TSTP Solution File: LCL898+1 by Twee---2.5.0
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.5.0
% Problem : LCL898+1 : TPTP v8.2.0. Released v5.5.0.
% Transfm : none
% Format : tptp:raw
% Command : parallel-twee /export/starexec/sandbox/benchmark/theBenchmark.p --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding
% Computer : n029.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 : Mon Jun 24 11:23:26 EDT 2024
% Result : Theorem 54.59s 7.30s
% Output : Proof 55.37s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.03/0.13 % Problem : LCL898+1 : TPTP v8.2.0. Released v5.5.0.
% 0.03/0.13 % Command : parallel-twee /export/starexec/sandbox/benchmark/theBenchmark.p --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding
% 0.13/0.34 % Computer : n029.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 : Sat Jun 22 11:08:39 EDT 2024
% 0.13/0.34 % CPUTime :
% 54.59/7.30 Command-line arguments: --no-flatten-goal
% 54.59/7.30
% 54.59/7.30 % SZS status Theorem
% 54.59/7.30
% 55.37/7.32 % SZS output start Proof
% 55.37/7.32 Take the following subset of the input axioms:
% 55.37/7.32 fof(goals_15, conjecture, ![X17, X18]: '==>'('==>'(X17, X18), X18)='==>'('==>'(X18, X17), X17)).
% 55.37/7.32 fof(sos_01, axiom, ![A, B, C]: '+'('+'(A, B), C)='+'(A, '+'(B, C))).
% 55.37/7.32 fof(sos_02, axiom, ![A2, B2]: '+'(A2, B2)='+'(B2, A2)).
% 55.37/7.32 fof(sos_03, axiom, ![A2]: '+'(A2, '0')=A2).
% 55.37/7.32 fof(sos_04, axiom, ![A2]: '>='(A2, A2)).
% 55.37/7.32 fof(sos_05, axiom, ![X0, X1, X2]: (('>='(X0, X1) & '>='(X1, X2)) => '>='(X0, X2))).
% 55.37/7.32 fof(sos_06, axiom, ![X3, X4]: (('>='(X3, X4) & '>='(X4, X3)) => X3=X4)).
% 55.37/7.32 fof(sos_07, axiom, ![X5, X6, X7]: ('>='('+'(X5, X6), X7) <=> '>='(X6, '==>'(X5, X7)))).
% 55.37/7.32 fof(sos_08, axiom, ![A2]: '>='(A2, '0')).
% 55.37/7.32 fof(sos_09, axiom, ![X8, X9, X10]: ('>='(X8, X9) => '>='('+'(X8, X10), '+'(X9, X10)))).
% 55.37/7.32 fof(sos_12, axiom, ![A2, B2]: '+'(A2, '==>'(A2, B2))='+'(B2, '==>'(B2, A2))).
% 55.37/7.32 fof(sos_13, axiom, ![A2]: '==>'('==>'(A2, '1'), '1')=A2).
% 55.37/7.32 fof(sos_14, axiom, ![A2]: '+'(A2, '1')='1').
% 55.37/7.32
% 55.37/7.32 Now clausify the problem and encode Horn clauses using encoding 3 of
% 55.37/7.32 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 55.37/7.32 We repeatedly replace C & s=t => u=v by the two clauses:
% 55.37/7.32 fresh(y, y, x1...xn) = u
% 55.37/7.32 C => fresh(s, t, x1...xn) = v
% 55.37/7.32 where fresh is a fresh function symbol and x1..xn are the free
% 55.37/7.32 variables of u and v.
% 55.37/7.32 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 55.37/7.32 input problem has no model of domain size 1).
% 55.37/7.32
% 55.37/7.32 The encoding turns the above axioms into the following unit equations and goals:
% 55.37/7.32
% 55.37/7.32 Axiom 1 (sos_02): X + Y = Y + X.
% 55.37/7.32 Axiom 2 (sos_03): X + 0 = X.
