TSTP Solution File: LCL461+1 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : LCL461+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 : n001.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 08:19:06 EDT 2023
% Result : Theorem 21.20s 3.13s
% Output : Proof 22.52s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : LCL461+1 : TPTP v8.1.2. Released v3.3.0.
% 0.00/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.13/0.34 % Computer : n001.cluster.edu
% 0.13/0.34 % Model : x86_64 x86_64
% 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34 % Memory : 8042.1875MB
% 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.34 % CPULimit : 300
% 0.13/0.34 % WCLimit : 300
% 0.13/0.34 % DateTime : Fri Aug 25 04:15:27 EDT 2023
% 0.13/0.34 % CPUTime :
% 21.20/3.13 Command-line arguments: --no-flatten-goal
% 21.20/3.13
% 21.20/3.13 % SZS status Theorem
% 21.20/3.13
% 22.23/3.21 % SZS output start Proof
% 22.23/3.21 Take the following subset of the input axioms:
% 22.23/3.21 fof(and_1, axiom, and_1 <=> ![X, Y]: is_a_theorem(implies(and(X, Y), X))).
% 22.23/3.21 fof(and_3, axiom, and_3 <=> ![X2, Y2]: is_a_theorem(implies(X2, implies(Y2, and(X2, Y2))))).
% 22.23/3.21 fof(cn3, axiom, cn3 <=> ![P]: is_a_theorem(implies(implies(not(P), P), P))).
% 22.23/3.21 fof(equivalence_3, axiom, equivalence_3 <=> ![X2, Y2]: is_a_theorem(implies(implies(X2, Y2), implies(implies(Y2, X2), equiv(X2, Y2))))).
% 22.23/3.21 fof(hilbert_and_1, axiom, and_1).
% 22.23/3.21 fof(hilbert_and_3, axiom, and_3).
% 22.23/3.21 fof(hilbert_equivalence_3, axiom, equivalence_3).
% 22.23/3.21 fof(hilbert_implies_1, axiom, implies_1).
% 22.23/3.21 fof(hilbert_implies_2, axiom, implies_2).
% 22.23/3.21 fof(hilbert_implies_3, axiom, implies_3).
% 22.23/3.21 fof(hilbert_modus_ponens, axiom, modus_ponens).
% 22.23/3.21 fof(hilbert_modus_tollens, axiom, modus_tollens).
% 22.23/3.21 fof(hilbert_op_implies_and, axiom, op_implies_and).
% 22.23/3.21 fof(hilbert_op_or, axiom, op_or).
% 22.23/3.21 fof(hilbert_or_3, axiom, or_3).
% 22.23/3.21 fof(implies_1, axiom, implies_1 <=> ![X2, Y2]: is_a_theorem(implies(X2, implies(Y2, X2)))).
% 22.23/3.21 fof(implies_2, axiom, implies_2 <=> ![X2, Y2]: is_a_theorem(implies(implies(X2, implies(X2, Y2)), implies(X2, Y2)))).
% 22.23/3.21 fof(implies_3, axiom, implies_3 <=> ![Z, X2, Y2]: is_a_theorem(implies(implies(X2, Y2), implies(implies(Y2, Z), implies(X2, Z))))).
% 22.23/3.21 fof(kn1, axiom, kn1 <=> ![P2]: is_a_theorem(implies(P2, and(P2, P2)))).
% 22.23/3.21 fof(kn2, axiom, kn2 <=> ![Q, P2]: is_a_theorem(implies(and(P2, Q), P2))).
% 22.23/3.21 fof(kn3, axiom, kn3 <=> ![R, P2, Q2]: is_a_theorem(implies(implies(P2, Q2), implies(not(and(Q2, R)), not(and(R, P2)))))).
% 22.23/3.21 fof(modus_ponens, axiom, modus_ponens <=> ![X2, Y2]: ((is_a_theorem(X2) & is_a_theorem(implies(X2, Y2))) => is_a_theorem(Y2))).
% 22.23/3.21 fof(modus_tollens, axiom, modus_tollens <=> ![X2, Y2]: is_a_theorem(implies(implies(not(Y2), not(X2)), implies(X2, Y2)))).
% 22.23/3.21 fof(op_implies_and, axiom, op_implies_and => ![X2, Y2]: implies(X2, Y2)=not(and(X2, not(Y2)))).
% 22.23/3.21 fof(op_or, axiom, op_or => ![X2, Y2]: or(X2, Y2)=not(and(not(X2), not(Y2)))).
% 22.23/3.21 fof(or_3, axiom, or_3 <=> ![X2, Y2, Z2]: is_a_theorem(implies(implies(X2, Z2), implies(implies(Y2, Z2), implies(or(X2, Y2), Z2))))).
% 22.23/3.21 fof(r1, axiom, r1 <=> ![P2]: is_a_theorem(implies(or(P2, P2), P2))).
% 22.23/3.21 fof(r3, axiom, r3 <=> ![P2, Q2]: is_a_theorem(implies(or(P2, Q2), or(Q2, P2)))).
% 22.23/3.21 fof(rosser_kn3, conjecture, kn3).
% 22.23/3.21 fof(substitution_of_equivalents, axiom, substitution_of_equivalents <=> ![X2, Y2]: (is_a_theorem(equiv(X2, Y2)) => X2=Y2)).
% 22.23/3.21 fof(substitution_of_equivalents, axiom, substitution_of_equivalents).
% 22.23/3.21
% 22.23/3.21 Now clausify the problem and encode Horn clauses using encoding 3 of
% 22.23/3.21 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 22.23/3.21 We repeatedly replace C & s=t => u=v by the two clauses:
% 22.23/3.21 fresh(y, y, x1...xn) = u
% 22.23/3.21 C => fresh(s, t, x1...xn) = v
% 22.23/3.21 where fresh is a fresh function symbol and x1..xn are the free
% 22.23/3.21 variables of u and v.
% 22.23/3.21 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 22.23/3.21 input problem has no model of domain size 1).
% 22.23/3.21
% 22.23/3.21 The encoding turns the above axioms into the following unit equations and goals:
% 22.23/3.21
% 22.23/3.21 Axiom 1 (hilbert_modus_ponens): modus_ponens = true.
% 22.23/3.21 Axiom 2 (substitution_of_equivalents): substitution_of_equivalents = true.
% 22.23/3.21 Axiom 3 (hilbert_modus_tollens): modus_tollens = true.
% 22.23/3.21 Axiom 4 (hilbert_implies_1): implies_1 = true.
% 22.23/3.21 Axiom 5 (hilbert_implies_2): implies_2 = true.
% 22.23/3.21 Axiom 6 (hilbert_implies_3): implies_3 = true.
% 22.23/3.21 Axiom 7 (hilbert_and_1): and_1 = true.
% 22.23/3.21 Axiom 8 (hilbert_and_3): and_3 = true.
% 22.23/3.21 Axiom 9 (hilbert_or_3): or_3 = true.
% 22.23/3.21 Axiom 10 (hilbert_equivalence_3): equivalence_3 = true.
% 22.23/3.21 Axiom 11 (hilbert_op_or): op_or = true.
% 22.23/3.21 Axiom 12 (hilbert_op_implies_and): op_implies_and = true.
% 22.23/3.21 Axiom 13 (cn3): fresh48(X, X) = true.
% 22.23/3.21 Axiom 14 (kn1): fresh34(X, X) = true.
% 22.23/3.21 Axiom 15 (kn2): fresh32(X, X) = true.
% 22.23/3.21 Axiom 16 (kn3): fresh30(X, X) = true.
% 22.23/3.21 Axiom 17 (r3): fresh9(X, X) = true.
% 22.23/3.21 Axiom 18 (modus_ponens_2): fresh60(X, X, Y) = true.
% 22.23/3.21 Axiom 19 (cn3_1): fresh47(X, X, Y) = true.
% 22.23/3.21 Axiom 20 (modus_ponens_2): fresh28(X, X, Y) = is_a_theorem(Y).
% 22.23/3.21 Axiom 21 (substitution_of_equivalents_2): fresh(X, X, Y, Z) = Z.
% 22.23/3.21 Axiom 22 (modus_ponens_2): fresh59(X, X, Y, Z) = fresh60(modus_ponens, true, Z).
% 22.23/3.21 Axiom 23 (and_1_1): fresh58(X, X, Y, Z) = true.
% 22.23/3.21 Axiom 24 (and_3_1): fresh53(X, X, Y, Z) = true.
% 22.23/3.21 Axiom 25 (equivalence_3_1): fresh41(X, X, Y, Z) = true.
% 22.23/3.21 Axiom 26 (implies_1_1): fresh39(X, X, Y, Z) = true.
% 22.23/3.21 Axiom 27 (implies_2_1): fresh37(X, X, Y, Z) = true.
% 22.23/3.21 Axiom 28 (modus_tollens_1): fresh25(X, X, Y, Z) = true.
% 22.23/3.21 Axiom 29 (op_implies_and): fresh22(X, X, Y, Z) = implies(Y, Z).
% 22.23/3.21 Axiom 30 (op_or): fresh20(X, X, Y, Z) = or(Y, Z).
% 22.23/3.21 Axiom 31 (r3_1): fresh8(X, X, Y, Z) = true.
% 22.23/3.21 Axiom 32 (substitution_of_equivalents_2): fresh2(X, X, Y, Z) = Y.
% 22.23/3.21 Axiom 33 (op_implies_and): fresh22(op_implies_and, true, X, Y) = not(and(X, not(Y))).
% 22.23/3.21 Axiom 34 (implies_3_1): fresh35(X, X, Y, Z, W) = true.
% 22.23/3.21 Axiom 35 (or_3_1): fresh14(X, X, Y, Z, W) = true.
% 22.23/3.21 Axiom 36 (op_or): fresh20(op_or, true, X, Y) = not(and(not(X), not(Y))).
% 22.23/3.21 Axiom 37 (kn1_1): fresh33(kn1, true, X) = is_a_theorem(implies(X, and(X, X))).
% 22.23/3.21 Axiom 38 (implies_1_1): fresh39(implies_1, true, X, Y) = is_a_theorem(implies(X, implies(Y, X))).
% 22.23/3.21 Axiom 39 (and_1_1): fresh58(and_1, true, X, Y) = is_a_theorem(implies(and(X, Y), X)).
% 22.23/3.21 Axiom 40 (r1_1): fresh12(r1, true, X) = is_a_theorem(implies(or(X, X), X)).
% 22.23/3.21 Axiom 41 (cn3_1): fresh47(cn3, true, X) = is_a_theorem(implies(implies(not(X), X), X)).
% 22.23/3.22 Axiom 42 (substitution_of_equivalents_2): fresh2(substitution_of_equivalents, true, X, Y) = fresh(is_a_theorem(equiv(X, Y)), true, X, Y).
% 22.23/3.22 Axiom 43 (modus_ponens_2): fresh59(is_a_theorem(implies(X, Y)), true, X, Y) = fresh28(is_a_theorem(X), true, Y).
% 22.23/3.22 Axiom 44 (kn1): fresh34(is_a_theorem(implies(p11, and(p11, p11))), true) = kn1.
% 22.23/3.22 Axiom 45 (kn2): fresh32(is_a_theorem(implies(and(p10, q8), p10)), true) = kn2.
% 22.23/3.22 Axiom 46 (and_3_1): fresh53(and_3, true, X, Y) = is_a_theorem(implies(X, implies(Y, and(X, Y)))).
