/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.pl /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- YES proof of /export/starexec/sandbox2/benchmark/theBenchmark.pl # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern totient_phi(g,a) w.r.t. the given Prolog program could successfully be proven: (0) Prolog (1) PrologToIRSwTTransformerProof [SOUND, 134 ms] (2) AND (3) IRSwT (4) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 1 ms] (5) TRUE (6) IRSwT (7) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (8) IRSwT (9) IntTRSCompressionProof [EQUIVALENT, 0 ms] (10) IRSwT (11) IRSFormatTransformerProof [EQUIVALENT, 1 ms] (12) IRSwT (13) IRSwTTerminationDigraphProof [EQUIVALENT, 31 ms] (14) IRSwT (15) IntTRSCompressionProof [EQUIVALENT, 0 ms] (16) IRSwT (17) TempFilterProof [SOUND, 40 ms] (18) IntTRS (19) RankingReductionPairProof [EQUIVALENT, 0 ms] (20) YES ---------------------------------------- (0) Obligation: Clauses: totient_phi(1, 1) :- !. totient_phi(M, Phi) :- t_phi(M, Phi, 1, 0). t_phi(M, Phi, M, Phi) :- !. t_phi(M, Phi, K, C) :- ','(<(K, M), ','(coprime(K, M), ','(!, ','(is(C1, +(C, 1)), ','(is(K1, +(K, 1)), t_phi(M, Phi, K1, C1)))))). t_phi(M, Phi, K, C) :- ','(<(K, M), ','(is(K1, +(K, 1)), t_phi(M, Phi, K1, C))). coprime(X, Y) :- gcd(X, Y, 1). gcd(X, 0, X) :- >(X, 0). gcd(X, Y, G) :- ','(>(Y, 0), ','(is(Z, mod(X, Y)), gcd(Y, Z, G))). Query: totient_phi(g,a) ---------------------------------------- (1) PrologToIRSwTTransformerProof (SOUND) Transformed Prolog program to IRSwT according to method in Master Thesis of A. Weinert { "root": 1, "program": { "directives": [], "clauses": [ [ "(totient_phi (1) (1))", "(!)" ], [ "(totient_phi M Phi)", "(t_phi M Phi (1) (0))" ], [ "(t_phi M Phi M Phi)", "(!)" ], [ "(t_phi M Phi K C)", "(',' (< K M) (',' (coprime K M) (',' (!) 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{ "arguments": [ { "type": "PlainIntegerConstant", "value": "0" }, { "type": "PlainIntegerConstant", "value": "1" } ], "type": "PlainIntegerOperation", "operation": "+" }, "operation": "=" }, { "lhs": { "arguments": [ { "type": "PlainIntegerConstant", "value": "1" }, { "type": "PlainIntegerConstant", "value": "1" } ], "type": "PlainIntegerOperation", "operation": "+" }, "type": "PlainIntegerRelation", "rhs": { "arguments": [ { "type": "PlainIntegerConstant", "value": "1" }, { "type": "PlainIntegerConstant", "value": "1" } ], "type": "PlainIntegerOperation", "operation": "+" }, "operation": "=" } ] }, "ground": [], "free": [], "exprvars": [ "T42", "T51", "T50", "T43" ] } } }, "edges": [ { "from": 1, "to": 5, "label": "CASE" }, { "from": 5, "to": 35, "label": "EVAL with clause\ntotient_phi(1, 1) :- !_1.\nand substitutionT1 -> 1,\nT2 -> 1" }, { "from": 5, "to": 36, "label": "EVAL-BACKTRACK" }, { "from": 35, "to": 37, "label": "CUT" }, { "from": 36, "to": 39, "label": "ONLY EVAL with clause\ntotient_phi(X7, X8) :- t_phi(X7, X8, 1, 0).