TSTP Solution File: LCL898+1 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : LCL898+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 : n002.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 26.26s 3.72s
% Output : Proof 26.55s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : LCL898+1 : TPTP v8.1.2. Released v5.5.0.
% 0.13/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.13/0.34 % Computer : n002.cluster.edu
% 0.13/0.34 % Model : x86_64 x86_64
% 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34 % Memory : 8042.1875MB
% 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.34 % CPULimit : 300
% 0.13/0.34 % WCLimit : 300
% 0.13/0.34 % DateTime : Thu Aug 24 17:40:01 EDT 2023
% 0.13/0.35 % CPUTime :
% 26.26/3.72 Command-line arguments: --no-flatten-goal
% 26.26/3.72
% 26.26/3.72 % SZS status Theorem
% 26.26/3.72
% 26.55/3.76 % SZS output start Proof
% 26.55/3.76 Take the following subset of the input axioms:
% 26.55/3.76 fof(goals_15, conjecture, ![X17, X18]: '==>'('==>'(X17, X18), X18)='==>'('==>'(X18, X17), X17)).
% 26.55/3.76 fof(sos_01, axiom, ![A, B, C]: '+'('+'(A, B), C)='+'(A, '+'(B, C))).
% 26.55/3.76 fof(sos_02, axiom, ![A2, B2]: '+'(A2, B2)='+'(B2, A2)).
% 26.55/3.76 fof(sos_03, axiom, ![A2]: '+'(A2, '0')=A2).
% 26.55/3.76 fof(sos_04, axiom, ![A2]: '>='(A2, A2)).
% 26.55/3.76 fof(sos_06, axiom, ![X3, X4]: (('>='(X3, X4) & '>='(X4, X3)) => X3=X4)).
% 26.55/3.76 fof(sos_07, axiom, ![X5, X6, X7]: ('>='('+'(X5, X6), X7) <=> '>='(X6, '==>'(X5, X7)))).
% 26.55/3.76 fof(sos_08, axiom, ![A2]: '>='(A2, '0')).
% 26.55/3.76 fof(sos_09, axiom, ![X8, X9, X10]: ('>='(X8, X9) => '>='('+'(X8, X10), '+'(X9, X10)))).
% 26.55/3.76 fof(sos_11, axiom, ![X14, X15, X16]: ('>='(X14, X15) => '>='('==>'(X16, X14), '==>'(X16, X15)))).
% 26.55/3.76 fof(sos_12, axiom, ![A2, B2]: '+'(A2, '==>'(A2, B2))='+'(B2, '==>'(B2, A2))).
% 26.55/3.76 fof(sos_13, axiom, ![A2]: '==>'('==>'(A2, '1'), '1')=A2).
% 26.55/3.76 fof(sos_14, axiom, ![A2]: '+'(A2, '1')='1').
% 26.55/3.76
% 26.55/3.76 Now clausify the problem and encode Horn clauses using encoding 3 of
% 26.55/3.76 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 26.55/3.76 We repeatedly replace C & s=t => u=v by the two clauses:
% 26.55/3.76 fresh(y, y, x1...xn) = u
% 26.55/3.76 C => fresh(s, t, x1...xn) = v
% 26.55/3.76 where fresh is a fresh function symbol and x1..xn are the free
% 26.55/3.76 variables of u and v.
% 26.55/3.76 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 26.55/3.76 input problem has no model of domain size 1).
% 26.55/3.76
% 26.55/3.76 The encoding turns the above axioms into the following unit equations and goals:
% 26.55/3.76
% 26.55/3.76 Axiom 1 (sos_02): X + Y = Y + X.
% 26.55/3.76 Axiom 2 (sos_03): X + 0 = X.
% 26.55/3.76 Axiom 3 (sos_14): X + 1 = 1.
% 26.55/3.76 Axiom 4 (sos_04): X >= X = true.
% 26.55/3.76 Axiom 5 (sos_08): X >= 0 = true.
% 26.55/3.76 Axiom 6 (sos_06): fresh(X, X, Y, Z) = Z.
% 26.55/3.76 Axiom 7 (sos_06): fresh2(X, X, Y, Z) = Y.
