/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern test_fun(g,g,g) w.r.t. the given Prolog program could successfully be proven: (0) Prolog (1) PrologToIRSwTTransformerProof [SOUND, 55 ms] (2) IRSwT (3) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (4) IRSwT (5) IntTRSCompressionProof [EQUIVALENT, 37 ms] (6) IRSwT (7) IRSFormatTransformerProof [EQUIVALENT, 0 ms] (8) IRSwT (9) IRSwTTerminationDigraphProof [EQUIVALENT, 36 ms] (10) IRSwT (11) IntTRSCompressionProof [EQUIVALENT, 0 ms] (12) IRSwT (13) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms] (14) IRSwT (15) TempFilterProof [SOUND, 44 ms] (16) IntTRS (17) RankingReductionPairProof [EQUIVALENT, 16 ms] (18) YES ---------------------------------------- (0) Obligation: Clauses: test_fun(X, Y, Z) :- loop(X, Y, Z, 0). loop(X, Y, Z, C) :- ','(>(X, +(Y, Z)), ','(is(Y1, +(Y, 1)), ','(is(Z1, +(Z, 1)), ','(is(C1, +(C, 1)), loop(X, Y1, Z1, C1))))). loop(X, Y, Z, C) :- =<(X, +(Y, Z)). Query: test_fun(g,g,g) ---------------------------------------- (1) PrologToIRSwTTransformerProof (SOUND) Transformed Prolog program to IRSwT according to method in Master Thesis of A. 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"exprvars": [ "T31", "T36", "T35", "T34", "T33", "T32" ] } }, "2208": { "goal": [{ "clause": 1, "scope": 3, "term": "(loop T31 T34 T35 T36)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [ { "lhs": { "name": "T34", "type": "PlainIntegerVariable" }, "type": "PlainIntegerRelation", "rhs": { "arguments": [ { "name": "T32", "type": "PlainIntegerVariable" }, { "type": "PlainIntegerConstant", "value": "1" } ], "type": "PlainIntegerOperation", "operation": "+" }, "operation": "=" }, { "lhs": { "name": "T35", "type": "PlainIntegerVariable" }, "type": "PlainIntegerRelation", "rhs": { "arguments": [ { "name": "T33", "type": "PlainIntegerVariable" }, { "type": "PlainIntegerConstant", "value": "1" } ], "type": "PlainIntegerOperation", "operation": "+" }, "operation": "=" }, { "lhs": { "name": "T36", "type": "PlainIntegerVariable" }, "type": "PlainIntegerRelation", "rhs": { "arguments": [ { "type": "PlainIntegerConstant", "value": "0" }, { "type": "PlainIntegerConstant", "value": "1" } ], "type": "PlainIntegerOperation", "operation": "+" }, "operation": "=" }, { "lhs": { "arguments": [ { "name": "T32", "type": "PlainIntegerVariable" }, { "name": "T33", "type": "PlainIntegerVariable" } ], "type": "PlainIntegerOperation", "operation": "+" }, "type": "PlainIntegerRelation", "rhs": { "name": "T31", "type": "PlainIntegerVariable" }, "operation": "<" } ] }, "ground": [ "T31", "T36", "T35", "T34" ], "free": [], "exprvars": [ "T31", "T36", "T35", "T34", "T33", "T32" ] } }, "13": { "goal": [ { "clause": 1, "scope": 2, "term": "(loop T13 T14 T15 (0))" }, { "clause": 2, "scope": 2, "term": "(loop T13 T14 T15 (0))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T13", "T14", "T15" ], "free": [], "exprvars": [] } }, "16": { "goal": [{ "clause": 1, "scope": 2, "term": "(loop T13 T14 T15 (0))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T13", "T14", "T15" ], "free": [], "exprvars": [] } }, "17": { "goal": [{ "clause": 2, "scope": 2, "term": "(loop T13 T14 T15 (0))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T13", "T14", "T15" ], "free": [], "exprvars": [] } }, "19": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (> T31 (+ T32 T33)) (',' (is X50 (+ T32 (1))) (',' (is X51 (+ T33 (1))) (',' (is X52 (+ (0) (1))) (loop T31 X50 X51 X52)))))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T31", "T32", "T33" ], "free": [ "X50", "X51", "X52" ], "exprvars": [] } }, "2287": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [{ "lhs": { "name": "T86", "type": "PlainIntegerVariable" }, "type": "PlainIntegerRelation", "rhs": { "arguments": [ { "name": "T87", "type": "PlainIntegerVariable" }, { "name": "T88", "type": "PlainIntegerVariable" } ], "type": "PlainIntegerOperation", "operation": "+" }, "operation": "<=" }] }, "ground": [], "free": [], "exprvars": [ "T86", "T88", "T87" ] } }, "2286": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [{ "lhs": { "name": "T86", "type": "PlainIntegerVariable" }, "type": "PlainIntegerRelation", "rhs": { "arguments": [ { "name": "T87", "type": "PlainIntegerVariable" }, { "name": "T88", "type": "PlainIntegerVariable" } ], "type": "PlainIntegerOperation", "operation": "+" }, "operation": ">" }] }, "ground": [ "T86", "T88", "T87" ], "free": [], "exprvars": [ "T86", "T88", "T87" ] } }, "2285": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [{ "lhs": { "name": "T86", "type": "PlainIntegerVariable" }, "type": "PlainIntegerRelation", "rhs": { "arguments": [ { "name": "T87", "type": "PlainIntegerVariable" }, { "name": "T88", "type": "PlainIntegerVariable" } ], "type": "PlainIntegerOperation", "operation": "+" }, "operation": "<=" }] }, "ground": [ "T86", "T88", "T87" ], "free": [], "exprvars": [ "T86", "T88", "T87" ] } }, "2": { "goal": [{ "clause": -1, "scope": -1, "term": "(test_fun T1 T2 T3)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T1", "T2", "T3" ], "free": [], "exprvars": [] } }, "5": { "goal": [{ "clause": 0, "scope": 1, "term": "(test_fun T1 T2 T3)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T1", "T2", "T3" ], "free": [], "exprvars": [] } }, "2256": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "2255": { "goal": [{ "clause": -1, "scope": -1, "term": "(=< T86 (+ T87 T88))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T86", "T87", "T88" ], "free": [], "exprvars": [] } }, "9": { "goal": [{ "clause": -1, "scope": -1, "term": "(loop T13 T14 T15 (0))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T13", "T14", "T15" ], "free": [], "exprvars": [] } }, "20": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } } }, "edges": [ { "from": 2, "to": 5, "label": "CASE" }, { "from": 5, "to": 9, "label": "ONLY EVAL with clause\ntest_fun(X14, X15, X16) :- loop(X14, X15, X16, 0).\nand substitutionT1 -> T13,\nX14 -> T13,\nT2 -> T14,\nX15 -> T14,\nT3 -> T15,\nX16 -> T15" }, { "from": 9, "to": 13, "label": "CASE" }, { "from": 13, "to": 16, "label": "PARALLEL" }, { "from": 13, "to": 17, "label": "PARALLEL" }, { "from": 16, "to": 19, "label": "ONLY EVAL with clause\nloop(X46, X47, X48, X49) :- ','(>(X46, +(X47, X48)), ','(is(X50, +(X47, 1)), ','(is(X51, +(X48, 1)), ','(is(X52, +(X49, 1)), loop(X46, X50, X51, X52))))).\nand substitutionT13 -> T31,\nX46 -> T31,\nT14 -> T32,\nX47 -> T32,\nT15 -> T33,\nX48 -> T33,\nX49 -> 0" }, { "from": 17, "to": 2255, "label": "ONLY EVAL with clause\nloop(X116, X117, X118, X119) :- =<(X116, +(X117, X118)).