\nand substitutionT1 -> T10,\nX7 -> T10,\nT2 -> T12,\nX8 -> T12,\nT11 -> T12" }, { "from": 37, "to": 38, "label": "SUCCESS" }, { "from": 39, "to": 40, "label": "CASE" }, { "from": 40, "to": 41, "label": "EVAL with clause\nt_phi(X15, X16, X15, X16) :- !_2.\nand substitutionT10 -> 1,\nX15 -> 1,\nT12 -> 0,\nX16 -> 0,\nT19 -> 1,\nT20 -> 0" }, { "from": 40, "to": 42, "label": "EVAL-BACKTRACK" }, { "from": 41, "to": 43, "label": "CUT" }, { "from": 42, "to": 45, "label": "PARALLEL" }, { "from": 42, "to": 46, "label": "PARALLEL" }, { "from": 43, "to": 44, "label": "SUCCESS" }, { "from": 45, "to": 47, "label": "ONLY EVAL with clause\nt_phi(X37, X38, X39, X40) :- ','(<(X39, X37), ','(coprime(X39, X37), ','(!_2, ','(is(X41, +(X40, 1)), ','(is(X42, +(X39, 1)), t_phi(X37, X38, X42, X41)))))).\nand substitutionT10 -> T31,\nX37 -> T31,\nT12 -> T33,\nX38 -> T33,\nX39 -> 1,\nX40 -> 0,\nT32 -> T33" }, { "from": 46, "to": 4139, "label": "ONLY EVAL with clause\nt_phi(X160, X161, X162, X163) :- ','(<(X162, X160), ','(is(X164, +(X162, 1)), t_phi(X160, X161, X164, X163))).\nand substitutionT10 -> T124,\nX160 -> T124,\nT12 -> T126,\nX161 -> T126,\nX162 -> 1,\nX163 -> 0,\nT125 -> T126" }, { "from": 47, "to": 48, "label": "IS ERROR" }, { "from": 47, "to": 1016, "label": "ARITHCOMP SUCCESS" }, { "from": 47, "to": 1018, "label": "ARITHCOMP FAIL" }, { "from": 1016, "to": 1020, "label": "SPLIT 1" }, { "from": 1016, "to": 1024, "label": "SPLIT 2\nnew knowledge:\nT31 is ground" }, { "from": 1020, "to": 1026, "label": "CASE" }, { "from": 1024, "to": 3604, "label": "CUT" }, { "from": 1026, "to": 1073, "label": "ONLY EVAL with clause\ncoprime(X52, X53) :- gcd(X52, X53, 1).\nand substitutionX52 -> 1,\nT31 -> T38,\nX53 -> T38" }, { "from": 1073, "to": 2151, "label": "CASE" }, { "from": 2151, "to": 2174, "label": "PARALLEL" }, { "from": 2151, "to": 2181, "label": "PARALLEL" }, { "from": 2174, "to": 2202, "label": "EVAL with clause\ngcd(X59, 0, X59) :- >(X59, 0).\nand substitutionX59 -> 1,\nT38 -> 0" }, { "from": 2174, "to": 2208, "label": "EVAL-BACKTRACK" }, { "from": 2181, "to": 3587, "label": "ONLY EVAL with clause\ngcd(X67, X68, X69) :- ','(>(X68, 0), ','(is(X70, mod(X67, X68)), gcd(X68, X70, X69))).\nand substitutionX67 -> 1,\nT38 -> T41,\nX68 -> T41,\nX69 -> 1" }, { "from": 3604, "to": 3605, "label": "\nX41 -> T42" }, { "from": 3605, "to": 3606, "label": "\nX42 -> T43" }, { "from": 3606, "to": 3607, "label": "CASE" }, { "from": 3607, "to": 3701, "label": "EVAL with clause\nt_phi(X77, X78, X77, X78) :- !_5.\nand substitutionT31 -> T50,\nX77 -> T50,\nT33 -> T51,\nX78 -> T51,\nT43 -> T50,\nT42 -> T51" }, { "from": 3607, "to": 3702, "label": "EVAL-BACKTRACK" }, { "from": 3701, "to": 3703, "label": "CUT" }, { "from": 3702, "to": 3705, "label": "PARALLEL" }, { "from": 3702, "to": 3706, "label": "PARALLEL" }, { "from": 3703, "to": 3704, "label": "SUCCESS" }, { "from": 3705, "to": 3707, "label": "ONLY EVAL with clause\nt_phi(X99, X100, X101, X102) :- ','(<(X101, X99), ','(coprime(X101, X99), ','(!_5, ','(is(X103, +(X102, 1)), ','(is(X104, +(X101, 1)), t_phi(X99, X100, X104, X103)))))).