TSTP Solution File: LCL519+1 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : LCL519+1 : TPTP v8.1.2. Released v3.3.0.
% Transfm : none
% Format : tptp:raw
% Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% Computer : n027.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Thu Aug 31 08:19:18 EDT 2023
% Result : Theorem 7.74s 1.42s
% Output : Proof 8.75s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.13 % Problem : LCL519+1 : TPTP v8.1.2. Released v3.3.0.
% 0.00/0.14 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.15/0.35 % Computer : n027.cluster.edu
% 0.15/0.35 % Model : x86_64 x86_64
% 0.15/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.35 % Memory : 8042.1875MB
% 0.15/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.15/0.35 % CPULimit : 300
% 0.15/0.35 % WCLimit : 300
% 0.15/0.35 % DateTime : Fri Aug 25 03:24:33 EDT 2023
% 0.15/0.36 % CPUTime :
% 7.74/1.42 Command-line arguments: --lhs-weight 1 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 7.74/1.42
% 7.74/1.42 % SZS status Theorem
% 7.74/1.42
% 8.75/1.50 % SZS output start Proof
% 8.75/1.50 Take the following subset of the input axioms:
% 8.75/1.51 fof(and_1, axiom, and_1 <=> ![X, Y]: is_a_theorem(implies(and(X, Y), X))).
% 8.75/1.51 fof(and_2, axiom, and_2 <=> ![X2, Y2]: is_a_theorem(implies(and(X2, Y2), Y2))).
% 8.75/1.51 fof(implies_1, axiom, implies_1 <=> ![X2, Y2]: is_a_theorem(implies(X2, implies(Y2, X2)))).
% 8.75/1.51 fof(kn1, axiom, kn1 <=> ![P]: is_a_theorem(implies(P, and(P, P)))).
% 8.75/1.51 fof(kn2, axiom, kn2 <=> ![Q, P2]: is_a_theorem(implies(and(P2, Q), P2))).
% 8.75/1.51 fof(kn3, axiom, kn3 <=> ![R, P2, Q2]: is_a_theorem(implies(implies(P2, Q2), implies(not(and(Q2, R)), not(and(R, P2)))))).
% 8.75/1.51 fof(modus_ponens, axiom, modus_ponens <=> ![X2, Y2]: ((is_a_theorem(X2) & is_a_theorem(implies(X2, Y2))) => is_a_theorem(Y2))).
% 8.75/1.51 fof(op_and, axiom, op_and => ![X2, Y2]: and(X2, Y2)=not(or(not(X2), not(Y2)))).
% 8.75/1.51 fof(op_equiv, axiom, op_equiv => ![X2, Y2]: equiv(X2, Y2)=and(implies(X2, Y2), implies(Y2, X2))).
% 8.75/1.51 fof(op_implies_and, axiom, op_implies_and => ![X2, Y2]: implies(X2, Y2)=not(and(X2, not(Y2)))).
% 8.75/1.51 fof(op_implies_or, axiom, op_implies_or => ![X2, Y2]: implies(X2, Y2)=or(not(X2), Y2)).
% 8.75/1.51 fof(op_or, axiom, op_or => ![X2, Y2]: or(X2, Y2)=not(and(not(X2), not(Y2)))).
% 8.75/1.51 fof(principia_op_and, axiom, op_and).
% 8.75/1.51 fof(principia_op_implies_or, axiom, op_implies_or).
% 8.75/1.51 fof(principia_r2, conjecture, r2).
% 8.75/1.51 fof(r2, axiom, r2 <=> ![P2, Q2]: is_a_theorem(implies(Q2, or(P2, Q2)))).
% 8.75/1.51 fof(rosser_kn1, axiom, kn1).
% 8.75/1.51 fof(rosser_kn2, axiom, kn2).
% 8.75/1.51 fof(rosser_kn3, axiom, kn3).
% 8.75/1.51 fof(rosser_modus_ponens, axiom, modus_ponens).
% 8.75/1.51 fof(rosser_op_equiv, axiom, op_equiv).
% 8.75/1.51 fof(rosser_op_implies_and, axiom, op_implies_and).
% 8.75/1.51 fof(rosser_op_or, axiom, op_or).
% 8.75/1.51 fof(substitution_of_equivalents, axiom, substitution_of_equivalents <=> ![X2, Y2]: (is_a_theorem(equiv(X2, Y2)) => X2=Y2)).
% 8.75/1.51 fof(substitution_of_equivalents, axiom, substitution_of_equivalents).
% 8.75/1.51
% 8.75/1.51 Now clausify the problem and encode Horn clauses using encoding 3 of
% 8.75/1.51 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 8.75/1.51 We repeatedly replace C & s=t => u=v by the two clauses:
% 8.75/1.51 fresh(y, y, x1...xn) = u
% 8.75/1.51 C => fresh(s, t, x1...xn) = v
% 8.75/1.51 where fresh is a fresh function symbol and x1..xn are the free
% 8.75/1.51 variables of u and v.
% 8.75/1.51 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 8.75/1.51 input problem has no model of domain size 1).
% 8.75/1.51
% 8.75/1.51 The encoding turns the above axioms into the following unit equations and goals:
% 8.75/1.51
% 8.75/1.51 Axiom 1 (rosser_modus_ponens): modus_ponens = true.
% 8.75/1.51 Axiom 2 (substitution_of_equivalents): substitution_of_equivalents = true.
% 8.75/1.51 Axiom 3 (rosser_kn1): kn1 = true.
% 8.75/1.51 Axiom 4 (rosser_kn2): kn2 = true.
% 8.75/1.51 Axiom 5 (rosser_kn3): kn3 = true.
% 8.75/1.51 Axiom 6 (rosser_op_equiv): op_equiv = true.
% 8.75/1.51 Axiom 7 (rosser_op_or): op_or = true.
% 8.75/1.51 Axiom 8 (principia_op_and): op_and = true.
% 8.75/1.51 Axiom 9 (rosser_op_implies_and): op_implies_and = true.
% 8.75/1.51 Axiom 10 (principia_op_implies_or): op_implies_or = true.
% 8.75/1.51 Axiom 11 (r2): fresh11(X, X) = true.
% 8.75/1.51 Axiom 12 (modus_ponens_2): fresh60(X, X, Y) = true.
% 8.75/1.51 Axiom 13 (kn1_1): fresh33(X, X, Y) = true.
% 8.75/1.51 Axiom 14 (modus_ponens_2): fresh28(X, X, Y) = is_a_theorem(Y).
% 8.75/1.51 Axiom 15 (substitution_of_equivalents_2): fresh(X, X, Y, Z) = Z.
% 8.75/1.51 Axiom 16 (modus_ponens_2): fresh59(X, X, Y, Z) = fresh60(modus_ponens, true, Z).
% 8.75/1.51 Axiom 17 (kn2_1): fresh31(X, X, Y, Z) = true.
% 8.75/1.51 Axiom 18 (op_and): fresh24(X, X, Y, Z) = and(Y, Z).
% 8.75/1.51 Axiom 19 (op_equiv): fresh23(X, X, Y, Z) = equiv(Y, Z).
% 8.75/1.51 Axiom 20 (op_implies_and): fresh22(X, X, Y, Z) = implies(Y, Z).
% 8.75/1.51 Axiom 21 (op_implies_and): fresh22(op_implies_and, true, X, Y) = not(and(X, not(Y))).
% 8.75/1.51 Axiom 22 (op_implies_or): fresh21(X, X, Y, Z) = implies(Y, Z).
% 8.75/1.51 Axiom 23 (op_implies_or): fresh21(op_implies_or, true, X, Y) = or(not(X), Y).
% 8.75/1.51 Axiom 24 (op_or): fresh20(X, X, Y, Z) = or(Y, Z).
% 8.75/1.51 Axiom 25 (substitution_of_equivalents_2): fresh2(X, X, Y, Z) = Y.
