TSTP Solution File: LCL541+1 by Twee---2.4.2
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : LCL541+1 : TPTP v8.1.2. Bugfixed v4.0.0.
% 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 : n002.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 08:19:21 EDT 2023
% Result : Theorem 79.56s 10.54s
% Output : Proof 80.12s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.12 % Problem : LCL541+1 : TPTP v8.1.2. Bugfixed v4.0.0.
% 0.12/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.12/0.33 % Computer : n002.cluster.edu
% 0.12/0.33 % Model : x86_64 x86_64
% 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.33 % Memory : 8042.1875MB
% 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.33 % CPULimit : 300
% 0.12/0.33 % WCLimit : 300
% 0.12/0.33 % DateTime : Thu Aug 24 18:06:46 EDT 2023
% 0.12/0.34 % CPUTime :
% 79.56/10.54 Command-line arguments: --lhs-weight 1 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 79.56/10.54
% 79.56/10.54 % SZS status Theorem
% 79.56/10.54
% 79.56/10.57 % SZS output start Proof
% 79.56/10.57 Take the following subset of the input axioms:
% 79.56/10.57 fof(and_1, axiom, and_1 <=> ![X, Y]: is_a_theorem(implies(and(X, Y), X))).
% 79.56/10.57 fof(and_3, axiom, and_3 <=> ![X2, Y2]: is_a_theorem(implies(X2, implies(Y2, and(X2, Y2))))).
% 79.56/10.57 fof(axiom_m1, axiom, axiom_m1 <=> ![X2, Y2]: is_a_theorem(strict_implies(and(X2, Y2), and(Y2, X2)))).
% 79.56/10.57 fof(equivalence_3, axiom, equivalence_3 <=> ![X2, Y2]: is_a_theorem(implies(implies(X2, Y2), implies(implies(Y2, X2), equiv(X2, Y2))))).
% 79.56/10.57 fof(hilbert_and_1, axiom, and_1).
% 79.56/10.57 fof(hilbert_and_3, axiom, and_3).
% 79.56/10.57 fof(hilbert_equivalence_3, axiom, equivalence_3).
% 79.56/10.57 fof(hilbert_implies_1, axiom, implies_1).
% 79.56/10.57 fof(hilbert_implies_2, axiom, implies_2).
% 79.56/10.57 fof(hilbert_modus_ponens, axiom, modus_ponens).
% 79.56/10.57 fof(hilbert_modus_tollens, axiom, modus_tollens).
% 79.56/10.57 fof(hilbert_op_implies_and, axiom, op_implies_and).
% 79.56/10.57 fof(hilbert_op_or, axiom, op_or).
% 79.56/10.57 fof(hilbert_or_1, axiom, or_1).
% 79.56/10.57 fof(hilbert_or_3, axiom, or_3).
% 79.56/10.57 fof(implies_1, axiom, implies_1 <=> ![X2, Y2]: is_a_theorem(implies(X2, implies(Y2, X2)))).
% 79.56/10.57 fof(implies_2, axiom, implies_2 <=> ![X2, Y2]: is_a_theorem(implies(implies(X2, implies(X2, Y2)), implies(X2, Y2)))).
% 79.56/10.57 fof(km4b_necessitation, axiom, necessitation).
% 79.56/10.57 fof(modus_ponens, axiom, modus_ponens <=> ![X2, Y2]: ((is_a_theorem(X2) & is_a_theorem(implies(X2, Y2))) => is_a_theorem(Y2))).
% 79.56/10.57 fof(modus_tollens, axiom, modus_tollens <=> ![X2, Y2]: is_a_theorem(implies(implies(not(Y2), not(X2)), implies(X2, Y2)))).
% 79.56/10.57 fof(necessitation, axiom, necessitation <=> ![X2]: (is_a_theorem(X2) => is_a_theorem(necessarily(X2)))).
% 79.56/10.57 fof(op_implies_and, axiom, op_implies_and => ![X2, Y2]: implies(X2, Y2)=not(and(X2, not(Y2)))).
% 79.56/10.57 fof(op_or, axiom, op_or => ![X2, Y2]: or(X2, Y2)=not(and(not(X2), not(Y2)))).
% 79.56/10.57 fof(op_strict_implies, axiom, op_strict_implies => ![X2, Y2]: strict_implies(X2, Y2)=necessarily(implies(X2, Y2))).
% 79.56/10.57 fof(or_1, axiom, or_1 <=> ![X2, Y2]: is_a_theorem(implies(X2, or(X2, Y2)))).
% 79.56/10.57 fof(or_3, axiom, or_3 <=> ![Z, X2, Y2]: is_a_theorem(implies(implies(X2, Z), implies(implies(Y2, Z), implies(or(X2, Y2), Z))))).
% 79.56/10.57 fof(s1_0_axiom_m1, conjecture, axiom_m1).
% 79.56/10.57 fof(s1_0_op_strict_implies, axiom, op_strict_implies).
% 79.56/10.57 fof(substitution_of_equivalents, axiom, substitution_of_equivalents <=> ![X2, Y2]: (is_a_theorem(equiv(X2, Y2)) => X2=Y2)).
% 79.56/10.57 fof(substitution_of_equivalents, axiom, substitution_of_equivalents).
% 79.56/10.57
% 79.56/10.57 Now clausify the problem and encode Horn clauses using encoding 3 of
% 79.56/10.57 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 79.56/10.57 We repeatedly replace C & s=t => u=v by the two clauses:
% 79.56/10.57 fresh(y, y, x1...xn) = u
% 79.56/10.57 C => fresh(s, t, x1...xn) = v
% 79.56/10.57 where fresh is a fresh function symbol and x1..xn are the free
% 79.56/10.57 variables of u and v.
