TSTP Solution File: ITP328_1 by Princess---230619
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%------------------------------------------------------------------------------
% File : Princess---230619
% Problem : ITP001_1 : TPTP v8.1.2. Released v8.1.0.
% Transfm : none
% Format : tptp
% Command : princess -inputFormat=tptp +threads -portfolio=casc +printProof -timeoutSec=%d %s
% Computer : n007.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 04:11:46 EDT 2023
% Result : Theorem 71.90s 10.37s
% Output : Proof 111.54s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.08/0.13 % Problem : ITP001_1 : TPTP v8.1.2. Released v8.1.0.
% 0.08/0.13 % Command : princess -inputFormat=tptp +threads -portfolio=casc +printProof -timeoutSec=%d %s
% 0.14/0.35 % Computer : n007.cluster.edu
% 0.14/0.35 % Model : x86_64 x86_64
% 0.14/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.35 % Memory : 8042.1875MB
% 0.14/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.35 % CPULimit : 300
% 0.14/0.35 % WCLimit : 300
% 0.14/0.35 % DateTime : Sun Aug 27 11:12:42 EDT 2023
% 0.14/0.35 % CPUTime :
% 0.20/0.63 ________ _____
% 0.20/0.63 ___ __ \_________(_)________________________________
% 0.20/0.63 __ /_/ /_ ___/_ /__ __ \ ___/ _ \_ ___/_ ___/
% 0.20/0.63 _ ____/_ / _ / _ / / / /__ / __/(__ )_(__ )
% 0.20/0.63 /_/ /_/ /_/ /_/ /_/\___/ \___//____/ /____/
% 0.20/0.63
% 0.20/0.63 A Theorem Prover for First-Order Logic modulo Linear Integer Arithmetic
% 0.20/0.63 (2023-06-19)
% 0.20/0.63
% 0.20/0.63 (c) Philipp Rümmer, 2009-2023
% 0.20/0.63 Contributors: Peter Backeman, Peter Baumgartner, Angelo Brillout, Zafer Esen,
% 0.20/0.63 Amanda Stjerna.
% 0.20/0.63 Free software under BSD-3-Clause.
% 0.20/0.63
% 0.20/0.63 For more information, visit http://www.philipp.ruemmer.org/princess.shtml
% 0.20/0.63
% 0.20/0.63 Loading /export/starexec/sandbox/benchmark/theBenchmark.p ...
% 0.20/0.64 Running up to 7 provers in parallel.
% 0.20/0.66 Prover 2: Options: +triggersInConjecture +genTotalityAxioms -tightFunctionScopes -clausifier=simple +reverseFunctionalityPropagation +boolFunsAsPreds -triggerStrategy=allMinimalAndEmpty -realRatSaturationRounds=1 -ignoreQuantifiers -constructProofs=never -generateTriggers=all -randomSeed=-1065072994
% 0.20/0.66 Prover 1: Options: +triggersInConjecture -genTotalityAxioms -tightFunctionScopes -clausifier=none -reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=maximal -realRatSaturationRounds=0 +ignoreQuantifiers -constructProofs=always -generateTriggers=all -randomSeed=-1571432423
% 0.20/0.66 Prover 0: Options: +triggersInConjecture +genTotalityAxioms +tightFunctionScopes -clausifier=simple -reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=allUni -realRatSaturationRounds=0 -ignoreQuantifiers -constructProofs=never -generateTriggers=all -randomSeed=1042961893
% 0.20/0.66 Prover 3: Options: +triggersInConjecture -genTotalityAxioms -tightFunctionScopes -clausifier=none -reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=maximal -realRatSaturationRounds=1 +ignoreQuantifiers -constructProofs=never -generateTriggers=all -randomSeed=1922548996
% 0.20/0.66 Prover 4: Options: +triggersInConjecture -genTotalityAxioms -tightFunctionScopes -clausifier=simple -reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=allUni -realRatSaturationRounds=0 +ignoreQuantifiers -constructProofs=always -generateTriggers=all -randomSeed=1868514696
% 0.20/0.66 Prover 5: Options: +triggersInConjecture -genTotalityAxioms +tightFunctionScopes -clausifier=none +reverseFunctionalityPropagation +boolFunsAsPreds -triggerStrategy=allMaximal -realRatSaturationRounds=1 -ignoreQuantifiers -constructProofs=never -generateTriggers=complete -randomSeed=1259561288
% 0.20/0.66 Prover 6: Options: -triggersInConjecture -genTotalityAxioms +tightFunctionScopes -clausifier=none +reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=maximalOutermost -realRatSaturationRounds=0 -ignoreQuantifiers -constructProofs=never -generateTriggers=all -randomSeed=-1399714365
% 23.67/4.00 Prover 1: Preprocessing ...
% 24.55/4.15 Prover 2: Preprocessing ...
% 24.55/4.15 Prover 6: Preprocessing ...
% 24.55/4.16 Prover 3: Preprocessing ...
% 24.55/4.19 Prover 0: Preprocessing ...
% 24.55/4.19 Prover 5: Preprocessing ...
% 25.80/4.30 Prover 4: Preprocessing ...
% 56.18/8.41 Prover 3: Warning: ignoring some quantifiers
% 56.18/8.48 Prover 3: Constructing countermodel ...
% 56.18/8.50 Prover 6: Proving ...
% 57.23/8.52 Prover 1: Warning: ignoring some quantifiers
% 58.59/8.65 Prover 1: Constructing countermodel ...
% 65.58/9.57 Prover 4: Warning: ignoring some quantifiers
% 67.48/9.79 Prover 5: Proving ...
% 67.48/9.81 Prover 4: Constructing countermodel ...
% 70.81/10.27 Prover 0: Proving ...
% 71.27/10.31 Prover 2: Proving ...
% 71.82/10.36 Prover 3: proved (9710ms)
% 71.90/10.37
% 71.90/10.37 % SZS status Theorem for /export/starexec/sandbox/benchmark/theBenchmark.p
% 71.90/10.37
% 71.90/10.37 Prover 7: Options: +triggersInConjecture -genTotalityAxioms +tightFunctionScopes -clausifier=simple +reverseFunctionalityPropagation +boolFunsAsPreds -triggerStrategy=allUni -realRatSaturationRounds=1 +ignoreQuantifiers -constructProofs=always -generateTriggers=all -randomSeed=-236303470
% 71.90/10.39 Prover 6: stopped
% 71.90/10.41 Prover 0: stopped
% 72.21/10.42 Prover 8: Options: +triggersInConjecture +genTotalityAxioms -tightFunctionScopes -clausifier=none -reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=maximal -realRatSaturationRounds=0 +ignoreQuantifiers -constructProofs=always -generateTriggers=all -randomSeed=-200781089
% 72.21/10.42 Prover 2: stopped
% 72.21/10.43 Prover 10: Options: +triggersInConjecture -genTotalityAxioms +tightFunctionScopes -clausifier=simple -reverseFunctionalityPropagation +boolFunsAsPreds -triggerStrategy=maximal -realRatSaturationRounds=1 +ignoreQuantifiers -constructProofs=always -generateTriggers=all -randomSeed=919308125
% 72.21/10.43 Prover 5: stopped
% 72.21/10.43 Prover 11: Options: +triggersInConjecture -genTotalityAxioms +tightFunctionScopes -clausifier=simple -reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=allUni -realRatSaturationRounds=1 +ignoreQuantifiers -constructProofs=always -generateTriggers=all -randomSeed=-1509710984
% 72.21/10.43 Prover 13: Options: +triggersInConjecture -genTotalityAxioms -tightFunctionScopes -clausifier=simple -reverseFunctionalityPropagation +boolFunsAsPreds -triggerStrategy=maximal -realRatSaturationRounds=0 +ignoreQuantifiers -constructProofs=always -generateTriggers=complete -randomSeed=1138197443
% 86.49/12.35 Prover 7: Preprocessing ...
% 86.49/12.35 Prover 13: Preprocessing ...
% 87.30/12.44 Prover 8: Preprocessing ...
% 87.60/12.50 Prover 10: Preprocessing ...
% 88.31/12.61 Prover 11: Preprocessing ...
% 99.86/14.15 Prover 10: Warning: ignoring some quantifiers
% 99.86/14.18 Prover 7: Warning: ignoring some quantifiers
% 101.63/14.31 Prover 7: Constructing countermodel ...
% 101.63/14.31 Prover 8: Warning: ignoring some quantifiers
% 101.63/14.31 Prover 10: Constructing countermodel ...
% 102.44/14.44 Prover 8: Constructing countermodel ...
% 106.60/15.02 Prover 1: Found proof (size 122)
% 106.60/15.02 Prover 1: proved (14369ms)
% 106.60/15.02 Prover 8: stopped
% 106.60/15.02 Prover 7: stopped
% 106.60/15.02 Prover 10: stopped
% 106.60/15.03 Prover 4: stopped
% 108.10/15.20 Prover 11: Warning: ignoring some quantifiers
% 108.81/15.36 Prover 11: Constructing countermodel ...
% 108.81/15.37 Prover 13: Warning: ignoring some quantifiers
% 109.37/15.49 Prover 11: stopped
% 109.37/15.53 Prover 13: Constructing countermodel ...
% 110.13/15.66 Prover 13: stopped
% 110.13/15.67
% 110.13/15.67 % SZS status Theorem for /export/starexec/sandbox/benchmark/theBenchmark.p
% 110.13/15.67
% 110.13/15.69 % SZS output start Proof for theBenchmark
% 110.40/15.71 Assumptions after simplification:
% 110.40/15.71 ---------------------------------
% 110.40/15.71
% 110.40/15.71 (axiom107)
% 110.40/15.73 N_n_bool_fun_fun$(less_eq$) & ! [v0: N$] : ! [v1: N_bool_fun$] : ! [v2:
% 110.40/15.73 int] : (v2 = 0 | ~ (fun_app$g(less_eq$, v0) = v1) | ~ (fun_app$d(v1, v0) =
% 110.40/15.73 v2) | ~ N$(v0))
% 110.40/15.73
% 110.40/15.73 (axiom109)
% 110.40/15.73 N_n_bool_fun_fun$(less_eq$) & ! [v0: N$] : ! [v1: N_bool_fun$] : ! [v2:
% 110.40/15.73 int] : (v2 = 0 | ~ (fun_app$g(less_eq$, v0) = v1) | ~ (fun_app$d(v1, v0) =
% 110.40/15.73 v2) | ~ N$(v0))
% 110.40/15.73
% 110.40/15.73 (axiom111)
% 110.40/15.74 N_n_bool_fun_fun$(less_eq$) & ! [v0: N$] : ! [v1: N_bool_fun$] : ! [v2:
% 110.40/15.74 int] : (v2 = 0 | ~ (fun_app$g(less_eq$, v0) = v1) | ~ (fun_app$d(v1, v0) =
% 110.40/15.74 v2) | ~ N$(v0))
% 110.40/15.74
% 110.40/15.74 (axiom114)
% 110.40/15.74 N_n_bool_fun_fun$(less_eq$) & N$(x$) & Nat$(a$) & Nat_n_fun$(from_nat$) & ?
% 110.40/15.74 [v0: N$] : ? [v1: N_bool_fun$] : ? [v2: int] : ( ~ (v2 = 0) &
% 110.40/15.74 fun_app$i(from_nat$, a$) = v0 & fun_app$g(less_eq$, v0) = v1 & fun_app$d(v1,
% 110.40/15.74 x$) = v2 & N_bool_fun$(v1) & N$(v0))
% 110.40/15.74
% 110.40/15.74 (axiom118)
% 110.40/15.74 Nat_int_fun$(of_nat$) & N$(i$) & Nat$(a$) & ? [v0: int] : ? [v1: Nat$] : ?
% 110.40/15.74 [v2: int] : ($lesseq(v2, v0) & to_nat$(i$) = v1 & fun_app$e(of_nat$, v1) = v2
% 110.40/15.74 & fun_app$e(of_nat$, a$) = v0 & Nat$(v1))
% 110.40/15.74
% 110.40/15.74 (axiom119)
% 110.40/15.74 N_n_bool_fun_fun$(less_eq$) & N$(i$) & Nat$(a$) & Nat_n_fun$(from_nat$) & ?