% 8.75/1.51 Axiom 26 (implies_1_1): fresh39(implies_1, true, X, Y) = is_a_theorem(implies(X, implies(Y, X))).
% 8.75/1.51 Axiom 27 (kn1_1): fresh33(kn1, true, X) = is_a_theorem(implies(X, and(X, X))).
% 8.75/1.51 Axiom 28 (and_1_1): fresh58(and_1, true, X, Y) = is_a_theorem(implies(and(X, Y), X)).
% 8.75/1.51 Axiom 29 (kn2_1): fresh31(kn2, true, X, Y) = is_a_theorem(implies(and(X, Y), X)).
% 8.75/1.51 Axiom 30 (and_2_1): fresh55(and_2, true, X, Y) = is_a_theorem(implies(and(X, Y), Y)).
% 8.75/1.51 Axiom 31 (op_and): fresh24(op_and, true, X, Y) = not(or(not(X), not(Y))).
% 8.75/1.51 Axiom 32 (op_or): fresh20(op_or, true, X, Y) = not(and(not(X), not(Y))).
% 8.75/1.51 Axiom 33 (kn3_1): fresh29(X, X, Y, Z, W) = true.
% 8.75/1.51 Axiom 34 (op_equiv): fresh23(op_equiv, true, X, Y) = and(implies(X, Y), implies(Y, X)).
% 8.75/1.51 Axiom 35 (substitution_of_equivalents_2): fresh2(substitution_of_equivalents, true, X, Y) = fresh(is_a_theorem(equiv(X, Y)), true, X, Y).
% 8.75/1.51 Axiom 36 (modus_ponens_2): fresh59(is_a_theorem(implies(X, Y)), true, X, Y) = fresh28(is_a_theorem(X), true, Y).
% 8.75/1.51 Axiom 37 (r2): fresh11(is_a_theorem(implies(q4, or(p4, q4))), true) = r2.
% 8.75/1.51 Axiom 38 (kn3_1): fresh29(kn3, true, X, Y, Z) = is_a_theorem(implies(implies(X, Y), implies(not(and(Y, Z)), not(and(Z, X))))).
% 8.75/1.51
% 8.75/1.51 Lemma 39: fresh(is_a_theorem(equiv(X, Y)), true, X, Y) = X.
% 8.75/1.51 Proof:
% 8.75/1.51 fresh(is_a_theorem(equiv(X, Y)), true, X, Y)
% 8.75/1.51 = { by axiom 35 (substitution_of_equivalents_2) R->L }
% 8.75/1.51 fresh2(substitution_of_equivalents, true, X, Y)
% 8.75/1.51 = { by axiom 2 (substitution_of_equivalents) }
% 8.75/1.51 fresh2(true, true, X, Y)
% 8.75/1.51 = { by axiom 25 (substitution_of_equivalents_2) }
% 8.75/1.51 X
% 8.75/1.51
% 8.75/1.51 Lemma 40: and(implies(X, Y), implies(Y, X)) = equiv(X, Y).
% 8.75/1.51 Proof:
% 8.75/1.51 and(implies(X, Y), implies(Y, X))
% 8.75/1.51 = { by axiom 34 (op_equiv) R->L }
% 8.75/1.51 fresh23(op_equiv, true, X, Y)
% 8.75/1.51 = { by axiom 6 (rosser_op_equiv) }
% 8.75/1.51 fresh23(true, true, X, Y)
% 8.75/1.51 = { by axiom 19 (op_equiv) }
% 8.75/1.51 equiv(X, Y)
% 8.75/1.51
% 8.75/1.51 Lemma 41: not(and(X, not(Y))) = implies(X, Y).
% 8.75/1.51 Proof:
% 8.75/1.51 not(and(X, not(Y)))
% 8.75/1.51 = { by axiom 21 (op_implies_and) R->L }
% 8.75/1.51 fresh22(op_implies_and, true, X, Y)
% 8.75/1.51 = { by axiom 9 (rosser_op_implies_and) }
% 8.75/1.51 fresh22(true, true, X, Y)
% 8.75/1.51 = { by axiom 20 (op_implies_and) }
% 8.75/1.51 implies(X, Y)
% 8.75/1.51
% 8.75/1.51 Lemma 42: implies(not(X), Y) = or(X, Y).
% 8.75/1.51 Proof:
% 8.75/1.51 implies(not(X), Y)
% 8.75/1.51 = { by lemma 41 R->L }
% 8.75/1.51 not(and(not(X), not(Y)))
% 8.75/1.51 = { by axiom 32 (op_or) R->L }
% 8.75/1.51 fresh20(op_or, true, X, Y)
% 8.75/1.51 = { by axiom 7 (rosser_op_or) }
% 8.75/1.51 fresh20(true, true, X, Y)
% 8.75/1.51 = { by axiom 24 (op_or) }
% 8.75/1.51 or(X, Y)
% 8.75/1.51
% 8.75/1.51 Lemma 43: or(not(X), Y) = implies(X, Y).
% 8.75/1.51 Proof:
% 8.75/1.51 or(not(X), Y)
% 8.75/1.51 = { by axiom 23 (op_implies_or) R->L }
% 8.75/1.51 fresh21(op_implies_or, true, X, Y)
% 8.75/1.51 = { by axiom 10 (principia_op_implies_or) }
% 8.75/1.51 fresh21(true, true, X, Y)
% 8.75/1.51 = { by axiom 22 (op_implies_or) }
% 8.75/1.51 implies(X, Y)
% 8.75/1.51
% 8.75/1.51 Lemma 44: not(implies(X, not(Y))) = and(X, Y).
% 8.75/1.51 Proof:
% 8.75/1.51 not(implies(X, not(Y)))
% 8.75/1.51 = { by lemma 43 R->L }
% 8.75/1.51 not(or(not(X), not(Y)))
% 8.75/1.51 = { by axiom 31 (op_and) R->L }
% 8.75/1.51 fresh24(op_and, true, X, Y)
% 8.75/1.51 = { by axiom 8 (principia_op_and) }
% 8.75/1.51 fresh24(true, true, X, Y)
% 8.75/1.51 = { by axiom 18 (op_and) }
% 8.75/1.51 and(X, Y)
% 8.75/1.51
% 8.75/1.51 Lemma 45: and(X, implies(Y, not(Z))) = not(implies(X, and(Y, Z))).
% 8.75/1.51 Proof:
% 8.75/1.51 and(X, implies(Y, not(Z)))
% 8.75/1.51 = { by lemma 44 R->L }
% 8.75/1.51 not(implies(X, not(implies(Y, not(Z)))))
% 8.75/1.51 = { by lemma 44 }
% 8.75/1.51 not(implies(X, and(Y, Z)))
% 8.75/1.51
% 8.75/1.51 Lemma 46: implies(implies(X, Y), and(Y, not(X))) = not(equiv(X, Y)).
% 8.75/1.51 Proof:
% 8.75/1.51 implies(implies(X, Y), and(Y, not(X)))
% 8.75/1.51 = { by lemma 41 R->L }
% 8.75/1.51 not(and(implies(X, Y), not(and(Y, not(X)))))
% 8.75/1.51 = { by lemma 41 }
% 8.75/1.51 not(and(implies(X, Y), implies(Y, X)))
% 8.75/1.51 = { by lemma 40 }
% 8.75/1.51 not(equiv(X, Y))
% 8.75/1.51
% 8.75/1.51 Lemma 47: equiv(not(not(X)), Y) = not(not(equiv(X, Y))).
