TSTP Solution File: LCL503+1 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : LCL503+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 : n016.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 08:19:15 EDT 2023
% Result : Theorem 59.46s 8.09s
% Output : Proof 60.20s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.13 % Problem : LCL503+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
% 0.13/0.35 % Computer : n016.cluster.edu
% 0.13/0.35 % Model : x86_64 x86_64
% 0.13/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.35 % Memory : 8042.1875MB
% 0.13/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.35 % CPULimit : 300
% 0.13/0.35 % WCLimit : 300
% 0.13/0.35 % DateTime : Thu Aug 24 17:43:09 EDT 2023
% 0.13/0.35 % CPUTime :
% 59.46/8.09 Command-line arguments: --set-join --lhs-weight 1 --no-flatten-goal --complete-subsets --goal-heuristic
% 59.46/8.09
% 59.46/8.09 % SZS status Theorem
% 59.46/8.09
% 60.20/8.14 % SZS output start Proof
% 60.20/8.14 Take the following subset of the input axioms:
% 60.20/8.15 fof(and_1, axiom, and_1 <=> ![X, Y]: is_a_theorem(implies(and(X, Y), X))).
% 60.20/8.15 fof(cn3, axiom, cn3 <=> ![P]: is_a_theorem(implies(implies(not(P), P), P))).
% 60.20/8.15 fof(hilbert_implies_1, conjecture, implies_1).
% 60.20/8.15 fof(implies_1, axiom, implies_1 <=> ![X2, Y2]: is_a_theorem(implies(X2, implies(Y2, X2)))).
% 60.20/8.15 fof(kn1, axiom, kn1 <=> ![P2]: is_a_theorem(implies(P2, and(P2, P2)))).
% 60.20/8.15 fof(kn2, axiom, kn2 <=> ![Q, P2]: is_a_theorem(implies(and(P2, Q), P2))).
% 60.20/8.15 fof(kn3, axiom, kn3 <=> ![R, P2, Q2]: is_a_theorem(implies(implies(P2, Q2), implies(not(and(Q2, R)), not(and(R, P2)))))).
% 60.20/8.15 fof(modus_ponens, axiom, modus_ponens <=> ![X2, Y2]: ((is_a_theorem(X2) & is_a_theorem(implies(X2, Y2))) => is_a_theorem(Y2))).
% 60.20/8.15 fof(op_equiv, axiom, op_equiv => ![X2, Y2]: equiv(X2, Y2)=and(implies(X2, Y2), implies(Y2, X2))).
% 60.20/8.15 fof(op_implies_and, axiom, op_implies_and => ![X2, Y2]: implies(X2, Y2)=not(and(X2, not(Y2)))).
% 60.20/8.15 fof(op_or, axiom, op_or => ![X2, Y2]: or(X2, Y2)=not(and(not(X2), not(Y2)))).
% 60.20/8.15 fof(r1, axiom, r1 <=> ![P2]: is_a_theorem(implies(or(P2, P2), P2))).
% 60.20/8.15 fof(r3, axiom, r3 <=> ![P2, Q2]: is_a_theorem(implies(or(P2, Q2), or(Q2, P2)))).
% 60.20/8.15 fof(rosser_kn1, axiom, kn1).
% 60.20/8.15 fof(rosser_kn2, axiom, kn2).
% 60.20/8.15 fof(rosser_kn3, axiom, kn3).
% 60.20/8.15 fof(rosser_modus_ponens, axiom, modus_ponens).
% 60.20/8.15 fof(rosser_op_equiv, axiom, op_equiv).
% 60.20/8.15 fof(rosser_op_implies_and, axiom, op_implies_and).
% 60.20/8.15 fof(rosser_op_or, axiom, op_or).
% 60.20/8.15
% 60.20/8.15 Now clausify the problem and encode Horn clauses using encoding 3 of
% 60.20/8.15 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 60.20/8.15 We repeatedly replace C & s=t => u=v by the two clauses:
% 60.20/8.15 fresh(y, y, x1...xn) = u
% 60.20/8.15 C => fresh(s, t, x1...xn) = v
% 60.20/8.15 where fresh is a fresh function symbol and x1..xn are the free
% 60.20/8.15 variables of u and v.
% 60.20/8.15 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 60.20/8.15 input problem has no model of domain size 1).
% 60.20/8.15
% 60.20/8.15 The encoding turns the above axioms into the following unit equations and goals:
% 60.20/8.15
% 60.20/8.15 Axiom 1 (rosser_modus_ponens): modus_ponens = true.
% 60.20/8.15 Axiom 2 (rosser_kn1): kn1 = true.
% 60.20/8.15 Axiom 3 (rosser_kn2): kn2 = true.
% 60.20/8.15 Axiom 4 (rosser_kn3): kn3 = true.
% 60.20/8.15 Axiom 5 (rosser_op_or): op_or = true.
% 60.20/8.15 Axiom 6 (rosser_op_implies_and): op_implies_and = true.
% 60.20/8.15 Axiom 7 (rosser_op_equiv): op_equiv = true.
% 60.20/8.15 Axiom 8 (and_1): fresh57(X, X) = true.
% 60.20/8.15 Axiom 9 (implies_1): fresh40(X, X) = true.
% 60.20/8.15 Axiom 10 (modus_ponens_2): fresh60(X, X, Y) = true.
% 60.20/8.15 Axiom 11 (kn1_1): fresh33(X, X, Y) = true.
% 60.20/8.15 Axiom 12 (modus_ponens_2): fresh28(X, X, Y) = is_a_theorem(Y).
% 60.20/8.15 Axiom 13 (modus_ponens_2): fresh59(X, X, Y, Z) = fresh60(modus_ponens, true, Z).
% 60.20/8.15 Axiom 14 (kn2_1): fresh31(X, X, Y, Z) = true.
% 60.20/8.15 Axiom 15 (op_equiv): fresh23(X, X, Y, Z) = equiv(Y, Z).
% 60.20/8.15 Axiom 16 (op_implies_and): fresh22(X, X, Y, Z) = implies(Y, Z).
% 60.20/8.15 Axiom 17 (op_or): fresh20(X, X, Y, Z) = or(Y, Z).
% 60.20/8.15 Axiom 18 (op_implies_and): fresh22(op_implies_and, true, X, Y) = not(and(X, not(Y))).
% 60.20/8.15 Axiom 19 (kn3_1): fresh29(X, X, Y, Z, W) = true.
% 60.20/8.15 Axiom 20 (op_or): fresh20(op_or, true, X, Y) = not(and(not(X), not(Y))).
% 60.20/8.15 Axiom 21 (kn1_1): fresh33(kn1, true, X) = is_a_theorem(implies(X, and(X, X))).
% 60.20/8.15 Axiom 22 (implies_1_1): fresh39(implies_1, true, X, Y) = is_a_theorem(implies(X, implies(Y, X))).
% 60.20/8.15 Axiom 23 (and_1_1): fresh58(and_1, true, X, Y) = is_a_theorem(implies(and(X, Y), X)).
% 60.20/8.15 Axiom 24 (kn2_1): fresh31(kn2, true, X, Y) = is_a_theorem(implies(and(X, Y), X)).
% 60.20/8.15 Axiom 25 (r1_1): fresh12(r1, true, X) = is_a_theorem(implies(or(X, X), X)).
% 60.20/8.15 Axiom 26 (op_equiv): fresh23(op_equiv, true, X, Y) = and(implies(X, Y), implies(Y, X)).
% 60.20/8.15 Axiom 27 (cn3_1): fresh47(cn3, true, X) = is_a_theorem(implies(implies(not(X), X), X)).
% 60.20/8.15 Axiom 28 (modus_ponens_2): fresh59(is_a_theorem(implies(X, Y)), true, X, Y) = fresh28(is_a_theorem(X), true, Y).
% 60.20/8.15 Axiom 29 (and_1): fresh57(is_a_theorem(implies(and(x9, y9), x9)), true) = and_1.
% 60.20/8.15 Axiom 30 (implies_1): fresh40(is_a_theorem(implies(x12, implies(y12, x12))), true) = implies_1.
% 60.20/8.15 Axiom 31 (r3_1): fresh8(r3, true, X, Y) = is_a_theorem(implies(or(X, Y), or(Y, X))).
% 60.20/8.15 Axiom 32 (kn3_1): fresh29(kn3, true, X, Y, Z) = is_a_theorem(implies(implies(X, Y), implies(not(and(Y, Z)), not(and(Z, X))))).
% 60.20/8.15
% 60.20/8.15 Lemma 33: is_a_theorem(implies(and(X, Y), X)) = fresh58(and_1, modus_ponens, X, Y).
% 60.20/8.15 Proof:
% 60.20/8.15 is_a_theorem(implies(and(X, Y), X))
% 60.20/8.15 = { by axiom 23 (and_1_1) R->L }
% 60.20/8.15 fresh58(and_1, true, X, Y)
% 60.20/8.15 = { by axiom 1 (rosser_modus_ponens) R->L }
% 60.20/8.15 fresh58(and_1, modus_ponens, X, Y)
% 60.20/8.15
% 60.20/8.15 Lemma 34: fresh58(and_1, modus_ponens, X, Y) = modus_ponens.
% 60.20/8.15 Proof:
% 60.20/8.15 fresh58(and_1, modus_ponens, X, Y)
% 60.20/8.15 = { by lemma 33 R->L }
% 60.20/8.15 is_a_theorem(implies(and(X, Y), X))
% 60.20/8.15 = { by axiom 24 (kn2_1) R->L }
% 60.20/8.15 fresh31(kn2, true, X, Y)
% 60.20/8.15 = { by axiom 3 (rosser_kn2) }
% 60.20/8.15 fresh31(true, true, X, Y)
% 60.20/8.15 = { by axiom 1 (rosser_modus_ponens) R->L }
% 60.20/8.15 fresh31(modus_ponens, true, X, Y)
% 60.20/8.15 = { by axiom 1 (rosser_modus_ponens) R->L }
% 60.20/8.15 fresh31(modus_ponens, modus_ponens, X, Y)
% 60.20/8.15 = { by axiom 14 (kn2_1) }
% 60.20/8.15 true
% 60.20/8.15 = { by axiom 1 (rosser_modus_ponens) R->L }
% 60.20/8.15 modus_ponens
% 60.20/8.15
% 60.20/8.15 Lemma 35: modus_ponens = and_1.
