TSTP Solution File: LCL458+1 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : LCL458+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 : n029.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 08:19:05 EDT 2023
% Result : Theorem 12.82s 2.27s
% Output : Proof 13.90s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : LCL458+1 : TPTP v8.1.2. Released v3.3.0.
% 0.00/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.12/0.34 % Computer : n029.cluster.edu
% 0.12/0.34 % Model : x86_64 x86_64
% 0.12/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.34 % Memory : 8042.1875MB
% 0.12/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.34 % CPULimit : 300
% 0.12/0.34 % WCLimit : 300
% 0.12/0.34 % DateTime : Thu Aug 24 18:29:53 EDT 2023
% 0.19/0.34 % CPUTime :
% 12.82/2.27 Command-line arguments: --lhs-weight 1 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 12.82/2.27
% 12.82/2.27 % SZS status Theorem
% 12.82/2.27
% 13.90/2.30 % SZS output start Proof
% 13.90/2.30 Take the following subset of the input axioms:
% 13.90/2.30 fof(equivalence_3, axiom, equivalence_3 <=> ![X, Y]: is_a_theorem(implies(implies(X, Y), implies(implies(Y, X), equiv(X, Y))))).
% 13.90/2.30 fof(hilbert_equivalence_3, axiom, equivalence_3).
% 13.90/2.30 fof(hilbert_implies_1, axiom, implies_1).
% 13.90/2.30 fof(hilbert_implies_2, axiom, implies_2).
% 13.90/2.30 fof(hilbert_implies_3, axiom, implies_3).
% 13.90/2.30 fof(hilbert_modus_ponens, axiom, modus_ponens).
% 13.90/2.30 fof(hilbert_modus_tollens, axiom, modus_tollens).
% 13.90/2.30 fof(hilbert_op_equiv, axiom, op_equiv).
% 13.90/2.30 fof(hilbert_op_implies_and, axiom, op_implies_and).
% 13.90/2.30 fof(hilbert_op_or, axiom, op_or).
% 13.90/2.30 fof(implies_1, axiom, implies_1 <=> ![X2, Y2]: is_a_theorem(implies(X2, implies(Y2, X2)))).
% 13.90/2.30 fof(implies_2, axiom, implies_2 <=> ![X2, Y2]: is_a_theorem(implies(implies(X2, implies(X2, Y2)), implies(X2, Y2)))).
% 13.90/2.30 fof(implies_3, axiom, implies_3 <=> ![Z, X2, Y2]: is_a_theorem(implies(implies(X2, Y2), implies(implies(Y2, Z), implies(X2, Z))))).
% 13.90/2.30 fof(modus_ponens, axiom, modus_ponens <=> ![X2, Y2]: ((is_a_theorem(X2) & is_a_theorem(implies(X2, Y2))) => is_a_theorem(Y2))).
% 13.90/2.30 fof(modus_tollens, axiom, modus_tollens <=> ![X2, Y2]: is_a_theorem(implies(implies(not(Y2), not(X2)), implies(X2, Y2)))).
% 13.90/2.30 fof(op_and, axiom, op_and => ![X2, Y2]: and(X2, Y2)=not(or(not(X2), not(Y2)))).
% 13.90/2.30 fof(op_equiv, axiom, op_equiv => ![X2, Y2]: equiv(X2, Y2)=and(implies(X2, Y2), implies(Y2, X2))).
% 13.90/2.30 fof(op_implies_and, axiom, op_implies_and => ![X2, Y2]: implies(X2, Y2)=not(and(X2, not(Y2)))).
% 13.90/2.30 fof(op_implies_or, axiom, op_implies_or => ![X2, Y2]: implies(X2, Y2)=or(not(X2), Y2)).
% 13.90/2.30 fof(op_or, axiom, op_or => ![X2, Y2]: or(X2, Y2)=not(and(not(X2), not(Y2)))).
% 13.90/2.30 fof(principia_op_and, axiom, op_and).
% 13.90/2.30 fof(principia_op_implies_or, axiom, op_implies_or).
% 13.90/2.30 fof(principia_r5, conjecture, r5).
% 13.90/2.30 fof(r5, axiom, r5 <=> ![P, Q, R]: is_a_theorem(implies(implies(Q, R), implies(or(P, Q), or(P, R))))).
% 13.90/2.30 fof(substitution_of_equivalents, axiom, substitution_of_equivalents <=> ![X2, Y2]: (is_a_theorem(equiv(X2, Y2)) => X2=Y2)).
% 13.90/2.30 fof(substitution_of_equivalents, axiom, substitution_of_equivalents).