% 8.75/1.51 Proof:
% 8.75/1.51 equiv(not(not(X)), Y)
% 8.75/1.51 = { by lemma 40 R->L }
% 8.75/1.51 and(implies(not(not(X)), Y), implies(Y, not(not(X))))
% 8.75/1.51 = { by lemma 42 }
% 8.75/1.51 and(or(not(X), Y), implies(Y, not(not(X))))
% 8.75/1.51 = { by lemma 45 }
% 8.75/1.51 not(implies(or(not(X), Y), and(Y, not(X))))
% 8.75/1.51 = { by lemma 43 }
% 8.75/1.51 not(implies(implies(X, Y), and(Y, not(X))))
% 8.75/1.51 = { by lemma 46 }
% 8.75/1.51 not(not(equiv(X, Y)))
% 8.75/1.51
% 8.75/1.51 Lemma 48: or(and(X, not(Y)), Z) = implies(implies(X, Y), Z).
% 8.75/1.51 Proof:
% 8.75/1.51 or(and(X, not(Y)), Z)
% 8.75/1.51 = { by lemma 42 R->L }
% 8.75/1.51 implies(not(and(X, not(Y))), Z)
% 8.75/1.51 = { by lemma 41 }
% 8.75/1.51 implies(implies(X, Y), Z)
% 8.75/1.51
% 8.75/1.51 Lemma 49: implies(implies(X, not(Y)), Z) = or(and(X, Y), Z).
% 8.75/1.51 Proof:
% 8.75/1.51 implies(implies(X, not(Y)), Z)
% 8.75/1.51 = { by lemma 43 R->L }
% 8.75/1.51 or(not(implies(X, not(Y))), Z)
% 8.75/1.51 = { by lemma 44 }
% 8.75/1.51 or(and(X, Y), Z)
% 8.75/1.51
% 8.75/1.51 Lemma 50: not(or(X, not(Y))) = and(not(X), Y).
% 8.75/1.51 Proof:
% 8.75/1.51 not(or(X, not(Y)))
% 8.75/1.51 = { by lemma 42 R->L }
% 8.75/1.51 not(implies(not(X), not(Y)))
% 8.75/1.51 = { by lemma 44 }
% 8.75/1.51 and(not(X), Y)
% 8.75/1.51
% 8.75/1.51 Lemma 51: fresh59(X, X, Y, Z) = true.
% 8.75/1.51 Proof:
% 8.75/1.51 fresh59(X, X, Y, Z)
% 8.75/1.51 = { by axiom 16 (modus_ponens_2) }
% 8.75/1.51 fresh60(modus_ponens, true, Z)
% 8.75/1.51 = { by axiom 1 (rosser_modus_ponens) }
% 8.75/1.51 fresh60(true, true, Z)
% 8.75/1.51 = { by axiom 12 (modus_ponens_2) }
% 8.75/1.51 true
% 8.75/1.51
% 8.75/1.51 Lemma 52: fresh58(and_1, true, X, Y) = true.
% 8.75/1.51 Proof:
% 8.75/1.51 fresh58(and_1, true, X, Y)
% 8.75/1.51 = { by axiom 28 (and_1_1) }
% 8.75/1.51 is_a_theorem(implies(and(X, Y), X))
% 8.75/1.51 = { by axiom 29 (kn2_1) R->L }
% 8.75/1.51 fresh31(kn2, true, X, Y)
% 8.75/1.51 = { by axiom 4 (rosser_kn2) }
% 8.75/1.51 fresh31(true, true, X, Y)
% 8.75/1.51 = { by axiom 17 (kn2_1) }
% 8.75/1.51 true
% 8.75/1.51
% 8.75/1.51 Lemma 53: is_a_theorem(implies(X, and(X, X))) = true.
% 8.75/1.51 Proof:
% 8.75/1.51 is_a_theorem(implies(X, and(X, X)))
% 8.75/1.51 = { by axiom 27 (kn1_1) R->L }
% 8.75/1.51 fresh33(kn1, true, X)
% 8.75/1.51 = { by axiom 3 (rosser_kn1) }
% 8.75/1.52 fresh33(true, true, X)
% 8.75/1.52 = { by axiom 13 (kn1_1) }
% 8.75/1.52 true
% 8.75/1.52
% 8.75/1.52 Lemma 54: fresh28(is_a_theorem(X), true, and(X, X)) = true.
% 8.75/1.52 Proof:
% 8.75/1.52 fresh28(is_a_theorem(X), true, and(X, X))
% 8.75/1.52 = { by axiom 36 (modus_ponens_2) R->L }
% 8.75/1.52 fresh59(is_a_theorem(implies(X, and(X, X))), true, X, and(X, X))
% 8.75/1.52 = { by lemma 53 }
% 8.75/1.52 fresh59(true, true, X, and(X, X))
% 8.75/1.52 = { by lemma 51 }
% 8.75/1.52 true
% 8.75/1.52
% 8.75/1.52 Lemma 55: fresh59(is_a_theorem(or(X, Y)), true, not(X), Y) = fresh28(is_a_theorem(not(X)), true, Y).
% 8.75/1.52 Proof:
% 8.75/1.52 fresh59(is_a_theorem(or(X, Y)), true, not(X), Y)
% 8.75/1.52 = { by lemma 42 R->L }
% 8.75/1.52 fresh59(is_a_theorem(implies(not(X), Y)), true, not(X), Y)
% 8.75/1.52 = { by axiom 36 (modus_ponens_2) }
% 8.75/1.52 fresh28(is_a_theorem(not(X)), true, Y)
% 8.75/1.52
% 8.75/1.52 Lemma 56: is_a_theorem(implies(implies(X, Y), or(and(Y, Z), not(and(Z, X))))) = true.
% 8.75/1.52 Proof:
% 8.75/1.52 is_a_theorem(implies(implies(X, Y), or(and(Y, Z), not(and(Z, X)))))
% 8.75/1.52 = { by lemma 42 R->L }
% 8.75/1.52 is_a_theorem(implies(implies(X, Y), implies(not(and(Y, Z)), not(and(Z, X)))))
% 8.75/1.52 = { by axiom 38 (kn3_1) R->L }
% 8.75/1.52 fresh29(kn3, true, X, Y, Z)
% 8.75/1.52 = { by axiom 5 (rosser_kn3) }
% 8.75/1.52 fresh29(true, true, X, Y, Z)
% 8.75/1.52 = { by axiom 33 (kn3_1) }
% 8.75/1.52 true
% 8.75/1.52
% 8.75/1.52 Lemma 57: fresh28(is_a_theorem(implies(X, Y)), true, or(and(Y, Z), not(and(Z, X)))) = true.
% 8.75/1.52 Proof:
% 8.75/1.52 fresh28(is_a_theorem(implies(X, Y)), true, or(and(Y, Z), not(and(Z, X))))
% 8.75/1.52 = { by axiom 36 (modus_ponens_2) R->L }
% 8.75/1.52 fresh59(is_a_theorem(implies(implies(X, Y), or(and(Y, Z), not(and(Z, X))))), true, implies(X, Y), or(and(Y, Z), not(and(Z, X))))
% 8.75/1.52 = { by lemma 56 }
% 8.75/1.52 fresh59(true, true, implies(X, Y), or(and(Y, Z), not(and(Z, X))))
% 8.75/1.52 = { by lemma 51 }
% 8.75/1.52 true
% 8.75/1.52
% 8.75/1.52 Lemma 58: is_a_theorem(implies(X, X)) = true.