% 60.20/8.15 Proof:
% 60.20/8.15 modus_ponens
% 60.20/8.15 = { by axiom 1 (rosser_modus_ponens) }
% 60.20/8.15 true
% 60.20/8.15 = { by axiom 8 (and_1) R->L }
% 60.20/8.15 fresh57(modus_ponens, modus_ponens)
% 60.20/8.15 = { by axiom 1 (rosser_modus_ponens) }
% 60.20/8.15 fresh57(modus_ponens, true)
% 60.20/8.15 = { by lemma 34 R->L }
% 60.20/8.15 fresh57(fresh58(and_1, modus_ponens, x9, y9), true)
% 60.20/8.15 = { by lemma 33 R->L }
% 60.20/8.15 fresh57(is_a_theorem(implies(and(x9, y9), x9)), true)
% 60.20/8.15 = { by axiom 29 (and_1) }
% 60.20/8.15 and_1
% 60.20/8.15
% 60.20/8.15 Lemma 36: and(implies(X, Y), implies(Y, X)) = equiv(X, Y).
% 60.20/8.15 Proof:
% 60.20/8.15 and(implies(X, Y), implies(Y, X))
% 60.20/8.15 = { by axiom 26 (op_equiv) R->L }
% 60.20/8.15 fresh23(op_equiv, true, X, Y)
% 60.20/8.15 = { by axiom 7 (rosser_op_equiv) }
% 60.20/8.15 fresh23(true, true, X, Y)
% 60.20/8.15 = { by axiom 1 (rosser_modus_ponens) R->L }
% 60.20/8.15 fresh23(modus_ponens, true, X, Y)
% 60.20/8.15 = { by axiom 1 (rosser_modus_ponens) R->L }
% 60.20/8.15 fresh23(modus_ponens, modus_ponens, X, Y)
% 60.20/8.15 = { by axiom 15 (op_equiv) }
% 60.20/8.15 equiv(X, Y)
% 60.20/8.15
% 60.20/8.15 Lemma 37: fresh60(X, X, Y) = modus_ponens.
% 60.20/8.15 Proof:
% 60.20/8.15 fresh60(X, X, Y)
% 60.20/8.15 = { by axiom 10 (modus_ponens_2) }
% 60.20/8.15 true
% 60.20/8.15 = { by axiom 1 (rosser_modus_ponens) R->L }
% 60.20/8.15 modus_ponens
% 60.20/8.15
% 60.20/8.15 Lemma 38: fresh59(X, X, Y, Z) = modus_ponens.
% 60.20/8.15 Proof:
% 60.20/8.15 fresh59(X, X, Y, Z)
% 60.20/8.15 = { by axiom 13 (modus_ponens_2) }
% 60.20/8.15 fresh60(modus_ponens, true, Z)
% 60.20/8.15 = { by axiom 1 (rosser_modus_ponens) R->L }
% 60.20/8.15 fresh60(modus_ponens, modus_ponens, Z)
% 60.20/8.15 = { by lemma 37 }
% 60.20/8.15 modus_ponens
% 60.20/8.15
% 60.20/8.15 Lemma 39: not(and(X, not(Y))) = implies(X, Y).
% 60.20/8.15 Proof:
% 60.20/8.15 not(and(X, not(Y)))
% 60.20/8.15 = { by axiom 18 (op_implies_and) R->L }
% 60.20/8.15 fresh22(op_implies_and, true, X, Y)
% 60.20/8.15 = { by axiom 6 (rosser_op_implies_and) }
% 60.20/8.15 fresh22(true, true, X, Y)
% 60.20/8.15 = { by axiom 1 (rosser_modus_ponens) R->L }
% 60.20/8.15 fresh22(modus_ponens, true, X, Y)
% 60.20/8.15 = { by axiom 1 (rosser_modus_ponens) R->L }
% 60.20/8.15 fresh22(modus_ponens, modus_ponens, X, Y)
% 60.20/8.15 = { by axiom 16 (op_implies_and) }
% 60.20/8.15 implies(X, Y)
% 60.20/8.15
% 60.20/8.15 Lemma 40: implies(not(X), Y) = or(X, Y).
% 60.20/8.15 Proof:
% 60.20/8.15 implies(not(X), Y)
% 60.20/8.15 = { by lemma 39 R->L }
% 60.20/8.15 not(and(not(X), not(Y)))
% 60.20/8.15 = { by axiom 20 (op_or) R->L }
% 60.20/8.15 fresh20(op_or, true, X, Y)
% 60.20/8.15 = { by axiom 5 (rosser_op_or) }
% 60.20/8.15 fresh20(true, true, X, Y)
% 60.20/8.15 = { by axiom 1 (rosser_modus_ponens) R->L }
% 60.20/8.15 fresh20(modus_ponens, true, X, Y)
% 60.20/8.15 = { by axiom 1 (rosser_modus_ponens) R->L }
% 60.20/8.15 fresh20(modus_ponens, modus_ponens, X, Y)
% 60.20/8.15 = { by axiom 17 (op_or) }
% 60.20/8.15 or(X, Y)
% 60.20/8.15
% 60.20/8.15 Lemma 41: is_a_theorem(implies(implies(X, Y), or(and(Y, Z), not(and(Z, X))))) = and_1.
% 60.20/8.15 Proof:
% 60.20/8.15 is_a_theorem(implies(implies(X, Y), or(and(Y, Z), not(and(Z, X)))))
% 60.20/8.15 = { by lemma 40 R->L }
% 60.20/8.15 is_a_theorem(implies(implies(X, Y), implies(not(and(Y, Z)), not(and(Z, X)))))
% 60.20/8.15 = { by axiom 32 (kn3_1) R->L }
% 60.20/8.15 fresh29(kn3, true, X, Y, Z)
% 60.20/8.15 = { by axiom 4 (rosser_kn3) }
% 60.20/8.15 fresh29(true, true, X, Y, Z)
% 60.20/8.15 = { by axiom 1 (rosser_modus_ponens) R->L }
% 60.20/8.15 fresh29(modus_ponens, true, X, Y, Z)
% 60.20/8.15 = { by lemma 35 }
% 60.20/8.15 fresh29(and_1, true, X, Y, Z)
% 60.20/8.15 = { by axiom 1 (rosser_modus_ponens) R->L }
% 60.20/8.15 fresh29(and_1, modus_ponens, X, Y, Z)
% 60.20/8.15 = { by lemma 35 }
% 60.20/8.15 fresh29(and_1, and_1, X, Y, Z)
% 60.20/8.15 = { by axiom 19 (kn3_1) }
% 60.20/8.15 true
% 60.20/8.15 = { by axiom 1 (rosser_modus_ponens) R->L }
% 60.20/8.15 modus_ponens
% 60.20/8.15 = { by lemma 35 }
% 60.20/8.15 and_1
% 60.20/8.15
% 60.20/8.15 Lemma 42: is_a_theorem(implies(or(X, Y), or(and(Y, Z), implies(Z, X)))) = and_1.
% 60.20/8.15 Proof:
% 60.20/8.15 is_a_theorem(implies(or(X, Y), or(and(Y, Z), implies(Z, X))))
% 60.20/8.15 = { by lemma 40 R->L }
% 60.20/8.15 is_a_theorem(implies(implies(not(X), Y), or(and(Y, Z), implies(Z, X))))
% 60.20/8.15 = { by lemma 39 R->L }
% 60.20/8.15 is_a_theorem(implies(implies(not(X), Y), or(and(Y, Z), not(and(Z, not(X))))))
% 60.20/8.15 = { by lemma 41 }
% 60.20/8.15 and_1
% 60.20/8.15
% 60.20/8.15 Lemma 43: fresh59(is_a_theorem(implies(X, Y)), modus_ponens, X, Y) = fresh28(is_a_theorem(X), modus_ponens, Y).
% 60.20/8.15 Proof:
% 60.20/8.15 fresh59(is_a_theorem(implies(X, Y)), modus_ponens, X, Y)
% 60.20/8.15 = { by axiom 1 (rosser_modus_ponens) }
% 60.20/8.15 fresh59(is_a_theorem(implies(X, Y)), true, X, Y)
% 60.20/8.15 = { by axiom 28 (modus_ponens_2) }
% 60.20/8.15 fresh28(is_a_theorem(X), true, Y)
% 60.20/8.15 = { by axiom 1 (rosser_modus_ponens) R->L }
% 60.20/8.15 fresh28(is_a_theorem(X), modus_ponens, Y)
% 60.20/8.15
% 60.20/8.15 Lemma 44: fresh28(is_a_theorem(implies(X, Y)), and_1, or(and(Y, Z), not(and(Z, X)))) = and_1.
% 60.20/8.15 Proof:
% 60.20/8.15 fresh28(is_a_theorem(implies(X, Y)), and_1, or(and(Y, Z), not(and(Z, X))))
% 60.20/8.15 = { by lemma 35 R->L }
% 60.20/8.15 fresh28(is_a_theorem(implies(X, Y)), modus_ponens, or(and(Y, Z), not(and(Z, X))))
% 60.20/8.15 = { by lemma 43 R->L }
% 60.20/8.15 fresh59(is_a_theorem(implies(implies(X, Y), or(and(Y, Z), not(and(Z, X))))), modus_ponens, implies(X, Y), or(and(Y, Z), not(and(Z, X))))
% 60.20/8.15 = { by lemma 41 }
% 60.20/8.15 fresh59(and_1, modus_ponens, implies(X, Y), or(and(Y, Z), not(and(Z, X))))
% 60.20/8.15 = { by lemma 35 }
% 60.20/8.15 fresh59(and_1, and_1, implies(X, Y), or(and(Y, Z), not(and(Z, X))))
% 60.20/8.15 = { by lemma 38 }
% 60.20/8.15 modus_ponens
% 60.20/8.15 = { by lemma 35 }
% 60.20/8.15 and_1
% 60.20/8.15
% 60.20/8.15 Lemma 45: is_a_theorem(implies(X, and(X, X))) = modus_ponens.
