TSTP Solution File: LCL888+1 by Twee---2.4.2
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : LCL888+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 : n024.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 08:21:04 EDT 2023
% Result : Theorem 26.85s 3.81s
% Output : Proof 27.24s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.06/0.12 % Problem : LCL888+1 : TPTP v8.1.2. Released v5.5.0.
% 0.06/0.13 % 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 : n024.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 : Fri Aug 25 00:07:09 EDT 2023
% 0.12/0.34 % CPUTime :
% 26.85/3.81 Command-line arguments: --lhs-weight 9 --flip-ordering --complete-subsets --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 26.85/3.81
% 26.85/3.81 % SZS status Theorem
% 26.85/3.81
% 27.24/3.85 % SZS output start Proof
% 27.24/3.85 Take the following subset of the input axioms:
% 27.24/3.85 fof(goals_13, conjecture, ![X17, X18, X19]: ((X17='==>'(X17, X18) & X19='==>'(X19, X18)) => X17=X19)).
% 27.24/3.85 fof(sos_01, axiom, ![A, B, C]: '+'('+'(A, B), C)='+'(A, '+'(B, C))).
% 27.24/3.85 fof(sos_02, axiom, ![A2, B2]: '+'(A2, B2)='+'(B2, A2)).
% 27.24/3.85 fof(sos_03, axiom, ![A2]: '+'(A2, '0')=A2).
% 27.24/3.85 fof(sos_04, axiom, ![A2]: '>='(A2, A2)).
% 27.24/3.85 fof(sos_06, axiom, ![X3, X4]: (('>='(X3, X4) & '>='(X4, X3)) => X3=X4)).
% 27.24/3.85 fof(sos_07, axiom, ![X5, X6, X7]: ('>='('+'(X5, X6), X7) <=> '>='(X6, '==>'(X5, X7)))).
% 27.24/3.85 fof(sos_08, axiom, ![A2]: '>='(A2, '0')).
% 27.24/3.85 fof(sos_09, axiom, ![X8, X9, X10]: ('>='(X8, X9) => '>='('+'(X8, X10), '+'(X9, X10)))).
% 27.24/3.85 fof(sos_11, axiom, ![X14, X15, X16]: ('>='(X14, X15) => '>='('==>'(X16, X14), '==>'(X16, X15)))).
% 27.24/3.85 fof(sos_12, axiom, ![A2, B2]: '+'(A2, '==>'(A2, B2))='+'(B2, '==>'(B2, A2))).
% 27.24/3.85
% 27.24/3.85 Now clausify the problem and encode Horn clauses using encoding 3 of
% 27.24/3.85 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 27.24/3.85 We repeatedly replace C & s=t => u=v by the two clauses:
% 27.24/3.85 fresh(y, y, x1...xn) = u
% 27.24/3.85 C => fresh(s, t, x1...xn) = v
% 27.24/3.85 where fresh is a fresh function symbol and x1..xn are the free
% 27.24/3.85 variables of u and v.
% 27.24/3.85 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 27.24/3.85 input problem has no model of domain size 1).
% 27.24/3.85
% 27.24/3.85 The encoding turns the above axioms into the following unit equations and goals:
% 27.24/3.85
% 27.24/3.85 Axiom 1 (sos_04): X >= X = true.
% 27.24/3.85 Axiom 2 (sos_08): X >= 0 = true.
% 27.24/3.85 Axiom 3 (sos_02): X + Y = Y + X.
% 27.24/3.85 Axiom 4 (sos_03): X + 0 = X.
% 27.24/3.85 Axiom 5 (goals_13): x19 = x19 ==> x18.
% 27.24/3.85 Axiom 6 (goals_13_1): x17 = x17 ==> x18.
% 27.24/3.85 Axiom 7 (sos_12): X + (X ==> Y) = Y + (Y ==> X).
% 27.24/3.85 Axiom 8 (sos_01): (X + Y) + Z = X + (Y + Z).
% 27.24/3.85 Axiom 9 (sos_06): fresh(X, X, Y, Z) = Z.
% 27.24/3.85 Axiom 10 (sos_06): fresh2(X, X, Y, Z) = Y.
% 27.24/3.85 Axiom 11 (sos_07_1): fresh6(X, X, Y, Z, W) = true.
% 27.24/3.85 Axiom 12 (sos_09): fresh5(X, X, Y, Z, W) = true.
% 27.24/3.85 Axiom 13 (sos_11): fresh3(X, X, Y, Z, W) = true.
