/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern test_fun(g,g) w.r.t. the given Prolog program could successfully be proven: (0) Prolog (1) PrologToIRSwTTransformerProof [SOUND, 84 ms] (2) AND (3) IRSwT (4) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 2 ms] (5) IRSwT (6) IntTRSCompressionProof [EQUIVALENT, 17 ms] (7) IRSwT (8) IRSFormatTransformerProof [EQUIVALENT, 0 ms] (9) IRSwT (10) IRSwTTerminationDigraphProof [EQUIVALENT, 10 ms] (11) IRSwT (12) IntTRSCompressionProof [EQUIVALENT, 0 ms] (13) IRSwT (14) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms] (15) IRSwT (16) TempFilterProof [SOUND, 40 ms] (17) IntTRS (18) RankingReductionPairProof [EQUIVALENT, 21 ms] (19) YES (20) IRSwT (21) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (22) IRSwT (23) IntTRSCompressionProof [EQUIVALENT, 3 ms] (24) IRSwT (25) IRSFormatTransformerProof [EQUIVALENT, 0 ms] (26) IRSwT (27) IRSwTTerminationDigraphProof [EQUIVALENT, 0 ms] (28) TRUE ---------------------------------------- (0) Obligation: Clauses: test_fun(I, J) :- loop(I, J, 0). loop(I, J, C) :- ','(>=(I, 0), ','(is(J1, 0), inner_loop(I, J1, C))). loop(I, J, C) :- <(I, 0). inner_loop(I, J, C) :- ','(=<(J, -(I, 1)), ','(is(J1, +(J, 1)), ','(is(C1, +(C, 1)), inner_loop(I, J1, C1)))). inner_loop(I, J, C) :- ','(>(J, -(I, 1)), ','(is(I1, -(I, 1)), loop(I1, J, C))). Query: test_fun(g,g) ---------------------------------------- (1) PrologToIRSwTTransformerProof (SOUND) Transformed Prolog program to IRSwT according to method in Master Thesis of A. Weinert { "root": 3, "program": { "directives": [], "clauses": [ [ "(test_fun I J)", "(loop I J (0))" ], [ "(loop I J C)", "(',' (>= I (0)) (',' (is J1 (0)) (inner_loop I J1 C)))" ], [ "(loop I J C)", "(< I (0))" ], [ "(inner_loop I J C)", "(',' (=< J (- I (1))) (',' (is J1 (+ J (1))) (',' (is C1 (+ C (1))) (inner_loop I J1 C1))))" ], [ "(inner_loop I J C)", "(',' (> J (- I (1))) (',' (is I1 (- I (1))) (loop I1 J C)))" ] ] }, "graph": { "nodes": { "3000": { "goal": [{ "clause": -1, "scope": -1, "term": "(inner_loop T53 T56 T57)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [ { "lhs": { "name": "T23", "type": "PlainIntegerVariable" }, "type": "PlainIntegerRelation", "rhs": { "type": "PlainIntegerConstant", "value": "0" }, "operation": "=" }, { "lhs": { "type": "PlainIntegerConstant", "value": "0" }, "type": "PlainIntegerRelation", "rhs": { "type": "PlainIntegerConstant", "value": "0" }, "operation": "=" }, { "lhs": { "name": "T36", "type": "PlainIntegerVariable" }, "type": "PlainIntegerRelation", "rhs": { "arguments": [ { "name": "T35", "type": "PlainIntegerVariable" }, { "type": "PlainIntegerConstant", "value": "1" } ], "type": "PlainIntegerOperation", "operation": "+" }, "operation": "=" }, { "lhs": { "name": "T37", "type": "PlainIntegerVariable" }, "type": "PlainIntegerRelation", "rhs": { "arguments": [ { "type": "PlainIntegerConstant", "value": "0" }, { "type": "PlainIntegerConstant", "value": "1" } ], "type": "PlainIntegerOperation", "operation": "+" }, "operation": "=" }, { "lhs": { "type": "PlainIntegerConstant", "value": "0" }, "type": "PlainIntegerRelation", "rhs": { "type": "PlainIntegerConstant", "value": "0" }, "operation": "=" }, { "lhs": { "type": "PlainIntegerConstant", "value": "0" }, "type": "PlainIntegerRelation", "rhs": { "type": "PlainIntegerConstant", "value": "0" }, "operation": "=" }, { "lhs": { "arguments": [ { "name": "T35", "type": "PlainIntegerVariable" }, { "type": 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"PlainIntegerRelation", "rhs": { "type": "PlainIntegerConstant", "value": "0" }, "operation": "=" }, { "lhs": { "type": "PlainIntegerConstant", "value": "0" }, "type": "PlainIntegerRelation", "rhs": { "type": "PlainIntegerConstant", "value": "0" }, "operation": "=" }, { "lhs": { "name": "T36", "type": "PlainIntegerVariable" }, "type": "PlainIntegerRelation", "rhs": { "arguments": [ { "name": "T35", "type": "PlainIntegerVariable" }, { "type": "PlainIntegerConstant", "value": "1" } ], "type": "PlainIntegerOperation", "operation": "+" }, "operation": "=" }, { "lhs": { "name": "T37", "type": "PlainIntegerVariable" }, "type": "PlainIntegerRelation", "rhs": { "arguments": [ { "type": "PlainIntegerConstant", "value": "0" }, { "type": "PlainIntegerConstant", "value": "1" } ], "type": "PlainIntegerOperation", "operation": "+" }, "operation": "=" }, { "lhs": { "type": "PlainIntegerConstant", "value": "0" }, "type": "PlainIntegerRelation", "rhs": { "type": "PlainIntegerConstant", "value": "0" }, "operation": "=" }, { "lhs": { "type": "PlainIntegerConstant", "value": "0" }, "type": "PlainIntegerRelation", "rhs": { "type": "PlainIntegerConstant", "value": "0" }, "operation": "=" }, { "lhs": { "arguments": [ { "name": "T35", "type": "PlainIntegerVariable" }, { "type": "PlainIntegerConstant", "value": "1" } ], "type": "PlainIntegerOperation", "operation": "+" }, "type": "PlainIntegerRelation", "rhs": { "arguments": [ { "name": "T35", "type": "PlainIntegerVariable" }, { "type": "PlainIntegerConstant", "value": "1" } ], "type": "PlainIntegerOperation", "operation": "+" }, "operation": "=" }, { "lhs": { "arguments": [ { "type": "PlainIntegerConstant", "value": "0" }, { "type": "PlainIntegerConstant", "value": "1" } ], "type": "PlainIntegerOperation", "operation": "+" }, "type": "PlainIntegerRelation", "rhs": { "arguments": [ { "type": "PlainIntegerConstant", "value": "0" }, { "type": "PlainIntegerConstant", "value": "1" } ], "type": "PlainIntegerOperation", "operation": "+" }, "operation": "=" }, { "lhs": { "type": "PlainIntegerConstant", "value": "0" }, "type": "PlainIntegerRelation", "rhs": { "name": "T21", "type": "PlainIntegerVariable" }, "operation": "<=" }, { "lhs": { "name": "T34", "type": "PlainIntegerVariable" }, "type": "PlainIntegerRelation", "rhs": { "name": "T21", "type": "PlainIntegerVariable" }, "operation": "<=" }, { "lhs": { "name": "T35", "type": "PlainIntegerVariable" }, "type": "PlainIntegerRelation", "rhs": { "arguments": [ { "name": "T34", "type": "PlainIntegerVariable" }, { "type": "PlainIntegerConstant", "value": "1" } ], "type": "PlainIntegerOperation", "operation": "-" }, "operation": "<=" }, { "lhs": { "name": "T21", "type": "PlainIntegerVariable" }, "type": "PlainIntegerRelation", "rhs": { "name": "T21", "type": "PlainIntegerVariable" }, "operation": "<=" }, { "lhs": { "name": "T21", "type": "PlainIntegerVariable" }, "type": "PlainIntegerRelation", "rhs": { "name": "T21", "type": "PlainIntegerVariable" }, "operation": "<=" }, { "lhs": { "arguments": [ { "name": "T53", "type": "PlainIntegerVariable" }, { "type": "PlainIntegerConstant", "value": "1" } ], "type": "PlainIntegerOperation", "operation": "-" }, "type": "PlainIntegerRelation", "rhs": { "arguments": [ { "name": "T34", "type": "PlainIntegerVariable" }, { "type": "PlainIntegerConstant", "value": "1" } ], "type": "PlainIntegerOperation", "operation": "-" }, "operation": "<=" } ] }, "ground": [ "T53", "T54", "T55" ], "free": [ "X83", "X84" ], "exprvars": [ "T53", "T36", "T35", "T23", "T34", "T55", "T21", "T54", "T37" ] } } }, "edges": [ { "from": 3, "to": 4, "label": "CASE" }, { "from": 4, "to": 8, "label": "ONLY EVAL with clause\ntest_fun(X9, X10) :- loop(X9, X10, 0).\nand substitutionT1 -> T9,\nX9 -> T9,\nT2 -> T10,\nX10 -> T10" }, { "from": 8, "to": 20, "label": "CASE" }, { "from": 20, "to": 21, "label": "PARALLEL" }, { "from": 20, "to": 22, "label": "PARALLEL" }, { "from": 21, "to": 23, "label": "ONLY EVAL with clause\nloop(X29, X30, X31) :- ','(>=(X29, 0), ','(is(X32, 0), inner_loop(X29, X32, X31))).\nand substitutionT9 -> T21,\nX29 -> T21,\nT10 -> T22,\nX30 -> T22,\nX31 -> 0" }, { "from": 22, "to": 3218, "label": "ONLY EVAL with clause\nloop(X190, X191, X192) :- <(X190, 0).\nand substitutionT9 -> T139,\nX190 -> T139,\nT10 -> T140,\nX191 -> T140,\nX192 -> 0" }, { "from": 23, "to": 24, "label": "IS ERROR" }, { "from": 23, "to": 25, "label": "ARITHCOMP SUCCESS" }, { "from": 23, "to": 27, "label": "ARITHCOMP FAIL" }, { "from": 25, "to": 28, "label": "\nX32 -> T23" }, { "from": 28, "to": 29, "label": "CASE" }, { "from": 29, "to": 30, "label": "PARALLEL" }, { "from": 29, "to": 31, "label": "PARALLEL" }, { "from": 30, "to": 32, "label": "ONLY EVAL with clause\ninner_loop(X54, X55, X56) :- ','(=<(X55, -(X54, 1)), ','(is(X57, +(X55, 1)), ','(is(X58, +(X56, 1)), inner_loop(X54, X57, X58)))).\nand substitutionT21 -> T34,\nX54 -> T34,\nT23 -> T35,\nX55 -> T35,\nX56 -> 0" }, { "from": 31, "to": 3209, "label": "ONLY EVAL with clause\ninner_loop(X176, X177, X178) :- ','(>(X177, -(X176, 1)), ','(is(X179, -(X176, 1)), loop(X179, X177, X178))).