TSTP Solution File: LCL900+1 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : LCL900+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 08:21:07 EDT 2023
% Result : Theorem 28.23s 3.96s
% Output : Proof 28.23s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : LCL900+1 : TPTP v8.1.2. Released v5.5.0.
% 0.00/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.13/0.33 % Computer : n024.cluster.edu
% 0.13/0.33 % Model : x86_64 x86_64
% 0.13/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.33 % Memory : 8042.1875MB
% 0.13/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.33 % CPULimit : 300
% 0.13/0.33 % WCLimit : 300
% 0.13/0.33 % DateTime : Fri Aug 25 06:04:24 EDT 2023
% 0.13/0.33 % CPUTime :
% 28.23/3.96 Command-line arguments: --no-flatten-goal
% 28.23/3.96
% 28.23/3.96 % SZS status Theorem
% 28.23/3.96
% 28.23/3.99 % SZS output start Proof
% 28.23/3.99 Take the following subset of the input axioms:
% 28.23/3.99 fof(goals_14, conjecture, ![X17, X18]: '+'(X17, '==>'(X17, X18))='+'(X18, '==>'(X18, X17))).
% 28.23/3.99 fof(sos_01, axiom, ![A, B, C]: '+'('+'(A, B), C)='+'(A, '+'(B, C))).
% 28.23/3.99 fof(sos_02, axiom, ![A2, B2]: '+'(A2, B2)='+'(B2, A2)).
% 28.23/3.99 fof(sos_03, axiom, ![A2]: '+'(A2, '0')=A2).
% 28.23/3.99 fof(sos_04, axiom, ![A2]: '>='(A2, A2)).
% 28.23/3.99 fof(sos_06, axiom, ![X3, X4]: (('>='(X3, X4) & '>='(X4, X3)) => X3=X4)).
% 28.23/3.99 fof(sos_07, axiom, ![X5, X6, X7]: ('>='('+'(X5, X6), X7) <=> '>='(X6, '==>'(X5, X7)))).
% 28.23/3.99 fof(sos_08, axiom, ![A2]: '>='(A2, '0')).
% 28.23/3.99 fof(sos_09, axiom, ![X8, X9, X10]: ('>='(X8, X9) => '>='('+'(X8, X10), '+'(X9, X10)))).
% 28.23/3.99 fof(sos_10, axiom, ![X11, X12, X13]: ('>='(X11, X12) => '>='('==>'(X12, X13), '==>'(X11, X13)))).
% 28.23/3.99 fof(sos_12, axiom, ![A2]: '+'(A2, '1')='1').
% 28.23/3.99 fof(sos_13, axiom, ![A2, B2]: '==>'('==>'(A2, B2), B2)='==>'('==>'(B2, A2), A2)).
% 28.23/3.99
% 28.23/3.99 Now clausify the problem and encode Horn clauses using encoding 3 of
% 28.23/3.99 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 28.23/3.99 We repeatedly replace C & s=t => u=v by the two clauses:
% 28.23/3.99 fresh(y, y, x1...xn) = u
% 28.23/3.99 C => fresh(s, t, x1...xn) = v
% 28.23/3.99 where fresh is a fresh function symbol and x1..xn are the free
% 28.23/3.99 variables of u and v.
% 28.23/3.99 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 28.23/3.99 input problem has no model of domain size 1).
% 28.23/3.99
% 28.23/3.99 The encoding turns the above axioms into the following unit equations and goals:
% 28.23/3.99
% 28.23/3.99 Axiom 1 (sos_02): X + Y = Y + X.
% 28.23/3.99 Axiom 2 (sos_03): X + 0 = X.
% 28.23/3.99 Axiom 3 (sos_12): X + 1 = 1.
% 28.23/3.99 Axiom 4 (sos_04): X >= X = true.
% 28.23/3.99 Axiom 5 (sos_08): X >= 0 = true.
% 28.23/3.99 Axiom 6 (sos_06): fresh(X, X, Y, Z) = Z.
% 28.23/3.99 Axiom 7 (sos_06): fresh2(X, X, Y, Z) = Y.
% 28.23/3.99 Axiom 8 (sos_13): (X ==> Y) ==> Y = (Y ==> X) ==> X.
% 28.23/3.99 Axiom 9 (sos_01): (X + Y) + Z = X + (Y + Z).
