TSTP Solution File: LCL488+1 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : LCL488+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 : n013.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 08:19:11 EDT 2023
% Result : Theorem 0.19s 0.47s
% Output : Proof 0.19s
% Verified :
% SZS Type : -
% Comments :
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%----WARNING: Could not form TPTP format derivation
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%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : LCL488+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 : n013.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 19:36:02 EDT 2023
% 0.12/0.34 % CPUTime :
% 0.19/0.47 Command-line arguments: --flip-ordering --lhs-weight 1 --depth-weight 60 --distributivity-heuristic
% 0.19/0.47
% 0.19/0.47 % SZS status Theorem
% 0.19/0.47
% 0.19/0.47 % SZS output start Proof
% 0.19/0.47 Take the following subset of the input axioms:
% 0.19/0.47 fof(and_2, axiom, and_2 <=> ![X, Y]: is_a_theorem(implies(and(X, Y), Y))).
% 0.19/0.47 fof(hilbert_and_2, conjecture, and_2).
% 0.19/0.47 fof(hilbert_op_implies_and, axiom, op_implies_and).
% 0.19/0.47 fof(hilbert_op_or, axiom, op_or).
% 0.19/0.47 fof(modus_ponens, axiom, modus_ponens <=> ![X2, Y2]: ((is_a_theorem(X2) & is_a_theorem(implies(X2, Y2))) => is_a_theorem(Y2))).
% 0.19/0.47 fof(op_and, axiom, op_and => ![X2, Y2]: and(X2, Y2)=not(or(not(X2), not(Y2)))).
% 0.19/0.47 fof(op_implies_and, axiom, op_implies_and => ![X2, Y2]: implies(X2, Y2)=not(and(X2, not(Y2)))).
% 0.19/0.47 fof(op_implies_or, axiom, op_implies_or => ![X2, Y2]: implies(X2, Y2)=or(not(X2), Y2)).
% 0.19/0.47 fof(op_or, axiom, op_or => ![X2, Y2]: or(X2, Y2)=not(and(not(X2), not(Y2)))).
% 0.19/0.47 fof(principia_modus_ponens, axiom, modus_ponens).
% 0.19/0.47 fof(principia_op_and, axiom, op_and).
% 0.19/0.47 fof(principia_op_implies_or, axiom, op_implies_or).
% 0.19/0.47 fof(principia_r2, axiom, r2).
% 0.19/0.47 fof(principia_r3, axiom, r3).
% 0.19/0.47 fof(r2, axiom, r2 <=> ![P, Q]: is_a_theorem(implies(Q, or(P, Q)))).
% 0.19/0.47 fof(r3, axiom, r3 <=> ![P2, Q2]: is_a_theorem(implies(or(P2, Q2), or(Q2, P2)))).
% 0.19/0.47
% 0.19/0.47 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.19/0.47 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.19/0.47 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.19/0.47 fresh(y, y, x1...xn) = u
% 0.19/0.47 C => fresh(s, t, x1...xn) = v
% 0.19/0.47 where fresh is a fresh function symbol and x1..xn are the free
% 0.19/0.47 variables of u and v.
% 0.19/0.47 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.19/0.47 input problem has no model of domain size 1).
% 0.19/0.47
% 0.19/0.47 The encoding turns the above axioms into the following unit equations and goals:
% 0.19/0.47
% 0.19/0.47 Axiom 1 (principia_modus_ponens): modus_ponens = true.
% 0.19/0.47 Axiom 2 (principia_r2): r2 = true.
% 0.19/0.47 Axiom 3 (principia_r3): r3 = true.
% 0.19/0.47 Axiom 4 (hilbert_op_or): op_or = true.
% 0.19/0.47 Axiom 5 (principia_op_and): op_and = true.
% 0.19/0.47 Axiom 6 (hilbert_op_implies_and): op_implies_and = true.
% 0.19/0.47 Axiom 7 (principia_op_implies_or): op_implies_or = true.
% 0.19/0.47 Axiom 8 (and_2): fresh56(X, X) = true.
% 0.19/0.47 Axiom 9 (modus_ponens_2): fresh60(X, X, Y) = true.
% 0.19/0.47 Axiom 10 (modus_ponens_2): fresh28(X, X, Y) = is_a_theorem(Y).