% 8.75/1.52 Proof:
% 8.75/1.52 is_a_theorem(implies(X, X))
% 8.75/1.52 = { by lemma 41 R->L }
% 8.75/1.52 is_a_theorem(not(and(X, not(X))))
% 8.75/1.52 = { by axiom 14 (modus_ponens_2) R->L }
% 8.75/1.52 fresh28(true, true, not(and(X, not(X))))
% 8.75/1.52 = { by lemma 51 R->L }
% 8.75/1.52 fresh28(fresh59(true, true, and(not(and(and(not(X), not(X)), X)), implies(and(not(X), not(X)), not(X))), not(and(and(not(X), not(X)), X))), true, not(and(X, not(X))))
% 8.75/1.52 = { by lemma 52 R->L }
% 8.75/1.52 fresh28(fresh59(fresh58(and_1, true, not(and(and(not(X), not(X)), X)), implies(and(not(X), not(X)), not(X))), true, and(not(and(and(not(X), not(X)), X)), implies(and(not(X), not(X)), not(X))), not(and(and(not(X), not(X)), X))), true, not(and(X, not(X))))
% 8.75/1.52 = { by axiom 28 (and_1_1) }
% 8.75/1.52 fresh28(fresh59(is_a_theorem(implies(and(not(and(and(not(X), not(X)), X)), implies(and(not(X), not(X)), not(X))), not(and(and(not(X), not(X)), X)))), true, and(not(and(and(not(X), not(X)), X)), implies(and(not(X), not(X)), not(X))), not(and(and(not(X), not(X)), X))), true, not(and(X, not(X))))
% 8.75/1.52 = { by axiom 36 (modus_ponens_2) }
% 8.75/1.52 fresh28(fresh28(is_a_theorem(and(not(and(and(not(X), not(X)), X)), implies(and(not(X), not(X)), not(X)))), true, not(and(and(not(X), not(X)), X))), true, not(and(X, not(X))))
% 8.75/1.52 = { by lemma 45 }
% 8.75/1.52 fresh28(fresh28(is_a_theorem(not(implies(not(and(and(not(X), not(X)), X)), and(and(not(X), not(X)), X)))), true, not(and(and(not(X), not(X)), X))), true, not(and(X, not(X))))
% 8.75/1.52 = { by lemma 42 }
% 8.75/1.52 fresh28(fresh28(is_a_theorem(not(or(and(and(not(X), not(X)), X), and(and(not(X), not(X)), X)))), true, not(and(and(not(X), not(X)), X))), true, not(and(X, not(X))))
% 8.75/1.52 = { by lemma 49 R->L }
% 8.75/1.52 fresh28(fresh28(is_a_theorem(not(implies(implies(and(not(X), not(X)), not(X)), and(and(not(X), not(X)), X)))), true, not(and(and(not(X), not(X)), X))), true, not(and(X, not(X))))
% 8.75/1.52 = { by lemma 45 R->L }
% 8.75/1.52 fresh28(fresh28(is_a_theorem(and(implies(and(not(X), not(X)), not(X)), implies(and(not(X), not(X)), not(X)))), true, not(and(and(not(X), not(X)), X))), true, not(and(X, not(X))))
% 8.75/1.52 = { by axiom 14 (modus_ponens_2) R->L }
% 8.75/1.52 fresh28(fresh28(fresh28(true, true, and(implies(and(not(X), not(X)), not(X)), implies(and(not(X), not(X)), not(X)))), true, not(and(and(not(X), not(X)), X))), true, not(and(X, not(X))))
% 8.75/1.52 = { by lemma 52 R->L }
% 8.75/1.52 fresh28(fresh28(fresh28(fresh58(and_1, true, not(X), not(X)), true, and(implies(and(not(X), not(X)), not(X)), implies(and(not(X), not(X)), not(X)))), true, not(and(and(not(X), not(X)), X))), true, not(and(X, not(X))))
% 8.75/1.52 = { by axiom 28 (and_1_1) }
% 8.75/1.52 fresh28(fresh28(fresh28(is_a_theorem(implies(and(not(X), not(X)), not(X))), true, and(implies(and(not(X), not(X)), not(X)), implies(and(not(X), not(X)), not(X)))), true, not(and(and(not(X), not(X)), X))), true, not(and(X, not(X))))
% 8.75/1.52 = { by lemma 54 }
% 8.75/1.52 fresh28(fresh28(true, true, not(and(and(not(X), not(X)), X))), true, not(and(X, not(X))))
% 8.75/1.52 = { by axiom 14 (modus_ponens_2) }
% 8.75/1.52 fresh28(is_a_theorem(not(and(and(not(X), not(X)), X))), true, not(and(X, not(X))))
% 8.75/1.52 = { by lemma 55 R->L }
% 8.75/1.52 fresh59(is_a_theorem(or(and(and(not(X), not(X)), X), not(and(X, not(X))))), true, not(and(and(not(X), not(X)), X)), not(and(X, not(X))))
% 8.75/1.52 = { by axiom 14 (modus_ponens_2) R->L }
% 8.75/1.52 fresh59(fresh28(true, true, or(and(and(not(X), not(X)), X), not(and(X, not(X))))), true, not(and(and(not(X), not(X)), X)), not(and(X, not(X))))
% 8.75/1.52 = { by lemma 53 R->L }
% 8.75/1.52 fresh59(fresh28(is_a_theorem(implies(not(X), and(not(X), not(X)))), true, or(and(and(not(X), not(X)), X), not(and(X, not(X))))), true, not(and(and(not(X), not(X)), X)), not(and(X, not(X))))
% 8.75/1.52 = { by lemma 57 }
% 8.75/1.52 fresh59(true, true, not(and(and(not(X), not(X)), X)), not(and(X, not(X))))
% 8.75/1.52 = { by lemma 51 }
% 8.75/1.52 true
% 8.75/1.52
% 8.75/1.52 Lemma 59: is_a_theorem(not(not(equiv(X, X)))) = true.
% 8.75/1.52 Proof:
% 8.75/1.52 is_a_theorem(not(not(equiv(X, X))))
% 8.75/1.52 = { by lemma 46 R->L }
% 8.75/1.52 is_a_theorem(not(implies(implies(X, X), and(X, not(X)))))
% 8.75/1.52 = { by lemma 48 R->L }
% 8.75/1.52 is_a_theorem(not(or(and(X, not(X)), and(X, not(X)))))
% 8.75/1.52 = { by lemma 49 R->L }
% 8.75/1.52 is_a_theorem(not(implies(implies(X, not(not(X))), and(X, not(X)))))
% 8.75/1.52 = { by lemma 44 R->L }
% 8.75/1.52 is_a_theorem(not(implies(implies(X, not(not(X))), not(implies(X, not(not(X)))))))
% 8.75/1.52 = { by lemma 43 R->L }
% 8.75/1.52 is_a_theorem(not(implies(implies(X, not(not(X))), not(or(not(X), not(not(X)))))))
% 8.75/1.52 = { by lemma 50 }
% 8.75/1.52 is_a_theorem(not(implies(implies(X, not(not(X))), and(not(not(X)), not(X)))))
% 8.75/1.52 = { by lemma 46 }
% 8.75/1.52 is_a_theorem(not(not(equiv(X, not(not(X))))))
% 8.75/1.52 = { by axiom 14 (modus_ponens_2) R->L }
% 8.75/1.52 fresh28(true, true, not(not(equiv(X, not(not(X))))))
% 8.75/1.52 = { by lemma 58 R->L }
% 8.75/1.52 fresh28(is_a_theorem(implies(not(not(X)), not(not(X)))), true, not(not(equiv(X, not(not(X))))))
% 8.75/1.52 = { by lemma 47 R->L }
% 8.75/1.52 fresh28(is_a_theorem(implies(not(not(X)), not(not(X)))), true, equiv(not(not(X)), not(not(X))))
% 8.75/1.52 = { by lemma 40 R->L }
% 8.75/1.52 fresh28(is_a_theorem(implies(not(not(X)), not(not(X)))), true, and(implies(not(not(X)), not(not(X))), implies(not(not(X)), not(not(X)))))
% 8.75/1.52 = { by lemma 54 }
% 8.75/1.52 true
% 8.75/1.52
% 8.75/1.52 Lemma 60: not(not(X)) = X.