% 13.90/2.30
% 13.90/2.30 Now clausify the problem and encode Horn clauses using encoding 3 of
% 13.90/2.30 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 13.90/2.30 We repeatedly replace C & s=t => u=v by the two clauses:
% 13.90/2.30 fresh(y, y, x1...xn) = u
% 13.90/2.30 C => fresh(s, t, x1...xn) = v
% 13.90/2.30 where fresh is a fresh function symbol and x1..xn are the free
% 13.90/2.30 variables of u and v.
% 13.90/2.30 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 13.90/2.30 input problem has no model of domain size 1).
% 13.90/2.30
% 13.90/2.30 The encoding turns the above axioms into the following unit equations and goals:
% 13.90/2.30
% 13.90/2.30 Axiom 1 (hilbert_modus_ponens): modus_ponens = true.
% 13.90/2.30 Axiom 2 (substitution_of_equivalents): substitution_of_equivalents = true.
% 13.90/2.30 Axiom 3 (hilbert_modus_tollens): modus_tollens = true.
% 13.90/2.30 Axiom 4 (hilbert_implies_1): implies_1 = true.
% 13.90/2.30 Axiom 5 (hilbert_implies_2): implies_2 = true.
% 13.90/2.30 Axiom 6 (hilbert_implies_3): implies_3 = true.
% 13.90/2.30 Axiom 7 (hilbert_equivalence_3): equivalence_3 = true.
% 13.90/2.30 Axiom 8 (hilbert_op_equiv): op_equiv = true.
% 13.90/2.30 Axiom 9 (hilbert_op_or): op_or = true.
% 13.90/2.30 Axiom 10 (principia_op_and): op_and = true.
% 13.90/2.30 Axiom 11 (hilbert_op_implies_and): op_implies_and = true.
% 13.90/2.30 Axiom 12 (principia_op_implies_or): op_implies_or = true.
% 13.90/2.30 Axiom 13 (r5): fresh5(X, X) = true.
% 13.90/2.30 Axiom 14 (modus_ponens_2): fresh60(X, X, Y) = true.
% 13.90/2.30 Axiom 15 (modus_ponens_2): fresh28(X, X, Y) = is_a_theorem(Y).
% 13.90/2.30 Axiom 16 (substitution_of_equivalents_2): fresh(X, X, Y, Z) = Z.
% 13.90/2.30 Axiom 17 (modus_ponens_2): fresh59(X, X, Y, Z) = fresh60(modus_ponens, true, Z).
% 13.90/2.30 Axiom 18 (equivalence_3_1): fresh41(X, X, Y, Z) = true.
% 13.90/2.30 Axiom 19 (implies_1_1): fresh39(X, X, Y, Z) = true.
% 13.90/2.30 Axiom 20 (implies_2_1): fresh37(X, X, Y, Z) = true.
% 13.90/2.30 Axiom 21 (modus_tollens_1): fresh25(X, X, Y, Z) = true.
% 13.90/2.30 Axiom 22 (op_and): fresh24(X, X, Y, Z) = and(Y, Z).
% 13.90/2.30 Axiom 23 (op_equiv): fresh23(X, X, Y, Z) = equiv(Y, Z).
% 13.90/2.30 Axiom 24 (op_implies_and): fresh22(X, X, Y, Z) = implies(Y, Z).
% 13.90/2.30 Axiom 25 (op_implies_and): fresh22(op_implies_and, true, X, Y) = not(and(X, not(Y))).
% 13.90/2.30 Axiom 26 (op_implies_or): fresh21(X, X, Y, Z) = implies(Y, Z).
% 13.90/2.30 Axiom 27 (op_implies_or): fresh21(op_implies_or, true, X, Y) = or(not(X), Y).
% 13.90/2.30 Axiom 28 (op_or): fresh20(X, X, Y, Z) = or(Y, Z).
% 13.90/2.30 Axiom 29 (substitution_of_equivalents_2): fresh2(X, X, Y, Z) = Y.
% 13.90/2.30 Axiom 30 (implies_1_1): fresh39(implies_1, true, X, Y) = is_a_theorem(implies(X, implies(Y, X))).
% 13.90/2.30 Axiom 31 (op_and): fresh24(op_and, true, X, Y) = not(or(not(X), not(Y))).
% 13.90/2.30 Axiom 32 (op_or): fresh20(op_or, true, X, Y) = not(and(not(X), not(Y))).
% 13.90/2.30 Axiom 33 (implies_3_1): fresh35(X, X, Y, Z, W) = true.
% 13.90/2.30 Axiom 34 (op_equiv): fresh23(op_equiv, true, X, Y) = and(implies(X, Y), implies(Y, X)).
% 13.90/2.30 Axiom 35 (substitution_of_equivalents_2): fresh2(substitution_of_equivalents, true, X, Y) = fresh(is_a_theorem(equiv(X, Y)), true, X, Y).