% 27.24/3.85 Axiom 14 (sos_06): fresh2(X >= Y, true, Y, X) = fresh(Y >= X, true, Y, X).
% 27.24/3.85 Axiom 15 (sos_09): fresh5(X >= Y, true, X, Y, Z) = (X + Z) >= (Y + Z).
% 27.24/3.85 Axiom 16 (sos_11): fresh3(X >= Y, true, X, Y, Z) = (Z ==> X) >= (Z ==> Y).
% 27.24/3.85 Axiom 17 (sos_07_1): fresh6((X + Y) >= Z, true, X, Y, Z) = Y >= (X ==> Z).
% 27.24/3.85
% 27.24/3.85 Lemma 18: 0 + X = X.
% 27.24/3.85 Proof:
% 27.24/3.85 0 + X
% 27.24/3.85 = { by axiom 3 (sos_02) R->L }
% 27.24/3.85 X + 0
% 27.24/3.85 = { by axiom 4 (sos_03) }
% 27.24/3.85 X
% 27.24/3.85
% 27.24/3.85 Lemma 19: X + (Y + Z) = Y + (X + Z).
% 27.24/3.85 Proof:
% 27.24/3.85 X + (Y + Z)
% 27.24/3.85 = { by axiom 3 (sos_02) R->L }
% 27.24/3.85 (Y + Z) + X
% 27.24/3.85 = { by axiom 8 (sos_01) }
% 27.24/3.85 Y + (Z + X)
% 27.24/3.85 = { by axiom 3 (sos_02) }
% 27.24/3.85 Y + (X + Z)
% 27.24/3.85
% 27.24/3.85 Lemma 20: (X + Y) >= X = true.
% 27.24/3.85 Proof:
% 27.24/3.85 (X + Y) >= X
% 27.24/3.85 = { by axiom 3 (sos_02) R->L }
% 27.24/3.85 (Y + X) >= X
% 27.24/3.85 = { by lemma 18 R->L }
% 27.24/3.85 (Y + X) >= (0 + X)
% 27.24/3.85 = { by axiom 15 (sos_09) R->L }
% 27.24/3.85 fresh5(Y >= 0, true, Y, 0, X)
% 27.24/3.85 = { by axiom 2 (sos_08) }
% 27.24/3.85 fresh5(true, true, Y, 0, X)
% 27.24/3.85 = { by axiom 12 (sos_09) }
% 27.24/3.85 true
% 27.24/3.85
% 27.24/3.85 Lemma 21: (X + Y) >= (Z ==> X) = true.
% 27.24/3.85 Proof:
% 27.24/3.85 (X + Y) >= (Z ==> X)
% 27.24/3.85 = { by axiom 17 (sos_07_1) R->L }
% 27.24/3.85 fresh6((Z + (X + Y)) >= X, true, Z, X + Y, X)
% 27.24/3.85 = { by lemma 19 R->L }
% 27.24/3.85 fresh6((X + (Z + Y)) >= X, true, Z, X + Y, X)
% 27.24/3.85 = { by lemma 20 }
% 27.24/3.85 fresh6(true, true, Z, X + Y, X)
% 27.24/3.85 = { by axiom 11 (sos_07_1) }
% 27.24/3.85 true
% 27.24/3.85
% 27.24/3.85 Lemma 22: fresh(0 >= X, true, 0, X) = 0.
% 27.24/3.85 Proof:
% 27.24/3.85 fresh(0 >= X, true, 0, X)
% 27.24/3.85 = { by axiom 14 (sos_06) R->L }
% 27.24/3.85 fresh2(X >= 0, true, 0, X)
% 27.24/3.85 = { by axiom 2 (sos_08) }
% 27.24/3.85 fresh2(true, true, 0, X)
% 27.24/3.85 = { by axiom 10 (sos_06) }
% 27.24/3.85 0
% 27.24/3.85
% 27.24/3.85 Lemma 23: x17 + x17 = x18.