% 55.37/7.32 Axiom 3 (sos_14): X + 1 = 1.
% 55.37/7.32 Axiom 4 (sos_04): X >= X = true.
% 55.37/7.32 Axiom 5 (sos_08): X >= 0 = true.
% 55.37/7.32 Axiom 6 (sos_06): fresh(X, X, Y, Z) = Z.
% 55.37/7.32 Axiom 7 (sos_05): fresh9(X, X, Y, Z) = true.
% 55.37/7.32 Axiom 8 (sos_06): fresh2(X, X, Y, Z) = Y.
% 55.37/7.32 Axiom 9 (sos_12): X + (X ==> Y) = Y + (Y ==> X).
% 55.37/7.32 Axiom 10 (sos_01): (X + Y) + Z = X + (Y + Z).
% 55.37/7.32 Axiom 11 (sos_13): (X ==> 1) ==> 1 = X.
% 55.37/7.32 Axiom 12 (sos_05): fresh8(X, X, Y, Z, W) = Y >= W.
% 55.37/7.32 Axiom 13 (sos_07_1): fresh6(X, X, Y, Z, W) = true.
% 55.37/7.32 Axiom 14 (sos_09): fresh5(X, X, Y, Z, W) = true.
% 55.37/7.32 Axiom 15 (sos_06): fresh2(X >= Y, true, Y, X) = fresh(Y >= X, true, Y, X).
% 55.37/7.32 Axiom 16 (sos_05): fresh8(X >= Y, true, Z, X, Y) = fresh9(Z >= X, true, Z, Y).
% 55.37/7.32 Axiom 17 (sos_09): fresh5(X >= Y, true, X, Y, Z) = (X + Z) >= (Y + Z).
% 55.37/7.32 Axiom 18 (sos_07_1): fresh6((X + Y) >= Z, true, X, Y, Z) = Y >= (X ==> Z).
% 55.37/7.32
% 55.37/7.32 Lemma 19: 1 + X = 1.
% 55.37/7.32 Proof:
% 55.37/7.32 1 + X
% 55.37/7.32 = { by axiom 1 (sos_02) R->L }
% 55.37/7.32 X + 1
% 55.37/7.32 = { by axiom 3 (sos_14) }
% 55.37/7.32 1
% 55.37/7.33
% 55.37/7.33 Lemma 20: (X + Y) >= X = true.
% 55.37/7.33 Proof:
% 55.37/7.33 (X + Y) >= X
% 55.37/7.33 = { by axiom 1 (sos_02) R->L }
% 55.37/7.33 (Y + X) >= X
% 55.37/7.33 = { by axiom 2 (sos_03) R->L }
% 55.37/7.33 (Y + X) >= (X + 0)
% 55.37/7.33 = { by axiom 1 (sos_02) }
% 55.37/7.33 (Y + X) >= (0 + X)
% 55.37/7.33 = { by axiom 17 (sos_09) R->L }
% 55.37/7.33 fresh5(Y >= 0, true, Y, 0, X)
% 55.37/7.33 = { by axiom 5 (sos_08) }
% 55.37/7.33 fresh5(true, true, Y, 0, X)
% 55.37/7.33 = { by axiom 14 (sos_09) }
% 55.37/7.33 true
% 55.37/7.33
% 55.37/7.33 Lemma 21: X + (Y + Z) = Y + (X + Z).
% 55.37/7.33 Proof:
% 55.37/7.33 X + (Y + Z)
% 55.37/7.33 = { by axiom 1 (sos_02) R->L }
% 55.37/7.33 (Y + Z) + X
% 55.37/7.33 = { by axiom 10 (sos_01) }
% 55.37/7.33 Y + (Z + X)
% 55.37/7.33 = { by axiom 1 (sos_02) }
% 55.37/7.33 Y + (X + Z)
% 55.37/7.33
% 55.37/7.33 Lemma 22: Z + ((Z ==> X) + Y) = X + (Y + (X ==> Z)).