% 8.75/1.52 Proof:
% 8.75/1.52 not(not(X))
% 8.75/1.52 = { by lemma 39 R->L }
% 8.75/1.52 fresh(is_a_theorem(equiv(not(not(X)), X)), true, not(not(X)), X)
% 8.75/1.52 = { by lemma 47 }
% 8.75/1.52 fresh(is_a_theorem(not(not(equiv(X, X)))), true, not(not(X)), X)
% 8.75/1.52 = { by lemma 59 }
% 8.75/1.52 fresh(true, true, not(not(X)), X)
% 8.75/1.52 = { by axiom 15 (substitution_of_equivalents_2) }
% 8.75/1.52 X
% 8.75/1.52
% 8.75/1.52 Lemma 61: not(and(X, Y)) = implies(X, not(Y)).
% 8.75/1.52 Proof:
% 8.75/1.52 not(and(X, Y))
% 8.75/1.52 = { by lemma 39 R->L }
% 8.75/1.53 fresh(is_a_theorem(equiv(not(and(X, Y)), implies(X, not(Y)))), true, not(and(X, Y)), implies(X, not(Y)))
% 8.75/1.53 = { by lemma 44 R->L }
% 8.75/1.53 fresh(is_a_theorem(equiv(not(not(implies(X, not(Y)))), implies(X, not(Y)))), true, not(and(X, Y)), implies(X, not(Y)))
% 8.75/1.53 = { by lemma 47 }
% 8.75/1.53 fresh(is_a_theorem(not(not(equiv(implies(X, not(Y)), implies(X, not(Y)))))), true, not(and(X, Y)), implies(X, not(Y)))
% 8.75/1.53 = { by lemma 59 }
% 8.75/1.53 fresh(true, true, not(and(X, Y)), implies(X, not(Y)))
% 8.75/1.53 = { by axiom 15 (substitution_of_equivalents_2) }
% 8.75/1.53 implies(X, not(Y))
% 8.75/1.53
% 8.75/1.53 Lemma 62: implies(and(X, not(Y)), Z) = or(implies(X, Y), Z).
% 8.75/1.53 Proof:
% 8.75/1.53 implies(and(X, not(Y)), Z)
% 8.75/1.53 = { by lemma 43 R->L }
% 8.75/1.53 or(not(and(X, not(Y))), Z)
% 8.75/1.53 = { by lemma 41 }
% 8.75/1.53 or(implies(X, Y), Z)
% 8.75/1.53
% 8.75/1.53 Lemma 63: is_a_theorem(implies(or(X, Y), or(and(Y, Z), implies(Z, X)))) = true.
% 8.75/1.53 Proof:
% 8.75/1.53 is_a_theorem(implies(or(X, Y), or(and(Y, Z), implies(Z, X))))
% 8.75/1.53 = { by lemma 42 R->L }
% 8.75/1.53 is_a_theorem(implies(implies(not(X), Y), or(and(Y, Z), implies(Z, X))))
% 8.75/1.53 = { by lemma 41 R->L }
% 8.75/1.53 is_a_theorem(implies(implies(not(X), Y), or(and(Y, Z), not(and(Z, not(X))))))
% 8.75/1.53 = { by lemma 56 }
% 8.75/1.53 true
% 8.75/1.53
% 8.75/1.53 Goal 1 (principia_r2): r2 = true.
% 8.75/1.53 Proof:
% 8.75/1.53 r2
% 8.75/1.53 = { by axiom 37 (r2) R->L }
% 8.75/1.53 fresh11(is_a_theorem(implies(q4, or(p4, q4))), true)
% 8.75/1.53 = { by lemma 42 R->L }
% 8.75/1.53 fresh11(is_a_theorem(implies(q4, implies(not(p4), q4))), true)
% 8.75/1.53 = { by axiom 26 (implies_1_1) R->L }
% 8.75/1.53 fresh11(fresh39(implies_1, true, q4, not(p4)), true)
% 8.75/1.53 = { by lemma 60 R->L }
% 8.75/1.53 fresh11(fresh39(implies_1, true, not(not(q4)), not(p4)), true)
% 8.75/1.53 = { by axiom 26 (implies_1_1) }
% 8.75/1.53 fresh11(is_a_theorem(implies(not(not(q4)), implies(not(p4), not(not(q4))))), true)
% 8.75/1.53 = { by lemma 42 }
% 8.75/1.53 fresh11(is_a_theorem(or(not(q4), implies(not(p4), not(not(q4))))), true)
% 8.75/1.53 = { by lemma 43 }
% 8.75/1.53 fresh11(is_a_theorem(implies(q4, implies(not(p4), not(not(q4))))), true)
% 8.75/1.53 = { by axiom 14 (modus_ponens_2) R->L }
% 8.75/1.53 fresh11(fresh28(true, true, implies(q4, implies(not(p4), not(not(q4))))), true)
% 8.75/1.53 = { by lemma 58 R->L }
% 8.75/1.53 fresh11(fresh28(is_a_theorem(implies(not(q4), not(q4))), true, implies(q4, implies(not(p4), not(not(q4))))), true)
% 8.75/1.53 = { by lemma 61 R->L }
% 8.75/1.53 fresh11(fresh28(is_a_theorem(not(and(not(q4), q4))), true, implies(q4, implies(not(p4), not(not(q4))))), true)
% 8.75/1.53 = { by lemma 55 R->L }
% 8.75/1.53 fresh11(fresh59(is_a_theorem(or(and(not(q4), q4), implies(q4, implies(not(p4), not(not(q4)))))), true, not(and(not(q4), q4)), implies(q4, implies(not(p4), not(not(q4))))), true)
% 8.75/1.53 = { by lemma 41 R->L }
% 8.75/1.53 fresh11(fresh59(is_a_theorem(or(and(not(q4), q4), not(and(q4, not(implies(not(p4), not(not(q4)))))))), true, not(and(not(q4), q4)), implies(q4, implies(not(p4), not(not(q4))))), true)
% 8.75/1.53 = { by lemma 44 }
% 8.75/1.53 fresh11(fresh59(is_a_theorem(or(and(not(q4), q4), not(and(q4, and(not(p4), not(q4)))))), true, not(and(not(q4), q4)), implies(q4, implies(not(p4), not(not(q4))))), true)
% 8.75/1.53 = { by axiom 14 (modus_ponens_2) R->L }
% 8.75/1.53 fresh11(fresh59(fresh28(true, true, or(and(not(q4), q4), not(and(q4, and(not(p4), not(q4)))))), true, not(and(not(q4), q4)), implies(q4, implies(not(p4), not(not(q4))))), true)
% 8.75/1.53 = { by lemma 51 R->L }
% 8.75/1.53 fresh11(fresh59(fresh28(fresh59(true, true, implies(or(not(not(q4)), not(not(p4))), implies(not(p4), not(not(q4)))), or(implies(not(p4), not(not(q4))), not(not(not(q4))))), true, or(and(not(q4), q4), not(and(q4, and(not(p4), not(q4)))))), true, not(and(not(q4), q4)), implies(q4, implies(not(p4), not(not(q4))))), true)
% 8.75/1.53 = { by lemma 57 R->L }
% 8.75/1.53 fresh11(fresh59(fresh28(fresh59(fresh28(is_a_theorem(implies(not(not(q4)), or(not(not(q4)), not(not(p4))))), true, or(and(or(not(not(q4)), not(not(p4))), not(implies(not(p4), not(not(q4))))), not(and(not(implies(not(p4), not(not(q4)))), not(not(q4)))))), true, implies(or(not(not(q4)), not(not(p4))), implies(not(p4), not(not(q4)))), or(implies(not(p4), not(not(q4))), not(not(not(q4))))), true, or(and(not(q4), q4), not(and(q4, and(not(p4), not(q4)))))), true, not(and(not(q4), q4)), implies(q4, implies(not(p4), not(not(q4))))), true)
% 8.75/1.53 = { by lemma 42 R->L }
% 8.75/1.54 fresh11(fresh59(fresh28(fresh59(fresh28(is_a_theorem(implies(not(not(q4)), implies(not(not(not(q4))), not(not(p4))))), true, or(and(or(not(not(q4)), not(not(p4))), not(implies(not(p4), not(not(q4))))), not(and(not(implies(not(p4), not(not(q4)))), not(not(q4)))))), true, implies(or(not(not(q4)), not(not(p4))), implies(not(p4), not(not(q4)))), or(implies(not(p4), not(not(q4))), not(not(not(q4))))), true, or(and(not(q4), q4), not(and(q4, and(not(p4), not(q4)))))), true, not(and(not(q4), q4)), implies(q4, implies(not(p4), not(not(q4))))), true)
