or(and(q5, not(not(r7))), implies(not(not(r7)), not(p7))))), true)
337.50/43.00	= { by lemma 45 }
337.50/43.00	  fresh30(is_a_theorem(implies(or(not(p7), q5), or(and(q5, not(not(r7))), implies(not(not(r7)), not(p7))))), true)
337.50/43.00	= { by lemma 45 }
337.50/43.00	  fresh30(is_a_theorem(implies(or(not(p7), q5), or(and(q5, not(not(r7))), or(not(r7), not(p7))))), true)
337.50/43.00	= { by lemma 55 }
337.50/43.00	  fresh30(is_a_theorem(implies(or(not(p7), q5), implies(implies(q5, not(r7)), or(not(r7), not(p7))))), true)
337.50/43.00	= { by lemma 58 R->L }
337.50/43.00	  fresh30(fresh60(is_a_theorem(implies(or(q5, not(p7)), implies(implies(q5, not(r7)), or(not(r7), not(p7))))), true, implies(or(not(p7), q5), implies(implies(q5, not(r7)), or(not(r7), not(p7))))), true)
337.50/43.00	= { by axiom 18 (modus_ponens_2) R->L }
337.50/43.00	  fresh30(fresh60(fresh28(true, true, implies(implies(q5, not(r7)), implies(or(q5, not(p7)), or(not(r7), not(p7)))), implies(or(q5, not(p7)), implies(implies(q5, not(r7)), or(not(r7), not(p7))))), true, implies(or(not(p7), q5), implies(implies(q5, not(r7)), or(not(r7), not(p7))))), true)
337.50/43.00	= { by lemma 53 R->L }
337.50/43.00	  fresh30(fresh60(fresh28(is_a_theorem(implies(or(not(implies(q5, not(r7))), or(not(or(q5, not(p7))), or(not(r7), not(p7)))), or(not(or(q5, not(p7))), or(not(implies(q5, not(r7))), or(not(r7), not(p7)))))), true, implies(implies(q5, not(r7)), implies(or(q5, not(p7)), or(not(r7), not(p7)))), implies(or(q5, not(p7)), implies(implies(q5, not(r7)), or(not(r7), not(p7))))), true, implies(or(not(p7), q5), implies(implies(q5, not(r7)), or(not(r7), not(p7))))), true)
337.50/43.00	= { by lemma 46 }
337.50/43.00	  fresh30(fresh60(fresh28(is_a_theorem(implies(implies(implies(q5, not(r7)), or(not(or(q5, not(p7))), or(not(r7), not(p7)))), or(not(or(q5, not(p7))), or(not(implies(q5, not(r7))), or(not(r7), not(p7)))))), true, implies(implies(q5, not(r7)), implies(or(q5, not(p7)), or(not(r7), not(p7)))), implies(or(q5, not(p7)), implies(implies(q5, not(r7)), or(not(r7), not(p7))))), true, implies(or(not(p7), q5), implies(implies(q5, not(r7)), or(not(r7), not(p7))))), true)
337.50/43.00	= { by lemma 46 }
337.50/43.00	  fresh30(fresh60(fresh28(is_a_theorem(implies(implies(implies(q5, not(r7)), or(not(or(q5, not(p7))), or(not(r7), not(p7)))), or(not(or(q5, not(p7))), implies(implies(q5, not(r7)), or(not(r7), not(p7)))))), true, implies(implies(q5, not(r7)), implies(or(q5, not(p7)), or(not(r7), not(p7)))), implies(or(q5, not(p7)), implies(implies(q5, not(r7)), or(not(r7), not(p7))))), true, implies(or(not(p7), q5), implies(implies(q5, not(r7)), or(not(r7), not(p7))))), true)
337.50/43.00	= { by lemma 46 }
337.50/43.00	  fresh30(fresh60(fresh28(is_a_theorem(implies(implies(implies(q5, not(r7)), implies(or(q5, not(p7)), or(not(r7), not(p7)))), or(not(or(q5, not(p7))), implies(implies(q5, not(r7)), or(not(r7), not(p7)))))), true, implies(implies(q5, not(r7)), implies(or(q5, not(p7)), or(not(r7), not(p7)))), implies(or(q5, not(p7)), implies(implies(q5, not(r7)), or(not(r7), not(p7))))), true, implies(or(not(p7), q5), implies(implies(q5, not(r7)), or(not(r7), not(p7))))), true)
337.50/43.00	= { by lemma 46 }
337.50/43.00	  fresh30(fresh60(fresh28(is_a_theorem(implies(implies(implies(q5, not(r7)), implies(or(q5, not(p7)), or(not(r7), not(p7)))), implies(or(q5, not(p7)), implies(implies(q5, not(r7)), or(not(r7), not(p7)))))), true, implies(implies(q5, not(r7)), implies(or(q5, not(p7)), or(not(r7), not(p7)))), implies(or(q5, not(p7)), implies(implies(q5, not(r7)), or(not(r7), not(p7))))), true, implies(or(not(p7), q5), implies(implies(q5, not(r7)), or(not(r7), not(p7))))), true)
337.50/43.00	= { by lemma 51 }