% 55.37/7.33 Proof:
% 55.37/7.33 Z + ((Z ==> X) + Y)
% 55.37/7.33 = { by axiom 10 (sos_01) R->L }
% 55.37/7.33 (Z + (Z ==> X)) + Y
% 55.37/7.33 = { by axiom 9 (sos_12) R->L }
% 55.37/7.33 (X + (X ==> Z)) + Y
% 55.37/7.33 = { by axiom 10 (sos_01) }
% 55.37/7.33 X + ((X ==> Z) + Y)
% 55.37/7.33 = { by axiom 1 (sos_02) }
% 55.37/7.33 X + (Y + (X ==> Z))
% 55.37/7.33
% 55.37/7.33 Lemma 23: X + ((X ==> 1) + Y) = 1.
% 55.37/7.33 Proof:
% 55.37/7.33 X + ((X ==> 1) + Y)
% 55.37/7.33 = { by lemma 22 }
% 55.37/7.33 1 + (Y + (1 ==> X))
% 55.37/7.33 = { by lemma 19 }
% 55.37/7.33 1
% 55.37/7.33
% 55.37/7.33 Lemma 24: X >= (Y ==> (X + Y)) = true.
% 55.37/7.33 Proof:
% 55.37/7.33 X >= (Y ==> (X + Y))
% 55.37/7.33 = { by axiom 1 (sos_02) R->L }
% 55.37/7.33 X >= (Y ==> (Y + X))
% 55.37/7.33 = { by axiom 18 (sos_07_1) R->L }
% 55.37/7.33 fresh6((Y + X) >= (Y + X), true, Y, X, Y + X)
% 55.37/7.33 = { by axiom 4 (sos_04) }
% 55.37/7.33 fresh6(true, true, Y, X, Y + X)
% 55.37/7.33 = { by axiom 13 (sos_07_1) }
% 55.37/7.33 true
% 55.37/7.33
% 55.37/7.33 Lemma 25: (X ==> (Y ==> 1)) >= (Y ==> (X ==> 1)) = true.
% 55.37/7.33 Proof:
% 55.37/7.33 (X ==> (Y ==> 1)) >= (Y ==> (X ==> 1))
% 55.37/7.33 = { by lemma 23 R->L }
% 55.37/7.33 (X ==> (Y ==> 1)) >= (Y ==> (X ==> (Y + ((Y ==> 1) + ((Y ==> 1) ==> X)))))
% 55.37/7.33 = { by axiom 9 (sos_12) }
% 55.37/7.33 (X ==> (Y ==> 1)) >= (Y ==> (X ==> (Y + (X + (X ==> (Y ==> 1))))))
% 55.37/7.33 = { by axiom 10 (sos_01) R->L }
% 55.37/7.33 (X ==> (Y ==> 1)) >= (Y ==> (X ==> ((Y + X) + (X ==> (Y ==> 1)))))
% 55.37/7.33 = { by axiom 1 (sos_02) R->L }
% 55.37/7.33 (X ==> (Y ==> 1)) >= (Y ==> (X ==> ((X ==> (Y ==> 1)) + (Y + X))))
% 55.37/7.33 = { by lemma 21 R->L }
% 55.37/7.33 (X ==> (Y ==> 1)) >= (Y ==> (X ==> (Y + ((X ==> (Y ==> 1)) + X))))
% 55.37/7.33 = { by axiom 10 (sos_01) R->L }
% 55.37/7.33 (X ==> (Y ==> 1)) >= (Y ==> (X ==> ((Y + (X ==> (Y ==> 1))) + X)))
% 55.37/7.33 = { by axiom 18 (sos_07_1) R->L }
% 55.37/7.33 fresh6((Y + (X ==> (Y ==> 1))) >= (X ==> ((Y + (X ==> (Y ==> 1))) + X)), true, Y, X ==> (Y ==> 1), X ==> ((Y + (X ==> (Y ==> 1))) + X))
% 55.37/7.33 = { by lemma 24 }
% 55.37/7.33 fresh6(true, true, Y, X ==> (Y ==> 1), X ==> ((Y + (X ==> (Y ==> 1))) + X))
% 55.37/7.33 = { by axiom 13 (sos_07_1) }
% 55.37/7.33 true
% 55.37/7.33
% 55.37/7.33 Lemma 26: X ==> (Y ==> 1) = Y ==> (X ==> 1).