% 27.24/3.85 Proof:
% 27.24/3.85 x17 + x17
% 27.24/3.85 = { by axiom 6 (goals_13_1) }
% 27.24/3.85 x17 + (x17 ==> x18)
% 27.24/3.85 = { by axiom 7 (sos_12) R->L }
% 27.24/3.85 x18 + (x18 ==> x17)
% 27.24/3.85 = { by axiom 9 (sos_06) R->L }
% 27.24/3.85 x18 + fresh(true, true, 0, x18 ==> x17)
% 27.24/3.85 = { by axiom 11 (sos_07_1) R->L }
% 27.24/3.85 x18 + fresh(fresh6(true, true, x18, 0, x17), true, 0, x18 ==> x17)
% 27.24/3.85 = { by lemma 21 R->L }
% 27.24/3.85 x18 + fresh(fresh6((x18 + 0) >= (x17 ==> x18), true, x18, 0, x17), true, 0, x18 ==> x17)
% 27.24/3.85 = { by axiom 6 (goals_13_1) R->L }
% 27.24/3.85 x18 + fresh(fresh6((x18 + 0) >= x17, true, x18, 0, x17), true, 0, x18 ==> x17)
% 27.24/3.85 = { by axiom 17 (sos_07_1) }
% 27.24/3.85 x18 + fresh(0 >= (x18 ==> x17), true, 0, x18 ==> x17)
% 27.24/3.85 = { by lemma 22 }
% 27.24/3.85 x18 + 0
% 27.24/3.85 = { by axiom 4 (sos_03) }
% 27.24/3.85 x18
% 27.24/3.85
% 27.24/3.85 Lemma 24: (X + (Y ==> Z)) >= (Y ==> (X + Z)) = true.
% 27.24/3.85 Proof:
% 27.24/3.85 (X + (Y ==> Z)) >= (Y ==> (X + Z))
% 27.24/3.85 = { by axiom 3 (sos_02) R->L }
% 27.24/3.85 (X + (Y ==> Z)) >= (Y ==> (Z + X))
% 27.24/3.85 = { by axiom 3 (sos_02) R->L }
% 27.24/3.85 ((Y ==> Z) + X) >= (Y ==> (Z + X))
% 27.24/3.85 = { by axiom 17 (sos_07_1) R->L }
% 27.24/3.85 fresh6((Y + ((Y ==> Z) + X)) >= (Z + X), true, Y, (Y ==> Z) + X, Z + X)
% 27.24/3.85 = { by axiom 8 (sos_01) R->L }
% 27.24/3.85 fresh6(((Y + (Y ==> Z)) + X) >= (Z + X), true, Y, (Y ==> Z) + X, Z + X)
% 27.24/3.85 = { by axiom 15 (sos_09) R->L }
% 27.24/3.85 fresh6(fresh5((Y + (Y ==> Z)) >= Z, true, Y + (Y ==> Z), Z, X), true, Y, (Y ==> Z) + X, Z + X)
% 27.24/3.86 = { by axiom 7 (sos_12) R->L }
% 27.24/3.86 fresh6(fresh5((Z + (Z ==> Y)) >= Z, true, Y + (Y ==> Z), Z, X), true, Y, (Y ==> Z) + X, Z + X)
% 27.24/3.86 = { by lemma 20 }
% 27.24/3.86 fresh6(fresh5(true, true, Y + (Y ==> Z), Z, X), true, Y, (Y ==> Z) + X, Z + X)
% 27.24/3.86 = { by axiom 12 (sos_09) }
% 27.24/3.86 fresh6(true, true, Y, (Y ==> Z) + X, Z + X)
% 27.24/3.86 = { by axiom 11 (sos_07_1) }
% 27.24/3.86 true
% 27.24/3.86
% 27.24/3.86 Lemma 25: x17 ==> x19 = x19 ==> x17.
% 27.24/3.86 Proof:
% 27.24/3.86 x17 ==> x19
% 27.24/3.86 = { by axiom 10 (sos_06) R->L }
% 27.24/3.86 fresh2(true, true, x17 ==> x19, x19 ==> x17)
% 27.24/3.86 = { by axiom 11 (sos_07_1) R->L }
% 27.24/3.86 fresh2(fresh6(true, true, x17, x19 ==> x17, x19), true, x17 ==> x19, x19 ==> x17)
% 27.24/3.86 = { by lemma 24 R->L }
% 27.24/3.86 fresh2(fresh6((x17 + (x19 ==> x17)) >= (x19 ==> (x17 + x17)), true, x17, x19 ==> x17, x19), true, x17 ==> x19, x19 ==> x17)
% 27.24/3.86 = { by lemma 23 }
% 27.24/3.86 fresh2(fresh6((x17 + (x19 ==> x17)) >= (x19 ==> x18), true, x17, x19 ==> x17, x19), true, x17 ==> x19, x19 ==> x17)
% 27.24/3.86 = { by axiom 5 (goals_13) R->L }
% 27.24/3.86 fresh2(fresh6((x17 + (x19 ==> x17)) >= x19, true, x17, x19 ==> x17, x19), true, x17 ==> x19, x19 ==> x17)