% 79.56/10.57 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 79.56/10.57 input problem has no model of domain size 1).
% 79.56/10.57
% 79.56/10.57 The encoding turns the above axioms into the following unit equations and goals:
% 79.56/10.57
% 79.56/10.57 Axiom 1 (hilbert_modus_ponens): modus_ponens = true.
% 79.56/10.57 Axiom 2 (substitution_of_equivalents): substitution_of_equivalents = true.
% 79.56/10.57 Axiom 3 (hilbert_modus_tollens): modus_tollens = true.
% 79.56/10.57 Axiom 4 (hilbert_implies_1): implies_1 = true.
% 79.56/10.57 Axiom 5 (hilbert_implies_2): implies_2 = true.
% 79.56/10.57 Axiom 6 (hilbert_and_1): and_1 = true.
% 79.56/10.57 Axiom 7 (hilbert_and_3): and_3 = true.
% 79.56/10.57 Axiom 8 (hilbert_or_1): or_1 = true.
% 79.56/10.57 Axiom 9 (hilbert_or_3): or_3 = true.
% 79.56/10.57 Axiom 10 (hilbert_equivalence_3): equivalence_3 = true.
% 79.56/10.57 Axiom 11 (hilbert_op_or): op_or = true.
% 79.56/10.57 Axiom 12 (km4b_necessitation): necessitation = true.
% 79.56/10.57 Axiom 13 (hilbert_op_implies_and): op_implies_and = true.
% 79.56/10.57 Axiom 14 (s1_0_op_strict_implies): op_strict_implies = true.
% 79.56/10.57 Axiom 15 (axiom_m1): fresh92(X, X) = true.
% 79.56/10.57 Axiom 16 (modus_ponens_2): fresh116(X, X, Y) = true.
% 79.56/10.57 Axiom 17 (modus_ponens_2): fresh40(X, X, Y) = is_a_theorem(Y).
% 79.56/10.57 Axiom 18 (necessitation_1): fresh34(X, X, Y) = is_a_theorem(necessarily(Y)).
% 79.56/10.57 Axiom 19 (necessitation_1): fresh33(X, X, Y) = true.
% 79.56/10.57 Axiom 20 (modus_ponens_2): fresh115(X, X, Y, Z) = fresh116(modus_ponens, true, Z).
% 79.56/10.57 Axiom 21 (and_1_1): fresh107(X, X, Y, Z) = true.
% 79.56/10.57 Axiom 22 (and_3_1): fresh103(X, X, Y, Z) = true.
% 79.56/10.57 Axiom 23 (equivalence_3_1): fresh53(X, X, Y, Z) = true.
% 79.56/10.57 Axiom 24 (implies_1_1): fresh51(X, X, Y, Z) = true.
% 79.56/10.57 Axiom 25 (implies_2_1): fresh49(X, X, Y, Z) = true.
% 79.56/10.57 Axiom 26 (modus_tollens_1): fresh35(X, X, Y, Z) = true.
% 79.56/10.57 Axiom 27 (necessitation_1): fresh34(necessitation, true, X) = fresh33(is_a_theorem(X), true, X).
% 79.56/10.57 Axiom 28 (op_implies_and): fresh29(X, X, Y, Z) = implies(Y, Z).
% 79.56/10.57 Axiom 29 (op_implies_and): fresh29(op_implies_and, true, X, Y) = not(and(X, not(Y))).
% 79.56/10.57 Axiom 30 (op_or): fresh26(X, X, Y, Z) = or(Y, Z).
% 79.56/10.57 Axiom 31 (op_strict_implies): fresh23(X, X, Y, Z) = strict_implies(Y, Z).
% 79.56/10.57 Axiom 32 (op_strict_implies): fresh23(op_strict_implies, true, X, Y) = necessarily(implies(X, Y)).
% 79.56/10.57 Axiom 33 (or_1_1): fresh21(X, X, Y, Z) = true.
% 79.56/10.57 Axiom 34 (substitution_of_equivalents_2): fresh4(X, X, Y, Z) = Y.
% 79.56/10.57 Axiom 35 (substitution_of_equivalents_2): fresh3(X, X, Y, Z) = Z.
% 79.56/10.57 Axiom 36 (implies_1_1): fresh51(implies_1, true, X, Y) = is_a_theorem(implies(X, implies(Y, X))).
% 79.56/10.57 Axiom 37 (or_1_1): fresh21(or_1, true, X, Y) = is_a_theorem(implies(X, or(X, Y))).
% 79.56/10.57 Axiom 38 (and_1_1): fresh107(and_1, true, X, Y) = is_a_theorem(implies(and(X, Y), X)).
% 79.56/10.57 Axiom 39 (op_or): fresh26(op_or, true, X, Y) = not(and(not(X), not(Y))).
% 79.56/10.57 Axiom 40 (or_3_1): fresh17(X, X, Y, Z, W) = true.
% 79.56/10.57 Axiom 41 (and_3_1): fresh103(and_3, true, X, Y) = is_a_theorem(implies(X, implies(Y, and(X, Y)))).
% 79.56/10.57 Axiom 42 (axiom_m1_1): fresh89(axiom_m1, true, X, Y) = is_a_theorem(strict_implies(and(X, Y), and(Y, X))).
% 79.56/10.57 Axiom 43 (modus_ponens_2): fresh115(is_a_theorem(implies(X, Y)), true, X, Y) = fresh40(is_a_theorem(X), true, Y).