% 13.90/2.30 Axiom 36 (modus_ponens_2): fresh59(is_a_theorem(implies(X, Y)), true, X, Y) = fresh28(is_a_theorem(X), true, Y).
% 13.90/2.30 Axiom 37 (implies_2_1): fresh37(implies_2, true, X, Y) = is_a_theorem(implies(implies(X, implies(X, Y)), implies(X, Y))).
% 13.90/2.30 Axiom 38 (modus_tollens_1): fresh25(modus_tollens, true, X, Y) = is_a_theorem(implies(implies(not(Y), not(X)), implies(X, Y))).
% 13.90/2.30 Axiom 39 (equivalence_3_1): fresh41(equivalence_3, true, X, Y) = is_a_theorem(implies(implies(X, Y), implies(implies(Y, X), equiv(X, Y)))).
% 13.90/2.30 Axiom 40 (implies_3_1): fresh35(implies_3, true, X, Y, Z) = is_a_theorem(implies(implies(X, Y), implies(implies(Y, Z), implies(X, Z)))).
% 13.90/2.30 Axiom 41 (r5): fresh5(is_a_theorem(implies(implies(q, r), implies(or(p, q), or(p, r)))), true) = r5.
% 13.90/2.30
% 13.90/2.30 Lemma 42: fresh59(X, X, Y, Z) = true.
% 13.90/2.30 Proof:
% 13.90/2.30 fresh59(X, X, Y, Z)
% 13.90/2.30 = { by axiom 17 (modus_ponens_2) }
% 13.90/2.30 fresh60(modus_ponens, true, Z)
% 13.90/2.30 = { by axiom 1 (hilbert_modus_ponens) }
% 13.90/2.30 fresh60(true, true, Z)
% 13.90/2.30 = { by axiom 14 (modus_ponens_2) }
% 13.90/2.30 true
% 13.90/2.30
% 13.90/2.30 Lemma 43: fresh28(is_a_theorem(implies(X, implies(X, Y))), true, implies(X, Y)) = true.
% 13.90/2.30 Proof:
% 13.90/2.30 fresh28(is_a_theorem(implies(X, implies(X, Y))), true, implies(X, Y))
% 13.90/2.30 = { by axiom 36 (modus_ponens_2) R->L }
% 13.90/2.30 fresh59(is_a_theorem(implies(implies(X, implies(X, Y)), implies(X, Y))), true, implies(X, implies(X, Y)), implies(X, Y))
% 13.90/2.30 = { by axiom 37 (implies_2_1) R->L }
% 13.90/2.30 fresh59(fresh37(implies_2, true, X, Y), true, implies(X, implies(X, Y)), implies(X, Y))
% 13.90/2.30 = { by axiom 5 (hilbert_implies_2) }
% 13.90/2.30 fresh59(fresh37(true, true, X, Y), true, implies(X, implies(X, Y)), implies(X, Y))
% 13.90/2.30 = { by axiom 20 (implies_2_1) }
% 13.90/2.30 fresh59(true, true, implies(X, implies(X, Y)), implies(X, Y))
% 13.90/2.30 = { by lemma 42 }
% 13.90/2.30 true
% 13.90/2.30
% 13.90/2.30 Lemma 44: is_a_theorem(implies(implies(X, Y), implies(implies(Y, X), equiv(X, Y)))) = true.
% 13.90/2.30 Proof:
% 13.90/2.30 is_a_theorem(implies(implies(X, Y), implies(implies(Y, X), equiv(X, Y))))
% 13.90/2.30 = { by axiom 39 (equivalence_3_1) R->L }
% 13.90/2.30 fresh41(equivalence_3, true, X, Y)
% 13.90/2.30 = { by axiom 7 (hilbert_equivalence_3) }
% 13.90/2.30 fresh41(true, true, X, Y)
% 13.90/2.30 = { by axiom 18 (equivalence_3_1) }
% 13.90/2.31 true
% 13.90/2.31
% 13.90/2.31 Lemma 45: not(and(X, not(Y))) = implies(X, Y).
% 13.90/2.31 Proof:
% 13.90/2.31 not(and(X, not(Y)))
% 13.90/2.31 = { by axiom 25 (op_implies_and) R->L }
% 13.90/2.31 fresh22(op_implies_and, true, X, Y)
% 13.90/2.31 = { by axiom 11 (hilbert_op_implies_and) }
% 13.90/2.31 fresh22(true, true, X, Y)
% 13.90/2.31 = { by axiom 24 (op_implies_and) }
% 13.90/2.31 implies(X, Y)
% 13.90/2.31
% 13.90/2.31 Lemma 46: implies(not(X), Y) = or(X, Y).