% 28.23/3.99 Axiom 10 (sos_07): fresh7(X, X, Y, Z, W) = true.
% 28.23/3.99 Axiom 11 (sos_07_1): fresh6(X, X, Y, Z, W) = true.
% 28.23/3.99 Axiom 12 (sos_09): fresh5(X, X, Y, Z, W) = true.
% 28.23/3.99 Axiom 13 (sos_10): fresh4(X, X, Y, Z, W) = true.
% 28.23/3.99 Axiom 14 (sos_06): fresh2(X >= Y, true, Y, X) = fresh(Y >= X, true, Y, X).
% 28.23/3.99 Axiom 15 (sos_09): fresh5(X >= Y, true, X, Y, Z) = (X + Z) >= (Y + Z).
% 28.23/3.99 Axiom 16 (sos_10): fresh4(X >= Y, true, X, Y, Z) = (Y ==> Z) >= (X ==> Z).
% 28.23/3.99 Axiom 17 (sos_07): fresh7(X >= (Y ==> Z), true, Y, X, Z) = (Y + X) >= Z.
% 28.23/3.99 Axiom 18 (sos_07_1): fresh6((X + Y) >= Z, true, X, Y, Z) = Y >= (X ==> Z).
% 28.23/3.99
% 28.23/3.99 Lemma 19: 0 + X = X.
% 28.23/3.99 Proof:
% 28.23/3.99 0 + X
% 28.23/3.99 = { by axiom 1 (sos_02) R->L }
% 28.23/3.99 X + 0
% 28.23/3.99 = { by axiom 2 (sos_03) }
% 28.23/3.99 X
% 28.23/3.99
% 28.23/3.99 Lemma 20: (X + Y) >= X = true.
% 28.23/3.99 Proof:
% 28.23/3.99 (X + Y) >= X
% 28.23/3.99 = { by axiom 1 (sos_02) R->L }
% 28.23/3.99 (Y + X) >= X
% 28.23/3.99 = { by lemma 19 R->L }
% 28.23/3.99 (Y + X) >= (0 + X)
% 28.23/3.99 = { by axiom 15 (sos_09) R->L }
% 28.23/3.99 fresh5(Y >= 0, true, Y, 0, X)
% 28.23/3.99 = { by axiom 5 (sos_08) }
% 28.23/3.99 fresh5(true, true, Y, 0, X)
% 28.23/3.99 = { by axiom 12 (sos_09) }
% 28.23/3.99 true
% 28.23/3.99
% 28.23/3.99 Lemma 21: 1 >= X = true.
% 28.23/3.99 Proof:
% 28.23/3.99 1 >= X
% 28.23/3.99 = { by axiom 3 (sos_12) R->L }
% 28.23/3.99 (X + 1) >= X
% 28.23/3.99 = { by lemma 20 }
% 28.23/3.99 true
% 28.23/3.99
% 28.23/3.99 Lemma 22: (X + (X ==> Y)) >= Y = true.
% 28.23/3.99 Proof:
% 28.23/3.99 (X + (X ==> Y)) >= Y
% 28.23/3.99 = { by axiom 17 (sos_07) R->L }
% 28.23/3.99 fresh7((X ==> Y) >= (X ==> Y), true, X, X ==> Y, Y)
% 28.23/3.99 = { by axiom 4 (sos_04) }
% 28.23/3.99 fresh7(true, true, X, X ==> Y, Y)
% 28.23/3.99 = { by axiom 10 (sos_07) }
% 28.23/4.00 true
% 28.23/4.00
% 28.23/4.00 Lemma 23: 0 ==> X = X.