% 79.56/10.57 Axiom 44 (substitution_of_equivalents_2): fresh4(substitution_of_equivalents, true, X, Y) = fresh3(is_a_theorem(equiv(X, Y)), true, X, Y).
% 79.56/10.57 Axiom 45 (implies_2_1): fresh49(implies_2, true, X, Y) = is_a_theorem(implies(implies(X, implies(X, Y)), implies(X, Y))).
% 79.56/10.57 Axiom 46 (modus_tollens_1): fresh35(modus_tollens, true, X, Y) = is_a_theorem(implies(implies(not(Y), not(X)), implies(X, Y))).
% 79.56/10.58 Axiom 47 (axiom_m1): fresh92(is_a_theorem(strict_implies(and(x8, y4), and(y4, x8))), true) = axiom_m1.
% 79.56/10.58 Axiom 48 (equivalence_3_1): fresh53(equivalence_3, true, X, Y) = is_a_theorem(implies(implies(X, Y), implies(implies(Y, X), equiv(X, Y)))).
% 79.56/10.58 Axiom 49 (or_3_1): fresh17(or_3, true, X, Y, Z) = is_a_theorem(implies(implies(X, Z), implies(implies(Y, Z), implies(or(X, Y), Z)))).
% 79.56/10.58
% 79.56/10.58 Lemma 50: fresh3(is_a_theorem(equiv(X, Y)), true, X, Y) = X.
% 79.56/10.58 Proof:
% 79.56/10.58 fresh3(is_a_theorem(equiv(X, Y)), true, X, Y)
% 79.56/10.58 = { by axiom 44 (substitution_of_equivalents_2) R->L }
% 79.56/10.58 fresh4(substitution_of_equivalents, true, X, Y)
% 79.56/10.58 = { by axiom 2 (substitution_of_equivalents) }
% 79.56/10.58 fresh4(true, true, X, Y)
% 79.56/10.58 = { by axiom 34 (substitution_of_equivalents_2) }
% 79.56/10.58 X
% 79.56/10.58
% 79.56/10.58 Lemma 51: fresh115(X, X, Y, Z) = true.
% 79.56/10.58 Proof:
% 79.56/10.58 fresh115(X, X, Y, Z)
% 79.56/10.58 = { by axiom 20 (modus_ponens_2) }
% 79.56/10.58 fresh116(modus_ponens, true, Z)
% 79.56/10.58 = { by axiom 1 (hilbert_modus_ponens) }
% 79.56/10.58 fresh116(true, true, Z)
% 79.56/10.58 = { by axiom 16 (modus_ponens_2) }
% 79.56/10.58 true
% 79.56/10.58
% 79.56/10.58 Lemma 52: fresh40(is_a_theorem(implies(X, implies(X, Y))), true, implies(X, Y)) = true.
% 79.56/10.58 Proof:
% 79.56/10.58 fresh40(is_a_theorem(implies(X, implies(X, Y))), true, implies(X, Y))
% 79.56/10.58 = { by axiom 43 (modus_ponens_2) R->L }
% 79.56/10.58 fresh115(is_a_theorem(implies(implies(X, implies(X, Y)), implies(X, Y))), true, implies(X, implies(X, Y)), implies(X, Y))
% 79.56/10.58 = { by axiom 45 (implies_2_1) R->L }
% 79.56/10.58 fresh115(fresh49(implies_2, true, X, Y), true, implies(X, implies(X, Y)), implies(X, Y))
% 79.56/10.58 = { by axiom 5 (hilbert_implies_2) }
% 79.56/10.58 fresh115(fresh49(true, true, X, Y), true, implies(X, implies(X, Y)), implies(X, Y))
% 79.56/10.58 = { by axiom 25 (implies_2_1) }
% 79.56/10.58 fresh115(true, true, implies(X, implies(X, Y)), implies(X, Y))
% 79.56/10.58 = { by lemma 51 }
% 79.56/10.58 true
% 79.56/10.58
% 79.56/10.58 Lemma 53: fresh40(is_a_theorem(implies(X, Y)), true, implies(implies(Y, X), equiv(X, Y))) = true.
% 79.56/10.58 Proof:
% 79.56/10.58 fresh40(is_a_theorem(implies(X, Y)), true, implies(implies(Y, X), equiv(X, Y)))
% 79.56/10.58 = { by axiom 43 (modus_ponens_2) R->L }
% 79.56/10.58 fresh115(is_a_theorem(implies(implies(X, Y), implies(implies(Y, X), equiv(X, Y)))), true, implies(X, Y), implies(implies(Y, X), equiv(X, Y)))
% 79.56/10.58 = { by axiom 48 (equivalence_3_1) R->L }
% 79.56/10.58 fresh115(fresh53(equivalence_3, true, X, Y), true, implies(X, Y), implies(implies(Y, X), equiv(X, Y)))
% 79.56/10.58 = { by axiom 10 (hilbert_equivalence_3) }
% 79.56/10.58 fresh115(fresh53(true, true, X, Y), true, implies(X, Y), implies(implies(Y, X), equiv(X, Y)))
% 79.56/10.58 = { by axiom 23 (equivalence_3_1) }
% 79.56/10.58 fresh115(true, true, implies(X, Y), implies(implies(Y, X), equiv(X, Y)))
% 79.56/10.58 = { by lemma 51 }
% 79.56/10.58 true
% 79.56/10.58
% 79.56/10.58 Lemma 54: or(X, X) = X.