% 110.40/15.74 [v0: N_bool_fun$] : ? [v1: N$] : (fun_app$i(from_nat$, a$) = v1 &
% 110.40/15.74 fun_app$g(less_eq$, i$) = v0 & fun_app$d(v0, v1) = 0 & N_bool_fun$(v0) &
% 110.40/15.74 N$(v1))
% 110.40/15.74
% 110.40/15.74 (axiom135)
% 110.40/15.75 Nat_int_fun$(of_nat$) & Nat_n_fun$(from_nat$) & ! [v0: N$] : ! [v1: Nat$] :
% 110.40/15.75 ! [v2: Nat$] : ! [v3: N$] : (v3 = v0 | ~ (fun_app$i(from_nat$, v1) = v3) |
% 110.40/15.75 ~ (to_nat$(v0) = v2) | ~ N$(v0) | ~ Nat$(v1) | ? [v4: int] : ? [v5: int]
% 110.40/15.75 : ( ~ (v5 = v4) & fun_app$e(of_nat$, v2) = v4 & fun_app$e(of_nat$, v1) =
% 110.40/15.75 v5))
% 110.40/15.75
% 110.40/15.75 (axiom16)
% 110.40/15.75 N_n_bool_fun_fun$(less_eq$) & N$(x$) & N$(i$) & ? [v0: N_bool_fun$] :
% 110.40/15.75 (fun_app$g(less_eq$, i$) = v0 & fun_app$d(v0, x$) = 0 & N_bool_fun$(v0))
% 110.40/15.75
% 110.40/15.75 (axiom186)
% 110.40/15.75 N_n_bool_fun_fun$(less_eq$) & ! [v0: N$] : ! [v1: N_bool_fun$] : ! [v2:
% 110.40/15.75 int] : (v2 = 0 | ~ (fun_app$g(less_eq$, v0) = v1) | ~ (fun_app$d(v1, v0) =
% 110.40/15.75 v2) | ~ N$(v0))
% 110.40/15.75
% 110.40/15.75 (axiom212)
% 110.40/15.75 N_n_bool_fun_fun$(less_eq$) & ! [v0: N_n_bool_fun_fun$] : ! [v1: N$] : !
% 110.40/15.75 [v2: N$] : ! [v3: N_bool_fun$] : ! [v4: int] : (v4 = 0 | ~ (fun_app$g(v0,
% 110.40/15.75 v1) = v3) | ~ (fun_app$d(v3, v2) = v4) | ~ N_n_bool_fun_fun$(v0) | ~
% 110.40/15.75 N$(v2) | ~ N$(v1) | ? [v5: N$] : ? [v6: N$] : ? [v7: N_bool_fun$] : ?
% 110.40/15.75 [v8: N_bool_fun$] : ? [v9: int] : ( ~ (v9 = 0) & fun_app$g(v0, v6) = v7 &
% 110.40/15.75 fun_app$g(v0, v5) = v8 & fun_app$d(v8, v6) = v9 & fun_app$d(v7, v5) = 0 &
% 110.40/15.76 N_bool_fun$(v8) & N_bool_fun$(v7) & N$(v6) & N$(v5)) | ? [v5: N$] : ?
% 110.40/15.76 [v6: N$] : ? [v7: N_bool_fun$] : ? [v8: N_bool_fun$] : ? [v9: int] : ( ~
% 110.40/15.76 (v9 = 0) & fun_app$g(v0, v5) = v8 & fun_app$g(less_eq$, v6) = v7 &
% 110.40/15.76 fun_app$d(v8, v6) = v9 & fun_app$d(v7, v5) = 0 & N_bool_fun$(v8) &
% 110.40/15.76 N_bool_fun$(v7) & N$(v6) & N$(v5)))
% 110.40/15.76
% 110.40/15.76 (axiom234)
% 110.40/15.76 N_n_bool_fun_fun$(less_eq$) & ! [v0: N$] : ! [v1: N_bool_fun$] : ! [v2:
% 110.40/15.76 int] : (v2 = 0 | ~ (fun_app$g(less_eq$, v0) = v1) | ~ (fun_app$d(v1, v0) =
% 110.40/15.76 v2) | ~ N$(v0))
% 110.40/15.76
% 110.40/15.76 (axiom24)
% 110.68/15.76 Nat_int_fun$(of_nat$) & N_n_bool_fun_fun$(less_eq$) & ! [v0: N$] : ! [v1:
% 110.68/15.76 N$] : ! [v2: N_bool_fun$] : ( ~ (fun_app$g(less_eq$, v0) = v2) | ~
% 110.68/15.76 (fun_app$d(v2, v1) = 0) | ~ N$(v1) | ~ N$(v0) | ? [v3: Nat$] : ? [v4:
% 110.68/15.76 int] : ? [v5: Nat$] : ? [v6: int] : ($lesseq(v6, v4) & to_nat$(v1) = v3
% 110.68/15.76 & to_nat$(v0) = v5 & fun_app$e(of_nat$, v5) = v6 & fun_app$e(of_nat$, v3)
% 110.68/15.76 = v4 & Nat$(v5) & Nat$(v3)))
% 110.68/15.76
% 110.68/15.76 (axiom242)
% 110.68/15.76 N_n_bool_fun_fun$(less$) & N$(x$) & Nat$(a$) & Nat_n_fun$(from_nat$) & ? [v0:
% 110.68/15.76 N_bool_fun$] : ? [v1: N$] : (fun_app$i(from_nat$, a$) = v1 &
% 110.68/15.76 fun_app$g(less$, x$) = v0 & fun_app$d(v0, v1) = 0 & N_bool_fun$(v0) &
% 110.68/15.76 N$(v1))
% 110.68/15.76
% 110.68/15.76 (axiom374)
% 110.68/15.77 Nat_int_fun$(of_nat$) & N_n_bool_fun_fun$(less$) & ! [v0: N$] : ! [v1: N$] :
% 110.68/15.77 ! [v2: N_bool_fun$] : ( ~ (fun_app$g(less$, v0) = v2) | ~ (fun_app$d(v2, v1)
% 110.68/15.77 = 0) | ~ N$(v1) | ~ N$(v0) | ? [v3: Nat$] : ? [v4: int] : ? [v5:
% 110.68/15.77 Nat$] : ? [v6: int] : ($lesseq(1, $difference(v4, v6)) & to_nat$(v1) = v3
% 110.68/15.77 & to_nat$(v0) = v5 & fun_app$e(of_nat$, v5) = v6 & fun_app$e(of_nat$, v3)
% 110.68/15.77 = v4 & Nat$(v5) & Nat$(v3)))
% 110.68/15.77
% 110.68/15.77 (axiom406)
% 110.68/15.77 Nat_int_fun$(of_nat$) & N_n_bool_fun_fun$(less$) & Nat_n_fun$(from_nat$) & !
% 110.68/15.77 [v0: N$] : ! [v1: Nat$] : ! [v2: N_bool_fun$] : ! [v3: N$] : ( ~
% 110.68/15.77 (fun_app$i(from_nat$, v1) = v3) | ~ (fun_app$g(less$, v0) = v2) | ~
% 110.68/15.77 (fun_app$d(v2, v3) = 0) | ~ N$(v0) | ~ Nat$(v1) | ? [v4: int] : ? [v5:
% 110.68/15.77 Nat$] : ? [v6: int] : ($lesseq(1, $difference(v4, v6)) & to_nat$(v0) = v5
% 110.68/15.77 & fun_app$e(of_nat$, v5) = v6 & fun_app$e(of_nat$, v1) = v4 & Nat$(v5)))
% 110.68/15.77
% 110.68/15.77 (axiom413)
% 110.68/15.77 N_n_bool_fun_fun$(less_eq$) & N_n_bool_fun_fun$(less$) & ! [v0: N$] : ! [v1:
% 110.68/15.77 N$] : ! [v2: N_bool_fun$] : ! [v3: int] : (v3 = 0 | v1 = v0 | ~
% 110.68/15.77 (fun_app$g(less$, v0) = v2) | ~ (fun_app$d(v2, v1) = v3) | ~ N$(v1) | ~
% 110.68/15.77 N$(v0) | ? [v4: N_bool_fun$] : ? [v5: int] : ( ~ (v5 = 0) &
% 110.68/15.77 fun_app$g(less_eq$, v0) = v4 & fun_app$d(v4, v1) = v5 & N_bool_fun$(v4)))
% 110.68/15.77 & ! [v0: N$] : ! [v1: N$] : ! [v2: N_bool_fun$] : ( ~ (fun_app$g(less$, v0)
% 110.68/15.77 = v2) | ~ (fun_app$d(v2, v1) = 0) | ~ N$(v1) | ~ N$(v0) | ( ~ (v1 = v0)
% 110.68/15.77 & ? [v3: N_bool_fun$] : (fun_app$g(less_eq$, v0) = v3 & fun_app$d(v3, v1)
% 110.68/15.78 = 0 & N_bool_fun$(v3))))
% 110.68/15.78
% 110.68/15.78 (axiom417)
% 110.68/15.78 N_n_bool_fun_fun$(less_eq$) & N_n_bool_fun_fun$(less$) & ! [v0: N$] : ! [v1:
% 110.68/15.78 N$] : ! [v2: N_bool_fun$] : ! [v3: int] : (v3 = 0 | v1 = v0 | ~
% 110.68/15.78 (fun_app$g(less$, v0) = v2) | ~ (fun_app$d(v2, v1) = v3) | ~ N$(v1) | ~
% 110.68/15.78 N$(v0) | ? [v4: N_bool_fun$] : ? [v5: int] : ( ~ (v5 = 0) &
% 110.68/15.78 fun_app$g(less_eq$, v0) = v4 & fun_app$d(v4, v1) = v5 & N_bool_fun$(v4)))
% 110.68/15.78 & ! [v0: N$] : ! [v1: N$] : ! [v2: N_bool_fun$] : ( ~ (fun_app$g(less$, v0)
% 110.68/15.78 = v2) | ~ (fun_app$d(v2, v1) = 0) | ~ N$(v1) | ~ N$(v0) | ( ~ (v1 = v0)
% 110.68/15.78 & ? [v3: N_bool_fun$] : (fun_app$g(less_eq$, v0) = v3 & fun_app$d(v3, v1)
% 110.68/15.78 = 0 & N_bool_fun$(v3))))
% 110.68/15.78
% 110.68/15.78 (axiom418)
% 110.68/15.78 N_n_bool_fun_fun$(less_eq$) & N_n_bool_fun_fun$(less$) & ! [v0: N$] : ! [v1:
% 110.68/15.78 N$] : ! [v2: N_bool_fun$] : ! [v3: int] : (v3 = 0 | v1 = v0 | ~
% 110.68/15.78 (fun_app$g(less$, v0) = v2) | ~ (fun_app$d(v2, v1) = v3) | ~ N$(v1) | ~
% 110.68/15.78 N$(v0) | ? [v4: N_bool_fun$] : ? [v5: int] : ( ~ (v5 = 0) &
% 110.68/15.78 fun_app$g(less_eq$, v0) = v4 & fun_app$d(v4, v1) = v5 & N_bool_fun$(v4)))
% 110.68/15.78 & ! [v0: N$] : ! [v1: N$] : ! [v2: N_bool_fun$] : ( ~ (fun_app$g(less$, v0)
% 110.68/15.78 = v2) | ~ (fun_app$d(v2, v1) = 0) | ~ N$(v1) | ~ N$(v0) | ( ~ (v1 = v0)
% 110.68/15.78 & ? [v3: N_bool_fun$] : (fun_app$g(less_eq$, v0) = v3 & fun_app$d(v3, v1)
% 110.68/15.78 = 0 & N_bool_fun$(v3))))
% 110.68/15.78
% 110.68/15.78 (axiom441)
% 110.68/15.79 N_n_bool_fun_fun$(less_eq$) & N_n_bool_fun_fun$(less$) & ! [v0: N$] : ! [v1:
% 110.68/15.79 N$] : ! [v2: N_bool_fun$] : ! [v3: int] : (v3 = 0 | v1 = v0 | ~
% 110.68/15.79 (fun_app$g(less$, v0) = v2) | ~ (fun_app$d(v2, v1) = v3) | ~ N$(v1) | ~
% 110.68/15.79 N$(v0) | ? [v4: N_bool_fun$] : ? [v5: int] : ( ~ (v5 = 0) &