% 26.55/3.76 Axiom 8 (sos_12): X + (X ==> Y) = Y + (Y ==> X).
% 26.55/3.76 Axiom 9 (sos_01): (X + Y) + Z = X + (Y + Z).
% 26.55/3.76 Axiom 10 (sos_13): (X ==> 1) ==> 1 = X.
% 26.55/3.76 Axiom 11 (sos_07_1): fresh6(X, X, Y, Z, W) = true.
% 26.55/3.76 Axiom 12 (sos_09): fresh5(X, X, Y, Z, W) = true.
% 26.55/3.76 Axiom 13 (sos_11): fresh3(X, X, Y, Z, W) = true.
% 26.55/3.76 Axiom 14 (sos_06): fresh2(X >= Y, true, Y, X) = fresh(Y >= X, true, Y, X).
% 26.55/3.76 Axiom 15 (sos_09): fresh5(X >= Y, true, X, Y, Z) = (X + Z) >= (Y + Z).
% 26.55/3.76 Axiom 16 (sos_11): fresh3(X >= Y, true, X, Y, Z) = (Z ==> X) >= (Z ==> Y).
% 26.55/3.76 Axiom 17 (sos_07_1): fresh6((X + Y) >= Z, true, X, Y, Z) = Y >= (X ==> Z).
% 26.55/3.76
% 26.55/3.76 Lemma 18: 0 + X = X.
% 26.55/3.76 Proof:
% 26.55/3.76 0 + X
% 26.55/3.76 = { by axiom 1 (sos_02) R->L }
% 26.55/3.76 X + 0
% 26.55/3.76 = { by axiom 2 (sos_03) }
% 26.55/3.76 X
% 26.55/3.76
% 26.55/3.76 Lemma 19: X ==> 0 = 0.
% 26.55/3.76 Proof:
% 26.55/3.76 X ==> 0
% 26.55/3.76 = { by axiom 6 (sos_06) R->L }
% 26.55/3.76 fresh(true, true, 0, X ==> 0)
% 26.55/3.76 = { by axiom 11 (sos_07_1) R->L }
% 26.55/3.76 fresh(fresh6(true, true, X, 0, 0), true, 0, X ==> 0)
% 26.55/3.76 = { by axiom 5 (sos_08) R->L }
% 26.55/3.76 fresh(fresh6((X + 0) >= 0, true, X, 0, 0), true, 0, X ==> 0)
% 26.55/3.76 = { by axiom 17 (sos_07_1) }
% 26.55/3.76 fresh(0 >= (X ==> 0), true, 0, X ==> 0)
% 26.55/3.76 = { by axiom 14 (sos_06) R->L }
% 26.55/3.76 fresh2((X ==> 0) >= 0, true, 0, X ==> 0)
% 26.55/3.76 = { by axiom 5 (sos_08) }
% 26.55/3.76 fresh2(true, true, 0, X ==> 0)
% 26.55/3.76 = { by axiom 7 (sos_06) }
% 26.55/3.76 0
% 26.55/3.76
% 26.55/3.76 Lemma 20: 0 ==> X = X.
% 26.55/3.76 Proof:
% 26.55/3.76 0 ==> X
% 26.55/3.76 = { by lemma 18 R->L }
% 26.55/3.76 0 + (0 ==> X)
% 26.55/3.76 = { by axiom 8 (sos_12) R->L }
% 26.55/3.76 X + (X ==> 0)
% 26.55/3.76 = { by lemma 19 }
% 26.55/3.76 X + 0
% 26.55/3.76 = { by axiom 2 (sos_03) }
% 26.55/3.76 X
% 26.55/3.76
% 26.55/3.76 Lemma 21: (X + Y) >= X = true.