% 79.56/10.58 Proof:
% 79.56/10.58 or(X, X)
% 79.56/10.58 = { by lemma 50 R->L }
% 79.56/10.58 fresh3(is_a_theorem(equiv(or(X, X), X)), true, or(X, X), X)
% 79.56/10.58 = { by axiom 17 (modus_ponens_2) R->L }
% 79.56/10.58 fresh3(fresh40(true, true, equiv(or(X, X), X)), true, or(X, X), X)
% 79.56/10.58 = { by axiom 33 (or_1_1) R->L }
% 79.56/10.58 fresh3(fresh40(fresh21(true, true, X, X), true, equiv(or(X, X), X)), true, or(X, X), X)
% 79.56/10.58 = { by axiom 8 (hilbert_or_1) R->L }
% 79.56/10.58 fresh3(fresh40(fresh21(or_1, true, X, X), true, equiv(or(X, X), X)), true, or(X, X), X)
% 79.56/10.58 = { by axiom 37 (or_1_1) }
% 79.56/10.58 fresh3(fresh40(is_a_theorem(implies(X, or(X, X))), true, equiv(or(X, X), X)), true, or(X, X), X)
% 79.56/10.58 = { by axiom 43 (modus_ponens_2) R->L }
% 79.56/10.58 fresh3(fresh115(is_a_theorem(implies(implies(X, or(X, X)), equiv(or(X, X), X))), true, implies(X, or(X, X)), equiv(or(X, X), X)), true, or(X, X), X)
% 79.56/10.58 = { by axiom 17 (modus_ponens_2) R->L }
% 79.56/10.58 fresh3(fresh115(fresh40(true, true, implies(implies(X, or(X, X)), equiv(or(X, X), X))), true, implies(X, or(X, X)), equiv(or(X, X), X)), true, or(X, X), X)
% 79.56/10.58 = { by lemma 51 R->L }
% 79.56/10.58 fresh3(fresh115(fresh40(fresh115(true, true, implies(X, X), implies(or(X, X), X)), true, implies(implies(X, or(X, X)), equiv(or(X, X), X))), true, implies(X, or(X, X)), equiv(or(X, X), X)), true, or(X, X), X)
% 79.56/10.58 = { by lemma 52 R->L }
% 79.56/10.58 fresh3(fresh115(fresh40(fresh115(fresh40(is_a_theorem(implies(implies(X, X), implies(implies(X, X), implies(or(X, X), X)))), true, implies(implies(X, X), implies(or(X, X), X))), true, implies(X, X), implies(or(X, X), X)), true, implies(implies(X, or(X, X)), equiv(or(X, X), X))), true, implies(X, or(X, X)), equiv(or(X, X), X)), true, or(X, X), X)
% 79.56/10.58 = { by axiom 49 (or_3_1) R->L }
% 79.56/10.58 fresh3(fresh115(fresh40(fresh115(fresh40(fresh17(or_3, true, X, X, X), true, implies(implies(X, X), implies(or(X, X), X))), true, implies(X, X), implies(or(X, X), X)), true, implies(implies(X, or(X, X)), equiv(or(X, X), X))), true, implies(X, or(X, X)), equiv(or(X, X), X)), true, or(X, X), X)
% 79.56/10.58 = { by axiom 9 (hilbert_or_3) }
% 79.56/10.58 fresh3(fresh115(fresh40(fresh115(fresh40(fresh17(true, true, X, X, X), true, implies(implies(X, X), implies(or(X, X), X))), true, implies(X, X), implies(or(X, X), X)), true, implies(implies(X, or(X, X)), equiv(or(X, X), X))), true, implies(X, or(X, X)), equiv(or(X, X), X)), true, or(X, X), X)
% 79.56/10.58 = { by axiom 40 (or_3_1) }
% 79.56/10.58 fresh3(fresh115(fresh40(fresh115(fresh40(true, true, implies(implies(X, X), implies(or(X, X), X))), true, implies(X, X), implies(or(X, X), X)), true, implies(implies(X, or(X, X)), equiv(or(X, X), X))), true, implies(X, or(X, X)), equiv(or(X, X), X)), true, or(X, X), X)
% 79.56/10.58 = { by axiom 17 (modus_ponens_2) }
% 79.56/10.58 fresh3(fresh115(fresh40(fresh115(is_a_theorem(implies(implies(X, X), implies(or(X, X), X))), true, implies(X, X), implies(or(X, X), X)), true, implies(implies(X, or(X, X)), equiv(or(X, X), X))), true, implies(X, or(X, X)), equiv(or(X, X), X)), true, or(X, X), X)
% 79.56/10.58 = { by axiom 43 (modus_ponens_2) }
% 79.56/10.58 fresh3(fresh115(fresh40(fresh40(is_a_theorem(implies(X, X)), true, implies(or(X, X), X)), true, implies(implies(X, or(X, X)), equiv(or(X, X), X))), true, implies(X, or(X, X)), equiv(or(X, X), X)), true, or(X, X), X)
% 79.56/10.58 = { by axiom 17 (modus_ponens_2) R->L }