% 22.23/3.22 Axiom 47 (r3_1): fresh8(r3, true, X, Y) = is_a_theorem(implies(or(X, Y), or(Y, X))).
% 22.23/3.22 Axiom 48 (cn3): fresh48(is_a_theorem(implies(implies(not(p6), p6), p6)), true) = cn3.
% 22.23/3.22 Axiom 49 (r3): fresh9(is_a_theorem(implies(or(p3, q3), or(q3, p3))), true) = r3.
% 22.23/3.22 Axiom 50 (implies_2_1): fresh37(implies_2, true, X, Y) = is_a_theorem(implies(implies(X, implies(X, Y)), implies(X, Y))).
% 22.23/3.22 Axiom 51 (modus_tollens_1): fresh25(modus_tollens, true, X, Y) = is_a_theorem(implies(implies(not(Y), not(X)), implies(X, Y))).
% 22.23/3.22 Axiom 52 (equivalence_3_1): fresh41(equivalence_3, true, X, Y) = is_a_theorem(implies(implies(X, Y), implies(implies(Y, X), equiv(X, Y)))).
% 22.23/3.22 Axiom 53 (implies_3_1): fresh35(implies_3, true, X, Y, Z) = is_a_theorem(implies(implies(X, Y), implies(implies(Y, Z), implies(X, Z)))).
% 22.23/3.22 Axiom 54 (kn3_1): fresh29(kn3, true, X, Y, Z) = is_a_theorem(implies(implies(X, Y), implies(not(and(Y, Z)), not(and(Z, X))))).
% 22.23/3.22 Axiom 55 (or_3_1): fresh14(or_3, true, X, Y, Z) = is_a_theorem(implies(implies(X, Z), implies(implies(Y, Z), implies(or(X, Y), Z)))).
% 22.23/3.22 Axiom 56 (kn3): fresh30(is_a_theorem(implies(implies(p9, q7), implies(not(and(q7, r8)), not(and(r8, p9))))), true) = kn3.
% 22.23/3.22
% 22.23/3.22 Lemma 57: is_a_theorem(implies(and(X, Y), X)) = modus_ponens.
% 22.23/3.22 Proof:
% 22.23/3.22 is_a_theorem(implies(and(X, Y), X))
% 22.23/3.22 = { by axiom 39 (and_1_1) R->L }
% 22.23/3.22 fresh58(and_1, true, X, Y)
% 22.23/3.22 = { by axiom 7 (hilbert_and_1) }
% 22.23/3.22 fresh58(true, true, X, Y)
% 22.23/3.22 = { by axiom 1 (hilbert_modus_ponens) R->L }
% 22.23/3.22 fresh58(modus_ponens, true, X, Y)
% 22.23/3.22 = { by axiom 1 (hilbert_modus_ponens) R->L }
% 22.23/3.22 fresh58(modus_ponens, modus_ponens, X, Y)
% 22.23/3.22 = { by axiom 23 (and_1_1) }
% 22.23/3.22 true
% 22.23/3.22 = { by axiom 1 (hilbert_modus_ponens) R->L }
% 22.23/3.22 modus_ponens
% 22.23/3.22
% 22.23/3.22 Lemma 58: modus_ponens = kn2.
% 22.23/3.22 Proof:
% 22.23/3.22 modus_ponens
% 22.23/3.22 = { by axiom 1 (hilbert_modus_ponens) }
% 22.23/3.22 true
% 22.23/3.22 = { by axiom 15 (kn2) R->L }
% 22.23/3.22 fresh32(modus_ponens, modus_ponens)
% 22.23/3.22 = { by axiom 1 (hilbert_modus_ponens) }
% 22.23/3.22 fresh32(modus_ponens, true)
% 22.23/3.22 = { by lemma 57 R->L }
% 22.23/3.22 fresh32(is_a_theorem(implies(and(p10, q8), p10)), true)
% 22.23/3.22 = { by axiom 45 (kn2) }
% 22.23/3.22 kn2
% 22.23/3.22
% 22.23/3.22 Lemma 59: is_a_theorem(implies(X, and(X, X))) = fresh33(kn1, modus_ponens, X).
% 22.23/3.22 Proof:
% 22.23/3.22 is_a_theorem(implies(X, and(X, X)))
% 22.23/3.22 = { by axiom 37 (kn1_1) R->L }
% 22.23/3.22 fresh33(kn1, true, X)
% 22.23/3.22 = { by axiom 1 (hilbert_modus_ponens) R->L }
% 22.23/3.22 fresh33(kn1, modus_ponens, X)
% 22.23/3.22
% 22.23/3.22 Lemma 60: fresh59(is_a_theorem(implies(X, Y)), modus_ponens, X, Y) = fresh28(is_a_theorem(X), modus_ponens, Y).
% 22.23/3.22 Proof:
% 22.23/3.22 fresh59(is_a_theorem(implies(X, Y)), modus_ponens, X, Y)
% 22.23/3.22 = { by axiom 1 (hilbert_modus_ponens) }
% 22.23/3.22 fresh59(is_a_theorem(implies(X, Y)), true, X, Y)
% 22.23/3.22 = { by axiom 43 (modus_ponens_2) }
% 22.23/3.22 fresh28(is_a_theorem(X), true, Y)
% 22.23/3.22 = { by axiom 1 (hilbert_modus_ponens) R->L }
% 22.23/3.22 fresh28(is_a_theorem(X), modus_ponens, Y)
% 22.23/3.22
% 22.23/3.22 Lemma 61: fresh60(X, X, Y) = modus_ponens.
% 22.23/3.22 Proof:
% 22.23/3.22 fresh60(X, X, Y)
% 22.23/3.22 = { by axiom 18 (modus_ponens_2) }
% 22.23/3.22 true
% 22.23/3.22 = { by axiom 1 (hilbert_modus_ponens) R->L }
% 22.23/3.22 modus_ponens
% 22.23/3.22
% 22.23/3.22 Lemma 62: fresh59(X, X, Y, Z) = modus_ponens.
% 22.23/3.22 Proof:
% 22.23/3.22 fresh59(X, X, Y, Z)
% 22.23/3.22 = { by axiom 22 (modus_ponens_2) }
% 22.23/3.22 fresh60(modus_ponens, true, Z)
% 22.23/3.22 = { by axiom 1 (hilbert_modus_ponens) R->L }
% 22.23/3.22 fresh60(modus_ponens, modus_ponens, Z)
% 22.23/3.22 = { by lemma 61 }
% 22.23/3.22 modus_ponens
% 22.23/3.22
% 22.23/3.22 Lemma 63: fresh28(is_a_theorem(implies(X, implies(X, Y))), kn2, implies(X, Y)) = kn2.
% 22.23/3.22 Proof:
% 22.23/3.22 fresh28(is_a_theorem(implies(X, implies(X, Y))), kn2, implies(X, Y))
% 22.23/3.22 = { by lemma 58 R->L }
% 22.23/3.22 fresh28(is_a_theorem(implies(X, implies(X, Y))), modus_ponens, implies(X, Y))
% 22.23/3.22 = { by lemma 60 R->L }
% 22.23/3.22 fresh59(is_a_theorem(implies(implies(X, implies(X, Y)), implies(X, Y))), modus_ponens, implies(X, implies(X, Y)), implies(X, Y))
% 22.23/3.22 = { by axiom 50 (implies_2_1) R->L }
% 22.23/3.22 fresh59(fresh37(implies_2, true, X, Y), modus_ponens, implies(X, implies(X, Y)), implies(X, Y))
% 22.23/3.22 = { by axiom 5 (hilbert_implies_2) }
% 22.23/3.22 fresh59(fresh37(true, true, X, Y), modus_ponens, implies(X, implies(X, Y)), implies(X, Y))
% 22.23/3.22 = { by axiom 1 (hilbert_modus_ponens) R->L }
% 22.23/3.22 fresh59(fresh37(modus_ponens, true, X, Y), modus_ponens, implies(X, implies(X, Y)), implies(X, Y))
% 22.23/3.22 = { by lemma 58 }
% 22.23/3.22 fresh59(fresh37(kn2, true, X, Y), modus_ponens, implies(X, implies(X, Y)), implies(X, Y))
% 22.23/3.22 = { by axiom 1 (hilbert_modus_ponens) R->L }
% 22.23/3.22 fresh59(fresh37(kn2, modus_ponens, X, Y), modus_ponens, implies(X, implies(X, Y)), implies(X, Y))
% 22.23/3.22 = { by lemma 58 }
% 22.23/3.22 fresh59(fresh37(kn2, kn2, X, Y), modus_ponens, implies(X, implies(X, Y)), implies(X, Y))
% 22.23/3.22 = { by axiom 27 (implies_2_1) }
% 22.23/3.22 fresh59(true, modus_ponens, implies(X, implies(X, Y)), implies(X, Y))
% 22.23/3.22 = { by axiom 1 (hilbert_modus_ponens) R->L }
% 22.23/3.22 fresh59(modus_ponens, modus_ponens, implies(X, implies(X, Y)), implies(X, Y))
% 22.23/3.22 = { by lemma 58 }
% 22.23/3.22 fresh59(kn2, modus_ponens, implies(X, implies(X, Y)), implies(X, Y))
% 22.23/3.22 = { by lemma 58 }
% 22.23/3.22 fresh59(kn2, kn2, implies(X, implies(X, Y)), implies(X, Y))
% 22.23/3.22 = { by lemma 62 }
% 22.23/3.22 modus_ponens
% 22.23/3.22 = { by lemma 58 }
% 22.23/3.22 kn2
% 22.23/3.22
% 22.23/3.22 Lemma 64: fresh33(kn1, kn2, X) = kn2.
% 22.23/3.22 Proof:
% 22.23/3.22 fresh33(kn1, kn2, X)
% 22.23/3.22 = { by lemma 58 R->L }
% 22.23/3.22 fresh33(kn1, modus_ponens, X)
% 22.23/3.22 = { by lemma 59 R->L }
% 22.23/3.22 is_a_theorem(implies(X, and(X, X)))
% 22.23/3.22 = { by axiom 20 (modus_ponens_2) R->L }
% 22.23/3.22 fresh28(kn2, kn2, implies(X, and(X, X)))
% 22.23/3.22 = { by lemma 58 R->L }
% 22.23/3.22 fresh28(modus_ponens, kn2, implies(X, and(X, X)))
% 22.23/3.22 = { by axiom 1 (hilbert_modus_ponens) }
% 22.23/3.22 fresh28(true, kn2, implies(X, and(X, X)))
% 22.23/3.22 = { by axiom 24 (and_3_1) R->L }
% 22.23/3.22 fresh28(fresh53(kn2, kn2, X, X), kn2, implies(X, and(X, X)))
% 22.23/3.22 = { by lemma 58 R->L }
% 22.23/3.22 fresh28(fresh53(kn2, modus_ponens, X, X), kn2, implies(X, and(X, X)))
% 22.23/3.22 = { by axiom 1 (hilbert_modus_ponens) }
% 22.23/3.22 fresh28(fresh53(kn2, true, X, X), kn2, implies(X, and(X, X)))
% 22.23/3.22 = { by lemma 58 R->L }
% 22.23/3.22 fresh28(fresh53(modus_ponens, true, X, X), kn2, implies(X, and(X, X)))
% 22.23/3.22 = { by axiom 1 (hilbert_modus_ponens) }
% 22.23/3.22 fresh28(fresh53(true, true, X, X), kn2, implies(X, and(X, X)))
% 22.23/3.22 = { by axiom 8 (hilbert_and_3) R->L }
% 22.23/3.22 fresh28(fresh53(and_3, true, X, X), kn2, implies(X, and(X, X)))
% 22.23/3.22 = { by axiom 46 (and_3_1) }
% 22.23/3.22 fresh28(is_a_theorem(implies(X, implies(X, and(X, X)))), kn2, implies(X, and(X, X)))
% 22.23/3.22 = { by lemma 63 }
% 22.23/3.22 kn2
% 22.23/3.22
% 22.23/3.22 Lemma 65: kn2 = kn1.
