TSTP Solution File: LCL509+1 by Twee---2.4.2
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : LCL509+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 : n011.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 08:19:16 EDT 2023
% Result : Theorem 57.44s 7.77s
% Output : Proof 58.23s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.12 % Problem : LCL509+1 : TPTP v8.1.2. Released v3.3.0.
% 0.07/0.13 % 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 : n011.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 : Thu Aug 24 18:58:37 EDT 2023
% 0.13/0.34 % CPUTime :
% 57.44/7.77 Command-line arguments: --set-join --lhs-weight 1 --no-flatten-goal --complete-subsets --goal-heuristic
% 57.44/7.77
% 57.44/7.77 % SZS status Theorem
% 57.44/7.77
% 58.23/7.84 % SZS output start Proof
% 58.23/7.84 Take the following subset of the input axioms:
% 58.23/7.84 fof(and_1, axiom, and_1 <=> ![X, Y]: is_a_theorem(implies(and(X, Y), X))).
% 58.23/7.84 fof(cn2, axiom, cn2 <=> ![P, Q]: is_a_theorem(implies(P, implies(not(P), Q)))).
% 58.23/7.84 fof(cn3, axiom, cn3 <=> ![P2]: is_a_theorem(implies(implies(not(P2), P2), P2))).
% 58.23/7.84 fof(hilbert_or_1, conjecture, or_1).
% 58.23/7.84 fof(kn1, axiom, kn1 <=> ![P2]: is_a_theorem(implies(P2, and(P2, P2)))).
% 58.23/7.84 fof(kn2, axiom, kn2 <=> ![P2, Q2]: is_a_theorem(implies(and(P2, Q2), P2))).
% 58.23/7.84 fof(kn3, axiom, kn3 <=> ![R, P2, Q2]: is_a_theorem(implies(implies(P2, Q2), implies(not(and(Q2, R)), not(and(R, P2)))))).
% 58.23/7.84 fof(modus_ponens, axiom, modus_ponens <=> ![X2, Y2]: ((is_a_theorem(X2) & is_a_theorem(implies(X2, Y2))) => is_a_theorem(Y2))).
% 58.23/7.84 fof(op_equiv, axiom, op_equiv => ![X2, Y2]: equiv(X2, Y2)=and(implies(X2, Y2), implies(Y2, X2))).
% 58.23/7.84 fof(op_implies_and, axiom, op_implies_and => ![X2, Y2]: implies(X2, Y2)=not(and(X2, not(Y2)))).
% 58.23/7.84 fof(op_or, axiom, op_or => ![X2, Y2]: or(X2, Y2)=not(and(not(X2), not(Y2)))).
% 58.23/7.84 fof(or_1, axiom, or_1 <=> ![X2, Y2]: is_a_theorem(implies(X2, or(X2, Y2)))).
% 58.23/7.84 fof(r1, axiom, r1 <=> ![P2]: is_a_theorem(implies(or(P2, P2), P2))).
% 58.23/7.84 fof(r3, axiom, r3 <=> ![P2, Q2]: is_a_theorem(implies(or(P2, Q2), or(Q2, P2)))).
% 58.23/7.85 fof(rosser_kn1, axiom, kn1).
% 58.23/7.85 fof(rosser_kn2, axiom, kn2).
% 58.23/7.85 fof(rosser_kn3, axiom, kn3).
% 58.23/7.85 fof(rosser_modus_ponens, axiom, modus_ponens).
% 58.23/7.85 fof(rosser_op_equiv, axiom, op_equiv).
% 58.23/7.85 fof(rosser_op_implies_and, axiom, op_implies_and).
% 58.23/7.85 fof(rosser_op_or, axiom, op_or).
% 58.23/7.85
% 58.23/7.85 Now clausify the problem and encode Horn clauses using encoding 3 of
% 58.23/7.85 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 58.23/7.85 We repeatedly replace C & s=t => u=v by the two clauses:
% 58.23/7.85 fresh(y, y, x1...xn) = u
% 58.23/7.85 C => fresh(s, t, x1...xn) = v
% 58.23/7.85 where fresh is a fresh function symbol and x1..xn are the free
% 58.23/7.85 variables of u and v.
% 58.23/7.85 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 58.23/7.85 input problem has no model of domain size 1).
% 58.23/7.85
% 58.23/7.85 The encoding turns the above axioms into the following unit equations and goals:
% 58.23/7.85
% 58.23/7.85 Axiom 1 (rosser_modus_ponens): modus_ponens = true.
% 58.23/7.85 Axiom 2 (rosser_kn1): kn1 = true.
% 58.23/7.85 Axiom 3 (rosser_kn2): kn2 = true.
% 58.23/7.85 Axiom 4 (rosser_kn3): kn3 = true.
% 58.23/7.85 Axiom 5 (rosser_op_or): op_or = true.
% 58.23/7.85 Axiom 6 (rosser_op_implies_and): op_implies_and = true.
% 58.23/7.85 Axiom 7 (rosser_op_equiv): op_equiv = true.
% 58.23/7.85 Axiom 8 (and_1): fresh57(X, X) = true.
% 58.23/7.85 Axiom 9 (or_1): fresh19(X, X) = true.
% 58.23/7.85 Axiom 10 (modus_ponens_2): fresh60(X, X, Y) = true.
% 58.23/7.85 Axiom 11 (kn1_1): fresh33(X, X, Y) = true.
% 58.23/7.85 Axiom 12 (modus_ponens_2): fresh28(X, X, Y) = is_a_theorem(Y).
% 58.23/7.85 Axiom 13 (modus_ponens_2): fresh59(X, X, Y, Z) = fresh60(modus_ponens, true, Z).
% 58.23/7.85 Axiom 14 (kn2_1): fresh31(X, X, Y, Z) = true.
% 58.23/7.85 Axiom 15 (op_equiv): fresh23(X, X, Y, Z) = equiv(Y, Z).
% 58.23/7.85 Axiom 16 (op_implies_and): fresh22(X, X, Y, Z) = implies(Y, Z).
% 58.23/7.85 Axiom 17 (op_or): fresh20(X, X, Y, Z) = or(Y, Z).
% 58.23/7.85 Axiom 18 (op_implies_and): fresh22(op_implies_and, true, X, Y) = not(and(X, not(Y))).
% 58.23/7.85 Axiom 19 (kn3_1): fresh29(X, X, Y, Z, W) = true.
% 58.23/7.85 Axiom 20 (op_or): fresh20(op_or, true, X, Y) = not(and(not(X), not(Y))).
% 58.23/7.85 Axiom 21 (kn1_1): fresh33(kn1, true, X) = is_a_theorem(implies(X, and(X, X))).
% 58.23/7.85 Axiom 22 (or_1_1): fresh18(or_1, true, X, Y) = is_a_theorem(implies(X, or(X, Y))).
% 58.23/7.85 Axiom 23 (and_1_1): fresh58(and_1, true, X, Y) = is_a_theorem(implies(and(X, Y), X)).
% 58.23/7.85 Axiom 24 (kn2_1): fresh31(kn2, true, X, Y) = is_a_theorem(implies(and(X, Y), X)).
% 58.23/7.85 Axiom 25 (r1_1): fresh12(r1, true, X) = is_a_theorem(implies(or(X, X), X)).
% 58.23/7.85 Axiom 26 (op_equiv): fresh23(op_equiv, true, X, Y) = and(implies(X, Y), implies(Y, X)).
% 58.23/7.85 Axiom 27 (cn2_1): fresh49(cn2, true, X, Y) = is_a_theorem(implies(X, implies(not(X), Y))).
% 58.23/7.85 Axiom 28 (cn3_1): fresh47(cn3, true, X) = is_a_theorem(implies(implies(not(X), X), X)).
% 58.23/7.85 Axiom 29 (modus_ponens_2): fresh59(is_a_theorem(implies(X, Y)), true, X, Y) = fresh28(is_a_theorem(X), true, Y).
% 58.23/7.85 Axiom 30 (and_1): fresh57(is_a_theorem(implies(and(x9, y9), x9)), true) = and_1.
% 58.23/7.85 Axiom 31 (or_1): fresh19(is_a_theorem(implies(x6, or(x6, y6))), true) = or_1.
% 58.23/7.85 Axiom 32 (r3_1): fresh8(r3, true, X, Y) = is_a_theorem(implies(or(X, Y), or(Y, X))).
% 58.23/7.85 Axiom 33 (kn3_1): fresh29(kn3, true, X, Y, Z) = is_a_theorem(implies(implies(X, Y), implies(not(and(Y, Z)), not(and(Z, X))))).
% 58.23/7.85
% 58.23/7.85 Lemma 34: is_a_theorem(implies(and(X, Y), X)) = fresh58(and_1, modus_ponens, X, Y).