337.50/43.00	  fresh30(fresh60(fresh60(is_a_theorem(implies(implies(q5, not(r7)), implies(or(q5, not(p7)), or(not(r7), not(p7))))), true, implies(or(q5, not(p7)), implies(implies(q5, not(r7)), or(not(r7), not(p7))))), true, implies(or(not(p7), q5), implies(implies(q5, not(r7)), or(not(r7), not(p7))))), true)
337.50/43.00	= { by lemma 46 R->L }
337.50/43.00	  fresh30(fresh60(fresh60(is_a_theorem(implies(or(not(q5), not(r7)), implies(or(q5, not(p7)), or(not(r7), not(p7))))), true, implies(or(q5, not(p7)), implies(implies(q5, not(r7)), or(not(r7), not(p7))))), true, implies(or(not(p7), q5), implies(implies(q5, not(r7)), or(not(r7), not(p7))))), true)
337.50/43.00	= { by lemma 58 R->L }
337.50/43.00	  fresh30(fresh60(fresh60(fresh60(is_a_theorem(implies(or(not(r7), not(q5)), implies(or(q5, not(p7)), or(not(r7), not(p7))))), true, implies(or(not(q5), not(r7)), implies(or(q5, not(p7)), or(not(r7), not(p7))))), true, implies(or(q5, not(p7)), implies(implies(q5, not(r7)), or(not(r7), not(p7))))), true, implies(or(not(p7), q5), implies(implies(q5, not(r7)), or(not(r7), not(p7))))), true)
337.50/43.00	= { by lemma 45 R->L }
337.50/43.00	  fresh30(fresh60(fresh60(fresh60(is_a_theorem(implies(or(not(r7), not(q5)), implies(implies(not(q5), not(p7)), or(not(r7), not(p7))))), true, implies(or(not(q5), not(r7)), implies(or(q5, not(p7)), or(not(r7), not(p7))))), true, implies(or(q5, not(p7)), implies(implies(q5, not(r7)), or(not(r7), not(p7))))), true, implies(or(not(p7), q5), implies(implies(q5, not(r7)), or(not(r7), not(p7))))), true)
337.50/43.00	= { by lemma 45 R->L }
337.50/43.00	  fresh30(fresh60(fresh60(fresh60(is_a_theorem(implies(or(not(r7), not(q5)), implies(implies(not(q5), not(p7)), implies(not(not(r7)), not(p7))))), true, implies(or(not(q5), not(r7)), implies(or(q5, not(p7)), or(not(r7), not(p7))))), true, implies(or(q5, not(p7)), implies(implies(q5, not(r7)), or(not(r7), not(p7))))), true, implies(or(not(p7), q5), implies(implies(q5, not(r7)), or(not(r7), not(p7))))), true)
337.50/43.00	= { by lemma 45 R->L }
337.50/43.00	  fresh30(fresh60(fresh60(fresh60(is_a_theorem(implies(implies(not(not(r7)), not(q5)), implies(implies(not(q5), not(p7)), implies(not(not(r7)), not(p7))))), true, implies(or(not(q5), not(r7)), implies(or(q5, not(p7)), or(not(r7), not(p7))))), true, implies(or(q5, not(p7)), implies(implies(q5, not(r7)), or(not(r7), not(p7))))), true, implies(or(not(p7), q5), implies(implies(q5, not(r7)), or(not(r7), not(p7))))), true)
337.50/43.00	= { by axiom 39 (cn1_1) R->L }
337.50/43.00	  fresh30(fresh60(fresh60(fresh60(fresh51(cn1, true, not(not(r7)), not(q5), not(p7)), true, implies(or(not(q5), not(r7)), implies(or(q5, not(p7)), or(not(r7), not(p7))))), true, implies(or(q5, not(p7)), implies(implies(q5, not(r7)), or(not(r7), not(p7))))), true, implies(or(not(p7), q5), implies(implies(q5, not(r7)), or(not(r7), not(p7))))), true)
337.50/43.00	= { by lemma 56 }
337.50/43.00	  fresh30(fresh60(fresh60(fresh60(true, true, implies(or(not(q5), not(r7)), implies(or(q5, not(p7)), or(not(r7), not(p7))))), true, implies(or(q5, not(p7)), implies(implies(q5, not(r7)), or(not(r7), not(p7))))), true, implies(or(not(p7), q5), implies(implies(q5, not(r7)), or(not(r7), not(p7))))), true)
337.50/43.00	= { by axiom 14 (modus_ponens_2) }
337.50/43.00	  fresh30(fresh60(fresh60(true, true, implies(or(q5, not(p7)), implies(implies(q5, not(r7)), or(not(r7), not(p7))))), true, implies(or(not(p7), q5), implies(implies(q5, not(r7)), or(not(r7), not(p7))))), true)
337.50/43.00	= { by axiom 14 (modus_ponens_2) }
337.50/43.00	  fresh30(fresh60(true, true, implies(or(not(p7), q5), implies(implies(q5, not(r7)), or(not(r7), not(p7))))), true)
337.50/43.00	= { by axiom 14 (modus_ponens_2) }
337.50/43.00	  fresh30(true, true)
337.50/43.00	= { by axiom 13 (kn3) }
337.50/43.00	  true
337.50/43.00	% SZS output end Proof
337.50/43.00	
337.50/43.00	RESULT: Theorem (the conjecture is true).
337.50/43.03	EOF