% 55.37/7.33 Proof:
% 55.37/7.33 X ==> (Y ==> 1)
% 55.37/7.33 = { by axiom 6 (sos_06) R->L }
% 55.37/7.33 fresh(true, true, Y ==> (X ==> 1), X ==> (Y ==> 1))
% 55.37/7.33 = { by lemma 25 R->L }
% 55.37/7.33 fresh((Y ==> (X ==> 1)) >= (X ==> (Y ==> 1)), true, Y ==> (X ==> 1), X ==> (Y ==> 1))
% 55.37/7.33 = { by axiom 15 (sos_06) R->L }
% 55.37/7.33 fresh2((X ==> (Y ==> 1)) >= (Y ==> (X ==> 1)), true, Y ==> (X ==> 1), X ==> (Y ==> 1))
% 55.37/7.33 = { by lemma 25 }
% 55.37/7.33 fresh2(true, true, Y ==> (X ==> 1), X ==> (Y ==> 1))
% 55.37/7.33 = { by axiom 8 (sos_06) }
% 55.37/7.33 Y ==> (X ==> 1)
% 55.37/7.33
% 55.37/7.33 Lemma 27: (X + Y) ==> 1 = X ==> (Y ==> 1).
% 55.37/7.33 Proof:
% 55.37/7.33 (X + Y) ==> 1
% 55.37/7.33 = { by axiom 1 (sos_02) R->L }
% 55.37/7.33 (Y + X) ==> 1
% 55.37/7.33 = { by axiom 11 (sos_13) R->L }
% 55.37/7.33 (Y + ((X ==> 1) ==> 1)) ==> 1
% 55.37/7.33 = { by axiom 6 (sos_06) R->L }
% 55.37/7.33 fresh(true, true, Y ==> (X ==> 1), (Y + ((X ==> 1) ==> 1)) ==> 1)
% 55.37/7.33 = { by axiom 7 (sos_05) R->L }
% 55.37/7.33 fresh(fresh9(true, true, Y ==> (X ==> 1), (Y + ((X ==> 1) ==> 1)) ==> 1), true, Y ==> (X ==> 1), (Y + ((X ==> 1) ==> 1)) ==> 1)
% 55.37/7.33 = { by lemma 24 R->L }
% 55.37/7.33 fresh(fresh9((Y ==> (X ==> 1)) >= ((Y + ((X ==> 1) ==> 1)) ==> ((Y ==> (X ==> 1)) + (Y + ((X ==> 1) ==> 1)))), true, Y ==> (X ==> 1), (Y + ((X ==> 1) ==> 1)) ==> 1), true, Y ==> (X ==> 1), (Y + ((X ==> 1) ==> 1)) ==> 1)
% 55.37/7.33 = { by lemma 21 }
% 55.37/7.33 fresh(fresh9((Y ==> (X ==> 1)) >= ((Y + ((X ==> 1) ==> 1)) ==> (Y + ((Y ==> (X ==> 1)) + ((X ==> 1) ==> 1)))), true, Y ==> (X ==> 1), (Y + ((X ==> 1) ==> 1)) ==> 1), true, Y ==> (X ==> 1), (Y + ((X ==> 1) ==> 1)) ==> 1)
% 55.37/7.33 = { by lemma 22 }
% 55.37/7.33 fresh(fresh9((Y ==> (X ==> 1)) >= ((Y + ((X ==> 1) ==> 1)) ==> ((X ==> 1) + (((X ==> 1) ==> 1) + ((X ==> 1) ==> Y)))), true, Y ==> (X ==> 1), (Y + ((X ==> 1) ==> 1)) ==> 1), true, Y ==> (X ==> 1), (Y + ((X ==> 1) ==> 1)) ==> 1)