% 58.23/7.85 Proof:
% 58.23/7.85 is_a_theorem(implies(and(X, Y), X))
% 58.23/7.85 = { by axiom 23 (and_1_1) R->L }
% 58.23/7.85 fresh58(and_1, true, X, Y)
% 58.23/7.85 = { by axiom 1 (rosser_modus_ponens) R->L }
% 58.23/7.85 fresh58(and_1, modus_ponens, X, Y)
% 58.23/7.85
% 58.23/7.85 Lemma 35: fresh58(and_1, modus_ponens, X, Y) = modus_ponens.
% 58.23/7.85 Proof:
% 58.23/7.85 fresh58(and_1, modus_ponens, X, Y)
% 58.23/7.85 = { by lemma 34 R->L }
% 58.23/7.85 is_a_theorem(implies(and(X, Y), X))
% 58.23/7.85 = { by axiom 24 (kn2_1) R->L }
% 58.23/7.85 fresh31(kn2, true, X, Y)
% 58.23/7.85 = { by axiom 3 (rosser_kn2) }
% 58.23/7.85 fresh31(true, true, X, Y)
% 58.23/7.85 = { by axiom 1 (rosser_modus_ponens) R->L }
% 58.23/7.85 fresh31(modus_ponens, true, X, Y)
% 58.23/7.85 = { by axiom 1 (rosser_modus_ponens) R->L }
% 58.23/7.85 fresh31(modus_ponens, modus_ponens, X, Y)
% 58.23/7.85 = { by axiom 14 (kn2_1) }
% 58.23/7.85 true
% 58.23/7.85 = { by axiom 1 (rosser_modus_ponens) R->L }
% 58.23/7.85 modus_ponens
% 58.23/7.85
% 58.23/7.85 Lemma 36: modus_ponens = and_1.
% 58.23/7.85 Proof:
% 58.23/7.85 modus_ponens
% 58.23/7.85 = { by axiom 1 (rosser_modus_ponens) }
% 58.23/7.85 true
% 58.23/7.85 = { by axiom 8 (and_1) R->L }
% 58.23/7.85 fresh57(modus_ponens, modus_ponens)
% 58.23/7.85 = { by axiom 1 (rosser_modus_ponens) }
% 58.23/7.85 fresh57(modus_ponens, true)
% 58.23/7.85 = { by lemma 35 R->L }
% 58.23/7.85 fresh57(fresh58(and_1, modus_ponens, x9, y9), true)
% 58.23/7.85 = { by lemma 34 R->L }
% 58.23/7.85 fresh57(is_a_theorem(implies(and(x9, y9), x9)), true)
% 58.23/7.85 = { by axiom 30 (and_1) }
% 58.23/7.85 and_1
% 58.23/7.85
% 58.23/7.85 Lemma 37: and(implies(X, Y), implies(Y, X)) = equiv(X, Y).
% 58.23/7.85 Proof:
% 58.23/7.85 and(implies(X, Y), implies(Y, X))
% 58.23/7.85 = { by axiom 26 (op_equiv) R->L }
% 58.23/7.85 fresh23(op_equiv, true, X, Y)
% 58.23/7.85 = { by axiom 7 (rosser_op_equiv) }
% 58.23/7.85 fresh23(true, true, X, Y)
% 58.23/7.85 = { by axiom 1 (rosser_modus_ponens) R->L }
% 58.23/7.85 fresh23(modus_ponens, true, X, Y)
% 58.23/7.85 = { by axiom 1 (rosser_modus_ponens) R->L }
% 58.23/7.85 fresh23(modus_ponens, modus_ponens, X, Y)
% 58.23/7.85 = { by axiom 15 (op_equiv) }
% 58.23/7.85 equiv(X, Y)
% 58.23/7.85
% 58.23/7.85 Lemma 38: fresh59(is_a_theorem(implies(X, Y)), modus_ponens, X, Y) = fresh28(is_a_theorem(X), modus_ponens, Y).
% 58.23/7.85 Proof:
% 58.23/7.85 fresh59(is_a_theorem(implies(X, Y)), modus_ponens, X, Y)
% 58.23/7.85 = { by axiom 1 (rosser_modus_ponens) }
% 58.23/7.85 fresh59(is_a_theorem(implies(X, Y)), true, X, Y)
% 58.23/7.85 = { by axiom 29 (modus_ponens_2) }
% 58.23/7.85 fresh28(is_a_theorem(X), true, Y)
% 58.23/7.85 = { by axiom 1 (rosser_modus_ponens) R->L }
% 58.23/7.85 fresh28(is_a_theorem(X), modus_ponens, Y)
% 58.23/7.85
% 58.23/7.85 Lemma 39: not(and(X, not(Y))) = implies(X, Y).
% 58.23/7.85 Proof:
% 58.23/7.85 not(and(X, not(Y)))
% 58.23/7.85 = { by axiom 18 (op_implies_and) R->L }
% 58.23/7.85 fresh22(op_implies_and, true, X, Y)
% 58.23/7.85 = { by axiom 6 (rosser_op_implies_and) }
% 58.23/7.85 fresh22(true, true, X, Y)
% 58.23/7.85 = { by axiom 1 (rosser_modus_ponens) R->L }
% 58.23/7.85 fresh22(modus_ponens, true, X, Y)
% 58.23/7.85 = { by axiom 1 (rosser_modus_ponens) R->L }
% 58.23/7.85 fresh22(modus_ponens, modus_ponens, X, Y)
% 58.23/7.85 = { by axiom 16 (op_implies_and) }
% 58.23/7.85 implies(X, Y)
% 58.23/7.85
% 58.23/7.85 Lemma 40: implies(not(X), Y) = or(X, Y).
% 58.23/7.85 Proof:
% 58.23/7.85 implies(not(X), Y)
% 58.23/7.85 = { by lemma 39 R->L }
% 58.23/7.85 not(and(not(X), not(Y)))
% 58.23/7.85 = { by axiom 20 (op_or) R->L }
% 58.23/7.85 fresh20(op_or, true, X, Y)
% 58.23/7.85 = { by axiom 5 (rosser_op_or) }
% 58.23/7.85 fresh20(true, true, X, Y)
% 58.23/7.85 = { by axiom 1 (rosser_modus_ponens) R->L }
% 58.23/7.85 fresh20(modus_ponens, true, X, Y)
% 58.23/7.85 = { by axiom 1 (rosser_modus_ponens) R->L }
% 58.23/7.85 fresh20(modus_ponens, modus_ponens, X, Y)
% 58.23/7.85 = { by axiom 17 (op_or) }
% 58.23/7.85 or(X, Y)
% 58.23/7.85
% 58.23/7.85 Lemma 41: fresh60(X, X, Y) = modus_ponens.
% 58.23/7.85 Proof:
% 58.23/7.85 fresh60(X, X, Y)
% 58.23/7.85 = { by axiom 10 (modus_ponens_2) }
% 58.23/7.85 true
% 58.23/7.85 = { by axiom 1 (rosser_modus_ponens) R->L }
% 58.23/7.85 modus_ponens
% 58.23/7.85
% 58.23/7.85 Lemma 42: fresh59(X, X, Y, Z) = modus_ponens.
% 58.23/7.85 Proof:
% 58.23/7.85 fresh59(X, X, Y, Z)
% 58.23/7.85 = { by axiom 13 (modus_ponens_2) }
% 58.23/7.85 fresh60(modus_ponens, true, Z)
% 58.23/7.85 = { by axiom 1 (rosser_modus_ponens) R->L }
% 58.23/7.85 fresh60(modus_ponens, modus_ponens, Z)
% 58.23/7.85 = { by lemma 41 }
% 58.23/7.85 modus_ponens
% 58.23/7.85
% 58.23/7.85 Lemma 43: is_a_theorem(implies(implies(X, Y), or(and(Y, Z), not(and(Z, X))))) = and_1.
% 58.23/7.85 Proof:
% 58.23/7.85 is_a_theorem(implies(implies(X, Y), or(and(Y, Z), not(and(Z, X)))))
% 58.23/7.85 = { by lemma 40 R->L }
% 58.23/7.85 is_a_theorem(implies(implies(X, Y), implies(not(and(Y, Z)), not(and(Z, X)))))
% 58.23/7.85 = { by axiom 33 (kn3_1) R->L }
% 58.23/7.85 fresh29(kn3, true, X, Y, Z)
% 58.23/7.85 = { by axiom 4 (rosser_kn3) }
% 58.23/7.85 fresh29(true, true, X, Y, Z)
% 58.23/7.85 = { by axiom 1 (rosser_modus_ponens) R->L }
% 58.23/7.85 fresh29(modus_ponens, true, X, Y, Z)
% 58.23/7.85 = { by lemma 36 }
% 58.23/7.85 fresh29(and_1, true, X, Y, Z)
% 58.23/7.85 = { by axiom 1 (rosser_modus_ponens) R->L }
% 58.23/7.85 fresh29(and_1, modus_ponens, X, Y, Z)
% 58.23/7.85 = { by lemma 36 }
% 58.23/7.85 fresh29(and_1, and_1, X, Y, Z)
% 58.23/7.85 = { by axiom 19 (kn3_1) }
% 58.23/7.85 true
% 58.23/7.85 = { by axiom 1 (rosser_modus_ponens) R->L }
% 58.23/7.85 modus_ponens
% 58.23/7.85 = { by lemma 36 }
% 58.23/7.85 and_1
% 58.23/7.85
% 58.23/7.85 Lemma 44: fresh28(is_a_theorem(implies(X, Y)), and_1, or(and(Y, Z), not(and(Z, X)))) = and_1.