% 22.23/3.22 Proof:
% 22.23/3.22 kn2
% 22.23/3.22 = { by lemma 58 R->L }
% 22.23/3.22 modus_ponens
% 22.23/3.22 = { by axiom 1 (hilbert_modus_ponens) }
% 22.23/3.22 true
% 22.23/3.22 = { by axiom 14 (kn1) R->L }
% 22.23/3.22 fresh34(kn2, kn2)
% 22.23/3.22 = { by lemma 64 R->L }
% 22.23/3.22 fresh34(fresh33(kn1, kn2, p11), kn2)
% 22.23/3.22 = { by lemma 58 R->L }
% 22.23/3.22 fresh34(fresh33(kn1, kn2, p11), modus_ponens)
% 22.23/3.22 = { by lemma 58 R->L }
% 22.23/3.22 fresh34(fresh33(kn1, modus_ponens, p11), modus_ponens)
% 22.23/3.22 = { by axiom 1 (hilbert_modus_ponens) }
% 22.23/3.22 fresh34(fresh33(kn1, modus_ponens, p11), true)
% 22.23/3.22 = { by lemma 59 R->L }
% 22.23/3.22 fresh34(is_a_theorem(implies(p11, and(p11, p11))), true)
% 22.23/3.22 = { by axiom 44 (kn1) }
% 22.23/3.22 kn1
% 22.23/3.22
% 22.23/3.22 Lemma 66: not(and(X, not(Y))) = implies(X, Y).
% 22.23/3.22 Proof:
% 22.23/3.22 not(and(X, not(Y)))
% 22.23/3.22 = { by axiom 33 (op_implies_and) R->L }
% 22.23/3.22 fresh22(op_implies_and, true, X, Y)
% 22.23/3.22 = { by axiom 12 (hilbert_op_implies_and) }
% 22.23/3.22 fresh22(true, true, X, Y)
% 22.23/3.22 = { by axiom 1 (hilbert_modus_ponens) R->L }
% 22.23/3.22 fresh22(modus_ponens, true, X, Y)
% 22.23/3.22 = { by axiom 1 (hilbert_modus_ponens) R->L }
% 22.23/3.22 fresh22(modus_ponens, modus_ponens, X, Y)
% 22.23/3.22 = { by axiom 29 (op_implies_and) }
% 22.23/3.22 implies(X, Y)
% 22.23/3.22
% 22.23/3.22 Lemma 67: implies(not(X), Y) = or(X, Y).
% 22.23/3.22 Proof:
% 22.23/3.22 implies(not(X), Y)
% 22.23/3.22 = { by lemma 66 R->L }
% 22.23/3.22 not(and(not(X), not(Y)))
% 22.23/3.22 = { by axiom 36 (op_or) R->L }
% 22.23/3.22 fresh20(op_or, true, X, Y)
% 22.23/3.22 = { by axiom 11 (hilbert_op_or) }
% 22.23/3.22 fresh20(true, true, X, Y)
% 22.23/3.22 = { by axiom 1 (hilbert_modus_ponens) R->L }
% 22.23/3.22 fresh20(modus_ponens, true, X, Y)
% 22.23/3.22 = { by axiom 1 (hilbert_modus_ponens) R->L }
% 22.23/3.22 fresh20(modus_ponens, modus_ponens, X, Y)
% 22.23/3.22 = { by axiom 30 (op_or) }
% 22.23/3.22 or(X, Y)
% 22.23/3.22
% 22.23/3.22 Lemma 68: fresh28(is_a_theorem(implies(X, Y)), kn1, implies(implies(Y, X), equiv(X, Y))) = kn1.
% 22.23/3.22 Proof:
% 22.23/3.22 fresh28(is_a_theorem(implies(X, Y)), kn1, implies(implies(Y, X), equiv(X, Y)))
% 22.23/3.22 = { by lemma 65 R->L }
% 22.23/3.22 fresh28(is_a_theorem(implies(X, Y)), kn2, implies(implies(Y, X), equiv(X, Y)))
% 22.23/3.22 = { by lemma 58 R->L }
% 22.23/3.22 fresh28(is_a_theorem(implies(X, Y)), modus_ponens, implies(implies(Y, X), equiv(X, Y)))
% 22.23/3.22 = { by lemma 60 R->L }
% 22.23/3.22 fresh59(is_a_theorem(implies(implies(X, Y), implies(implies(Y, X), equiv(X, Y)))), modus_ponens, implies(X, Y), implies(implies(Y, X), equiv(X, Y)))
% 22.23/3.22 = { by axiom 52 (equivalence_3_1) R->L }
% 22.23/3.22 fresh59(fresh41(equivalence_3, true, X, Y), modus_ponens, implies(X, Y), implies(implies(Y, X), equiv(X, Y)))
% 22.23/3.22 = { by axiom 10 (hilbert_equivalence_3) }
% 22.23/3.22 fresh59(fresh41(true, true, X, Y), modus_ponens, implies(X, Y), implies(implies(Y, X), equiv(X, Y)))
% 22.23/3.22 = { by axiom 1 (hilbert_modus_ponens) R->L }
% 22.23/3.22 fresh59(fresh41(modus_ponens, true, X, Y), modus_ponens, implies(X, Y), implies(implies(Y, X), equiv(X, Y)))
% 22.23/3.22 = { by lemma 58 }
% 22.23/3.22 fresh59(fresh41(kn2, true, X, Y), modus_ponens, implies(X, Y), implies(implies(Y, X), equiv(X, Y)))
% 22.23/3.22 = { by axiom 1 (hilbert_modus_ponens) R->L }
% 22.23/3.22 fresh59(fresh41(kn2, modus_ponens, X, Y), modus_ponens, implies(X, Y), implies(implies(Y, X), equiv(X, Y)))
% 22.23/3.22 = { by lemma 58 }
% 22.23/3.22 fresh59(fresh41(kn2, kn2, X, Y), modus_ponens, implies(X, Y), implies(implies(Y, X), equiv(X, Y)))
% 22.23/3.22 = { by axiom 25 (equivalence_3_1) }
% 22.23/3.22 fresh59(true, modus_ponens, implies(X, Y), implies(implies(Y, X), equiv(X, Y)))
% 22.23/3.22 = { by axiom 1 (hilbert_modus_ponens) R->L }
% 22.23/3.22 fresh59(modus_ponens, modus_ponens, implies(X, Y), implies(implies(Y, X), equiv(X, Y)))
% 22.23/3.22 = { by lemma 58 }
% 22.23/3.22 fresh59(kn2, modus_ponens, implies(X, Y), implies(implies(Y, X), equiv(X, Y)))
% 22.23/3.22 = { by lemma 65 }
% 22.23/3.22 fresh59(kn1, modus_ponens, implies(X, Y), implies(implies(Y, X), equiv(X, Y)))
% 22.23/3.22 = { by lemma 58 }
% 22.23/3.22 fresh59(kn1, kn2, implies(X, Y), implies(implies(Y, X), equiv(X, Y)))
% 22.23/3.22 = { by lemma 65 }
% 22.23/3.22 fresh59(kn1, kn1, implies(X, Y), implies(implies(Y, X), equiv(X, Y)))
% 22.23/3.22 = { by lemma 62 }
% 22.23/3.22 modus_ponens
% 22.23/3.22 = { by lemma 58 }
% 22.23/3.22 kn2
% 22.23/3.22 = { by lemma 65 }
% 22.23/3.22 kn1
% 22.23/3.22
% 22.23/3.22 Lemma 69: fresh(is_a_theorem(equiv(X, Y)), modus_ponens, X, Y) = X.
% 22.23/3.22 Proof:
% 22.23/3.22 fresh(is_a_theorem(equiv(X, Y)), modus_ponens, X, Y)
% 22.23/3.22 = { by axiom 1 (hilbert_modus_ponens) }
% 22.23/3.22 fresh(is_a_theorem(equiv(X, Y)), true, X, Y)
% 22.23/3.22 = { by axiom 42 (substitution_of_equivalents_2) R->L }
% 22.23/3.22 fresh2(substitution_of_equivalents, true, X, Y)
% 22.23/3.22 = { by axiom 2 (substitution_of_equivalents) }
% 22.23/3.22 fresh2(true, true, X, Y)
% 22.23/3.22 = { by axiom 1 (hilbert_modus_ponens) R->L }
% 22.23/3.22 fresh2(modus_ponens, true, X, Y)
% 22.23/3.22 = { by axiom 1 (hilbert_modus_ponens) R->L }
% 22.23/3.22 fresh2(modus_ponens, modus_ponens, X, Y)
% 22.23/3.22 = { by axiom 32 (substitution_of_equivalents_2) }
% 22.23/3.22 X
% 22.23/3.22
% 22.23/3.22 Lemma 70: implies(not(X), X) = not(not(X)).