\nand substitutionT21 -> T130,\nX176 -> T130,\nT23 -> T131,\nX177 -> T131,\nX178 -> 0" }, { "from": 32, "to": 1695, "label": "ARITHCOMP SUCCESS" }, { "from": 32, "to": 1696, "label": "ARITHCOMP FAIL" }, { "from": 1695, "to": 1697, "label": "\nX57 -> T36" }, { "from": 1697, "to": 1711, "label": "\nX58 -> T37" }, { "from": 1711, "to": 2243, "label": "CASE" }, { "from": 2243, "to": 2413, "label": "PARALLEL" }, { "from": 2243, "to": 2423, "label": "PARALLEL" }, { "from": 2413, "to": 2619, "label": "ONLY EVAL with clause\ninner_loop(X80, X81, X82) :- ','(=<(X81, -(X80, 1)), ','(is(X83, +(X81, 1)), ','(is(X84, +(X82, 1)), inner_loop(X80, X83, X84)))).\nand substitutionT34 -> T53,\nX80 -> T53,\nT36 -> T54,\nX81 -> T54,\nT37 -> T55,\nX82 -> T55" }, { "from": 2423, "to": 3107, "label": "ONLY EVAL with clause\ninner_loop(X97, X98, X99) :- ','(>(X98, -(X97, 1)), ','(is(X100, -(X97, 1)), loop(X100, X98, X99))).\nand substitutionT34 -> T67,\nX97 -> T67,\nT36 -> T68,\nX98 -> T68,\nT37 -> T69,\nX99 -> T69" }, { "from": 2619, "to": 2997, "label": "ARITHCOMP SUCCESS" }, { "from": 2619, "to": 2998, "label": "ARITHCOMP FAIL" }, { "from": 2997, "to": 2999, "label": "\nX83 -> T56" }, { "from": 2999, "to": 3000, "label": "\nX84 -> T57" }, { "from": 3000, "to": 1711, "label": "INSTANCE with matching:\nT34 -> T53\nT36 -> T56\nT37 -> T57\nX57 -> X83\nX58 -> X84" }, { "from": 3107, "to": 3110, "label": "ARITHCOMP SUCCESS" }, { "from": 3107, "to": 3111, "label": "ARITHCOMP FAIL" }, { "from": 3110, "to": 3112, "label": "\nX100 -> T70" }, { "from": 3112, "to": 3113, "label": "CASE" }, { "from": 3113, "to": 3114, "label": "PARALLEL" }, { "from": 3113, "to": 3115, "label": "PARALLEL" }, { "from": 3114, "to": 3116, "label": "ONLY EVAL with clause\nloop(X119, X120, X121) :- ','(>=(X119, 0), ','(is(X122, 0), inner_loop(X119, X122, X121))).\nand substitutionT70 -> T86,\nX119 -> T86,\nT68 -> T87,\nX120 -> T87,\nT69 -> T88,\nX121 -> T88" }, { "from": 3115, "to": 3203, "label": "ONLY EVAL with clause\nloop(X166, X167, X168) :- <(X166, 0).\nand substitutionT70 -> T123,\nX166 -> T123,\nT68 -> T124,\nX167 -> T124,\nT69 -> T125,\nX168 -> T125" }, { "from": 3116, "to": 3127, "label": "ARITHCOMP SUCCESS" }, { "from": 3116, "to": 3128, "label": "ARITHCOMP FAIL" }, { "from": 3127, "to": 3129, "label": "\nX122 -> T89" }, { "from": 3129, "to": 3130, "label": "CASE" }, { "from": 3130, "to": 3131, "label": "PARALLEL" }, { "from": 3130, "to": 3132, "label": "PARALLEL" }, { "from": 3131, "to": 3133, "label": "ONLY EVAL with clause\ninner_loop(X144, X145, X146) :- ','(=<(X145, -(X144, 1)), ','(is(X147, +(X145, 1)), ','(is(X148, +(X146, 1)), inner_loop(X144, X147, X148)))).