% 58.23/7.85 Proof:
% 58.23/7.85 fresh28(is_a_theorem(implies(X, Y)), and_1, or(and(Y, Z), not(and(Z, X))))
% 58.23/7.85 = { by lemma 36 R->L }
% 58.23/7.85 fresh28(is_a_theorem(implies(X, Y)), modus_ponens, or(and(Y, Z), not(and(Z, X))))
% 58.23/7.85 = { by lemma 38 R->L }
% 58.23/7.85 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))))
% 58.23/7.85 = { by lemma 43 }
% 58.23/7.85 fresh59(and_1, modus_ponens, implies(X, Y), or(and(Y, Z), not(and(Z, X))))
% 58.23/7.85 = { by lemma 36 }
% 58.23/7.85 fresh59(and_1, and_1, implies(X, Y), or(and(Y, Z), not(and(Z, X))))
% 58.23/7.85 = { by lemma 42 }
% 58.23/7.85 modus_ponens
% 58.23/7.85 = { by lemma 36 }
% 58.23/7.85 and_1
% 58.23/7.85
% 58.23/7.85 Lemma 45: is_a_theorem(implies(X, and(X, X))) = modus_ponens.
% 58.23/7.85 Proof:
% 58.23/7.85 is_a_theorem(implies(X, and(X, X)))
% 58.23/7.85 = { by axiom 21 (kn1_1) R->L }
% 58.23/7.85 fresh33(kn1, true, X)
% 58.23/7.85 = { by axiom 2 (rosser_kn1) }
% 58.23/7.85 fresh33(true, true, X)
% 58.23/7.85 = { by axiom 1 (rosser_modus_ponens) R->L }
% 58.23/7.85 fresh33(modus_ponens, true, X)
% 58.23/7.85 = { by axiom 1 (rosser_modus_ponens) R->L }
% 58.23/7.85 fresh33(modus_ponens, modus_ponens, X)
% 58.23/7.85 = { by axiom 11 (kn1_1) }
% 58.23/7.85 true
% 58.23/7.85 = { by axiom 1 (rosser_modus_ponens) R->L }
% 58.23/7.85 modus_ponens
% 58.23/7.85
% 58.23/7.85 Lemma 46: or(and(X, not(Y)), Z) = implies(implies(X, Y), Z).
% 58.23/7.85 Proof:
% 58.23/7.85 or(and(X, not(Y)), Z)
% 58.23/7.85 = { by lemma 40 R->L }
% 58.23/7.85 implies(not(and(X, not(Y))), Z)
% 58.23/7.85 = { by lemma 39 }
% 58.23/7.85 implies(implies(X, Y), Z)
% 58.23/7.85
% 58.23/7.85 Lemma 47: is_a_theorem(implies(or(X, Y), or(and(Y, Z), implies(Z, X)))) = and_1.
% 58.23/7.85 Proof:
% 58.23/7.85 is_a_theorem(implies(or(X, Y), or(and(Y, Z), implies(Z, X))))
% 58.23/7.85 = { by lemma 40 R->L }
% 58.23/7.85 is_a_theorem(implies(implies(not(X), Y), or(and(Y, Z), implies(Z, X))))
% 58.23/7.85 = { by lemma 39 R->L }
% 58.23/7.85 is_a_theorem(implies(implies(not(X), Y), or(and(Y, Z), not(and(Z, not(X))))))
% 58.23/7.85 = { by lemma 43 }
% 58.23/7.85 and_1
% 58.23/7.85
% 58.23/7.85 Lemma 48: fresh28(is_a_theorem(or(X, Y)), and_1, implies(implies(Y, Z), or(Z, X))) = and_1.
% 58.23/7.85 Proof:
% 58.23/7.85 fresh28(is_a_theorem(or(X, Y)), and_1, implies(implies(Y, Z), or(Z, X)))
% 58.23/7.85 = { by lemma 36 R->L }
% 58.23/7.85 fresh28(is_a_theorem(or(X, Y)), modus_ponens, implies(implies(Y, Z), or(Z, X)))
% 58.23/7.85 = { by lemma 38 R->L }
% 58.23/7.85 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)))
% 58.23/7.85 = { by lemma 46 R->L }
% 58.23/7.85 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)))
% 58.23/7.85 = { by lemma 40 R->L }
% 58.23/7.85 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)))
% 58.23/7.85 = { by lemma 47 }
% 58.23/7.85 fresh59(and_1, modus_ponens, or(X, Y), implies(implies(Y, Z), or(Z, X)))
% 58.23/7.85 = { by lemma 36 }
% 58.23/7.85 fresh59(and_1, and_1, or(X, Y), implies(implies(Y, Z), or(Z, X)))
% 58.23/7.85 = { by lemma 42 }
% 58.23/7.85 modus_ponens
% 58.23/7.85 = { by lemma 36 }
% 58.23/7.85 and_1
% 58.23/7.85
% 58.23/7.85 Lemma 49: fresh28(is_a_theorem(implies(X, Y)), and_1, or(Y, not(X))) = and_1.
% 58.23/7.85 Proof:
% 58.23/7.85 fresh28(is_a_theorem(implies(X, Y)), and_1, or(Y, not(X)))
% 58.23/7.85 = { by lemma 36 R->L }
% 58.23/7.85 fresh28(is_a_theorem(implies(X, Y)), modus_ponens, or(Y, not(X)))
% 58.23/7.85 = { by lemma 38 R->L }
% 58.23/7.85 fresh59(is_a_theorem(implies(implies(X, Y), or(Y, not(X)))), modus_ponens, implies(X, Y), or(Y, not(X)))
% 58.23/7.85 = { by axiom 12 (modus_ponens_2) R->L }
% 58.23/7.85 fresh59(fresh28(and_1, and_1, implies(implies(X, Y), or(Y, not(X)))), modus_ponens, implies(X, Y), or(Y, not(X)))
% 58.23/7.85 = { by lemma 36 R->L }
% 58.23/7.85 fresh59(fresh28(modus_ponens, and_1, implies(implies(X, Y), or(Y, not(X)))), modus_ponens, implies(X, Y), or(Y, not(X)))
% 58.23/7.85 = { by lemma 42 R->L }
% 58.23/7.85 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)))
% 58.23/7.85 = { by lemma 44 R->L }
% 58.23/7.85 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)))
% 58.23/7.85 = { by lemma 45 }
% 58.23/7.85 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)))
% 58.23/7.85 = { by lemma 36 }
% 58.23/7.85 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)))
% 58.23/7.85 = { by axiom 12 (modus_ponens_2) }
% 58.23/7.85 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)))
% 58.23/7.86 = { by lemma 36 R->L }
% 58.23/7.86 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)))
% 58.23/7.86 = { by lemma 40 R->L }
% 58.23/7.86 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)))
% 58.23/7.86 = { by lemma 38 }
% 58.23/7.86 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)))
% 58.23/7.86 = { by lemma 36 }
% 58.23/7.86 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)))
% 58.23/7.86 = { by lemma 39 }
% 58.23/7.86 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)))
% 58.23/7.86 = { by lemma 34 }
% 58.23/7.86 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)))
% 58.23/7.86 = { by lemma 35 }
% 58.23/7.86 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)))
% 58.23/7.86 = { by lemma 36 }
% 58.23/7.86 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)))
% 58.23/7.86 = { by axiom 12 (modus_ponens_2) }
% 58.23/7.86 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)))
% 58.23/7.86 = { by lemma 39 }
% 58.23/7.86 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)))
% 58.23/7.86 = { by lemma 40 }
% 58.23/7.86 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)))
% 58.23/7.86 = { by lemma 48 }
% 58.23/7.86 fresh59(and_1, modus_ponens, implies(X, Y), or(Y, not(X)))
% 58.23/7.86 = { by lemma 36 }
% 58.23/7.86 fresh59(and_1, and_1, implies(X, Y), or(Y, not(X)))
% 58.23/7.86 = { by lemma 42 }
% 58.23/7.86 modus_ponens
% 58.23/7.86 = { by lemma 36 }
% 58.23/7.86 and_1
% 58.23/7.86
% 58.23/7.86 Lemma 50: is_a_theorem(or(X, not(and(X, Y)))) = and_1.