% 27.24/3.86 = { by axiom 17 (sos_07_1) }
% 27.24/3.86 fresh2((x19 ==> x17) >= (x17 ==> x19), true, x17 ==> x19, x19 ==> x17)
% 27.24/3.86 = { by axiom 14 (sos_06) }
% 27.24/3.86 fresh((x17 ==> x19) >= (x19 ==> x17), true, x17 ==> x19, x19 ==> x17)
% 27.24/3.86 = { by axiom 17 (sos_07_1) R->L }
% 27.24/3.86 fresh(fresh6((x19 + (x17 ==> x19)) >= x17, true, x19, x17 ==> x19, x17), true, x17 ==> x19, x19 ==> x17)
% 27.24/3.86 = { by axiom 6 (goals_13_1) }
% 27.24/3.86 fresh(fresh6((x19 + (x17 ==> x19)) >= (x17 ==> x18), true, x19, x17 ==> x19, x17), true, x17 ==> x19, x19 ==> x17)
% 27.24/3.86 = { by axiom 4 (sos_03) R->L }
% 27.24/3.86 fresh(fresh6((x19 + (x17 ==> x19)) >= (x17 ==> (x18 + 0)), true, x19, x17 ==> x19, x17), true, x17 ==> x19, x19 ==> x17)
% 27.24/3.86 = { by lemma 22 R->L }
% 27.24/3.86 fresh(fresh6((x19 + (x17 ==> x19)) >= (x17 ==> (x18 + fresh(0 >= (x18 ==> x19), true, 0, x18 ==> x19))), true, x19, x17 ==> x19, x17), true, x17 ==> x19, x19 ==> x17)
% 27.24/3.86 = { by axiom 17 (sos_07_1) R->L }
% 27.24/3.86 fresh(fresh6((x19 + (x17 ==> x19)) >= (x17 ==> (x18 + fresh(fresh6((x18 + 0) >= x19, true, x18, 0, x19), true, 0, x18 ==> x19))), true, x19, x17 ==> x19, x17), true, x17 ==> x19, x19 ==> x17)
% 27.24/3.86 = { by axiom 5 (goals_13) }
% 27.24/3.86 fresh(fresh6((x19 + (x17 ==> x19)) >= (x17 ==> (x18 + fresh(fresh6((x18 + 0) >= (x19 ==> x18), true, x18, 0, x19), true, 0, x18 ==> x19))), true, x19, x17 ==> x19, x17), true, x17 ==> x19, x19 ==> x17)
% 27.24/3.86 = { by lemma 21 }
% 27.24/3.86 fresh(fresh6((x19 + (x17 ==> x19)) >= (x17 ==> (x18 + fresh(fresh6(true, true, x18, 0, x19), true, 0, x18 ==> x19))), true, x19, x17 ==> x19, x17), true, x17 ==> x19, x19 ==> x17)
% 27.24/3.86 = { by axiom 11 (sos_07_1) }
% 27.24/3.86 fresh(fresh6((x19 + (x17 ==> x19)) >= (x17 ==> (x18 + fresh(true, true, 0, x18 ==> x19))), true, x19, x17 ==> x19, x17), true, x17 ==> x19, x19 ==> x17)
% 27.24/3.86 = { by axiom 9 (sos_06) }
% 27.24/3.86 fresh(fresh6((x19 + (x17 ==> x19)) >= (x17 ==> (x18 + (x18 ==> x19))), true, x19, x17 ==> x19, x17), true, x17 ==> x19, x19 ==> x17)
% 27.24/3.86 = { by axiom 7 (sos_12) }
% 27.24/3.86 fresh(fresh6((x19 + (x17 ==> x19)) >= (x17 ==> (x19 + (x19 ==> x18))), true, x19, x17 ==> x19, x17), true, x17 ==> x19, x19 ==> x17)
% 27.24/3.86 = { by axiom 5 (goals_13) R->L }
% 27.24/3.86 fresh(fresh6((x19 + (x17 ==> x19)) >= (x17 ==> (x19 + x19)), true, x19, x17 ==> x19, x17), true, x17 ==> x19, x19 ==> x17)
% 27.24/3.86 = { by lemma 24 }
% 27.24/3.86 fresh(fresh6(true, true, x19, x17 ==> x19, x17), true, x17 ==> x19, x19 ==> x17)
% 27.24/3.86 = { by axiom 11 (sos_07_1) }
% 27.24/3.86 fresh(true, true, x17 ==> x19, x19 ==> x17)
% 27.24/3.86 = { by axiom 9 (sos_06) }
% 27.24/3.86 x19 ==> x17
% 27.24/3.86
% 27.24/3.86 Lemma 26: Z + ((Z ==> X) + Y) = X + (Y + (X ==> Z)).