% 26.55/3.76 Proof:
% 26.55/3.76 (X + Y) >= X
% 26.55/3.76 = { by axiom 1 (sos_02) R->L }
% 26.55/3.76 (Y + X) >= X
% 26.55/3.76 = { by lemma 18 R->L }
% 26.55/3.76 (Y + X) >= (0 + X)
% 26.55/3.76 = { by axiom 15 (sos_09) R->L }
% 26.55/3.76 fresh5(Y >= 0, true, Y, 0, X)
% 26.55/3.76 = { by axiom 5 (sos_08) }
% 26.55/3.76 fresh5(true, true, Y, 0, X)
% 26.55/3.76 = { by axiom 12 (sos_09) }
% 26.55/3.76 true
% 26.55/3.76
% 26.55/3.76 Lemma 22: 1 ==> X = 0.
% 26.55/3.76 Proof:
% 26.55/3.76 1 ==> X
% 26.55/3.76 = { by axiom 7 (sos_06) R->L }
% 26.55/3.76 fresh2(true, true, 1 ==> X, 0)
% 26.55/3.76 = { by axiom 13 (sos_11) R->L }
% 26.55/3.76 fresh2(fresh3(true, true, 1, X, 1), true, 1 ==> X, 0)
% 26.55/3.76 = { by lemma 21 R->L }
% 26.55/3.76 fresh2(fresh3((X + 1) >= X, true, 1, X, 1), true, 1 ==> X, 0)
% 26.55/3.76 = { by axiom 3 (sos_14) }
% 26.55/3.76 fresh2(fresh3(1 >= X, true, 1, X, 1), true, 1 ==> X, 0)
% 26.55/3.76 = { by axiom 16 (sos_11) }
% 26.55/3.76 fresh2((1 ==> 1) >= (1 ==> X), true, 1 ==> X, 0)
% 26.55/3.76 = { by lemma 20 R->L }
% 26.55/3.76 fresh2(((0 ==> 1) ==> 1) >= (1 ==> X), true, 1 ==> X, 0)
% 26.55/3.76 = { by axiom 10 (sos_13) }
% 26.55/3.76 fresh2(0 >= (1 ==> X), true, 1 ==> X, 0)
% 26.55/3.76 = { by axiom 14 (sos_06) }
% 26.55/3.76 fresh((1 ==> X) >= 0, true, 1 ==> X, 0)
% 26.55/3.76 = { by axiom 5 (sos_08) }
% 26.55/3.76 fresh(true, true, 1 ==> X, 0)
% 26.55/3.76 = { by axiom 6 (sos_06) }
% 26.55/3.76 0
% 26.55/3.76
% 26.55/3.76 Lemma 23: X + (Y + Z) = Y + (X + Z).
% 26.55/3.76 Proof:
% 26.55/3.76 X + (Y + Z)
% 26.55/3.76 = { by axiom 1 (sos_02) R->L }
% 26.55/3.76 (Y + Z) + X
% 26.55/3.76 = { by axiom 9 (sos_01) }
% 26.55/3.76 Y + (Z + X)
% 26.55/3.76 = { by axiom 1 (sos_02) }
% 26.55/3.76 Y + (X + Z)
% 26.55/3.76
% 26.55/3.76 Lemma 24: Z + ((Z ==> X) + Y) = X + (Y + (X ==> Z)).
% 26.55/3.76 Proof:
% 26.55/3.76 Z + ((Z ==> X) + Y)
% 26.55/3.76 = { by axiom 9 (sos_01) R->L }
% 26.55/3.76 (Z + (Z ==> X)) + Y
% 26.55/3.76 = { by axiom 8 (sos_12) R->L }
% 26.55/3.76 (X + (X ==> Z)) + Y
% 26.55/3.76 = { by axiom 9 (sos_01) }
% 26.55/3.76 X + ((X ==> Z) + Y)
% 26.55/3.76 = { by axiom 1 (sos_02) }
% 26.55/3.76 X + (Y + (X ==> Z))
% 26.55/3.76
% 26.55/3.76 Lemma 25: X + (Y + (Y ==> (X ==> 1))) = 1.