% 58.23/7.86 Proof:
% 58.23/7.86 is_a_theorem(or(X, not(and(X, Y))))
% 58.23/7.86 = { by axiom 12 (modus_ponens_2) R->L }
% 58.23/7.86 fresh28(and_1, and_1, or(X, not(and(X, Y))))
% 58.23/7.86 = { by lemma 36 R->L }
% 58.23/7.86 fresh28(modus_ponens, and_1, or(X, not(and(X, Y))))
% 58.23/7.86 = { by lemma 35 R->L }
% 58.23/7.86 fresh28(fresh58(and_1, modus_ponens, X, Y), and_1, or(X, not(and(X, Y))))
% 58.23/7.86 = { by lemma 34 R->L }
% 58.23/7.86 fresh28(is_a_theorem(implies(and(X, Y), X)), and_1, or(X, not(and(X, Y))))
% 58.23/7.86 = { by lemma 49 }
% 58.23/7.86 and_1
% 58.23/7.86
% 58.23/7.86 Lemma 51: fresh28(is_a_theorem(or(and(X, Y), Z)), and_1, or(Z, X)) = and_1.
% 58.23/7.86 Proof:
% 58.23/7.86 fresh28(is_a_theorem(or(and(X, Y), Z)), and_1, or(Z, X))
% 58.23/7.86 = { by lemma 36 R->L }
% 58.23/7.86 fresh28(is_a_theorem(or(and(X, Y), Z)), modus_ponens, or(Z, X))
% 58.23/7.86 = { by lemma 38 R->L }
% 58.23/7.86 fresh59(is_a_theorem(implies(or(and(X, Y), Z), or(Z, X))), modus_ponens, or(and(X, Y), Z), or(Z, X))
% 58.23/7.86 = { by lemma 40 R->L }
% 58.23/7.86 fresh59(is_a_theorem(implies(implies(not(and(X, Y)), Z), or(Z, X))), modus_ponens, or(and(X, Y), Z), or(Z, X))
% 58.23/7.86 = { by axiom 12 (modus_ponens_2) R->L }
% 58.23/7.86 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))
% 58.23/7.86 = { by lemma 50 R->L }
% 58.23/7.86 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))
% 58.23/7.86 = { by lemma 48 }
% 58.23/7.86 fresh59(and_1, modus_ponens, or(and(X, Y), Z), or(Z, X))
% 58.23/7.86 = { by lemma 36 }
% 58.23/7.86 fresh59(and_1, and_1, or(and(X, Y), Z), or(Z, X))
% 58.23/7.86 = { by lemma 42 }
% 58.23/7.86 modus_ponens
% 58.23/7.86 = { by lemma 36 }
% 58.23/7.86 and_1
% 58.23/7.86
% 58.23/7.86 Lemma 52: is_a_theorem(or(X, implies(X, Y))) = and_1.
% 58.23/7.86 Proof:
% 58.23/7.86 is_a_theorem(or(X, implies(X, Y)))
% 58.23/7.86 = { by lemma 39 R->L }
% 58.23/7.86 is_a_theorem(or(X, not(and(X, not(Y)))))
% 58.23/7.86 = { by lemma 50 }
% 58.23/7.86 and_1
% 58.23/7.86
% 58.23/7.86 Lemma 53: is_a_theorem(or(X, and(not(X), not(X)))) = and_1.
% 58.23/7.86 Proof:
% 58.23/7.86 is_a_theorem(or(X, and(not(X), not(X))))
% 58.23/7.86 = { by lemma 40 R->L }
% 58.23/7.86 is_a_theorem(implies(not(X), and(not(X), not(X))))
% 58.23/7.86 = { by lemma 45 }
% 58.23/7.86 modus_ponens
% 58.23/7.86 = { by lemma 36 }
% 58.23/7.86 and_1
% 58.23/7.86
% 58.23/7.86 Lemma 54: is_a_theorem(equiv(X, X)) = and_1.
% 58.23/7.86 Proof:
% 58.23/7.86 is_a_theorem(equiv(X, X))
% 58.23/7.86 = { by lemma 37 R->L }
% 58.23/7.86 is_a_theorem(and(implies(X, X), implies(X, X)))
% 58.23/7.86 = { by axiom 12 (modus_ponens_2) R->L }
% 58.23/7.86 fresh28(and_1, and_1, and(implies(X, X), implies(X, X)))
% 58.23/7.86 = { by lemma 51 R->L }
% 58.23/7.86 fresh28(fresh28(is_a_theorem(or(and(implies(X, X), X), implies(X, X))), and_1, or(implies(X, X), implies(X, X))), and_1, and(implies(X, X), implies(X, X)))
% 58.23/7.86 = { by axiom 12 (modus_ponens_2) R->L }
% 58.23/7.86 fresh28(fresh28(fresh28(and_1, and_1, or(and(implies(X, X), X), implies(X, X))), and_1, or(implies(X, X), implies(X, X))), and_1, and(implies(X, X), implies(X, X)))
% 58.23/7.86 = { by lemma 52 R->L }
% 58.23/7.86 fresh28(fresh28(fresh28(is_a_theorem(or(X, implies(X, X))), and_1, or(and(implies(X, X), X), implies(X, X))), and_1, or(implies(X, X), implies(X, X))), and_1, and(implies(X, X), implies(X, X)))
% 58.23/7.86 = { by lemma 36 R->L }
% 58.23/7.86 fresh28(fresh28(fresh28(is_a_theorem(or(X, implies(X, X))), modus_ponens, or(and(implies(X, X), X), implies(X, X))), and_1, or(implies(X, X), implies(X, X))), and_1, and(implies(X, X), implies(X, X)))
% 58.23/7.86 = { by lemma 38 R->L }
% 58.23/7.86 fresh28(fresh28(fresh59(is_a_theorem(implies(or(X, implies(X, X)), or(and(implies(X, X), X), implies(X, X)))), modus_ponens, or(X, implies(X, X)), or(and(implies(X, X), X), implies(X, X))), and_1, or(implies(X, X), implies(X, X))), and_1, and(implies(X, X), implies(X, X)))
% 58.23/7.86 = { by lemma 47 }
% 58.23/7.86 fresh28(fresh28(fresh59(and_1, modus_ponens, or(X, implies(X, X)), or(and(implies(X, X), X), implies(X, X))), and_1, or(implies(X, X), implies(X, X))), and_1, and(implies(X, X), implies(X, X)))
% 58.23/7.86 = { by lemma 36 }
% 58.23/7.86 fresh28(fresh28(fresh59(and_1, and_1, or(X, implies(X, X)), or(and(implies(X, X), X), implies(X, X))), and_1, or(implies(X, X), implies(X, X))), and_1, and(implies(X, X), implies(X, X)))
% 58.23/7.86 = { by lemma 42 }
% 58.23/7.86 fresh28(fresh28(modus_ponens, and_1, or(implies(X, X), implies(X, X))), and_1, and(implies(X, X), implies(X, X)))
% 58.23/7.86 = { by lemma 36 }
% 58.23/7.86 fresh28(fresh28(and_1, and_1, or(implies(X, X), implies(X, X))), and_1, and(implies(X, X), implies(X, X)))
% 58.23/7.86 = { by axiom 12 (modus_ponens_2) }
% 58.23/7.86 fresh28(is_a_theorem(or(implies(X, X), implies(X, X))), and_1, and(implies(X, X), implies(X, X)))
% 58.23/7.86 = { by lemma 36 R->L }
% 58.23/7.86 fresh28(is_a_theorem(or(implies(X, X), implies(X, X))), modus_ponens, and(implies(X, X), implies(X, X)))
% 58.23/7.86 = { by lemma 38 R->L }
% 58.23/7.86 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)))
% 58.23/7.86 = { by lemma 40 R->L }
% 58.23/7.86 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)))
% 58.23/7.86 = { by lemma 46 R->L }
% 58.23/7.86 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)))
% 58.23/7.86 = { by axiom 12 (modus_ponens_2) R->L }
% 58.23/7.86 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)))
% 58.23/7.86 = { by lemma 53 R->L }
% 58.23/7.86 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)))
% 58.23/7.86 = { by lemma 36 R->L }
% 58.23/7.86 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)))
% 58.23/7.86 = { by lemma 38 R->L }
% 58.23/7.86 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)))
% 58.23/7.86 = { by lemma 40 R->L }
% 58.23/7.86 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)))
% 58.23/7.86 = { by axiom 12 (modus_ponens_2) R->L }
% 58.23/7.86 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)))
% 58.23/7.86 = { by lemma 49 R->L }
% 58.23/7.86 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)))
% 58.23/7.86 = { by lemma 45 }
% 58.23/7.86 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)))
% 58.23/7.86 = { by lemma 36 }
% 58.23/7.86 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)))
% 58.23/7.86 = { by axiom 12 (modus_ponens_2) }
% 58.23/7.86 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)))
% 58.23/7.86 = { by lemma 48 }
% 58.23/7.86 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)))
% 58.23/7.86 = { by lemma 36 }
% 58.23/7.86 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)))
% 58.23/7.86 = { by lemma 42 }
% 58.23/7.86 fresh59(modus_ponens, modus_ponens, or(implies(X, X), implies(X, X)), and(implies(X, X), implies(X, X)))
% 58.23/7.86 = { by lemma 36 }
% 58.23/7.86 fresh59(and_1, modus_ponens, or(implies(X, X), implies(X, X)), and(implies(X, X), implies(X, X)))
% 58.23/7.86 = { by lemma 36 }
% 58.23/7.86 fresh59(and_1, and_1, or(implies(X, X), implies(X, X)), and(implies(X, X), implies(X, X)))
% 58.23/7.86 = { by lemma 42 }
% 58.23/7.86 modus_ponens
% 58.23/7.86 = { by lemma 36 }
% 58.23/7.86 and_1
% 58.23/7.86
% 58.23/7.86 Lemma 55: is_a_theorem(implies(or(X, X), X)) = fresh12(r1, modus_ponens, X).