% 28.23/4.00 Proof:
% 28.23/4.00 0 ==> X
% 28.23/4.00 = { by axiom 6 (sos_06) R->L }
% 28.23/4.00 fresh(true, true, X, 0 ==> X)
% 28.23/4.00 = { by axiom 11 (sos_07_1) R->L }
% 28.23/4.00 fresh(fresh6(true, true, 0, X, X), true, X, 0 ==> X)
% 28.23/4.00 = { by lemma 20 R->L }
% 28.23/4.00 fresh(fresh6((X + 0) >= X, true, 0, X, X), true, X, 0 ==> X)
% 28.23/4.00 = { by axiom 1 (sos_02) }
% 28.23/4.00 fresh(fresh6((0 + X) >= X, true, 0, X, X), true, X, 0 ==> X)
% 28.23/4.00 = { by axiom 18 (sos_07_1) }
% 28.23/4.00 fresh(X >= (0 ==> X), true, X, 0 ==> X)
% 28.23/4.00 = { by axiom 14 (sos_06) R->L }
% 28.23/4.00 fresh2((0 ==> X) >= X, true, X, 0 ==> X)
% 28.23/4.00 = { by lemma 19 R->L }
% 28.23/4.00 fresh2((0 + (0 ==> X)) >= X, true, X, 0 ==> X)
% 28.23/4.00 = { by lemma 22 }
% 28.23/4.00 fresh2(true, true, X, 0 ==> X)
% 28.23/4.00 = { by axiom 7 (sos_06) }
% 28.23/4.00 X
% 28.23/4.00
% 28.23/4.00 Lemma 24: X >= (Y ==> (X + Y)) = true.
% 28.23/4.00 Proof:
% 28.23/4.00 X >= (Y ==> (X + Y))
% 28.23/4.00 = { by axiom 1 (sos_02) R->L }
% 28.23/4.00 X >= (Y ==> (Y + X))
% 28.23/4.00 = { by axiom 18 (sos_07_1) R->L }
% 28.23/4.00 fresh6((Y + X) >= (Y + X), true, Y, X, Y + X)
% 28.23/4.00 = { by axiom 4 (sos_04) }
% 28.23/4.00 fresh6(true, true, Y, X, Y + X)
% 28.23/4.00 = { by axiom 11 (sos_07_1) }
% 28.23/4.00 true
% 28.23/4.00
% 28.23/4.00 Lemma 25: 1 ==> X = 0.
% 28.23/4.00 Proof:
% 28.23/4.00 1 ==> X
% 28.23/4.00 = { by axiom 7 (sos_06) R->L }
% 28.23/4.00 fresh2(true, true, 1 ==> X, 0)
% 28.23/4.00 = { by axiom 13 (sos_10) R->L }
% 28.23/4.00 fresh2(fresh4(true, true, 1, X, X), true, 1 ==> X, 0)
% 28.23/4.00 = { by lemma 21 R->L }
% 28.23/4.00 fresh2(fresh4(1 >= X, true, 1, X, X), true, 1 ==> X, 0)
% 28.23/4.00 = { by axiom 16 (sos_10) }
% 28.23/4.00 fresh2((X ==> X) >= (1 ==> X), true, 1 ==> X, 0)
% 28.23/4.00 = { by lemma 19 R->L }
% 28.23/4.00 fresh2((X ==> (0 + X)) >= (1 ==> X), true, 1 ==> X, 0)
% 28.23/4.00 = { by axiom 6 (sos_06) R->L }
% 28.23/4.00 fresh2(fresh(true, true, 0, X ==> (0 + X)) >= (1 ==> X), true, 1 ==> X, 0)
% 28.23/4.00 = { by lemma 24 R->L }
% 28.23/4.00 fresh2(fresh(0 >= (X ==> (0 + X)), true, 0, X ==> (0 + X)) >= (1 ==> X), true, 1 ==> X, 0)
% 28.23/4.00 = { by axiom 14 (sos_06) R->L }
% 28.23/4.00 fresh2(fresh2((X ==> (0 + X)) >= 0, true, 0, X ==> (0 + X)) >= (1 ==> X), true, 1 ==> X, 0)
% 28.23/4.00 = { by axiom 5 (sos_08) }
% 28.23/4.00 fresh2(fresh2(true, true, 0, X ==> (0 + X)) >= (1 ==> X), true, 1 ==> X, 0)
% 28.23/4.00 = { by axiom 7 (sos_06) }
% 28.23/4.00 fresh2(0 >= (1 ==> X), true, 1 ==> X, 0)
% 28.23/4.00 = { by axiom 14 (sos_06) }
% 28.23/4.00 fresh((1 ==> X) >= 0, true, 1 ==> X, 0)
% 28.23/4.00 = { by axiom 5 (sos_08) }
% 28.23/4.00 fresh(true, true, 1 ==> X, 0)
% 28.23/4.00 = { by axiom 6 (sos_06) }
% 28.23/4.00 0
% 28.23/4.00
% 28.23/4.00 Lemma 26: (X + (Y + (Y ==> (X ==> Z)))) >= Z = true.