% 26.55/3.76 Proof:
% 26.55/3.76 X + (Y + (Y ==> (X ==> 1)))
% 26.55/3.76 = { by axiom 8 (sos_12) R->L }
% 26.55/3.76 X + ((X ==> 1) + ((X ==> 1) ==> Y))
% 26.55/3.76 = { by lemma 24 }
% 26.55/3.76 1 + (((X ==> 1) ==> Y) + (1 ==> X))
% 26.55/3.76 = { by axiom 1 (sos_02) R->L }
% 26.55/3.76 (((X ==> 1) ==> Y) + (1 ==> X)) + 1
% 26.55/3.76 = { by axiom 3 (sos_14) }
% 26.55/3.76 1
% 26.55/3.76
% 26.55/3.76 Lemma 26: X >= (Y ==> (X + Y)) = true.
% 26.55/3.76 Proof:
% 26.55/3.76 X >= (Y ==> (X + Y))
% 26.55/3.76 = { by axiom 1 (sos_02) R->L }
% 26.55/3.76 X >= (Y ==> (Y + X))
% 26.55/3.76 = { by axiom 17 (sos_07_1) R->L }
% 26.55/3.76 fresh6((Y + X) >= (Y + X), true, Y, X, Y + X)
% 26.55/3.76 = { by axiom 4 (sos_04) }
% 26.55/3.76 fresh6(true, true, Y, X, Y + X)
% 26.55/3.76 = { by axiom 11 (sos_07_1) }
% 26.55/3.76 true
% 26.55/3.76
% 26.55/3.76 Lemma 27: (X ==> (Y ==> 1)) >= (Y ==> (X ==> 1)) = true.
% 26.55/3.76 Proof:
% 26.55/3.76 (X ==> (Y ==> 1)) >= (Y ==> (X ==> 1))
% 26.55/3.76 = { by lemma 25 R->L }
% 26.55/3.76 (X ==> (Y ==> 1)) >= (Y ==> (X ==> (Y + (X + (X ==> (Y ==> 1))))))
% 26.55/3.76 = { by axiom 9 (sos_01) R->L }
% 26.55/3.76 (X ==> (Y ==> 1)) >= (Y ==> (X ==> ((Y + X) + (X ==> (Y ==> 1)))))
% 26.55/3.76 = { by axiom 1 (sos_02) R->L }
% 26.55/3.76 (X ==> (Y ==> 1)) >= (Y ==> (X ==> ((X ==> (Y ==> 1)) + (Y + X))))
% 26.55/3.76 = { by lemma 23 R->L }
% 26.55/3.76 (X ==> (Y ==> 1)) >= (Y ==> (X ==> (Y + ((X ==> (Y ==> 1)) + X))))
% 26.55/3.76 = { by axiom 9 (sos_01) R->L }
% 26.55/3.76 (X ==> (Y ==> 1)) >= (Y ==> (X ==> ((Y + (X ==> (Y ==> 1))) + X)))
% 26.55/3.76 = { by axiom 17 (sos_07_1) R->L }
% 26.55/3.76 fresh6((Y + (X ==> (Y ==> 1))) >= (X ==> ((Y + (X ==> (Y ==> 1))) + X)), true, Y, X ==> (Y ==> 1), X ==> ((Y + (X ==> (Y ==> 1))) + X))
% 26.55/3.76 = { by lemma 26 }
% 26.55/3.76 fresh6(true, true, Y, X ==> (Y ==> 1), X ==> ((Y + (X ==> (Y ==> 1))) + X))
% 26.55/3.76 = { by axiom 11 (sos_07_1) }
% 26.55/3.76 true
% 26.55/3.76
% 26.55/3.76 Lemma 28: X ==> (Y ==> 1) = Y ==> (X ==> 1).
% 26.55/3.76 Proof:
% 26.55/3.76 X ==> (Y ==> 1)
% 26.55/3.76 = { by axiom 6 (sos_06) R->L }
% 26.55/3.76 fresh(true, true, Y ==> (X ==> 1), X ==> (Y ==> 1))
% 26.55/3.76 = { by lemma 27 R->L }
% 26.55/3.76 fresh((Y ==> (X ==> 1)) >= (X ==> (Y ==> 1)), true, Y ==> (X ==> 1), X ==> (Y ==> 1))
% 26.55/3.76 = { by axiom 14 (sos_06) R->L }
% 26.55/3.76 fresh2((X ==> (Y ==> 1)) >= (Y ==> (X ==> 1)), true, Y ==> (X ==> 1), X ==> (Y ==> 1))
% 26.55/3.76 = { by lemma 27 }
% 26.55/3.76 fresh2(true, true, Y ==> (X ==> 1), X ==> (Y ==> 1))
% 26.55/3.76 = { by axiom 7 (sos_06) }
% 26.55/3.76 Y ==> (X ==> 1)
% 26.55/3.76
% 26.55/3.76 Lemma 29: X ==> (Y ==> (X + Y)) = 0.
