TSTP Solution File: LCL896+1 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : LCL896+1 : TPTP v8.1.2. Released v5.5.0.
% 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 08:21:06 EDT 2023
% Result : Theorem 0.19s 0.43s
% Output : Proof 0.19s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : LCL896+1 : TPTP v8.1.2. Released v5.5.0.
% 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 : n013.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
% 0.12/0.34 % CPULimit : 300
% 0.12/0.34 % WCLimit : 300
% 0.12/0.34 % DateTime : Fri Aug 25 04:18:32 EDT 2023
% 0.12/0.34 % CPUTime :
% 0.19/0.43 Command-line arguments: --set-join --lhs-weight 1 --no-flatten-goal --complete-subsets --goal-heuristic
% 0.19/0.43
% 0.19/0.43 % SZS status Theorem
% 0.19/0.43
% 0.19/0.45 % SZS output start Proof
% 0.19/0.45 Take the following subset of the input axioms:
% 0.19/0.45 fof(goals_13, conjecture, ![X17, X18]: '+'(X17, '==>'(X17, X18))='+'(X18, '==>'(X18, X17))).
% 0.19/0.45 fof(sos_01, axiom, ![A, B, C]: '+'('+'(A, B), C)='+'(A, '+'(B, C))).
% 0.19/0.45 fof(sos_02, axiom, ![A2, B2]: '+'(A2, B2)='+'(B2, A2)).
% 0.19/0.45 fof(sos_03, axiom, ![A2]: '+'(A2, '0')=A2).
% 0.19/0.45 fof(sos_04, axiom, ![A2]: '>='(A2, A2)).
% 0.19/0.45 fof(sos_06, axiom, ![X3, X4]: (('>='(X3, X4) & '>='(X4, X3)) => X3=X4)).
% 0.19/0.45 fof(sos_07, axiom, ![X5, X6, X7]: ('>='('+'(X5, X6), X7) <=> '>='(X6, '==>'(X5, X7)))).
% 0.19/0.45 fof(sos_08, axiom, ![A2]: '>='(A2, '0')).
% 0.19/0.45 fof(sos_09, axiom, ![X8, X9, X10]: ('>='(X8, X9) => '>='('+'(X8, X10), '+'(X9, X10)))).
% 0.19/0.45 fof(sos_10, axiom, ![X11, X12, X13]: ('>='(X11, X12) => '>='('==>'(X12, X13), '==>'(X11, X13)))).
% 0.19/0.45 fof(sos_12, axiom, ![A2, B2, C2]: '+'('+'(A2, '==>'(A2, B2)), '==>'('+'(A2, '==>'(A2, B2)), C2))='+'(A2, '==>'(A2, '+'(B2, '==>'(B2, C2))))).
% 0.19/0.45
% 0.19/0.45 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.19/0.45 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.19/0.45 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.19/0.45 fresh(y, y, x1...xn) = u
% 0.19/0.45 C => fresh(s, t, x1...xn) = v
% 0.19/0.45 where fresh is a fresh function symbol and x1..xn are the free
% 0.19/0.45 variables of u and v.
% 0.19/0.45 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.19/0.45 input problem has no model of domain size 1).
% 0.19/0.45
% 0.19/0.45 The encoding turns the above axioms into the following unit equations and goals:
% 0.19/0.45
% 0.19/0.45 Axiom 1 (sos_04): X >= X = true.
% 0.19/0.45 Axiom 2 (sos_08): X >= 0 = true.
% 0.19/0.45 Axiom 3 (sos_02): X + Y = Y + X.
% 0.19/0.45 Axiom 4 (sos_03): X + 0 = X.
% 0.19/0.45 Axiom 5 (sos_06): fresh(X, X, Y, Z) = Z.
% 0.19/0.45 Axiom 6 (sos_06): fresh2(X, X, Y, Z) = Y.
% 0.19/0.45 Axiom 7 (sos_01): (X + Y) + Z = X + (Y + Z).
% 0.19/0.45 Axiom 8 (sos_07): fresh7(X, X, Y, Z, W) = true.
% 0.19/0.45 Axiom 9 (sos_07_1): fresh6(X, X, Y, Z, W) = true.
% 0.19/0.45 Axiom 10 (sos_09): fresh5(X, X, Y, Z, W) = true.
% 0.19/0.45 Axiom 11 (sos_10): fresh4(X, X, Y, Z, W) = true.
% 0.19/0.45 Axiom 12 (sos_06): fresh2(X >= Y, true, Y, X) = fresh(Y >= X, true, Y, X).
% 0.19/0.45 Axiom 13 (sos_09): fresh5(X >= Y, true, X, Y, Z) = (X + Z) >= (Y + Z).
% 0.19/0.45 Axiom 14 (sos_10): fresh4(X >= Y, true, X, Y, Z) = (Y ==> Z) >= (X ==> Z).
% 0.19/0.45 Axiom 15 (sos_07): fresh7(X >= (Y ==> Z), true, Y, X, Z) = (Y + X) >= Z.