% 79.56/10.58 fresh3(fresh115(fresh40(fresh40(fresh40(true, true, implies(X, X)), true, implies(or(X, X), X)), true, implies(implies(X, or(X, X)), equiv(or(X, X), X))), true, implies(X, or(X, X)), equiv(or(X, X), X)), true, or(X, X), X)
% 79.56/10.58 = { by axiom 24 (implies_1_1) R->L }
% 79.56/10.58 fresh3(fresh115(fresh40(fresh40(fresh40(fresh51(true, true, X, X), true, implies(X, X)), true, implies(or(X, X), X)), true, implies(implies(X, or(X, X)), equiv(or(X, X), X))), true, implies(X, or(X, X)), equiv(or(X, X), X)), true, or(X, X), X)
% 79.56/10.58 = { by axiom 4 (hilbert_implies_1) R->L }
% 79.56/10.58 fresh3(fresh115(fresh40(fresh40(fresh40(fresh51(implies_1, true, X, X), true, implies(X, X)), true, implies(or(X, X), X)), true, implies(implies(X, or(X, X)), equiv(or(X, X), X))), true, implies(X, or(X, X)), equiv(or(X, X), X)), true, or(X, X), X)
% 79.56/10.58 = { by axiom 36 (implies_1_1) }
% 79.56/10.58 fresh3(fresh115(fresh40(fresh40(fresh40(is_a_theorem(implies(X, implies(X, X))), true, implies(X, X)), true, implies(or(X, X), X)), true, implies(implies(X, or(X, X)), equiv(or(X, X), X))), true, implies(X, or(X, X)), equiv(or(X, X), X)), true, or(X, X), X)
% 79.56/10.58 = { by lemma 52 }
% 79.56/10.58 fresh3(fresh115(fresh40(fresh40(true, true, implies(or(X, X), X)), true, implies(implies(X, or(X, X)), equiv(or(X, X), X))), true, implies(X, or(X, X)), equiv(or(X, X), X)), true, or(X, X), X)
% 79.56/10.58 = { by axiom 17 (modus_ponens_2) }
% 79.56/10.58 fresh3(fresh115(fresh40(is_a_theorem(implies(or(X, X), X)), true, implies(implies(X, or(X, X)), equiv(or(X, X), X))), true, implies(X, or(X, X)), equiv(or(X, X), X)), true, or(X, X), X)
% 79.56/10.58 = { by lemma 53 }
% 79.56/10.58 fresh3(fresh115(true, true, implies(X, or(X, X)), equiv(or(X, X), X)), true, or(X, X), X)
% 79.56/10.58 = { by lemma 51 }
% 79.56/10.58 fresh3(true, true, or(X, X), X)
% 79.56/10.58 = { by axiom 35 (substitution_of_equivalents_2) }
% 79.56/10.58 X
% 79.56/10.58
% 79.56/10.58 Lemma 55: not(and(X, not(Y))) = implies(X, Y).
% 79.56/10.58 Proof:
% 79.56/10.58 not(and(X, not(Y)))
% 79.56/10.58 = { by axiom 29 (op_implies_and) R->L }
% 79.56/10.58 fresh29(op_implies_and, true, X, Y)
% 79.56/10.58 = { by axiom 13 (hilbert_op_implies_and) }
% 79.56/10.58 fresh29(true, true, X, Y)
% 79.56/10.58 = { by axiom 28 (op_implies_and) }
% 79.56/10.58 implies(X, Y)
% 79.56/10.58
% 79.56/10.58 Lemma 56: implies(not(X), Y) = or(X, Y).
% 79.56/10.58 Proof:
% 79.56/10.58 implies(not(X), Y)
% 79.56/10.58 = { by lemma 55 R->L }
% 79.56/10.58 not(and(not(X), not(Y)))
% 79.56/10.58 = { by axiom 39 (op_or) R->L }
% 79.56/10.58 fresh26(op_or, true, X, Y)
% 79.56/10.58 = { by axiom 11 (hilbert_op_or) }
% 79.56/10.58 fresh26(true, true, X, Y)
% 79.56/10.58 = { by axiom 30 (op_or) }
% 79.56/10.58 or(X, Y)
% 79.56/10.58
% 79.56/10.58 Lemma 57: or(and(X, not(Y)), Z) = implies(implies(X, Y), Z).
% 79.56/10.58 Proof:
% 79.56/10.58 or(and(X, not(Y)), Z)
% 79.56/10.58 = { by lemma 56 R->L }
% 79.56/10.58 implies(not(and(X, not(Y))), Z)
% 79.56/10.58 = { by lemma 55 }
% 79.56/10.58 implies(implies(X, Y), Z)
% 79.56/10.58
% 79.56/10.58 Lemma 58: is_a_theorem(implies(or(X, not(Y)), implies(Y, X))) = true.
% 79.56/10.58 Proof:
% 79.56/10.58 is_a_theorem(implies(or(X, not(Y)), implies(Y, X)))
% 79.56/10.58 = { by lemma 56 R->L }
% 79.56/10.58 is_a_theorem(implies(implies(not(X), not(Y)), implies(Y, X)))
% 79.56/10.58 = { by axiom 46 (modus_tollens_1) R->L }
% 79.56/10.58 fresh35(modus_tollens, true, Y, X)
% 79.56/10.58 = { by axiom 3 (hilbert_modus_tollens) }
% 79.56/10.58 fresh35(true, true, Y, X)
% 79.56/10.58 = { by axiom 26 (modus_tollens_1) }
% 79.56/10.58 true
% 79.56/10.58
% 79.56/10.58 Goal 1 (s1_0_axiom_m1): axiom_m1 = true.