% 60.20/8.15 Proof:
% 60.20/8.15 is_a_theorem(implies(X, and(X, X)))
% 60.20/8.15 = { by axiom 21 (kn1_1) R->L }
% 60.20/8.15 fresh33(kn1, true, X)
% 60.20/8.15 = { by axiom 2 (rosser_kn1) }
% 60.20/8.15 fresh33(true, true, X)
% 60.20/8.15 = { by axiom 1 (rosser_modus_ponens) R->L }
% 60.20/8.15 fresh33(modus_ponens, true, X)
% 60.20/8.15 = { by axiom 1 (rosser_modus_ponens) R->L }
% 60.20/8.15 fresh33(modus_ponens, modus_ponens, X)
% 60.20/8.15 = { by axiom 11 (kn1_1) }
% 60.20/8.15 true
% 60.20/8.15 = { by axiom 1 (rosser_modus_ponens) R->L }
% 60.20/8.15 modus_ponens
% 60.20/8.15
% 60.20/8.15 Lemma 46: or(and(X, not(Y)), Z) = implies(implies(X, Y), Z).
% 60.20/8.15 Proof:
% 60.20/8.15 or(and(X, not(Y)), Z)
% 60.20/8.15 = { by lemma 40 R->L }
% 60.20/8.15 implies(not(and(X, not(Y))), Z)
% 60.20/8.15 = { by lemma 39 }
% 60.20/8.15 implies(implies(X, Y), Z)
% 60.20/8.15
% 60.20/8.15 Lemma 47: fresh28(is_a_theorem(or(X, Y)), and_1, implies(implies(Y, Z), or(Z, X))) = and_1.
% 60.20/8.15 Proof:
% 60.20/8.15 fresh28(is_a_theorem(or(X, Y)), and_1, implies(implies(Y, Z), or(Z, X)))
% 60.20/8.15 = { by lemma 35 R->L }
% 60.20/8.15 fresh28(is_a_theorem(or(X, Y)), modus_ponens, implies(implies(Y, Z), or(Z, X)))
% 60.20/8.15 = { by lemma 43 R->L }
% 60.20/8.15 fresh59(is_a_theorem(implies(or(X, Y), implies(implies(Y, Z), or(Z, X)))), modus_ponens, or(X, Y), implies(implies(Y, Z), or(Z, X)))
% 60.20/8.15 = { by lemma 46 R->L }
% 60.20/8.15 fresh59(is_a_theorem(implies(or(X, Y), or(and(Y, not(Z)), or(Z, X)))), modus_ponens, or(X, Y), implies(implies(Y, Z), or(Z, X)))
% 60.20/8.15 = { by lemma 40 R->L }
% 60.20/8.15 fresh59(is_a_theorem(implies(or(X, Y), or(and(Y, not(Z)), implies(not(Z), X)))), modus_ponens, or(X, Y), implies(implies(Y, Z), or(Z, X)))
% 60.20/8.15 = { by lemma 42 }
% 60.20/8.15 fresh59(and_1, modus_ponens, or(X, Y), implies(implies(Y, Z), or(Z, X)))
% 60.20/8.15 = { by lemma 35 }
% 60.20/8.15 fresh59(and_1, and_1, or(X, Y), implies(implies(Y, Z), or(Z, X)))
% 60.20/8.15 = { by lemma 38 }
% 60.20/8.15 modus_ponens
% 60.20/8.15 = { by lemma 35 }
% 60.20/8.15 and_1
% 60.20/8.15
% 60.20/8.15 Lemma 48: fresh28(is_a_theorem(implies(X, Y)), and_1, or(Y, not(X))) = and_1.
% 60.20/8.15 Proof:
% 60.20/8.15 fresh28(is_a_theorem(implies(X, Y)), and_1, or(Y, not(X)))
% 60.20/8.15 = { by lemma 35 R->L }
% 60.20/8.15 fresh28(is_a_theorem(implies(X, Y)), modus_ponens, or(Y, not(X)))
% 60.20/8.15 = { by lemma 43 R->L }
% 60.20/8.15 fresh59(is_a_theorem(implies(implies(X, Y), or(Y, not(X)))), modus_ponens, implies(X, Y), or(Y, not(X)))
% 60.20/8.15 = { by axiom 12 (modus_ponens_2) R->L }
% 60.20/8.15 fresh59(fresh28(and_1, and_1, implies(implies(X, Y), or(Y, not(X)))), modus_ponens, implies(X, Y), or(Y, not(X)))
% 60.20/8.15 = { by lemma 35 R->L }
% 60.20/8.15 fresh59(fresh28(modus_ponens, and_1, implies(implies(X, Y), or(Y, not(X)))), modus_ponens, implies(X, Y), or(Y, not(X)))
% 60.20/8.15 = { by lemma 38 R->L }
% 60.20/8.15 fresh59(fresh28(fresh59(and_1, and_1, not(and(and(not(X), not(X)), not(not(X)))), not(and(not(not(X)), not(X)))), and_1, implies(implies(X, Y), or(Y, not(X)))), modus_ponens, implies(X, Y), or(Y, not(X)))
% 60.20/8.15 = { by lemma 44 R->L }
% 60.20/8.15 fresh59(fresh28(fresh59(fresh28(is_a_theorem(implies(not(X), and(not(X), not(X)))), and_1, or(and(and(not(X), not(X)), not(not(X))), not(and(not(not(X)), not(X))))), and_1, not(and(and(not(X), not(X)), not(not(X)))), not(and(not(not(X)), not(X)))), and_1, implies(implies(X, Y), or(Y, not(X)))), modus_ponens, implies(X, Y), or(Y, not(X)))
% 60.20/8.16 = { by lemma 45 }
% 60.20/8.16 fresh59(fresh28(fresh59(fresh28(modus_ponens, and_1, or(and(and(not(X), not(X)), not(not(X))), not(and(not(not(X)), not(X))))), and_1, not(and(and(not(X), not(X)), not(not(X)))), not(and(not(not(X)), not(X)))), and_1, implies(implies(X, Y), or(Y, not(X)))), modus_ponens, implies(X, Y), or(Y, not(X)))
% 60.20/8.16 = { by lemma 35 }
% 60.20/8.16 fresh59(fresh28(fresh59(fresh28(and_1, and_1, or(and(and(not(X), not(X)), not(not(X))), not(and(not(not(X)), not(X))))), and_1, not(and(and(not(X), not(X)), not(not(X)))), not(and(not(not(X)), not(X)))), and_1, implies(implies(X, Y), or(Y, not(X)))), modus_ponens, implies(X, Y), or(Y, not(X)))
% 60.20/8.16 = { by axiom 12 (modus_ponens_2) }
% 60.20/8.16 fresh59(fresh28(fresh59(is_a_theorem(or(and(and(not(X), not(X)), not(not(X))), not(and(not(not(X)), not(X))))), and_1, not(and(and(not(X), not(X)), not(not(X)))), not(and(not(not(X)), not(X)))), and_1, implies(implies(X, Y), or(Y, not(X)))), modus_ponens, implies(X, Y), or(Y, not(X)))
% 60.20/8.16 = { by lemma 35 R->L }
% 60.20/8.16 fresh59(fresh28(fresh59(is_a_theorem(or(and(and(not(X), not(X)), not(not(X))), not(and(not(not(X)), not(X))))), modus_ponens, not(and(and(not(X), not(X)), not(not(X)))), not(and(not(not(X)), not(X)))), and_1, implies(implies(X, Y), or(Y, not(X)))), modus_ponens, implies(X, Y), or(Y, not(X)))
% 60.20/8.16 = { by lemma 40 R->L }
% 60.20/8.16 fresh59(fresh28(fresh59(is_a_theorem(implies(not(and(and(not(X), not(X)), not(not(X)))), not(and(not(not(X)), not(X))))), modus_ponens, not(and(and(not(X), not(X)), not(not(X)))), not(and(not(not(X)), not(X)))), and_1, implies(implies(X, Y), or(Y, not(X)))), modus_ponens, implies(X, Y), or(Y, not(X)))
% 60.20/8.16 = { by lemma 43 }
% 60.20/8.16 fresh59(fresh28(fresh28(is_a_theorem(not(and(and(not(X), not(X)), not(not(X))))), modus_ponens, not(and(not(not(X)), not(X)))), and_1, implies(implies(X, Y), or(Y, not(X)))), modus_ponens, implies(X, Y), or(Y, not(X)))
% 60.20/8.16 = { by lemma 35 }
% 60.20/8.16 fresh59(fresh28(fresh28(is_a_theorem(not(and(and(not(X), not(X)), not(not(X))))), and_1, not(and(not(not(X)), not(X)))), and_1, implies(implies(X, Y), or(Y, not(X)))), modus_ponens, implies(X, Y), or(Y, not(X)))
% 60.20/8.16 = { by lemma 39 }
% 60.20/8.16 fresh59(fresh28(fresh28(is_a_theorem(implies(and(not(X), not(X)), not(X))), and_1, not(and(not(not(X)), not(X)))), and_1, implies(implies(X, Y), or(Y, not(X)))), modus_ponens, implies(X, Y), or(Y, not(X)))
% 60.20/8.16 = { by lemma 33 }
% 60.20/8.16 fresh59(fresh28(fresh28(fresh58(and_1, modus_ponens, not(X), not(X)), and_1, not(and(not(not(X)), not(X)))), and_1, implies(implies(X, Y), or(Y, not(X)))), modus_ponens, implies(X, Y), or(Y, not(X)))
% 60.20/8.16 = { by lemma 34 }
% 60.20/8.16 fresh59(fresh28(fresh28(modus_ponens, and_1, not(and(not(not(X)), not(X)))), and_1, implies(implies(X, Y), or(Y, not(X)))), modus_ponens, implies(X, Y), or(Y, not(X)))
% 60.20/8.16 = { by lemma 35 }
% 60.20/8.16 fresh59(fresh28(fresh28(and_1, and_1, not(and(not(not(X)), not(X)))), and_1, implies(implies(X, Y), or(Y, not(X)))), modus_ponens, implies(X, Y), or(Y, not(X)))
% 60.20/8.16 = { by axiom 12 (modus_ponens_2) }
% 60.20/8.16 fresh59(fresh28(is_a_theorem(not(and(not(not(X)), not(X)))), and_1, implies(implies(X, Y), or(Y, not(X)))), modus_ponens, implies(X, Y), or(Y, not(X)))
% 60.20/8.16 = { by lemma 39 }
% 60.20/8.16 fresh59(fresh28(is_a_theorem(implies(not(not(X)), X)), and_1, implies(implies(X, Y), or(Y, not(X)))), modus_ponens, implies(X, Y), or(Y, not(X)))
% 60.20/8.16 = { by lemma 40 }
% 60.20/8.16 fresh59(fresh28(is_a_theorem(or(not(X), X)), and_1, implies(implies(X, Y), or(Y, not(X)))), modus_ponens, implies(X, Y), or(Y, not(X)))
% 60.20/8.16 = { by lemma 47 }
% 60.20/8.16 fresh59(and_1, modus_ponens, implies(X, Y), or(Y, not(X)))
% 60.20/8.16 = { by lemma 35 }
% 60.20/8.16 fresh59(and_1, and_1, implies(X, Y), or(Y, not(X)))
% 60.20/8.16 = { by lemma 38 }
% 60.20/8.16 modus_ponens
% 60.20/8.16 = { by lemma 35 }
% 60.20/8.16 and_1
% 60.20/8.16
% 60.20/8.16 Lemma 49: is_a_theorem(or(X, not(and(X, Y)))) = and_1.