% 110.68/15.79 fun_app$g(less_eq$, v0) = v4 & fun_app$d(v4, v1) = v5 & N_bool_fun$(v4)))
% 110.68/15.79 & ! [v0: N$] : ! [v1: N$] : ! [v2: N_bool_fun$] : ( ~ (fun_app$g(less$, v0)
% 110.68/15.79 = v2) | ~ (fun_app$d(v2, v1) = 0) | ~ N$(v1) | ~ N$(v0) | ( ~ (v1 = v0)
% 110.68/15.79 & ? [v3: N_bool_fun$] : (fun_app$g(less_eq$, v0) = v3 & fun_app$d(v3, v1)
% 110.68/15.79 = 0 & N_bool_fun$(v3))))
% 110.68/15.79
% 110.68/15.79 (axiom451)
% 110.68/15.79 N_n_bool_fun_fun$(less_eq$) & N_n_bool_fun_fun$(less$) & ! [v0: N$] : ! [v1:
% 110.68/15.79 N$] : ! [v2: N_bool_fun$] : ! [v3: int] : (v3 = 0 | v1 = v0 | ~
% 110.68/15.79 (fun_app$g(less$, v0) = v2) | ~ (fun_app$d(v2, v1) = v3) | ~ N$(v1) | ~
% 110.68/15.79 N$(v0) | ? [v4: N_bool_fun$] : ? [v5: int] : ( ~ (v5 = 0) &
% 110.68/15.79 fun_app$g(less_eq$, v0) = v4 & fun_app$d(v4, v1) = v5 & N_bool_fun$(v4)))
% 110.68/15.79 & ! [v0: N$] : ! [v1: N$] : ! [v2: N_bool_fun$] : ( ~ (fun_app$g(less$, v0)
% 110.68/15.79 = v2) | ~ (fun_app$d(v2, v1) = 0) | ~ N$(v1) | ~ N$(v0) | ( ~ (v1 = v0)
% 110.68/15.79 & ? [v3: N_bool_fun$] : (fun_app$g(less_eq$, v0) = v3 & fun_app$d(v3, v1)
% 110.68/15.79 = 0 & N_bool_fun$(v3))))
% 110.68/15.79
% 110.68/15.79 (axiom465)
% 110.68/15.79 N_n_bool_fun_fun$(less_eq$) & N_n_bool_fun_fun$(less$) & ! [v0: N$] : ! [v1:
% 110.68/15.79 N$] : ! [v2: N_bool_fun$] : ! [v3: int] : (v3 = 0 | v1 = v0 | ~
% 110.68/15.79 (fun_app$g(less$, v0) = v2) | ~ (fun_app$d(v2, v1) = v3) | ~ N$(v1) | ~
% 110.68/15.79 N$(v0) | ? [v4: N_bool_fun$] : ? [v5: int] : ( ~ (v5 = 0) &
% 110.68/15.79 fun_app$g(less_eq$, v0) = v4 & fun_app$d(v4, v1) = v5 & N_bool_fun$(v4)))
% 110.68/15.79 & ! [v0: N$] : ! [v1: N$] : ! [v2: N_bool_fun$] : ( ~ (fun_app$g(less$, v0)
% 110.68/15.79 = v2) | ~ (fun_app$d(v2, v1) = 0) | ~ N$(v1) | ~ N$(v0) | ( ~ (v1 = v0)
% 110.68/15.79 & ? [v3: N_bool_fun$] : (fun_app$g(less_eq$, v0) = v3 & fun_app$d(v3, v1)
% 110.68/15.79 = 0 & N_bool_fun$(v3))))
% 110.68/15.79
% 110.68/15.79 (axiom574)
% 110.68/15.80 N_n_bool_fun_fun$(less_eq$) & ! [v0: N$] : ! [v1: N_bool_fun$] : ! [v2:
% 110.68/15.80 int] : (v2 = 0 | ~ (fun_app$g(less_eq$, v0) = v1) | ~ (fun_app$d(v1, v0) =
% 110.68/15.80 v2) | ~ N$(v0))
% 110.68/15.80
% 110.68/15.80 (axiom6)
% 110.68/15.80 Nat_int_fun$(of_nat$) & N$(x$) & N$(i$) & Nat$(ja$) & ? [v0: Nat$] : ? [v1:
% 110.68/15.80 int] : ? [v2: Nat$] : ? [v3: int] : ? [v4: int] : (to_nat$(x$) = v2 &
% 110.68/15.80 to_nat$(i$) = v0 & fun_app$e(of_nat$, v2) = v3 & fun_app$e(of_nat$, v0) = v1
% 110.68/15.80 & fun_app$e(of_nat$, ja$) = v4 & Nat$(v2) & Nat$(v0) & ($sum($difference(v4,
% 110.68/15.80 v3), v1) = 0 | ~ ($lesseq(v1, v3))) & (v4 = 0 | ~ ($lesseq(1,
% 110.68/15.80 $difference(v1, v3)))))
% 110.68/15.80
% 110.68/15.80 (conjecture5)
% 110.68/15.80 Nat_int_fun$(of_nat$) & N$(x$) & N$(i$) & ? [v0: Nat$] : ? [v1: int] : ?
% 110.68/15.80 [v2: Nat$] : ? [v3: int] : ($lesseq(1, $difference(v1, v3)) & to_nat$(x$) =
% 110.68/15.80 v2 & to_nat$(i$) = v0 & fun_app$e(of_nat$, v2) = v3 & fun_app$e(of_nat$, v0)
% 110.68/15.80 = v1 & Nat$(v2) & Nat$(v0))
% 110.68/15.80
% 110.68/15.80 (function-axioms)
% 110.68/15.82 ! [v0: Nat$] : ! [v1: Nat$] : ! [v2: Nat$] : ! [v3: Nat_nat_fun$] : (v1 =
% 110.68/15.82 v0 | ~ (fun_app$r(v3, v2) = v1) | ~ (fun_app$r(v3, v2) = v0)) & ! [v0:
% 110.68/15.82 Nat_bool_fun$] : ! [v1: Nat_bool_fun$] : ! [v2: Nat$] : ! [v3:
% 110.68/15.82 Nat_nat_bool_fun_fun$] : (v1 = v0 | ~ (fun_app$q(v3, v2) = v1) | ~
% 110.68/15.82 (fun_app$q(v3, v2) = v0)) & ! [v0: N$] : ! [v1: N$] : ! [v2: int] : !
% 110.68/15.82 [v3: Int_n_fun$] : (v1 = v0 | ~ (fun_app$p(v3, v2) = v1) | ~ (fun_app$p(v3,
% 110.68/15.82 v2) = v0)) & ! [v0: int] : ! [v1: int] : ! [v2: N$] : ! [v3:
% 110.68/15.82 N_int_fun$] : (v1 = v0 | ~ (fun_app$o(v3, v2) = v1) | ~ (fun_app$o(v3, v2)
% 110.68/15.82 = v0)) & ! [v0: int] : ! [v1: int] : ! [v2: Int_set$] : ! [v3:
% 110.68/15.82 Int_set_int_fun$] : (v1 = v0 | ~ (fun_app$n(v3, v2) = v1) | ~
% 110.68/15.82 (fun_app$n(v3, v2) = v0)) & ! [v0: N$] : ! [v1: N$] : ! [v2: Int_set$] :
% 110.68/15.82 ! [v3: Int_set_n_fun$] : (v1 = v0 | ~ (fun_app$m(v3, v2) = v1) | ~
% 110.68/15.82 (fun_app$m(v3, v2) = v0)) & ! [v0: Int_set$] : ! [v1: Int_set$] : ! [v2:
% 110.68/15.82 int] : ! [v3: Int_int_set_fun$] : (v1 = v0 | ~ (fun_app$l(v3, v2) = v1) |
% 110.68/15.82 ~ (fun_app$l(v3, v2) = v0)) & ! [v0: Int_set$] : ! [v1: Int_set$] : !
% 110.68/15.82 [v2: N$] : ! [v3: N_int_set_fun$] : (v1 = v0 | ~ (fun_app$k(v3, v2) = v1) |
% 110.68/15.82 ~ (fun_app$k(v3, v2) = v0)) & ! [v0: MultipleValueBool] : ! [v1:
% 110.68/15.82 MultipleValueBool] : ! [v2: Nat$] : ! [v3: Nat_bool_fun$] : (v1 = v0 | ~
% 110.68/15.82 (fun_app$j(v3, v2) = v1) | ~ (fun_app$j(v3, v2) = v0)) & ! [v0:
% 110.68/15.82 MultipleValueBool] : ! [v1: MultipleValueBool] : ! [v2: Int_set_set$] : !
% 110.68/15.82 [v3: Int_set_set$] : (v1 = v0 | ~ (less_eq$c(v3, v2) = v1) | ~
% 110.68/15.82 (less_eq$c(v3, v2) = v0)) & ! [v0: MultipleValueBool] : ! [v1:
% 110.68/15.82 MultipleValueBool] : ! [v2: N_set$] : ! [v3: N_set$] : (v1 = v0 | ~
% 110.68/15.82 (less_eq$b(v3, v2) = v1) | ~ (less_eq$b(v3, v2) = v0)) & ! [v0: N$] : !
% 110.68/15.82 [v1: N$] : ! [v2: Nat$] : ! [v3: Nat_n_fun$] : (v1 = v0 | ~ (fun_app$i(v3,
% 110.68/15.82 v2) = v1) | ~ (fun_app$i(v3, v2) = v0)) & ! [v0: Int_set_set_set$] :
% 110.68/15.82 ! [v1: Int_set_set_set$] : ! [v2: Int_set_set_set$] : ! [v3:
% 110.68/15.82 Int_set_set_set$] : (v1 = v0 | ~ (plus$e(v3, v2) = v1) | ~ (plus$e(v3, v2)
% 110.68/15.82 = v0)) & ! [v0: MultipleValueBool] : ! [v1: MultipleValueBool] : ! [v2:
% 110.68/15.82 Int_set_set_set$] : ! [v3: Int_set_set$] : (v1 = v0 | ~ (member$d(v3, v2)
% 110.68/15.82 = v1) | ~ (member$d(v3, v2) = v0)) & ! [v0: N_set_set$] : ! [v1:
% 110.68/15.82 N_set_set$] : ! [v2: N_set_set$] : ! [v3: N_set_set$] : (v1 = v0 | ~
% 110.68/15.82 (plus$d(v3, v2) = v1) | ~ (plus$d(v3, v2) = v0)) & ! [v0:
% 110.68/15.82 MultipleValueBool] : ! [v1: MultipleValueBool] : ! [v2: N_set_set$] : !
% 110.68/15.82 [v3: N_set$] : (v1 = v0 | ~ (member$c(v3, v2) = v1) | ~ (member$c(v3, v2) =
% 110.68/15.82 v0)) & ! [v0: N_set$] : ! [v1: N_set$] : ! [v2: N_set$] : ! [v3:
% 110.68/15.82 N_set$] : (v1 = v0 | ~ (plus$c(v3, v2) = v1) | ~ (plus$c(v3, v2) = v0)) &
% 110.68/15.82 ! [v0: Int_set$] : ! [v1: Int_set$] : ! [v2: Int_set$] : ! [v3:
% 110.68/15.82 Int_set_int_set_fun$] : (v1 = v0 | ~ (fun_app$h(v3, v2) = v1) | ~
% 110.68/15.82 (fun_app$h(v3, v2) = v0)) & ! [v0: Int_set_set$] : ! [v1: Int_set_set$] :
% 110.68/15.82 ! [v2: Int_set_set$] : ! [v3: Int_set_set$] : (v1 = v0 | ~ (plus$a(v3, v2) =
% 110.68/15.83 v1) | ~ (plus$a(v3, v2) = v0)) & ! [v0: N_bool_fun$] : ! [v1:
% 110.68/15.83 N_bool_fun$] : ! [v2: N$] : ! [v3: N_n_bool_fun_fun$] : (v1 = v0 | ~
% 110.68/15.83 (fun_app$g(v3, v2) = v1) | ~ (fun_app$g(v3, v2) = v0)) & ! [v0: N$] : !
