/export/starexec/sandbox/solver/bin/starexec_run_standard /export/starexec/sandbox/benchmark/theBenchmark.pl /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- YES proof of /export/starexec/sandbox/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern test_fun(g,g) w.r.t. the given Prolog program could successfully be proven: (0) Prolog (1)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, 103 ms] (2) AND (3) IRSwT (4) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 2 ms] (5) IRSwT (6) IntTRSCompressionProof [EQUIVALENT, 0 ms] (7) IRSwT (8) IRSFormatTransformerProof [EQUIVALENT, 0 ms] (9) IRSwT (10) IRSwTTerminationDigraphProof [EQUIVALENT, 36 ms] (11) IRSwT (12) IntTRSCompressionProof [EQUIVALENT, 0 ms] (13) IRSwT (14) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms] (15) IRSwT (16) TempFilterProof [SOUND, 35 ms] (17) IntTRS (18) PolynomialOrderProcessor [EQUIVALENT, 14 ms] (19) YES (20) IRSwT (21) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (22) IRSwT (23) IntTRSCompressionProof [EQUIVALENT, 0 ms] (24) IRSwT (25) IRSFormatTransformerProof [EQUIVALENT, 0 ms] (26) IRSwT (27) IRSwTTerminationDigraphProof [EQUIVALENT, 9 ms] (28) TRUE ---------------------------------------- (0) Obligation: Clauses: test_fun(X, Y) :- loop1(X, Y, 0). loop1(X, Y, C) :- ','(>=(X, 0), ','(is(X1, +(X, 1)), ','(is(Y1, 1), loop2(X1, Y1, C)))). loop1(X, Y, C) :- <(X, 0). loop2(X, Y, C) :- ','(>(X, Y), ','(is(Y1, +(Y, 1)), ','(is(C1, +(C, 1)), loop2(X, Y1, C1)))). loop2(X, Y, C) :- ','(=<(X, Y), ','(is(X1, -(X, 2)), loop1(X1, Y, C))). Query: test_fun(g,g) ---------------------------------------- (1) PrologToIRSwTTransformerProof (SOUND) Transformed Prolog program to IRSwT according to method in Master Thesis of A. Weinert { "root": 4, "program": { "directives": [], "clauses": [ [ "(test_fun X Y)", "(loop1 X Y (0))" ], [ "(loop1 X Y C)", "(',' (>= X (0)) (',' (is X1 (+ X (1))) (',' (is Y1 (1)) (loop2 X1 Y1 C))))" ], [ "(loop1 X Y C)", "(< X (0))" ], [ "(loop2 X Y C)", "(',' (> X Y) (',' (is Y1 (+ Y (1))) (',' (is C1 (+ C (1))) (loop2 X Y1 C1))))" ], [ "(loop2 X Y C)", "(',' (=< X Y) (',' (is X1 (- X (2))) (loop1 X1 Y C)))" ] ] }, "graph": { "nodes": { "22": { "goal": [{ "clause": 2, "scope": 2, "term": "(loop1 T9 T10 (0))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T9", "T10" ], "free": [], "exprvars": [] } }, "23": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (>= T21 (0)) (',' (is X37 (+ T21 (1))) (',' (is X38 (1)) (loop2 X37 X38 (0)))))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T21"], "free": [ "X37", "X38" ], "exprvars": [] } }, "24": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "2892": { "goal": [{ "clause": 3, "scope": 3, "term": "(loop2 T23 T24 (0))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [ { "lhs": { "name": "T23", "type": "PlainIntegerVariable" }, "type": "PlainIntegerRelation", "rhs": { "arguments": [ { "name": "T21", "type": "PlainIntegerVariable" }, { "type": "PlainIntegerConstant", "value": "1" } ], "type": "PlainIntegerOperation", "operation": "+" }, "operation": "=" }, { "lhs": { "name": "T24", "type": "PlainIntegerVariable" }, "type": "PlainIntegerRelation", "rhs": { "type": "PlainIntegerConstant", "value": "1" }, "operation": "=" }, { "lhs": { "type": "PlainIntegerConstant", "value": "0" }, "type": "PlainIntegerRelation", "rhs": { "name": "T21", "type": "PlainIntegerVariable" }, "operation": "<=" } ] }, "ground": [ "T24", "T23" ], "free": [], "exprvars": [ "T24", "T23", "T21" ] } }, "2891": { "goal": [ { "clause": 3, "scope": 3, "term": "(loop2 T23 T24 (0))" }, { "clause": 4, "scope": 3, "term": "(loop2 T23 T24 (0))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [ { "lhs": { "name": "T23", "type": "PlainIntegerVariable" }, "type": "PlainIntegerRelation", "rhs": { "arguments": [ { "name": "T21", "type": "PlainIntegerVariable" }, { "type": "PlainIntegerConstant", "value": "1" } ], "type": "PlainIntegerOperation", "operation": "+" }, "operation": "=" }, { "lhs": { "name": "T24", "type": "PlainIntegerVariable" }, "type": "PlainIntegerRelation", "rhs": { "type": "PlainIntegerConstant", "value": "1" }, "operation": "=" }, { "lhs": { "type": "PlainIntegerConstant", "value": "0" }, "type": "PlainIntegerRelation", "rhs": { "name": "T21", "type": "PlainIntegerVariable" }, "operation": "<=" } ] }, 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"lhs": { "name": "T112", "type": "PlainIntegerVariable" }, "type": "PlainIntegerRelation", "rhs": { "type": "PlainIntegerConstant", "value": "0" }, "operation": "<" }] }, "ground": ["T112"], "free": [], "exprvars": ["T112"] } }, "2888": { "goal": [{ "clause": -1, "scope": -1, "term": "(loop2 T23 T24 (0))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [ { "lhs": { "name": "T23", "type": "PlainIntegerVariable" }, "type": "PlainIntegerRelation", "rhs": { "arguments": [ { "name": "T21", "type": "PlainIntegerVariable" }, { "type": "PlainIntegerConstant", "value": "1" } ], "type": "PlainIntegerOperation", "operation": "+" }, "operation": "=" }, { "lhs": { "name": "T24", "type": "PlainIntegerVariable" }, "type": "PlainIntegerRelation", "rhs": { "type": "PlainIntegerConstant", "value": "1" }, "operation": "=" }, { "lhs": { "type": "PlainIntegerConstant", "value": "0" }, "type": "PlainIntegerRelation", "rhs": { "name": "T21", "type": "PlainIntegerVariable" }, "operation": "<=" } ] }, "ground": [ "T24", "T23", "T21" ], "free": [ "X37", "X38" ], "exprvars": [ "T24", "T23", "T21" ] } }, "3337": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (=< T103 T104) (',' (is X152 (- T103 (2))) (loop1 X152 T104 (0))))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [ { "lhs": { "name": "T23", "type": "PlainIntegerVariable" }, "type": "PlainIntegerRelation", "rhs": { "arguments": [ { "name": "T21", "type": "PlainIntegerVariable" }, { "type": "PlainIntegerConstant", "value": "1" } ], "type": "PlainIntegerOperation", "operation": "+" }, "operation": "=" }, { "lhs": { "name": "T24", "type": "PlainIntegerVariable" }, "type": "PlainIntegerRelation", "rhs": { "type": "PlainIntegerConstant", "value": "1" }, "operation": "=" }, { "lhs": { "arguments": [ { "name": "T21", "type": "PlainIntegerVariable" }, { "type": "PlainIntegerConstant", "value": "1" } ], 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"lhs": { "name": "T103", "type": "PlainIntegerVariable" }, "type": "PlainIntegerRelation", "rhs": { "name": "T104", "type": "PlainIntegerVariable" }, "operation": "<=" } ] }, "ground": [ "T104", "T103" ], "free": ["X152"], "exprvars": [ "T104", "T103", "T24", "T23", "T21" ] } }, "6": { "goal": [{ "clause": 0, "scope": 1, "term": "(test_fun T1 T2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T1", "T2" ], "free": [], "exprvars": [] } }, "3169": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (is X63 (+ T36 (1))) (',' (is X64 (+ (0) (1))) (loop2 T35 X63 X64)))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [ { "lhs": { "name": "T23", "type": "PlainIntegerVariable" }, "type": "PlainIntegerRelation", "rhs": { "arguments": [ { "name": "T21", "type": "PlainIntegerVariable" }, { "type": "PlainIntegerConstant", "value": "1" } ], "type": 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"type": "PlainIntegerConstant", "value": "0" }, "type": "PlainIntegerRelation", "rhs": { "name": "T21", "type": "PlainIntegerVariable" }, "operation": "<=" }, { "lhs": { "name": "T21", "type": "PlainIntegerVariable" }, "type": "PlainIntegerRelation", "rhs": { "name": "T21", "type": "PlainIntegerVariable" }, "operation": "<=" } ] }, "ground": [ "T36", "T35" ], "free": [ "X63", "X64" ], "exprvars": [ "T36", "T24", "T35", "T23", "T21" ] } }, "2893": { "goal": [{ "clause": 4, "scope": 3, "term": "(loop2 T23 T24 (0))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [ { "lhs": { "name": "T23", "type": "PlainIntegerVariable" }, "type": "PlainIntegerRelation", "rhs": { "arguments": [ { "name": "T21", "type": "PlainIntegerVariable" }, { "type": "PlainIntegerConstant", "value": "1" } ], "type": "PlainIntegerOperation", "operation": "+" }, "operation": "=" }, { "lhs": { "name": "T24", "type": "PlainIntegerVariable" }, "type": "PlainIntegerRelation", "rhs": { "type": "PlainIntegerConstant", "value": "1" }, "operation": "=" }, { "lhs": { "type": "PlainIntegerConstant", "value": "0" }, "type": "PlainIntegerRelation", "rhs": { "name": "T21", "type": "PlainIntegerVariable" }, "operation": "<=" } ] }, "ground": [ "T24", "T23" ], "free": [], "exprvars": [ "T24", "T23", "T21" ] } }, "8": { "goal": [{ "clause": -1, "scope": -1, "term": "(loop1 T9 T10 (0))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T9", "T10" ], "free": [], "exprvars": [] } }, "20": { "goal": [ { "clause": 1, "scope": 2, "term": "(loop1 T9 T10 (0))" }, { "clause": 2, "scope": 2, "term": "(loop1 T9 T10 (0))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T9", "T10" ], "free": [], "exprvars": [] } }, "21": { "goal": [{ "clause": 1, "scope": 2, "term": "(loop1 T9 T10 (0))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T9", "T10" ], "free": [], "exprvars": [] } } }, "edges": [ { "from": 4, "to": 6, "label": "CASE" }, { "from": 6, "to": 8, "label": "ONLY EVAL with clause\ntest_fun(X11, X12) :- loop1(X11, X12, 0).