% 79.56/10.58 Proof:
% 79.56/10.58 axiom_m1
% 79.56/10.58 = { by axiom 47 (axiom_m1) R->L }
% 79.56/10.58 fresh92(is_a_theorem(strict_implies(and(x8, y4), and(y4, x8))), true)
% 79.56/10.58 = { by axiom 42 (axiom_m1_1) R->L }
% 79.56/10.58 fresh92(fresh89(axiom_m1, true, x8, y4), true)
% 79.56/10.58 = { by lemma 54 R->L }
% 79.56/10.58 fresh92(fresh89(axiom_m1, true, x8, or(y4, y4)), true)
% 79.56/10.58 = { by axiom 42 (axiom_m1_1) }
% 79.56/10.58 fresh92(is_a_theorem(strict_implies(and(x8, or(y4, y4)), and(or(y4, y4), x8))), true)
% 79.56/10.58 = { by axiom 31 (op_strict_implies) R->L }
% 79.56/10.58 fresh92(is_a_theorem(fresh23(true, true, and(x8, or(y4, y4)), and(or(y4, y4), x8))), true)
% 79.56/10.58 = { by axiom 14 (s1_0_op_strict_implies) R->L }
% 79.56/10.58 fresh92(is_a_theorem(fresh23(op_strict_implies, true, and(x8, or(y4, y4)), and(or(y4, y4), x8))), true)
% 79.56/10.58 = { by axiom 32 (op_strict_implies) }
% 79.56/10.58 fresh92(is_a_theorem(necessarily(implies(and(x8, or(y4, y4)), and(or(y4, y4), x8)))), true)
% 79.56/10.58 = { by axiom 18 (necessitation_1) R->L }
% 79.56/10.58 fresh92(fresh34(true, true, implies(and(x8, or(y4, y4)), and(or(y4, y4), x8))), true)
% 79.56/10.58 = { by axiom 12 (km4b_necessitation) R->L }
% 79.56/10.58 fresh92(fresh34(necessitation, true, implies(and(x8, or(y4, y4)), and(or(y4, y4), x8))), true)
% 79.56/10.58 = { by axiom 27 (necessitation_1) }
% 79.56/10.58 fresh92(fresh33(is_a_theorem(implies(and(x8, or(y4, y4)), and(or(y4, y4), x8))), true, implies(and(x8, or(y4, y4)), and(or(y4, y4), x8))), true)
% 79.56/10.59 = { by lemma 54 R->L }
% 79.56/10.59 fresh92(fresh33(is_a_theorem(implies(and(x8, or(y4, y4)), and(or(y4, y4), or(x8, x8)))), true, implies(and(x8, or(y4, y4)), and(or(y4, y4), x8))), true)
% 79.56/10.59 = { by lemma 56 R->L }
% 79.56/10.59 fresh92(fresh33(is_a_theorem(implies(and(x8, or(y4, y4)), and(or(y4, y4), implies(not(x8), x8)))), true, implies(and(x8, or(y4, y4)), and(or(y4, y4), x8))), true)
% 79.56/10.59 = { by lemma 55 R->L }
% 79.56/10.59 fresh92(fresh33(is_a_theorem(implies(and(x8, or(y4, y4)), and(or(y4, y4), not(and(not(x8), not(x8)))))), true, implies(and(x8, or(y4, y4)), and(or(y4, y4), x8))), true)
% 79.56/10.59 = { by axiom 35 (substitution_of_equivalents_2) R->L }
% 79.56/10.59 fresh92(fresh33(is_a_theorem(implies(and(x8, or(y4, y4)), and(or(y4, y4), not(fresh3(true, true, not(x8), and(not(x8), not(x8))))))), true, implies(and(x8, or(y4, y4)), and(or(y4, y4), x8))), true)
% 79.56/10.59 = { by lemma 51 R->L }
% 80.12/10.59 fresh92(fresh33(is_a_theorem(implies(and(x8, or(y4, y4)), and(or(y4, y4), not(fresh3(fresh115(true, true, implies(and(not(x8), not(x8)), not(x8)), equiv(not(x8), and(not(x8), not(x8)))), true, not(x8), and(not(x8), not(x8))))))), true, implies(and(x8, or(y4, y4)), and(or(y4, y4), x8))), true)
% 80.12/10.59 = { by lemma 53 R->L }
% 80.12/10.59 fresh92(fresh33(is_a_theorem(implies(and(x8, or(y4, y4)), and(or(y4, y4), not(fresh3(fresh115(fresh40(is_a_theorem(implies(not(x8), and(not(x8), not(x8)))), true, implies(implies(and(not(x8), not(x8)), not(x8)), equiv(not(x8), and(not(x8), not(x8))))), true, implies(and(not(x8), not(x8)), not(x8)), equiv(not(x8), and(not(x8), not(x8)))), true, not(x8), and(not(x8), not(x8))))))), true, implies(and(x8, or(y4, y4)), and(or(y4, y4), x8))), true)
% 80.12/10.59 = { by axiom 17 (modus_ponens_2) R->L }
% 80.12/10.59 fresh92(fresh33(is_a_theorem(implies(and(x8, or(y4, y4)), and(or(y4, y4), not(fresh3(fresh115(fresh40(fresh40(true, true, implies(not(x8), and(not(x8), not(x8)))), true, implies(implies(and(not(x8), not(x8)), not(x8)), equiv(not(x8), and(not(x8), not(x8))))), true, implies(and(not(x8), not(x8)), not(x8)), equiv(not(x8), and(not(x8), not(x8)))), true, not(x8), and(not(x8), not(x8))))))), true, implies(and(x8, or(y4, y4)), and(or(y4, y4), x8))), true)
% 80.12/10.59 = { by axiom 22 (and_3_1) R->L }