% 22.23/3.22 Proof:
% 22.23/3.22 implies(not(X), X)
% 22.23/3.22 = { by lemma 66 R->L }
% 22.23/3.22 not(and(not(X), not(X)))
% 22.23/3.22 = { by axiom 21 (substitution_of_equivalents_2) R->L }
% 22.23/3.22 not(fresh(kn1, kn1, not(X), and(not(X), not(X))))
% 22.23/3.22 = { by lemma 65 R->L }
% 22.23/3.22 not(fresh(kn1, kn2, not(X), and(not(X), not(X))))
% 22.23/3.22 = { by lemma 58 R->L }
% 22.23/3.22 not(fresh(kn1, modus_ponens, not(X), and(not(X), not(X))))
% 22.23/3.22 = { by lemma 65 R->L }
% 22.23/3.22 not(fresh(kn2, modus_ponens, not(X), and(not(X), not(X))))
% 22.23/3.22 = { by lemma 58 R->L }
% 22.23/3.22 not(fresh(modus_ponens, modus_ponens, not(X), and(not(X), not(X))))
% 22.23/3.22 = { by lemma 62 R->L }
% 22.23/3.22 not(fresh(fresh59(kn1, kn1, implies(and(not(X), not(X)), not(X)), equiv(not(X), and(not(X), not(X)))), modus_ponens, not(X), and(not(X), not(X))))
% 22.23/3.22 = { by lemma 65 R->L }
% 22.23/3.22 not(fresh(fresh59(kn1, kn2, implies(and(not(X), not(X)), not(X)), equiv(not(X), and(not(X), not(X)))), modus_ponens, not(X), and(not(X), not(X))))
% 22.23/3.22 = { by lemma 58 R->L }
% 22.23/3.22 not(fresh(fresh59(kn1, modus_ponens, implies(and(not(X), not(X)), not(X)), equiv(not(X), and(not(X), not(X)))), modus_ponens, not(X), and(not(X), not(X))))
% 22.23/3.22 = { by lemma 68 R->L }
% 22.23/3.22 not(fresh(fresh59(fresh28(is_a_theorem(implies(not(X), and(not(X), not(X)))), kn1, implies(implies(and(not(X), not(X)), not(X)), equiv(not(X), and(not(X), not(X))))), modus_ponens, implies(and(not(X), not(X)), not(X)), equiv(not(X), and(not(X), not(X)))), modus_ponens, not(X), and(not(X), not(X))))
% 22.23/3.22 = { by lemma 59 }
% 22.23/3.22 not(fresh(fresh59(fresh28(fresh33(kn1, modus_ponens, not(X)), kn1, implies(implies(and(not(X), not(X)), not(X)), equiv(not(X), and(not(X), not(X))))), modus_ponens, implies(and(not(X), not(X)), not(X)), equiv(not(X), and(not(X), not(X)))), modus_ponens, not(X), and(not(X), not(X))))
% 22.23/3.22 = { by lemma 58 }
% 22.23/3.22 not(fresh(fresh59(fresh28(fresh33(kn1, kn2, not(X)), kn1, implies(implies(and(not(X), not(X)), not(X)), equiv(not(X), and(not(X), not(X))))), modus_ponens, implies(and(not(X), not(X)), not(X)), equiv(not(X), and(not(X), not(X)))), modus_ponens, not(X), and(not(X), not(X))))
% 22.23/3.22 = { by lemma 64 }
% 22.23/3.22 not(fresh(fresh59(fresh28(kn2, kn1, implies(implies(and(not(X), not(X)), not(X)), equiv(not(X), and(not(X), not(X))))), modus_ponens, implies(and(not(X), not(X)), not(X)), equiv(not(X), and(not(X), not(X)))), modus_ponens, not(X), and(not(X), not(X))))
% 22.23/3.22 = { by lemma 65 }
% 22.23/3.22 not(fresh(fresh59(fresh28(kn1, kn1, implies(implies(and(not(X), not(X)), not(X)), equiv(not(X), and(not(X), not(X))))), modus_ponens, implies(and(not(X), not(X)), not(X)), equiv(not(X), and(not(X), not(X)))), modus_ponens, not(X), and(not(X), not(X))))
% 22.23/3.22 = { by axiom 20 (modus_ponens_2) }
% 22.23/3.22 not(fresh(fresh59(is_a_theorem(implies(implies(and(not(X), not(X)), not(X)), equiv(not(X), and(not(X), not(X))))), modus_ponens, implies(and(not(X), not(X)), not(X)), equiv(not(X), and(not(X), not(X)))), modus_ponens, not(X), and(not(X), not(X))))
% 22.23/3.22 = { by lemma 60 }
% 22.23/3.22 not(fresh(fresh28(is_a_theorem(implies(and(not(X), not(X)), not(X))), modus_ponens, equiv(not(X), and(not(X), not(X)))), modus_ponens, not(X), and(not(X), not(X))))
% 22.23/3.22 = { by lemma 57 }
% 22.23/3.22 not(fresh(fresh28(modus_ponens, modus_ponens, equiv(not(X), and(not(X), not(X)))), modus_ponens, not(X), and(not(X), not(X))))
% 22.23/3.22 = { by axiom 20 (modus_ponens_2) }
% 22.23/3.22 not(fresh(is_a_theorem(equiv(not(X), and(not(X), not(X)))), modus_ponens, not(X), and(not(X), not(X))))
% 22.23/3.22 = { by lemma 69 }
% 22.23/3.22 not(not(X))
% 22.23/3.23
% 22.23/3.23 Lemma 71: or(X, X) = not(not(X)).
% 22.23/3.23 Proof:
% 22.23/3.23 or(X, X)
% 22.23/3.23 = { by lemma 67 R->L }
% 22.23/3.23 implies(not(X), X)
% 22.23/3.23 = { by lemma 70 }
% 22.23/3.23 not(not(X))
% 22.23/3.23
% 22.23/3.23 Lemma 72: is_a_theorem(implies(X, implies(Y, X))) = modus_ponens.
% 22.23/3.23 Proof:
% 22.23/3.23 is_a_theorem(implies(X, implies(Y, X)))
% 22.23/3.23 = { by axiom 38 (implies_1_1) R->L }
% 22.23/3.23 fresh39(implies_1, true, X, Y)
% 22.23/3.23 = { by axiom 4 (hilbert_implies_1) }
% 22.23/3.23 fresh39(true, true, X, Y)
% 22.23/3.23 = { by axiom 1 (hilbert_modus_ponens) R->L }
% 22.23/3.23 fresh39(modus_ponens, true, X, Y)
% 22.23/3.23 = { by axiom 1 (hilbert_modus_ponens) R->L }
% 22.23/3.23 fresh39(modus_ponens, modus_ponens, X, Y)
% 22.23/3.23 = { by axiom 26 (implies_1_1) }
% 22.23/3.23 true
% 22.23/3.23 = { by axiom 1 (hilbert_modus_ponens) R->L }
% 22.23/3.23 modus_ponens
% 22.23/3.23
% 22.23/3.23 Lemma 73: is_a_theorem(implies(or(X, X), X)) = fresh12(r1, modus_ponens, X).
% 22.23/3.23 Proof:
% 22.23/3.23 is_a_theorem(implies(or(X, X), X))
% 22.23/3.23 = { by axiom 40 (r1_1) R->L }
% 22.23/3.23 fresh12(r1, true, X)
% 22.23/3.23 = { by axiom 1 (hilbert_modus_ponens) R->L }
% 22.23/3.23 fresh12(r1, modus_ponens, X)
% 22.23/3.23
% 22.23/3.23 Lemma 74: fresh12(r1, modus_ponens, X) = fresh47(cn3, modus_ponens, X).
% 22.23/3.23 Proof:
% 22.23/3.23 fresh12(r1, modus_ponens, X)
% 22.23/3.23 = { by lemma 73 R->L }
% 22.23/3.23 is_a_theorem(implies(or(X, X), X))
% 22.23/3.23 = { by lemma 67 R->L }
% 22.23/3.23 is_a_theorem(implies(implies(not(X), X), X))
% 22.23/3.23 = { by axiom 41 (cn3_1) R->L }
% 22.23/3.23 fresh47(cn3, true, X)
% 22.23/3.23 = { by axiom 1 (hilbert_modus_ponens) R->L }
% 22.23/3.23 fresh47(cn3, modus_ponens, X)
% 22.23/3.23
% 22.23/3.23 Lemma 75: not(not(X)) = X.
% 22.51/3.23 Proof:
% 22.51/3.23 not(not(X))
% 22.51/3.23 = { by lemma 69 R->L }
% 22.51/3.23 fresh(is_a_theorem(equiv(not(not(X)), X)), modus_ponens, not(not(X)), X)
% 22.51/3.23 = { by axiom 20 (modus_ponens_2) R->L }
% 22.51/3.23 fresh(fresh28(kn1, kn1, equiv(not(not(X)), X)), modus_ponens, not(not(X)), X)
% 22.51/3.23 = { by lemma 65 R->L }
% 22.51/3.23 fresh(fresh28(kn1, kn2, equiv(not(not(X)), X)), modus_ponens, not(not(X)), X)
% 22.51/3.23 = { by lemma 58 R->L }
% 22.51/3.23 fresh(fresh28(kn1, modus_ponens, equiv(not(not(X)), X)), modus_ponens, not(not(X)), X)
% 22.51/3.23 = { by lemma 65 R->L }
% 22.51/3.23 fresh(fresh28(kn2, modus_ponens, equiv(not(not(X)), X)), modus_ponens, not(not(X)), X)
% 22.51/3.23 = { by lemma 58 R->L }
% 22.51/3.23 fresh(fresh28(modus_ponens, modus_ponens, equiv(not(not(X)), X)), modus_ponens, not(not(X)), X)
% 22.51/3.23 = { by lemma 72 R->L }
% 22.51/3.23 fresh(fresh28(is_a_theorem(implies(X, implies(not(X), X))), modus_ponens, equiv(not(not(X)), X)), modus_ponens, not(not(X)), X)
% 22.51/3.23 = { by lemma 70 }
% 22.51/3.23 fresh(fresh28(is_a_theorem(implies(X, not(not(X)))), modus_ponens, equiv(not(not(X)), X)), modus_ponens, not(not(X)), X)
% 22.51/3.23 = { by lemma 60 R->L }
% 22.51/3.23 fresh(fresh59(is_a_theorem(implies(implies(X, not(not(X))), equiv(not(not(X)), X))), modus_ponens, implies(X, not(not(X))), equiv(not(not(X)), X)), modus_ponens, not(not(X)), X)
% 22.51/3.23 = { by lemma 71 R->L }
% 22.51/3.23 fresh(fresh59(is_a_theorem(implies(implies(X, or(X, X)), equiv(not(not(X)), X))), modus_ponens, implies(X, not(not(X))), equiv(not(not(X)), X)), modus_ponens, not(not(X)), X)
% 22.51/3.23 = { by lemma 71 R->L }
% 22.51/3.23 fresh(fresh59(is_a_theorem(implies(implies(X, or(X, X)), equiv(or(X, X), X))), modus_ponens, implies(X, not(not(X))), equiv(not(not(X)), X)), modus_ponens, not(not(X)), X)
% 22.51/3.23 = { by axiom 20 (modus_ponens_2) R->L }
% 22.51/3.23 fresh(fresh59(fresh28(kn1, kn1, implies(implies(X, or(X, X)), equiv(or(X, X), X))), modus_ponens, implies(X, not(not(X))), equiv(not(not(X)), X)), modus_ponens, not(not(X)), X)
% 22.51/3.23 = { by lemma 65 R->L }
% 22.51/3.23 fresh(fresh59(fresh28(kn2, kn1, implies(implies(X, or(X, X)), equiv(or(X, X), X))), modus_ponens, implies(X, not(not(X))), equiv(not(not(X)), X)), modus_ponens, not(not(X)), X)
% 22.51/3.23 = { by lemma 58 R->L }
% 22.51/3.23 fresh(fresh59(fresh28(modus_ponens, kn1, implies(implies(X, or(X, X)), equiv(or(X, X), X))), modus_ponens, implies(X, not(not(X))), equiv(not(not(X)), X)), modus_ponens, not(not(X)), X)
% 22.51/3.23 = { by axiom 1 (hilbert_modus_ponens) }