\nand substitutionT13 -> T86,\nX116 -> T86,\nT14 -> T87,\nX117 -> T87,\nT15 -> T88,\nX118 -> T88,\nX119 -> 0" }, { "from": 19, "to": 20, "label": "IS ERROR" }, { "from": 19, "to": 2150, "label": "ARITHCOMP SUCCESS" }, { "from": 19, "to": 2151, "label": "ARITHCOMP FAIL" }, { "from": 2150, "to": 2204, "label": "\nX50 -> T34" }, { "from": 2204, "to": 2205, "label": "\nX51 -> T35" }, { "from": 2205, "to": 2206, "label": "\nX52 -> T36" }, { "from": 2206, "to": 2207, "label": "CASE" }, { "from": 2207, "to": 2208, "label": "PARALLEL" }, { "from": 2207, "to": 2209, "label": "PARALLEL" }, { "from": 2208, "to": 2210, "label": "ONLY EVAL with clause\nloop(X82, X83, X84, X85) :- ','(>(X82, +(X83, X84)), ','(is(X86, +(X83, 1)), ','(is(X87, +(X84, 1)), ','(is(X88, +(X85, 1)), loop(X82, X86, X87, X88))))).\nand substitutionT31 -> T57,\nX82 -> T57,\nT34 -> T58,\nX83 -> T58,\nT35 -> T59,\nX84 -> T59,\nT36 -> T60,\nX85 -> T60" }, { "from": 2209, "to": 2248, "label": "ONLY EVAL with clause\nloop(X104, X105, X106, X107) :- =<(X104, +(X105, X106)).\nand substitutionT31 -> T76,\nX104 -> T76,\nT34 -> T77,\nX105 -> T77,\nT35 -> T78,\nX106 -> T78,\nT36 -> T79,\nX107 -> T79" }, { "from": 2210, "to": 2227, "label": "ARITHCOMP SUCCESS" }, { "from": 2210, "to": 2228, "label": "ARITHCOMP FAIL" }, { "from": 2227, "to": 2229, "label": "\nX86 -> T61" }, { "from": 2229, "to": 2230, "label": "\nX87 -> T62" }, { "from": 2230, "to": 2231, "label": "\nX88 -> T63" }, { "from": 2231, "to": 2206, "label": "INSTANCE with matching:\nT31 -> T57\nT34 -> T61\nT35 -> T62\nT36 -> T63\nX50 -> X86\nX51 -> X87\nX52 -> X88" }, { "from": 2248, "to": 2252, "label": "ARITHCOMP SUCCESS" }, { "from": 2248, "to": 2253, "label": "ARITHCOMP FAIL" }, { "from": 2252, "to": 2254, "label": "SUCCESS" }, { "from": 2255, "to": 2256, "label": "IS ERROR" }, { "from": 2255, "to": 2285, "label": "ARITHCOMP SUCCESS" }, { "from": 2255, "to": 2286, "label": "ARITHCOMP FAIL" }, { "from": 2285, "to": 2287, "label": "SUCCESS" } ], "type": "Graph" } } ---------------------------------------- (2) Obligation: Rules: f2210_in(T57, T58, T59, T60) -> f2228_in(T58, T57, T60, T59) :|: T57 <= T58 + T59 f2227_out(x, x1, x2, x3) -> f2210_out(x3, x, x1, x2) :|: x3 > x + x1 f2210_in(x4, x5, x6, x7) -> f2227_in(x5, x6, x7, x4) :|: x4 > x5 + x6 f2228_out(x8, x9, x10, x11) -> f2210_out(x9, x8, x11, x10) :|: x9 <= x8 + x11 f2206_out(x12, x13, x14, x15, x16, x17) -> f2231_out(x12, x13, x14, x15, x18, x19, x20) :|: TRUE f2231_in(x21, x22, x23, x24, x25, x26, x27) -> f2206_in(x21, x22, x23, x24, x28, x29) :|: TRUE f2207_in(T31, T34, T35, T36) -> f2208_in(T31, T34, T35, T36) :|: TRUE f2208_out(x30, x31, x32, x33) -> f2207_out(x30, x31, x32, x33) :|: TRUE f2209_out(x34, x35, x36, x37) -> f2207_out(x34, x35, x36, x37) :|: TRUE f2207_in(x38, x39, x40, x41) -> f2209_in(x38, x39, x40, x41) :|: TRUE f2208_in(x42, x43, x44, x45) -> f2210_in(x42, x43, x44, x45) :|: TRUE f2210_out(x46, x47, x48, x49) -> f2208_out(x46, x47, x48, x49) :|: TRUE f2231_out(x50, x51, x52, x53, x54, x55, x56) -> f2230_out(x55, x50, x51, x52, x54, x56) :|: TRUE f2230_in(x57, x58, x59, x60, x61, x62) -> f2231_in(x58, x59, x60, x63, x61, x57, x62) :|: x63 = x57 + 1 f2227_in(x64, x65, x66, x67) -> f2229_in(x65, x66, x67, x68, x64) :|: x68 = x64 + 1 f2229_out(x69, x70, x71, x72, x73) -> f2227_out(x73, x69, x70, x71) :|: TRUE f2230_out(x74, x75, x76, x77, x78, x79) -> f2229_out(x79, x74, x75, x76, x78) :|: TRUE f2229_in(x80, x81, x82, x83, x84) -> f2230_in(x81, x82, x83, x85, x84, x80) :|: x85 = x80 + 1 f2207_out(x86, x87, x88, x89) -> f2206_out(x86, x87, x88, x89, x90, x91) :|: TRUE f2206_in(x92, x93, x94, x95, x96, x97) -> f2207_in(x92, x93, x94, x95) :|: TRUE f5_out(T1, T2, T3) -> f2_out(T1, T2, T3) :|: TRUE f2_in(x98, x99, x100) -> f5_in(x98, x99, x100) :|: TRUE f9_out(T13, T14, T15) -> f5_out(T13, T14, T15) :|: TRUE f5_in(x101, x102, x103) -> f9_in(x101, x102, x103) :|: TRUE f9_in(x104, x105, x106) -> f13_in(x104, x105, x106) :|: TRUE f13_out(x107, x108, x109) -> f9_out(x107, x108, x109) :|: TRUE f16_out(x110, x111, x112) -> f13_out(x110, x111, x112) :|: TRUE f17_out(x113, x114, x115) -> f13_out(x113, x114, x115) :|: TRUE f13_in(x116, x117, x118) -> f16_in(x116, x117, x118) :|: TRUE f13_in(x119, x120, x121) -> f17_in(x119, x120, x121) :|: TRUE f19_out(x122, x123, x124) -> f16_out(x122, x123, x124) :|: TRUE f16_in(x125, x126, x127) -> f19_in(x125, x126, x127) :|: TRUE f20_out -> f19_out(x128, x129, x130) :|: TRUE f19_in(x131, x132, x133) -> f20_in :|: TRUE f19_in(x134, x135, x136) -> f2151_in(x134, x136, x135) :|: x134 <= x135 + x136 f19_in(x137, x138, x139) -> f2150_in(x138, x139, x137) :|: x137 > x138 + x139 f2150_out(x140, x141, x142) -> f19_out(x142, x140, x141) :|: x142 > x140 + x141 f2151_out(x143, x144, x145) -> f19_out(x143, x145, x144) :|: x143 <= x145 + x144 f2204_out(x146, x147, x148, x149) -> f2150_out(x149, x146, x147) :|: TRUE f2150_in(x150, x151, x152) -> f2204_in(x151, x152, x153, x150) :|: x153 = x150 + 1 f2204_in(x154, x155, x156, x157) -> f2205_in(x155, x156, x158, x154, x157) :|: x158 = x154 + 1 f2205_out(x159, x160, x161, x162, x163) -> f2204_out(x162, x159, x160, x163) :|: TRUE f2205_in(x164, x165, x166, x167, x168) -> f2206_in(x164, x165, x166, x169, x167, x168) :|: x169 = 0 + 1 f2206_out(x170, x171, x172, x173, x174, x175) -> f2205_out(x170, x171, x172, x174, x175) :|: TRUE Start term: f2_in(T1, T2, T3) ---------------------------------------- (3) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: intTRSProblem: f2210_in(x4, x5, x6, x7) -> f2227_in(x5, x6, x7, x4) :|: x4 > x5 + x6 f2231_in(x21, x22, x23, x24, x25, x26, x27) -> f2206_in(x21, x22, x23, x24, x28, x29) :|: TRUE f2207_in(T31, T34, T35, T36) -> f2208_in(T31, T34, T35, T36) :|: TRUE f2208_in(x42, x43, x44, x45) -> f2210_in(x42, x43, x44, x45) :|: TRUE f2230_in(x57, x58, x59, x60, x61, x62) -> f2231_in(x58, x59, x60, x63, x61, x57, x62) :|: x63 = x57 + 1 f2227_in(x64, x65, x66, x67) -> f2229_in(x65, x66, x67, x68, x64) :|: x68 = x64 + 1 f2229_in(x80, x81, x82, x83, x84) -> f2230_in(x81, x82, x83, x85, x84, x80) :|: x85 = x80 + 1 f2206_in(x92, x93, x94, x95, x96, x97) -> f2207_in(x92, x93, x94, x95) :|: TRUE ---------------------------------------- (4) Obligation: Rules: f2210_in(x4, x5, x6, x7) -> f2227_in(x5, x6, x7, x4) :|: x4 > x5 + x6 f2231_in(x21, x22, x23, x24, x25, x26, x27) -> f2206_in(x21, x22, x23, x24, x28, x29) :|: TRUE