TSTP Solution File: LCL531+1 by Twee---2.4.2
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : LCL531+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 : n027.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 08:19:20 EDT 2023
% Result : Theorem 0.21s 0.63s
% Output : Proof 0.21s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.13 % Problem : LCL531+1 : TPTP v8.1.2. Released v3.3.0.
% 0.14/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.15/0.34 % Computer : n027.cluster.edu
% 0.15/0.34 % Model : x86_64 x86_64
% 0.15/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.34 % Memory : 8042.1875MB
% 0.15/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.15/0.34 % CPULimit : 300
% 0.15/0.34 % WCLimit : 300
% 0.15/0.34 % DateTime : Fri Aug 25 01:24:33 EDT 2023
% 0.15/0.35 % CPUTime :
% 0.21/0.63 Command-line arguments: --flip-ordering --lhs-weight 1 --depth-weight 60 --distributivity-heuristic
% 0.21/0.63
% 0.21/0.63 % SZS status Theorem
% 0.21/0.63
% 0.21/0.63 % SZS output start Proof
% 0.21/0.63 Take the following subset of the input axioms:
% 0.21/0.63 fof(and_3, axiom, and_3 <=> ![X, Y]: is_a_theorem(implies(X, implies(Y, and(X, Y))))).
% 0.21/0.63 fof(axiom_m4, axiom, axiom_m4 <=> ![X2]: is_a_theorem(strict_implies(X2, and(X2, X2)))).
% 0.21/0.63 fof(hilbert_and_3, axiom, and_3).
% 0.21/0.63 fof(hilbert_implies_2, axiom, implies_2).
% 0.21/0.63 fof(hilbert_modus_ponens, axiom, modus_ponens).
% 0.21/0.63 fof(implies_2, axiom, implies_2 <=> ![X2, Y2]: is_a_theorem(implies(implies(X2, implies(X2, Y2)), implies(X2, Y2)))).
% 0.21/0.63 fof(km5_necessitation, axiom, necessitation).
% 0.21/0.63 fof(modus_ponens, axiom, modus_ponens <=> ![X2, Y2]: ((is_a_theorem(X2) & is_a_theorem(implies(X2, Y2))) => is_a_theorem(Y2))).
% 0.21/0.63 fof(necessitation, axiom, necessitation <=> ![X2]: (is_a_theorem(X2) => is_a_theorem(necessarily(X2)))).
% 0.21/0.63 fof(op_strict_implies, axiom, op_strict_implies => ![X2, Y2]: strict_implies(X2, Y2)=necessarily(implies(X2, Y2))).
% 0.21/0.63 fof(s1_0_axiom_m4, conjecture, axiom_m4).
% 0.21/0.63 fof(s1_0_op_strict_implies, axiom, op_strict_implies).
% 0.21/0.63
% 0.21/0.63 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.21/0.63 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.21/0.63 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.21/0.63 fresh(y, y, x1...xn) = u
% 0.21/0.63 C => fresh(s, t, x1...xn) = v
% 0.21/0.63 where fresh is a fresh function symbol and x1..xn are the free
% 0.21/0.63 variables of u and v.
% 0.21/0.63 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.21/0.63 input problem has no model of domain size 1).
% 0.21/0.63
% 0.21/0.63 The encoding turns the above axioms into the following unit equations and goals:
% 0.21/0.63
% 0.21/0.63 Axiom 1 (hilbert_modus_ponens): modus_ponens = true.
% 0.21/0.63 Axiom 2 (hilbert_implies_2): implies_2 = true.
% 0.21/0.63 Axiom 3 (hilbert_and_3): and_3 = true.
% 0.21/0.63 Axiom 4 (km5_necessitation): necessitation = true.
% 0.21/0.63 Axiom 5 (s1_0_op_strict_implies): op_strict_implies = true.
% 0.21/0.63 Axiom 6 (axiom_m4): fresh84(X, X) = true.
% 0.21/0.63 Axiom 7 (modus_ponens_2): fresh116(X, X, Y) = true.
% 0.21/0.63 Axiom 8 (modus_ponens_2): fresh40(X, X, Y) = is_a_theorem(Y).
% 0.21/0.63 Axiom 9 (necessitation_1): fresh34(X, X, Y) = is_a_theorem(necessarily(Y)).
% 0.21/0.63 Axiom 10 (necessitation_1): fresh33(X, X, Y) = true.
% 0.21/0.63 Axiom 11 (modus_ponens_2): fresh115(X, X, Y, Z) = fresh116(modus_ponens, true, Z).
% 0.21/0.63 Axiom 12 (and_3_1): fresh103(X, X, Y, Z) = true.
% 0.21/0.63 Axiom 13 (implies_2_1): fresh49(X, X, Y, Z) = true.
% 0.21/0.63 Axiom 14 (necessitation_1): fresh34(necessitation, true, X) = fresh33(is_a_theorem(X), true, X).
% 0.21/0.63 Axiom 15 (op_strict_implies): fresh23(X, X, Y, Z) = strict_implies(Y, Z).
% 0.21/0.63 Axiom 16 (op_strict_implies): fresh23(op_strict_implies, true, X, Y) = necessarily(implies(X, Y)).
% 0.21/0.63 Axiom 17 (and_3_1): fresh103(and_3, true, X, Y) = is_a_theorem(implies(X, implies(Y, and(X, Y)))).
% 0.21/0.63 Axiom 18 (modus_ponens_2): fresh115(is_a_theorem(implies(X, Y)), true, X, Y) = fresh40(is_a_theorem(X), true, Y).