% 110.68/15.83 [v1: N$] : ! [v2: N$] : ! [v3: N_n_fun$] : (v1 = v0 | ~ (fun_app$f(v3, v2)
% 110.68/15.83 = v1) | ~ (fun_app$f(v3, v2) = v0)) & ! [v0: int] : ! [v1: int] : !
% 110.68/15.83 [v2: Nat$] : ! [v3: Nat_int_fun$] : (v1 = v0 | ~ (fun_app$e(v3, v2) = v1) |
% 110.68/15.83 ~ (fun_app$e(v3, v2) = v0)) & ! [v0: MultipleValueBool] : ! [v1:
% 110.68/15.83 MultipleValueBool] : ! [v2: N$] : ! [v3: N_bool_fun$] : (v1 = v0 | ~
% 110.68/15.83 (fun_app$d(v3, v2) = v1) | ~ (fun_app$d(v3, v2) = v0)) & ! [v0:
% 110.68/15.83 MultipleValueBool] : ! [v1: MultipleValueBool] : ! [v2: N_set$] : ! [v3:
% 110.68/15.83 N$] : (v1 = v0 | ~ (member$b(v3, v2) = v1) | ~ (member$b(v3, v2) = v0)) &
% 110.68/15.83 ! [v0: MultipleValueBool] : ! [v1: MultipleValueBool] : ! [v2: Int_set$] :
% 110.68/15.83 ! [v3: Int_set_bool_fun$] : (v1 = v0 | ~ (fun_app$c(v3, v2) = v1) | ~
% 110.68/15.83 (fun_app$c(v3, v2) = v0)) & ! [v0: MultipleValueBool] : ! [v1:
% 110.68/15.83 MultipleValueBool] : ! [v2: Int_set_set$] : ! [v3: Int_set$] : (v1 = v0 |
% 110.68/15.83 ~ (member$(v3, v2) = v1) | ~ (member$(v3, v2) = v0)) & ! [v0: int] : !
% 110.68/15.83 [v1: int] : ! [v2: int] : ! [v3: Int_int_fun$] : (v1 = v0 | ~
% 110.68/15.83 (fun_app$b(v3, v2) = v1) | ~ (fun_app$b(v3, v2) = v0)) & ! [v0:
% 110.68/15.83 Int_bool_fun$] : ! [v1: Int_bool_fun$] : ! [v2: int] : ! [v3:
% 110.68/15.83 Int_int_bool_fun_fun$] : (v1 = v0 | ~ (fun_app$a(v3, v2) = v1) | ~
% 110.68/15.83 (fun_app$a(v3, v2) = v0)) & ! [v0: MultipleValueBool] : ! [v1:
% 110.68/15.83 MultipleValueBool] : ! [v2: int] : ! [v3: Int_bool_fun$] : (v1 = v0 | ~
% 110.68/15.83 (fun_app$(v3, v2) = v1) | ~ (fun_app$(v3, v2) = v0)) & ! [v0:
% 110.68/15.83 Int_set_bool_fun$] : ! [v1: Int_set_bool_fun$] : ! [v2: Int_set$] : (v1 =
% 110.68/15.83 v0 | ~ (less$a(v2) = v1) | ~ (less$a(v2) = v0)) & ! [v0:
% 110.68/15.83 Int_set_int_set_fun$] : ! [v1: Int_set_int_set_fun$] : ! [v2: Int_set$] :
% 110.68/15.83 (v1 = v0 | ~ (minus$a(v2) = v1) | ~ (minus$a(v2) = v0)) & ! [v0:
% 110.68/15.83 Int_int_fun$] : ! [v1: Int_int_fun$] : ! [v2: Nat$] : (v1 = v0 | ~
% 110.68/15.83 (iterate_add$(v2) = v1) | ~ (iterate_add$(v2) = v0)) & ! [v0: Nat$] : !
% 110.68/15.83 [v1: Nat$] : ! [v2: int] : (v1 = v0 | ~ (nat$(v2) = v1) | ~ (nat$(v2) =
% 110.68/15.83 v0)) & ! [v0: Int_set_bool_fun$] : ! [v1: Int_set_bool_fun$] : ! [v2:
% 110.68/15.83 Int_set$] : (v1 = v0 | ~ (less_eq$a(v2) = v1) | ~ (less_eq$a(v2) = v0)) &
% 110.68/15.83 ! [v0: Int_set$] : ! [v1: Int_set$] : ! [v2: Int_bool_fun$] : (v1 = v0 | ~
% 110.68/15.83 (collect$b(v2) = v1) | ~ (collect$b(v2) = v0)) & ! [v0: Int_set_set$] : !
% 110.68/15.83 [v1: Int_set_set$] : ! [v2: Int_set_bool_fun$] : (v1 = v0 | ~ (collect$a(v2)
% 110.68/15.83 = v1) | ~ (collect$a(v2) = v0)) & ! [v0: N_set$] : ! [v1: N_set$] : !
% 110.68/15.83 [v2: N_bool_fun$] : (v1 = v0 | ~ (collect$(v2) = v1) | ~ (collect$(v2) =
% 110.68/15.83 v0)) & ! [v0: Int_set_int_set_fun$] : ! [v1: Int_set_int_set_fun$] : !
% 110.68/15.83 [v2: Int_set$] : (v1 = v0 | ~ (plus$b(v2) = v1) | ~ (plus$b(v2) = v0)) & !
% 110.68/15.83 [v0: N_n_fun$] : ! [v1: N_n_fun$] : ! [v2: N$] : (v1 = v0 | ~ (plus$(v2) =
% 110.68/15.83 v1) | ~ (plus$(v2) = v0)) & ! [v0: N_n_fun$] : ! [v1: N_n_fun$] : !
% 110.68/15.83 [v2: N$] : (v1 = v0 | ~ (minus$(v2) = v1) | ~ (minus$(v2) = v0)) & ! [v0:
% 110.68/15.83 Nat$] : ! [v1: Nat$] : ! [v2: N$] : (v1 = v0 | ~ (to_nat$(v2) = v1) | ~
% 110.68/15.83 (to_nat$(v2) = v0)) & ! [v0: N_bool_fun$] : ! [v1: N_bool_fun$] : ! [v2:
% 110.68/15.83 N_set$] : (v1 = v0 | ~ (uu$(v2) = v1) | ~ (uu$(v2) = v0)) & ! [v0:
% 110.68/15.83 Int_bool_fun$] : ! [v1: Int_bool_fun$] : ! [v2: Int_set$] : (v1 = v0 | ~
% 110.68/15.83 (uub$(v2) = v1) | ~ (uub$(v2) = v0)) & ! [v0: Int_set_bool_fun$] : ! [v1:
% 110.68/15.83 Int_set_bool_fun$] : ! [v2: int] : (v1 = v0 | ~ (member$a(v2) = v1) | ~
% 110.68/15.83 (member$a(v2) = v0)) & ! [v0: Int_set_bool_fun$] : ! [v1:
% 110.68/15.83 Int_set_bool_fun$] : ! [v2: Int_set_set$] : (v1 = v0 | ~ (uua$(v2) = v1) |
% 110.68/15.83 ~ (uua$(v2) = v0)) & ! [v0: Int_int_fun$] : ! [v1: Int_int_fun$] : !
% 110.68/15.83 [v2: int] : (v1 = v0 | ~ (uud$(v2) = v1) | ~ (uud$(v2) = v0))
% 110.68/15.83
% 110.68/15.83 Further assumptions not needed in the proof:
% 110.68/15.83 --------------------------------------------
% 110.68/15.83 axiom0, axiom1, axiom10, axiom100, axiom101, axiom102, axiom103, axiom104,
% 110.68/15.83 axiom105, axiom106, axiom108, axiom11, axiom110, axiom112, axiom113, axiom115,
% 110.68/15.83 axiom116, axiom117, axiom12, axiom120, axiom121, axiom122, axiom123, axiom124,
% 110.68/15.83 axiom125, axiom126, axiom127, axiom128, axiom129, axiom13, axiom130, axiom131,
% 110.68/15.83 axiom132, axiom133, axiom134, axiom136, axiom137, axiom138, axiom139, axiom14,
% 110.68/15.83 axiom140, axiom141, axiom142, axiom143, axiom144, axiom145, axiom146, axiom147,
% 110.68/15.83 axiom148, axiom149, axiom15, axiom150, axiom151, axiom152, axiom153, axiom154,
% 110.68/15.83 axiom155, axiom156, axiom157, axiom158, axiom159, axiom160, axiom161, axiom162,
% 110.68/15.83 axiom163, axiom164, axiom165, axiom166, axiom167, axiom168, axiom169, axiom17,
% 110.68/15.83 axiom170, axiom171, axiom172, axiom173, axiom174, axiom175, axiom176, axiom177,
% 110.68/15.83 axiom178, axiom179, axiom18, axiom180, axiom181, axiom182, axiom183, axiom184,
% 110.68/15.83 axiom185, axiom187, axiom188, axiom189, axiom19, axiom190, axiom191, axiom192,
% 110.68/15.83 axiom193, axiom194, axiom195, axiom196, axiom197, axiom198, axiom199, axiom2,
% 110.68/15.83 axiom20, axiom200, axiom201, axiom202, axiom203, axiom204, axiom205, axiom206,
% 110.68/15.83 axiom207, axiom208, axiom209, axiom21, axiom210, axiom211, axiom213, axiom214,
% 110.68/15.83 axiom215, axiom216, axiom217, axiom218, axiom219, axiom22, axiom220, axiom221,
% 110.68/15.83 axiom222, axiom223, axiom224, axiom225, axiom226, axiom227, axiom228, axiom229,
% 110.68/15.83 axiom23, axiom230, axiom231, axiom232, axiom233, axiom235, axiom236, axiom237,
% 110.68/15.83 axiom238, axiom239, axiom240, axiom241, axiom243, axiom244, axiom245, axiom246,
% 110.68/15.83 axiom247, axiom248, axiom249, axiom25, axiom250, axiom251, axiom252, axiom253,
% 110.68/15.83 axiom254, axiom255, axiom256, axiom257, axiom258, axiom259, axiom26, axiom260,
% 110.68/15.83 axiom261, axiom262, axiom263, axiom264, axiom265, axiom266, axiom267, axiom268,
% 110.68/15.83 axiom269, axiom27, axiom270, axiom271, axiom272, axiom273, axiom274, axiom275,
% 110.68/15.83 axiom276, axiom277, axiom278, axiom279, axiom28, axiom280, axiom281, axiom282,
% 110.68/15.83 axiom283, axiom284, axiom285, axiom286, axiom287, axiom288, axiom289, axiom29,
% 110.68/15.83 axiom290, axiom291, axiom292, axiom293, axiom294, axiom295, axiom296, axiom297,
% 110.68/15.83 axiom298, axiom299, axiom3, axiom30, axiom300, axiom301, axiom302, axiom303,
% 110.68/15.83 axiom304, axiom305, axiom306, axiom307, axiom308, axiom309, axiom31, axiom310,
% 110.68/15.83 axiom311, axiom312, axiom313, axiom314, axiom315, axiom316, axiom317, axiom318,
% 110.68/15.83 axiom319, axiom32, axiom320, axiom321, axiom322, axiom323, axiom324, axiom325,
% 110.68/15.83 axiom326, axiom327, axiom328, axiom329, axiom33, axiom330, axiom331, axiom332,
% 110.68/15.83 axiom333, axiom334, axiom335, axiom336, axiom337, axiom338, axiom339, axiom34,
% 110.68/15.83 axiom340, axiom341, axiom342, axiom343, axiom344, axiom345, axiom346, axiom347,
% 110.68/15.83 axiom348, axiom349, axiom35, axiom350, axiom351, axiom352, axiom353, axiom354,
% 110.68/15.83 axiom355, axiom356, axiom357, axiom358, axiom359, axiom36, axiom360, axiom361,
% 110.68/15.83 axiom362, axiom363, axiom364, axiom365, axiom366, axiom367, axiom368, axiom369,
% 110.68/15.83 axiom37, axiom370, axiom371, axiom372, axiom373, axiom375, axiom376, axiom377,
% 110.68/15.83 axiom378, axiom379, axiom38, axiom380, axiom381, axiom382, axiom383, axiom384,
% 110.68/15.83 axiom385, axiom386, axiom387, axiom388, axiom389, axiom39, axiom390, axiom391,
% 110.68/15.83 axiom392, axiom393, axiom394, axiom395, axiom396, axiom397, axiom398, axiom399,