% 8.75/1.54 = { by axiom 14 (modus_ponens_2) R->L }
% 8.75/1.54 fresh11(fresh59(fresh28(fresh59(fresh28(fresh28(true, true, implies(not(not(q4)), implies(not(not(not(q4))), not(not(p4))))), true, or(and(or(not(not(q4)), not(not(p4))), not(implies(not(p4), not(not(q4))))), not(and(not(implies(not(p4), not(not(q4)))), not(not(q4)))))), true, implies(or(not(not(q4)), not(not(p4))), implies(not(p4), not(not(q4)))), or(implies(not(p4), not(not(q4))), not(not(not(q4))))), true, or(and(not(q4), q4), not(and(q4, and(not(p4), not(q4)))))), true, not(and(not(q4), q4)), implies(q4, implies(not(p4), not(not(q4))))), true)
% 8.75/1.54 = { by lemma 58 R->L }
% 8.75/1.54 fresh11(fresh59(fresh28(fresh59(fresh28(fresh28(is_a_theorem(implies(not(not(not(q4))), not(not(not(q4))))), true, implies(not(not(q4)), implies(not(not(not(q4))), not(not(p4))))), true, or(and(or(not(not(q4)), not(not(p4))), not(implies(not(p4), not(not(q4))))), not(and(not(implies(not(p4), not(not(q4)))), not(not(q4)))))), true, implies(or(not(not(q4)), not(not(p4))), implies(not(p4), not(not(q4)))), or(implies(not(p4), not(not(q4))), not(not(not(q4))))), true, or(and(not(q4), q4), not(and(q4, and(not(p4), not(q4)))))), true, not(and(not(q4), q4)), implies(q4, implies(not(p4), not(not(q4))))), true)
% 8.75/1.54 = { by lemma 61 R->L }
% 8.75/1.54 fresh11(fresh59(fresh28(fresh59(fresh28(fresh28(is_a_theorem(not(and(not(not(not(q4))), not(not(q4))))), true, implies(not(not(q4)), implies(not(not(not(q4))), not(not(p4))))), true, or(and(or(not(not(q4)), not(not(p4))), not(implies(not(p4), not(not(q4))))), not(and(not(implies(not(p4), not(not(q4)))), not(not(q4)))))), true, implies(or(not(not(q4)), not(not(p4))), implies(not(p4), not(not(q4)))), or(implies(not(p4), not(not(q4))), not(not(not(q4))))), true, or(and(not(q4), q4), not(and(q4, and(not(p4), not(q4)))))), true, not(and(not(q4), q4)), implies(q4, implies(not(p4), not(not(q4))))), true)
% 8.75/1.54 = { by lemma 55 R->L }
% 8.75/1.54 fresh11(fresh59(fresh28(fresh59(fresh28(fresh59(is_a_theorem(or(and(not(not(not(q4))), not(not(q4))), implies(not(not(q4)), implies(not(not(not(q4))), not(not(p4)))))), true, not(and(not(not(not(q4))), not(not(q4)))), implies(not(not(q4)), implies(not(not(not(q4))), not(not(p4))))), true, or(and(or(not(not(q4)), not(not(p4))), not(implies(not(p4), not(not(q4))))), not(and(not(implies(not(p4), not(not(q4)))), not(not(q4)))))), true, implies(or(not(not(q4)), not(not(p4))), implies(not(p4), not(not(q4)))), or(implies(not(p4), not(not(q4))), not(not(not(q4))))), true, or(and(not(q4), q4), not(and(q4, and(not(p4), not(q4)))))), true, not(and(not(q4), q4)), implies(q4, implies(not(p4), not(not(q4))))), true)
% 8.75/1.54 = { by axiom 14 (modus_ponens_2) R->L }
% 8.75/1.54 fresh11(fresh59(fresh28(fresh59(fresh28(fresh59(fresh28(true, true, or(and(not(not(not(q4))), not(not(q4))), implies(not(not(q4)), implies(not(not(not(q4))), not(not(p4)))))), true, not(and(not(not(not(q4))), not(not(q4)))), implies(not(not(q4)), implies(not(not(not(q4))), not(not(p4))))), true, or(and(or(not(not(q4)), not(not(p4))), not(implies(not(p4), not(not(q4))))), not(and(not(implies(not(p4), not(not(q4)))), not(not(q4)))))), true, implies(or(not(not(q4)), not(not(p4))), implies(not(p4), not(not(q4)))), or(implies(not(p4), not(not(q4))), not(not(not(q4))))), true, or(and(not(q4), q4), not(and(q4, and(not(p4), not(q4)))))), true, not(and(not(q4), q4)), implies(q4, implies(not(p4), not(not(q4))))), true)
% 8.75/1.54 = { by lemma 52 R->L }
% 8.75/1.55 fresh11(fresh59(fresh28(fresh59(fresh28(fresh59(fresh28(fresh58(and_1, true, not(not(not(q4))), not(not(not(p4)))), true, or(and(not(not(not(q4))), not(not(q4))), implies(not(not(q4)), implies(not(not(not(q4))), not(not(p4)))))), true, not(and(not(not(not(q4))), not(not(q4)))), implies(not(not(q4)), implies(not(not(not(q4))), not(not(p4))))), true, or(and(or(not(not(q4)), not(not(p4))), not(implies(not(p4), not(not(q4))))), not(and(not(implies(not(p4), not(not(q4)))), not(not(q4)))))), true, implies(or(not(not(q4)), not(not(p4))), implies(not(p4), not(not(q4)))), or(implies(not(p4), not(not(q4))), not(not(not(q4))))), true, or(and(not(q4), q4), not(and(q4, and(not(p4), not(q4)))))), true, not(and(not(q4), q4)), implies(q4, implies(not(p4), not(not(q4))))), true)
% 8.75/1.55 = { by axiom 28 (and_1_1) }
% 8.75/1.55 fresh11(fresh59(fresh28(fresh59(fresh28(fresh59(fresh28(is_a_theorem(implies(and(not(not(not(q4))), not(not(not(p4)))), not(not(not(q4))))), true, or(and(not(not(not(q4))), not(not(q4))), implies(not(not(q4)), implies(not(not(not(q4))), not(not(p4)))))), true, not(and(not(not(not(q4))), not(not(q4)))), implies(not(not(q4)), implies(not(not(not(q4))), not(not(p4))))), true, or(and(or(not(not(q4)), not(not(p4))), not(implies(not(p4), not(not(q4))))), not(and(not(implies(not(p4), not(not(q4)))), not(not(q4)))))), true, implies(or(not(not(q4)), not(not(p4))), implies(not(p4), not(not(q4)))), or(implies(not(p4), not(not(q4))), not(not(not(q4))))), true, or(and(not(q4), q4), not(and(q4, and(not(p4), not(q4)))))), true, not(and(not(q4), q4)), implies(q4, implies(not(p4), not(not(q4))))), true)
% 8.75/1.55 = { by lemma 62 }
% 8.75/1.55 fresh11(fresh59(fresh28(fresh59(fresh28(fresh59(fresh28(is_a_theorem(or(implies(not(not(not(q4))), not(not(p4))), not(not(not(q4))))), true, or(and(not(not(not(q4))), not(not(q4))), implies(not(not(q4)), implies(not(not(not(q4))), not(not(p4)))))), true, not(and(not(not(not(q4))), not(not(q4)))), implies(not(not(q4)), implies(not(not(not(q4))), not(not(p4))))), true, or(and(or(not(not(q4)), not(not(p4))), not(implies(not(p4), not(not(q4))))), not(and(not(implies(not(p4), not(not(q4)))), not(not(q4)))))), true, implies(or(not(not(q4)), not(not(p4))), implies(not(p4), not(not(q4)))), or(implies(not(p4), not(not(q4))), not(not(not(q4))))), true, or(and(not(q4), q4), not(and(q4, and(not(p4), not(q4)))))), true, not(and(not(q4), q4)), implies(q4, implies(not(p4), not(not(q4))))), true)