\nand substitutionT31 -> T70,\nX99 -> T70,\nT33 -> T74,\nX100 -> T74,\nT43 -> T72,\nX101 -> T72,\nT42 -> T73,\nX102 -> T73,\nT71 -> T74" }, { "from": 3706, "to": 4110, "label": "ONLY EVAL with clause\nt_phi(X144, X145, X146, X147) :- ','(<(X146, X144), ','(is(X148, +(X146, 1)), t_phi(X144, X145, X148, X147))).\nand substitutionT31 -> T111,\nX144 -> T111,\nT33 -> T115,\nX145 -> T115,\nT43 -> T113,\nX146 -> T113,\nT42 -> T114,\nX147 -> T114,\nT112 -> T115" }, { "from": 3707, "to": 3708, "label": "IS ERROR" }, { "from": 3707, "to": 4050, "label": "ARITHCOMP SUCCESS" }, { "from": 3707, "to": 4052, "label": "ARITHCOMP FAIL" }, { "from": 4050, "to": 4054, "label": "SPLIT 1" }, { "from": 4050, "to": 4055, "label": "SPLIT 2\nnew knowledge:\nT72 is ground\nT70 is ground" }, { "from": 4054, "to": 4056, "label": "CASE" }, { "from": 4055, "to": 4092, "label": "CUT" }, { "from": 4056, "to": 4057, "label": "ONLY EVAL with clause\ncoprime(X114, X115) :- gcd(X114, X115, 1).\nand substitutionT72 -> T84,\nX114 -> T84,\nT70 -> T85,\nX115 -> T85" }, { "from": 4057, "to": 4058, "label": "CASE" }, { "from": 4058, "to": 4066, "label": "PARALLEL" }, { "from": 4058, "to": 4067, "label": "PARALLEL" }, { "from": 4066, "to": 4071, "label": "EVAL with clause\ngcd(X121, 0, X121) :- >(X121, 0).\nand substitutionT84 -> 1,\nX121 -> 1,\nT85 -> 0,\nT91 -> 1" }, { "from": 4066, "to": 4072, "label": "EVAL-BACKTRACK" }, { "from": 4067, "to": 4077, "label": "ONLY EVAL with clause\ngcd(X129, X130, X131) :- ','(>(X130, 0), ','(is(X132, mod(X129, X130)), gcd(X130, X132, X131))).\nand substitutionT84 -> T96,\nX129 -> T96,\nT85 -> T97,\nX130 -> T97,\nX131 -> 1" }, { "from": 4092, "to": 4093, "label": "IS ERROR" }, { "from": 4092, "to": 4094, "label": "\nX103 -> T98" }, { "from": 4094, "to": 4095, "label": "IS ERROR" }, { "from": 4094, "to": 4096, "label": "\nX104 -> T99" }, { "from": 4096, "to": 3606, "label": "INSTANCE with matching:\nT31 -> T70\nT33 -> T74\nT43 -> T99\nT42 -> T98\nX41 -> X77\nX42 -> X78" }, { "from": 4110, "to": 4111, "label": "IS ERROR" }, { "from": 4110, "to": 4123, "label": "ARITHCOMP SUCCESS" }, { "from": 4110, "to": 4124, "label": "ARITHCOMP FAIL" }, { "from": 4123, "to": 4125, "label": "\nX148 -> T116" }, { "from": 4125, "to": 3606, "label": "INSTANCE with matching:\nT31 -> T111\nT33 -> T115\nT43 -> T116\nT42 -> T114\nX41 -> X77\nX42 -> X78" }, { "from": 4139, "to": 4140, "label": "IS ERROR" }, { "from": 4139, "to": 4150, "label": "ARITHCOMP SUCCESS" }, { "from": 4139, "to": 4151, "label": "ARITHCOMP FAIL" }, { "from": 4150, "to": 4152, "label": "\nX164 -> T127" }, { "from": 4152, "to": 4153, "label": "CASE" }, { "from": 4153, "to": 4154, "label": "EVAL with clause\nt_phi(X171, X172, X171, X172) :- !_8.\nand substitutionT124 -> T134,\nX171 -> T134,\nT126 -> 0,\nX172 -> 0,\nT127 -> T134,\nT135 -> 0" }, { "from": 4153, "to": 4155, "label": "EVAL-BACKTRACK" }, { "from": 4154, "to": 4156, "label": "CUT" }, { "from": 4155, "to": 4158, "label": "PARALLEL" }, { "from": 4155, "to": 4159, "label": "PARALLEL" }, { "from": 4156, "to": 4157, "label": "SUCCESS" }, { "from": 4158, "to": 4160, "label": "ONLY EVAL with clause\nt_phi(X193, X194, X195, X196) :- ','(<(X195, X193), ','(coprime(X195, X193), ','(!