% 0.19/0.47 Axiom 11 (modus_ponens_2): fresh59(X, X, Y, Z) = fresh60(modus_ponens, true, Z).
% 0.19/0.47 Axiom 12 (op_and): fresh24(X, X, Y, Z) = and(Y, Z).
% 0.19/0.47 Axiom 13 (op_implies_and): fresh22(X, X, Y, Z) = implies(Y, Z).
% 0.19/0.47 Axiom 14 (op_implies_and): fresh22(op_implies_and, true, X, Y) = not(and(X, not(Y))).
% 0.19/0.47 Axiom 15 (op_implies_or): fresh21(X, X, Y, Z) = implies(Y, Z).
% 0.19/0.47 Axiom 16 (op_implies_or): fresh21(op_implies_or, true, X, Y) = or(not(X), Y).
% 0.19/0.47 Axiom 17 (op_or): fresh20(X, X, Y, Z) = or(Y, Z).
% 0.19/0.47 Axiom 18 (r2_1): fresh10(X, X, Y, Z) = true.
% 0.19/0.47 Axiom 19 (r3_1): fresh8(X, X, Y, Z) = true.
% 0.19/0.47 Axiom 20 (r2_1): fresh10(r2, true, X, Y) = is_a_theorem(implies(Y, or(X, Y))).
% 0.19/0.47 Axiom 21 (op_and): fresh24(op_and, true, X, Y) = not(or(not(X), not(Y))).
% 0.19/0.47 Axiom 22 (op_or): fresh20(op_or, true, X, Y) = not(and(not(X), not(Y))).
% 0.19/0.47 Axiom 23 (r3_1): fresh8(r3, true, X, Y) = is_a_theorem(implies(or(X, Y), or(Y, X))).
% 0.19/0.47 Axiom 24 (modus_ponens_2): fresh59(is_a_theorem(implies(X, Y)), true, X, Y) = fresh28(is_a_theorem(X), true, Y).
% 0.19/0.47 Axiom 25 (and_2): fresh56(is_a_theorem(implies(and(x8, y8), y8)), true) = and_2.
% 0.19/0.47
% 0.19/0.47 Lemma 26: implies(not(X), Y) = or(X, Y).
% 0.19/0.47 Proof:
% 0.19/0.47 implies(not(X), Y)
% 0.19/0.47 = { by axiom 13 (op_implies_and) R->L }
% 0.19/0.47 fresh22(true, true, not(X), Y)
% 0.19/0.47 = { by axiom 6 (hilbert_op_implies_and) R->L }
% 0.19/0.47 fresh22(op_implies_and, true, not(X), Y)
% 0.19/0.47 = { by axiom 14 (op_implies_and) }
% 0.19/0.47 not(and(not(X), not(Y)))
% 0.19/0.47 = { by axiom 22 (op_or) R->L }
% 0.19/0.47 fresh20(op_or, true, X, Y)
% 0.19/0.47 = { by axiom 4 (hilbert_op_or) }
% 0.19/0.47 fresh20(true, true, X, Y)
% 0.19/0.47 = { by axiom 17 (op_or) }
% 0.19/0.47 or(X, Y)
% 0.19/0.47
% 0.19/0.47 Lemma 27: or(not(X), Y) = implies(X, Y).
% 0.19/0.47 Proof:
% 0.19/0.47 or(not(X), Y)
% 0.19/0.47 = { by axiom 16 (op_implies_or) R->L }
% 0.19/0.47 fresh21(op_implies_or, true, X, Y)
% 0.19/0.47 = { by axiom 7 (principia_op_implies_or) }
% 0.19/0.47 fresh21(true, true, X, Y)
% 0.19/0.47 = { by axiom 15 (op_implies_or) }
% 0.19/0.47 implies(X, Y)
% 0.19/0.47
% 0.19/0.47 Goal 1 (hilbert_and_2): and_2 = true.