% 55.37/7.33 = { by lemma 23 }
% 55.37/7.33 fresh(fresh9((Y ==> (X ==> 1)) >= ((Y + ((X ==> 1) ==> 1)) ==> 1), true, Y ==> (X ==> 1), (Y + ((X ==> 1) ==> 1)) ==> 1), true, Y ==> (X ==> 1), (Y + ((X ==> 1) ==> 1)) ==> 1)
% 55.37/7.33 = { by axiom 16 (sos_05) R->L }
% 55.37/7.33 fresh(fresh8(((Y + ((X ==> 1) ==> 1)) ==> 1) >= ((Y + ((X ==> 1) ==> 1)) ==> 1), true, Y ==> (X ==> 1), (Y + ((X ==> 1) ==> 1)) ==> 1, (Y + ((X ==> 1) ==> 1)) ==> 1), true, Y ==> (X ==> 1), (Y + ((X ==> 1) ==> 1)) ==> 1)
% 55.37/7.33 = { by axiom 18 (sos_07_1) R->L }
% 55.37/7.33 fresh(fresh8(fresh6(((Y + ((X ==> 1) ==> 1)) + ((Y + ((X ==> 1) ==> 1)) ==> 1)) >= 1, true, Y + ((X ==> 1) ==> 1), (Y + ((X ==> 1) ==> 1)) ==> 1, 1), true, Y ==> (X ==> 1), (Y + ((X ==> 1) ==> 1)) ==> 1, (Y + ((X ==> 1) ==> 1)) ==> 1), true, Y ==> (X ==> 1), (Y + ((X ==> 1) ==> 1)) ==> 1)
% 55.37/7.33 = { by axiom 9 (sos_12) R->L }
% 55.37/7.33 fresh(fresh8(fresh6((1 + (1 ==> (Y + ((X ==> 1) ==> 1)))) >= 1, true, Y + ((X ==> 1) ==> 1), (Y + ((X ==> 1) ==> 1)) ==> 1, 1), true, Y ==> (X ==> 1), (Y + ((X ==> 1) ==> 1)) ==> 1, (Y + ((X ==> 1) ==> 1)) ==> 1), true, Y ==> (X ==> 1), (Y + ((X ==> 1) ==> 1)) ==> 1)
% 55.37/7.33 = { by lemma 19 }
% 55.37/7.33 fresh(fresh8(fresh6(1 >= 1, true, Y + ((X ==> 1) ==> 1), (Y + ((X ==> 1) ==> 1)) ==> 1, 1), true, Y ==> (X ==> 1), (Y + ((X ==> 1) ==> 1)) ==> 1, (Y + ((X ==> 1) ==> 1)) ==> 1), true, Y ==> (X ==> 1), (Y + ((X ==> 1) ==> 1)) ==> 1)
% 55.37/7.33 = { by axiom 3 (sos_14) R->L }
% 55.37/7.33 fresh(fresh8(fresh6((1 + 1) >= 1, true, Y + ((X ==> 1) ==> 1), (Y + ((X ==> 1) ==> 1)) ==> 1, 1), true, Y ==> (X ==> 1), (Y + ((X ==> 1) ==> 1)) ==> 1, (Y + ((X ==> 1) ==> 1)) ==> 1), true, Y ==> (X ==> 1), (Y + ((X ==> 1) ==> 1)) ==> 1)
% 55.37/7.33 = { by lemma 20 }
% 55.37/7.33 fresh(fresh8(fresh6(true, true, Y + ((X ==> 1) ==> 1), (Y + ((X ==> 1) ==> 1)) ==> 1, 1), true, Y ==> (X ==> 1), (Y + ((X ==> 1) ==> 1)) ==> 1, (Y + ((X ==> 1) ==> 1)) ==> 1), true, Y ==> (X ==> 1), (Y + ((X ==> 1) ==> 1)) ==> 1)