% 58.23/7.86 Proof:
% 58.23/7.86 is_a_theorem(implies(or(X, X), X))
% 58.23/7.86 = { by axiom 25 (r1_1) R->L }
% 58.23/7.86 fresh12(r1, true, X)
% 58.23/7.86 = { by axiom 1 (rosser_modus_ponens) R->L }
% 58.23/7.86 fresh12(r1, modus_ponens, X)
% 58.23/7.86
% 58.23/7.86 Lemma 56: fresh12(r1, modus_ponens, X) = fresh47(cn3, modus_ponens, X).
% 58.23/7.86 Proof:
% 58.23/7.86 fresh12(r1, modus_ponens, X)
% 58.23/7.86 = { by lemma 55 R->L }
% 58.23/7.86 is_a_theorem(implies(or(X, X), X))
% 58.23/7.86 = { by lemma 40 R->L }
% 58.23/7.86 is_a_theorem(implies(implies(not(X), X), X))
% 58.23/7.86 = { by axiom 28 (cn3_1) R->L }
% 58.23/7.86 fresh47(cn3, true, X)
% 58.23/7.86 = { by axiom 1 (rosser_modus_ponens) R->L }
% 58.23/7.86 fresh47(cn3, modus_ponens, X)
% 58.23/7.86
% 58.23/7.86 Lemma 57: is_a_theorem(implies(X, or(X, Y))) = fresh18(or_1, modus_ponens, X, Y).
% 58.23/7.86 Proof:
% 58.23/7.86 is_a_theorem(implies(X, or(X, Y)))
% 58.23/7.86 = { by axiom 22 (or_1_1) R->L }
% 58.23/7.86 fresh18(or_1, true, X, Y)
% 58.23/7.86 = { by axiom 1 (rosser_modus_ponens) R->L }
% 58.23/7.86 fresh18(or_1, modus_ponens, X, Y)
% 58.23/7.86
% 58.23/7.86 Lemma 58: fresh18(or_1, modus_ponens, X, Y) = fresh49(cn2, modus_ponens, X, Y).
% 58.23/7.86 Proof:
% 58.23/7.86 fresh18(or_1, modus_ponens, X, Y)
% 58.23/7.86 = { by lemma 57 R->L }
% 58.23/7.86 is_a_theorem(implies(X, or(X, Y)))
% 58.23/7.86 = { by lemma 40 R->L }
% 58.23/7.86 is_a_theorem(implies(X, implies(not(X), Y)))
% 58.23/7.86 = { by axiom 27 (cn2_1) R->L }
% 58.23/7.86 fresh49(cn2, true, X, Y)
% 58.23/7.86 = { by axiom 1 (rosser_modus_ponens) R->L }
% 58.23/7.87 fresh49(cn2, modus_ponens, X, Y)
% 58.23/7.87
% 58.23/7.87 Lemma 59: is_a_theorem(implies(or(X, Y), or(Y, X))) = fresh8(r3, and_1, X, Y).
% 58.23/7.87 Proof:
% 58.23/7.87 is_a_theorem(implies(or(X, Y), or(Y, X)))
% 58.23/7.87 = { by axiom 32 (r3_1) R->L }
% 58.23/7.87 fresh8(r3, true, X, Y)
% 58.23/7.87 = { by axiom 1 (rosser_modus_ponens) R->L }
% 58.23/7.87 fresh8(r3, modus_ponens, X, Y)
% 58.23/7.87 = { by lemma 36 }
% 58.23/7.87 fresh8(r3, and_1, X, Y)
% 58.23/7.87
% 58.23/7.87 Lemma 60: fresh28(is_a_theorem(equiv(X, Y)), and_1, implies(X, Y)) = and_1.
% 58.23/7.87 Proof:
% 58.23/7.87 fresh28(is_a_theorem(equiv(X, Y)), and_1, implies(X, Y))
% 58.23/7.87 = { by lemma 37 R->L }
% 58.23/7.87 fresh28(is_a_theorem(and(implies(X, Y), implies(Y, X))), and_1, implies(X, Y))
% 58.23/7.87 = { by lemma 36 R->L }
% 58.23/7.87 fresh28(is_a_theorem(and(implies(X, Y), implies(Y, X))), modus_ponens, implies(X, Y))
% 58.23/7.87 = { by lemma 38 R->L }
% 58.23/7.87 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))
% 58.23/7.87 = { by lemma 34 }
% 58.23/7.87 fresh59(fresh58(and_1, modus_ponens, implies(X, Y), implies(Y, X)), modus_ponens, and(implies(X, Y), implies(Y, X)), implies(X, Y))
% 58.23/7.87 = { by lemma 35 }
% 58.23/7.87 fresh59(modus_ponens, modus_ponens, and(implies(X, Y), implies(Y, X)), implies(X, Y))
% 58.23/7.87 = { by lemma 42 }
% 58.23/7.87 modus_ponens
% 58.23/7.87 = { by lemma 36 }
% 58.23/7.87 and_1
% 58.23/7.87
% 58.23/7.87 Lemma 61: fresh28(is_a_theorem(or(X, Y)), and_1, or(Y, X)) = and_1.
% 58.23/7.87 Proof:
% 58.23/7.87 fresh28(is_a_theorem(or(X, Y)), and_1, or(Y, X))
% 58.23/7.87 = { by lemma 36 R->L }
% 58.23/7.87 fresh28(is_a_theorem(or(X, Y)), modus_ponens, or(Y, X))
% 58.23/7.87 = { by lemma 38 R->L }
% 58.23/7.87 fresh59(is_a_theorem(implies(or(X, Y), or(Y, X))), modus_ponens, or(X, Y), or(Y, X))
% 58.23/7.87 = { by lemma 59 }
% 58.23/7.87 fresh59(fresh8(r3, and_1, X, Y), modus_ponens, or(X, Y), or(Y, X))
% 58.23/7.87 = { by lemma 36 }
% 58.23/7.87 fresh59(fresh8(r3, and_1, X, Y), and_1, or(X, Y), or(Y, X))
% 58.23/7.87 = { by lemma 59 R->L }
% 58.23/7.87 fresh59(is_a_theorem(implies(or(X, Y), or(Y, X))), and_1, or(X, Y), or(Y, X))
% 58.23/7.87 = { by lemma 40 R->L }
% 58.23/7.87 fresh59(is_a_theorem(implies(implies(not(X), Y), or(Y, X))), and_1, or(X, Y), or(Y, X))
% 58.23/7.87 = { by axiom 12 (modus_ponens_2) R->L }
% 58.23/7.87 fresh59(fresh28(and_1, and_1, implies(implies(not(X), Y), or(Y, X))), and_1, or(X, Y), or(Y, X))
% 58.23/7.87 = { by lemma 60 R->L }
% 58.23/7.87 fresh59(fresh28(fresh28(is_a_theorem(equiv(not(X), not(X))), and_1, implies(not(X), not(X))), and_1, implies(implies(not(X), Y), or(Y, X))), and_1, or(X, Y), or(Y, X))
% 58.23/7.87 = { by lemma 40 }
% 58.23/7.87 fresh59(fresh28(fresh28(is_a_theorem(equiv(not(X), not(X))), and_1, or(X, not(X))), and_1, implies(implies(not(X), Y), or(Y, X))), and_1, or(X, Y), or(Y, X))
% 58.23/7.87 = { by lemma 54 }
% 58.23/7.87 fresh59(fresh28(fresh28(and_1, and_1, or(X, not(X))), and_1, implies(implies(not(X), Y), or(Y, X))), and_1, or(X, Y), or(Y, X))
% 58.23/7.87 = { by axiom 12 (modus_ponens_2) }
% 58.23/7.87 fresh59(fresh28(is_a_theorem(or(X, not(X))), and_1, implies(implies(not(X), Y), or(Y, X))), and_1, or(X, Y), or(Y, X))
% 58.23/7.87 = { by lemma 48 }
% 58.23/7.87 fresh59(and_1, and_1, or(X, Y), or(Y, X))
% 58.23/7.87 = { by lemma 42 }
% 58.23/7.87 modus_ponens
% 58.23/7.87 = { by lemma 36 }
% 58.23/7.87 and_1
% 58.23/7.87
% 58.23/7.87 Lemma 62: fresh28(is_a_theorem(or(X, not(Y))), and_1, implies(or(Y, Z), or(Z, X))) = and_1.