% 60.20/8.16 Proof:
% 60.20/8.16 is_a_theorem(or(X, not(and(X, Y))))
% 60.20/8.16 = { by axiom 12 (modus_ponens_2) R->L }
% 60.20/8.16 fresh28(and_1, and_1, or(X, not(and(X, Y))))
% 60.20/8.16 = { by lemma 35 R->L }
% 60.20/8.16 fresh28(modus_ponens, and_1, or(X, not(and(X, Y))))
% 60.20/8.16 = { by lemma 34 R->L }
% 60.20/8.16 fresh28(fresh58(and_1, modus_ponens, X, Y), and_1, or(X, not(and(X, Y))))
% 60.20/8.16 = { by lemma 33 R->L }
% 60.20/8.16 fresh28(is_a_theorem(implies(and(X, Y), X)), and_1, or(X, not(and(X, Y))))
% 60.20/8.16 = { by lemma 48 }
% 60.20/8.16 and_1
% 60.20/8.16
% 60.20/8.16 Lemma 50: fresh28(is_a_theorem(or(and(X, Y), Z)), and_1, or(Z, X)) = and_1.
% 60.20/8.16 Proof:
% 60.20/8.16 fresh28(is_a_theorem(or(and(X, Y), Z)), and_1, or(Z, X))
% 60.20/8.16 = { by lemma 35 R->L }
% 60.20/8.16 fresh28(is_a_theorem(or(and(X, Y), Z)), modus_ponens, or(Z, X))
% 60.20/8.16 = { by lemma 43 R->L }
% 60.20/8.16 fresh59(is_a_theorem(implies(or(and(X, Y), Z), or(Z, X))), modus_ponens, or(and(X, Y), Z), or(Z, X))
% 60.20/8.16 = { by lemma 40 R->L }
% 60.20/8.16 fresh59(is_a_theorem(implies(implies(not(and(X, Y)), Z), or(Z, X))), modus_ponens, or(and(X, Y), Z), or(Z, X))
% 60.20/8.16 = { by axiom 12 (modus_ponens_2) R->L }
% 60.20/8.16 fresh59(fresh28(and_1, and_1, implies(implies(not(and(X, Y)), Z), or(Z, X))), modus_ponens, or(and(X, Y), Z), or(Z, X))
% 60.20/8.16 = { by lemma 49 R->L }
% 60.20/8.16 fresh59(fresh28(is_a_theorem(or(X, not(and(X, Y)))), and_1, implies(implies(not(and(X, Y)), Z), or(Z, X))), modus_ponens, or(and(X, Y), Z), or(Z, X))
% 60.20/8.16 = { by lemma 47 }
% 60.20/8.16 fresh59(and_1, modus_ponens, or(and(X, Y), Z), or(Z, X))
% 60.20/8.16 = { by lemma 35 }
% 60.20/8.16 fresh59(and_1, and_1, or(and(X, Y), Z), or(Z, X))
% 60.20/8.16 = { by lemma 38 }
% 60.20/8.16 modus_ponens
% 60.20/8.16 = { by lemma 35 }
% 60.20/8.16 and_1
% 60.20/8.16
% 60.20/8.16 Lemma 51: is_a_theorem(or(implies(X, Y), implies(Y, Z))) = and_1.
% 60.20/8.16 Proof:
% 60.20/8.16 is_a_theorem(or(implies(X, Y), implies(Y, Z)))
% 60.20/8.16 = { by axiom 12 (modus_ponens_2) R->L }
% 60.20/8.16 fresh28(and_1, and_1, or(implies(X, Y), implies(Y, Z)))
% 60.20/8.16 = { by lemma 35 R->L }
% 60.20/8.16 fresh28(modus_ponens, and_1, or(implies(X, Y), implies(Y, Z)))
% 60.20/8.16 = { by lemma 38 R->L }
% 60.20/8.16 fresh28(fresh59(and_1, and_1, or(Y, implies(Y, Z)), or(and(implies(Y, Z), X), implies(X, Y))), and_1, or(implies(X, Y), implies(Y, Z)))
% 60.20/8.16 = { by lemma 35 R->L }
% 60.20/8.16 fresh28(fresh59(and_1, modus_ponens, or(Y, implies(Y, Z)), or(and(implies(Y, Z), X), implies(X, Y))), and_1, or(implies(X, Y), implies(Y, Z)))
% 60.20/8.16 = { by lemma 42 R->L }
% 60.20/8.16 fresh28(fresh59(is_a_theorem(implies(or(Y, implies(Y, Z)), or(and(implies(Y, Z), X), implies(X, Y)))), modus_ponens, or(Y, implies(Y, Z)), or(and(implies(Y, Z), X), implies(X, Y))), and_1, or(implies(X, Y), implies(Y, Z)))
% 60.20/8.16 = { by lemma 43 }
% 60.20/8.16 fresh28(fresh28(is_a_theorem(or(Y, implies(Y, Z))), modus_ponens, or(and(implies(Y, Z), X), implies(X, Y))), and_1, or(implies(X, Y), implies(Y, Z)))
% 60.20/8.16 = { by lemma 35 }
% 60.20/8.16 fresh28(fresh28(is_a_theorem(or(Y, implies(Y, Z))), and_1, or(and(implies(Y, Z), X), implies(X, Y))), and_1, or(implies(X, Y), implies(Y, Z)))
% 60.20/8.16 = { by lemma 39 R->L }
% 60.20/8.16 fresh28(fresh28(is_a_theorem(or(Y, not(and(Y, not(Z))))), and_1, or(and(implies(Y, Z), X), implies(X, Y))), and_1, or(implies(X, Y), implies(Y, Z)))
% 60.20/8.16 = { by lemma 49 }
% 60.20/8.16 fresh28(fresh28(and_1, and_1, or(and(implies(Y, Z), X), implies(X, Y))), and_1, or(implies(X, Y), implies(Y, Z)))
% 60.20/8.16 = { by axiom 12 (modus_ponens_2) }
% 60.20/8.16 fresh28(is_a_theorem(or(and(implies(Y, Z), X), implies(X, Y))), and_1, or(implies(X, Y), implies(Y, Z)))
% 60.20/8.16 = { by lemma 50 }
% 60.20/8.16 and_1
% 60.20/8.16
% 60.20/8.16 Lemma 52: is_a_theorem(or(X, and(not(X), not(X)))) = and_1.
% 60.20/8.16 Proof:
% 60.20/8.16 is_a_theorem(or(X, and(not(X), not(X))))
% 60.20/8.16 = { by lemma 40 R->L }
% 60.20/8.16 is_a_theorem(implies(not(X), and(not(X), not(X))))
% 60.20/8.16 = { by lemma 45 }
% 60.20/8.16 modus_ponens
% 60.20/8.16 = { by lemma 35 }
% 60.20/8.16 and_1
% 60.20/8.16
% 60.20/8.16 Lemma 53: is_a_theorem(equiv(X, X)) = and_1.