f2207_in(T31, T34, T35, T36) -> f2208_in(T31, T34, T35, T36) :|: TRUE f2208_in(x42, x43, x44, x45) -> f2210_in(x42, x43, x44, x45) :|: TRUE f2230_in(x57, x58, x59, x60, x61, x62) -> f2231_in(x58, x59, x60, x63, x61, x57, x62) :|: x63 = x57 + 1 f2227_in(x64, x65, x66, x67) -> f2229_in(x65, x66, x67, x68, x64) :|: x68 = x64 + 1 f2229_in(x80, x81, x82, x83, x84) -> f2230_in(x81, x82, x83, x85, x84, x80) :|: x85 = x80 + 1 f2206_in(x92, x93, x94, x95, x96, x97) -> f2207_in(x92, x93, x94, x95) :|: TRUE ---------------------------------------- (5) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (6) Obligation: Rules: f2207_in(T31:0, T34:0, T35:0, T36:0) -> f2207_in(T31:0, T34:0 + 1, T35:0 + 1, T36:0 + 1) :|: T34:0 + T35:0 < T31:0 ---------------------------------------- (7) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (8) Obligation: Rules: f2207_in(T31:0, T34:0, T35:0, T36:0) -> f2207_in(T31:0, arith, arith1, arith2) :|: T34:0 + T35:0 < T31:0 && arith = T34:0 + 1 && arith1 = T35:0 + 1 && arith2 = T36:0 + 1 ---------------------------------------- (9) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) f2207_in(T31:0, T34:0, T35:0, T36:0) -> f2207_in(T31:0, arith, arith1, arith2) :|: T34:0 + T35:0 < T31:0 && arith = T34:0 + 1 && arith1 = T35:0 + 1 && arith2 = T36:0 + 1 Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (10) Obligation: Termination digraph: Nodes: (1) f2207_in(T31:0, T34:0, T35:0, T36:0) -> f2207_in(T31:0, arith, arith1, arith2) :|: T34:0 + T35:0 < T31:0 && arith = T34:0 + 1 && arith1 = T35:0 + 1 && arith2 = T36:0 + 1 Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (11) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (12) Obligation: Rules: f2207_in(T31:0:0, T34:0:0, T35:0:0, T36:0:0) -> f2207_in(T31:0:0, T34:0:0 + 1, T35:0:0 + 1, T36:0:0 + 1) :|: T34:0:0 + T35:0:0 < T31:0:0 ---------------------------------------- (13) IntTRSUnneededArgumentFilterProof (EQUIVALENT) Some arguments are removed because they cannot influence termination. We removed arguments according to the following replacements: f2207_in(x1, x2, x3, x4) -> f2207_in(x1, x2, x3) ---------------------------------------- (14) Obligation: Rules: f2207_in(T31:0:0, T34:0:0, T35:0:0) -> f2207_in(T31:0:0, T34:0:0 + 1, T35:0:0 + 1) :|: T34:0:0 + T35:0:0 < T31:0:0 ---------------------------------------- (15) TempFilterProof (SOUND) Used the following sort dictionary for filtering: f2207_in(INTEGER, INTEGER, INTEGER) Replaced non-predefined constructor symbols by 0. ---------------------------------------- (16) Obligation: Rules: f2207_in(T31:0:0, T34:0:0, T35:0:0) -> f2207_in(T31:0:0, c, c1) :|: c1 = T35:0:0 + 1 && c = T34:0:0 + 1 && T34:0:0 + T35:0:0 < T31:0:0 ---------------------------------------- (17) RankingReductionPairProof (EQUIVALENT) Interpretation: [ f2207_in ] = -1/2*f2207_in_2 + -1/2*f2207_in_3 + 1/2*f2207_in_1 The following rules are decreasing: f2207_in(T31:0:0, T34:0:0, T35:0:0) -> f2207_in(T31:0:0, c, c1) :|: c1 = T35:0:0 + 1 && c = T34:0:0 + 1 && T34:0:0 + T35:0:0 < T31:0:0 The following rules are bounded: f2207_in(T31:0:0, T34:0:0, T35:0:0) -> f2207_in(T31:0:0, c, c1) :|: c1 = T35:0:0 + 1 && c = T34:0:0 + 1 && T34:0:0 + T35:0:0 < T31:0:0 ---------------------------------------- (18) YES