% 26.55/3.76 Proof:
% 26.55/3.76 X ==> (Y ==> (X + Y))
% 26.55/3.76 = { by axiom 7 (sos_06) R->L }
% 26.55/3.76 fresh2(true, true, X ==> (Y ==> (X + Y)), 0)
% 26.55/3.76 = { by axiom 11 (sos_07_1) R->L }
% 26.55/3.76 fresh2(fresh6(true, true, X, 0, Y ==> (X + Y)), true, X ==> (Y ==> (X + Y)), 0)
% 26.55/3.76 = { by lemma 26 R->L }
% 26.55/3.76 fresh2(fresh6(X >= (Y ==> (X + Y)), true, X, 0, Y ==> (X + Y)), true, X ==> (Y ==> (X + Y)), 0)
% 26.55/3.76 = { by axiom 2 (sos_03) R->L }
% 26.55/3.76 fresh2(fresh6((X + 0) >= (Y ==> (X + Y)), true, X, 0, Y ==> (X + Y)), true, X ==> (Y ==> (X + Y)), 0)
% 26.55/3.76 = { by axiom 17 (sos_07_1) }
% 26.55/3.76 fresh2(0 >= (X ==> (Y ==> (X + Y))), true, X ==> (Y ==> (X + Y)), 0)
% 26.55/3.76 = { by axiom 14 (sos_06) }
% 26.55/3.76 fresh((X ==> (Y ==> (X + Y))) >= 0, true, X ==> (Y ==> (X + Y)), 0)
% 26.55/3.76 = { by axiom 5 (sos_08) }
% 26.55/3.76 fresh(true, true, X ==> (Y ==> (X + Y)), 0)
% 26.55/3.76 = { by axiom 6 (sos_06) }
% 26.55/3.76 0
% 26.55/3.76
% 26.55/3.76 Lemma 30: (X ==> 1) ==> (Y ==> 1) = Y ==> X.
% 26.55/3.76 Proof:
% 26.55/3.76 (X ==> 1) ==> (Y ==> 1)
% 26.55/3.76 = { by lemma 28 R->L }
% 26.55/3.76 Y ==> ((X ==> 1) ==> 1)
% 26.55/3.76 = { by axiom 10 (sos_13) }
% 26.55/3.76 Y ==> X
% 26.55/3.76
% 26.55/3.76 Lemma 31: (X ==> 1) ==> Y = (Y ==> 1) ==> X.
% 26.55/3.76 Proof:
% 26.55/3.76 (X ==> 1) ==> Y
% 26.55/3.76 = { by lemma 30 R->L }
% 26.55/3.76 (Y ==> 1) ==> ((X ==> 1) ==> 1)
% 26.55/3.76 = { by axiom 10 (sos_13) }
% 26.55/3.76 (Y ==> 1) ==> X
% 26.55/3.76
% 26.55/3.76 Lemma 32: (X ==> (Y ==> Z)) >= (Y ==> (X ==> Z)) = true.