% 0.19/0.45 Axiom 16 (sos_07_1): fresh6((X + Y) >= Z, true, X, Y, Z) = Y >= (X ==> Z).
% 0.19/0.46 Axiom 17 (sos_12): (X + (X ==> Y)) + ((X + (X ==> Y)) ==> Z) = X + (X ==> (Y + (Y ==> Z))).
% 0.19/0.46
% 0.19/0.46 Lemma 18: 0 + X = X.
% 0.19/0.46 Proof:
% 0.19/0.46 0 + X
% 0.19/0.46 = { by axiom 3 (sos_02) R->L }
% 0.19/0.46 X + 0
% 0.19/0.46 = { by axiom 4 (sos_03) }
% 0.19/0.46 X
% 0.19/0.46
% 0.19/0.46 Lemma 19: (X + (X ==> Y)) >= (Y + (Y ==> X)) = true.
% 0.19/0.46 Proof:
% 0.19/0.46 (X + (X ==> Y)) >= (Y + (Y ==> X))
% 0.19/0.46 = { by axiom 4 (sos_03) R->L }
% 0.19/0.46 (X + ((X ==> Y) + 0)) >= (Y + (Y ==> X))
% 0.19/0.46 = { by axiom 5 (sos_06) R->L }
% 0.19/0.46 (X + ((X ==> Y) + fresh(true, true, (X + (X ==> Y)) ==> X, 0))) >= (Y + (Y ==> X))
% 0.19/0.46 = { by axiom 2 (sos_08) R->L }
% 0.19/0.46 (X + ((X ==> Y) + fresh(((X + (X ==> Y)) ==> X) >= 0, true, (X + (X ==> Y)) ==> X, 0))) >= (Y + (Y ==> X))
% 0.19/0.46 = { by axiom 12 (sos_06) R->L }
% 0.19/0.46 (X + ((X ==> Y) + fresh2(0 >= ((X + (X ==> Y)) ==> X), true, (X + (X ==> Y)) ==> X, 0))) >= (Y + (Y ==> X))
% 0.19/0.46 = { by axiom 3 (sos_02) R->L }
% 0.19/0.46 (X + ((X ==> Y) + fresh2(0 >= (((X ==> Y) + X) ==> X), true, (X + (X ==> Y)) ==> X, 0))) >= (Y + (Y ==> X))
% 0.19/0.46 = { by axiom 5 (sos_06) R->L }
% 0.19/0.46 (X + ((X ==> Y) + fresh2(fresh(true, true, X ==> X, 0) >= (((X ==> Y) + X) ==> X), true, (X + (X ==> Y)) ==> X, 0))) >= (Y + (Y ==> X))
% 0.19/0.46 = { by axiom 2 (sos_08) R->L }
% 0.19/0.46 (X + ((X ==> Y) + fresh2(fresh((X ==> X) >= 0, true, X ==> X, 0) >= (((X ==> Y) + X) ==> X), true, (X + (X ==> Y)) ==> X, 0))) >= (Y + (Y ==> X))
% 0.19/0.46 = { by axiom 12 (sos_06) R->L }
% 0.19/0.46 (X + ((X ==> Y) + fresh2(fresh2(0 >= (X ==> X), true, X ==> X, 0) >= (((X ==> Y) + X) ==> X), true, (X + (X ==> Y)) ==> X, 0))) >= (Y + (Y ==> X))
% 0.19/0.46 = { by lemma 18 R->L }
% 0.19/0.46 (X + ((X ==> Y) + fresh2(fresh2(0 >= (X ==> (0 + X)), true, X ==> X, 0) >= (((X ==> Y) + X) ==> X), true, (X + (X ==> Y)) ==> X, 0))) >= (Y + (Y ==> X))
% 0.19/0.46 = { by axiom 3 (sos_02) R->L }
% 0.19/0.46 (X + ((X ==> Y) + fresh2(fresh2(0 >= (X ==> (X + 0)), true, X ==> X, 0) >= (((X ==> Y) + X) ==> X), true, (X + (X ==> Y)) ==> X, 0))) >= (Y + (Y ==> X))
% 0.19/0.46 = { by axiom 16 (sos_07_1) R->L }
% 0.19/0.46 (X + ((X ==> Y) + fresh2(fresh2(fresh6((X + 0) >= (X + 0), true, X, 0, X + 0), true, X ==> X, 0) >= (((X ==> Y) + X) ==> X), true, (X + (X ==> Y)) ==> X, 0))) >= (Y + (Y ==> X))
% 0.19/0.46 = { by axiom 1 (sos_04) }
% 0.19/0.46 (X + ((X ==> Y) + fresh2(fresh2(fresh6(true, true, X, 0, X + 0), true, X ==> X, 0) >= (((X ==> Y) + X) ==> X), true, (X + (X ==> Y)) ==> X, 0))) >= (Y + (Y ==> X))
% 0.19/0.46 = { by axiom 9 (sos_07_1) }
% 0.19/0.46 (X + ((X ==> Y) + fresh2(fresh2(true, true, X ==> X, 0) >= (((X ==> Y) + X) ==> X), true, (X + (X ==> Y)) ==> X, 0))) >= (Y + (Y ==> X))