_8, ','(is(X197, +(X196, 1)), ','(is(X198, +(X195, 1)), t_phi(X193, X194, X198, X197)))))).\nand substitutionT124 -> T150,\nX193 -> T150,\nT126 -> T153,\nX194 -> T153,\nT127 -> T152,\nX195 -> T152,\nX196 -> 0,\nT151 -> T153" }, { "from": 4159, "to": 4200, "label": "ONLY EVAL with clause\nt_phi(X238, X239, X240, X241) :- ','(<(X240, X238), ','(is(X242, +(X240, 1)), t_phi(X238, X239, X242, X241))).\nand substitutionT124 -> T188,\nX238 -> T188,\nT126 -> T191,\nX239 -> T191,\nT127 -> T190,\nX240 -> T190,\nX241 -> 0,\nT189 -> T191" }, { "from": 4160, "to": 4168, "label": "ARITHCOMP SUCCESS" }, { "from": 4160, "to": 4169, "label": "ARITHCOMP FAIL" }, { "from": 4168, "to": 4170, "label": "SPLIT 1" }, { "from": 4168, "to": 4171, "label": "SPLIT 2\nnew knowledge:\nT152 is ground\nT150 is ground" }, { "from": 4170, "to": 4172, "label": "CASE" }, { "from": 4171, "to": 4196, "label": "CUT" }, { "from": 4172, "to": 4173, "label": "ONLY EVAL with clause\ncoprime(X208, X209) :- gcd(X208, X209, 1).\nand substitutionT152 -> T163,\nX208 -> T163,\nT150 -> T164,\nX209 -> T164" }, { "from": 4173, "to": 4177, "label": "CASE" }, { "from": 4177, "to": 4178, "label": "PARALLEL" }, { "from": 4177, "to": 4180, "label": "PARALLEL" }, { "from": 4178, "to": 4181, "label": "EVAL with clause\ngcd(X215, 0, X215) :- >(X215, 0).\nand substitutionT163 -> 1,\nX215 -> 1,\nT164 -> 0,\nT170 -> 1" }, { "from": 4178, "to": 4182, "label": "EVAL-BACKTRACK" }, { "from": 4180, "to": 4192, "label": "ONLY EVAL with clause\ngcd(X223, X224, X225) :- ','(>(X224, 0), ','(is(X226, mod(X223, X224)), gcd(X224, X226, X225))).\nand substitutionT163 -> T175,\nX223 -> T175,\nT164 -> T176,\nX224 -> T176,\nX225 -> 1" }, { "from": 4196, "to": 4197, "label": "\nX197 -> T177" }, { "from": 4197, "to": 4198, "label": "IS ERROR" }, { "from": 4197, "to": 4199, "label": "\nX198 -> T178" }, { "from": 4199, "to": 3606, "label": "INSTANCE with matching:\nT31 -> T150\nT33 -> T153\nT43 -> T178\nT42 -> T177\nX41 -> X171\nX42 -> X172" }, { "from": 4200, "to": 4201, "label": "ARITHCOMP SUCCESS" }, { "from": 4200, "to": 4202, "label": "ARITHCOMP FAIL" }, { "from": 4201, "to": 4203, "label": "\nX242 -> T192" }, { "from": 4203, "to": 4152, "label": "INSTANCE with matching:\nT124 -> T188\nT126 -> T191\nT127 -> T192\nX164 -> X171" } ], "type": "Graph" } } ---------------------------------------- (2) Complex Obligation (AND) ---------------------------------------- (3) Obligation: Rules: f4095_out -> f4094_out(T72, T70, T98, T73) :|: TRUE f4096_out(x, x1, x2, x3, x4) -> f4094_out(x4, x, x2, x3) :|: TRUE f4094_in(x5, x6, x7, x8) -> f4096_in(x6, x9, x7, x8, x5) :|: x9 = x5 + 1 f4094_in(x10, x11, x12, x13) -> f4095_in :|: TRUE f4066_out(x14, x15, x16) -> f4058_out(x14, x15, x16) :|: TRUE f4067_out(x17, x18, x19) -> f4058_out(x17, x18, x19) :|: TRUE f4058_in(x20, x21, x22) -> f4067_in(x20, x21, x22) :|: TRUE