% 60.20/8.16 Proof:
% 60.20/8.16 is_a_theorem(equiv(X, X))
% 60.20/8.16 = { by lemma 36 R->L }
% 60.20/8.16 is_a_theorem(and(implies(X, X), implies(X, X)))
% 60.20/8.16 = { by axiom 12 (modus_ponens_2) R->L }
% 60.20/8.16 fresh28(and_1, and_1, and(implies(X, X), implies(X, X)))
% 60.20/8.16 = { by lemma 51 R->L }
% 60.20/8.16 fresh28(is_a_theorem(or(implies(X, X), implies(X, X))), and_1, and(implies(X, X), implies(X, X)))
% 60.20/8.16 = { by lemma 35 R->L }
% 60.20/8.16 fresh28(is_a_theorem(or(implies(X, X), implies(X, X))), modus_ponens, and(implies(X, X), implies(X, X)))
% 60.20/8.16 = { by lemma 43 R->L }
% 60.20/8.16 fresh59(is_a_theorem(implies(or(implies(X, X), implies(X, X)), and(implies(X, X), implies(X, X)))), modus_ponens, or(implies(X, X), implies(X, X)), and(implies(X, X), implies(X, X)))
% 60.20/8.16 = { by lemma 40 R->L }
% 60.20/8.16 fresh59(is_a_theorem(implies(implies(not(implies(X, X)), implies(X, X)), and(implies(X, X), implies(X, X)))), modus_ponens, or(implies(X, X), implies(X, X)), and(implies(X, X), implies(X, X)))
% 60.20/8.16 = { by lemma 46 R->L }
% 60.20/8.16 fresh59(is_a_theorem(or(and(not(implies(X, X)), not(implies(X, X))), and(implies(X, X), implies(X, X)))), modus_ponens, or(implies(X, X), implies(X, X)), and(implies(X, X), implies(X, X)))
% 60.20/8.16 = { by axiom 12 (modus_ponens_2) R->L }
% 60.20/8.16 fresh59(fresh28(and_1, and_1, or(and(not(implies(X, X)), not(implies(X, X))), and(implies(X, X), implies(X, X)))), modus_ponens, or(implies(X, X), implies(X, X)), and(implies(X, X), implies(X, X)))
% 60.20/8.16 = { by lemma 52 R->L }
% 60.20/8.16 fresh59(fresh28(is_a_theorem(or(implies(X, X), and(not(implies(X, X)), not(implies(X, X))))), and_1, or(and(not(implies(X, X)), not(implies(X, X))), and(implies(X, X), implies(X, X)))), modus_ponens, or(implies(X, X), implies(X, X)), and(implies(X, X), implies(X, X)))
% 60.20/8.16 = { by lemma 35 R->L }
% 60.20/8.16 fresh59(fresh28(is_a_theorem(or(implies(X, X), and(not(implies(X, X)), not(implies(X, X))))), modus_ponens, or(and(not(implies(X, X)), not(implies(X, X))), and(implies(X, X), implies(X, X)))), modus_ponens, or(implies(X, X), implies(X, X)), and(implies(X, X), implies(X, X)))
% 60.20/8.16 = { by lemma 43 R->L }
% 60.20/8.16 fresh59(fresh59(is_a_theorem(implies(or(implies(X, X), and(not(implies(X, X)), not(implies(X, X)))), or(and(not(implies(X, X)), not(implies(X, X))), and(implies(X, X), implies(X, X))))), modus_ponens, or(implies(X, X), and(not(implies(X, X)), not(implies(X, X)))), or(and(not(implies(X, X)), not(implies(X, X))), and(implies(X, X), implies(X, X)))), modus_ponens, or(implies(X, X), implies(X, X)), and(implies(X, X), implies(X, X)))
% 60.20/8.16 = { by lemma 40 R->L }
% 60.20/8.16 fresh59(fresh59(is_a_theorem(implies(implies(not(implies(X, X)), and(not(implies(X, X)), not(implies(X, X)))), or(and(not(implies(X, X)), not(implies(X, X))), and(implies(X, X), implies(X, X))))), modus_ponens, or(implies(X, X), and(not(implies(X, X)), not(implies(X, X)))), or(and(not(implies(X, X)), not(implies(X, X))), and(implies(X, X), implies(X, X)))), modus_ponens, or(implies(X, X), implies(X, X)), and(implies(X, X), implies(X, X)))
% 60.20/8.16 = { by axiom 12 (modus_ponens_2) R->L }
% 60.20/8.16 fresh59(fresh59(fresh28(and_1, and_1, implies(implies(not(implies(X, X)), and(not(implies(X, X)), not(implies(X, X)))), or(and(not(implies(X, X)), not(implies(X, X))), and(implies(X, X), implies(X, X))))), modus_ponens, or(implies(X, X), and(not(implies(X, X)), not(implies(X, X)))), or(and(not(implies(X, X)), not(implies(X, X))), and(implies(X, X), implies(X, X)))), modus_ponens, or(implies(X, X), implies(X, X)), and(implies(X, X), implies(X, X)))
% 60.20/8.16 = { by lemma 48 R->L }
% 60.20/8.16 fresh59(fresh59(fresh28(fresh28(is_a_theorem(implies(implies(X, X), and(implies(X, X), implies(X, X)))), and_1, or(and(implies(X, X), implies(X, X)), not(implies(X, X)))), and_1, implies(implies(not(implies(X, X)), and(not(implies(X, X)), not(implies(X, X)))), or(and(not(implies(X, X)), not(implies(X, X))), and(implies(X, X), implies(X, X))))), modus_ponens, or(implies(X, X), and(not(implies(X, X)), not(implies(X, X)))), or(and(not(implies(X, X)), not(implies(X, X))), and(implies(X, X), implies(X, X)))), modus_ponens, or(implies(X, X), implies(X, X)), and(implies(X, X), implies(X, X)))
% 60.20/8.16 = { by lemma 45 }
% 60.20/8.16 fresh59(fresh59(fresh28(fresh28(modus_ponens, and_1, or(and(implies(X, X), implies(X, X)), not(implies(X, X)))), and_1, implies(implies(not(implies(X, X)), and(not(implies(X, X)), not(implies(X, X)))), or(and(not(implies(X, X)), not(implies(X, X))), and(implies(X, X), implies(X, X))))), modus_ponens, or(implies(X, X), and(not(implies(X, X)), not(implies(X, X)))), or(and(not(implies(X, X)), not(implies(X, X))), and(implies(X, X), implies(X, X)))), modus_ponens, or(implies(X, X), implies(X, X)), and(implies(X, X), implies(X, X)))
% 60.20/8.16 = { by lemma 35 }
% 60.20/8.16 fresh59(fresh59(fresh28(fresh28(and_1, and_1, or(and(implies(X, X), implies(X, X)), not(implies(X, X)))), and_1, implies(implies(not(implies(X, X)), and(not(implies(X, X)), not(implies(X, X)))), or(and(not(implies(X, X)), not(implies(X, X))), and(implies(X, X), implies(X, X))))), modus_ponens, or(implies(X, X), and(not(implies(X, X)), not(implies(X, X)))), or(and(not(implies(X, X)), not(implies(X, X))), and(implies(X, X), implies(X, X)))), modus_ponens, or(implies(X, X), implies(X, X)), and(implies(X, X), implies(X, X)))
% 60.20/8.16 = { by axiom 12 (modus_ponens_2) }
% 60.20/8.16 fresh59(fresh59(fresh28(is_a_theorem(or(and(implies(X, X), implies(X, X)), not(implies(X, X)))), and_1, implies(implies(not(implies(X, X)), and(not(implies(X, X)), not(implies(X, X)))), or(and(not(implies(X, X)), not(implies(X, X))), and(implies(X, X), implies(X, X))))), modus_ponens, or(implies(X, X), and(not(implies(X, X)), not(implies(X, X)))), or(and(not(implies(X, X)), not(implies(X, X))), and(implies(X, X), implies(X, X)))), modus_ponens, or(implies(X, X), implies(X, X)), and(implies(X, X), implies(X, X)))
% 60.20/8.16 = { by lemma 47 }
% 60.20/8.16 fresh59(fresh59(and_1, modus_ponens, or(implies(X, X), and(not(implies(X, X)), not(implies(X, X)))), or(and(not(implies(X, X)), not(implies(X, X))), and(implies(X, X), implies(X, X)))), modus_ponens, or(implies(X, X), implies(X, X)), and(implies(X, X), implies(X, X)))
% 60.20/8.16 = { by lemma 35 }
% 60.20/8.16 fresh59(fresh59(and_1, and_1, or(implies(X, X), and(not(implies(X, X)), not(implies(X, X)))), or(and(not(implies(X, X)), not(implies(X, X))), and(implies(X, X), implies(X, X)))), modus_ponens, or(implies(X, X), implies(X, X)), and(implies(X, X), implies(X, X)))
% 60.20/8.16 = { by lemma 38 }
% 60.20/8.16 fresh59(modus_ponens, modus_ponens, or(implies(X, X), implies(X, X)), and(implies(X, X), implies(X, X)))
% 60.20/8.16 = { by lemma 35 }
% 60.20/8.16 fresh59(and_1, modus_ponens, or(implies(X, X), implies(X, X)), and(implies(X, X), implies(X, X)))
% 60.20/8.16 = { by lemma 35 }
% 60.20/8.16 fresh59(and_1, and_1, or(implies(X, X), implies(X, X)), and(implies(X, X), implies(X, X)))
% 60.20/8.16 = { by lemma 38 }
% 60.20/8.16 modus_ponens
% 60.20/8.16 = { by lemma 35 }
% 60.20/8.16 and_1
% 60.20/8.16
% 60.20/8.16 Lemma 54: is_a_theorem(implies(or(X, X), X)) = fresh12(r1, modus_ponens, X).
% 60.20/8.16 Proof:
% 60.20/8.16 is_a_theorem(implies(or(X, X), X))
% 60.20/8.16 = { by axiom 25 (r1_1) R->L }
% 60.20/8.16 fresh12(r1, true, X)
% 60.20/8.16 = { by axiom 1 (rosser_modus_ponens) R->L }
% 60.20/8.16 fresh12(r1, modus_ponens, X)
% 60.20/8.16
% 60.20/8.16 Lemma 55: fresh12(r1, modus_ponens, X) = fresh47(cn3, modus_ponens, X).
% 60.20/8.16 Proof:
% 60.20/8.16 fresh12(r1, modus_ponens, X)
% 60.20/8.16 = { by lemma 54 R->L }
% 60.20/8.16 is_a_theorem(implies(or(X, X), X))
% 60.20/8.16 = { by lemma 40 R->L }
% 60.20/8.16 is_a_theorem(implies(implies(not(X), X), X))
% 60.20/8.16 = { by axiom 27 (cn3_1) R->L }
% 60.20/8.16 fresh47(cn3, true, X)
% 60.20/8.16 = { by axiom 1 (rosser_modus_ponens) R->L }
% 60.20/8.16 fresh47(cn3, modus_ponens, X)
% 60.20/8.16
% 60.20/8.16 Lemma 56: is_a_theorem(implies(or(X, Y), or(Y, X))) = fresh8(r3, and_1, X, Y).
% 60.20/8.16 Proof:
% 60.20/8.16 is_a_theorem(implies(or(X, Y), or(Y, X)))
% 60.20/8.17 = { by axiom 31 (r3_1) R->L }
% 60.20/8.17 fresh8(r3, true, X, Y)
% 60.20/8.17 = { by axiom 1 (rosser_modus_ponens) R->L }
% 60.20/8.17 fresh8(r3, modus_ponens, X, Y)
% 60.20/8.17 = { by lemma 35 }
% 60.20/8.17 fresh8(r3, and_1, X, Y)
% 60.20/8.17
% 60.20/8.17 Lemma 57: fresh28(is_a_theorem(equiv(X, Y)), and_1, implies(X, Y)) = and_1.