% 55.37/7.33 = { by axiom 13 (sos_07_1) }
% 55.37/7.33 fresh(fresh8(true, true, Y ==> (X ==> 1), (Y + ((X ==> 1) ==> 1)) ==> 1, (Y + ((X ==> 1) ==> 1)) ==> 1), true, Y ==> (X ==> 1), (Y + ((X ==> 1) ==> 1)) ==> 1)
% 55.37/7.33 = { by axiom 12 (sos_05) }
% 55.37/7.33 fresh((Y ==> (X ==> 1)) >= ((Y + ((X ==> 1) ==> 1)) ==> 1), true, Y ==> (X ==> 1), (Y + ((X ==> 1) ==> 1)) ==> 1)
% 55.37/7.33 = { by axiom 15 (sos_06) R->L }
% 55.37/7.33 fresh2(((Y + ((X ==> 1) ==> 1)) ==> 1) >= (Y ==> (X ==> 1)), true, Y ==> (X ==> 1), (Y + ((X ==> 1) ==> 1)) ==> 1)
% 55.37/7.33 = { by axiom 18 (sos_07_1) R->L }
% 55.37/7.33 fresh2(fresh6((Y + ((Y + ((X ==> 1) ==> 1)) ==> 1)) >= (X ==> 1), true, Y, (Y + ((X ==> 1) ==> 1)) ==> 1, X ==> 1), true, Y ==> (X ==> 1), (Y + ((X ==> 1) ==> 1)) ==> 1)
% 55.37/7.33 = { by axiom 11 (sos_13) R->L }
% 55.37/7.33 fresh2(fresh6((Y + ((Y + ((X ==> 1) ==> 1)) ==> 1)) >= (((X ==> 1) ==> 1) ==> 1), true, Y, (Y + ((X ==> 1) ==> 1)) ==> 1, X ==> 1), true, Y ==> (X ==> 1), (Y + ((X ==> 1) ==> 1)) ==> 1)
% 55.37/7.33 = { by axiom 1 (sos_02) R->L }
% 55.37/7.33 fresh2(fresh6((Y + ((((X ==> 1) ==> 1) + Y) ==> 1)) >= (((X ==> 1) ==> 1) ==> 1), true, Y, (Y + ((X ==> 1) ==> 1)) ==> 1, X ==> 1), true, Y ==> (X ==> 1), (Y + ((X ==> 1) ==> 1)) ==> 1)
% 55.37/7.34 = { by axiom 18 (sos_07_1) R->L }
% 55.37/7.34 fresh2(fresh6(fresh6((((X ==> 1) ==> 1) + (Y + ((((X ==> 1) ==> 1) + Y) ==> 1))) >= 1, true, (X ==> 1) ==> 1, Y + ((((X ==> 1) ==> 1) + Y) ==> 1), 1), true, Y, (Y + ((X ==> 1) ==> 1)) ==> 1, X ==> 1), true, Y ==> (X ==> 1), (Y + ((X ==> 1) ==> 1)) ==> 1)
% 55.37/7.34 = { by axiom 10 (sos_01) R->L }
% 55.37/7.34 fresh2(fresh6(fresh6(((((X ==> 1) ==> 1) + Y) + ((((X ==> 1) ==> 1) + Y) ==> 1)) >= 1, true, (X ==> 1) ==> 1, Y + ((((X ==> 1) ==> 1) + Y) ==> 1), 1), true, Y, (Y + ((X ==> 1) ==> 1)) ==> 1, X ==> 1), true, Y ==> (X ==> 1), (Y + ((X ==> 1) ==> 1)) ==> 1)
% 55.37/7.34 = { by axiom 9 (sos_12) }