% 110.68/15.83 axiom4, axiom40, axiom400, axiom401, axiom402, axiom403, axiom404, axiom405,
% 110.68/15.83 axiom407, axiom408, axiom409, axiom41, axiom410, axiom411, axiom412, axiom414,
% 110.68/15.83 axiom415, axiom416, axiom419, axiom42, axiom420, axiom421, axiom422, axiom423,
% 110.68/15.83 axiom424, axiom425, axiom426, axiom427, axiom428, axiom429, axiom43, axiom430,
% 110.68/15.83 axiom431, axiom432, axiom433, axiom434, axiom435, axiom436, axiom437, axiom438,
% 110.68/15.83 axiom439, axiom44, axiom440, axiom442, axiom443, axiom444, axiom445, axiom446,
% 110.68/15.83 axiom447, axiom448, axiom449, axiom45, axiom450, axiom452, axiom453, axiom454,
% 110.68/15.83 axiom455, axiom456, axiom457, axiom458, axiom459, axiom46, axiom460, axiom461,
% 110.68/15.83 axiom462, axiom463, axiom464, axiom466, axiom467, axiom468, axiom469, axiom47,
% 110.68/15.83 axiom470, axiom471, axiom472, axiom473, axiom474, axiom475, axiom476, axiom477,
% 110.68/15.83 axiom478, axiom479, axiom48, axiom480, axiom481, axiom482, axiom483, axiom484,
% 110.68/15.83 axiom485, axiom486, axiom487, axiom488, axiom489, axiom49, axiom490, axiom491,
% 110.68/15.83 axiom492, axiom493, axiom494, axiom495, axiom496, axiom497, axiom498, axiom499,
% 110.68/15.83 axiom50, axiom500, axiom501, axiom502, axiom503, axiom504, axiom505, axiom506,
% 110.68/15.83 axiom507, axiom508, axiom509, axiom51, axiom510, axiom511, axiom512, axiom513,
% 110.68/15.83 axiom514, axiom515, axiom516, axiom517, axiom518, axiom519, axiom52, axiom520,
% 110.68/15.83 axiom521, axiom522, axiom523, axiom524, axiom525, axiom526, axiom527, axiom528,
% 110.68/15.83 axiom529, axiom53, axiom530, axiom531, axiom532, axiom533, axiom534, axiom535,
% 110.68/15.83 axiom536, axiom537, axiom538, axiom539, axiom54, axiom540, axiom541, axiom542,
% 110.68/15.83 axiom543, axiom544, axiom545, axiom546, axiom547, axiom548, axiom549, axiom55,
% 110.68/15.83 axiom550, axiom551, axiom552, axiom553, axiom554, axiom555, axiom556, axiom557,
% 110.68/15.83 axiom558, axiom559, axiom56, axiom560, axiom561, axiom562, axiom563, axiom564,
% 110.68/15.83 axiom565, axiom566, axiom567, axiom568, axiom569, axiom57, axiom570, axiom571,
% 110.68/15.83 axiom572, axiom573, axiom575, axiom576, axiom577, axiom578, axiom579, axiom58,
% 110.68/15.83 axiom580, axiom581, axiom582, axiom583, axiom584, axiom585, axiom586, axiom587,
% 110.68/15.83 axiom588, axiom589, axiom59, axiom590, axiom591, axiom592, axiom593, axiom594,
% 110.68/15.83 axiom595, axiom596, axiom597, axiom598, axiom599, axiom60, axiom600, axiom601,
% 110.68/15.83 axiom602, axiom603, axiom604, axiom605, axiom606, axiom607, axiom608, axiom609,
% 110.68/15.83 axiom61, axiom610, axiom611, axiom612, axiom613, axiom614, axiom615, axiom616,
% 110.68/15.83 axiom617, axiom62, axiom63, axiom64, axiom65, axiom66, axiom67, axiom68,
% 110.68/15.83 axiom69, axiom7, axiom70, axiom71, axiom72, axiom73, axiom74, axiom75, axiom76,
% 110.68/15.83 axiom77, axiom78, axiom79, axiom8, axiom80, axiom81, axiom82, axiom83, axiom84,
% 110.68/15.83 axiom85, axiom86, axiom87, axiom88, axiom89, axiom9, axiom90, axiom91, axiom92,
% 110.68/15.83 axiom93, axiom94, axiom95, axiom96, axiom97, axiom98, axiom99, formula_619,
% 110.68/15.83 formula_620
% 110.68/15.83
% 110.68/15.83 Those formulas are unsatisfiable:
% 110.68/15.83 ---------------------------------
% 110.68/15.83
% 110.68/15.83 Begin of proof
% 110.68/15.84 |
% 110.68/15.84 | ALPHA: (axiom6) implies:
% 110.68/15.84 | (1) ? [v0: Nat$] : ? [v1: int] : ? [v2: Nat$] : ? [v3: int] : ? [v4:
% 110.68/15.84 | int] : (to_nat$(x$) = v2 & to_nat$(i$) = v0 & fun_app$e(of_nat$, v2)
% 110.68/15.84 | = v3 & fun_app$e(of_nat$, v0) = v1 & fun_app$e(of_nat$, ja$) = v4 &
% 110.68/15.84 | Nat$(v2) & Nat$(v0) & ($sum($difference(v4, v3), v1) = 0 | ~
% 110.68/15.84 | ($lesseq(v1, v3))) & (v4 = 0 | ~ ($lesseq(1, $difference(v1,
% 110.68/15.84 | v3)))))
% 110.68/15.84 |
% 110.68/15.84 | ALPHA: (axiom16) implies:
% 110.68/15.84 | (2) ? [v0: N_bool_fun$] : (fun_app$g(less_eq$, i$) = v0 & fun_app$d(v0,
% 110.68/15.84 | x$) = 0 & N_bool_fun$(v0))
% 110.68/15.84 |
% 110.68/15.84 | ALPHA: (axiom24) implies:
% 110.68/15.85 | (3) ! [v0: N$] : ! [v1: N$] : ! [v2: N_bool_fun$] : ( ~
% 110.68/15.85 | (fun_app$g(less_eq$, v0) = v2) | ~ (fun_app$d(v2, v1) = 0) | ~
% 110.68/15.85 | N$(v1) | ~ N$(v0) | ? [v3: Nat$] : ? [v4: int] : ? [v5: Nat$] :
% 110.68/15.85 | ? [v6: int] : ($lesseq(v6, v4) & to_nat$(v1) = v3 & to_nat$(v0) = v5
% 110.68/15.85 | & fun_app$e(of_nat$, v5) = v6 & fun_app$e(of_nat$, v3) = v4 &
% 110.68/15.85 | Nat$(v5) & Nat$(v3)))
% 110.68/15.85 |
% 110.68/15.85 | ALPHA: (axiom114) implies:
% 110.68/15.85 | (4) ? [v0: N$] : ? [v1: N_bool_fun$] : ? [v2: int] : ( ~ (v2 = 0) &
% 110.68/15.85 | fun_app$i(from_nat$, a$) = v0 & fun_app$g(less_eq$, v0) = v1 &
% 110.68/15.85 | fun_app$d(v1, x$) = v2 & N_bool_fun$(v1) & N$(v0))
% 110.68/15.85 |
% 110.68/15.85 | ALPHA: (axiom118) implies:
% 110.68/15.85 | (5) ? [v0: int] : ? [v1: Nat$] : ? [v2: int] : ($lesseq(v2, v0) &
% 110.68/15.85 | to_nat$(i$) = v1 & fun_app$e(of_nat$, v1) = v2 & fun_app$e(of_nat$,
% 110.68/15.85 | a$) = v0 & Nat$(v1))
% 110.68/15.85 |
% 110.68/15.85 | ALPHA: (axiom119) implies:
% 110.68/15.85 | (6) ? [v0: N_bool_fun$] : ? [v1: N$] : (fun_app$i(from_nat$, a$) = v1 &
% 110.68/15.85 | fun_app$g(less_eq$, i$) = v0 & fun_app$d(v0, v1) = 0 &
% 110.68/15.85 | N_bool_fun$(v0) & N$(v1))
% 110.68/15.85 |
% 110.68/15.85 | ALPHA: (axiom135) implies:
% 110.68/15.85 | (7) ! [v0: N$] : ! [v1: Nat$] : ! [v2: Nat$] : ! [v3: N$] : (v3 = v0 |
% 110.68/15.85 | ~ (fun_app$i(from_nat$, v1) = v3) | ~ (to_nat$(v0) = v2) | ~ N$(v0)
% 110.68/15.85 | | ~ Nat$(v1) | ? [v4: int] : ? [v5: int] : ( ~ (v5 = v4) &
% 110.68/15.85 | fun_app$e(of_nat$, v2) = v4 & fun_app$e(of_nat$, v1) = v5))
% 110.68/15.85 |
% 110.68/15.85 | ALPHA: (axiom212) implies:
% 111.15/15.86 | (8) ! [v0: N_n_bool_fun_fun$] : ! [v1: N$] : ! [v2: N$] : ! [v3:
% 111.15/15.86 | N_bool_fun$] : ! [v4: int] : (v4 = 0 | ~ (fun_app$g(v0, v1) = v3) |
% 111.15/15.86 | ~ (fun_app$d(v3, v2) = v4) | ~ N_n_bool_fun_fun$(v0) | ~ N$(v2) |
% 111.15/15.86 | ~ N$(v1) | ? [v5: N$] : ? [v6: N$] : ? [v7: N_bool_fun$] : ? [v8:
% 111.15/15.86 | N_bool_fun$] : ? [v9: int] : ( ~ (v9 = 0) & fun_app$g(v0, v6) = v7
% 111.15/15.86 | & fun_app$g(v0, v5) = v8 & fun_app$d(v8, v6) = v9 & fun_app$d(v7,
% 111.15/15.86 | v5) = 0 & N_bool_fun$(v8) & N_bool_fun$(v7) & N$(v6) & N$(v5)) |
% 111.15/15.86 | ? [v5: N$] : ? [v6: N$] : ? [v7: N_bool_fun$] : ? [v8:
% 111.15/15.86 | N_bool_fun$] : ? [v9: int] : ( ~ (v9 = 0) & fun_app$g(v0, v5) = v8
% 111.15/15.86 | & fun_app$g(less_eq$, v6) = v7 & fun_app$d(v8, v6) = v9 &
% 111.15/15.86 | fun_app$d(v7, v5) = 0 & N_bool_fun$(v8) & N_bool_fun$(v7) & N$(v6)
% 111.15/15.86 | & N$(v5)))
% 111.15/15.86 |
% 111.15/15.86 | ALPHA: (axiom242) implies:
% 111.15/15.86 | (9) Nat$(a$)
% 111.15/15.86 | (10) ? [v0: N_bool_fun$] : ? [v1: N$] : (fun_app$i(from_nat$, a$) = v1 &
% 111.15/15.86 | fun_app$g(less$, x$) = v0 & fun_app$d(v0, v1) = 0 & N_bool_fun$(v0)
% 111.15/15.86 | & N$(v1))
% 111.15/15.86 |
% 111.15/15.86 | ALPHA: (axiom374) implies:
% 111.15/15.86 | (11) ! [v0: N$] : ! [v1: N$] : ! [v2: N_bool_fun$] : ( ~
% 111.15/15.86 | (fun_app$g(less$, v0) = v2) | ~ (fun_app$d(v2, v1) = 0) | ~ N$(v1)
% 111.15/15.86 | | ~ N$(v0) | ? [v3: Nat$] : ? [v4: int] : ? [v5: Nat$] : ? [v6:
% 111.15/15.86 | int] : ($lesseq(1, $difference(v4, v6)) & to_nat$(v1) = v3 &
% 111.15/15.86 | to_nat$(v0) = v5 & fun_app$e(of_nat$, v5) = v6 &
% 111.15/15.86 | fun_app$e(of_nat$, v3) = v4 & Nat$(v5) & Nat$(v3)))
% 111.15/15.86 |
% 111.15/15.86 | ALPHA: (axiom406) implies:
% 111.15/15.87 | (12) ! [v0: N$] : ! [v1: Nat$] : ! [v2: N_bool_fun$] : ! [v3: N$] : ( ~
% 111.15/15.87 | (fun_app$i(from_nat$, v1) = v3) | ~ (fun_app$g(less$, v0) = v2) |
% 111.15/15.87 | ~ (fun_app$d(v2, v3) = 0) | ~ N$(v0) | ~ Nat$(v1) | ? [v4: int] :
% 111.15/15.87 | ? [v5: Nat$] : ? [v6: int] : ($lesseq(1, $difference(v4, v6)) &
% 111.15/15.87 | to_nat$(v0) = v5 & fun_app$e(of_nat$, v5) = v6 &