% 27.24/3.86 Proof:
% 27.24/3.86 Z + ((Z ==> X) + Y)
% 27.24/3.86 = { by axiom 8 (sos_01) R->L }
% 27.24/3.86 (Z + (Z ==> X)) + Y
% 27.24/3.86 = { by axiom 7 (sos_12) R->L }
% 27.24/3.86 (X + (X ==> Z)) + Y
% 27.24/3.86 = { by axiom 8 (sos_01) }
% 27.24/3.86 X + ((X ==> Z) + Y)
% 27.24/3.86 = { by axiom 3 (sos_02) }
% 27.24/3.86 X + (Y + (X ==> Z))
% 27.24/3.86
% 27.24/3.86 Lemma 27: fresh6(X >= Y, true, X, 0, Y) = 0 >= (X ==> Y).
% 27.24/3.86 Proof:
% 27.24/3.86 fresh6(X >= Y, true, X, 0, Y)
% 27.24/3.86 = { by axiom 4 (sos_03) R->L }
% 27.24/3.86 fresh6((X + 0) >= Y, true, X, 0, Y)
% 27.24/3.86 = { by axiom 17 (sos_07_1) }
% 27.24/3.86 0 >= (X ==> Y)
% 27.24/3.86
% 27.24/3.86 Lemma 28: (X + ((X ==> Y) + Z)) ==> Y = 0.
% 27.24/3.86 Proof:
% 27.24/3.86 (X + ((X ==> Y) + Z)) ==> Y
% 27.24/3.86 = { by lemma 26 }
% 27.24/3.86 (Y + (Z + (Y ==> X))) ==> Y
% 27.24/3.86 = { by axiom 10 (sos_06) R->L }
% 27.24/3.86 fresh2(true, true, (Y + (Z + (Y ==> X))) ==> Y, 0)
% 27.24/3.86 = { by axiom 11 (sos_07_1) R->L }
% 27.24/3.86 fresh2(fresh6(true, true, Y + (Z + (Y ==> X)), 0, Y), true, (Y + (Z + (Y ==> X))) ==> Y, 0)
% 27.24/3.86 = { by lemma 20 R->L }
% 27.24/3.86 fresh2(fresh6((Y + (Z + (Y ==> X))) >= Y, true, Y + (Z + (Y ==> X)), 0, Y), true, (Y + (Z + (Y ==> X))) ==> Y, 0)
% 27.24/3.86 = { by lemma 27 }
% 27.24/3.86 fresh2(0 >= ((Y + (Z + (Y ==> X))) ==> Y), true, (Y + (Z + (Y ==> X))) ==> Y, 0)
% 27.24/3.86 = { by axiom 14 (sos_06) }
% 27.24/3.86 fresh(((Y + (Z + (Y ==> X))) ==> Y) >= 0, true, (Y + (Z + (Y ==> X))) ==> Y, 0)
% 27.24/3.86 = { by axiom 2 (sos_08) }
% 27.24/3.86 fresh(true, true, (Y + (Z + (Y ==> X))) ==> Y, 0)
% 27.24/3.86 = { by axiom 9 (sos_06) }
% 27.24/3.86 0
% 27.24/3.86
% 27.24/3.86 Lemma 29: x19 ==> x17 = 0.