% 22.51/3.23 fresh(fresh59(fresh28(true, kn1, implies(implies(X, or(X, X)), equiv(or(X, X), X))), modus_ponens, implies(X, not(not(X))), equiv(not(not(X)), X)), modus_ponens, not(not(X)), X)
% 22.51/3.23 = { by axiom 19 (cn3_1) R->L }
% 22.51/3.23 fresh(fresh59(fresh28(fresh47(kn1, kn1, X), kn1, implies(implies(X, or(X, X)), equiv(or(X, X), X))), modus_ponens, implies(X, not(not(X))), equiv(not(not(X)), X)), modus_ponens, not(not(X)), X)
% 22.51/3.23 = { by lemma 65 R->L }
% 22.51/3.23 fresh(fresh59(fresh28(fresh47(kn1, kn2, X), kn1, implies(implies(X, or(X, X)), equiv(or(X, X), X))), modus_ponens, implies(X, not(not(X))), equiv(not(not(X)), X)), modus_ponens, not(not(X)), X)
% 22.51/3.23 = { by lemma 58 R->L }
% 22.51/3.23 fresh(fresh59(fresh28(fresh47(kn1, modus_ponens, X), kn1, implies(implies(X, or(X, X)), equiv(or(X, X), X))), modus_ponens, implies(X, not(not(X))), equiv(not(not(X)), X)), modus_ponens, not(not(X)), X)
% 22.51/3.23 = { by lemma 65 R->L }
% 22.51/3.23 fresh(fresh59(fresh28(fresh47(kn2, modus_ponens, X), kn1, implies(implies(X, or(X, X)), equiv(or(X, X), X))), modus_ponens, implies(X, not(not(X))), equiv(not(not(X)), X)), modus_ponens, not(not(X)), X)
% 22.51/3.23 = { by lemma 58 R->L }
% 22.51/3.23 fresh(fresh59(fresh28(fresh47(modus_ponens, modus_ponens, X), kn1, implies(implies(X, or(X, X)), equiv(or(X, X), X))), modus_ponens, implies(X, not(not(X))), equiv(not(not(X)), X)), modus_ponens, not(not(X)), X)
% 22.51/3.23 = { by axiom 1 (hilbert_modus_ponens) }
% 22.51/3.23 fresh(fresh59(fresh28(fresh47(true, modus_ponens, X), kn1, implies(implies(X, or(X, X)), equiv(or(X, X), X))), modus_ponens, implies(X, not(not(X))), equiv(not(not(X)), X)), modus_ponens, not(not(X)), X)
% 22.51/3.23 = { by axiom 13 (cn3) R->L }
% 22.51/3.23 fresh(fresh59(fresh28(fresh47(fresh48(kn1, kn1), modus_ponens, X), kn1, implies(implies(X, or(X, X)), equiv(or(X, X), X))), modus_ponens, implies(X, not(not(X))), equiv(not(not(X)), X)), modus_ponens, not(not(X)), X)
% 22.51/3.23 = { by lemma 65 R->L }
% 22.51/3.23 fresh(fresh59(fresh28(fresh47(fresh48(kn2, kn1), modus_ponens, X), kn1, implies(implies(X, or(X, X)), equiv(or(X, X), X))), modus_ponens, implies(X, not(not(X))), equiv(not(not(X)), X)), modus_ponens, not(not(X)), X)
% 22.51/3.23 = { by lemma 58 R->L }
% 22.51/3.23 fresh(fresh59(fresh28(fresh47(fresh48(modus_ponens, kn1), modus_ponens, X), kn1, implies(implies(X, or(X, X)), equiv(or(X, X), X))), modus_ponens, implies(X, not(not(X))), equiv(not(not(X)), X)), modus_ponens, not(not(X)), X)
% 22.51/3.23 = { by lemma 62 R->L }
% 22.51/3.23 fresh(fresh59(fresh28(fresh47(fresh48(fresh59(kn1, kn1, implies(p6, p6), implies(or(p6, p6), p6)), kn1), modus_ponens, X), kn1, implies(implies(X, or(X, X)), equiv(or(X, X), X))), modus_ponens, implies(X, not(not(X))), equiv(not(not(X)), X)), modus_ponens, not(not(X)), X)
% 22.51/3.23 = { by lemma 65 R->L }
% 22.52/3.23 fresh(fresh59(fresh28(fresh47(fresh48(fresh59(kn1, kn2, implies(p6, p6), implies(or(p6, p6), p6)), kn1), modus_ponens, X), kn1, implies(implies(X, or(X, X)), equiv(or(X, X), X))), modus_ponens, implies(X, not(not(X))), equiv(not(not(X)), X)), modus_ponens, not(not(X)), X)
% 22.52/3.23 = { by lemma 58 R->L }
% 22.52/3.23 fresh(fresh59(fresh28(fresh47(fresh48(fresh59(kn1, modus_ponens, implies(p6, p6), implies(or(p6, p6), p6)), kn1), modus_ponens, X), kn1, implies(implies(X, or(X, X)), equiv(or(X, X), X))), modus_ponens, implies(X, not(not(X))), equiv(not(not(X)), X)), modus_ponens, not(not(X)), X)
% 22.52/3.23 = { by lemma 65 R->L }
% 22.52/3.23 fresh(fresh59(fresh28(fresh47(fresh48(fresh59(kn2, modus_ponens, implies(p6, p6), implies(or(p6, p6), p6)), kn1), modus_ponens, X), kn1, implies(implies(X, or(X, X)), equiv(or(X, X), X))), modus_ponens, implies(X, not(not(X))), equiv(not(not(X)), X)), modus_ponens, not(not(X)), X)
% 22.52/3.23 = { by lemma 63 R->L }
% 22.52/3.23 fresh(fresh59(fresh28(fresh47(fresh48(fresh59(fresh28(is_a_theorem(implies(implies(p6, p6), implies(implies(p6, p6), implies(or(p6, p6), p6)))), kn2, implies(implies(p6, p6), implies(or(p6, p6), p6))), modus_ponens, implies(p6, p6), implies(or(p6, p6), p6)), kn1), modus_ponens, X), kn1, implies(implies(X, or(X, X)), equiv(or(X, X), X))), modus_ponens, implies(X, not(not(X))), equiv(not(not(X)), X)), modus_ponens, not(not(X)), X)
% 22.52/3.23 = { by axiom 55 (or_3_1) R->L }
% 22.52/3.23 fresh(fresh59(fresh28(fresh47(fresh48(fresh59(fresh28(fresh14(or_3, true, p6, p6, p6), kn2, implies(implies(p6, p6), implies(or(p6, p6), p6))), modus_ponens, implies(p6, p6), implies(or(p6, p6), p6)), kn1), modus_ponens, X), kn1, implies(implies(X, or(X, X)), equiv(or(X, X), X))), modus_ponens, implies(X, not(not(X))), equiv(not(not(X)), X)), modus_ponens, not(not(X)), X)
% 22.52/3.23 = { by axiom 9 (hilbert_or_3) }
% 22.52/3.23 fresh(fresh59(fresh28(fresh47(fresh48(fresh59(fresh28(fresh14(true, true, p6, p6, p6), kn2, implies(implies(p6, p6), implies(or(p6, p6), p6))), modus_ponens, implies(p6, p6), implies(or(p6, p6), p6)), kn1), modus_ponens, X), kn1, implies(implies(X, or(X, X)), equiv(or(X, X), X))), modus_ponens, implies(X, not(not(X))), equiv(not(not(X)), X)), modus_ponens, not(not(X)), X)
% 22.52/3.23 = { by axiom 1 (hilbert_modus_ponens) R->L }
% 22.52/3.23 fresh(fresh59(fresh28(fresh47(fresh48(fresh59(fresh28(fresh14(modus_ponens, true, p6, p6, p6), kn2, implies(implies(p6, p6), implies(or(p6, p6), p6))), modus_ponens, implies(p6, p6), implies(or(p6, p6), p6)), kn1), modus_ponens, X), kn1, implies(implies(X, or(X, X)), equiv(or(X, X), X))), modus_ponens, implies(X, not(not(X))), equiv(not(not(X)), X)), modus_ponens, not(not(X)), X)
% 22.52/3.23 = { by lemma 58 }
% 22.52/3.23 fresh(fresh59(fresh28(fresh47(fresh48(fresh59(fresh28(fresh14(kn2, true, p6, p6, p6), kn2, implies(implies(p6, p6), implies(or(p6, p6), p6))), modus_ponens, implies(p6, p6), implies(or(p6, p6), p6)), kn1), modus_ponens, X), kn1, implies(implies(X, or(X, X)), equiv(or(X, X), X))), modus_ponens, implies(X, not(not(X))), equiv(not(not(X)), X)), modus_ponens, not(not(X)), X)
% 22.52/3.23 = { by axiom 1 (hilbert_modus_ponens) R->L }
% 22.52/3.23 fresh(fresh59(fresh28(fresh47(fresh48(fresh59(fresh28(fresh14(kn2, modus_ponens, p6, p6, p6), kn2, implies(implies(p6, p6), implies(or(p6, p6), p6))), modus_ponens, implies(p6, p6), implies(or(p6, p6), p6)), kn1), modus_ponens, X), kn1, implies(implies(X, or(X, X)), equiv(or(X, X), X))), modus_ponens, implies(X, not(not(X))), equiv(not(not(X)), X)), modus_ponens, not(not(X)), X)
% 22.52/3.23 = { by lemma 58 }
% 22.52/3.23 fresh(fresh59(fresh28(fresh47(fresh48(fresh59(fresh28(fresh14(kn2, kn2, p6, p6, p6), kn2, implies(implies(p6, p6), implies(or(p6, p6), p6))), modus_ponens, implies(p6, p6), implies(or(p6, p6), p6)), kn1), modus_ponens, X), kn1, implies(implies(X, or(X, X)), equiv(or(X, X), X))), modus_ponens, implies(X, not(not(X))), equiv(not(not(X)), X)), modus_ponens, not(not(X)), X)
% 22.52/3.23 = { by axiom 35 (or_3_1) }
% 22.52/3.23 fresh(fresh59(fresh28(fresh47(fresh48(fresh59(fresh28(true, kn2, implies(implies(p6, p6), implies(or(p6, p6), p6))), modus_ponens, implies(p6, p6), implies(or(p6, p6), p6)), kn1), modus_ponens, X), kn1, implies(implies(X, or(X, X)), equiv(or(X, X), X))), modus_ponens, implies(X, not(not(X))), equiv(not(not(X)), X)), modus_ponens, not(not(X)), X)
% 22.52/3.23 = { by axiom 1 (hilbert_modus_ponens) R->L }
% 22.52/3.23 fresh(fresh59(fresh28(fresh47(fresh48(fresh59(fresh28(modus_ponens, kn2, implies(implies(p6, p6), implies(or(p6, p6), p6))), modus_ponens, implies(p6, p6), implies(or(p6, p6), p6)), kn1), modus_ponens, X), kn1, implies(implies(X, or(X, X)), equiv(or(X, X), X))), modus_ponens, implies(X, not(not(X))), equiv(not(not(X)), X)), modus_ponens, not(not(X)), X)
% 22.52/3.23 = { by lemma 58 }
% 22.52/3.23 fresh(fresh59(fresh28(fresh47(fresh48(fresh59(fresh28(kn2, kn2, implies(implies(p6, p6), implies(or(p6, p6), p6))), modus_ponens, implies(p6, p6), implies(or(p6, p6), p6)), kn1), modus_ponens, X), kn1, implies(implies(X, or(X, X)), equiv(or(X, X), X))), modus_ponens, implies(X, not(not(X))), equiv(not(not(X)), X)), modus_ponens, not(not(X)), X)
% 22.52/3.23 = { by axiom 20 (modus_ponens_2) }
% 22.52/3.23 fresh(fresh59(fresh28(fresh47(fresh48(fresh59(is_a_theorem(implies(implies(p6, p6), implies(or(p6, p6), p6))), modus_ponens, implies(p6, p6), implies(or(p6, p6), p6)), kn1), modus_ponens, X), kn1, implies(implies(X, or(X, X)), equiv(or(X, X), X))), modus_ponens, implies(X, not(not(X))), equiv(not(not(X)), X)), modus_ponens, not(not(X)), X)