f4058_in(x23, x24, x25) -> f4066_in(x23, x24, x25) :|: TRUE f4096_in(x26, x27, x28, x29, x30) -> f3606_in(x26, x27, x28) :|: TRUE f3606_out(x31, x32, x33) -> f4096_out(x31, x32, x33, x34, x35) :|: TRUE f3707_in(x36, x37, x38) -> f3708_in :|: TRUE f3707_in(x39, x40, x41) -> f4050_in(x39, x40, x41) :|: x39 < x40 f4050_out(x42, x43, x44) -> f3707_out(x42, x43, x44) :|: x42 < x43 f3708_out -> f3707_out(x45, x46, x47) :|: TRUE f4052_out(x48, x49, x50) -> f3707_out(x50, x48, x49) :|: x50 >= x48 f3707_in(x51, x52, x53) -> f4052_in(x52, x53, x51) :|: x51 >= x52 f4094_out(x54, x55, x56, x57) -> f4092_out(x57, x54, x55) :|: TRUE f4092_in(x58, x59, x60) -> f4093_in :|: TRUE f4093_out -> f4092_out(x61, x62, x63) :|: TRUE f4092_in(x64, x65, x66) -> f4094_in(x65, x66, x67, x64) :|: x67 = x64 + 1 f4067_in(x68, x69, x70) -> f4077_in(x69, x68, x70) :|: TRUE f4077_out(x71, x72, x73) -> f4067_out(x72, x71, x73) :|: TRUE f4056_in(x74, x75, x76) -> f4057_in(x74, x75, x76) :|: TRUE f4057_out(x77, x78, x79) -> f4056_out(x77, x78, x79) :|: TRUE f3702_out(T31, T43, T42) -> f3607_out(T31, T43, T42) :|: TRUE f3701_out(T50, T51) -> f3607_out(T50, T50, T51) :|: TRUE f3607_in(x80, x81, x82) -> f3702_in(x80, x81, x82) :|: TRUE f3607_in(x83, x83, x84) -> f3701_in(x83, x84) :|: TRUE f4071_out -> f4066_out(1, 0, x85) :|: TRUE f4066_in(1, 0, x86) -> f4071_in :|: TRUE f4066_in(x87, x88, x89) -> f4072_in :|: TRUE f4072_out -> f4066_out(x90, x91, x92) :|: TRUE f4058_out(x93, x94, x95) -> f4057_out(x93, x94, x95) :|: TRUE f4057_in(x96, x97, x98) -> f4058_in(x96, x97, x98) :|: TRUE f4055_in(x99, x100, x101) -> f4092_in(x99, x100, x101) :|: TRUE f4092_out(x102, x103, x104) -> f4055_out(x102, x103, x104) :|: TRUE f4123_in(T113, T111, T114) -> f4125_in(T111, T116, T114, T113) :|: T116 = T113 + 1 f4125_out(x105, x106, x107, x108) -> f4123_out(x108, x105, x107) :|: TRUE f4050_in(x109, x110, x111) -> f4054_in(x109, x110, x111) :|: TRUE f4054_out(x112, x113, x114) -> f4055_in(x114, x112, x113) :|: TRUE f4055_out(x115, x116, x117) -> f4050_out(x116, x117, x115) :|: TRUE f3705_in(x118, x119, x120) -> f3707_in(x119, x118, x120) :|: TRUE f3707_out(x121, x122, x123) -> f3705_out(x122, x121, x123) :|: TRUE f3606_in(x124, x125, x126) -> f3607_in(x124, x125, x126) :|: TRUE f3607_out(x127, x128, x129) -> f3606_out(x127, x128, x129) :|: TRUE f3706_in(x130, x131, x132) -> f4110_in(x131, x130, x132) :|: TRUE f4110_out(x133, x134, x135) -> f3706_out(x134, x133, x135) :|: TRUE f4125_in(x136, x137, x138, x139) -> f3606_in(x136, x137, x138) :|: TRUE f3606_out(x140, x141, x142) -> f4125_out(x140, x141, x142, x143) :|: TRUE f4054_in(x144, x145, x146) -> f4056_in(x144, x145, x146) :|: TRUE f4056_out(x147, x148, x149) -> f4054_out(x147, x148, x149) :|: TRUE f3702_in(x150, x151, x152) -> f3706_in(x150, x151, x152) :|: TRUE f3705_out(x153, x154, x155) -> f3702_out(x153, x154, x155) :|: TRUE f3702_in(x156, x157, x158) -> f3705_in(x156, x157, x158) :|: TRUE f3706_out(x159, x160, x161) -> f3702_out(x159, x160, x161) :|: TRUE f4111_out -> f4110_out(x162, x163, x164) :|: TRUE f4110_in(x165, x166, x167) -> f4111_in :|: TRUE f4110_in(x168, x169, x170) -> f4124_in(x170, x169, x168) :|: x168 >= x169 f4124_out(x171, x172, x173) -> f4110_out(x173, x172, x171) :|: x173 >= x172 f4110_in(x174, x175, x176) -> f4123_in(x174, x175, x176) :|: x174 < x175 f4123_out(x177, x178, x179) -> f4110_out(x177, x178, x179) :|: x177 < x178 f1_in(T1) -> f5_in(T1) :|: TRUE f5_out(x180) -> f1_out(x180) :|: TRUE f35_out -> f5_out(1) :|: TRUE f5_in(x181) -> f36_in(x181) :|: TRUE f5_in(1) -> f35_in :|: TRUE f36_out(x182) -> f5_out(x182) :|: TRUE f39_out(T10) -> f36_out(T10) :|: TRUE f36_in(x183) -> f39_in(x183) :|: TRUE f39_in(x184) -> f40_in(x184) :|: TRUE f40_out(x185) -> f39_out(x185) :|: TRUE f41_out -> f40_out(1) :|: TRUE f40_in(1) -> f41_in :|: TRUE f42_out(x186) -> f40_out(x186) :|: TRUE f40_in(x187) -> f42_in(x187) :|: TRUE f42_in(x188) -> f45_in(x188) :|: TRUE f42_in(x189) -> f46_in(x189) :|: TRUE f45_out(x190) -> f42_out(x190) :|: TRUE f46_out(x191) -> f42_out(x191) :|: TRUE f46_in(T124) -> f4139_in(T124) :|: TRUE f4139_out(x192) -> f46_out(x192) :|: TRUE f4139_in(x193) -> f4150_in(x193) :|: 1 < x193 f4139_in(x194) -> f4151_in(x194) :|: 1 >= x194 f4139_in(x195) -> f4140_in :|: TRUE f4151_out(x196) -> f4139_out(x196) :|: 1 >= x196 f4140_out -> f4139_out(x197) :|: TRUE f4150_out(x198) -> f4139_out(x198) :|: 1 < x198 f4152_out(x199, x200) -> f4150_out(x199) :|: TRUE f4150_in(x201) -> f4152_in(x201, x202) :|: x202 = 1 + 1 f4153_out(x203, x204) -> f4152_out(x203, x204) :|: TRUE f4152_in(x205, x206) -> f4153_in(x205, x206) :|: TRUE f4153_in(T134, T134) -> f4154_in(T134) :|: TRUE f4155_out(x207, x208) -> f4153_out(x207, x208) :|: TRUE f4154_out(x209) -> f4153_out(x209, x209) :|: TRUE f4153_in(x210, x211) -> f4155_in(x210, x211) :|: TRUE f4155_in(x212, x213) -> f4159_in(x212, x213) :|: TRUE f4159_out(x214, x215) -> f4155_out(x214, x215) :|: TRUE f4158_out(x216, x217) -> f4155_out(x216, x217) :|: TRUE f4155_in(x218, x219) -> f4158_in(x218, x219) :|: TRUE f4160_out(T152, T150) -> f4158_out(T150, T152) :|: TRUE f4158_in(x220, x221) -> f4160_in(x221, x220) :|: TRUE f4168_out(x222, x223) -> f4160_out(x222, x223) :|: x222 < x223 f4169_out(x224, x225) -> f4160_out(x225, x224) :|: x225 >= x224 f4160_in(x226, x227) -> f4168_in(x226, x227) :|: x226 < x227 f4160_in(x228, x229) -> f4169_in(x229, x228) :|: x228 >= x229 f4171_out(x230, x231) -> f4168_out(x230, x231) :|: TRUE f4168_in(x232, x233) -> f4170_in(x232, x233) :|: TRUE f4170_out(x234, x235) -> f4171_in(x234, x235) :|: TRUE f4196_out(x236, x237) -> f4171_out(x236, x237) :|: TRUE f4171_in(x238, x239) -> f4196_in(x238, x239) :|: TRUE f4196_in(x240, x241) -> f4197_in(x240, x241, x242) :|: x242 = 0 + 1 f4197_out(x243, x244, x245) -> f4196_out(x243, x244) :|: TRUE f4197_in(x246, x247, x248) -> f4198_in :|: TRUE f4198_out -> f4197_out(x249, x250, x251) :|: TRUE f4199_out(x252, x253, x254, x255) -> f4197_out(x255, x252, x254) :|: TRUE f4197_in(x256, x257, x258) -> f4199_in(x257, x259, x258, x256) :|: x259 = x256 + 1 