% 27.24/3.86 Proof:
% 27.24/3.86 x19 ==> x17
% 27.24/3.86 = { by lemma 25 R->L }
% 27.24/3.86 x17 ==> x19
% 27.24/3.86 = { by axiom 5 (goals_13) }
% 27.24/3.86 x17 ==> (x19 ==> x18)
% 27.24/3.86 = { by lemma 18 R->L }
% 27.24/3.86 x17 ==> (x19 ==> (0 + x18))
% 27.24/3.86 = { by lemma 23 R->L }
% 27.24/3.86 x17 ==> (x19 ==> (0 + (x17 + x17)))
% 27.24/3.86 = { by lemma 19 R->L }
% 27.24/3.86 x17 ==> (x19 ==> (x17 + (0 + x17)))
% 27.24/3.86 = { by axiom 4 (sos_03) R->L }
% 27.24/3.86 x17 ==> (x19 ==> (x17 + (0 + (x17 + 0))))
% 27.24/3.86 = { by axiom 8 (sos_01) R->L }
% 27.24/3.86 x17 ==> (x19 ==> (x17 + ((0 + x17) + 0)))
% 27.24/3.86 = { by lemma 28 R->L }
% 27.24/3.86 x17 ==> (x19 ==> (x17 + ((0 + x17) + ((x17 + ((x17 ==> (x17 ==> x19)) + 0)) ==> (x17 ==> x19)))))
% 27.24/3.86 = { by axiom 10 (sos_06) R->L }
% 27.24/3.86 x17 ==> (x19 ==> (x17 + ((0 + x17) + ((x17 + (fresh2(true, true, x17 ==> (x17 ==> x19), 0) + 0)) ==> (x17 ==> x19)))))
% 27.24/3.86 = { by axiom 11 (sos_07_1) R->L }
% 27.24/3.86 x17 ==> (x19 ==> (x17 + ((0 + x17) + ((x17 + (fresh2(fresh6(true, true, x17, 0, x17 ==> x19), true, x17 ==> (x17 ==> x19), 0) + 0)) ==> (x17 ==> x19)))))
% 27.24/3.86 = { by axiom 13 (sos_11) R->L }
% 27.24/3.86 x17 ==> (x19 ==> (x17 + ((0 + x17) + ((x17 + (fresh2(fresh6(fresh3(true, true, x18, x19, x17), true, x17, 0, x17 ==> x19), true, x17 ==> (x17 ==> x19), 0) + 0)) ==> (x17 ==> x19)))))
% 27.24/3.86 = { by axiom 11 (sos_07_1) R->L }
% 27.24/3.86 x17 ==> (x19 ==> (x17 + ((0 + x17) + ((x17 + (fresh2(fresh6(fresh3(fresh6(true, true, x19, x18, x18), true, x18, x19, x17), true, x17, 0, x17 ==> x19), true, x17 ==> (x17 ==> x19), 0) + 0)) ==> (x17 ==> x19)))))
% 27.24/3.86 = { by lemma 20 R->L }
% 27.24/3.86 x17 ==> (x19 ==> (x17 + ((0 + x17) + ((x17 + (fresh2(fresh6(fresh3(fresh6((x18 + x19) >= x18, true, x19, x18, x18), true, x18, x19, x17), true, x17, 0, x17 ==> x19), true, x17 ==> (x17 ==> x19), 0) + 0)) ==> (x17 ==> x19)))))
% 27.24/3.86 = { by axiom 3 (sos_02) }
% 27.24/3.86 x17 ==> (x19 ==> (x17 + ((0 + x17) + ((x17 + (fresh2(fresh6(fresh3(fresh6((x19 + x18) >= x18, true, x19, x18, x18), true, x18, x19, x17), true, x17, 0, x17 ==> x19), true, x17 ==> (x17 ==> x19), 0) + 0)) ==> (x17 ==> x19)))))
% 27.24/3.86 = { by axiom 17 (sos_07_1) }
% 27.24/3.86 x17 ==> (x19 ==> (x17 + ((0 + x17) + ((x17 + (fresh2(fresh6(fresh3(x18 >= (x19 ==> x18), true, x18, x19, x17), true, x17, 0, x17 ==> x19), true, x17 ==> (x17 ==> x19), 0) + 0)) ==> (x17 ==> x19)))))
% 27.24/3.86 = { by axiom 5 (goals_13) R->L }
% 27.24/3.86 x17 ==> (x19 ==> (x17 + ((0 + x17) + ((x17 + (fresh2(fresh6(fresh3(x18 >= x19, true, x18, x19, x17), true, x17, 0, x17 ==> x19), true, x17 ==> (x17 ==> x19), 0) + 0)) ==> (x17 ==> x19)))))
% 27.24/3.86 = { by axiom 16 (sos_11) }