% 13.90/2.31 Proof:
% 13.90/2.31 implies(not(X), Y)
% 13.90/2.31 = { by lemma 45 R->L }
% 13.90/2.31 not(and(not(X), not(Y)))
% 13.90/2.31 = { by axiom 32 (op_or) R->L }
% 13.90/2.31 fresh20(op_or, true, X, Y)
% 13.90/2.31 = { by axiom 9 (hilbert_op_or) }
% 13.90/2.31 fresh20(true, true, X, Y)
% 13.90/2.31 = { by axiom 28 (op_or) }
% 13.90/2.31 or(X, Y)
% 13.90/2.31
% 13.90/2.31 Lemma 47: fresh(is_a_theorem(equiv(X, Y)), true, X, Y) = X.
% 13.90/2.31 Proof:
% 13.90/2.31 fresh(is_a_theorem(equiv(X, Y)), true, X, Y)
% 13.90/2.31 = { by axiom 35 (substitution_of_equivalents_2) R->L }
% 13.90/2.31 fresh2(substitution_of_equivalents, true, X, Y)
% 13.90/2.31 = { by axiom 2 (substitution_of_equivalents) }
% 13.90/2.31 fresh2(true, true, X, Y)
% 13.90/2.31 = { by axiom 29 (substitution_of_equivalents_2) }
% 13.90/2.31 X
% 13.90/2.31
% 13.90/2.31 Lemma 48: is_a_theorem(implies(or(X, not(Y)), implies(Y, X))) = true.
% 13.90/2.31 Proof:
% 13.90/2.31 is_a_theorem(implies(or(X, not(Y)), implies(Y, X)))
% 13.90/2.31 = { by lemma 46 R->L }
% 13.90/2.31 is_a_theorem(implies(implies(not(X), not(Y)), implies(Y, X)))
% 13.90/2.31 = { by axiom 38 (modus_tollens_1) R->L }
% 13.90/2.31 fresh25(modus_tollens, true, Y, X)
% 13.90/2.31 = { by axiom 3 (hilbert_modus_tollens) }
% 13.90/2.31 fresh25(true, true, Y, X)
% 13.90/2.31 = { by axiom 21 (modus_tollens_1) }
% 13.90/2.31 true
% 13.90/2.31
% 13.90/2.31 Lemma 49: or(not(X), Y) = implies(X, Y).
% 13.90/2.31 Proof:
% 13.90/2.31 or(not(X), Y)
% 13.90/2.31 = { by axiom 27 (op_implies_or) R->L }
% 13.90/2.31 fresh21(op_implies_or, true, X, Y)
% 13.90/2.31 = { by axiom 12 (principia_op_implies_or) }
% 13.90/2.31 fresh21(true, true, X, Y)
% 13.90/2.31 = { by axiom 26 (op_implies_or) }
% 13.90/2.31 implies(X, Y)
% 13.90/2.31
% 13.90/2.31 Lemma 50: not(implies(X, not(Y))) = and(X, Y).
% 13.90/2.31 Proof:
% 13.90/2.31 not(implies(X, not(Y)))
% 13.90/2.31 = { by lemma 49 R->L }
% 13.90/2.31 not(or(not(X), not(Y)))
% 13.90/2.31 = { by axiom 31 (op_and) R->L }
% 13.90/2.31 fresh24(op_and, true, X, Y)
% 13.90/2.31 = { by axiom 10 (principia_op_and) }
% 13.90/2.31 fresh24(true, true, X, Y)
% 13.90/2.31 = { by axiom 22 (op_and) }
% 13.90/2.31 and(X, Y)
% 13.90/2.31
% 13.90/2.31 Lemma 51: implies(implies(X, not(Y)), Z) = or(and(X, Y), Z).
% 13.90/2.31 Proof:
% 13.90/2.31 implies(implies(X, not(Y)), Z)
% 13.90/2.31 = { by lemma 49 R->L }
% 13.90/2.31 or(not(implies(X, not(Y))), Z)
% 13.90/2.31 = { by lemma 50 }
% 13.90/2.31 or(and(X, Y), Z)
% 13.90/2.31
% 13.90/2.31 Lemma 52: or(X, not(Y)) = implies(Y, X).