% 26.55/3.76 Proof:
% 26.55/3.76 (X ==> (Y ==> Z)) >= (Y ==> (X ==> Z))
% 26.55/3.76 = { by axiom 17 (sos_07_1) R->L }
% 26.55/3.76 fresh6((Y + (X ==> (Y ==> Z))) >= (X ==> Z), true, Y, X ==> (Y ==> Z), X ==> Z)
% 26.55/3.76 = { by axiom 17 (sos_07_1) R->L }
% 26.55/3.76 fresh6(fresh6((X + (Y + (X ==> (Y ==> Z)))) >= Z, true, X, Y + (X ==> (Y ==> Z)), Z), true, Y, X ==> (Y ==> Z), X ==> Z)
% 26.55/3.76 = { by lemma 23 R->L }
% 26.55/3.76 fresh6(fresh6((Y + (X + (X ==> (Y ==> Z)))) >= Z, true, X, Y + (X ==> (Y ==> Z)), Z), true, Y, X ==> (Y ==> Z), X ==> Z)
% 26.55/3.76 = { by axiom 8 (sos_12) }
% 26.55/3.76 fresh6(fresh6((Y + ((Y ==> Z) + ((Y ==> Z) ==> X))) >= Z, true, X, Y + (X ==> (Y ==> Z)), Z), true, Y, X ==> (Y ==> Z), X ==> Z)
% 26.55/3.76 = { by lemma 24 }
% 26.55/3.77 fresh6(fresh6((Z + (((Y ==> Z) ==> X) + (Z ==> Y))) >= Z, true, X, Y + (X ==> (Y ==> Z)), Z), true, Y, X ==> (Y ==> Z), X ==> Z)
% 26.55/3.77 = { by lemma 21 }
% 26.55/3.77 fresh6(fresh6(true, true, X, Y + (X ==> (Y ==> Z)), Z), true, Y, X ==> (Y ==> Z), X ==> Z)
% 26.55/3.77 = { by axiom 11 (sos_07_1) }
% 26.55/3.77 fresh6(true, true, Y, X ==> (Y ==> Z), X ==> Z)
% 26.55/3.77 = { by axiom 11 (sos_07_1) }
% 26.55/3.77 true
% 26.55/3.77
% 26.55/3.77 Lemma 33: ((X ==> Y) ==> 1) ==> Z = X ==> ((Y ==> 1) ==> Z).
% 26.55/3.77 Proof:
% 26.55/3.77 ((X ==> Y) ==> 1) ==> Z
% 26.55/3.77 = { by lemma 31 R->L }
% 26.55/3.77 (Z ==> 1) ==> (X ==> Y)
% 26.55/3.77 = { by axiom 7 (sos_06) R->L }
% 26.55/3.77 fresh2(true, true, (Z ==> 1) ==> (X ==> Y), X ==> ((Z ==> 1) ==> Y))
% 26.55/3.77 = { by lemma 32 R->L }
% 26.55/3.77 fresh2((X ==> ((Z ==> 1) ==> Y)) >= ((Z ==> 1) ==> (X ==> Y)), true, (Z ==> 1) ==> (X ==> Y), X ==> ((Z ==> 1) ==> Y))
% 26.55/3.77 = { by axiom 14 (sos_06) }
% 26.55/3.77 fresh(((Z ==> 1) ==> (X ==> Y)) >= (X ==> ((Z ==> 1) ==> Y)), true, (Z ==> 1) ==> (X ==> Y), X ==> ((Z ==> 1) ==> Y))
% 26.55/3.77 = { by lemma 32 }
% 26.55/3.77 fresh(true, true, (Z ==> 1) ==> (X ==> Y), X ==> ((Z ==> 1) ==> Y))
% 26.55/3.77 = { by axiom 6 (sos_06) }
% 26.55/3.77 X ==> ((Z ==> 1) ==> Y)
% 26.55/3.77 = { by lemma 31 }
% 26.55/3.77 X ==> ((Y ==> 1) ==> Z)
% 26.55/3.77
% 26.55/3.77 Lemma 34: (X + Y) ==> 1 = X ==> (Y ==> 1).