% 28.23/4.00 Proof:
% 28.23/4.00 (X + (Y + (Y ==> (X ==> Z)))) >= Z
% 28.23/4.00 = { by axiom 17 (sos_07) R->L }
% 28.23/4.00 fresh7((Y + (Y ==> (X ==> Z))) >= (X ==> Z), true, X, Y + (Y ==> (X ==> Z)), Z)
% 28.23/4.00 = { by lemma 22 }
% 28.23/4.00 fresh7(true, true, X, Y + (Y ==> (X ==> Z)), Z)
% 28.23/4.00 = { by axiom 10 (sos_07) }
% 28.23/4.00 true
% 28.23/4.00
% 28.23/4.00 Lemma 27: (X ==> (Y ==> Z)) >= (Y ==> (X ==> Z)) = true.
% 28.23/4.00 Proof:
% 28.23/4.00 (X ==> (Y ==> Z)) >= (Y ==> (X ==> Z))
% 28.23/4.00 = { by axiom 18 (sos_07_1) R->L }
% 28.23/4.00 fresh6((Y + (X ==> (Y ==> Z))) >= (X ==> Z), true, Y, X ==> (Y ==> Z), X ==> Z)
% 28.23/4.00 = { by axiom 18 (sos_07_1) R->L }
% 28.23/4.00 fresh6(fresh6((X + (Y + (X ==> (Y ==> Z)))) >= Z, true, X, Y + (X ==> (Y ==> Z)), Z), true, Y, X ==> (Y ==> Z), X ==> Z)
% 28.23/4.00 = { by axiom 1 (sos_02) R->L }
% 28.23/4.00 fresh6(fresh6((X + ((X ==> (Y ==> Z)) + Y)) >= Z, true, X, Y + (X ==> (Y ==> Z)), Z), true, Y, X ==> (Y ==> Z), X ==> Z)
% 28.23/4.00 = { by axiom 9 (sos_01) R->L }
% 28.23/4.00 fresh6(fresh6(((X + (X ==> (Y ==> Z))) + Y) >= Z, true, X, Y + (X ==> (Y ==> Z)), Z), true, Y, X ==> (Y ==> Z), X ==> Z)
% 28.23/4.00 = { by axiom 1 (sos_02) R->L }
% 28.23/4.00 fresh6(fresh6((Y + (X + (X ==> (Y ==> Z)))) >= Z, true, X, Y + (X ==> (Y ==> Z)), Z), true, Y, X ==> (Y ==> Z), X ==> Z)
% 28.23/4.00 = { by lemma 26 }
% 28.23/4.00 fresh6(fresh6(true, true, X, Y + (X ==> (Y ==> Z)), Z), true, Y, X ==> (Y ==> Z), X ==> Z)
% 28.23/4.00 = { by axiom 11 (sos_07_1) }
% 28.23/4.00 fresh6(true, true, Y, X ==> (Y ==> Z), X ==> Z)
% 28.23/4.00 = { by axiom 11 (sos_07_1) }
% 28.23/4.00 true
% 28.23/4.00
% 28.23/4.00 Lemma 28: X ==> (Y ==> Z) = Y ==> (X ==> Z).
% 28.23/4.00 Proof:
% 28.23/4.00 X ==> (Y ==> Z)
% 28.23/4.00 = { by axiom 6 (sos_06) R->L }
% 28.23/4.00 fresh(true, true, Y ==> (X ==> Z), X ==> (Y ==> Z))
% 28.23/4.00 = { by lemma 27 R->L }
% 28.23/4.00 fresh((Y ==> (X ==> Z)) >= (X ==> (Y ==> Z)), true, Y ==> (X ==> Z), X ==> (Y ==> Z))
% 28.23/4.00 = { by axiom 14 (sos_06) R->L }
% 28.23/4.00 fresh2((X ==> (Y ==> Z)) >= (Y ==> (X ==> Z)), true, Y ==> (X ==> Z), X ==> (Y ==> Z))
% 28.23/4.00 = { by lemma 27 }
% 28.23/4.00 fresh2(true, true, Y ==> (X ==> Z), X ==> (Y ==> Z))
% 28.23/4.00 = { by axiom 7 (sos_06) }
% 28.23/4.00 Y ==> (X ==> Z)
% 28.23/4.00
% 28.23/4.00 Lemma 29: fresh(X >= 1, true, X, 1) = X.