% 111.15/15.87 | fun_app$e(of_nat$, v1) = v4 & Nat$(v5)))
% 111.15/15.87 |
% 111.15/15.87 | ALPHA: (axiom465) implies:
% 111.15/15.87 | (13) ! [v0: N$] : ! [v1: N$] : ! [v2: N_bool_fun$] : ( ~
% 111.15/15.87 | (fun_app$g(less$, v0) = v2) | ~ (fun_app$d(v2, v1) = 0) | ~ N$(v1)
% 111.15/15.87 | | ~ N$(v0) | ( ~ (v1 = v0) & ? [v3: N_bool_fun$] :
% 111.15/15.87 | (fun_app$g(less_eq$, v0) = v3 & fun_app$d(v3, v1) = 0 &
% 111.15/15.87 | N_bool_fun$(v3))))
% 111.15/15.87 |
% 111.15/15.87 | ALPHA: (axiom574) implies:
% 111.15/15.87 | (14) N_n_bool_fun_fun$(less_eq$)
% 111.15/15.87 |
% 111.15/15.87 | ALPHA: (conjecture5) implies:
% 111.15/15.87 | (15) N$(i$)
% 111.15/15.87 | (16) N$(x$)
% 111.15/15.87 | (17) ? [v0: Nat$] : ? [v1: int] : ? [v2: Nat$] : ? [v3: int] :
% 111.15/15.87 | ($lesseq(1, $difference(v1, v3)) & to_nat$(x$) = v2 & to_nat$(i$) = v0
% 111.15/15.87 | & fun_app$e(of_nat$, v2) = v3 & fun_app$e(of_nat$, v0) = v1 &
% 111.15/15.87 | Nat$(v2) & Nat$(v0))
% 111.15/15.87 |
% 111.15/15.87 | ALPHA: (function-axioms) implies:
% 111.15/15.87 | (18) ! [v0: Nat$] : ! [v1: Nat$] : ! [v2: N$] : (v1 = v0 | ~
% 111.15/15.87 | (to_nat$(v2) = v1) | ~ (to_nat$(v2) = v0))
% 111.15/15.87 | (19) ! [v0: int] : ! [v1: int] : ! [v2: Nat$] : ! [v3: Nat_int_fun$] :
% 111.15/15.87 | (v1 = v0 | ~ (fun_app$e(v3, v2) = v1) | ~ (fun_app$e(v3, v2) = v0))
% 111.15/15.87 | (20) ! [v0: N_bool_fun$] : ! [v1: N_bool_fun$] : ! [v2: N$] : ! [v3:
% 111.15/15.87 | N_n_bool_fun_fun$] : (v1 = v0 | ~ (fun_app$g(v3, v2) = v1) | ~
% 111.15/15.87 | (fun_app$g(v3, v2) = v0))
% 111.15/15.88 | (21) ! [v0: N$] : ! [v1: N$] : ! [v2: Nat$] : ! [v3: Nat_n_fun$] : (v1
% 111.15/15.88 | = v0 | ~ (fun_app$i(v3, v2) = v1) | ~ (fun_app$i(v3, v2) = v0))
% 111.15/15.88 |
% 111.15/15.88 | DELTA: instantiating (2) with fresh symbol all_387_0 gives:
% 111.15/15.88 | (22) fun_app$g(less_eq$, i$) = all_387_0 & fun_app$d(all_387_0, x$) = 0 &
% 111.15/15.88 | N_bool_fun$(all_387_0)
% 111.15/15.88 |
% 111.15/15.88 | ALPHA: (22) implies:
% 111.15/15.88 | (23) fun_app$d(all_387_0, x$) = 0
% 111.15/15.88 | (24) fun_app$g(less_eq$, i$) = all_387_0
% 111.15/15.88 |
% 111.15/15.88 | DELTA: instantiating (10) with fresh symbols all_390_0, all_390_1 gives:
% 111.15/15.88 | (25) fun_app$i(from_nat$, a$) = all_390_0 & fun_app$g(less$, x$) =
% 111.15/15.88 | all_390_1 & fun_app$d(all_390_1, all_390_0) = 0 &
% 111.15/15.88 | N_bool_fun$(all_390_1) & N$(all_390_0)
% 111.15/15.88 |
% 111.15/15.88 | ALPHA: (25) implies:
% 111.15/15.88 | (26) fun_app$d(all_390_1, all_390_0) = 0
% 111.15/15.88 | (27) fun_app$g(less$, x$) = all_390_1
% 111.15/15.88 | (28) fun_app$i(from_nat$, a$) = all_390_0
% 111.15/15.88 |
% 111.15/15.88 | DELTA: instantiating (5) with fresh symbols all_392_0, all_392_1, all_392_2
% 111.15/15.88 | gives:
% 111.15/15.88 | (29) $lesseq(all_392_0, all_392_2) & to_nat$(i$) = all_392_1 &
% 111.15/15.88 | fun_app$e(of_nat$, all_392_1) = all_392_0 & fun_app$e(of_nat$, a$) =
% 111.15/15.88 | all_392_2 & Nat$(all_392_1)
% 111.15/15.88 |
% 111.15/15.88 | ALPHA: (29) implies:
% 111.15/15.88 | (30) fun_app$e(of_nat$, all_392_1) = all_392_0
% 111.15/15.88 | (31) to_nat$(i$) = all_392_1
% 111.15/15.88 |
% 111.15/15.88 | DELTA: instantiating (6) with fresh symbols all_401_0, all_401_1 gives:
% 111.15/15.88 | (32) fun_app$i(from_nat$, a$) = all_401_0 & fun_app$g(less_eq$, i$) =
% 111.15/15.88 | all_401_1 & fun_app$d(all_401_1, all_401_0) = 0 &
% 111.15/15.88 | N_bool_fun$(all_401_1) & N$(all_401_0)
% 111.15/15.89 |
% 111.15/15.89 | ALPHA: (32) implies:
% 111.15/15.89 | (33) N$(all_401_0)
% 111.15/15.89 | (34) fun_app$d(all_401_1, all_401_0) = 0
% 111.15/15.89 | (35) fun_app$g(less_eq$, i$) = all_401_1
% 111.15/15.89 | (36) fun_app$i(from_nat$, a$) = all_401_0
% 111.15/15.89 |
% 111.15/15.89 | DELTA: instantiating (4) with fresh symbols all_403_0, all_403_1, all_403_2
% 111.15/15.89 | gives:
% 111.15/15.89 | (37) ~ (all_403_0 = 0) & fun_app$i(from_nat$, a$) = all_403_2 &
% 111.15/15.89 | fun_app$g(less_eq$, all_403_2) = all_403_1 & fun_app$d(all_403_1, x$)
% 111.15/15.89 | = all_403_0 & N_bool_fun$(all_403_1) & N$(all_403_2)
% 111.15/15.89 |
% 111.15/15.89 | ALPHA: (37) implies:
% 111.15/15.89 | (38) ~ (all_403_0 = 0)
% 111.15/15.89 | (39) fun_app$d(all_403_1, x$) = all_403_0
% 111.15/15.89 | (40) fun_app$g(less_eq$, all_403_2) = all_403_1
% 111.15/15.89 | (41) fun_app$i(from_nat$, a$) = all_403_2
% 111.15/15.89 |
% 111.15/15.89 | DELTA: instantiating (17) with fresh symbols all_405_0, all_405_1, all_405_2,
% 111.15/15.89 | all_405_3 gives:
% 111.32/15.89 | (42) $lesseq(1, $difference(all_405_2, all_405_0)) & to_nat$(x$) =
% 111.32/15.89 | all_405_1 & to_nat$(i$) = all_405_3 & fun_app$e(of_nat$, all_405_1) =
% 111.32/15.89 | all_405_0 & fun_app$e(of_nat$, all_405_3) = all_405_2 &
% 111.32/15.89 | Nat$(all_405_1) & Nat$(all_405_3)
% 111.32/15.89 |
% 111.32/15.89 | ALPHA: (42) implies:
% 111.32/15.89 | (43) $lesseq(1, $difference(all_405_2, all_405_0))
% 111.32/15.89 | (44) fun_app$e(of_nat$, all_405_3) = all_405_2
% 111.32/15.89 | (45) fun_app$e(of_nat$, all_405_1) = all_405_0
% 111.32/15.89 | (46) to_nat$(i$) = all_405_3
% 111.32/15.89 | (47) to_nat$(x$) = all_405_1
% 111.32/15.89 |
% 111.32/15.89 | DELTA: instantiating (1) with fresh symbols all_416_0, all_416_1, all_416_2,
% 111.32/15.89 | all_416_3, all_416_4 gives:
% 111.32/15.90 | (48) to_nat$(x$) = all_416_2 & to_nat$(i$) = all_416_4 & fun_app$e(of_nat$,
% 111.32/15.90 | all_416_2) = all_416_1 & fun_app$e(of_nat$, all_416_4) = all_416_3 &
% 111.32/15.90 | fun_app$e(of_nat$, ja$) = all_416_0 & Nat$(all_416_2) &
% 111.32/15.90 | Nat$(all_416_4) & ($sum($difference(all_416_0, all_416_1), all_416_3)
% 111.32/15.90 | = 0 | ~ ($lesseq(all_416_3, all_416_1))) & (all_416_0 = 0 | ~
% 111.32/15.90 | ($lesseq(1, $difference(all_416_3, all_416_1))))
% 111.32/15.90 |
% 111.32/15.90 | ALPHA: (48) implies:
% 111.32/15.90 | (49) fun_app$e(of_nat$, all_416_4) = all_416_3
% 111.32/15.90 | (50) fun_app$e(of_nat$, all_416_2) = all_416_1
% 111.32/15.90 | (51) to_nat$(i$) = all_416_4
% 111.32/15.90 | (52) to_nat$(x$) = all_416_2
% 111.32/15.90 | (53) all_416_0 = 0 | ~ ($lesseq(1, $difference(all_416_3, all_416_1)))
% 111.32/15.90 |
% 111.32/15.90 | GROUND_INST: instantiating (18) with all_405_3, all_416_4, i$, simplifying
% 111.32/15.90 | with (46), (51) gives:
% 111.32/15.90 | (54) all_416_4 = all_405_3
% 111.32/15.90 |
% 111.32/15.90 | GROUND_INST: instantiating (18) with all_392_1, all_416_4, i$, simplifying
% 111.32/15.90 | with (31), (51) gives:
% 111.32/15.90 | (55) all_416_4 = all_392_1
% 111.32/15.90 |
% 111.32/15.90 | GROUND_INST: instantiating (18) with all_405_1, all_416_2, x$, simplifying
% 111.32/15.90 | with (47), (52) gives:
% 111.32/15.90 | (56) all_416_2 = all_405_1
% 111.32/15.90 |
% 111.32/15.90 | GROUND_INST: instantiating (20) with all_387_0, all_401_1, i$, less_eq$,
% 111.32/15.90 | simplifying with (24), (35) gives:
% 111.32/15.90 | (57) all_401_1 = all_387_0
% 111.32/15.90 |
% 111.32/15.90 | GROUND_INST: instantiating (21) with all_401_0, all_403_2, a$, from_nat$,
% 111.32/15.90 | simplifying with (36), (41) gives:
% 111.32/15.90 | (58) all_403_2 = all_401_0
% 111.32/15.90 |
% 111.32/15.90 | GROUND_INST: instantiating (21) with all_390_0, all_403_2, a$, from_nat$,
% 111.32/15.90 | simplifying with (28), (41) gives:
% 111.32/15.90 | (59) all_403_2 = all_390_0
% 111.32/15.90 |
% 111.32/15.90 | COMBINE_EQS: (54), (55) imply:
% 111.32/15.91 | (60) all_405_3 = all_392_1
% 111.32/15.91 |
% 111.32/15.91 | SIMP: (60) implies:
% 111.32/15.91 | (61) all_405_3 = all_392_1
% 111.32/15.91 |
% 111.32/15.91 | COMBINE_EQS: (58), (59) imply:
% 111.32/15.91 | (62) all_401_0 = all_390_0
% 111.32/15.91 |
% 111.32/15.91 | SIMP: (62) implies:
% 111.32/15.91 | (63) all_401_0 = all_390_0
% 111.32/15.91 |
% 111.32/15.91 | REDUCE: (40), (59) imply:
% 111.32/15.91 | (64) fun_app$g(less_eq$, all_390_0) = all_403_1
% 111.32/15.91 |
% 111.32/15.91 | REDUCE: (50), (56) imply:
% 111.32/15.91 | (65) fun_app$e(of_nat$, all_405_1) = all_416_1
% 111.32/15.91 |