\nand substitutionT1 -> T9,\nX11 -> T9,\nT2 -> T10,\nX12 -> 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\nloop1(X34, X35, X36) :- ','(>=(X34, 0), ','(is(X37, +(X34, 1)), ','(is(X38, 1), loop2(X37, X38, X36)))).\nand substitutionT9 -> T21,\nX34 -> T21,\nT10 -> T22,\nX35 -> T22,\nX36 -> 0" }, { "from": 22, "to": 3352, "label": "ONLY EVAL with clause\nloop1(X164, X165, X166) :- <(X164, 0).\nand substitutionT9 -> T112,\nX164 -> T112,\nT10 -> T113,\nX165 -> T113,\nX166 -> 0" }, { "from": 23, "to": 24, "label": "IS ERROR" }, { "from": 23, "to": 2747, "label": "ARITHCOMP SUCCESS" }, { "from": 23, "to": 2752, "label": "ARITHCOMP FAIL" }, { "from": 2747, "to": 2848, "label": "\nX37 -> T23" }, { "from": 2848, "to": 2888, "label": "\nX38 -> T24" }, { "from": 2888, "to": 2891, "label": "CASE" }, { "from": 2891, "to": 2892, "label": "PARALLEL" }, { "from": 2891, "to": 2893, "label": "PARALLEL" }, { "from": 2892, "to": 2895, "label": "ONLY EVAL with clause\nloop2(X60, X61, X62) :- ','(>(X60, X61), ','(is(X63, +(X61, 1)), ','(is(X64, +(X62, 1)), loop2(X60, X63, X64)))).\nand substitutionT23 -> T35,\nX60 -> T35,\nT24 -> T36,\nX61 -> T36,\nX62 -> 0" }, { "from": 2893, "to": 3337, "label": "ONLY EVAL with clause\nloop2(X149, X150, X151) :- ','(=<(X149, X150), ','(is(X152, -(X149, 2)), loop1(X152, X150, X151))).\nand substitutionT23 -> T103,\nX149 -> T103,\nT24 -> T104,\nX150 -> T104,\nX151 -> 0" }, { "from": 2895, "to": 3169, "label": "ARITHCOMP SUCCESS" }, { "from": 2895, "to": 3170, "label": "ARITHCOMP FAIL" }, { "from": 3169, "to": 3171, "label": "\nX63 -> T37" }, { "from": 3171, "to": 3172, "label": "\nX64 -> T38" }, { "from": 3172, "to": 3173, "label": "CASE" }, { "from": 3173, "to": 3174, "label": "PARALLEL" }, { "from": 3173, "to": 3175, "label": "PARALLEL" }, { "from": 3174, "to": 3176, "label": "ONLY EVAL with clause\nloop2(X86, X87, X88) :- ','(>(X86, X87), ','(is(X89, +(X87, 1)), ','(is(X90, +(X88, 1)), loop2(X86, X89, X90)))).\nand substitutionT35 -> T54,\nX86 -> T54,\nT37 -> T55,\nX87 -> T55,\nT38 -> T56,\nX88 -> T56" }, { "from": 3175, "to": 3279, "label": "ONLY EVAL with clause\nloop2(X103, X104, X105) :- ','(=<(X103, X104), ','(is(X106, -(X103, 2)), loop1(X106, X104, X105))).