% 28.23/4.00 Proof:
% 28.23/4.00 fresh(X >= 1, true, X, 1)
% 28.23/4.00 = { by axiom 14 (sos_06) R->L }
% 28.23/4.00 fresh2(1 >= X, true, X, 1)
% 28.23/4.00 = { by lemma 21 }
% 28.23/4.00 fresh2(true, true, X, 1)
% 28.23/4.00 = { by axiom 7 (sos_06) }
% 28.23/4.00 X
% 28.23/4.00
% 28.23/4.00 Lemma 30: (X ==> 1) ==> (Y ==> 1) = Y ==> X.
% 28.23/4.00 Proof:
% 28.23/4.00 (X ==> 1) ==> (Y ==> 1)
% 28.23/4.00 = { by lemma 28 R->L }
% 28.23/4.00 Y ==> ((X ==> 1) ==> 1)
% 28.23/4.00 = { by axiom 8 (sos_13) R->L }
% 28.23/4.00 Y ==> ((1 ==> X) ==> X)
% 28.23/4.00 = { by lemma 25 }
% 28.23/4.00 Y ==> (0 ==> X)
% 28.23/4.00 = { by lemma 23 }
% 28.23/4.00 Y ==> X
% 28.23/4.00
% 28.23/4.00 Lemma 31: (X ==> (Y ==> 1)) ==> 1 = X + Y.
% 28.23/4.00 Proof:
% 28.23/4.00 (X ==> (Y ==> 1)) ==> 1
% 28.23/4.00 = { by axiom 6 (sos_06) R->L }
% 28.23/4.00 fresh(true, true, (X + Y) ==> 1, X ==> (Y ==> 1)) ==> 1
% 28.23/4.00 = { by axiom 11 (sos_07_1) R->L }
% 28.23/4.00 fresh(fresh6(true, true, X, (X + Y) ==> 1, Y ==> 1), true, (X + Y) ==> 1, X ==> (Y ==> 1)) ==> 1
% 28.23/4.00 = { by axiom 11 (sos_07_1) R->L }
% 28.23/4.00 fresh(fresh6(fresh6(true, true, Y, X + ((Y + X) ==> 1), 1), true, X, (X + Y) ==> 1, Y ==> 1), true, (X + Y) ==> 1, X ==> (Y ==> 1)) ==> 1
% 28.23/4.00 = { by lemma 21 R->L }
% 28.23/4.00 fresh(fresh6(fresh6(1 >= 1, true, Y, X + ((Y + X) ==> 1), 1), true, X, (X + Y) ==> 1, Y ==> 1), true, (X + Y) ==> 1, X ==> (Y ==> 1)) ==> 1
% 28.23/4.00 = { by axiom 6 (sos_06) R->L }
% 28.23/4.00 fresh(fresh6(fresh6(fresh(true, true, Y + (X + ((Y + X) ==> 1)), 1) >= 1, true, Y, X + ((Y + X) ==> 1), 1), true, X, (X + Y) ==> 1, Y ==> 1), true, (X + Y) ==> 1, X ==> (Y ==> 1)) ==> 1
% 28.23/4.00 = { by lemma 22 R->L }
% 28.23/4.00 fresh(fresh6(fresh6(fresh(((Y + X) + ((Y + X) ==> 1)) >= 1, true, Y + (X + ((Y + X) ==> 1)), 1) >= 1, true, Y, X + ((Y + X) ==> 1), 1), true, X, (X + Y) ==> 1, Y ==> 1), true, (X + Y) ==> 1, X ==> (Y ==> 1)) ==> 1
% 28.23/4.00 = { by axiom 9 (sos_01) }
% 28.23/4.00 fresh(fresh6(fresh6(fresh((Y + (X + ((Y + X) ==> 1))) >= 1, true, Y + (X + ((Y + X) ==> 1)), 1) >= 1, true, Y, X + ((Y + X) ==> 1), 1), true, X, (X + Y) ==> 1, Y ==> 1), true, (X + Y) ==> 1, X ==> (Y ==> 1)) ==> 1
% 28.23/4.00 = { by lemma 29 }
% 28.23/4.00 fresh(fresh6(fresh6((Y + (X + ((Y + X) ==> 1))) >= 1, true, Y, X + ((Y + X) ==> 1), 1), true, X, (X + Y) ==> 1, Y ==> 1), true, (X + Y) ==> 1, X ==> (Y ==> 1)) ==> 1
% 28.23/4.00 = { by axiom 18 (sos_07_1) }