% 80.12/10.59 fresh92(fresh33(is_a_theorem(implies(and(x8, or(y4, y4)), and(or(y4, y4), not(fresh3(fresh115(fresh40(fresh40(fresh103(true, true, not(x8), not(x8)), true, implies(not(x8), and(not(x8), not(x8)))), true, implies(implies(and(not(x8), not(x8)), not(x8)), equiv(not(x8), and(not(x8), not(x8))))), true, implies(and(not(x8), not(x8)), not(x8)), equiv(not(x8), and(not(x8), not(x8)))), true, not(x8), and(not(x8), not(x8))))))), true, implies(and(x8, or(y4, y4)), and(or(y4, y4), x8))), true)
% 80.12/10.59 = { by axiom 7 (hilbert_and_3) R->L }
% 80.12/10.59 fresh92(fresh33(is_a_theorem(implies(and(x8, or(y4, y4)), and(or(y4, y4), not(fresh3(fresh115(fresh40(fresh40(fresh103(and_3, true, not(x8), not(x8)), true, implies(not(x8), and(not(x8), not(x8)))), true, implies(implies(and(not(x8), not(x8)), not(x8)), equiv(not(x8), and(not(x8), not(x8))))), true, implies(and(not(x8), not(x8)), not(x8)), equiv(not(x8), and(not(x8), not(x8)))), true, not(x8), and(not(x8), not(x8))))))), true, implies(and(x8, or(y4, y4)), and(or(y4, y4), x8))), true)
% 80.12/10.59 = { by axiom 41 (and_3_1) }
% 80.12/10.59 fresh92(fresh33(is_a_theorem(implies(and(x8, or(y4, y4)), and(or(y4, y4), not(fresh3(fresh115(fresh40(fresh40(is_a_theorem(implies(not(x8), implies(not(x8), and(not(x8), not(x8))))), true, implies(not(x8), and(not(x8), not(x8)))), true, implies(implies(and(not(x8), not(x8)), not(x8)), equiv(not(x8), and(not(x8), not(x8))))), true, implies(and(not(x8), not(x8)), not(x8)), equiv(not(x8), and(not(x8), not(x8)))), true, not(x8), and(not(x8), not(x8))))))), true, implies(and(x8, or(y4, y4)), and(or(y4, y4), x8))), true)
% 80.12/10.59 = { by lemma 52 }
% 80.12/10.59 fresh92(fresh33(is_a_theorem(implies(and(x8, or(y4, y4)), and(or(y4, y4), not(fresh3(fresh115(fresh40(true, true, implies(implies(and(not(x8), not(x8)), not(x8)), equiv(not(x8), and(not(x8), not(x8))))), true, implies(and(not(x8), not(x8)), not(x8)), equiv(not(x8), and(not(x8), not(x8)))), true, not(x8), and(not(x8), not(x8))))))), true, implies(and(x8, or(y4, y4)), and(or(y4, y4), x8))), true)
% 80.12/10.59 = { by axiom 17 (modus_ponens_2) }
% 80.12/10.59 fresh92(fresh33(is_a_theorem(implies(and(x8, or(y4, y4)), and(or(y4, y4), not(fresh3(fresh115(is_a_theorem(implies(implies(and(not(x8), not(x8)), not(x8)), equiv(not(x8), and(not(x8), not(x8))))), true, implies(and(not(x8), not(x8)), not(x8)), equiv(not(x8), and(not(x8), not(x8)))), true, not(x8), and(not(x8), not(x8))))))), true, implies(and(x8, or(y4, y4)), and(or(y4, y4), x8))), true)
% 80.12/10.59 = { by axiom 43 (modus_ponens_2) }
% 80.12/10.59 fresh92(fresh33(is_a_theorem(implies(and(x8, or(y4, y4)), and(or(y4, y4), not(fresh3(fresh40(is_a_theorem(implies(and(not(x8), not(x8)), not(x8))), true, equiv(not(x8), and(not(x8), not(x8)))), true, not(x8), and(not(x8), not(x8))))))), true, implies(and(x8, or(y4, y4)), and(or(y4, y4), x8))), true)
% 80.12/10.59 = { by axiom 38 (and_1_1) R->L }
% 80.12/10.59 fresh92(fresh33(is_a_theorem(implies(and(x8, or(y4, y4)), and(or(y4, y4), not(fresh3(fresh40(fresh107(and_1, true, not(x8), not(x8)), true, equiv(not(x8), and(not(x8), not(x8)))), true, not(x8), and(not(x8), not(x8))))))), true, implies(and(x8, or(y4, y4)), and(or(y4, y4), x8))), true)
% 80.12/10.60 = { by axiom 6 (hilbert_and_1) }
% 80.12/10.60 fresh92(fresh33(is_a_theorem(implies(and(x8, or(y4, y4)), and(or(y4, y4), not(fresh3(fresh40(fresh107(true, true, not(x8), not(x8)), true, equiv(not(x8), and(not(x8), not(x8)))), true, not(x8), and(not(x8), not(x8))))))), true, implies(and(x8, or(y4, y4)), and(or(y4, y4), x8))), true)
% 80.12/10.60 = { by axiom 21 (and_1_1) }
% 80.12/10.60 fresh92(fresh33(is_a_theorem(implies(and(x8, or(y4, y4)), and(or(y4, y4), not(fresh3(fresh40(true, true, equiv(not(x8), and(not(x8), not(x8)))), true, not(x8), and(not(x8), not(x8))))))), true, implies(and(x8, or(y4, y4)), and(or(y4, y4), x8))), true)