% 0.19/0.47 Proof:
% 0.19/0.47 and_2
% 0.19/0.48 = { by axiom 25 (and_2) R->L }
% 0.19/0.48 fresh56(is_a_theorem(implies(and(x8, y8), y8)), true)
% 0.19/0.48 = { by axiom 12 (op_and) R->L }
% 0.19/0.48 fresh56(is_a_theorem(implies(fresh24(true, true, x8, y8), y8)), true)
% 0.19/0.48 = { by axiom 5 (principia_op_and) R->L }
% 0.19/0.48 fresh56(is_a_theorem(implies(fresh24(op_and, true, x8, y8), y8)), true)
% 0.19/0.48 = { by axiom 21 (op_and) }
% 0.19/0.48 fresh56(is_a_theorem(implies(not(or(not(x8), not(y8))), y8)), true)
% 0.19/0.48 = { by lemma 27 }
% 0.19/0.48 fresh56(is_a_theorem(implies(not(implies(x8, not(y8))), y8)), true)
% 0.19/0.48 = { by lemma 26 }
% 0.19/0.48 fresh56(is_a_theorem(or(implies(x8, not(y8)), y8)), true)
% 0.19/0.48 = { by axiom 10 (modus_ponens_2) R->L }
% 0.19/0.48 fresh56(fresh28(true, true, or(implies(x8, not(y8)), y8)), true)
% 0.19/0.48 = { by axiom 18 (r2_1) R->L }
% 0.19/0.48 fresh56(fresh28(fresh10(true, true, not(x8), not(y8)), true, or(implies(x8, not(y8)), y8)), true)
% 0.19/0.48 = { by axiom 2 (principia_r2) R->L }
% 0.19/0.48 fresh56(fresh28(fresh10(r2, true, not(x8), not(y8)), true, or(implies(x8, not(y8)), y8)), true)
% 0.19/0.48 = { by axiom 20 (r2_1) }
% 0.19/0.48 fresh56(fresh28(is_a_theorem(implies(not(y8), or(not(x8), not(y8)))), true, or(implies(x8, not(y8)), y8)), true)
% 0.19/0.48 = { by lemma 27 }
% 0.19/0.48 fresh56(fresh28(is_a_theorem(implies(not(y8), implies(x8, not(y8)))), true, or(implies(x8, not(y8)), y8)), true)
% 0.19/0.48 = { by lemma 26 }
% 0.19/0.48 fresh56(fresh28(is_a_theorem(or(y8, implies(x8, not(y8)))), true, or(implies(x8, not(y8)), y8)), true)
% 0.19/0.48 = { by axiom 24 (modus_ponens_2) R->L }
% 0.19/0.48 fresh56(fresh59(is_a_theorem(implies(or(y8, implies(x8, not(y8))), or(implies(x8, not(y8)), y8))), true, or(y8, implies(x8, not(y8))), or(implies(x8, not(y8)), y8)), true)
% 0.19/0.48 = { by axiom 23 (r3_1) R->L }
% 0.19/0.48 fresh56(fresh59(fresh8(r3, true, y8, implies(x8, not(y8))), true, or(y8, implies(x8, not(y8))), or(implies(x8, not(y8)), y8)), true)
% 0.19/0.48 = { by axiom 3 (principia_r3) }
% 0.19/0.48 fresh56(fresh59(fresh8(true, true, y8, implies(x8, not(y8))), true, or(y8, implies(x8, not(y8))), or(implies(x8, not(y8)), y8)), true)
% 0.19/0.48 = { by axiom 19 (r3_1) }
% 0.19/0.48 fresh56(fresh59(true, true, or(y8, implies(x8, not(y8))), or(implies(x8, not(y8)), y8)), true)
% 0.19/0.48 = { by axiom 11 (modus_ponens_2) }
% 0.19/0.48 fresh56(fresh60(modus_ponens, true, or(implies(x8, not(y8)), y8)), true)
% 0.19/0.48 = { by axiom 1 (principia_modus_ponens) }
% 0.19/0.48 fresh56(fresh60(true, true, or(implies(x8, not(y8)), y8)), true)
% 0.19/0.48 = { by axiom 9 (modus_ponens_2) }
% 0.19/0.48 fresh56(true, true)
% 0.19/0.48 = { by axiom 8 (and_2) }
% 0.19/0.48 true
% 0.19/0.48 % SZS output end Proof
% 0.19/0.48
% 0.19/0.48 RESULT: Theorem (the conjecture is true).
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