f3606_out(x260, x261, x262) -> f4199_out(x260, x261, x262, x263) :|: TRUE f4199_in(x264, x265, x266, x267) -> f3606_in(x264, x265, x266) :|: TRUE f47_out(x268) -> f45_out(x268) :|: TRUE f45_in(x269) -> f47_in(x269) :|: TRUE f1016_out(x270) -> f47_out(x270) :|: 1 < x270 f48_out -> f47_out(x271) :|: TRUE f47_in(x272) -> f48_in :|: TRUE f47_in(x273) -> f1016_in(x273) :|: 1 < x273 f47_in(x274) -> f1018_in(x274) :|: 1 >= x274 f1018_out(x275) -> f47_out(x275) :|: 1 >= x275 f1020_out(x276) -> f1024_in(x276) :|: TRUE f1024_out(x277) -> f1016_out(x277) :|: TRUE f1016_in(x278) -> f1020_in(x278) :|: TRUE f3604_out(x279) -> f1024_out(x279) :|: TRUE f1024_in(x280) -> f3604_in(x280) :|: TRUE f3605_out(x281, x282) -> f3604_out(x281) :|: TRUE f3604_in(x283) -> f3605_in(x283, x284) :|: x284 = 0 + 1 f3605_in(x285, x286) -> f3606_in(x285, x287, x286) :|: x287 = 1 + 1 f3606_out(x288, x289, x290) -> f3605_out(x288, x290) :|: TRUE Start term: f1_in(T1) ---------------------------------------- (4) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: ---------------------------------------- (5) TRUE ---------------------------------------- (6) Obligation: Rules: f4153_out(T124, T127) -> f4152_out(T124, T127) :|: TRUE f4152_in(x, x1) -> f4153_in(x, x1) :|: TRUE f4153_in(T134, T134) -> f4154_in(T134) :|: TRUE f4155_out(x2, x3) -> f4153_out(x2, x3) :|: TRUE f4154_out(x4) -> f4153_out(x4, x4) :|: TRUE f4153_in(x5, x6) -> f4155_in(x5, x6) :|: TRUE f4201_in(T190, T188) -> f4203_in(T188, T192, T190) :|: T192 = T190 + 1 f4203_out(x7, x8, x9) -> f4201_out(x9, x7) :|: TRUE f4200_out(x10, x11) -> f4159_out(x11, x10) :|: TRUE f4159_in(x12, x13) -> f4200_in(x13, x12) :|: TRUE f4152_out(x14, x15) -> f4203_out(x14, x15, x16) :|: TRUE f4203_in(x17, x18, x19) -> f4152_in(x17, x18) :|: TRUE f4155_in(x20, x21) -> f4159_in(x20, x21) :|: TRUE f4159_out(x22, x23) -> f4155_out(x22, x23) :|: TRUE f4158_out(x24, x25) -> f4155_out(x24, x25) :|: TRUE f4155_in(x26, x27) -> f4158_in(x26, x27) :|: TRUE f4200_in(x28, x29) -> f4202_in(x29, x28) :|: x28 >= x29 f4202_out(x30, x31) -> f4200_out(x31, x30) :|: x31 >= x30 f4200_in(x32, x33) -> f4201_in(x32, x33) :|: x32 < x33 f4201_out(x34, x35) -> f4200_out(x34, x35) :|: x34 < x35 f1_in(T1) -> f5_in(T1) :|: TRUE f5_out(x36) -> f1_out(x36) :|: TRUE f35_out -> f5_out(1) :|: TRUE f5_in(x37) -> f36_in(x37) :|: TRUE f5_in(1) -> f35_in :|: TRUE f36_out(x38) -> f5_out(x38) :|: TRUE f39_out(T10) -> f36_out(T10) :|: TRUE f36_in(x39) -> f39_in(x39) :|: TRUE f39_in(x40) -> f40_in(x40) :|: TRUE f40_out(x41) -> f39_out(x41) :|: TRUE f41_out -> f40_out(1) :|: TRUE f40_in(1) -> f41_in :|: TRUE f42_out(x42) -> f40_out(x42) :|: TRUE f40_in(x43) -> f42_in(x43) :|: TRUE f42_in(x44) -> f45_in(x44) :|: TRUE f42_in(x45) -> f46_in(x45) :|: TRUE f45_out(x46) -> f42_out(x46) :|: TRUE f46_out(x47) -> f42_out(x47) :|: TRUE f46_in(x48) -> f4139_in(x48) :|: TRUE f4139_out(x49) -> f46_out(x49) :|: TRUE f4139_in(x50) -> f4150_in(x50) :|: 1 < x50 f4139_in(x51) -> f4151_in(x51) :|: 1 >= x51 f4139_in(x52) -> f4140_in :|: TRUE