% 60.20/8.17 Proof:
% 60.20/8.17 fresh28(is_a_theorem(equiv(X, Y)), and_1, implies(X, Y))
% 60.20/8.17 = { by lemma 36 R->L }
% 60.20/8.17 fresh28(is_a_theorem(and(implies(X, Y), implies(Y, X))), and_1, implies(X, Y))
% 60.20/8.17 = { by lemma 35 R->L }
% 60.20/8.17 fresh28(is_a_theorem(and(implies(X, Y), implies(Y, X))), modus_ponens, implies(X, Y))
% 60.20/8.17 = { by lemma 43 R->L }
% 60.20/8.17 fresh59(is_a_theorem(implies(and(implies(X, Y), implies(Y, X)), implies(X, Y))), modus_ponens, and(implies(X, Y), implies(Y, X)), implies(X, Y))
% 60.20/8.17 = { by lemma 33 }
% 60.20/8.17 fresh59(fresh58(and_1, modus_ponens, implies(X, Y), implies(Y, X)), modus_ponens, and(implies(X, Y), implies(Y, X)), implies(X, Y))
% 60.20/8.17 = { by lemma 34 }
% 60.20/8.17 fresh59(modus_ponens, modus_ponens, and(implies(X, Y), implies(Y, X)), implies(X, Y))
% 60.20/8.17 = { by lemma 38 }
% 60.20/8.17 modus_ponens
% 60.20/8.17 = { by lemma 35 }
% 60.20/8.17 and_1
% 60.20/8.17
% 60.20/8.17 Goal 1 (hilbert_implies_1): implies_1 = true.
% 60.20/8.17 Proof:
% 60.20/8.17 implies_1
% 60.20/8.17 = { by axiom 30 (implies_1) R->L }
% 60.20/8.17 fresh40(is_a_theorem(implies(x12, implies(y12, x12))), true)
% 60.20/8.17 = { by axiom 22 (implies_1_1) R->L }
% 60.20/8.17 fresh40(fresh39(implies_1, true, x12, y12), true)
% 60.20/8.17 = { by axiom 1 (rosser_modus_ponens) R->L }
% 60.20/8.17 fresh40(fresh39(implies_1, modus_ponens, x12, y12), true)
% 60.20/8.17 = { by lemma 35 }
% 60.20/8.17 fresh40(fresh39(implies_1, and_1, x12, y12), true)
% 60.20/8.17 = { by axiom 1 (rosser_modus_ponens) R->L }
% 60.20/8.17 fresh40(fresh39(implies_1, and_1, x12, y12), modus_ponens)
% 60.20/8.17 = { by lemma 35 }
% 60.20/8.17 fresh40(fresh39(implies_1, and_1, x12, y12), and_1)
% 60.20/8.17 = { by lemma 35 R->L }
% 60.20/8.17 fresh40(fresh39(implies_1, modus_ponens, x12, y12), and_1)
% 60.20/8.17 = { by axiom 1 (rosser_modus_ponens) }
% 60.20/8.17 fresh40(fresh39(implies_1, true, x12, y12), and_1)
% 60.20/8.17 = { by axiom 22 (implies_1_1) }
% 60.20/8.17 fresh40(is_a_theorem(implies(x12, implies(y12, x12))), and_1)
% 60.20/8.17 = { by axiom 12 (modus_ponens_2) R->L }
% 60.20/8.17 fresh40(fresh28(and_1, and_1, implies(x12, implies(y12, x12))), and_1)
% 60.20/8.17 = { by lemma 35 R->L }
% 60.20/8.17 fresh40(fresh28(modus_ponens, and_1, implies(x12, implies(y12, x12))), and_1)
% 60.20/8.17 = { by lemma 37 R->L }
% 60.20/8.17 fresh40(fresh28(fresh60(modus_ponens, modus_ponens, or(implies(x12, implies(y12, x12)), implies(x12, implies(y12, x12)))), and_1, implies(x12, implies(y12, x12))), and_1)
% 60.20/8.17 = { by axiom 1 (rosser_modus_ponens) }
% 60.20/8.17 fresh40(fresh28(fresh60(modus_ponens, true, or(implies(x12, implies(y12, x12)), implies(x12, implies(y12, x12)))), and_1, implies(x12, implies(y12, x12))), and_1)
% 60.20/8.17 = { by axiom 13 (modus_ponens_2) R->L }
% 60.20/8.17 fresh40(fresh28(fresh59(and_1, and_1, or(implies(y12, x12), implies(x12, implies(y12, x12))), or(implies(x12, implies(y12, x12)), implies(x12, implies(y12, x12)))), and_1, implies(x12, implies(y12, x12))), and_1)
% 60.20/8.17 = { by lemma 47 R->L }
% 60.20/8.17 fresh40(fresh28(fresh59(fresh28(is_a_theorem(or(not(and(x12, not(implies(y12, x12)))), not(implies(y12, x12)))), and_1, implies(implies(not(implies(y12, x12)), implies(x12, implies(y12, x12))), or(implies(x12, implies(y12, x12)), not(and(x12, not(implies(y12, x12))))))), and_1, or(implies(y12, x12), implies(x12, implies(y12, x12))), or(implies(x12, implies(y12, x12)), implies(x12, implies(y12, x12)))), and_1, implies(x12, implies(y12, x12))), and_1)
% 60.20/8.17 = { by lemma 40 }
% 60.20/8.17 fresh40(fresh28(fresh59(fresh28(is_a_theorem(or(not(and(x12, not(implies(y12, x12)))), not(implies(y12, x12)))), and_1, implies(or(implies(y12, x12), implies(x12, implies(y12, x12))), or(implies(x12, implies(y12, x12)), not(and(x12, not(implies(y12, x12))))))), and_1, or(implies(y12, x12), implies(x12, implies(y12, x12))), or(implies(x12, implies(y12, x12)), implies(x12, implies(y12, x12)))), and_1, implies(x12, implies(y12, x12))), and_1)
% 60.20/8.17 = { by axiom 12 (modus_ponens_2) R->L }
% 60.20/8.17 fresh40(fresh28(fresh59(fresh28(fresh28(and_1, and_1, or(not(and(x12, not(implies(y12, x12)))), not(implies(y12, x12)))), and_1, implies(or(implies(y12, x12), implies(x12, implies(y12, x12))), or(implies(x12, implies(y12, x12)), not(and(x12, not(implies(y12, x12))))))), and_1, or(implies(y12, x12), implies(x12, implies(y12, x12))), or(implies(x12, implies(y12, x12)), implies(x12, implies(y12, x12)))), and_1, implies(x12, implies(y12, x12))), and_1)
% 60.20/8.17 = { by lemma 44 R->L }
% 60.20/8.17 fresh40(fresh28(fresh59(fresh28(fresh28(fresh28(is_a_theorem(implies(not(implies(y12, x12)), not(implies(y12, x12)))), and_1, or(and(not(implies(y12, x12)), x12), not(and(x12, not(implies(y12, x12)))))), and_1, or(not(and(x12, not(implies(y12, x12)))), not(implies(y12, x12)))), and_1, implies(or(implies(y12, x12), implies(x12, implies(y12, x12))), or(implies(x12, implies(y12, x12)), not(and(x12, not(implies(y12, x12))))))), and_1, or(implies(y12, x12), implies(x12, implies(y12, x12))), or(implies(x12, implies(y12, x12)), implies(x12, implies(y12, x12)))), and_1, implies(x12, implies(y12, x12))), and_1)
% 60.20/8.17 = { by axiom 12 (modus_ponens_2) R->L }
% 60.20/8.17 fresh40(fresh28(fresh59(fresh28(fresh28(fresh28(fresh28(and_1, and_1, implies(not(implies(y12, x12)), not(implies(y12, x12)))), and_1, or(and(not(implies(y12, x12)), x12), not(and(x12, not(implies(y12, x12)))))), and_1, or(not(and(x12, not(implies(y12, x12)))), not(implies(y12, x12)))), and_1, implies(or(implies(y12, x12), implies(x12, implies(y12, x12))), or(implies(x12, implies(y12, x12)), not(and(x12, not(implies(y12, x12))))))), and_1, or(implies(y12, x12), implies(x12, implies(y12, x12))), or(implies(x12, implies(y12, x12)), implies(x12, implies(y12, x12)))), and_1, implies(x12, implies(y12, x12))), and_1)
% 60.20/8.17 = { by lemma 53 R->L }
% 60.20/8.17 fresh40(fresh28(fresh59(fresh28(fresh28(fresh28(fresh28(is_a_theorem(equiv(not(implies(y12, x12)), not(implies(y12, x12)))), and_1, implies(not(implies(y12, x12)), not(implies(y12, x12)))), and_1, or(and(not(implies(y12, x12)), x12), not(and(x12, not(implies(y12, x12)))))), and_1, or(not(and(x12, not(implies(y12, x12)))), not(implies(y12, x12)))), and_1, implies(or(implies(y12, x12), implies(x12, implies(y12, x12))), or(implies(x12, implies(y12, x12)), not(and(x12, not(implies(y12, x12))))))), and_1, or(implies(y12, x12), implies(x12, implies(y12, x12))), or(implies(x12, implies(y12, x12)), implies(x12, implies(y12, x12)))), and_1, implies(x12, implies(y12, x12))), and_1)
% 60.20/8.17 = { by lemma 57 }
% 60.20/8.17 fresh40(fresh28(fresh59(fresh28(fresh28(fresh28(and_1, and_1, or(and(not(implies(y12, x12)), x12), not(and(x12, not(implies(y12, x12)))))), and_1, or(not(and(x12, not(implies(y12, x12)))), not(implies(y12, x12)))), and_1, implies(or(implies(y12, x12), implies(x12, implies(y12, x12))), or(implies(x12, implies(y12, x12)), not(and(x12, not(implies(y12, x12))))))), and_1, or(implies(y12, x12), implies(x12, implies(y12, x12))), or(implies(x12, implies(y12, x12)), implies(x12, implies(y12, x12)))), and_1, implies(x12, implies(y12, x12))), and_1)
% 60.20/8.17 = { by axiom 12 (modus_ponens_2) }