% 28.23/4.00 fresh(fresh6((X + ((Y + X) ==> 1)) >= (Y ==> 1), true, X, (X + Y) ==> 1, Y ==> 1), true, (X + Y) ==> 1, X ==> (Y ==> 1)) ==> 1
% 28.23/4.00 = { by axiom 1 (sos_02) }
% 28.23/4.00 fresh(fresh6((X + ((X + Y) ==> 1)) >= (Y ==> 1), true, X, (X + Y) ==> 1, Y ==> 1), true, (X + Y) ==> 1, X ==> (Y ==> 1)) ==> 1
% 28.23/4.00 = { by axiom 18 (sos_07_1) }
% 28.23/4.00 fresh(((X + Y) ==> 1) >= (X ==> (Y ==> 1)), true, (X + Y) ==> 1, X ==> (Y ==> 1)) ==> 1
% 28.23/4.00 = { by axiom 14 (sos_06) R->L }
% 28.23/4.00 fresh2((X ==> (Y ==> 1)) >= ((X + Y) ==> 1), true, (X + Y) ==> 1, X ==> (Y ==> 1)) ==> 1
% 28.23/4.00 = { by lemma 28 R->L }
% 28.23/4.00 fresh2((Y ==> (X ==> 1)) >= ((X + Y) ==> 1), true, (X + Y) ==> 1, X ==> (Y ==> 1)) ==> 1
% 28.23/4.00 = { by axiom 6 (sos_06) R->L }
% 28.23/4.00 fresh2((Y ==> (X ==> 1)) >= ((X + Y) ==> fresh(true, true, X + (Y + (Y ==> (X ==> 1))), 1)), true, (X + Y) ==> 1, X ==> (Y ==> 1)) ==> 1
% 28.23/4.00 = { by lemma 26 R->L }
% 28.23/4.00 fresh2((Y ==> (X ==> 1)) >= ((X + Y) ==> fresh((X + (Y + (Y ==> (X ==> 1)))) >= 1, true, X + (Y + (Y ==> (X ==> 1))), 1)), true, (X + Y) ==> 1, X ==> (Y ==> 1)) ==> 1
% 28.23/4.00 = { by lemma 29 }
% 28.23/4.00 fresh2((Y ==> (X ==> 1)) >= ((X + Y) ==> (X + (Y + (Y ==> (X ==> 1))))), true, (X + Y) ==> 1, X ==> (Y ==> 1)) ==> 1
% 28.23/4.00 = { by axiom 1 (sos_02) R->L }
% 28.23/4.00 fresh2((Y ==> (X ==> 1)) >= ((X + Y) ==> (X + ((Y ==> (X ==> 1)) + Y))), true, (X + Y) ==> 1, X ==> (Y ==> 1)) ==> 1
% 28.23/4.00 = { by axiom 1 (sos_02) R->L }
% 28.23/4.00 fresh2((Y ==> (X ==> 1)) >= ((X + Y) ==> (((Y ==> (X ==> 1)) + Y) + X)), true, (X + Y) ==> 1, X ==> (Y ==> 1)) ==> 1
% 28.23/4.00 = { by axiom 9 (sos_01) }
% 28.23/4.00 fresh2((Y ==> (X ==> 1)) >= ((X + Y) ==> ((Y ==> (X ==> 1)) + (Y + X))), true, (X + Y) ==> 1, X ==> (Y ==> 1)) ==> 1
% 28.23/4.00 = { by axiom 1 (sos_02) }
% 28.23/4.00 fresh2((Y ==> (X ==> 1)) >= ((X + Y) ==> ((Y ==> (X ==> 1)) + (X + Y))), true, (X + Y) ==> 1, X ==> (Y ==> 1)) ==> 1
% 28.23/4.00 = { by lemma 24 }
% 28.23/4.00 fresh2(true, true, (X + Y) ==> 1, X ==> (Y ==> 1)) ==> 1
% 28.23/4.00 = { by axiom 7 (sos_06) }
% 28.23/4.00 ((X + Y) ==> 1) ==> 1
% 28.23/4.00 = { by axiom 1 (sos_02) }
% 28.23/4.00 ((Y + X) ==> 1) ==> 1
% 28.23/4.00 = { by axiom 8 (sos_13) }
% 28.23/4.00 (1 ==> (Y + X)) ==> (Y + X)
% 28.23/4.00 = { by lemma 25 }
% 28.23/4.00 0 ==> (Y + X)
% 28.23/4.00 = { by lemma 23 }
% 28.23/4.00 Y + X
% 28.23/4.00 = { by axiom 1 (sos_02) }
% 28.23/4.00 X + Y
% 28.23/4.00
% 28.23/4.00 Lemma 32: ((X ==> Y) ==> 1) ==> Z = X ==> ((Z ==> 1) ==> Y).