% 55.37/7.34 fresh2(fresh6(fresh6((1 + (1 ==> (((X ==> 1) ==> 1) + Y))) >= 1, true, (X ==> 1) ==> 1, Y + ((((X ==> 1) ==> 1) + Y) ==> 1), 1), true, Y, (Y + ((X ==> 1) ==> 1)) ==> 1, X ==> 1), true, Y ==> (X ==> 1), (Y + ((X ==> 1) ==> 1)) ==> 1)
% 55.37/7.34 = { by lemma 20 }
% 55.37/7.34 fresh2(fresh6(fresh6(true, true, (X ==> 1) ==> 1, Y + ((((X ==> 1) ==> 1) + Y) ==> 1), 1), true, Y, (Y + ((X ==> 1) ==> 1)) ==> 1, X ==> 1), true, Y ==> (X ==> 1), (Y + ((X ==> 1) ==> 1)) ==> 1)
% 55.37/7.34 = { by axiom 13 (sos_07_1) }
% 55.37/7.34 fresh2(fresh6(true, true, Y, (Y + ((X ==> 1) ==> 1)) ==> 1, X ==> 1), true, Y ==> (X ==> 1), (Y + ((X ==> 1) ==> 1)) ==> 1)
% 55.37/7.34 = { by axiom 13 (sos_07_1) }
% 55.37/7.34 fresh2(true, true, Y ==> (X ==> 1), (Y + ((X ==> 1) ==> 1)) ==> 1)
% 55.37/7.34 = { by axiom 8 (sos_06) }
% 55.37/7.34 Y ==> (X ==> 1)
% 55.37/7.34 = { by lemma 26 }
% 55.37/7.34 X ==> (Y ==> 1)
% 55.37/7.34
% 55.37/7.34 Lemma 28: (X ==> 1) ==> (Y ==> 1) = Y ==> X.
% 55.37/7.34 Proof:
% 55.37/7.34 (X ==> 1) ==> (Y ==> 1)
% 55.37/7.34 = { by lemma 26 R->L }
% 55.37/7.34 Y ==> ((X ==> 1) ==> 1)
% 55.37/7.34 = { by axiom 11 (sos_13) }
% 55.37/7.34 Y ==> X
% 55.37/7.34
% 55.37/7.34 Goal 1 (goals_15): (x17 ==> x18) ==> x18 = (x18 ==> x17) ==> x17.
% 55.37/7.34 Proof:
% 55.37/7.34 (x17 ==> x18) ==> x18
% 55.37/7.34 = { by lemma 28 R->L }
% 55.37/7.34 (x18 ==> 1) ==> ((x17 ==> x18) ==> 1)
% 55.37/7.34 = { by lemma 27 R->L }
% 55.37/7.34 ((x18 ==> 1) + (x17 ==> x18)) ==> 1
% 55.37/7.34 = { by lemma 28 R->L }
% 55.37/7.34 ((x18 ==> 1) + ((x18 ==> 1) ==> (x17 ==> 1))) ==> 1
% 55.37/7.34 = { by axiom 9 (sos_12) R->L }
% 55.37/7.34 ((x17 ==> 1) + ((x17 ==> 1) ==> (x18 ==> 1))) ==> 1
% 55.37/7.34 = { by lemma 28 }
% 55.37/7.34 ((x17 ==> 1) + (x18 ==> x17)) ==> 1
% 55.37/7.34 = { by lemma 27 }
% 55.37/7.34 (x17 ==> 1) ==> ((x18 ==> x17) ==> 1)
% 55.37/7.34 = { by lemma 28 }
% 55.37/7.34 (x18 ==> x17) ==> x17
% 55.37/7.34 % SZS output end Proof
% 55.37/7.34
% 55.37/7.34 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------