% 22.52/3.23 = { by lemma 60 }
% 22.52/3.23 fresh(fresh59(fresh28(fresh47(fresh48(fresh28(is_a_theorem(implies(p6, p6)), modus_ponens, implies(or(p6, p6), p6)), kn1), modus_ponens, X), kn1, implies(implies(X, or(X, X)), equiv(or(X, X), X))), modus_ponens, implies(X, not(not(X))), equiv(not(not(X)), X)), modus_ponens, not(not(X)), X)
% 22.52/3.23 = { by lemma 58 }
% 22.52/3.23 fresh(fresh59(fresh28(fresh47(fresh48(fresh28(is_a_theorem(implies(p6, p6)), kn2, implies(or(p6, p6), p6)), kn1), modus_ponens, X), kn1, implies(implies(X, or(X, X)), equiv(or(X, X), X))), modus_ponens, implies(X, not(not(X))), equiv(not(not(X)), X)), modus_ponens, not(not(X)), X)
% 22.52/3.23 = { by lemma 65 }
% 22.52/3.24 fresh(fresh59(fresh28(fresh47(fresh48(fresh28(is_a_theorem(implies(p6, p6)), kn1, implies(or(p6, p6), p6)), kn1), modus_ponens, X), kn1, implies(implies(X, or(X, X)), equiv(or(X, X), X))), modus_ponens, implies(X, not(not(X))), equiv(not(not(X)), X)), modus_ponens, not(not(X)), X)
% 22.52/3.24 = { by axiom 20 (modus_ponens_2) R->L }
% 22.52/3.24 fresh(fresh59(fresh28(fresh47(fresh48(fresh28(fresh28(kn2, kn2, implies(p6, p6)), kn1, implies(or(p6, p6), p6)), kn1), modus_ponens, X), kn1, implies(implies(X, or(X, X)), equiv(or(X, X), X))), modus_ponens, implies(X, not(not(X))), equiv(not(not(X)), X)), modus_ponens, not(not(X)), X)
% 22.52/3.24 = { by lemma 58 R->L }
% 22.52/3.24 fresh(fresh59(fresh28(fresh47(fresh48(fresh28(fresh28(modus_ponens, kn2, implies(p6, p6)), kn1, implies(or(p6, p6), p6)), kn1), modus_ponens, X), kn1, implies(implies(X, or(X, X)), equiv(or(X, X), X))), modus_ponens, implies(X, not(not(X))), equiv(not(not(X)), X)), modus_ponens, not(not(X)), X)
% 22.52/3.24 = { by lemma 72 R->L }
% 22.52/3.24 fresh(fresh59(fresh28(fresh47(fresh48(fresh28(fresh28(is_a_theorem(implies(p6, implies(p6, p6))), kn2, implies(p6, p6)), kn1, implies(or(p6, p6), p6)), kn1), modus_ponens, X), kn1, implies(implies(X, or(X, X)), equiv(or(X, X), X))), modus_ponens, implies(X, not(not(X))), equiv(not(not(X)), X)), modus_ponens, not(not(X)), X)
% 22.52/3.24 = { by lemma 63 }
% 22.52/3.24 fresh(fresh59(fresh28(fresh47(fresh48(fresh28(kn2, kn1, implies(or(p6, p6), p6)), kn1), modus_ponens, X), kn1, implies(implies(X, or(X, X)), equiv(or(X, X), X))), modus_ponens, implies(X, not(not(X))), equiv(not(not(X)), X)), modus_ponens, not(not(X)), X)
% 22.52/3.24 = { by lemma 65 }
% 22.52/3.24 fresh(fresh59(fresh28(fresh47(fresh48(fresh28(kn1, kn1, implies(or(p6, p6), p6)), kn1), modus_ponens, X), kn1, implies(implies(X, or(X, X)), equiv(or(X, X), X))), modus_ponens, implies(X, not(not(X))), equiv(not(not(X)), X)), modus_ponens, not(not(X)), X)
% 22.52/3.24 = { by axiom 20 (modus_ponens_2) }
% 22.52/3.24 fresh(fresh59(fresh28(fresh47(fresh48(is_a_theorem(implies(or(p6, p6), p6)), kn1), modus_ponens, X), kn1, implies(implies(X, or(X, X)), equiv(or(X, X), X))), modus_ponens, implies(X, not(not(X))), equiv(not(not(X)), X)), modus_ponens, not(not(X)), X)
% 22.52/3.24 = { by lemma 73 }
% 22.52/3.24 fresh(fresh59(fresh28(fresh47(fresh48(fresh12(r1, modus_ponens, p6), kn1), modus_ponens, X), kn1, implies(implies(X, or(X, X)), equiv(or(X, X), X))), modus_ponens, implies(X, not(not(X))), equiv(not(not(X)), X)), modus_ponens, not(not(X)), X)
% 22.52/3.24 = { by lemma 74 }
% 22.52/3.24 fresh(fresh59(fresh28(fresh47(fresh48(fresh47(cn3, modus_ponens, p6), kn1), modus_ponens, X), kn1, implies(implies(X, or(X, X)), equiv(or(X, X), X))), modus_ponens, implies(X, not(not(X))), equiv(not(not(X)), X)), modus_ponens, not(not(X)), X)
% 22.52/3.24 = { by lemma 58 }
% 22.52/3.24 fresh(fresh59(fresh28(fresh47(fresh48(fresh47(cn3, kn2, p6), kn1), modus_ponens, X), kn1, implies(implies(X, or(X, X)), equiv(or(X, X), X))), modus_ponens, implies(X, not(not(X))), equiv(not(not(X)), X)), modus_ponens, not(not(X)), X)
% 22.52/3.24 = { by lemma 65 }
% 22.52/3.24 fresh(fresh59(fresh28(fresh47(fresh48(fresh47(cn3, kn1, p6), kn1), modus_ponens, X), kn1, implies(implies(X, or(X, X)), equiv(or(X, X), X))), modus_ponens, implies(X, not(not(X))), equiv(not(not(X)), X)), modus_ponens, not(not(X)), X)
% 22.52/3.24 = { by lemma 65 R->L }
% 22.52/3.24 fresh(fresh59(fresh28(fresh47(fresh48(fresh47(cn3, kn1, p6), kn2), modus_ponens, X), kn1, implies(implies(X, or(X, X)), equiv(or(X, X), X))), modus_ponens, implies(X, not(not(X))), equiv(not(not(X)), X)), modus_ponens, not(not(X)), X)
% 22.52/3.24 = { by lemma 65 R->L }
% 22.52/3.24 fresh(fresh59(fresh28(fresh47(fresh48(fresh47(cn3, kn2, p6), kn2), modus_ponens, X), kn1, implies(implies(X, or(X, X)), equiv(or(X, X), X))), modus_ponens, implies(X, not(not(X))), equiv(not(not(X)), X)), modus_ponens, not(not(X)), X)
% 22.52/3.24 = { by lemma 58 R->L }
% 22.52/3.24 fresh(fresh59(fresh28(fresh47(fresh48(fresh47(cn3, kn2, p6), modus_ponens), modus_ponens, X), kn1, implies(implies(X, or(X, X)), equiv(or(X, X), X))), modus_ponens, implies(X, not(not(X))), equiv(not(not(X)), X)), modus_ponens, not(not(X)), X)
% 22.52/3.24 = { by axiom 1 (hilbert_modus_ponens) }
% 22.52/3.24 fresh(fresh59(fresh28(fresh47(fresh48(fresh47(cn3, kn2, p6), true), modus_ponens, X), kn1, implies(implies(X, or(X, X)), equiv(or(X, X), X))), modus_ponens, implies(X, not(not(X))), equiv(not(not(X)), X)), modus_ponens, not(not(X)), X)
% 22.52/3.24 = { by lemma 58 R->L }
% 22.52/3.24 fresh(fresh59(fresh28(fresh47(fresh48(fresh47(cn3, modus_ponens, p6), true), modus_ponens, X), kn1, implies(implies(X, or(X, X)), equiv(or(X, X), X))), modus_ponens, implies(X, not(not(X))), equiv(not(not(X)), X)), modus_ponens, not(not(X)), X)
% 22.52/3.24 = { by lemma 74 R->L }
% 22.52/3.24 fresh(fresh59(fresh28(fresh47(fresh48(fresh12(r1, modus_ponens, p6), true), modus_ponens, X), kn1, implies(implies(X, or(X, X)), equiv(or(X, X), X))), modus_ponens, implies(X, not(not(X))), equiv(not(not(X)), X)), modus_ponens, not(not(X)), X)
% 22.52/3.24 = { by lemma 73 R->L }
% 22.52/3.24 fresh(fresh59(fresh28(fresh47(fresh48(is_a_theorem(implies(or(p6, p6), p6)), true), modus_ponens, X), kn1, implies(implies(X, or(X, X)), equiv(or(X, X), X))), modus_ponens, implies(X, not(not(X))), equiv(not(not(X)), X)), modus_ponens, not(not(X)), X)
% 22.52/3.24 = { by lemma 67 R->L }
% 22.52/3.24 fresh(fresh59(fresh28(fresh47(fresh48(is_a_theorem(implies(implies(not(p6), p6), p6)), true), modus_ponens, X), kn1, implies(implies(X, or(X, X)), equiv(or(X, X), X))), modus_ponens, implies(X, not(not(X))), equiv(not(not(X)), X)), modus_ponens, not(not(X)), X)
% 22.52/3.24 = { by axiom 48 (cn3) }
% 22.52/3.24 fresh(fresh59(fresh28(fresh47(cn3, modus_ponens, X), kn1, implies(implies(X, or(X, X)), equiv(or(X, X), X))), modus_ponens, implies(X, not(not(X))), equiv(not(not(X)), X)), modus_ponens, not(not(X)), X)
% 22.52/3.24 = { by lemma 74 R->L }
% 22.52/3.24 fresh(fresh59(fresh28(fresh12(r1, modus_ponens, X), kn1, implies(implies(X, or(X, X)), equiv(or(X, X), X))), modus_ponens, implies(X, not(not(X))), equiv(not(not(X)), X)), modus_ponens, not(not(X)), X)
% 22.52/3.24 = { by lemma 73 R->L }
% 22.52/3.24 fresh(fresh59(fresh28(is_a_theorem(implies(or(X, X), X)), kn1, implies(implies(X, or(X, X)), equiv(or(X, X), X))), modus_ponens, implies(X, not(not(X))), equiv(not(not(X)), X)), modus_ponens, not(not(X)), X)
% 22.52/3.24 = { by lemma 68 }
% 22.52/3.24 fresh(fresh59(kn1, modus_ponens, implies(X, not(not(X))), equiv(not(not(X)), X)), modus_ponens, not(not(X)), X)
% 22.52/3.24 = { by lemma 58 }
% 22.52/3.24 fresh(fresh59(kn1, kn2, implies(X, not(not(X))), equiv(not(not(X)), X)), modus_ponens, not(not(X)), X)
% 22.52/3.24 = { by lemma 65 }
% 22.52/3.24 fresh(fresh59(kn1, kn1, implies(X, not(not(X))), equiv(not(not(X)), X)), modus_ponens, not(not(X)), X)
% 22.52/3.24 = { by lemma 62 }
% 22.52/3.24 fresh(modus_ponens, modus_ponens, not(not(X)), X)
% 22.52/3.24 = { by lemma 58 }
% 22.52/3.24 fresh(kn2, modus_ponens, not(not(X)), X)
% 22.52/3.24 = { by lemma 65 }
% 22.52/3.24 fresh(kn1, modus_ponens, not(not(X)), X)
% 22.52/3.24 = { by lemma 58 }
% 22.52/3.24 fresh(kn1, kn2, not(not(X)), X)
% 22.52/3.24 = { by lemma 65 }
% 22.52/3.24 fresh(kn1, kn1, not(not(X)), X)
% 22.52/3.24 = { by axiom 21 (substitution_of_equivalents_2) }
% 22.52/3.24 X
% 22.52/3.24
% 22.52/3.24 Lemma 76: is_a_theorem(implies(or(X, Y), or(Y, X))) = fresh8(r3, kn2, X, Y).