% 27.24/3.86 x17 ==> (x19 ==> (x17 + ((0 + x17) + ((x17 + (fresh2(fresh6((x17 ==> x18) >= (x17 ==> x19), true, x17, 0, x17 ==> x19), true, x17 ==> (x17 ==> x19), 0) + 0)) ==> (x17 ==> x19)))))
% 27.24/3.86 = { by axiom 6 (goals_13_1) R->L }
% 27.24/3.86 x17 ==> (x19 ==> (x17 + ((0 + x17) + ((x17 + (fresh2(fresh6(x17 >= (x17 ==> x19), true, x17, 0, x17 ==> x19), true, x17 ==> (x17 ==> x19), 0) + 0)) ==> (x17 ==> x19)))))
% 27.24/3.86 = { by lemma 27 }
% 27.24/3.86 x17 ==> (x19 ==> (x17 + ((0 + x17) + ((x17 + (fresh2(0 >= (x17 ==> (x17 ==> x19)), true, x17 ==> (x17 ==> x19), 0) + 0)) ==> (x17 ==> x19)))))
% 27.24/3.86 = { by axiom 14 (sos_06) }
% 27.24/3.86 x17 ==> (x19 ==> (x17 + ((0 + x17) + ((x17 + (fresh((x17 ==> (x17 ==> x19)) >= 0, true, x17 ==> (x17 ==> x19), 0) + 0)) ==> (x17 ==> x19)))))
% 27.24/3.86 = { by axiom 2 (sos_08) }
% 27.24/3.86 x17 ==> (x19 ==> (x17 + ((0 + x17) + ((x17 + (fresh(true, true, x17 ==> (x17 ==> x19), 0) + 0)) ==> (x17 ==> x19)))))
% 27.24/3.86 = { by axiom 9 (sos_06) }
% 27.24/3.86 x17 ==> (x19 ==> (x17 + ((0 + x17) + ((x17 + (0 + 0)) ==> (x17 ==> x19)))))
% 27.24/3.86 = { by lemma 18 }
% 27.24/3.86 x17 ==> (x19 ==> (x17 + ((0 + x17) + ((x17 + 0) ==> (x17 ==> x19)))))
% 27.24/3.86 = { by axiom 3 (sos_02) }
% 27.24/3.86 x17 ==> (x19 ==> (x17 + ((0 + x17) + ((0 + x17) ==> (x17 ==> x19)))))
% 27.24/3.86 = { by axiom 7 (sos_12) R->L }
% 27.24/3.86 x17 ==> (x19 ==> (x17 + ((x17 ==> x19) + ((x17 ==> x19) ==> (0 + x17)))))
% 27.24/3.86 = { by lemma 26 }
% 27.24/3.86 x17 ==> (x19 ==> (x19 + (((x17 ==> x19) ==> (0 + x17)) + (x19 ==> x17))))
% 27.24/3.86 = { by axiom 3 (sos_02) }
% 27.24/3.86 x17 ==> (x19 ==> (x19 + ((x19 ==> x17) + ((x17 ==> x19) ==> (0 + x17)))))
% 27.24/3.86 = { by lemma 25 }
% 27.24/3.86 x17 ==> (x19 ==> (x19 + ((x19 ==> x17) + ((x19 ==> x17) ==> (0 + x17)))))
% 27.24/3.86 = { by axiom 7 (sos_12) R->L }
% 27.24/3.86 x17 ==> (x19 ==> (x19 + ((0 + x17) + ((0 + x17) ==> (x19 ==> x17)))))
% 27.24/3.86 = { by axiom 8 (sos_01) }
% 27.24/3.86 x17 ==> (x19 ==> (x19 + (0 + (x17 + ((0 + x17) ==> (x19 ==> x17))))))
% 27.24/3.86 = { by axiom 3 (sos_02) R->L }
% 27.24/3.86 x17 ==> (x19 ==> (x19 + (0 + (x17 + ((x17 + 0) ==> (x19 ==> x17))))))
% 27.24/3.86 = { by lemma 18 R->L }
% 27.24/3.86 x17 ==> (x19 ==> (x19 + (0 + (x17 + ((x17 + (0 + 0)) ==> (x19 ==> x17))))))
% 27.24/3.86 = { by lemma 22 R->L }
% 27.24/3.86 x17 ==> (x19 ==> (x19 + (0 + (x17 + ((x17 + (fresh(0 >= (x17 ==> (x19 ==> x17)), true, 0, x17 ==> (x19 ==> x17)) + 0)) ==> (x19 ==> x17))))))
% 27.24/3.86 = { by axiom 17 (sos_07_1) R->L }
% 27.24/3.86 x17 ==> (x19 ==> (x19 + (0 + (x17 + ((x17 + (fresh(fresh6((x17 + 0) >= (x19 ==> x17), true, x17, 0, x19 ==> x17), true, 0, x17 ==> (x19 ==> x17)) + 0)) ==> (x19 ==> x17))))))
% 27.24/3.86 = { by lemma 21 }
% 27.24/3.86 x17 ==> (x19 ==> (x19 + (0 + (x17 + ((x17 + (fresh(fresh6(true, true, x17, 0, x19 ==> x17), true, 0, x17 ==> (x19 ==> x17)) + 0)) ==> (x19 ==> x17))))))