\nand substitutionT86 -> T105,\nX144 -> T105,\nT89 -> T106,\nX145 -> T106,\nT88 -> T107,\nX146 -> T107" }, { "from": 3132, "to": 3183, "label": "ONLY EVAL with clause\ninner_loop(X156, X157, X158) :- ','(>(X157, -(X156, 1)), ','(is(X159, -(X156, 1)), loop(X159, X157, X158))).\nand substitutionT86 -> T114,\nX156 -> T114,\nT89 -> T115,\nX157 -> T115,\nT88 -> T116,\nX158 -> T116" }, { "from": 3203, "to": 3206, "label": "ARITHCOMP SUCCESS" }, { "from": 3203, "to": 3207, "label": "ARITHCOMP FAIL" }, { "from": 3206, "to": 3208, "label": "SUCCESS" }, { "from": 3209, "to": 3212, "label": "ARITHCOMP SUCCESS" }, { "from": 3209, "to": 3213, "label": "ARITHCOMP FAIL" }, { "from": 3212, "to": 3214, "label": "\nX179 -> T132" }, { "from": 3214, "to": 8, "label": "INSTANCE with matching:\nT9 -> T132\nT10 -> T131" }, { "from": 3218, "to": 3219, "label": "IS ERROR" }, { "from": 3218, "to": 3230, "label": "ARITHCOMP SUCCESS" }, { "from": 3218, "to": 3231, "label": "ARITHCOMP FAIL" }, { "from": 3230, "to": 3232, "label": "SUCCESS" } ], "type": "Graph" } } ---------------------------------------- (2) Complex Obligation (AND) ---------------------------------------- (3) Obligation: Rules: f2243_out(T34, T36, T37) -> f1711_out(T34, T36, T37, T35) :|: TRUE f1711_in(x, x1, x2, x3) -> f2243_in(x, x1, x2) :|: TRUE f2413_in(T53, T54, T55) -> f2619_in(T54, T53, T55) :|: TRUE f2619_out(x4, x5, x6) -> f2413_out(x5, x4, x6) :|: TRUE f2619_in(x7, x8, x9) -> f2998_in(x8, x9, x7) :|: x7 > x8 - 1 f2998_out(x10, x11, x12) -> f2619_out(x12, x10, x11) :|: x12 > x10 - 1 f2997_out(x13, x14, x15) -> f2619_out(x13, x15, x14) :|: x13 <= x15 - 1 f2619_in(x16, x17, x18) -> f2997_in(x16, x18, x17) :|: x16 <= x17 - 1 f3000_in(x19, x20, x21, x22, x23) -> f1711_in(x19, x20, x21, x24) :|: TRUE f1711_out(x25, x26, x27, x28) -> f3000_out(x25, x26, x27, x29, x30) :|: TRUE f2997_in(x31, x32, x33) -> f2999_in(x32, x33, x34, x31) :|: x34 = x31 + 1 f2999_out(x35, x36, x37, x38) -> f2997_out(x38, x35, x36) :|: TRUE f2999_in(x39, x40, x41, x42) -> f3000_in(x40, x41, x43, x39, x42) :|: x43 = x39 + 1 f3000_out(x44, x45, x46, x47, x48) -> f2999_out(x47, x44, x45, x48) :|: TRUE f2243_in(x49, x50, x51) -> f2413_in(x49, x50, x51) :|: TRUE f2413_out(x52, x53, x54) -> f2243_out(x52, x53, x54) :|: TRUE f2423_out(x55, x56, x57) -> f2243_out(x55, x56, x57) :|: TRUE f2243_in(x58, x59, x60) -> f2423_in(x58, x59, x60) :|: TRUE f4_out(T1, T2) -> f3_out(T1, T2) :|: TRUE f3_in(x61, x62) -> f4_in(x61, x62) :|: TRUE f4_in(T9, T10) -> f8_in(T9, T10) :|: TRUE f8_out(x63, x64) -> f4_out(x63, x64) :|: TRUE f8_in(x65, x66) -> f20_in(x65, x66) :|: TRUE f20_out(x67, x68) -> f8_out(x67, x68) :|: TRUE f21_out(x69, x70) -> f20_out(x69, x70) :|: TRUE f22_out(x71, x72) -> f20_out(x71, x72) :|: TRUE f20_in(x73, x74) -> f21_in(x73, x74) :|: TRUE f20_in(x75, x76) -> f22_in(x75, x76) :|: TRUE f21_in(T21, T22) -> f23_in(T21) :|: TRUE f23_out(x77) -> f21_out(x77, x78) :|: TRUE