% 26.55/3.77 Proof:
% 26.55/3.77 (X + Y) ==> 1
% 26.55/3.77 = { by lemma 18 R->L }
% 26.55/3.77 (0 + (X + Y)) ==> 1
% 26.55/3.77 = { by lemma 23 R->L }
% 26.55/3.77 (X + (0 + Y)) ==> 1
% 26.55/3.77 = { by axiom 1 (sos_02) R->L }
% 26.55/3.77 (X + (Y + 0)) ==> 1
% 26.55/3.77 = { by lemma 20 R->L }
% 26.55/3.77 (X + (Y + (0 ==> 0))) ==> 1
% 26.55/3.77 = { by lemma 29 R->L }
% 26.55/3.77 (X + (Y + ((X ==> (Y ==> (X + Y))) ==> 0))) ==> 1
% 26.55/3.77 = { by lemma 20 R->L }
% 26.55/3.77 (X + (Y + ((X ==> (Y ==> (0 ==> (X + Y)))) ==> 0))) ==> 1
% 26.55/3.77 = { by lemma 22 R->L }
% 26.55/3.77 (X + (Y + ((X ==> (Y ==> ((1 ==> 1) ==> (X + Y)))) ==> 0))) ==> 1
% 26.55/3.77 = { by lemma 33 R->L }
% 26.55/3.77 (X + (Y + ((X ==> (((Y ==> 1) ==> 1) ==> (X + Y))) ==> 0))) ==> 1
% 26.55/3.77 = { by lemma 33 R->L }
% 26.55/3.77 (X + (Y + ((((X ==> (Y ==> 1)) ==> 1) ==> (X + Y)) ==> 0))) ==> 1
% 26.55/3.77 = { by lemma 18 R->L }
% 26.55/3.77 (X + (Y + (0 + ((((X ==> (Y ==> 1)) ==> 1) ==> (X + Y)) ==> 0)))) ==> 1
% 26.55/3.77 = { by axiom 9 (sos_01) R->L }
% 26.55/3.77 ((X + Y) + (0 + ((((X ==> (Y ==> 1)) ==> 1) ==> (X + Y)) ==> 0))) ==> 1
% 26.55/3.77 = { by lemma 29 R->L }
% 26.55/3.77 ((X + Y) + (((Y + X) ==> ((X ==> (Y ==> 1)) ==> ((Y + X) + (X ==> (Y ==> 1))))) + ((((X ==> (Y ==> 1)) ==> 1) ==> (X + Y)) ==> 0))) ==> 1
% 26.55/3.77 = { by axiom 9 (sos_01) }
% 26.55/3.77 ((X + Y) + (((Y + X) ==> ((X ==> (Y ==> 1)) ==> (Y + (X + (X ==> (Y ==> 1)))))) + ((((X ==> (Y ==> 1)) ==> 1) ==> (X + Y)) ==> 0))) ==> 1
% 26.55/3.77 = { by lemma 25 }
% 26.55/3.77 ((X + Y) + (((Y + X) ==> ((X ==> (Y ==> 1)) ==> 1)) + ((((X ==> (Y ==> 1)) ==> 1) ==> (X + Y)) ==> 0))) ==> 1
% 26.55/3.77 = { by axiom 1 (sos_02) }
% 26.55/3.77 ((X + Y) + (((X + Y) ==> ((X ==> (Y ==> 1)) ==> 1)) + ((((X ==> (Y ==> 1)) ==> 1) ==> (X + Y)) ==> 0))) ==> 1
% 26.55/3.77 = { by axiom 1 (sos_02) R->L }
% 26.55/3.77 ((X + Y) + (((((X ==> (Y ==> 1)) ==> 1) ==> (X + Y)) ==> 0) + ((X + Y) ==> ((X ==> (Y ==> 1)) ==> 1)))) ==> 1
% 26.55/3.77 = { by lemma 24 R->L }
% 26.55/3.77 (((X ==> (Y ==> 1)) ==> 1) + ((((X ==> (Y ==> 1)) ==> 1) ==> (X + Y)) + ((((X ==> (Y ==> 1)) ==> 1) ==> (X + Y)) ==> 0))) ==> 1
% 26.55/3.77 = { by axiom 8 (sos_12) }
% 26.55/3.77 (((X ==> (Y ==> 1)) ==> 1) + (0 + (0 ==> (((X ==> (Y ==> 1)) ==> 1) ==> (X + Y))))) ==> 1
% 26.55/3.77 = { by axiom 1 (sos_02) }