% 22.52/3.24 Proof:
% 22.52/3.24 is_a_theorem(implies(or(X, Y), or(Y, X)))
% 22.52/3.24 = { by axiom 47 (r3_1) R->L }
% 22.52/3.24 fresh8(r3, true, X, Y)
% 22.52/3.24 = { by axiom 1 (hilbert_modus_ponens) R->L }
% 22.52/3.24 fresh8(r3, modus_ponens, X, Y)
% 22.52/3.24 = { by lemma 58 }
% 22.52/3.24 fresh8(r3, kn2, X, Y)
% 22.52/3.24
% 22.52/3.24 Lemma 77: is_a_theorem(implies(or(X, Y), or(Y, X))) = kn1.
% 22.52/3.24 Proof:
% 22.52/3.24 is_a_theorem(implies(or(X, Y), or(Y, X)))
% 22.52/3.24 = { by axiom 47 (r3_1) R->L }
% 22.52/3.24 fresh8(r3, true, X, Y)
% 22.52/3.24 = { by axiom 1 (hilbert_modus_ponens) R->L }
% 22.52/3.24 fresh8(r3, modus_ponens, X, Y)
% 22.52/3.24 = { by lemma 58 }
% 22.52/3.24 fresh8(r3, kn2, X, Y)
% 22.52/3.24 = { by axiom 49 (r3) R->L }
% 22.52/3.24 fresh8(fresh9(is_a_theorem(implies(or(p3, q3), or(q3, p3))), true), kn2, X, Y)
% 22.52/3.24 = { by lemma 76 }
% 22.52/3.24 fresh8(fresh9(fresh8(r3, kn2, p3, q3), true), kn2, X, Y)
% 22.52/3.24 = { by axiom 1 (hilbert_modus_ponens) R->L }
% 22.52/3.24 fresh8(fresh9(fresh8(r3, kn2, p3, q3), modus_ponens), kn2, X, Y)
% 22.52/3.24 = { by lemma 58 }
% 22.52/3.24 fresh8(fresh9(fresh8(r3, kn2, p3, q3), kn2), kn2, X, Y)
% 22.52/3.24 = { by lemma 65 }
% 22.52/3.24 fresh8(fresh9(fresh8(r3, kn1, p3, q3), kn2), kn2, X, Y)
% 22.52/3.24 = { by lemma 65 }
% 22.52/3.24 fresh8(fresh9(fresh8(r3, kn1, p3, q3), kn1), kn2, X, Y)
% 22.52/3.24 = { by lemma 65 R->L }
% 22.52/3.24 fresh8(fresh9(fresh8(r3, kn2, p3, q3), kn1), kn2, X, Y)
% 22.52/3.24 = { by lemma 76 R->L }
% 22.52/3.24 fresh8(fresh9(is_a_theorem(implies(or(p3, q3), or(q3, p3))), kn1), kn2, X, Y)
% 22.52/3.24 = { by lemma 67 R->L }
% 22.52/3.24 fresh8(fresh9(is_a_theorem(implies(or(p3, q3), implies(not(q3), p3))), kn1), kn2, X, Y)
% 22.52/3.24 = { by lemma 75 R->L }
% 22.52/3.24 fresh8(fresh9(is_a_theorem(implies(or(p3, not(not(q3))), implies(not(q3), p3))), kn1), kn2, X, Y)
% 22.52/3.24 = { by lemma 67 R->L }
% 22.52/3.24 fresh8(fresh9(is_a_theorem(implies(implies(not(p3), not(not(q3))), implies(not(q3), p3))), kn1), kn2, X, Y)
% 22.52/3.24 = { by axiom 51 (modus_tollens_1) R->L }
% 22.52/3.24 fresh8(fresh9(fresh25(modus_tollens, true, not(q3), p3), kn1), kn2, X, Y)
% 22.52/3.24 = { by axiom 3 (hilbert_modus_tollens) }
% 22.52/3.24 fresh8(fresh9(fresh25(true, true, not(q3), p3), kn1), kn2, X, Y)
% 22.52/3.24 = { by axiom 1 (hilbert_modus_ponens) R->L }
% 22.52/3.24 fresh8(fresh9(fresh25(modus_ponens, true, not(q3), p3), kn1), kn2, X, Y)
% 22.52/3.24 = { by lemma 58 }
% 22.52/3.24 fresh8(fresh9(fresh25(kn2, true, not(q3), p3), kn1), kn2, X, Y)
% 22.52/3.24 = { by axiom 1 (hilbert_modus_ponens) R->L }
% 22.52/3.24 fresh8(fresh9(fresh25(kn2, modus_ponens, not(q3), p3), kn1), kn2, X, Y)
% 22.52/3.24 = { by lemma 58 }
% 22.52/3.24 fresh8(fresh9(fresh25(kn2, kn2, not(q3), p3), kn1), kn2, X, Y)
% 22.52/3.24 = { by axiom 28 (modus_tollens_1) }
% 22.52/3.24 fresh8(fresh9(true, kn1), kn2, X, Y)
% 22.52/3.24 = { by axiom 1 (hilbert_modus_ponens) R->L }
% 22.52/3.24 fresh8(fresh9(modus_ponens, kn1), kn2, X, Y)
% 22.52/3.24 = { by lemma 58 }
% 22.52/3.24 fresh8(fresh9(kn2, kn1), kn2, X, Y)
% 22.52/3.24 = { by lemma 65 }
% 22.52/3.24 fresh8(fresh9(kn1, kn1), kn2, X, Y)
% 22.52/3.24 = { by axiom 17 (r3) }
% 22.52/3.24 fresh8(true, kn2, X, Y)
% 22.52/3.24 = { by axiom 1 (hilbert_modus_ponens) R->L }
% 22.52/3.24 fresh8(modus_ponens, kn2, X, Y)
% 22.52/3.24 = { by lemma 58 }
% 22.52/3.24 fresh8(kn2, kn2, X, Y)
% 22.52/3.24 = { by lemma 65 }
% 22.52/3.24 fresh8(kn1, kn2, X, Y)
% 22.52/3.24 = { by lemma 65 }
% 22.52/3.24 fresh8(kn1, kn1, X, Y)
% 22.52/3.24 = { by axiom 31 (r3_1) }
% 22.52/3.24 true
% 22.52/3.24 = { by axiom 1 (hilbert_modus_ponens) R->L }
% 22.52/3.24 modus_ponens
% 22.52/3.24 = { by lemma 58 }
% 22.52/3.24 kn2
% 22.52/3.24 = { by lemma 65 }
% 22.52/3.24 kn1
% 22.52/3.24
% 22.52/3.24 Goal 1 (rosser_kn3): kn3 = true.
% 22.52/3.24 Proof:
% 22.52/3.24 kn3
% 22.52/3.24 = { by axiom 56 (kn3) R->L }
% 22.52/3.24 fresh30(is_a_theorem(implies(implies(p9, q7), implies(not(and(q7, r8)), not(and(r8, p9))))), true)
% 22.52/3.24 = { by axiom 54 (kn3_1) R->L }
% 22.52/3.24 fresh30(fresh29(kn3, true, p9, q7, r8), true)
% 22.52/3.24 = { by axiom 1 (hilbert_modus_ponens) R->L }
% 22.52/3.24 fresh30(fresh29(kn3, modus_ponens, p9, q7, r8), true)
% 22.52/3.24 = { by lemma 58 }
% 22.52/3.24 fresh30(fresh29(kn3, kn2, p9, q7, r8), true)
% 22.52/3.24 = { by axiom 1 (hilbert_modus_ponens) R->L }
% 22.52/3.24 fresh30(fresh29(kn3, kn2, p9, q7, r8), modus_ponens)
% 22.52/3.24 = { by lemma 58 }
% 22.52/3.24 fresh30(fresh29(kn3, kn2, p9, q7, r8), kn2)
% 22.52/3.24 = { by lemma 65 }
% 22.52/3.24 fresh30(fresh29(kn3, kn1, p9, q7, r8), kn2)
% 22.52/3.24 = { by lemma 65 }
% 22.52/3.24 fresh30(fresh29(kn3, kn1, p9, q7, r8), kn1)
% 22.52/3.24 = { by lemma 75 R->L }
% 22.52/3.24 fresh30(fresh29(kn3, kn1, p9, q7, not(not(r8))), kn1)
% 22.52/3.24 = { by lemma 75 R->L }
% 22.52/3.24 fresh30(fresh29(kn3, kn1, not(not(p9)), q7, not(not(r8))), kn1)
% 22.52/3.24 = { by lemma 65 R->L }
% 22.52/3.24 fresh30(fresh29(kn3, kn2, not(not(p9)), q7, not(not(r8))), kn1)
% 22.52/3.24 = { by lemma 58 R->L }
% 22.52/3.24 fresh30(fresh29(kn3, modus_ponens, not(not(p9)), q7, not(not(r8))), kn1)
% 22.52/3.24 = { by axiom 1 (hilbert_modus_ponens) }
% 22.52/3.24 fresh30(fresh29(kn3, true, not(not(p9)), q7, not(not(r8))), kn1)
% 22.52/3.24 = { by axiom 54 (kn3_1) }
% 22.52/3.24 fresh30(is_a_theorem(implies(implies(not(not(p9)), q7), implies(not(and(q7, not(not(r8)))), not(and(not(not(r8)), not(not(p9))))))), kn1)
% 22.52/3.24 = { by lemma 67 }
% 22.52/3.24 fresh30(is_a_theorem(implies(implies(not(not(p9)), q7), or(and(q7, not(not(r8))), not(and(not(not(r8)), not(not(p9))))))), kn1)
% 22.52/3.24 = { by lemma 66 }
% 22.52/3.24 fresh30(is_a_theorem(implies(implies(not(not(p9)), q7), or(and(q7, not(not(r8))), implies(not(not(r8)), not(p9))))), kn1)
% 22.52/3.24 = { by lemma 67 }
% 22.52/3.24 fresh30(is_a_theorem(implies(or(not(p9), q7), or(and(q7, not(not(r8))), implies(not(not(r8)), not(p9))))), kn1)
% 22.52/3.24 = { by lemma 67 }
% 22.52/3.24 fresh30(is_a_theorem(implies(or(not(p9), q7), or(and(q7, not(not(r8))), or(not(r8), not(p9))))), kn1)
% 22.52/3.24 = { by lemma 67 R->L }
% 22.52/3.24 fresh30(is_a_theorem(implies(or(not(p9), q7), implies(not(and(q7, not(not(r8)))), or(not(r8), not(p9))))), kn1)
% 22.52/3.24 = { by lemma 66 }
% 22.52/3.24 fresh30(is_a_theorem(implies(or(not(p9), q7), implies(implies(q7, not(r8)), or(not(r8), not(p9))))), kn1)
% 22.52/3.24 = { by lemma 69 R->L }
% 22.52/3.24 fresh30(is_a_theorem(implies(or(not(p9), q7), implies(implies(q7, not(r8)), fresh(is_a_theorem(equiv(or(not(r8), not(p9)), or(not(p9), not(r8)))), modus_ponens, or(not(r8), not(p9)), or(not(p9), not(r8)))))), kn1)
% 22.52/3.24 = { by lemma 58 }
% 22.52/3.24 fresh30(is_a_theorem(implies(or(not(p9), q7), implies(implies(q7, not(r8)), fresh(is_a_theorem(equiv(or(not(r8), not(p9)), or(not(p9), not(r8)))), kn2, or(not(r8), not(p9)), or(not(p9), not(r8)))))), kn1)