f25_out(x79) -> f23_out(x79) :|: x79 >= 0 f23_in(x80) -> f24_in :|: TRUE f23_in(x81) -> f25_in(x81) :|: x81 >= 0 f24_out -> f23_out(x82) :|: TRUE f23_in(x83) -> f27_in(x83) :|: x83 < 0 f27_out(x84) -> f23_out(x84) :|: x84 < 0 f25_in(x85) -> f28_in(x85, x86) :|: x86 = 0 f28_out(x87, x88) -> f25_out(x87) :|: TRUE f28_in(x89, x90) -> f29_in(x89, x90) :|: TRUE f29_out(x91, x92) -> f28_out(x91, x92) :|: TRUE f30_out(x93, x94) -> f29_out(x93, x94) :|: TRUE f29_in(x95, x96) -> f30_in(x95, x96) :|: TRUE f31_out(x97, x98) -> f29_out(x97, x98) :|: TRUE f29_in(x99, x100) -> f31_in(x99, x100) :|: TRUE f32_out(x101, x102) -> f30_out(x102, x101) :|: TRUE f30_in(x103, x104) -> f32_in(x104, x103) :|: TRUE f32_in(x105, x106) -> f1695_in(x105, x106) :|: x105 <= x106 - 1 f1695_out(x107, x108) -> f32_out(x107, x108) :|: x107 <= x108 - 1 f32_in(x109, x110) -> f1696_in(x109, x110) :|: x109 > x110 - 1 f1696_out(x111, x112) -> f32_out(x111, x112) :|: x111 > x112 - 1 f1697_out(x113, x114, x115) -> f1695_out(x115, x113) :|: TRUE f1695_in(x116, x117) -> f1697_in(x117, x118, x116) :|: x118 = x116 + 1 f1711_out(x119, x120, x121, x122) -> f1697_out(x119, x120, x122) :|: TRUE f1697_in(x123, x124, x125) -> f1711_in(x123, x124, x126, x125) :|: x126 = 0 + 1 Start term: f3_in(T1, T2) ---------------------------------------- (4) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: intTRSProblem: f1711_in(x, x1, x2, x3) -> f2243_in(x, x1, x2) :|: TRUE f2413_in(T53, T54, T55) -> f2619_in(T54, T53, T55) :|: TRUE f2619_in(x16, x17, x18) -> f2997_in(x16, x18, x17) :|: x16 <= x17 - 1 f3000_in(x19, x20, x21, x22, x23) -> f1711_in(x19, x20, x21, x24) :|: TRUE f2997_in(x31, x32, x33) -> f2999_in(x32, x33, x34, x31) :|: x34 = x31 + 1 f2999_in(x39, x40, x41, x42) -> f3000_in(x40, x41, x43, x39, x42) :|: x43 = x39 + 1 f2243_in(x49, x50, x51) -> f2413_in(x49, x50, x51) :|: TRUE ---------------------------------------- (5) Obligation: Rules: f1711_in(x, x1, x2, x3) -> f2243_in(x, x1, x2) :|: TRUE f2413_in(T53, T54, T55) -> f2619_in(T54, T53, T55) :|: TRUE f2619_in(x16, x17, x18) -> f2997_in(x16, x18, x17) :|: x16 <= x17 - 1 f3000_in(x19, x20, x21, x22, x23) -> f1711_in(x19, x20, x21, x24) :|: TRUE f2997_in(x31, x32, x33) -> f2999_in(x32, x33, x34, x31) :|: x34 = x31 + 1 f2999_in(x39, x40, x41, x42) -> f3000_in(x40, x41, x43, x39, x42) :|: x43 = x39 + 1 f2243_in(x49, x50, x51) -> f2413_in(x49, x50, x51) :|: TRUE ---------------------------------------- (6) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (7) Obligation: Rules: f2413_in(T53:0, T54:0, T55:0) -> f2413_in(T53:0, T54:0 + 1, T55:0 + 1) :|: T54:0 <= T53:0 - 1 ---------------------------------------- (8) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (9) Obligation: Rules: f2413_in(T53:0, T54:0, T55:0) -> f2413_in(T53:0, arith, arith1) :|: T54:0 <= T53:0 - 1 && arith = T54:0 + 1 && arith1 = T55:0 + 1 ---------------------------------------- (10) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) f2413_in(T53:0, T54:0, T55:0) -> f2413_in(T53:0, arith, arith1) :|: T54:0 <= T53:0 - 1 && arith = T54:0 + 1 && arith1 = T55:0 + 1 Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (11) Obligation: Termination digraph: Nodes: (1) f2413_in(T53:0, T54:0, T55:0) -> f2413_in(T53:0, arith, arith1) :|: T54:0 <= T53:0 - 1 && arith = T54:0 + 1 && arith1 = T55:0 + 1 Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (12) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (13) Obligation: Rules: f2413_in(T53:0:0, T54:0:0, T55:0:0) -> f2413_in(T53:0:0, T54:0:0 + 1, T55:0:0 + 1) :|: T54:0:0 <= T53:0:0 - 1 ---------------------------------------- (14) IntTRSUnneededArgumentFilterProof (EQUIVALENT) Some arguments are removed because they cannot influence termination. We removed arguments according to the following replacements: f2413_in(x1, x2, x3) -> f2413_in(x1, x2) ---------------------------------------- (15) Obligation: Rules: f2413_in(T53:0:0, T54:0:0) -> f2413_in(T53:0:0, T54:0:0 + 1) :|: T54:0:0 <= T53:0:0 - 1 ---------------------------------------- (16) TempFilterProof (SOUND) Used the following sort dictionary for filtering: f2413_in(INTEGER, INTEGER) Replaced non-predefined constructor symbols by 0. ---------------------------------------- (17) Obligation: Rules: f2413_in(T53:0:0, T54:0:0) -> f2413_in(T53:0:0, c) :|: c = T54:0:0 + 1 && T54:0:0 <= T53:0:0 - 1 ---------------------------------------- (18) RankingReductionPairProof (EQUIVALENT) Interpretation: [ f2413_in ] = -1*f2413_in_2 + f2413_in_1 The following rules are decreasing: f2413_in(T53:0:0, T54:0:0) -> f2413_in(T53:0:0, c) :|: c = T54:0:0 + 1 && T54:0:0 <= T53:0:0 - 1 The following rules are bounded: f2413_in(T53:0:0, T54:0:0) -> f2413_in(T53:0:0, c) :|: c = T54:0:0 + 1 && T54:0:0 <= T53:0:0 - 1 ---------------------------------------- (19) YES ---------------------------------------- (20) Obligation: Rules: f8_in(T9, T10) -> f20_in(T9, T10) :|: TRUE f20_out(x, x1) -> f8_out(x, x1) :|: TRUE f21_out(x2, x3) -> f20_out(x2, x3) :|: TRUE f22_out(x4, x5) -> f20_out(x4, x5) :|: TRUE f20_in(x6, x7) -> f21_in(x6, x7) :|: TRUE f20_in(x8, x9) -> f22_in(x8, x9) :|: TRUE f30_out(T21, T23) -> f29_out(T21, T23) :|: TRUE f29_in(x10, x11) -> f30_in(x10, x11) :|: TRUE f31_out(x12, x13) -> f29_out(x12, x13) :|: TRUE f29_in(x14, x15) -> f31_in(x14, x15) :|: TRUE f3214_out(T132, T131, T130) -> f3212_out(T130, T131) :|: TRUE f3212_in(x16, x17) -> f3214_in(x18, x17, x16) :|: x18 = x16 - 1 f21_in(x19, x20) -> f23_in(x19) :|: TRUE f23_out(x21) -> f21_out(x21, x22) :|: TRUE f25_in(x23) -> f28_in(x23, x24) :|: x24 = 0 f28_out(x25, x26) -> f25_out(x25) :|: TRUE f28_in(x27, x28) -> f29_in(x27, x28) :|: TRUE f29_out(x29, x30) -> f28_out(x29, x30) :|: TRUE f3209_in(x31, x32) -> f3212_in(x32, x31) :|: x31 > x32 - 1 f3213_out(x33, x34) -> f3209_out(x33, x34) :|: x33 <= x34 - 1 f3212_out(x35, x36) -> f3209_out(x36, x35) :|: x36 > x35 - 1 f3209_in(x37, x38) -> f3213_in(x37, x38) :|: x37 <= x38 - 1 f3214_in(x39, x40, x41) -> f8_in(x39, x40) :|: TRUE f8_out(x42, x43) -> f3214_out(x42, x43, x44) :|: TRUE f25_out(x45) -> f23_out(x45) :|: x45 >= 0 f23_in(x46) -> f24_in :|: TRUE f23_in(x47) -> f25_in(x47) :|: x47 >= 0 f24_out -> f23_out(x48) :|: TRUE f23_in(x49) -> f27_in(x49) :|: x49 < 0 f27_out(x50) -> f23_out(x50) :|: x50 < 0 f31_in(x51, x52) -> f3209_in(x52, x51) :|: TRUE f3209_out(x53, x54) -> f31_out(x54, x53) :|: TRUE f4_out(T1, T2) -> f3_out(T1, T2) :|: TRUE f3_in(x55, x56) -> f4_in(x55, x56) :|: TRUE f4_in(x57, x58) -> f8_in(x57, x58) :|: TRUE f8_out(x59, x60) -> f4_out(x59, x60) :|: TRUE Start term: f3_in(T1, T2) ---------------------------------------- (21) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: intTRSProblem: f8_in(T9, T10) -> f20_in(T9, T10) :|: TRUE f20_in(x6, x7) -> f21_in(x6, x7) :|: TRUE f29_in(x14, x15) -> f31_in(x14, x15) :|: TRUE f3212_in(x16, x17) -> f3214_in(x18, x17, x16) :|: x18 = x16 - 1 f21_in(x19, x20) -> f23_in(x19) :|: TRUE f25_in(x23) -> f28_in(x23, x24) :|: x24 = 0 f28_in(x27, x28) -> f29_in(x27, x28) :|: TRUE f3209_in(x31, x32) -> f3212_in(x32, x31) :|: x31 > x32 - 1 f3214_in(x39, x40, x41) -> f8_in(x39, x40) :|: TRUE f23_in(x47) -> f25_in(x47) :|: x47 >= 0 f31_in(x51, x52) -> f3209_in(x52, x51) :|: TRUE ---------------------------------------- (22) Obligation: Rules: f8_in(T9, T10) -> f20_in(T9, T10) :|: TRUE f20_in(x6, x7) -> f21_in(x6, x7) :|: TRUE f29_in(x14, x15) -> f31_in(x14, x15) :|: TRUE f3212_in(x16, x17) -> f3214_in(x18, x17, x16) :|: x18 = x16 - 1 f21_in(x19, x20) -> f23_in(x19) :|: TRUE f25_in(x23) -> f28_in(x23, x24) :|: x24 = 0 f28_in(x27, x28) -> f29_in(x27, x28) :|: TRUE f3209_in(x31, x32) -> f3212_in(x32, x31) :|: x31 > x32 - 1 f3214_in(x39, x40, x41) -> f8_in(x39, x40) :|: TRUE f23_in(x47) -> f25_in(x47) :|: x47 >= 0 f31_in(x51, x52) -> f3209_in(x52, x51) :|: TRUE ---------------------------------------- (23) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (24) Obligation: Rules: f3209_in(x31:0, x32:0) -> f3209_in(0, x32:0 - 1) :|: x32:0 - 1 < x31:0 && x32:0 > 0 ---------------------------------------- (25) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (26) Obligation: Rules: f3209_in(x31:0, x32:0) -> f3209_in(0, arith) :|: x32:0 - 1 < x31:0 && x32:0 > 0 && arith = x32:0 - 1 ---------------------------------------- (27) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) f3209_in(x31:0, x32:0) -> f3209_in(0, arith) :|: x32:0 - 1 < x31:0 && x32:0 > 0 && arith = x32:0 - 1 No arcs! This digraph is fully evaluated! ---------------------------------------- (28) TRUE