% 60.20/8.17 fresh40(fresh28(fresh59(fresh28(fresh28(is_a_theorem(or(and(not(implies(y12, x12)), x12), not(and(x12, not(implies(y12, x12)))))), and_1, or(not(and(x12, not(implies(y12, x12)))), not(implies(y12, x12)))), and_1, implies(or(implies(y12, x12), implies(x12, implies(y12, x12))), or(implies(x12, implies(y12, x12)), not(and(x12, not(implies(y12, x12))))))), and_1, or(implies(y12, x12), implies(x12, implies(y12, x12))), or(implies(x12, implies(y12, x12)), implies(x12, implies(y12, x12)))), and_1, implies(x12, implies(y12, x12))), and_1)
% 60.20/8.17 = { by lemma 50 }
% 60.20/8.17 fresh40(fresh28(fresh59(fresh28(and_1, and_1, implies(or(implies(y12, x12), implies(x12, implies(y12, x12))), or(implies(x12, implies(y12, x12)), not(and(x12, not(implies(y12, x12))))))), and_1, or(implies(y12, x12), implies(x12, implies(y12, x12))), or(implies(x12, implies(y12, x12)), implies(x12, implies(y12, x12)))), and_1, implies(x12, implies(y12, x12))), and_1)
% 60.20/8.17 = { by axiom 12 (modus_ponens_2) }
% 60.20/8.17 fresh40(fresh28(fresh59(is_a_theorem(implies(or(implies(y12, x12), implies(x12, implies(y12, x12))), or(implies(x12, implies(y12, x12)), not(and(x12, not(implies(y12, x12))))))), and_1, or(implies(y12, x12), implies(x12, implies(y12, x12))), or(implies(x12, implies(y12, x12)), implies(x12, implies(y12, x12)))), and_1, implies(x12, implies(y12, x12))), and_1)
% 60.20/8.17 = { by lemma 39 }
% 60.20/8.17 fresh40(fresh28(fresh59(is_a_theorem(implies(or(implies(y12, x12), implies(x12, implies(y12, x12))), or(implies(x12, implies(y12, x12)), implies(x12, implies(y12, x12))))), and_1, or(implies(y12, x12), implies(x12, implies(y12, x12))), or(implies(x12, implies(y12, x12)), implies(x12, implies(y12, x12)))), and_1, implies(x12, implies(y12, x12))), and_1)
% 60.20/8.17 = { by lemma 35 R->L }
% 60.20/8.17 fresh40(fresh28(fresh59(is_a_theorem(implies(or(implies(y12, x12), implies(x12, implies(y12, x12))), or(implies(x12, implies(y12, x12)), implies(x12, implies(y12, x12))))), modus_ponens, or(implies(y12, x12), implies(x12, implies(y12, x12))), or(implies(x12, implies(y12, x12)), implies(x12, implies(y12, x12)))), and_1, implies(x12, implies(y12, x12))), and_1)
% 60.20/8.17 = { by lemma 43 }
% 60.20/8.17 fresh40(fresh28(fresh28(is_a_theorem(or(implies(y12, x12), implies(x12, implies(y12, x12)))), modus_ponens, or(implies(x12, implies(y12, x12)), implies(x12, implies(y12, x12)))), and_1, implies(x12, implies(y12, x12))), and_1)
% 60.20/8.17 = { by lemma 35 }
% 60.20/8.17 fresh40(fresh28(fresh28(is_a_theorem(or(implies(y12, x12), implies(x12, implies(y12, x12)))), and_1, or(implies(x12, implies(y12, x12)), implies(x12, implies(y12, x12)))), and_1, implies(x12, implies(y12, x12))), and_1)
% 60.20/8.17 = { by lemma 51 }
% 60.20/8.17 fresh40(fresh28(fresh28(and_1, and_1, or(implies(x12, implies(y12, x12)), implies(x12, implies(y12, x12)))), and_1, implies(x12, implies(y12, x12))), and_1)
% 60.20/8.17 = { by axiom 12 (modus_ponens_2) }
% 60.20/8.17 fresh40(fresh28(is_a_theorem(or(implies(x12, implies(y12, x12)), implies(x12, implies(y12, x12)))), and_1, implies(x12, implies(y12, x12))), and_1)
% 60.20/8.17 = { by lemma 35 R->L }
% 60.20/8.17 fresh40(fresh28(is_a_theorem(or(implies(x12, implies(y12, x12)), implies(x12, implies(y12, x12)))), modus_ponens, implies(x12, implies(y12, x12))), and_1)
% 60.20/8.17 = { by lemma 43 R->L }
% 60.20/8.17 fresh40(fresh59(is_a_theorem(implies(or(implies(x12, implies(y12, x12)), implies(x12, implies(y12, x12))), implies(x12, implies(y12, x12)))), modus_ponens, or(implies(x12, implies(y12, x12)), implies(x12, implies(y12, x12))), implies(x12, implies(y12, x12))), and_1)
% 60.20/8.17 = { by lemma 54 }
% 60.20/8.17 fresh40(fresh59(fresh12(r1, modus_ponens, implies(x12, implies(y12, x12))), modus_ponens, or(implies(x12, implies(y12, x12)), implies(x12, implies(y12, x12))), implies(x12, implies(y12, x12))), and_1)
% 60.20/8.17 = { by lemma 55 }
% 60.20/8.17 fresh40(fresh59(fresh47(cn3, modus_ponens, implies(x12, implies(y12, x12))), modus_ponens, or(implies(x12, implies(y12, x12)), implies(x12, implies(y12, x12))), implies(x12, implies(y12, x12))), and_1)
% 60.20/8.17 = { by lemma 35 }
% 60.20/8.17 fresh40(fresh59(fresh47(cn3, modus_ponens, implies(x12, implies(y12, x12))), and_1, or(implies(x12, implies(y12, x12)), implies(x12, implies(y12, x12))), implies(x12, implies(y12, x12))), and_1)
% 60.20/8.17 = { by lemma 55 R->L }
% 60.20/8.17 fresh40(fresh59(fresh12(r1, modus_ponens, implies(x12, implies(y12, x12))), and_1, or(implies(x12, implies(y12, x12)), implies(x12, implies(y12, x12))), implies(x12, implies(y12, x12))), and_1)
% 60.20/8.17 = { by lemma 54 R->L }
% 60.20/8.17 fresh40(fresh59(is_a_theorem(implies(or(implies(x12, implies(y12, x12)), implies(x12, implies(y12, x12))), implies(x12, implies(y12, x12)))), and_1, or(implies(x12, implies(y12, x12)), implies(x12, implies(y12, x12))), implies(x12, implies(y12, x12))), and_1)
% 60.20/8.17 = { by lemma 40 R->L }
% 60.20/8.17 fresh40(fresh59(is_a_theorem(implies(implies(not(implies(x12, implies(y12, x12))), implies(x12, implies(y12, x12))), implies(x12, implies(y12, x12)))), and_1, or(implies(x12, implies(y12, x12)), implies(x12, implies(y12, x12))), implies(x12, implies(y12, x12))), and_1)
% 60.20/8.17 = { by lemma 46 R->L }
% 60.20/8.17 fresh40(fresh59(is_a_theorem(or(and(not(implies(x12, implies(y12, x12))), not(implies(x12, implies(y12, x12)))), implies(x12, implies(y12, x12)))), and_1, or(implies(x12, implies(y12, x12)), implies(x12, implies(y12, x12))), implies(x12, implies(y12, x12))), and_1)
% 60.20/8.18 = { by axiom 12 (modus_ponens_2) R->L }
% 60.20/8.18 fresh40(fresh59(fresh28(and_1, and_1, or(and(not(implies(x12, implies(y12, x12))), not(implies(x12, implies(y12, x12)))), implies(x12, implies(y12, x12)))), and_1, or(implies(x12, implies(y12, x12)), implies(x12, implies(y12, x12))), implies(x12, implies(y12, x12))), and_1)
% 60.20/8.18 = { by lemma 52 R->L }
% 60.20/8.18 fresh40(fresh59(fresh28(is_a_theorem(or(implies(x12, implies(y12, x12)), and(not(implies(x12, implies(y12, x12))), not(implies(x12, implies(y12, x12)))))), and_1, or(and(not(implies(x12, implies(y12, x12))), not(implies(x12, implies(y12, x12)))), implies(x12, implies(y12, x12)))), and_1, or(implies(x12, implies(y12, x12)), implies(x12, implies(y12, x12))), implies(x12, implies(y12, x12))), and_1)
% 60.20/8.18 = { by lemma 35 R->L }
% 60.20/8.18 fresh40(fresh59(fresh28(is_a_theorem(or(implies(x12, implies(y12, x12)), and(not(implies(x12, implies(y12, x12))), not(implies(x12, implies(y12, x12)))))), modus_ponens, or(and(not(implies(x12, implies(y12, x12))), not(implies(x12, implies(y12, x12)))), implies(x12, implies(y12, x12)))), and_1, or(implies(x12, implies(y12, x12)), implies(x12, implies(y12, x12))), implies(x12, implies(y12, x12))), and_1)
% 60.20/8.18 = { by lemma 43 R->L }
% 60.20/8.18 fresh40(fresh59(fresh59(is_a_theorem(implies(or(implies(x12, implies(y12, x12)), and(not(implies(x12, implies(y12, x12))), not(implies(x12, implies(y12, x12))))), or(and(not(implies(x12, implies(y12, x12))), not(implies(x12, implies(y12, x12)))), implies(x12, implies(y12, x12))))), modus_ponens, or(implies(x12, implies(y12, x12)), and(not(implies(x12, implies(y12, x12))), not(implies(x12, implies(y12, x12))))), or(and(not(implies(x12, implies(y12, x12))), not(implies(x12, implies(y12, x12)))), implies(x12, implies(y12, x12)))), and_1, or(implies(x12, implies(y12, x12)), implies(x12, implies(y12, x12))), implies(x12, implies(y12, x12))), and_1)
% 60.20/8.18 = { by lemma 56 }