% 27.24/3.86 = { by axiom 11 (sos_07_1) }
% 27.24/3.86 x17 ==> (x19 ==> (x19 + (0 + (x17 + ((x17 + (fresh(true, true, 0, x17 ==> (x19 ==> x17)) + 0)) ==> (x19 ==> x17))))))
% 27.24/3.86 = { by axiom 9 (sos_06) }
% 27.24/3.86 x17 ==> (x19 ==> (x19 + (0 + (x17 + ((x17 + ((x17 ==> (x19 ==> x17)) + 0)) ==> (x19 ==> x17))))))
% 27.24/3.86 = { by lemma 28 }
% 27.24/3.86 x17 ==> (x19 ==> (x19 + (0 + (x17 + 0))))
% 27.24/3.86 = { by axiom 4 (sos_03) }
% 27.24/3.86 x17 ==> (x19 ==> (x19 + (0 + x17)))
% 27.24/3.86 = { by lemma 19 }
% 27.24/3.86 x17 ==> (x19 ==> (0 + (x19 + x17)))
% 27.24/3.86 = { by lemma 18 }
% 27.24/3.86 x17 ==> (x19 ==> (x19 + x17))
% 27.24/3.86 = { by axiom 3 (sos_02) R->L }
% 27.24/3.86 x17 ==> (x19 ==> (x17 + x19))
% 27.24/3.86 = { by axiom 10 (sos_06) R->L }
% 27.24/3.86 fresh2(true, true, x17 ==> (x19 ==> (x17 + x19)), 0)
% 27.24/3.86 = { by axiom 11 (sos_07_1) R->L }
% 27.24/3.86 fresh2(fresh6(true, true, x17, 0, x19 ==> (x17 + x19)), true, x17 ==> (x19 ==> (x17 + x19)), 0)
% 27.24/3.86 = { by axiom 11 (sos_07_1) R->L }
% 27.24/3.86 fresh2(fresh6(fresh6(true, true, x19, x17, x19 + x17), true, x17, 0, x19 ==> (x17 + x19)), true, x17 ==> (x19 ==> (x17 + x19)), 0)
% 27.24/3.86 = { by axiom 1 (sos_04) R->L }
% 27.24/3.86 fresh2(fresh6(fresh6((x19 + x17) >= (x19 + x17), true, x19, x17, x19 + x17), true, x17, 0, x19 ==> (x17 + x19)), true, x17 ==> (x19 ==> (x17 + x19)), 0)
% 27.24/3.86 = { by axiom 17 (sos_07_1) }
% 27.24/3.86 fresh2(fresh6(x17 >= (x19 ==> (x19 + x17)), true, x17, 0, x19 ==> (x17 + x19)), true, x17 ==> (x19 ==> (x17 + x19)), 0)
% 27.24/3.87 = { by axiom 3 (sos_02) }
% 27.24/3.87 fresh2(fresh6(x17 >= (x19 ==> (x17 + x19)), true, x17, 0, x19 ==> (x17 + x19)), true, x17 ==> (x19 ==> (x17 + x19)), 0)
% 27.24/3.87 = { by lemma 27 }
% 27.24/3.87 fresh2(0 >= (x17 ==> (x19 ==> (x17 + x19))), true, x17 ==> (x19 ==> (x17 + x19)), 0)
% 27.24/3.87 = { by axiom 14 (sos_06) }
% 27.24/3.87 fresh((x17 ==> (x19 ==> (x17 + x19))) >= 0, true, x17 ==> (x19 ==> (x17 + x19)), 0)
% 27.24/3.87 = { by axiom 2 (sos_08) }
% 27.24/3.87 fresh(true, true, x17 ==> (x19 ==> (x17 + x19)), 0)
% 27.24/3.87 = { by axiom 9 (sos_06) }
% 27.24/3.87 0
% 27.24/3.87
% 27.24/3.87 Goal 1 (goals_13_2): x17 = x19.
% 27.24/3.87 Proof:
% 27.24/3.87 x17
% 27.24/3.87 = { by axiom 4 (sos_03) R->L }
% 27.24/3.87 x17 + 0
% 27.24/3.87 = { by lemma 29 R->L }
% 27.24/3.87 x17 + (x19 ==> x17)
% 27.24/3.87 = { by lemma 25 R->L }
% 27.24/3.87 x17 + (x17 ==> x19)
% 27.24/3.87 = { by axiom 7 (sos_12) }
% 27.24/3.87 x19 + (x19 ==> x17)
% 27.24/3.87 = { by lemma 29 }
% 27.24/3.87 x19 + 0
% 27.24/3.87 = { by axiom 4 (sos_03) }
% 27.24/3.87 x19
% 27.24/3.87 % SZS output end Proof
% 27.24/3.87
% 27.24/3.87 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------