% 111.32/15.91 | REDUCE: (49), (55) imply:
% 111.32/15.91 | (66) fun_app$e(of_nat$, all_392_1) = all_416_3
% 111.32/15.91 |
% 111.32/15.91 | REDUCE: (44), (61) imply:
% 111.32/15.91 | (67) fun_app$e(of_nat$, all_392_1) = all_405_2
% 111.32/15.91 |
% 111.32/15.91 | REDUCE: (34), (57), (63) imply:
% 111.32/15.91 | (68) fun_app$d(all_387_0, all_390_0) = 0
% 111.32/15.91 |
% 111.32/15.91 | REDUCE: (33), (63) imply:
% 111.32/15.91 | (69) N$(all_390_0)
% 111.32/15.91 |
% 111.32/15.91 | GROUND_INST: instantiating (19) with all_392_0, all_416_3, all_392_1, of_nat$,
% 111.32/15.91 | simplifying with (30), (66) gives:
% 111.32/15.91 | (70) all_416_3 = all_392_0
% 111.32/15.91 |
% 111.32/15.91 | GROUND_INST: instantiating (19) with all_405_2, all_416_3, all_392_1, of_nat$,
% 111.32/15.91 | simplifying with (66), (67) gives:
% 111.32/15.91 | (71) all_416_3 = all_405_2
% 111.32/15.91 |
% 111.32/15.91 | GROUND_INST: instantiating (19) with all_405_0, all_416_1, all_405_1, of_nat$,
% 111.32/15.91 | simplifying with (45), (65) gives:
% 111.32/15.91 | (72) all_416_1 = all_405_0
% 111.32/15.91 |
% 111.32/15.91 | COMBINE_EQS: (70), (71) imply:
% 111.32/15.91 | (73) all_405_2 = all_392_0
% 111.32/15.91 |
% 111.32/15.91 | SIMP: (73) implies:
% 111.32/15.91 | (74) all_405_2 = all_392_0
% 111.32/15.91 |
% 111.32/15.91 | REDUCE: (43), (74) imply:
% 111.32/15.91 | (75) $lesseq(1, $difference(all_392_0, all_405_0))
% 111.32/15.91 |
% 111.32/15.91 | BETA: splitting (53) gives:
% 111.32/15.91 |
% 111.32/15.91 | Case 1:
% 111.32/15.91 | |
% 111.32/15.91 | | (76) $lesseq(all_416_3, all_416_1)
% 111.32/15.91 | |
% 111.32/15.91 | | REDUCE: (70), (72), (76) imply:
% 111.32/15.91 | | (77) $lesseq(all_392_0, all_405_0)
% 111.32/15.91 | |
% 111.32/15.91 | | COMBINE_INEQS: (75), (77) imply:
% 111.32/15.91 | | (78) $false
% 111.32/15.91 | |
% 111.32/15.91 | | CLOSE: (78) is inconsistent.
% 111.32/15.91 | |
% 111.32/15.91 | Case 2:
% 111.32/15.91 | |
% 111.32/15.91 | | (79) $lesseq(1, $difference(all_416_3, all_416_1))
% 111.32/15.91 | |
% 111.32/15.92 | | GROUND_INST: instantiating (11) with x$, all_390_0, all_390_1, simplifying
% 111.32/15.92 | | with (16), (26), (27), (69) gives:
% 111.32/15.92 | | (80) ? [v0: Nat$] : ? [v1: int] : ? [v2: Nat$] : ? [v3: int] :
% 111.32/15.92 | | ($lesseq(1, $difference(v1, v3)) & to_nat$(all_390_0) = v0 &
% 111.32/15.92 | | to_nat$(x$) = v2 & fun_app$e(of_nat$, v2) = v3 &
% 111.32/15.92 | | fun_app$e(of_nat$, v0) = v1 & Nat$(v2) & Nat$(v0))
% 111.32/15.92 | |
% 111.32/15.92 | | GROUND_INST: instantiating (13) with x$, all_390_0, all_390_1, simplifying
% 111.32/15.92 | | with (16), (26), (27), (69) gives:
% 111.32/15.92 | | (81) ~ (all_390_0 = x$) & ? [v0: N_bool_fun$] : (fun_app$g(less_eq$,
% 111.32/15.92 | | x$) = v0 & fun_app$d(v0, all_390_0) = 0 & N_bool_fun$(v0))
% 111.32/15.92 | |
% 111.32/15.92 | | ALPHA: (81) implies:
% 111.32/15.92 | | (82) ~ (all_390_0 = x$)
% 111.32/15.92 | |
% 111.32/15.92 | | GROUND_INST: instantiating (3) with i$, all_390_0, all_387_0, simplifying
% 111.32/15.92 | | with (15), (24), (68), (69) gives:
% 111.32/15.92 | | (83) ? [v0: Nat$] : ? [v1: int] : ? [v2: Nat$] : ? [v3: int] :
% 111.32/15.92 | | ($lesseq(v3, v1) & to_nat$(all_390_0) = v0 & to_nat$(i$) = v2 &
% 111.32/15.92 | | fun_app$e(of_nat$, v2) = v3 & fun_app$e(of_nat$, v0) = v1 &
% 111.32/15.92 | | Nat$(v2) & Nat$(v0))
% 111.32/15.92 | |
% 111.32/15.92 | | GROUND_INST: instantiating (3) with i$, x$, all_387_0, simplifying with
% 111.32/15.92 | | (15), (16), (23), (24) gives:
% 111.32/15.92 | | (84) ? [v0: Nat$] : ? [v1: int] : ? [v2: Nat$] : ? [v3: int] :
% 111.32/15.92 | | ($lesseq(v3, v1) & to_nat$(x$) = v0 & to_nat$(i$) = v2 &
% 111.32/15.92 | | fun_app$e(of_nat$, v2) = v3 & fun_app$e(of_nat$, v0) = v1 &
% 111.32/15.92 | | Nat$(v2) & Nat$(v0))
% 111.32/15.92 | |
% 111.32/15.92 | | GROUND_INST: instantiating (8) with less_eq$, all_390_0, x$, all_403_1,
% 111.32/15.92 | | all_403_0, simplifying with (14), (16), (39), (64), (69) gives:
% 111.32/15.92 | | (85) all_403_0 = 0 | ? [v0: N$] : ? [v1: N$] : ? [v2: N_bool_fun$] :
% 111.32/15.92 | | ? [v3: N_bool_fun$] : ? [v4: int] : ( ~ (v4 = 0) &
% 111.32/15.92 | | fun_app$g(less_eq$, v1) = v2 & fun_app$g(less_eq$, v0) = v3 &
% 111.32/15.92 | | fun_app$d(v3, v1) = v4 & fun_app$d(v2, v0) = 0 & N_bool_fun$(v3) &
% 111.32/15.92 | | N_bool_fun$(v2) & N$(v1) & N$(v0))
% 111.32/15.92 | |
% 111.32/15.92 | | GROUND_INST: instantiating (7) with x$, a$, all_405_1, all_390_0,
% 111.32/15.92 | | simplifying with (9), (16), (28), (47) gives:
% 111.32/15.93 | | (86) all_390_0 = x$ | ? [v0: int] : ? [v1: int] : ( ~ (v1 = v0) &
% 111.32/15.93 | | fun_app$e(of_nat$, all_405_1) = v0 & fun_app$e(of_nat$, a$) = v1)
% 111.32/15.93 | |
% 111.32/15.93 | | GROUND_INST: instantiating (12) with x$, a$, all_390_1, all_390_0,
% 111.32/15.93 | | simplifying with (9), (16), (26), (27), (28) gives:
% 111.32/15.93 | | (87) ? [v0: int] : ? [v1: Nat$] : ? [v2: int] : ($lesseq(1,
% 111.32/15.93 | | $difference(v0, v2)) & to_nat$(x$) = v1 & fun_app$e(of_nat$, v1)
% 111.32/15.93 | | = v2 & fun_app$e(of_nat$, a$) = v0 & Nat$(v1))
% 111.32/15.93 | |
% 111.32/15.93 | | DELTA: instantiating (87) with fresh symbols all_506_0, all_506_1, all_506_2
% 111.32/15.93 | | gives:
% 111.32/15.93 | | (88) $lesseq(1, $difference(all_506_2, all_506_0)) & to_nat$(x$) =
% 111.32/15.93 | | all_506_1 & fun_app$e(of_nat$, all_506_1) = all_506_0 &
% 111.32/15.93 | | fun_app$e(of_nat$, a$) = all_506_2 & Nat$(all_506_1)
% 111.32/15.93 | |
% 111.32/15.93 | | ALPHA: (88) implies:
% 111.32/15.93 | | (89) fun_app$e(of_nat$, all_506_1) = all_506_0
% 111.32/15.93 | | (90) to_nat$(x$) = all_506_1
% 111.32/15.93 | |
% 111.32/15.93 | | DELTA: instantiating (80) with fresh symbols all_513_0, all_513_1,
% 111.32/15.93 | | all_513_2, all_513_3 gives:
% 111.32/15.93 | | (91) $lesseq(1, $difference(all_513_2, all_513_0)) & to_nat$(all_390_0) =
% 111.32/15.93 | | all_513_3 & to_nat$(x$) = all_513_1 & fun_app$e(of_nat$, all_513_1)
% 111.32/15.93 | | = all_513_0 & fun_app$e(of_nat$, all_513_3) = all_513_2 &
% 111.32/15.93 | | Nat$(all_513_1) & Nat$(all_513_3)
% 111.32/15.93 | |
% 111.32/15.93 | | ALPHA: (91) implies:
% 111.32/15.93 | | (92) fun_app$e(of_nat$, all_513_1) = all_513_0
% 111.32/15.93 | | (93) to_nat$(x$) = all_513_1
% 111.32/15.93 | |
% 111.32/15.93 | | DELTA: instantiating (84) with fresh symbols all_516_0, all_516_1,
% 111.32/15.93 | | all_516_2, all_516_3 gives:
% 111.32/15.93 | | (94) $lesseq(all_516_0, all_516_2) & to_nat$(x$) = all_516_3 &
% 111.32/15.93 | | to_nat$(i$) = all_516_1 & fun_app$e(of_nat$, all_516_1) = all_516_0
% 111.32/15.93 | | & fun_app$e(of_nat$, all_516_3) = all_516_2 & Nat$(all_516_1) &
% 111.32/15.93 | | Nat$(all_516_3)
% 111.32/15.93 | |
% 111.32/15.93 | | ALPHA: (94) implies:
% 111.32/15.93 | | (95) $lesseq(all_516_0, all_516_2)
% 111.52/15.93 | | (96) fun_app$e(of_nat$, all_516_3) = all_516_2
% 111.52/15.93 | | (97) fun_app$e(of_nat$, all_516_1) = all_516_0
% 111.52/15.93 | | (98) to_nat$(i$) = all_516_1
% 111.52/15.93 | | (99) to_nat$(x$) = all_516_3
% 111.52/15.93 | |
% 111.52/15.93 | | DELTA: instantiating (83) with fresh symbols all_521_0, all_521_1,
% 111.52/15.93 | | all_521_2, all_521_3 gives:
% 111.52/15.93 | | (100) $lesseq(all_521_0, all_521_2) & to_nat$(all_390_0) = all_521_3 &
% 111.52/15.93 | | to_nat$(i$) = all_521_1 & fun_app$e(of_nat$, all_521_1) = all_521_0
% 111.52/15.93 | | & fun_app$e(of_nat$, all_521_3) = all_521_2 & Nat$(all_521_1) &
% 111.52/15.93 | | Nat$(all_521_3)
% 111.52/15.93 | |
% 111.52/15.94 | | ALPHA: (100) implies:
% 111.52/15.94 | | (101) fun_app$e(of_nat$, all_521_1) = all_521_0
% 111.52/15.94 | | (102) to_nat$(i$) = all_521_1
% 111.52/15.94 | |
% 111.52/15.94 | | BETA: splitting (86) gives:
% 111.52/15.94 | |
% 111.52/15.94 | | Case 1:
% 111.52/15.94 | | |
% 111.52/15.94 | | | (103) all_390_0 = x$
% 111.52/15.94 | | |
% 111.52/15.94 | | | REDUCE: (82), (103) imply:
% 111.52/15.94 | | | (104) $false
% 111.52/15.94 | | |
% 111.52/15.94 | | | CLOSE: (104) is inconsistent.