% 0.19/0.46 = { by axiom 6 (sos_06) }
% 0.19/0.46 (X + ((X ==> Y) + fresh2((X ==> X) >= (((X ==> Y) + X) ==> X), true, (X + (X ==> Y)) ==> X, 0))) >= (Y + (Y ==> X))
% 0.19/0.46 = { by axiom 14 (sos_10) R->L }
% 0.19/0.46 (X + ((X ==> Y) + fresh2(fresh4(((X ==> Y) + X) >= X, true, (X ==> Y) + X, X, X), true, (X + (X ==> Y)) ==> X, 0))) >= (Y + (Y ==> X))
% 0.19/0.46 = { by lemma 18 R->L }
% 0.19/0.46 (X + ((X ==> Y) + fresh2(fresh4(((X ==> Y) + X) >= (0 + X), true, (X ==> Y) + X, X, X), true, (X + (X ==> Y)) ==> X, 0))) >= (Y + (Y ==> X))
% 0.19/0.46 = { by axiom 13 (sos_09) R->L }
% 0.19/0.46 (X + ((X ==> Y) + fresh2(fresh4(fresh5((X ==> Y) >= 0, true, X ==> Y, 0, X), true, (X ==> Y) + X, X, X), true, (X + (X ==> Y)) ==> X, 0))) >= (Y + (Y ==> X))
% 0.19/0.46 = { by axiom 2 (sos_08) }
% 0.19/0.46 (X + ((X ==> Y) + fresh2(fresh4(fresh5(true, true, X ==> Y, 0, X), true, (X ==> Y) + X, X, X), true, (X + (X ==> Y)) ==> X, 0))) >= (Y + (Y ==> X))
% 0.19/0.46 = { by axiom 10 (sos_09) }
% 0.19/0.46 (X + ((X ==> Y) + fresh2(fresh4(true, true, (X ==> Y) + X, X, X), true, (X + (X ==> Y)) ==> X, 0))) >= (Y + (Y ==> X))
% 0.19/0.46 = { by axiom 11 (sos_10) }
% 0.19/0.46 (X + ((X ==> Y) + fresh2(true, true, (X + (X ==> Y)) ==> X, 0))) >= (Y + (Y ==> X))
% 0.19/0.46 = { by axiom 6 (sos_06) }
% 0.19/0.46 (X + ((X ==> Y) + ((X + (X ==> Y)) ==> X))) >= (Y + (Y ==> X))
% 0.19/0.46 = { by axiom 7 (sos_01) R->L }
% 0.19/0.46 ((X + (X ==> Y)) + ((X + (X ==> Y)) ==> X)) >= (Y + (Y ==> X))
% 0.19/0.46 = { by axiom 17 (sos_12) }
% 0.19/0.46 (X + (X ==> (Y + (Y ==> X)))) >= (Y + (Y ==> X))
% 0.19/0.46 = { by axiom 15 (sos_07) R->L }
% 0.19/0.46 fresh7((X ==> (Y + (Y ==> X))) >= (X ==> (Y + (Y ==> X))), true, X, X ==> (Y + (Y ==> X)), Y + (Y ==> X))
% 0.19/0.46 = { by axiom 1 (sos_04) }
% 0.19/0.46 fresh7(true, true, X, X ==> (Y + (Y ==> X)), Y + (Y ==> X))
% 0.19/0.46 = { by axiom 8 (sos_07) }
% 0.19/0.46 true
% 0.19/0.46
% 0.19/0.46 Goal 1 (goals_13): x17 + (x17 ==> x18) = x18 + (x18 ==> x17).
% 0.19/0.46 Proof:
% 0.19/0.46 x17 + (x17 ==> x18)
% 0.19/0.46 = { by axiom 6 (sos_06) R->L }
% 0.19/0.46 fresh2(true, true, x17 + (x17 ==> x18), x18 + (x18 ==> x17))
% 0.19/0.46 = { by lemma 19 R->L }
% 0.19/0.46 fresh2((x18 + (x18 ==> x17)) >= (x17 + (x17 ==> x18)), true, x17 + (x17 ==> x18), x18 + (x18 ==> x17))
% 0.19/0.46 = { by axiom 12 (sos_06) }
% 0.19/0.46 fresh((x17 + (x17 ==> x18)) >= (x18 + (x18 ==> x17)), true, x17 + (x17 ==> x18), x18 + (x18 ==> x17))
% 0.19/0.46 = { by lemma 19 }
% 0.19/0.46 fresh(true, true, x17 + (x17 ==> x18), x18 + (x18 ==> x17))
% 0.19/0.46 = { by axiom 5 (sos_06) }
% 0.19/0.46 x18 + (x18 ==> x17)
% 0.19/0.46 % SZS output end Proof
% 0.19/0.46
% 0.19/0.46 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------