% 58.23/7.87 Proof:
% 58.23/7.87 fresh28(is_a_theorem(or(X, not(Y))), and_1, implies(or(Y, Z), or(Z, X)))
% 58.23/7.87 = { by lemma 40 R->L }
% 58.23/7.87 fresh28(is_a_theorem(or(X, not(Y))), and_1, implies(implies(not(Y), Z), or(Z, X)))
% 58.23/7.87 = { by lemma 48 }
% 58.23/7.87 and_1
% 58.23/7.87
% 58.23/7.87 Goal 1 (hilbert_or_1): or_1 = true.
% 58.23/7.87 Proof:
% 58.23/7.87 or_1
% 58.23/7.87 = { by axiom 31 (or_1) R->L }
% 58.23/7.87 fresh19(is_a_theorem(implies(x6, or(x6, y6))), true)
% 58.23/7.87 = { by lemma 57 }
% 58.23/7.87 fresh19(fresh18(or_1, modus_ponens, x6, y6), true)
% 58.23/7.87 = { by lemma 58 }
% 58.23/7.87 fresh19(fresh49(cn2, modus_ponens, x6, y6), true)
% 58.23/7.87 = { by lemma 36 }
% 58.23/7.87 fresh19(fresh49(cn2, and_1, x6, y6), true)
% 58.23/7.87 = { by axiom 1 (rosser_modus_ponens) R->L }
% 58.23/7.87 fresh19(fresh49(cn2, and_1, x6, y6), modus_ponens)
% 58.23/7.87 = { by lemma 36 }
% 58.23/7.87 fresh19(fresh49(cn2, and_1, x6, y6), and_1)
% 58.23/7.87 = { by lemma 36 R->L }
% 58.23/7.87 fresh19(fresh49(cn2, modus_ponens, x6, y6), and_1)
% 58.23/7.87 = { by lemma 58 R->L }
% 58.23/7.87 fresh19(fresh18(or_1, modus_ponens, x6, y6), and_1)
% 58.23/7.87 = { by axiom 1 (rosser_modus_ponens) }
% 58.23/7.87 fresh19(fresh18(or_1, true, x6, y6), and_1)
% 58.23/7.87 = { by axiom 22 (or_1_1) }
% 58.23/7.87 fresh19(is_a_theorem(implies(x6, or(x6, y6))), and_1)
% 58.23/7.87 = { by axiom 12 (modus_ponens_2) R->L }
% 58.23/7.87 fresh19(fresh28(and_1, and_1, implies(x6, or(x6, y6))), and_1)
% 58.23/7.87 = { by lemma 36 R->L }
% 58.23/7.87 fresh19(fresh28(modus_ponens, and_1, implies(x6, or(x6, y6))), and_1)
% 58.23/7.87 = { by lemma 41 R->L }
% 58.23/7.87 fresh19(fresh28(fresh60(modus_ponens, modus_ponens, or(implies(x6, or(x6, y6)), implies(x6, or(x6, y6)))), and_1, implies(x6, or(x6, y6))), and_1)
% 58.23/7.87 = { by axiom 1 (rosser_modus_ponens) }
% 58.23/7.87 fresh19(fresh28(fresh60(modus_ponens, true, or(implies(x6, or(x6, y6)), implies(x6, or(x6, y6)))), and_1, implies(x6, or(x6, y6))), and_1)
% 58.23/7.87 = { by axiom 13 (modus_ponens_2) R->L }
% 58.23/7.87 fresh19(fresh28(fresh59(and_1, and_1, or(or(x6, y6), implies(x6, or(x6, y6))), or(implies(x6, or(x6, y6)), implies(x6, or(x6, y6)))), and_1, implies(x6, or(x6, y6))), and_1)
% 58.23/7.87 = { by lemma 62 R->L }
% 58.23/7.87 fresh19(fresh28(fresh59(fresh28(is_a_theorem(or(not(and(x6, not(or(x6, y6)))), not(or(x6, y6)))), and_1, implies(or(or(x6, y6), implies(x6, or(x6, y6))), or(implies(x6, or(x6, y6)), not(and(x6, not(or(x6, y6))))))), and_1, or(or(x6, y6), implies(x6, or(x6, y6))), or(implies(x6, or(x6, y6)), implies(x6, or(x6, y6)))), and_1, implies(x6, or(x6, y6))), and_1)
% 58.23/7.87 = { by axiom 12 (modus_ponens_2) R->L }
% 58.23/7.87 fresh19(fresh28(fresh59(fresh28(fresh28(and_1, and_1, or(not(and(x6, not(or(x6, y6)))), not(or(x6, y6)))), and_1, implies(or(or(x6, y6), implies(x6, or(x6, y6))), or(implies(x6, or(x6, y6)), not(and(x6, not(or(x6, y6))))))), and_1, or(or(x6, y6), implies(x6, or(x6, y6))), or(implies(x6, or(x6, y6)), implies(x6, or(x6, y6)))), and_1, implies(x6, or(x6, y6))), and_1)
% 58.23/7.87 = { by lemma 44 R->L }
% 58.23/7.87 fresh19(fresh28(fresh59(fresh28(fresh28(fresh28(is_a_theorem(implies(not(or(x6, y6)), not(or(x6, y6)))), and_1, or(and(not(or(x6, y6)), x6), not(and(x6, not(or(x6, y6)))))), and_1, or(not(and(x6, not(or(x6, y6)))), not(or(x6, y6)))), and_1, implies(or(or(x6, y6), implies(x6, or(x6, y6))), or(implies(x6, or(x6, y6)), not(and(x6, not(or(x6, y6))))))), and_1, or(or(x6, y6), implies(x6, or(x6, y6))), or(implies(x6, or(x6, y6)), implies(x6, or(x6, y6)))), and_1, implies(x6, or(x6, y6))), and_1)
% 58.23/7.87 = { by axiom 12 (modus_ponens_2) R->L }
% 58.23/7.87 fresh19(fresh28(fresh59(fresh28(fresh28(fresh28(fresh28(and_1, and_1, implies(not(or(x6, y6)), not(or(x6, y6)))), and_1, or(and(not(or(x6, y6)), x6), not(and(x6, not(or(x6, y6)))))), and_1, or(not(and(x6, not(or(x6, y6)))), not(or(x6, y6)))), and_1, implies(or(or(x6, y6), implies(x6, or(x6, y6))), or(implies(x6, or(x6, y6)), not(and(x6, not(or(x6, y6))))))), and_1, or(or(x6, y6), implies(x6, or(x6, y6))), or(implies(x6, or(x6, y6)), implies(x6, or(x6, y6)))), and_1, implies(x6, or(x6, y6))), and_1)
% 58.23/7.87 = { by lemma 54 R->L }
% 58.23/7.87 fresh19(fresh28(fresh59(fresh28(fresh28(fresh28(fresh28(is_a_theorem(equiv(not(or(x6, y6)), not(or(x6, y6)))), and_1, implies(not(or(x6, y6)), not(or(x6, y6)))), and_1, or(and(not(or(x6, y6)), x6), not(and(x6, not(or(x6, y6)))))), and_1, or(not(and(x6, not(or(x6, y6)))), not(or(x6, y6)))), and_1, implies(or(or(x6, y6), implies(x6, or(x6, y6))), or(implies(x6, or(x6, y6)), not(and(x6, not(or(x6, y6))))))), and_1, or(or(x6, y6), implies(x6, or(x6, y6))), or(implies(x6, or(x6, y6)), implies(x6, or(x6, y6)))), and_1, implies(x6, or(x6, y6))), and_1)
% 58.23/7.87 = { by lemma 60 }
% 58.23/7.87 fresh19(fresh28(fresh59(fresh28(fresh28(fresh28(and_1, and_1, or(and(not(or(x6, y6)), x6), not(and(x6, not(or(x6, y6)))))), and_1, or(not(and(x6, not(or(x6, y6)))), not(or(x6, y6)))), and_1, implies(or(or(x6, y6), implies(x6, or(x6, y6))), or(implies(x6, or(x6, y6)), not(and(x6, not(or(x6, y6))))))), and_1, or(or(x6, y6), implies(x6, or(x6, y6))), or(implies(x6, or(x6, y6)), implies(x6, or(x6, y6)))), and_1, implies(x6, or(x6, y6))), and_1)
% 58.23/7.87 = { by axiom 12 (modus_ponens_2) }