% 80.12/10.60 = { by axiom 17 (modus_ponens_2) }
% 80.12/10.60 fresh92(fresh33(is_a_theorem(implies(and(x8, or(y4, y4)), and(or(y4, y4), not(fresh3(is_a_theorem(equiv(not(x8), and(not(x8), not(x8)))), true, not(x8), and(not(x8), not(x8))))))), true, implies(and(x8, or(y4, y4)), and(or(y4, y4), x8))), true)
% 80.12/10.60 = { by lemma 50 }
% 80.12/10.60 fresh92(fresh33(is_a_theorem(implies(and(x8, or(y4, y4)), and(or(y4, y4), not(not(x8))))), true, implies(and(x8, or(y4, y4)), and(or(y4, y4), x8))), true)
% 80.12/10.60 = { by axiom 17 (modus_ponens_2) R->L }
% 80.12/10.60 fresh92(fresh33(fresh40(true, true, implies(and(x8, or(y4, y4)), and(or(y4, y4), not(not(x8))))), true, implies(and(x8, or(y4, y4)), and(or(y4, y4), x8))), true)
% 80.12/10.60 = { by lemma 58 R->L }
% 80.12/10.60 fresh92(fresh33(fresh40(is_a_theorem(implies(or(and(not(y4), not(y4)), not(x8)), implies(x8, and(not(y4), not(y4))))), true, implies(and(x8, or(y4, y4)), and(or(y4, y4), not(not(x8))))), true, implies(and(x8, or(y4, y4)), and(or(y4, y4), x8))), true)
% 80.12/10.60 = { by lemma 55 R->L }
% 80.12/10.60 fresh92(fresh33(fresh40(is_a_theorem(implies(or(and(not(y4), not(y4)), not(x8)), not(and(x8, not(and(not(y4), not(y4))))))), true, implies(and(x8, or(y4, y4)), and(or(y4, y4), not(not(x8))))), true, implies(and(x8, or(y4, y4)), and(or(y4, y4), x8))), true)
% 80.12/10.60 = { by lemma 55 }
% 80.12/10.60 fresh92(fresh33(fresh40(is_a_theorem(implies(or(and(not(y4), not(y4)), not(x8)), not(and(x8, implies(not(y4), y4))))), true, implies(and(x8, or(y4, y4)), and(or(y4, y4), not(not(x8))))), true, implies(and(x8, or(y4, y4)), and(or(y4, y4), x8))), true)
% 80.12/10.60 = { by lemma 56 }
% 80.12/10.60 fresh92(fresh33(fresh40(is_a_theorem(implies(or(and(not(y4), not(y4)), not(x8)), not(and(x8, or(y4, y4))))), true, implies(and(x8, or(y4, y4)), and(or(y4, y4), not(not(x8))))), true, implies(and(x8, or(y4, y4)), and(or(y4, y4), x8))), true)
% 80.12/10.60 = { by lemma 57 }
% 80.12/10.60 fresh92(fresh33(fresh40(is_a_theorem(implies(implies(implies(not(y4), y4), not(x8)), not(and(x8, or(y4, y4))))), true, implies(and(x8, or(y4, y4)), and(or(y4, y4), not(not(x8))))), true, implies(and(x8, or(y4, y4)), and(or(y4, y4), x8))), true)
% 80.12/10.60 = { by lemma 56 }
% 80.12/10.60 fresh92(fresh33(fresh40(is_a_theorem(implies(implies(or(y4, y4), not(x8)), not(and(x8, or(y4, y4))))), true, implies(and(x8, or(y4, y4)), and(or(y4, y4), not(not(x8))))), true, implies(and(x8, or(y4, y4)), and(or(y4, y4), x8))), true)
% 80.12/10.60 = { by lemma 57 R->L }
% 80.12/10.60 fresh92(fresh33(fresh40(is_a_theorem(or(and(or(y4, y4), not(not(x8))), not(and(x8, or(y4, y4))))), true, implies(and(x8, or(y4, y4)), and(or(y4, y4), not(not(x8))))), true, implies(and(x8, or(y4, y4)), and(or(y4, y4), x8))), true)
% 80.12/10.60 = { by axiom 43 (modus_ponens_2) R->L }
% 80.12/10.60 fresh92(fresh33(fresh115(is_a_theorem(implies(or(and(or(y4, y4), not(not(x8))), not(and(x8, or(y4, y4)))), implies(and(x8, or(y4, y4)), and(or(y4, y4), not(not(x8)))))), true, or(and(or(y4, y4), not(not(x8))), not(and(x8, or(y4, y4)))), implies(and(x8, or(y4, y4)), and(or(y4, y4), not(not(x8))))), true, implies(and(x8, or(y4, y4)), and(or(y4, y4), x8))), true)
% 80.12/10.60 = { by lemma 58 }
% 80.12/10.60 fresh92(fresh33(fresh115(true, true, or(and(or(y4, y4), not(not(x8))), not(and(x8, or(y4, y4)))), implies(and(x8, or(y4, y4)), and(or(y4, y4), not(not(x8))))), true, implies(and(x8, or(y4, y4)), and(or(y4, y4), x8))), true)
% 80.12/10.60 = { by lemma 51 }
% 80.12/10.60 fresh92(fresh33(true, true, implies(and(x8, or(y4, y4)), and(or(y4, y4), x8))), true)
% 80.12/10.60 = { by axiom 19 (necessitation_1) }
% 80.12/10.60 fresh92(true, true)
% 80.12/10.60 = { by axiom 15 (axiom_m1) }
% 80.12/10.60 true
% 80.12/10.60 % SZS output end Proof
% 80.12/10.60
% 80.12/10.60 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------