% 13.90/2.31 Proof:
% 13.90/2.31 or(X, not(Y))
% 13.90/2.31 = { by lemma 47 R->L }
% 13.90/2.31 fresh(is_a_theorem(equiv(or(X, not(Y)), implies(Y, X))), true, or(X, not(Y)), implies(Y, X))
% 13.90/2.31 = { by axiom 15 (modus_ponens_2) R->L }
% 13.90/2.31 fresh(fresh28(true, true, equiv(or(X, not(Y)), implies(Y, X))), true, or(X, not(Y)), implies(Y, X))
% 13.90/2.31 = { by lemma 48 R->L }
% 13.90/2.31 fresh(fresh28(is_a_theorem(implies(or(not(Y), not(not(X))), implies(not(X), not(Y)))), true, equiv(or(X, not(Y)), implies(Y, X))), true, or(X, not(Y)), implies(Y, X))
% 13.90/2.31 = { by lemma 49 }
% 13.90/2.31 fresh(fresh28(is_a_theorem(implies(implies(Y, not(not(X))), implies(not(X), not(Y)))), true, equiv(or(X, not(Y)), implies(Y, X))), true, or(X, not(Y)), implies(Y, X))
% 13.90/2.31 = { by lemma 51 }
% 13.90/2.31 fresh(fresh28(is_a_theorem(or(and(Y, not(X)), implies(not(X), not(Y)))), true, equiv(or(X, not(Y)), implies(Y, X))), true, or(X, not(Y)), implies(Y, X))
% 13.90/2.31 = { by lemma 46 }
% 13.90/2.31 fresh(fresh28(is_a_theorem(or(and(Y, not(X)), or(X, not(Y)))), true, equiv(or(X, not(Y)), implies(Y, X))), true, or(X, not(Y)), implies(Y, X))
% 13.90/2.31 = { by lemma 46 R->L }
% 13.90/2.31 fresh(fresh28(is_a_theorem(implies(not(and(Y, not(X))), or(X, not(Y)))), true, equiv(or(X, not(Y)), implies(Y, X))), true, or(X, not(Y)), implies(Y, X))
% 13.90/2.31 = { by lemma 45 }
% 13.90/2.31 fresh(fresh28(is_a_theorem(implies(implies(Y, X), or(X, not(Y)))), true, equiv(or(X, not(Y)), implies(Y, X))), true, or(X, not(Y)), implies(Y, X))
% 13.90/2.31 = { by axiom 36 (modus_ponens_2) R->L }
% 13.90/2.31 fresh(fresh59(is_a_theorem(implies(implies(implies(Y, X), or(X, not(Y))), equiv(or(X, not(Y)), implies(Y, X)))), true, implies(implies(Y, X), or(X, not(Y))), equiv(or(X, not(Y)), implies(Y, X))), true, or(X, not(Y)), implies(Y, X))
% 13.90/2.31 = { by axiom 15 (modus_ponens_2) R->L }
% 13.90/2.31 fresh(fresh59(fresh28(true, true, implies(implies(implies(Y, X), or(X, not(Y))), equiv(or(X, not(Y)), implies(Y, X)))), true, implies(implies(Y, X), or(X, not(Y))), equiv(or(X, not(Y)), implies(Y, X))), true, or(X, not(Y)), implies(Y, X))
% 13.90/2.31 = { by lemma 48 R->L }
% 13.90/2.31 fresh(fresh59(fresh28(is_a_theorem(implies(or(X, not(Y)), implies(Y, X))), true, implies(implies(implies(Y, X), or(X, not(Y))), equiv(or(X, not(Y)), implies(Y, X)))), true, implies(implies(Y, X), or(X, not(Y))), equiv(or(X, not(Y)), implies(Y, X))), true, or(X, not(Y)), implies(Y, X))
% 13.90/2.31 = { by axiom 36 (modus_ponens_2) R->L }
% 13.90/2.31 fresh(fresh59(fresh59(is_a_theorem(implies(implies(or(X, not(Y)), implies(Y, X)), implies(implies(implies(Y, X), or(X, not(Y))), equiv(or(X, not(Y)), implies(Y, X))))), true, implies(or(X, not(Y)), implies(Y, X)), implies(implies(implies(Y, X), or(X, not(Y))), equiv(or(X, not(Y)), implies(Y, X)))), true, implies(implies(Y, X), or(X, not(Y))), equiv(or(X, not(Y)), implies(Y, X))), true, or(X, not(Y)), implies(Y, X))
% 13.90/2.31 = { by lemma 44 }
% 13.90/2.31 fresh(fresh59(fresh59(true, true, implies(or(X, not(Y)), implies(Y, X)), implies(implies(implies(Y, X), or(X, not(Y))), equiv(or(X, not(Y)), implies(Y, X)))), true, implies(implies(Y, X), or(X, not(Y))), equiv(or(X, not(Y)), implies(Y, X))), true, or(X, not(Y)), implies(Y, X))
% 13.90/2.31 = { by lemma 42 }
% 13.90/2.31 fresh(fresh59(true, true, implies(implies(Y, X), or(X, not(Y))), equiv(or(X, not(Y)), implies(Y, X))), true, or(X, not(Y)), implies(Y, X))
% 13.90/2.31 = { by lemma 42 }
% 13.90/2.31 fresh(true, true, or(X, not(Y)), implies(Y, X))
% 13.90/2.31 = { by axiom 16 (substitution_of_equivalents_2) }
% 13.90/2.31 implies(Y, X)
% 13.90/2.31
% 13.90/2.31 Lemma 53: not(or(X, not(Y))) = and(not(X), Y).