% 8.75/1.55 = { by axiom 36 (modus_ponens_2) R->L }
% 8.75/1.55 fresh11(fresh59(fresh28(fresh59(fresh28(fresh59(fresh59(is_a_theorem(implies(or(implies(not(not(not(q4))), not(not(p4))), not(not(not(q4)))), or(and(not(not(not(q4))), not(not(q4))), implies(not(not(q4)), implies(not(not(not(q4))), not(not(p4))))))), true, or(implies(not(not(not(q4))), not(not(p4))), not(not(not(q4)))), or(and(not(not(not(q4))), not(not(q4))), implies(not(not(q4)), implies(not(not(not(q4))), not(not(p4)))))), true, not(and(not(not(not(q4))), not(not(q4)))), implies(not(not(q4)), implies(not(not(not(q4))), not(not(p4))))), true, or(and(or(not(not(q4)), not(not(p4))), not(implies(not(p4), not(not(q4))))), not(and(not(implies(not(p4), not(not(q4)))), not(not(q4)))))), true, implies(or(not(not(q4)), not(not(p4))), implies(not(p4), not(not(q4)))), or(implies(not(p4), not(not(q4))), not(not(not(q4))))), true, or(and(not(q4), q4), not(and(q4, and(not(p4), not(q4)))))), true, not(and(not(q4), q4)), implies(q4, implies(not(p4), not(not(q4))))), true)
% 8.75/1.55 = { by lemma 63 }
% 8.75/1.55 fresh11(fresh59(fresh28(fresh59(fresh28(fresh59(fresh59(true, true, or(implies(not(not(not(q4))), not(not(p4))), not(not(not(q4)))), or(and(not(not(not(q4))), not(not(q4))), implies(not(not(q4)), implies(not(not(not(q4))), not(not(p4)))))), true, not(and(not(not(not(q4))), not(not(q4)))), implies(not(not(q4)), implies(not(not(not(q4))), not(not(p4))))), true, or(and(or(not(not(q4)), not(not(p4))), not(implies(not(p4), not(not(q4))))), not(and(not(implies(not(p4), not(not(q4)))), not(not(q4)))))), true, implies(or(not(not(q4)), not(not(p4))), implies(not(p4), not(not(q4)))), or(implies(not(p4), not(not(q4))), not(not(not(q4))))), true, or(and(not(q4), q4), not(and(q4, and(not(p4), not(q4)))))), true, not(and(not(q4), q4)), implies(q4, implies(not(p4), not(not(q4))))), true)
% 8.75/1.55 = { by lemma 51 }
% 8.75/1.55 fresh11(fresh59(fresh28(fresh59(fresh28(fresh59(true, true, not(and(not(not(not(q4))), not(not(q4)))), implies(not(not(q4)), implies(not(not(not(q4))), not(not(p4))))), true, or(and(or(not(not(q4)), not(not(p4))), not(implies(not(p4), not(not(q4))))), not(and(not(implies(not(p4), not(not(q4)))), not(not(q4)))))), true, implies(or(not(not(q4)), not(not(p4))), implies(not(p4), not(not(q4)))), or(implies(not(p4), not(not(q4))), not(not(not(q4))))), true, or(and(not(q4), q4), not(and(q4, and(not(p4), not(q4)))))), true, not(and(not(q4), q4)), implies(q4, implies(not(p4), not(not(q4))))), true)
% 8.75/1.55 = { by lemma 51 }
% 8.75/1.55 fresh11(fresh59(fresh28(fresh59(fresh28(true, true, or(and(or(not(not(q4)), not(not(p4))), not(implies(not(p4), not(not(q4))))), not(and(not(implies(not(p4), not(not(q4)))), not(not(q4)))))), true, implies(or(not(not(q4)), not(not(p4))), implies(not(p4), not(not(q4)))), or(implies(not(p4), not(not(q4))), not(not(not(q4))))), true, or(and(not(q4), q4), not(and(q4, and(not(p4), not(q4)))))), true, not(and(not(q4), q4)), implies(q4, implies(not(p4), not(not(q4))))), true)
% 8.75/1.56 = { by axiom 14 (modus_ponens_2) }
% 8.75/1.56 fresh11(fresh59(fresh28(fresh59(is_a_theorem(or(and(or(not(not(q4)), not(not(p4))), not(implies(not(p4), not(not(q4))))), not(and(not(implies(not(p4), not(not(q4)))), not(not(q4)))))), true, implies(or(not(not(q4)), not(not(p4))), implies(not(p4), not(not(q4)))), or(implies(not(p4), not(not(q4))), not(not(not(q4))))), true, or(and(not(q4), q4), not(and(q4, and(not(p4), not(q4)))))), true, not(and(not(q4), q4)), implies(q4, implies(not(p4), not(not(q4))))), true)
% 8.75/1.56 = { by lemma 61 }
% 8.75/1.56 fresh11(fresh59(fresh28(fresh59(is_a_theorem(or(and(or(not(not(q4)), not(not(p4))), not(implies(not(p4), not(not(q4))))), implies(not(implies(not(p4), not(not(q4)))), not(not(not(q4)))))), true, implies(or(not(not(q4)), not(not(p4))), implies(not(p4), not(not(q4)))), or(implies(not(p4), not(not(q4))), not(not(not(q4))))), true, or(and(not(q4), q4), not(and(q4, and(not(p4), not(q4)))))), true, not(and(not(q4), q4)), implies(q4, implies(not(p4), not(not(q4))))), true)
% 8.75/1.56 = { by lemma 42 }
% 8.75/1.56 fresh11(fresh59(fresh28(fresh59(is_a_theorem(or(and(or(not(not(q4)), not(not(p4))), not(implies(not(p4), not(not(q4))))), or(implies(not(p4), not(not(q4))), not(not(not(q4)))))), true, implies(or(not(not(q4)), not(not(p4))), implies(not(p4), not(not(q4)))), or(implies(not(p4), not(not(q4))), not(not(not(q4))))), true, or(and(not(q4), q4), not(and(q4, and(not(p4), not(q4)))))), true, not(and(not(q4), q4)), implies(q4, implies(not(p4), not(not(q4))))), true)
% 8.75/1.56 = { by lemma 48 }
% 8.75/1.56 fresh11(fresh59(fresh28(fresh59(is_a_theorem(implies(implies(or(not(not(q4)), not(not(p4))), implies(not(p4), not(not(q4)))), or(implies(not(p4), not(not(q4))), not(not(not(q4)))))), true, implies(or(not(not(q4)), not(not(p4))), implies(not(p4), not(not(q4)))), or(implies(not(p4), not(not(q4))), not(not(not(q4))))), true, or(and(not(q4), q4), not(and(q4, and(not(p4), not(q4)))))), true, not(and(not(q4), q4)), implies(q4, implies(not(p4), not(not(q4))))), true)
% 8.75/1.56 = { by axiom 36 (modus_ponens_2) }
% 8.75/1.56 fresh11(fresh59(fresh28(fresh28(is_a_theorem(implies(or(not(not(q4)), not(not(p4))), implies(not(p4), not(not(q4))))), true, or(implies(not(p4), not(not(q4))), not(not(not(q4))))), true, or(and(not(q4), q4), not(and(q4, and(not(p4), not(q4)))))), true, not(and(not(q4), q4)), implies(q4, implies(not(p4), not(not(q4))))), true)