% 26.55/3.77 ((0 + (0 ==> (((X ==> (Y ==> 1)) ==> 1) ==> (X + Y)))) + ((X ==> (Y ==> 1)) ==> 1)) ==> 1
% 26.55/3.77 = { by axiom 9 (sos_01) }
% 26.55/3.77 (0 + ((0 ==> (((X ==> (Y ==> 1)) ==> 1) ==> (X + Y))) + ((X ==> (Y ==> 1)) ==> 1))) ==> 1
% 26.55/3.77 = { by axiom 1 (sos_02) }
% 26.55/3.77 (0 + (((X ==> (Y ==> 1)) ==> 1) + (0 ==> (((X ==> (Y ==> 1)) ==> 1) ==> (X + Y))))) ==> 1
% 26.55/3.77 = { by lemma 33 }
% 26.55/3.77 (0 + (((X ==> (Y ==> 1)) ==> 1) + (0 ==> (X ==> (((Y ==> 1) ==> 1) ==> (X + Y)))))) ==> 1
% 26.55/3.77 = { by lemma 33 }
% 26.55/3.77 (0 + (((X ==> (Y ==> 1)) ==> 1) + (0 ==> (X ==> (Y ==> ((1 ==> 1) ==> (X + Y))))))) ==> 1
% 26.55/3.77 = { by lemma 22 }
% 26.55/3.77 (0 + (((X ==> (Y ==> 1)) ==> 1) + (0 ==> (X ==> (Y ==> (0 ==> (X + Y))))))) ==> 1
% 26.55/3.77 = { by lemma 20 }
% 26.55/3.77 (0 + (((X ==> (Y ==> 1)) ==> 1) + (0 ==> (X ==> (Y ==> (X + Y)))))) ==> 1
% 26.55/3.77 = { by lemma 29 }
% 26.55/3.77 (0 + (((X ==> (Y ==> 1)) ==> 1) + (0 ==> 0))) ==> 1
% 26.55/3.77 = { by lemma 19 }
% 26.55/3.77 (0 + (((X ==> (Y ==> 1)) ==> 1) + 0)) ==> 1
% 26.55/3.77 = { by axiom 2 (sos_03) }
% 26.55/3.77 (0 + ((X ==> (Y ==> 1)) ==> 1)) ==> 1
% 26.55/3.77 = { by lemma 18 }
% 26.55/3.77 ((X ==> (Y ==> 1)) ==> 1) ==> 1
% 26.55/3.77 = { by lemma 28 }
% 26.55/3.77 ((Y ==> (X ==> 1)) ==> 1) ==> 1
% 26.55/3.77 = { by axiom 10 (sos_13) }
% 26.55/3.77 Y ==> (X ==> 1)
% 26.55/3.77 = { by lemma 28 }
% 26.55/3.77 X ==> (Y ==> 1)
% 26.55/3.77
% 26.55/3.77 Goal 1 (goals_15): (x17 ==> x18) ==> x18 = (x18 ==> x17) ==> x17.
% 26.55/3.77 Proof:
% 26.55/3.77 (x17 ==> x18) ==> x18
% 26.55/3.77 = { by lemma 30 R->L }
% 26.55/3.77 (x18 ==> 1) ==> ((x17 ==> x18) ==> 1)
% 26.55/3.77 = { by lemma 34 R->L }
% 26.55/3.77 ((x18 ==> 1) + (x17 ==> x18)) ==> 1
% 26.55/3.77 = { by lemma 30 R->L }
% 26.55/3.77 ((x18 ==> 1) + ((x18 ==> 1) ==> (x17 ==> 1))) ==> 1
% 26.55/3.77 = { by axiom 8 (sos_12) R->L }
% 26.55/3.77 ((x17 ==> 1) + ((x17 ==> 1) ==> (x18 ==> 1))) ==> 1
% 26.55/3.77 = { by lemma 30 }
% 26.55/3.77 ((x17 ==> 1) + (x18 ==> x17)) ==> 1
% 26.55/3.77 = { by lemma 34 }
% 26.55/3.77 (x17 ==> 1) ==> ((x18 ==> x17) ==> 1)
% 26.55/3.77 = { by lemma 30 }
% 26.55/3.77 (x18 ==> x17) ==> x17
% 26.55/3.77 % SZS output end Proof
% 26.55/3.77
% 26.55/3.77 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------