% 111.52/15.94 | | |
% 111.52/15.94 | | Case 2:
% 111.52/15.94 | | |
% 111.52/15.94 | | | (105) ? [v0: int] : ? [v1: int] : ( ~ (v1 = v0) & fun_app$e(of_nat$,
% 111.52/15.94 | | | all_405_1) = v0 & fun_app$e(of_nat$, a$) = v1)
% 111.52/15.94 | | |
% 111.52/15.94 | | | DELTA: instantiating (105) with fresh symbols all_669_0, all_669_1 gives:
% 111.54/15.94 | | | (106) ~ (all_669_0 = all_669_1) & fun_app$e(of_nat$, all_405_1) =
% 111.54/15.94 | | | all_669_1 & fun_app$e(of_nat$, a$) = all_669_0
% 111.54/15.94 | | |
% 111.54/15.94 | | | ALPHA: (106) implies:
% 111.54/15.94 | | | (107) fun_app$e(of_nat$, all_405_1) = all_669_1
% 111.54/15.94 | | |
% 111.54/15.94 | | | BETA: splitting (85) gives:
% 111.54/15.94 | | |
% 111.54/15.94 | | | Case 1:
% 111.54/15.94 | | | |
% 111.54/15.94 | | | | (108) all_403_0 = 0
% 111.54/15.94 | | | |
% 111.54/15.94 | | | | REDUCE: (38), (108) imply:
% 111.54/15.94 | | | | (109) $false
% 111.54/15.94 | | | |
% 111.54/15.94 | | | | CLOSE: (109) is inconsistent.
% 111.54/15.94 | | | |
% 111.54/15.94 | | | Case 2:
% 111.54/15.94 | | | |
% 111.54/15.94 | | | |
% 111.54/15.94 | | | | GROUND_INST: instantiating (19) with all_405_0, all_669_1, all_405_1,
% 111.54/15.94 | | | | of_nat$, simplifying with (45), (107) gives:
% 111.54/15.94 | | | | (110) all_669_1 = all_405_0
% 111.54/15.94 | | | |
% 111.54/15.94 | | | | GROUND_INST: instantiating (18) with all_392_1, all_521_1, i$,
% 111.54/15.94 | | | | simplifying with (31), (102) gives:
% 111.54/15.94 | | | | (111) all_521_1 = all_392_1
% 111.54/15.94 | | | |
% 111.54/15.94 | | | | GROUND_INST: instantiating (18) with all_516_1, all_521_1, i$,
% 111.54/15.94 | | | | simplifying with (98), (102) gives:
% 111.54/15.94 | | | | (112) all_521_1 = all_516_1
% 111.54/15.94 | | | |
% 111.54/15.94 | | | | GROUND_INST: instantiating (18) with all_405_1, all_513_1, x$,
% 111.54/15.94 | | | | simplifying with (47), (93) gives:
% 111.54/15.94 | | | | (113) all_513_1 = all_405_1
% 111.54/15.94 | | | |
% 111.54/15.94 | | | | GROUND_INST: instantiating (18) with all_513_1, all_516_3, x$,
% 111.54/15.94 | | | | simplifying with (93), (99) gives:
% 111.54/15.94 | | | | (114) all_516_3 = all_513_1
% 111.54/15.94 | | | |
% 111.54/15.94 | | | | GROUND_INST: instantiating (18) with all_506_1, all_516_3, x$,
% 111.54/15.94 | | | | simplifying with (90), (99) gives:
% 111.54/15.94 | | | | (115) all_516_3 = all_506_1
% 111.54/15.94 | | | |
% 111.54/15.94 | | | | COMBINE_EQS: (111), (112) imply:
% 111.54/15.94 | | | | (116) all_516_1 = all_392_1
% 111.54/15.94 | | | |
% 111.54/15.94 | | | | COMBINE_EQS: (114), (115) imply:
% 111.54/15.94 | | | | (117) all_513_1 = all_506_1
% 111.54/15.94 | | | |
% 111.54/15.95 | | | | SIMP: (117) implies:
% 111.54/15.95 | | | | (118) all_513_1 = all_506_1
% 111.54/15.95 | | | |
% 111.54/15.95 | | | | COMBINE_EQS: (113), (118) imply:
% 111.54/15.95 | | | | (119) all_506_1 = all_405_1
% 111.54/15.95 | | | |
% 111.54/15.95 | | | | COMBINE_EQS: (115), (119) imply:
% 111.54/15.95 | | | | (120) all_516_3 = all_405_1
% 111.54/15.95 | | | |
% 111.54/15.95 | | | | REDUCE: (101), (111) imply:
% 111.54/15.95 | | | | (121) fun_app$e(of_nat$, all_392_1) = all_521_0
% 111.54/15.95 | | | |
% 111.54/15.95 | | | | REDUCE: (97), (116) imply:
% 111.54/15.95 | | | | (122) fun_app$e(of_nat$, all_392_1) = all_516_0
% 111.54/15.95 | | | |
% 111.54/15.95 | | | | REDUCE: (96), (120) imply:
% 111.54/15.95 | | | | (123) fun_app$e(of_nat$, all_405_1) = all_516_2
% 111.54/15.95 | | | |
% 111.54/15.95 | | | | REDUCE: (92), (113) imply:
% 111.54/15.95 | | | | (124) fun_app$e(of_nat$, all_405_1) = all_513_0
% 111.54/15.95 | | | |
% 111.54/15.95 | | | | REDUCE: (89), (119) imply:
% 111.54/15.95 | | | | (125) fun_app$e(of_nat$, all_405_1) = all_506_0
% 111.54/15.95 | | | |
% 111.54/15.95 | | | | GROUND_INST: instantiating (19) with all_392_0, all_521_0, all_392_1,
% 111.54/15.95 | | | | of_nat$, simplifying with (30), (121) gives:
% 111.54/15.95 | | | | (126) all_521_0 = all_392_0
% 111.54/15.95 | | | |
% 111.54/15.95 | | | | GROUND_INST: instantiating (19) with all_516_0, all_521_0, all_392_1,
% 111.54/15.95 | | | | of_nat$, simplifying with (121), (122) gives:
% 111.54/15.95 | | | | (127) all_521_0 = all_516_0
% 111.54/15.95 | | | |
% 111.54/15.95 | | | | GROUND_INST: instantiating (19) with all_405_0, all_513_0, all_405_1,
% 111.54/15.95 | | | | of_nat$, simplifying with (45), (124) gives:
% 111.54/15.95 | | | | (128) all_513_0 = all_405_0
% 111.54/15.95 | | | |
% 111.54/15.95 | | | | GROUND_INST: instantiating (19) with all_513_0, all_516_2, all_405_1,
% 111.54/15.95 | | | | of_nat$, simplifying with (123), (124) gives:
% 111.54/15.95 | | | | (129) all_516_2 = all_513_0
% 111.54/15.95 | | | |
% 111.54/15.95 | | | | GROUND_INST: instantiating (19) with all_506_0, all_516_2, all_405_1,
% 111.54/15.95 | | | | of_nat$, simplifying with (123), (125) gives:
% 111.54/15.95 | | | | (130) all_516_2 = all_506_0
% 111.54/15.95 | | | |
% 111.54/15.95 | | | | COMBINE_EQS: (126), (127) imply:
% 111.54/15.95 | | | | (131) all_516_0 = all_392_0
% 111.54/15.95 | | | |
% 111.54/15.95 | | | | COMBINE_EQS: (129), (130) imply:
% 111.54/15.95 | | | | (132) all_513_0 = all_506_0
% 111.54/15.95 | | | |
% 111.54/15.95 | | | | SIMP: (132) implies:
% 111.54/15.95 | | | | (133) all_513_0 = all_506_0
% 111.54/15.95 | | | |
% 111.54/15.95 | | | | COMBINE_EQS: (128), (133) imply:
% 111.54/15.95 | | | | (134) all_506_0 = all_405_0
% 111.54/15.95 | | | |
% 111.54/15.95 | | | | SIMP: (134) implies:
% 111.54/15.95 | | | | (135) all_506_0 = all_405_0
% 111.54/15.95 | | | |
% 111.54/15.95 | | | | COMBINE_EQS: (130), (135) imply:
% 111.54/15.95 | | | | (136) all_516_2 = all_405_0
% 111.54/15.95 | | | |
% 111.54/15.95 | | | | REDUCE: (95), (131), (136) imply:
% 111.54/15.95 | | | | (137) $lesseq(all_392_0, all_405_0)
% 111.54/15.95 | | | |
% 111.54/15.95 | | | | COMBINE_INEQS: (75), (137) imply:
% 111.54/15.95 | | | | (138) $false
% 111.54/15.95 | | | |
% 111.54/15.95 | | | | CLOSE: (138) is inconsistent.
% 111.54/15.95 | | | |
% 111.54/15.95 | | | End of split
% 111.54/15.95 | | |
% 111.54/15.95 | | End of split
% 111.54/15.95 | |
% 111.54/15.95 | End of split
% 111.54/15.95 |
% 111.54/15.95 End of proof
% 111.54/15.95 % SZS output end Proof for theBenchmark
% 111.54/15.95
% 111.54/15.95 15321ms
%------------------------------------------------------------------------------