% 13.90/2.31 Proof:
% 13.90/2.31 not(or(X, not(Y)))
% 13.90/2.31 = { by lemma 46 R->L }
% 13.90/2.31 not(implies(not(X), not(Y)))
% 13.90/2.31 = { by lemma 50 }
% 13.90/2.31 and(not(X), Y)
% 13.90/2.31
% 13.90/2.31 Lemma 54: and(implies(X, Y), implies(Y, X)) = equiv(X, Y).
% 13.90/2.31 Proof:
% 13.90/2.31 and(implies(X, Y), implies(Y, X))
% 13.90/2.31 = { by axiom 34 (op_equiv) R->L }
% 13.90/2.31 fresh23(op_equiv, true, X, Y)
% 13.90/2.31 = { by axiom 8 (hilbert_op_equiv) }
% 13.90/2.31 fresh23(true, true, X, Y)
% 13.90/2.31 = { by axiom 23 (op_equiv) }
% 13.90/2.31 equiv(X, Y)
% 13.90/2.31
% 13.90/2.31 Lemma 55: or(Y, X) = or(X, Y).
% 13.90/2.31 Proof:
% 13.90/2.31 or(Y, X)
% 13.90/2.31 = { by axiom 16 (substitution_of_equivalents_2) R->L }
% 13.90/2.31 fresh(true, true, or(X, Y), or(Y, X))
% 13.90/2.31 = { by lemma 42 R->L }
% 13.90/2.31 fresh(fresh59(true, true, implies(not(or(Y, X)), not(or(Y, X))), equiv(not(or(Y, X)), not(or(Y, X)))), true, or(X, Y), or(Y, X))
% 13.90/2.31 = { by lemma 43 R->L }
% 13.90/2.31 fresh(fresh59(fresh28(is_a_theorem(implies(implies(not(or(Y, X)), not(or(Y, X))), implies(implies(not(or(Y, X)), not(or(Y, X))), equiv(not(or(Y, X)), not(or(Y, X)))))), true, implies(implies(not(or(Y, X)), not(or(Y, X))), equiv(not(or(Y, X)), not(or(Y, X))))), true, implies(not(or(Y, X)), not(or(Y, X))), equiv(not(or(Y, X)), not(or(Y, X)))), true, or(X, Y), or(Y, X))
% 13.90/2.31 = { by lemma 44 }
% 13.90/2.31 fresh(fresh59(fresh28(true, true, implies(implies(not(or(Y, X)), not(or(Y, X))), equiv(not(or(Y, X)), not(or(Y, X))))), true, implies(not(or(Y, X)), not(or(Y, X))), equiv(not(or(Y, X)), not(or(Y, X)))), true, or(X, Y), or(Y, X))
% 13.90/2.31 = { by axiom 15 (modus_ponens_2) }
% 13.90/2.31 fresh(fresh59(is_a_theorem(implies(implies(not(or(Y, X)), not(or(Y, X))), equiv(not(or(Y, X)), not(or(Y, X))))), true, implies(not(or(Y, X)), not(or(Y, X))), equiv(not(or(Y, X)), not(or(Y, X)))), true, or(X, Y), or(Y, X))
% 13.90/2.31 = { by axiom 36 (modus_ponens_2) }
% 13.90/2.31 fresh(fresh28(is_a_theorem(implies(not(or(Y, X)), not(or(Y, X)))), true, equiv(not(or(Y, X)), not(or(Y, X)))), true, or(X, Y), or(Y, X))
% 13.90/2.31 = { by axiom 15 (modus_ponens_2) R->L }
% 13.90/2.31 fresh(fresh28(fresh28(true, true, implies(not(or(Y, X)), not(or(Y, X)))), true, equiv(not(or(Y, X)), not(or(Y, X)))), true, or(X, Y), or(Y, X))
% 13.90/2.31 = { by axiom 19 (implies_1_1) R->L }
% 13.90/2.31 fresh(fresh28(fresh28(fresh39(true, true, not(or(Y, X)), not(or(Y, X))), true, implies(not(or(Y, X)), not(or(Y, X)))), true, equiv(not(or(Y, X)), not(or(Y, X)))), true, or(X, Y), or(Y, X))
% 13.90/2.31 = { by axiom 4 (hilbert_implies_1) R->L }
% 13.90/2.31 fresh(fresh28(fresh28(fresh39(implies_1, true, not(or(Y, X)), not(or(Y, X))), true, implies(not(or(Y, X)), not(or(Y, X)))), true, equiv(not(or(Y, X)), not(or(Y, X)))), true, or(X, Y), or(Y, X))
% 13.90/2.31 = { by axiom 30 (implies_1_1) }