\nand substitutionT35 -> T68,\nX103 -> T68,\nT37 -> T69,\nX104 -> T69,\nT38 -> T70,\nX105 -> T70" }, { "from": 3176, "to": 3248, "label": "ARITHCOMP SUCCESS" }, { "from": 3176, "to": 3249, "label": "ARITHCOMP FAIL" }, { "from": 3248, "to": 3250, "label": "\nX89 -> T57" }, { "from": 3250, "to": 3251, "label": "\nX90 -> T58" }, { "from": 3251, "to": 3172, "label": "INSTANCE with matching:\nT35 -> T54\nT37 -> T57\nT38 -> T58\nX63 -> X89\nX64 -> X90" }, { "from": 3279, "to": 3326, "label": "ARITHCOMP SUCCESS" }, { "from": 3279, "to": 3327, "label": "ARITHCOMP FAIL" }, { "from": 3326, "to": 3328, "label": "\nX106 -> T71" }, { "from": 3328, "to": 3329, "label": "CASE" }, { "from": 3329, "to": 3330, "label": "PARALLEL" }, { "from": 3329, "to": 3331, "label": "PARALLEL" }, { "from": 3330, "to": 3332, "label": "ONLY EVAL with clause\nloop1(X128, X129, X130) :- ','(>=(X128, 0), ','(is(X131, +(X128, 1)), ','(is(X132, 1), loop2(X131, X132, X130)))).\nand substitutionT71 -> T87,\nX128 -> T87,\nT69 -> T88,\nX129 -> T88,\nT70 -> T89,\nX130 -> T89" }, { "from": 3331, "to": 3334, "label": "ONLY EVAL with clause\nloop1(X139, X140, X141) :- <(X139, 0).\nand substitutionT71 -> T96,\nX139 -> T96,\nT69 -> T97,\nX140 -> T97,\nT70 -> T98,\nX141 -> T98" }, { "from": 3337, "to": 3346, "label": "ARITHCOMP SUCCESS" }, { "from": 3337, "to": 3347, "label": "ARITHCOMP FAIL" }, { "from": 3346, "to": 3351, "label": "\nX152 -> T105" }, { "from": 3351, "to": 8, "label": "INSTANCE with matching:\nT9 -> T105\nT10 -> T104" }, { "from": 3352, "to": 3353, "label": "IS ERROR" }, { "from": 3352, "to": 3361, "label": "ARITHCOMP SUCCESS" }, { "from": 3352, "to": 3362, "label": "ARITHCOMP FAIL" }, { "from": 3361, "to": 3363, "label": "SUCCESS" } ], "type": "Graph" } } ---------------------------------------- (2) Complex Obligation (AND) ---------------------------------------- (3) Obligation: Rules: f3248_out(T55, T56, T54) -> f3176_out(T54, T55, T56) :|: T54 > T55 f3176_in(x, x1, x2) -> f3248_in(x1, x2, x) :|: x > x1 f3249_out(x3, x4, x5) -> f3176_out(x5, x4, x3) :|: x5 <= x4 f3176_in(x6, x7, x8) -> f3249_in(x8, x7, x6) :|: x6 <= x7 f3173_in(T35, T37, T38) -> f3175_in(T35, T37, T38) :|: TRUE f3173_in(x9, x10, x11) -> f3174_in(x9, x10, x11) :|: TRUE f3174_out(x12, x13, x14) -> f3173_out(x12, x13, x14) :|: TRUE f3175_out(x15, x16, x17) -> f3173_out(x15, x16, x17) :|: TRUE f3174_in(x18, x19, x20) -> f3176_in(x18, x19, x20) :|: TRUE f3176_out(x21, x22, x23) -> f3174_out(x21, x22, x23) :|: TRUE f3248_in(x24, x25, x26) -> f3250_in(x25, x26, x27, x24) :|: x27 = x24 + 1 f3250_out(x28, x29, x30, x31) -> f3248_out(x31, x28, x29) :|: TRUE f3251_in(x32, x33, x34, x35, x36) -> f3172_in(x32, x33, x34, x37) :|: TRUE f3172_out(x38, x39, x40, x41) -> f3251_out(x38, x39, x40, x42, x43) :|: TRUE f3173_out(x44, x45, x46) -> f3172_out(x44, x45, x46, x47) :|: TRUE f3172_in(x48, x49, x50, x51) -> f3173_in(x48, x49, x50) :|: TRUE f3250_in(x52, x53, x54, x55) -> f3251_in(x53, x54, x56, x52, x55) :|: x56 = x52 + 1 f3251_out(x57, x58, x59, x60, x61) -> f3250_out(x60, x57, x58, x61) :|: TRUE f4_in(T1, T2) -> f6_in(T1, T2) :|: TRUE f6_out(x62, x63) -> f4_out(x62, x63) :|: TRUE f6_in(T9, T10) -> f8_in(T9, T10) :|: TRUE f8_out(x64, x65) -> f6_out(x64, x65) :|: TRUE f8_in(x66, x67) -> f20_in(x66, x67) :|: TRUE f20_out(x68, x69) -> f8_out(x68, x69) :|: TRUE f21_out(x70, x71) -> f20_out(x70, x71) :|: TRUE f22_out(x72, x73) -> f20_out(x72, x73) :|: TRUE f20_in(x74, x75) -> f21_in(x74, x75) :|: TRUE f20_in(x76, x77) -> f22_in(x76, x77) :|: TRUE f21_in(T21, T22) -> f23_in(T21) :|: TRUE f23_out(x78) -> f21_out(x78, x79) :|: TRUE f23_in(x80) -> f24_in :|: TRUE f24_out -> f23_out(x81) :|: TRUE f23_in(x82) -> f2752_in(x82) :|: x82 < 0 f2752_out(x83) -> f23_out(x83) :|: x83 < 0 f23_in(x84) -> f2747_in(x84) :|: x84 >= 0 f2747_out(x85) -> f23_out(x85) :|: x85 >= 0 f2747_in(x86) -> f2848_in(x87, x86) :|: x87 = x86 + 1 f2848_out(x88, x89) -> f2747_out(x89) :|: TRUE f2888_out(x90, x91, x92) -> f2848_out(x90, x92) :|: TRUE f2848_in(x93, x94) -> f2888_in(x93, x95, x94) :|: x95 = 1 f2888_in(x96, x97, x98) -> f2891_in(x96, x97) :|: TRUE f2891_out(x99, x100) -> f2888_out(x99, x100, x101) :|: TRUE f2891_in(T23, T24) -> f2892_in(T23, T24) :|: TRUE f2891_in(x102, x103) -> f2893_in(x102, x103) :|: TRUE f2893_out(x104, x105) -> f2891_out(x104, x105) :|: TRUE f2892_out(x106, x107) -> f2891_out(x106, x107) :|: TRUE f2892_in(x108, x109) -> f2895_in(x108, x109) :|: TRUE f2895_out(x110, x111) -> f2892_out(x110, x111) :|: TRUE f2895_in(x112, x113) -> f3170_in(x113, x112) :|: x112 <= x113 f3169_out(x114, x115) -> f2895_out(x115, x114) :|: x115 > x114 f2895_in(x116, x117) -> f3169_in(x117, x116) :|: x116 > x117 f3170_out(x118, x119) -> f2895_out(x119, x118) :|: x119 <= x118 f3169_in(x120, x121) -> f3171_in(x121, x122, x120) :|: x122 = x120 + 1 f3171_out(x123, x124, x125) -> f3169_out(x125, x123) :|: TRUE f3172_out(x126, x127, x128, x129) -> f3171_out(x126, x127, x129) :|: TRUE f3171_in(x130, x131, x132) -> f3172_in(x130, x131, x133, x132) :|: x133 = 0 + 1 Start term: f4_in(T1, T2) ---------------------------------------- (4) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: intTRSProblem: f3176_in(x, x1, x2) -> f3248_in(x1, x2, x) :|: x > x1 f3173_in(x9, x10, x11) -> f3174_in(x9, x10, x11) :|: TRUE f3174_in(x18, x19, x20) -> f3176_in(x18, x19, x20) :|: TRUE f3248_in(x24, x25, x26) -> f3250_in(x25, x26, x27, x24) :|: x27 = x24 + 1 f3251_in(x32, x33, x34, x35, x36) -> f3172_in(x32, x33, x34, x37) :|: TRUE f3172_in(x48, x49, x50, x51) -> f3173_in(x48, x49, x50) :|: TRUE f3250_in(x52, x53, x54, x55) -> f3251_in(x53, x54, x56, x52, x55) :|: x56 = x52 + 1 ---------------------------------------- (5) Obligation: Rules: f3176_in(x, x1, x2) -> f3248_in(x1, x2, x) :|: x > x1 f3173_in(x9, x10, x11) -> f3174_in(x9, x10, x11) :|: TRUE f3174_in(x18, x19, x20) -> f3176_in(x18, x19, x20) :|: TRUE f3248_in(x24, x25, x26) -> f3250_in(x25, x26, x27, x24) :|: x27 = x24 + 1 f3251_in(x32, x33, x34, x35, x36) -> f3172_in(x32, x33, x34, x37) :|: TRUE f3172_in(x48, x49, x50, x51) -> f3173_in(x48, x49, x50) :|: TRUE f3250_in(x52, x53, x54, x55) -> f3251_in(x53, x54, x56, x52, x55) :|: x56 = x52 + 1 ---------------------------------------- (6) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (7) Obligation: Rules: f3251_in(x32:0, x33:0, x34:0, x35:0, x36:0) -> f3251_in(x32:0, x33:0 + 1, x34:0 + 1, x34:0, x33:0) :|: x33:0 < x32:0 ---------------------------------------- (8) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (9) Obligation: Rules: f3251_in(x32:0, x33:0, x34:0, x35:0, x36:0) -> f3251_in(x32:0, arith, arith1, x34:0, x33:0) :|: x33:0 < x32:0 && arith = x33:0 + 1 && arith1 = x34:0 + 1 ---------------------------------------- (10) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) f3251_in(x32:0, x33:0, x34:0, x35:0, x36:0) -> f3251_in(x32:0, arith, arith1, x34:0, x33:0) :|: x33:0 < x32:0 && arith = x33:0 + 1 && arith1 = x34:0 + 1 Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (11) Obligation: Termination digraph: Nodes: (1) f3251_in(x32:0, x33:0, x34:0, x35:0, x36:0) -> f3251_in(x32:0, arith, arith1, x34:0, x33:0) :|: x33:0 < x32:0 && arith = x33:0 + 1 && arith1 = x34:0 + 1 Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (12) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (13) Obligation: Rules: f3251_in(x32:0:0, x33:0:0, x34:0:0, x35:0:0, x36:0:0) -> f3251_in(x32:0:0, x33:0:0 + 1, x34:0:0 + 1, x34:0:0, x33:0:0) :|: x33:0:0 < x32:0:0 ---------------------------------------- (14) IntTRSUnneededArgumentFilterProof (EQUIVALENT) Some arguments are removed because they cannot influence termination. We removed arguments according to the following replacements: f3251_in(x1, x2, x3, x4, x5) -> f3251_in(x1, x2) ---------------------------------------- (15) Obligation: Rules: f3251_in(x32:0:0, x33:0:0) -> f3251_in(x32:0:0, x33:0:0 + 1) :|: x33:0:0 < x32:0:0 ---------------------------------------- (16) TempFilterProof (SOUND) Used the following sort dictionary for filtering: f3251_in(INTEGER, INTEGER) Replaced non-predefined constructor symbols by 0. ---------------------------------------- (17) Obligation: Rules: f3251_in(x32:0:0, x33:0:0) -> f3251_in(x32:0:0, c) :|: c = x33:0:0 + 1 && x33:0:0 < x32:0:0 ---------------------------------------- (18) PolynomialOrderProcessor (EQUIVALENT) Found the following polynomial interpretation: [f3251_in(x, x1)] = x - x1 The following rules are decreasing: f3251_in(x32:0:0, x33:0:0) -> f3251_in(x32:0:0, c) :|: c = x33:0:0 + 1 && x33:0:0 < x32:0:0 The following rules are bounded: f3251_in(x32:0:0, x33:0:0) -> f3251_in(x32:0:0, c) :|: c = x33:0:0 + 1 && x33:0:0 < x32:0:0 ---------------------------------------- (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 f3346_in(T103, T104) -> f3351_in(T105, T104, T103) :|: T105 = T103 - 2 f3351_out(x10, x11, x12) -> f3346_out(x12, x11) :|: TRUE f2747_in(T21) -> f2848_in(T23, T21) :|: T23 = T21 + 1 f2848_out(x13, x14) -> f2747_out(x14) :|: TRUE f2891_in(x15, x16) -> f2892_in(x15, x16) :|: TRUE f2891_in(x17, x18) -> f2893_in(x17, x18) :|: TRUE f2893_out(x19, x20) -> f2891_out(x19, x20) :|: TRUE f2892_out(x21, x22) -> f2891_out(x21, x22) :|: TRUE f3346_out(x23, x24) -> f3337_out(x23, x24) :|: x23 <= x24 f3337_in(x25, x26) -> f3346_in(x25, x26) :|: x25 <= x26 