% 0.21/0.63 Axiom 19 (axiom_m4): fresh84(is_a_theorem(strict_implies(x5, and(x5, x5))), true) = axiom_m4.
% 0.21/0.63 Axiom 20 (implies_2_1): fresh49(implies_2, true, X, Y) = is_a_theorem(implies(implies(X, implies(X, Y)), implies(X, Y))).
% 0.21/0.63
% 0.21/0.63 Goal 1 (s1_0_axiom_m4): axiom_m4 = true.
% 0.21/0.63 Proof:
% 0.21/0.63 axiom_m4
% 0.21/0.63 = { by axiom 19 (axiom_m4) R->L }
% 0.21/0.63 fresh84(is_a_theorem(strict_implies(x5, and(x5, x5))), true)
% 0.21/0.63 = { by axiom 15 (op_strict_implies) R->L }
% 0.21/0.63 fresh84(is_a_theorem(fresh23(true, true, x5, and(x5, x5))), true)
% 0.21/0.63 = { by axiom 5 (s1_0_op_strict_implies) R->L }
% 0.21/0.63 fresh84(is_a_theorem(fresh23(op_strict_implies, true, x5, and(x5, x5))), true)
% 0.21/0.63 = { by axiom 16 (op_strict_implies) }
% 0.21/0.63 fresh84(is_a_theorem(necessarily(implies(x5, and(x5, x5)))), true)
% 0.21/0.63 = { by axiom 9 (necessitation_1) R->L }
% 0.21/0.63 fresh84(fresh34(true, true, implies(x5, and(x5, x5))), true)
% 0.21/0.63 = { by axiom 4 (km5_necessitation) R->L }
% 0.21/0.63 fresh84(fresh34(necessitation, true, implies(x5, and(x5, x5))), true)
% 0.21/0.63 = { by axiom 14 (necessitation_1) }
% 0.21/0.63 fresh84(fresh33(is_a_theorem(implies(x5, and(x5, x5))), true, implies(x5, and(x5, x5))), true)
% 0.21/0.64 = { by axiom 8 (modus_ponens_2) R->L }
% 0.21/0.64 fresh84(fresh33(fresh40(true, true, implies(x5, and(x5, x5))), true, implies(x5, and(x5, x5))), true)
% 0.21/0.64 = { by axiom 12 (and_3_1) R->L }
% 0.21/0.64 fresh84(fresh33(fresh40(fresh103(true, true, x5, x5), true, implies(x5, and(x5, x5))), true, implies(x5, and(x5, x5))), true)
% 0.21/0.64 = { by axiom 3 (hilbert_and_3) R->L }
% 0.21/0.64 fresh84(fresh33(fresh40(fresh103(and_3, true, x5, x5), true, implies(x5, and(x5, x5))), true, implies(x5, and(x5, x5))), true)
% 0.21/0.64 = { by axiom 17 (and_3_1) }
% 0.21/0.64 fresh84(fresh33(fresh40(is_a_theorem(implies(x5, implies(x5, and(x5, x5)))), true, implies(x5, and(x5, x5))), true, implies(x5, and(x5, x5))), true)
% 0.21/0.64 = { by axiom 18 (modus_ponens_2) R->L }
% 0.21/0.64 fresh84(fresh33(fresh115(is_a_theorem(implies(implies(x5, implies(x5, and(x5, x5))), implies(x5, and(x5, x5)))), true, implies(x5, implies(x5, and(x5, x5))), implies(x5, and(x5, x5))), true, implies(x5, and(x5, x5))), true)
% 0.21/0.64 = { by axiom 20 (implies_2_1) R->L }
% 0.21/0.64 fresh84(fresh33(fresh115(fresh49(implies_2, true, x5, and(x5, x5)), true, implies(x5, implies(x5, and(x5, x5))), implies(x5, and(x5, x5))), true, implies(x5, and(x5, x5))), true)
% 0.21/0.64 = { by axiom 2 (hilbert_implies_2) }
% 0.21/0.64 fresh84(fresh33(fresh115(fresh49(true, true, x5, and(x5, x5)), true, implies(x5, implies(x5, and(x5, x5))), implies(x5, and(x5, x5))), true, implies(x5, and(x5, x5))), true)
% 0.21/0.64 = { by axiom 13 (implies_2_1) }
% 0.21/0.64 fresh84(fresh33(fresh115(true, true, implies(x5, implies(x5, and(x5, x5))), implies(x5, and(x5, x5))), true, implies(x5, and(x5, x5))), true)
% 0.21/0.64 = { by axiom 11 (modus_ponens_2) }
% 0.21/0.64 fresh84(fresh33(fresh116(modus_ponens, true, implies(x5, and(x5, x5))), true, implies(x5, and(x5, x5))), true)
% 0.21/0.64 = { by axiom 1 (hilbert_modus_ponens) }
% 0.21/0.64 fresh84(fresh33(fresh116(true, true, implies(x5, and(x5, x5))), true, implies(x5, and(x5, x5))), true)
% 0.21/0.64 = { by axiom 7 (modus_ponens_2) }
% 0.21/0.64 fresh84(fresh33(true, true, implies(x5, and(x5, x5))), true)
% 0.21/0.64 = { by axiom 10 (necessitation_1) }
% 0.21/0.64 fresh84(true, true)
% 0.21/0.64 = { by axiom 6 (axiom_m4) }
% 0.21/0.64 true
% 0.21/0.64 % SZS output end Proof
% 0.21/0.64
% 0.21/0.64 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------