% 22.52/3.24 = { by lemma 65 }
% 22.52/3.24 fresh30(is_a_theorem(implies(or(not(p9), q7), implies(implies(q7, not(r8)), fresh(is_a_theorem(equiv(or(not(r8), not(p9)), or(not(p9), not(r8)))), kn1, or(not(r8), not(p9)), or(not(p9), not(r8)))))), kn1)
% 22.52/3.25 = { by axiom 20 (modus_ponens_2) R->L }
% 22.52/3.25 fresh30(is_a_theorem(implies(or(not(p9), q7), implies(implies(q7, not(r8)), fresh(fresh28(kn1, kn1, equiv(or(not(r8), not(p9)), or(not(p9), not(r8)))), kn1, or(not(r8), not(p9)), or(not(p9), not(r8)))))), kn1)
% 22.52/3.25 = { by lemma 77 R->L }
% 22.52/3.25 fresh30(is_a_theorem(implies(or(not(p9), q7), implies(implies(q7, not(r8)), fresh(fresh28(is_a_theorem(implies(or(not(p9), not(r8)), or(not(r8), not(p9)))), kn1, equiv(or(not(r8), not(p9)), or(not(p9), not(r8)))), kn1, or(not(r8), not(p9)), or(not(p9), not(r8)))))), kn1)
% 22.52/3.25 = { by lemma 65 R->L }
% 22.52/3.25 fresh30(is_a_theorem(implies(or(not(p9), q7), implies(implies(q7, not(r8)), fresh(fresh28(is_a_theorem(implies(or(not(p9), not(r8)), or(not(r8), not(p9)))), kn2, equiv(or(not(r8), not(p9)), or(not(p9), not(r8)))), kn1, or(not(r8), not(p9)), or(not(p9), not(r8)))))), kn1)
% 22.52/3.25 = { by lemma 58 R->L }
% 22.52/3.25 fresh30(is_a_theorem(implies(or(not(p9), q7), implies(implies(q7, not(r8)), fresh(fresh28(is_a_theorem(implies(or(not(p9), not(r8)), or(not(r8), not(p9)))), modus_ponens, equiv(or(not(r8), not(p9)), or(not(p9), not(r8)))), kn1, or(not(r8), not(p9)), or(not(p9), not(r8)))))), kn1)
% 22.52/3.25 = { by lemma 60 R->L }
% 22.52/3.25 fresh30(is_a_theorem(implies(or(not(p9), q7), implies(implies(q7, not(r8)), fresh(fresh59(is_a_theorem(implies(implies(or(not(p9), not(r8)), or(not(r8), not(p9))), equiv(or(not(r8), not(p9)), or(not(p9), not(r8))))), modus_ponens, implies(or(not(p9), not(r8)), or(not(r8), not(p9))), equiv(or(not(r8), not(p9)), or(not(p9), not(r8)))), kn1, or(not(r8), not(p9)), or(not(p9), not(r8)))))), kn1)
% 22.52/3.25 = { by lemma 58 }
% 22.52/3.25 fresh30(is_a_theorem(implies(or(not(p9), q7), implies(implies(q7, not(r8)), fresh(fresh59(is_a_theorem(implies(implies(or(not(p9), not(r8)), or(not(r8), not(p9))), equiv(or(not(r8), not(p9)), or(not(p9), not(r8))))), kn2, implies(or(not(p9), not(r8)), or(not(r8), not(p9))), equiv(or(not(r8), not(p9)), or(not(p9), not(r8)))), kn1, or(not(r8), not(p9)), or(not(p9), not(r8)))))), kn1)
% 22.52/3.25 = { by lemma 65 }
% 22.52/3.25 fresh30(is_a_theorem(implies(or(not(p9), q7), implies(implies(q7, not(r8)), fresh(fresh59(is_a_theorem(implies(implies(or(not(p9), not(r8)), or(not(r8), not(p9))), equiv(or(not(r8), not(p9)), or(not(p9), not(r8))))), kn1, implies(or(not(p9), not(r8)), or(not(r8), not(p9))), equiv(or(not(r8), not(p9)), or(not(p9), not(r8)))), kn1, or(not(r8), not(p9)), or(not(p9), not(r8)))))), kn1)
% 22.52/3.25 = { by axiom 20 (modus_ponens_2) R->L }
% 22.52/3.25 fresh30(is_a_theorem(implies(or(not(p9), q7), implies(implies(q7, not(r8)), fresh(fresh59(fresh28(kn1, kn1, implies(implies(or(not(p9), not(r8)), or(not(r8), not(p9))), equiv(or(not(r8), not(p9)), or(not(p9), not(r8))))), kn1, implies(or(not(p9), not(r8)), or(not(r8), not(p9))), equiv(or(not(r8), not(p9)), or(not(p9), not(r8)))), kn1, or(not(r8), not(p9)), or(not(p9), not(r8)))))), kn1)
% 22.52/3.25 = { by lemma 77 R->L }
% 22.52/3.25 fresh30(is_a_theorem(implies(or(not(p9), q7), implies(implies(q7, not(r8)), fresh(fresh59(fresh28(is_a_theorem(implies(or(not(r8), not(p9)), or(not(p9), not(r8)))), kn1, implies(implies(or(not(p9), not(r8)), or(not(r8), not(p9))), equiv(or(not(r8), not(p9)), or(not(p9), not(r8))))), kn1, implies(or(not(p9), not(r8)), or(not(r8), not(p9))), equiv(or(not(r8), not(p9)), or(not(p9), not(r8)))), kn1, or(not(r8), not(p9)), or(not(p9), not(r8)))))), kn1)
% 22.52/3.25 = { by lemma 68 }
% 22.52/3.25 fresh30(is_a_theorem(implies(or(not(p9), q7), implies(implies(q7, not(r8)), fresh(fresh59(kn1, kn1, implies(or(not(p9), not(r8)), or(not(r8), not(p9))), equiv(or(not(r8), not(p9)), or(not(p9), not(r8)))), kn1, or(not(r8), not(p9)), or(not(p9), not(r8)))))), kn1)
% 22.52/3.25 = { by axiom 22 (modus_ponens_2) }
% 22.52/3.25 fresh30(is_a_theorem(implies(or(not(p9), q7), implies(implies(q7, not(r8)), fresh(fresh60(modus_ponens, true, equiv(or(not(r8), not(p9)), or(not(p9), not(r8)))), kn1, or(not(r8), not(p9)), or(not(p9), not(r8)))))), kn1)
% 22.52/3.25 = { by axiom 1 (hilbert_modus_ponens) R->L }
% 22.52/3.25 fresh30(is_a_theorem(implies(or(not(p9), q7), implies(implies(q7, not(r8)), fresh(fresh60(modus_ponens, modus_ponens, equiv(or(not(r8), not(p9)), or(not(p9), not(r8)))), kn1, or(not(r8), not(p9)), or(not(p9), not(r8)))))), kn1)
% 22.52/3.25 = { by lemma 61 }
% 22.52/3.25 fresh30(is_a_theorem(implies(or(not(p9), q7), implies(implies(q7, not(r8)), fresh(modus_ponens, kn1, or(not(r8), not(p9)), or(not(p9), not(r8)))))), kn1)
% 22.52/3.25 = { by lemma 58 }
% 22.52/3.25 fresh30(is_a_theorem(implies(or(not(p9), q7), implies(implies(q7, not(r8)), fresh(kn2, kn1, or(not(r8), not(p9)), or(not(p9), not(r8)))))), kn1)
% 22.52/3.25 = { by lemma 65 }
% 22.52/3.25 fresh30(is_a_theorem(implies(or(not(p9), q7), implies(implies(q7, not(r8)), fresh(kn1, kn1, or(not(r8), not(p9)), or(not(p9), not(r8)))))), kn1)
% 22.52/3.25 = { by axiom 21 (substitution_of_equivalents_2) }
% 22.52/3.25 fresh30(is_a_theorem(implies(or(not(p9), q7), implies(implies(q7, not(r8)), or(not(p9), not(r8))))), kn1)
% 22.52/3.25 = { by lemma 67 R->L }
% 22.52/3.25 fresh30(is_a_theorem(implies(or(not(p9), q7), implies(implies(q7, not(r8)), implies(not(not(p9)), not(r8))))), kn1)
% 22.52/3.25 = { by lemma 67 R->L }
% 22.52/3.25 fresh30(is_a_theorem(implies(implies(not(not(p9)), q7), implies(implies(q7, not(r8)), implies(not(not(p9)), not(r8))))), kn1)
% 22.52/3.25 = { by axiom 53 (implies_3_1) R->L }
% 22.52/3.25 fresh30(fresh35(implies_3, true, not(not(p9)), q7, not(r8)), kn1)
% 22.52/3.25 = { by axiom 6 (hilbert_implies_3) }
% 22.52/3.25 fresh30(fresh35(true, true, not(not(p9)), q7, not(r8)), kn1)
% 22.52/3.25 = { by axiom 1 (hilbert_modus_ponens) R->L }
% 22.52/3.25 fresh30(fresh35(modus_ponens, true, not(not(p9)), q7, not(r8)), kn1)
% 22.52/3.25 = { by lemma 58 }
% 22.52/3.25 fresh30(fresh35(kn2, true, not(not(p9)), q7, not(r8)), kn1)
% 22.52/3.25 = { by axiom 1 (hilbert_modus_ponens) R->L }
% 22.52/3.25 fresh30(fresh35(kn2, modus_ponens, not(not(p9)), q7, not(r8)), kn1)
% 22.52/3.25 = { by lemma 58 }
% 22.52/3.25 fresh30(fresh35(kn2, kn2, not(not(p9)), q7, not(r8)), kn1)
% 22.52/3.25 = { by axiom 34 (implies_3_1) }
% 22.52/3.25 fresh30(true, kn1)
% 22.52/3.25 = { by axiom 1 (hilbert_modus_ponens) R->L }
% 22.52/3.25 fresh30(modus_ponens, kn1)
% 22.52/3.25 = { by lemma 58 }
% 22.52/3.25 fresh30(kn2, kn1)
% 22.52/3.25 = { by lemma 65 }
% 22.52/3.25 fresh30(kn1, kn1)
% 22.52/3.25 = { by axiom 16 (kn3) }
% 22.52/3.25 true
% 22.52/3.25 % SZS output end Proof
% 22.52/3.25
% 22.52/3.25 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------