f3347_out(x27, x28) -> f3337_out(x28, x27) :|: x28 > x27 f3337_in(x29, x30) -> f3347_in(x30, x29) :|: x29 > x30 f2888_out(x31, x32, x33) -> f2848_out(x31, x33) :|: TRUE f2848_in(x34, x35) -> f2888_in(x34, x36, x35) :|: x36 = 1 f2893_in(x37, x38) -> f3337_in(x37, x38) :|: TRUE f3337_out(x39, x40) -> f2893_out(x39, x40) :|: TRUE f23_in(x41) -> f24_in :|: TRUE f24_out -> f23_out(x42) :|: TRUE f23_in(x43) -> f2752_in(x43) :|: x43 < 0 f2752_out(x44) -> f23_out(x44) :|: x44 < 0 f23_in(x45) -> f2747_in(x45) :|: x45 >= 0 f2747_out(x46) -> f23_out(x46) :|: x46 >= 0 f3351_in(x47, x48, x49) -> f8_in(x47, x48) :|: TRUE f8_out(x50, x51) -> f3351_out(x50, x51, x52) :|: TRUE f2888_in(x53, x54, x55) -> f2891_in(x53, x54) :|: TRUE f2891_out(x56, x57) -> f2888_out(x56, x57, x58) :|: TRUE f21_in(x59, x60) -> f23_in(x59) :|: TRUE f23_out(x61) -> f21_out(x61, x62) :|: TRUE f4_in(T1, T2) -> f6_in(T1, T2) :|: TRUE f6_out(x63, x64) -> f4_out(x63, x64) :|: TRUE f6_in(x65, x66) -> f8_in(x65, x66) :|: TRUE f8_out(x67, x68) -> f6_out(x67, x68) :|: TRUE Start term: f4_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 f3346_in(T103, T104) -> f3351_in(T105, T104, T103) :|: T105 = T103 - 2 f2747_in(T21) -> f2848_in(T23, T21) :|: T23 = T21 + 1 f2891_in(x17, x18) -> f2893_in(x17, x18) :|: TRUE f3337_in(x25, x26) -> f3346_in(x25, x26) :|: x25 <= x26 f2848_in(x34, x35) -> f2888_in(x34, x36, x35) :|: x36 = 1 f2893_in(x37, x38) -> f3337_in(x37, x38) :|: TRUE f23_in(x45) -> f2747_in(x45) :|: x45 >= 0 f3351_in(x47, x48, x49) -> f8_in(x47, x48) :|: TRUE f2888_in(x53, x54, x55) -> f2891_in(x53, x54) :|: TRUE f21_in(x59, x60) -> f23_in(x59) :|: TRUE ---------------------------------------- (22) Obligation: Rules: f8_in(T9, T10) -> f20_in(T9, T10) :|: TRUE f20_in(x6, x7) -> f21_in(x6, x7) :|: TRUE f3346_in(T103, T104) -> f3351_in(T105, T104, T103) :|: T105 = T103 - 2 f2747_in(T21) -> f2848_in(T23, T21) :|: T23 = T21 + 1 f2891_in(x17, x18) -> f2893_in(x17, x18) :|: TRUE f3337_in(x25, x26) -> f3346_in(x25, x26) :|: x25 <= x26 f2848_in(x34, x35) -> f2888_in(x34, x36, x35) :|: x36 = 1 f2893_in(x37, x38) -> f3337_in(x37, x38) :|: TRUE f23_in(x45) -> f2747_in(x45) :|: x45 >= 0 f3351_in(x47, x48, x49) -> f8_in(x47, x48) :|: TRUE f2888_in(x53, x54, x55) -> f2891_in(x53, x54) :|: TRUE f21_in(x59, x60) -> f23_in(x59) :|: TRUE ---------------------------------------- (23) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (24) Obligation: Rules: f23_in(x45:0) -> f23_in(x45:0 - 1) :|: x45:0 > -1 && x45:0 < 1 ---------------------------------------- (25) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (26) Obligation: Rules: f23_in(x45:0) -> f23_in(arith) :|: x45:0 > -1 && x45:0 < 1 && arith = x45:0 - 1 ---------------------------------------- (27) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) f23_in(x45:0) -> f23_in(arith) :|: x45:0 > -1 && x45:0 < 1 && arith = x45:0 - 1 No arcs! This digraph is fully evaluated! ---------------------------------------- (28) TRUE