% 60.20/8.18 fresh40(fresh59(fresh59(fresh8(r3, and_1, implies(x12, implies(y12, x12)), and(not(implies(x12, implies(y12, x12))), not(implies(x12, implies(y12, x12))))), modus_ponens, or(implies(x12, implies(y12, x12)), and(not(implies(x12, implies(y12, x12))), not(implies(x12, implies(y12, x12))))), or(and(not(implies(x12, implies(y12, x12))), not(implies(x12, implies(y12, x12)))), implies(x12, implies(y12, x12)))), and_1, or(implies(x12, implies(y12, x12)), implies(x12, implies(y12, x12))), implies(x12, implies(y12, x12))), and_1)
% 60.20/8.18 = { by lemma 35 }
% 60.20/8.18 fresh40(fresh59(fresh59(fresh8(r3, and_1, implies(x12, implies(y12, x12)), and(not(implies(x12, implies(y12, x12))), not(implies(x12, implies(y12, x12))))), and_1, or(implies(x12, implies(y12, x12)), and(not(implies(x12, implies(y12, x12))), not(implies(x12, implies(y12, x12))))), or(and(not(implies(x12, implies(y12, x12))), not(implies(x12, implies(y12, x12)))), implies(x12, implies(y12, x12)))), and_1, or(implies(x12, implies(y12, x12)), implies(x12, implies(y12, x12))), implies(x12, implies(y12, x12))), and_1)
% 60.20/8.18 = { by lemma 56 R->L }
% 60.20/8.18 fresh40(fresh59(fresh59(is_a_theorem(implies(or(implies(x12, implies(y12, x12)), and(not(implies(x12, implies(y12, x12))), not(implies(x12, implies(y12, x12))))), or(and(not(implies(x12, implies(y12, x12))), not(implies(x12, implies(y12, x12)))), implies(x12, implies(y12, x12))))), and_1, or(implies(x12, implies(y12, x12)), and(not(implies(x12, implies(y12, x12))), not(implies(x12, implies(y12, x12))))), or(and(not(implies(x12, implies(y12, x12))), not(implies(x12, implies(y12, x12)))), implies(x12, implies(y12, x12)))), and_1, or(implies(x12, implies(y12, x12)), implies(x12, implies(y12, x12))), implies(x12, implies(y12, x12))), and_1)
% 60.20/8.18 = { by lemma 40 R->L }
% 60.20/8.18 fresh40(fresh59(fresh59(is_a_theorem(implies(implies(not(implies(x12, implies(y12, x12))), and(not(implies(x12, implies(y12, x12))), not(implies(x12, implies(y12, x12))))), or(and(not(implies(x12, implies(y12, x12))), not(implies(x12, implies(y12, x12)))), implies(x12, implies(y12, x12))))), and_1, or(implies(x12, implies(y12, x12)), and(not(implies(x12, implies(y12, x12))), not(implies(x12, implies(y12, x12))))), or(and(not(implies(x12, implies(y12, x12))), not(implies(x12, implies(y12, x12)))), implies(x12, implies(y12, x12)))), and_1, or(implies(x12, implies(y12, x12)), implies(x12, implies(y12, x12))), implies(x12, implies(y12, x12))), and_1)
% 60.20/8.18 = { by axiom 12 (modus_ponens_2) R->L }
% 60.20/8.18 fresh40(fresh59(fresh59(fresh28(and_1, and_1, implies(implies(not(implies(x12, implies(y12, x12))), and(not(implies(x12, implies(y12, x12))), not(implies(x12, implies(y12, x12))))), or(and(not(implies(x12, implies(y12, x12))), not(implies(x12, implies(y12, x12)))), implies(x12, implies(y12, x12))))), and_1, or(implies(x12, implies(y12, x12)), and(not(implies(x12, implies(y12, x12))), not(implies(x12, implies(y12, x12))))), or(and(not(implies(x12, implies(y12, x12))), not(implies(x12, implies(y12, x12)))), implies(x12, implies(y12, x12)))), and_1, or(implies(x12, implies(y12, x12)), implies(x12, implies(y12, x12))), implies(x12, implies(y12, x12))), and_1)
% 60.20/8.18 = { by lemma 57 R->L }
% 60.20/8.18 fresh40(fresh59(fresh59(fresh28(fresh28(is_a_theorem(equiv(not(implies(x12, implies(y12, x12))), not(implies(x12, implies(y12, x12))))), and_1, implies(not(implies(x12, implies(y12, x12))), not(implies(x12, implies(y12, x12))))), and_1, implies(implies(not(implies(x12, implies(y12, x12))), and(not(implies(x12, implies(y12, x12))), not(implies(x12, implies(y12, x12))))), or(and(not(implies(x12, implies(y12, x12))), not(implies(x12, implies(y12, x12)))), implies(x12, implies(y12, x12))))), and_1, or(implies(x12, implies(y12, x12)), and(not(implies(x12, implies(y12, x12))), not(implies(x12, implies(y12, x12))))), or(and(not(implies(x12, implies(y12, x12))), not(implies(x12, implies(y12, x12)))), implies(x12, implies(y12, x12)))), and_1, or(implies(x12, implies(y12, x12)), implies(x12, implies(y12, x12))), implies(x12, implies(y12, x12))), and_1)
% 60.20/8.18 = { by lemma 40 }
% 60.20/8.18 fresh40(fresh59(fresh59(fresh28(fresh28(is_a_theorem(equiv(not(implies(x12, implies(y12, x12))), not(implies(x12, implies(y12, x12))))), and_1, or(implies(x12, implies(y12, x12)), not(implies(x12, implies(y12, x12))))), and_1, implies(implies(not(implies(x12, implies(y12, x12))), and(not(implies(x12, implies(y12, x12))), not(implies(x12, implies(y12, x12))))), or(and(not(implies(x12, implies(y12, x12))), not(implies(x12, implies(y12, x12)))), implies(x12, implies(y12, x12))))), and_1, or(implies(x12, implies(y12, x12)), and(not(implies(x12, implies(y12, x12))), not(implies(x12, implies(y12, x12))))), or(and(not(implies(x12, implies(y12, x12))), not(implies(x12, implies(y12, x12)))), implies(x12, implies(y12, x12)))), and_1, or(implies(x12, implies(y12, x12)), implies(x12, implies(y12, x12))), implies(x12, implies(y12, x12))), and_1)
% 60.20/8.18 = { by lemma 53 }
% 60.20/8.18 fresh40(fresh59(fresh59(fresh28(fresh28(and_1, and_1, or(implies(x12, implies(y12, x12)), not(implies(x12, implies(y12, x12))))), and_1, implies(implies(not(implies(x12, implies(y12, x12))), and(not(implies(x12, implies(y12, x12))), not(implies(x12, implies(y12, x12))))), or(and(not(implies(x12, implies(y12, x12))), not(implies(x12, implies(y12, x12)))), implies(x12, implies(y12, x12))))), and_1, or(implies(x12, implies(y12, x12)), and(not(implies(x12, implies(y12, x12))), not(implies(x12, implies(y12, x12))))), or(and(not(implies(x12, implies(y12, x12))), not(implies(x12, implies(y12, x12)))), implies(x12, implies(y12, x12)))), and_1, or(implies(x12, implies(y12, x12)), implies(x12, implies(y12, x12))), implies(x12, implies(y12, x12))), and_1)
% 60.20/8.18 = { by axiom 12 (modus_ponens_2) }
% 60.20/8.18 fresh40(fresh59(fresh59(fresh28(is_a_theorem(or(implies(x12, implies(y12, x12)), not(implies(x12, implies(y12, x12))))), and_1, implies(implies(not(implies(x12, implies(y12, x12))), and(not(implies(x12, implies(y12, x12))), not(implies(x12, implies(y12, x12))))), or(and(not(implies(x12, implies(y12, x12))), not(implies(x12, implies(y12, x12)))), implies(x12, implies(y12, x12))))), and_1, or(implies(x12, implies(y12, x12)), and(not(implies(x12, implies(y12, x12))), not(implies(x12, implies(y12, x12))))), or(and(not(implies(x12, implies(y12, x12))), not(implies(x12, implies(y12, x12)))), implies(x12, implies(y12, x12)))), and_1, or(implies(x12, implies(y12, x12)), implies(x12, implies(y12, x12))), implies(x12, implies(y12, x12))), and_1)
% 60.20/8.18 = { by lemma 47 }
% 60.20/8.18 fresh40(fresh59(fresh59(and_1, and_1, or(implies(x12, implies(y12, x12)), and(not(implies(x12, implies(y12, x12))), not(implies(x12, implies(y12, x12))))), or(and(not(implies(x12, implies(y12, x12))), not(implies(x12, implies(y12, x12)))), implies(x12, implies(y12, x12)))), and_1, or(implies(x12, implies(y12, x12)), implies(x12, implies(y12, x12))), implies(x12, implies(y12, x12))), and_1)
% 60.20/8.18 = { by lemma 38 }
% 60.20/8.18 fresh40(fresh59(modus_ponens, and_1, or(implies(x12, implies(y12, x12)), implies(x12, implies(y12, x12))), implies(x12, implies(y12, x12))), and_1)
% 60.20/8.18 = { by lemma 35 }
% 60.20/8.18 fresh40(fresh59(and_1, and_1, or(implies(x12, implies(y12, x12)), implies(x12, implies(y12, x12))), implies(x12, implies(y12, x12))), and_1)
% 60.20/8.18 = { by lemma 38 }
% 60.20/8.18 fresh40(modus_ponens, and_1)
% 60.20/8.18 = { by lemma 35 }
% 60.20/8.18 fresh40(and_1, and_1)
% 60.20/8.18 = { by axiom 9 (implies_1) }
% 60.20/8.18 true
% 60.20/8.18 % SZS output end Proof
% 60.20/8.18
% 60.20/8.18 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------