f4151_out(x53) -> f4139_out(x53) :|: 1 >= x53 f4140_out -> f4139_out(x54) :|: TRUE f4150_out(x55) -> f4139_out(x55) :|: 1 < x55 f4152_out(x56, x57) -> f4150_out(x56) :|: TRUE f4150_in(x58) -> f4152_in(x58, x59) :|: x59 = 1 + 1 Start term: f1_in(T1) ---------------------------------------- (7) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: intTRSProblem: f4152_in(x, x1) -> f4153_in(x, x1) :|: TRUE f4153_in(x5, x6) -> f4155_in(x5, x6) :|: TRUE f4201_in(T190, T188) -> f4203_in(T188, T192, T190) :|: T192 = T190 + 1 f4159_in(x12, x13) -> f4200_in(x13, x12) :|: TRUE f4203_in(x17, x18, x19) -> f4152_in(x17, x18) :|: TRUE f4155_in(x20, x21) -> f4159_in(x20, x21) :|: TRUE f4200_in(x32, x33) -> f4201_in(x32, x33) :|: x32 < x33 ---------------------------------------- (8) Obligation: Rules: f4152_in(x, x1) -> f4153_in(x, x1) :|: TRUE f4153_in(x5, x6) -> f4155_in(x5, x6) :|: TRUE f4201_in(T190, T188) -> f4203_in(T188, T192, T190) :|: T192 = T190 + 1 f4159_in(x12, x13) -> f4200_in(x13, x12) :|: TRUE f4203_in(x17, x18, x19) -> f4152_in(x17, x18) :|: TRUE f4155_in(x20, x21) -> f4159_in(x20, x21) :|: TRUE f4200_in(x32, x33) -> f4201_in(x32, x33) :|: x32 < x33 ---------------------------------------- (9) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (10) Obligation: Rules: f4201_in(T190:0, T188:0) -> f4201_in(T190:0 + 1, T188:0) :|: T190:0 + 1 < T188:0 ---------------------------------------- (11) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (12) Obligation: Rules: f4201_in(T190:0, T188:0) -> f4201_in(arith, T188:0) :|: T190:0 + 1 < T188:0 && arith = T190:0 + 1 ---------------------------------------- (13) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) f4201_in(T190:0, T188:0) -> f4201_in(arith, T188:0) :|: T190:0 + 1 < T188:0 && arith = T190:0 + 1 Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (14) Obligation: Termination digraph: Nodes: (1) f4201_in(T190:0, T188:0) -> f4201_in(arith, T188:0) :|: T190:0 + 1 < T188:0 && arith = T190:0 + 1 Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (15) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (16) Obligation: Rules: f4201_in(T190:0:0, T188:0:0) -> f4201_in(T190:0:0 + 1, T188:0:0) :|: T190:0:0 + 1 < T188:0:0 ---------------------------------------- (17) TempFilterProof (SOUND) Used the following sort dictionary for filtering: f4201_in(INTEGER, INTEGER) Replaced non-predefined constructor symbols by 0. ---------------------------------------- (18) Obligation: Rules: f4201_in(T190:0:0, T188:0:0) -> f4201_in(c, T188:0:0) :|: c = T190:0:0 + 1 && T190:0:0 + 1 < T188:0:0 ---------------------------------------- (19) RankingReductionPairProof (EQUIVALENT) Interpretation: [ f4201_in ] = -1*f4201_in_1 + f4201_in_2 The following rules are decreasing: f4201_in(T190:0:0, T188:0:0) -> f4201_in(c, T188:0:0) :|: c = T190:0:0 + 1 && T190:0:0 + 1 < T188:0:0 The following rules are bounded: f4201_in(T190:0:0, T188:0:0) -> f4201_in(c, T188:0:0) :|: c = T190:0:0 + 1 && T190:0:0 + 1 < T188:0:0 ---------------------------------------- (20) YES