% 28.23/4.00 Proof:
% 28.23/4.00 ((X ==> Y) ==> 1) ==> Z
% 28.23/4.00 = { by lemma 23 R->L }
% 28.23/4.00 ((X ==> Y) ==> 1) ==> (0 ==> Z)
% 28.23/4.00 = { by lemma 25 R->L }
% 28.23/4.00 ((X ==> Y) ==> 1) ==> ((1 ==> Z) ==> Z)
% 28.23/4.00 = { by axiom 8 (sos_13) R->L }
% 28.23/4.00 ((X ==> Y) ==> 1) ==> ((Z ==> 1) ==> 1)
% 28.23/4.00 = { by lemma 30 }
% 28.23/4.00 (Z ==> 1) ==> (X ==> Y)
% 28.23/4.00 = { by lemma 28 }
% 28.23/4.00 X ==> ((Z ==> 1) ==> Y)
% 28.23/4.00
% 28.23/4.00 Goal 1 (goals_14): x17 + (x17 ==> x18) = x18 + (x18 ==> x17).
% 28.23/4.00 Proof:
% 28.23/4.00 x17 + (x17 ==> x18)
% 28.23/4.00 = { by lemma 31 R->L }
% 28.23/4.00 (x17 ==> ((x17 ==> x18) ==> 1)) ==> 1
% 28.23/4.00 = { by lemma 30 R->L }
% 28.23/4.00 (x17 ==> ((1 ==> 1) ==> ((x17 ==> x18) ==> 1))) ==> 1
% 28.23/4.00 = { by lemma 32 R->L }
% 28.23/4.00 (((x17 ==> ((x17 ==> x18) ==> 1)) ==> 1) ==> 1) ==> 1
% 28.23/4.00 = { by lemma 28 R->L }
% 28.23/4.00 ((((x17 ==> x18) ==> (x17 ==> 1)) ==> 1) ==> 1) ==> 1
% 28.23/4.00 = { by lemma 30 R->L }
% 28.23/4.00 (((((x18 ==> 1) ==> (x17 ==> 1)) ==> (x17 ==> 1)) ==> 1) ==> 1) ==> 1
% 28.23/4.00 = { by axiom 8 (sos_13) }
% 28.23/4.00 (((((x17 ==> 1) ==> (x18 ==> 1)) ==> (x18 ==> 1)) ==> 1) ==> 1) ==> 1
% 28.23/4.00 = { by lemma 30 }
% 28.23/4.00 ((((x18 ==> x17) ==> (x18 ==> 1)) ==> 1) ==> 1) ==> 1
% 28.23/4.00 = { by lemma 28 }
% 28.23/4.00 (((x18 ==> ((x18 ==> x17) ==> 1)) ==> 1) ==> 1) ==> 1
% 28.23/4.00 = { by lemma 32 }
% 28.23/4.00 (x18 ==> ((1 ==> 1) ==> ((x18 ==> x17) ==> 1))) ==> 1
% 28.23/4.00 = { by lemma 30 }
% 28.23/4.00 (x18 ==> ((x18 ==> x17) ==> 1)) ==> 1
% 28.23/4.00 = { by lemma 31 }
% 28.23/4.00 x18 + (x18 ==> x17)
% 28.23/4.00 % SZS output end Proof
% 28.23/4.00
% 28.23/4.00 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------