% 13.90/2.31 fresh(fresh28(fresh28(is_a_theorem(implies(not(or(Y, X)), implies(not(or(Y, X)), not(or(Y, X))))), true, implies(not(or(Y, X)), not(or(Y, X)))), true, equiv(not(or(Y, X)), not(or(Y, X)))), true, or(X, Y), or(Y, X))
% 13.90/2.31 = { by lemma 43 }
% 13.90/2.31 fresh(fresh28(true, true, equiv(not(or(Y, X)), not(or(Y, X)))), true, or(X, Y), or(Y, X))
% 13.90/2.31 = { by axiom 15 (modus_ponens_2) }
% 13.90/2.31 fresh(is_a_theorem(equiv(not(or(Y, X)), not(or(Y, X)))), true, or(X, Y), or(Y, X))
% 13.90/2.31 = { by lemma 46 R->L }
% 13.90/2.31 fresh(is_a_theorem(equiv(not(or(Y, X)), not(implies(not(Y), X)))), true, or(X, Y), or(Y, X))
% 13.90/2.31 = { by lemma 52 R->L }
% 13.90/2.31 fresh(is_a_theorem(equiv(not(or(Y, X)), not(or(X, not(not(Y)))))), true, or(X, Y), or(Y, X))
% 13.90/2.31 = { by lemma 53 }
% 13.90/2.31 fresh(is_a_theorem(equiv(not(or(Y, X)), and(not(X), not(Y)))), true, or(X, Y), or(Y, X))
% 13.90/2.31 = { by lemma 54 R->L }
% 13.90/2.31 fresh(is_a_theorem(and(implies(not(or(Y, X)), and(not(X), not(Y))), implies(and(not(X), not(Y)), not(or(Y, X))))), true, or(X, Y), or(Y, X))
% 13.90/2.31 = { by lemma 45 R->L }
% 13.90/2.31 fresh(is_a_theorem(and(not(and(not(or(Y, X)), not(and(not(X), not(Y))))), implies(and(not(X), not(Y)), not(or(Y, X))))), true, or(X, Y), or(Y, X))
% 13.90/2.31 = { by lemma 45 }
% 13.90/2.31 fresh(is_a_theorem(and(not(and(not(or(Y, X)), implies(not(X), Y))), implies(and(not(X), not(Y)), not(or(Y, X))))), true, or(X, Y), or(Y, X))
% 13.90/2.31 = { by lemma 46 }
% 13.90/2.31 fresh(is_a_theorem(and(not(and(not(or(Y, X)), or(X, Y))), implies(and(not(X), not(Y)), not(or(Y, X))))), true, or(X, Y), or(Y, X))
% 13.90/2.31 = { by lemma 53 R->L }
% 13.90/2.31 fresh(is_a_theorem(not(or(and(not(or(Y, X)), or(X, Y)), not(implies(and(not(X), not(Y)), not(or(Y, X))))))), true, or(X, Y), or(Y, X))
% 13.90/2.31 = { by lemma 51 R->L }
% 13.90/2.31 fresh(is_a_theorem(not(implies(implies(not(or(Y, X)), not(or(X, Y))), not(implies(and(not(X), not(Y)), not(or(Y, X))))))), true, or(X, Y), or(Y, X))
% 13.90/2.31 = { by lemma 50 }
% 13.90/2.31 fresh(is_a_theorem(and(implies(not(or(Y, X)), not(or(X, Y))), implies(and(not(X), not(Y)), not(or(Y, X))))), true, or(X, Y), or(Y, X))
% 13.90/2.32 = { by lemma 49 R->L }
% 13.90/2.32 fresh(is_a_theorem(and(implies(not(or(Y, X)), not(or(X, Y))), or(not(and(not(X), not(Y))), not(or(Y, X))))), true, or(X, Y), or(Y, X))
% 13.90/2.32 = { by lemma 45 }
% 13.90/2.32 fresh(is_a_theorem(and(implies(not(or(Y, X)), not(or(X, Y))), or(implies(not(X), Y), not(or(Y, X))))), true, or(X, Y), or(Y, X))
% 13.90/2.32 = { by lemma 46 }
% 13.90/2.32 fresh(is_a_theorem(and(implies(not(or(Y, X)), not(or(X, Y))), or(or(X, Y), not(or(Y, X))))), true, or(X, Y), or(Y, X))
% 13.90/2.32 = { by lemma 46 R->L }
% 13.90/2.32 fresh(is_a_theorem(and(implies(not(or(Y, X)), not(or(X, Y))), implies(not(or(X, Y)), not(or(Y, X))))), true, or(X, Y), or(Y, X))