% 8.75/1.56 = { by lemma 43 R->L }
% 8.75/1.56 fresh11(fresh59(fresh28(fresh28(is_a_theorem(or(not(or(not(not(q4)), not(not(p4)))), implies(not(p4), not(not(q4))))), true, or(implies(not(p4), not(not(q4))), not(not(not(q4))))), true, or(and(not(q4), q4), not(and(q4, and(not(p4), not(q4)))))), true, not(and(not(q4), q4)), implies(q4, implies(not(p4), not(not(q4))))), true)
% 8.75/1.56 = { by lemma 50 }
% 8.75/1.56 fresh11(fresh59(fresh28(fresh28(is_a_theorem(or(and(not(not(not(q4))), not(p4)), implies(not(p4), not(not(q4))))), true, or(implies(not(p4), not(not(q4))), not(not(not(q4))))), true, or(and(not(q4), q4), not(and(q4, and(not(p4), not(q4)))))), true, not(and(not(q4), q4)), implies(q4, implies(not(p4), not(not(q4))))), true)
% 8.75/1.56 = { by lemma 60 R->L }
% 8.75/1.56 fresh11(fresh59(fresh28(fresh28(is_a_theorem(or(and(not(not(not(q4))), not(p4)), implies(not(p4), not(not(not(not(q4))))))), true, or(implies(not(p4), not(not(q4))), not(not(not(q4))))), true, or(and(not(q4), q4), not(and(q4, and(not(p4), not(q4)))))), true, not(and(not(q4), q4)), implies(q4, implies(not(p4), not(not(q4))))), true)
% 8.75/1.56 = { by axiom 14 (modus_ponens_2) R->L }
% 8.75/1.56 fresh11(fresh59(fresh28(fresh28(fresh28(true, true, or(and(not(not(not(q4))), not(p4)), implies(not(p4), not(not(not(not(q4))))))), true, or(implies(not(p4), not(not(q4))), not(not(not(q4))))), true, or(and(not(q4), q4), not(and(q4, and(not(p4), not(q4)))))), true, not(and(not(q4), q4)), implies(q4, implies(not(p4), not(not(q4))))), true)
% 8.75/1.56 = { by lemma 58 R->L }
% 8.75/1.56 fresh11(fresh59(fresh28(fresh28(fresh28(is_a_theorem(implies(not(not(not(q4))), not(not(not(q4))))), true, or(and(not(not(not(q4))), not(p4)), implies(not(p4), not(not(not(not(q4))))))), true, or(implies(not(p4), not(not(q4))), not(not(not(q4))))), true, or(and(not(q4), q4), not(and(q4, and(not(p4), not(q4)))))), true, not(and(not(q4), q4)), implies(q4, implies(not(p4), not(not(q4))))), true)
% 8.75/1.56 = { by axiom 36 (modus_ponens_2) R->L }
% 8.75/1.56 fresh11(fresh59(fresh28(fresh28(fresh59(is_a_theorem(implies(implies(not(not(not(q4))), not(not(not(q4)))), or(and(not(not(not(q4))), not(p4)), implies(not(p4), not(not(not(not(q4)))))))), true, implies(not(not(not(q4))), not(not(not(q4)))), or(and(not(not(not(q4))), not(p4)), implies(not(p4), not(not(not(not(q4))))))), true, or(implies(not(p4), not(not(q4))), not(not(not(q4))))), true, or(and(not(q4), q4), not(and(q4, and(not(p4), not(q4)))))), true, not(and(not(q4), q4)), implies(q4, implies(not(p4), not(not(q4))))), true)
% 8.75/1.56 = { by lemma 43 R->L }
% 8.75/1.56 fresh11(fresh59(fresh28(fresh28(fresh59(is_a_theorem(implies(or(not(not(not(not(q4)))), not(not(not(q4)))), or(and(not(not(not(q4))), not(p4)), implies(not(p4), not(not(not(not(q4)))))))), true, implies(not(not(not(q4))), not(not(not(q4)))), or(and(not(not(not(q4))), not(p4)), implies(not(p4), not(not(not(not(q4))))))), true, or(implies(not(p4), not(not(q4))), not(not(not(q4))))), true, or(and(not(q4), q4), not(and(q4, and(not(p4), not(q4)))))), true, not(and(not(q4), q4)), implies(q4, implies(not(p4), not(not(q4))))), true)
% 8.75/1.56 = { by lemma 63 }
% 8.75/1.56 fresh11(fresh59(fresh28(fresh28(fresh59(true, true, implies(not(not(not(q4))), not(not(not(q4)))), or(and(not(not(not(q4))), not(p4)), implies(not(p4), not(not(not(not(q4))))))), true, or(implies(not(p4), not(not(q4))), not(not(not(q4))))), true, or(and(not(q4), q4), not(and(q4, and(not(p4), not(q4)))))), true, not(and(not(q4), q4)), implies(q4, implies(not(p4), not(not(q4))))), true)
% 8.75/1.56 = { by lemma 51 }
% 8.75/1.56 fresh11(fresh59(fresh28(fresh28(true, true, or(implies(not(p4), not(not(q4))), not(not(not(q4))))), true, or(and(not(q4), q4), not(and(q4, and(not(p4), not(q4)))))), true, not(and(not(q4), q4)), implies(q4, implies(not(p4), not(not(q4))))), true)
% 8.75/1.56 = { by axiom 14 (modus_ponens_2) }
% 8.75/1.56 fresh11(fresh59(fresh28(is_a_theorem(or(implies(not(p4), not(not(q4))), not(not(not(q4))))), true, or(and(not(q4), q4), not(and(q4, and(not(p4), not(q4)))))), true, not(and(not(q4), q4)), implies(q4, implies(not(p4), not(not(q4))))), true)
% 8.75/1.56 = { by lemma 62 R->L }
% 8.75/1.56 fresh11(fresh59(fresh28(is_a_theorem(implies(and(not(p4), not(not(not(q4)))), not(not(not(q4))))), true, or(and(not(q4), q4), not(and(q4, and(not(p4), not(q4)))))), true, not(and(not(q4), q4)), implies(q4, implies(not(p4), not(not(q4))))), true)
% 8.75/1.56 = { by axiom 30 (and_2_1) R->L }
% 8.75/1.57 fresh11(fresh59(fresh28(fresh55(and_2, true, not(p4), not(not(not(q4)))), true, or(and(not(q4), q4), not(and(q4, and(not(p4), not(q4)))))), true, not(and(not(q4), q4)), implies(q4, implies(not(p4), not(not(q4))))), true)
% 8.75/1.57 = { by lemma 60 }
% 8.75/1.57 fresh11(fresh59(fresh28(fresh55(and_2, true, not(p4), not(q4)), true, or(and(not(q4), q4), not(and(q4, and(not(p4), not(q4)))))), true, not(and(not(q4), q4)), implies(q4, implies(not(p4), not(not(q4))))), true)
% 8.75/1.57 = { by axiom 30 (and_2_1) }
% 8.75/1.57 fresh11(fresh59(fresh28(is_a_theorem(implies(and(not(p4), not(q4)), not(q4))), true, or(and(not(q4), q4), not(and(q4, and(not(p4), not(q4)))))), true, not(and(not(q4), q4)), implies(q4, implies(not(p4), not(not(q4))))), true)
% 8.75/1.57 = { by lemma 57 }
% 8.75/1.57 fresh11(fresh59(true, true, not(and(not(q4), q4)), implies(q4, implies(not(p4), not(not(q4))))), true)
% 8.75/1.57 = { by lemma 51 }
% 8.75/1.57 fresh11(true, true)
% 8.75/1.57 = { by axiom 11 (r2) }
% 8.75/1.57 true
% 8.75/1.57 % SZS output end Proof
% 8.75/1.57
% 8.75/1.57 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------