% 58.23/7.87 fresh19(fresh28(fresh59(fresh28(fresh28(is_a_theorem(or(and(not(or(x6, y6)), x6), not(and(x6, not(or(x6, y6)))))), and_1, or(not(and(x6, not(or(x6, y6)))), not(or(x6, y6)))), and_1, implies(or(or(x6, y6), implies(x6, or(x6, y6))), or(implies(x6, or(x6, y6)), not(and(x6, not(or(x6, y6))))))), and_1, or(or(x6, y6), implies(x6, or(x6, y6))), or(implies(x6, or(x6, y6)), implies(x6, or(x6, y6)))), and_1, implies(x6, or(x6, y6))), and_1)
% 58.23/7.87 = { by lemma 51 }
% 58.23/7.87 fresh19(fresh28(fresh59(fresh28(and_1, and_1, implies(or(or(x6, y6), implies(x6, or(x6, y6))), or(implies(x6, or(x6, y6)), not(and(x6, not(or(x6, y6))))))), and_1, or(or(x6, y6), implies(x6, or(x6, y6))), or(implies(x6, or(x6, y6)), implies(x6, or(x6, y6)))), and_1, implies(x6, or(x6, y6))), and_1)
% 58.23/7.87 = { by axiom 12 (modus_ponens_2) }
% 58.23/7.87 fresh19(fresh28(fresh59(is_a_theorem(implies(or(or(x6, y6), implies(x6, or(x6, y6))), or(implies(x6, or(x6, y6)), not(and(x6, not(or(x6, y6))))))), and_1, or(or(x6, y6), implies(x6, or(x6, y6))), or(implies(x6, or(x6, y6)), implies(x6, or(x6, y6)))), and_1, implies(x6, or(x6, y6))), and_1)
% 58.23/7.87 = { by lemma 39 }
% 58.23/7.87 fresh19(fresh28(fresh59(is_a_theorem(implies(or(or(x6, y6), implies(x6, or(x6, y6))), or(implies(x6, or(x6, y6)), implies(x6, or(x6, y6))))), and_1, or(or(x6, y6), implies(x6, or(x6, y6))), or(implies(x6, or(x6, y6)), implies(x6, or(x6, y6)))), and_1, implies(x6, or(x6, y6))), and_1)
% 58.23/7.87 = { by lemma 36 R->L }
% 58.23/7.87 fresh19(fresh28(fresh59(is_a_theorem(implies(or(or(x6, y6), implies(x6, or(x6, y6))), or(implies(x6, or(x6, y6)), implies(x6, or(x6, y6))))), modus_ponens, or(or(x6, y6), implies(x6, or(x6, y6))), or(implies(x6, or(x6, y6)), implies(x6, or(x6, y6)))), and_1, implies(x6, or(x6, y6))), and_1)
% 58.23/7.87 = { by lemma 38 }
% 58.23/7.87 fresh19(fresh28(fresh28(is_a_theorem(or(or(x6, y6), implies(x6, or(x6, y6)))), modus_ponens, or(implies(x6, or(x6, y6)), implies(x6, or(x6, y6)))), and_1, implies(x6, or(x6, y6))), and_1)
% 58.23/7.87 = { by lemma 36 }
% 58.23/7.87 fresh19(fresh28(fresh28(is_a_theorem(or(or(x6, y6), implies(x6, or(x6, y6)))), and_1, or(implies(x6, or(x6, y6)), implies(x6, or(x6, y6)))), and_1, implies(x6, or(x6, y6))), and_1)
% 58.23/7.87 = { by axiom 12 (modus_ponens_2) R->L }
% 58.23/7.87 fresh19(fresh28(fresh28(fresh28(and_1, and_1, or(or(x6, y6), implies(x6, or(x6, y6)))), and_1, or(implies(x6, or(x6, y6)), implies(x6, or(x6, y6)))), and_1, implies(x6, or(x6, y6))), and_1)
% 58.23/7.87 = { by lemma 36 R->L }
% 58.23/7.87 fresh19(fresh28(fresh28(fresh28(modus_ponens, and_1, or(or(x6, y6), implies(x6, or(x6, y6)))), and_1, or(implies(x6, or(x6, y6)), implies(x6, or(x6, y6)))), and_1, implies(x6, or(x6, y6))), and_1)
% 58.23/7.88 = { by lemma 42 R->L }
% 58.23/7.88 fresh19(fresh28(fresh28(fresh28(fresh59(and_1, and_1, or(x6, implies(x6, or(x6, y6))), or(implies(x6, or(x6, y6)), or(x6, y6))), and_1, or(or(x6, y6), implies(x6, or(x6, y6)))), and_1, or(implies(x6, or(x6, y6)), implies(x6, or(x6, y6)))), and_1, implies(x6, or(x6, y6))), and_1)
% 58.23/7.88 = { by lemma 36 R->L }
% 58.23/7.88 fresh19(fresh28(fresh28(fresh28(fresh59(and_1, modus_ponens, or(x6, implies(x6, or(x6, y6))), or(implies(x6, or(x6, y6)), or(x6, y6))), and_1, or(or(x6, y6), implies(x6, or(x6, y6)))), and_1, or(implies(x6, or(x6, y6)), implies(x6, or(x6, y6)))), and_1, implies(x6, or(x6, y6))), and_1)
% 58.23/7.88 = { by lemma 62 R->L }
% 58.23/7.88 fresh19(fresh28(fresh28(fresh28(fresh59(fresh28(is_a_theorem(or(implies(not(x6), y6), not(x6))), and_1, implies(or(x6, implies(x6, or(x6, y6))), or(implies(x6, or(x6, y6)), implies(not(x6), y6)))), modus_ponens, or(x6, implies(x6, or(x6, y6))), or(implies(x6, or(x6, y6)), or(x6, y6))), and_1, or(or(x6, y6), implies(x6, or(x6, y6)))), and_1, or(implies(x6, or(x6, y6)), implies(x6, or(x6, y6)))), and_1, implies(x6, or(x6, y6))), and_1)
% 58.23/7.88 = { by axiom 12 (modus_ponens_2) R->L }
% 58.23/7.88 fresh19(fresh28(fresh28(fresh28(fresh59(fresh28(fresh28(and_1, and_1, or(implies(not(x6), y6), not(x6))), and_1, implies(or(x6, implies(x6, or(x6, y6))), or(implies(x6, or(x6, y6)), implies(not(x6), y6)))), modus_ponens, or(x6, implies(x6, or(x6, y6))), or(implies(x6, or(x6, y6)), or(x6, y6))), and_1, or(or(x6, y6), implies(x6, or(x6, y6)))), and_1, or(implies(x6, or(x6, y6)), implies(x6, or(x6, y6)))), and_1, implies(x6, or(x6, y6))), and_1)
% 58.23/7.88 = { by lemma 52 R->L }
% 58.23/7.88 fresh19(fresh28(fresh28(fresh28(fresh59(fresh28(fresh28(is_a_theorem(or(not(x6), implies(not(x6), y6))), and_1, or(implies(not(x6), y6), not(x6))), and_1, implies(or(x6, implies(x6, or(x6, y6))), or(implies(x6, or(x6, y6)), implies(not(x6), y6)))), modus_ponens, or(x6, implies(x6, or(x6, y6))), or(implies(x6, or(x6, y6)), or(x6, y6))), and_1, or(or(x6, y6), implies(x6, or(x6, y6)))), and_1, or(implies(x6, or(x6, y6)), implies(x6, or(x6, y6)))), and_1, implies(x6, or(x6, y6))), and_1)
% 58.23/7.88 = { by lemma 61 }
% 58.23/7.88 fresh19(fresh28(fresh28(fresh28(fresh59(fresh28(and_1, and_1, implies(or(x6, implies(x6, or(x6, y6))), or(implies(x6, or(x6, y6)), implies(not(x6), y6)))), modus_ponens, or(x6, implies(x6, or(x6, y6))), or(implies(x6, or(x6, y6)), or(x6, y6))), and_1, or(or(x6, y6), implies(x6, or(x6, y6)))), and_1, or(implies(x6, or(x6, y6)), implies(x6, or(x6, y6)))), and_1, implies(x6, or(x6, y6))), and_1)
% 58.23/7.88 = { by axiom 12 (modus_ponens_2) }