% 13.90/2.32 = { by lemma 46 }
% 13.90/2.32 fresh(is_a_theorem(and(or(or(Y, X), not(or(X, Y))), implies(not(or(X, Y)), not(or(Y, X))))), true, or(X, Y), or(Y, X))
% 13.90/2.32 = { by lemma 50 R->L }
% 13.90/2.32 fresh(is_a_theorem(not(implies(or(or(Y, X), not(or(X, Y))), not(implies(not(or(X, Y)), not(or(Y, X))))))), true, or(X, Y), or(Y, X))
% 13.90/2.32 = { by lemma 50 }
% 13.90/2.32 fresh(is_a_theorem(not(implies(or(or(Y, X), not(or(X, Y))), and(not(or(X, Y)), or(Y, X))))), true, or(X, Y), or(Y, X))
% 13.90/2.32 = { by lemma 52 }
% 13.90/2.32 fresh(is_a_theorem(not(implies(implies(or(X, Y), or(Y, X)), and(not(or(X, Y)), or(Y, X))))), true, or(X, Y), or(Y, X))
% 13.90/2.32 = { by lemma 53 R->L }
% 13.90/2.32 fresh(is_a_theorem(not(implies(implies(or(X, Y), or(Y, X)), not(or(or(X, Y), not(or(Y, X))))))), true, or(X, Y), or(Y, X))
% 13.90/2.32 = { by lemma 50 }
% 13.90/2.32 fresh(is_a_theorem(and(implies(or(X, Y), or(Y, X)), or(or(X, Y), not(or(Y, X))))), true, or(X, Y), or(Y, X))
% 13.90/2.32 = { by lemma 52 }
% 13.90/2.32 fresh(is_a_theorem(and(implies(or(X, Y), or(Y, X)), implies(or(Y, X), or(X, Y)))), true, or(X, Y), or(Y, X))
% 13.90/2.32 = { by lemma 54 }
% 13.90/2.32 fresh(is_a_theorem(equiv(or(X, Y), or(Y, X))), true, or(X, Y), or(Y, X))
% 13.90/2.32 = { by lemma 47 }
% 13.90/2.32 or(X, Y)
% 13.90/2.32
% 13.90/2.32 Goal 1 (principia_r5): r5 = true.
% 13.90/2.32 Proof:
% 13.90/2.32 r5
% 13.90/2.32 = { by axiom 41 (r5) R->L }
% 13.90/2.32 fresh5(is_a_theorem(implies(implies(q, r), implies(or(p, q), or(p, r)))), true)
% 13.90/2.32 = { by lemma 55 }
% 13.90/2.32 fresh5(is_a_theorem(implies(implies(q, r), implies(or(p, q), or(r, p)))), true)
% 13.90/2.32 = { by lemma 55 }
% 13.90/2.32 fresh5(is_a_theorem(implies(implies(q, r), implies(or(q, p), or(r, p)))), true)
% 13.90/2.32 = { by lemma 52 R->L }
% 13.90/2.32 fresh5(is_a_theorem(implies(or(r, not(q)), implies(or(q, p), or(r, p)))), true)
% 13.90/2.32 = { by lemma 46 R->L }
% 13.90/2.32 fresh5(is_a_theorem(implies(or(r, not(q)), implies(implies(not(q), p), or(r, p)))), true)
% 13.90/2.32 = { by lemma 46 R->L }
% 13.90/2.32 fresh5(is_a_theorem(implies(or(r, not(q)), implies(implies(not(q), p), implies(not(r), p)))), true)
% 13.90/2.32 = { by lemma 46 R->L }
% 13.90/2.32 fresh5(is_a_theorem(implies(implies(not(r), not(q)), implies(implies(not(q), p), implies(not(r), p)))), true)
% 13.90/2.32 = { by axiom 40 (implies_3_1) R->L }
% 13.90/2.32 fresh5(fresh35(implies_3, true, not(r), not(q), p), true)
% 13.90/2.32 = { by axiom 6 (hilbert_implies_3) }
% 13.90/2.32 fresh5(fresh35(true, true, not(r), not(q), p), true)
% 13.90/2.32 = { by axiom 33 (implies_3_1) }
% 13.90/2.32 fresh5(true, true)
% 13.90/2.32 = { by axiom 13 (r5) }
% 13.90/2.32 true
% 13.90/2.32 % SZS output end Proof
% 13.90/2.32
% 13.90/2.32 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------