% 58.23/7.88 fresh19(fresh28(fresh28(fresh28(fresh59(is_a_theorem(implies(or(x6, implies(x6, or(x6, y6))), or(implies(x6, or(x6, y6)), implies(not(x6), y6)))), modus_ponens, or(x6, implies(x6, or(x6, y6))), or(implies(x6, or(x6, y6)), or(x6, y6))), and_1, or(or(x6, y6), implies(x6, or(x6, y6)))), and_1, or(implies(x6, or(x6, y6)), implies(x6, or(x6, y6)))), and_1, implies(x6, or(x6, y6))), and_1)
% 58.23/7.88 = { by lemma 40 }
% 58.23/7.88 fresh19(fresh28(fresh28(fresh28(fresh59(is_a_theorem(implies(or(x6, implies(x6, or(x6, y6))), or(implies(x6, or(x6, y6)), or(x6, y6)))), modus_ponens, or(x6, implies(x6, or(x6, y6))), or(implies(x6, or(x6, y6)), or(x6, y6))), and_1, or(or(x6, y6), implies(x6, or(x6, y6)))), and_1, or(implies(x6, or(x6, y6)), implies(x6, or(x6, y6)))), and_1, implies(x6, or(x6, y6))), and_1)
% 58.23/7.88 = { by lemma 38 }
% 58.23/7.88 fresh19(fresh28(fresh28(fresh28(fresh28(is_a_theorem(or(x6, implies(x6, or(x6, y6)))), modus_ponens, or(implies(x6, or(x6, y6)), or(x6, y6))), and_1, or(or(x6, y6), implies(x6, or(x6, y6)))), and_1, or(implies(x6, or(x6, y6)), implies(x6, or(x6, y6)))), and_1, implies(x6, or(x6, y6))), and_1)
% 58.23/7.88 = { by lemma 36 }
% 58.23/7.88 fresh19(fresh28(fresh28(fresh28(fresh28(is_a_theorem(or(x6, implies(x6, or(x6, y6)))), and_1, or(implies(x6, or(x6, y6)), or(x6, y6))), and_1, or(or(x6, y6), implies(x6, or(x6, y6)))), and_1, or(implies(x6, or(x6, y6)), implies(x6, or(x6, y6)))), and_1, implies(x6, or(x6, y6))), and_1)
% 58.23/7.88 = { by lemma 52 }
% 58.23/7.88 fresh19(fresh28(fresh28(fresh28(fresh28(and_1, and_1, or(implies(x6, or(x6, y6)), or(x6, y6))), and_1, or(or(x6, y6), implies(x6, or(x6, y6)))), and_1, or(implies(x6, or(x6, y6)), implies(x6, or(x6, y6)))), and_1, implies(x6, or(x6, y6))), and_1)
% 58.23/7.88 = { by axiom 12 (modus_ponens_2) }
% 58.23/7.88 fresh19(fresh28(fresh28(fresh28(is_a_theorem(or(implies(x6, or(x6, y6)), or(x6, y6))), and_1, or(or(x6, y6), implies(x6, or(x6, y6)))), and_1, or(implies(x6, or(x6, y6)), implies(x6, or(x6, y6)))), and_1, implies(x6, or(x6, y6))), and_1)
% 58.23/7.88 = { by lemma 61 }
% 58.23/7.88 fresh19(fresh28(fresh28(and_1, and_1, or(implies(x6, or(x6, y6)), implies(x6, or(x6, y6)))), and_1, implies(x6, or(x6, y6))), and_1)
% 58.23/7.88 = { by axiom 12 (modus_ponens_2) }
% 58.23/7.88 fresh19(fresh28(is_a_theorem(or(implies(x6, or(x6, y6)), implies(x6, or(x6, y6)))), and_1, implies(x6, or(x6, y6))), and_1)
% 58.23/7.88 = { by lemma 36 R->L }
% 58.23/7.88 fresh19(fresh28(is_a_theorem(or(implies(x6, or(x6, y6)), implies(x6, or(x6, y6)))), modus_ponens, implies(x6, or(x6, y6))), and_1)
% 58.23/7.88 = { by lemma 38 R->L }
% 58.23/7.88 fresh19(fresh59(is_a_theorem(implies(or(implies(x6, or(x6, y6)), implies(x6, or(x6, y6))), implies(x6, or(x6, y6)))), modus_ponens, or(implies(x6, or(x6, y6)), implies(x6, or(x6, y6))), implies(x6, or(x6, y6))), and_1)
% 58.23/7.88 = { by lemma 55 }
% 58.23/7.88 fresh19(fresh59(fresh12(r1, modus_ponens, implies(x6, or(x6, y6))), modus_ponens, or(implies(x6, or(x6, y6)), implies(x6, or(x6, y6))), implies(x6, or(x6, y6))), and_1)
% 58.23/7.88 = { by lemma 56 }
% 58.23/7.88 fresh19(fresh59(fresh47(cn3, modus_ponens, implies(x6, or(x6, y6))), modus_ponens, or(implies(x6, or(x6, y6)), implies(x6, or(x6, y6))), implies(x6, or(x6, y6))), and_1)
% 58.23/7.88 = { by lemma 36 }
% 58.23/7.88 fresh19(fresh59(fresh47(cn3, modus_ponens, implies(x6, or(x6, y6))), and_1, or(implies(x6, or(x6, y6)), implies(x6, or(x6, y6))), implies(x6, or(x6, y6))), and_1)
% 58.23/7.88 = { by lemma 56 R->L }
% 58.23/7.88 fresh19(fresh59(fresh12(r1, modus_ponens, implies(x6, or(x6, y6))), and_1, or(implies(x6, or(x6, y6)), implies(x6, or(x6, y6))), implies(x6, or(x6, y6))), and_1)
% 58.23/7.88 = { by lemma 55 R->L }
% 58.23/7.88 fresh19(fresh59(is_a_theorem(implies(or(implies(x6, or(x6, y6)), implies(x6, or(x6, y6))), implies(x6, or(x6, y6)))), and_1, or(implies(x6, or(x6, y6)), implies(x6, or(x6, y6))), implies(x6, or(x6, y6))), and_1)
% 58.23/7.88 = { by lemma 40 R->L }
% 58.23/7.88 fresh19(fresh59(is_a_theorem(implies(implies(not(implies(x6, or(x6, y6))), implies(x6, or(x6, y6))), implies(x6, or(x6, y6)))), and_1, or(implies(x6, or(x6, y6)), implies(x6, or(x6, y6))), implies(x6, or(x6, y6))), and_1)
% 58.23/7.88 = { by lemma 46 R->L }
% 58.23/7.88 fresh19(fresh59(is_a_theorem(or(and(not(implies(x6, or(x6, y6))), not(implies(x6, or(x6, y6)))), implies(x6, or(x6, y6)))), and_1, or(implies(x6, or(x6, y6)), implies(x6, or(x6, y6))), implies(x6, or(x6, y6))), and_1)
% 58.23/7.88 = { by axiom 12 (modus_ponens_2) R->L }
% 58.23/7.88 fresh19(fresh59(fresh28(and_1, and_1, or(and(not(implies(x6, or(x6, y6))), not(implies(x6, or(x6, y6)))), implies(x6, or(x6, y6)))), and_1, or(implies(x6, or(x6, y6)), implies(x6, or(x6, y6))), implies(x6, or(x6, y6))), and_1)
% 58.23/7.88 = { by lemma 53 R->L }
% 58.23/7.88 fresh19(fresh59(fresh28(is_a_theorem(or(implies(x6, or(x6, y6)), and(not(implies(x6, or(x6, y6))), not(implies(x6, or(x6, y6)))))), and_1, or(and(not(implies(x6, or(x6, y6))), not(implies(x6, or(x6, y6)))), implies(x6, or(x6, y6)))), and_1, or(implies(x6, or(x6, y6)), implies(x6, or(x6, y6))), implies(x6, or(x6, y6))), and_1)
% 58.23/7.88 = { by lemma 61 }
% 58.23/7.88 fresh19(fresh59(and_1, and_1, or(implies(x6, or(x6, y6)), implies(x6, or(x6, y6))), implies(x6, or(x6, y6))), and_1)
% 58.23/7.88 = { by lemma 42 }
% 58.23/7.88 fresh19(modus_ponens, and_1)
% 58.23/7.88 = { by lemma 36 }
% 58.23/7.88 fresh19(and_1, and_1)
% 58.23/7.88 = { by axiom 9 (or_1) }
% 58.23/7.88 true
% 58.23/7.88 % SZS output end Proof
% 58.23/7.88
% 58.23/7.88 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------