6.68/2.70 YES 6.97/2.73 proof of /export/starexec/sandbox/benchmark/theBenchmark.pl 6.97/2.73 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 6.97/2.73 6.97/2.73 6.97/2.73 Left Termination of the query pattern 6.97/2.73 6.97/2.73 rem(g,g,a) 6.97/2.73 6.97/2.73 w.r.t. the given Prolog program could successfully be proven: 6.97/2.73 6.97/2.73 (0) Prolog 6.97/2.73 (1) PrologToDTProblemTransformerProof [SOUND, 44 ms] 6.97/2.73 (2) TRIPLES 6.97/2.73 (3) TriplesToPiDPProof [SOUND, 0 ms] 6.97/2.73 (4) PiDP 6.97/2.73 (5) DependencyGraphProof [EQUIVALENT, 0 ms] 6.97/2.73 (6) AND 6.97/2.73 (7) PiDP 6.97/2.73 (8) UsableRulesProof [EQUIVALENT, 0 ms] 6.97/2.73 (9) PiDP 6.97/2.73 (10) PiDPToQDPProof [EQUIVALENT, 0 ms] 6.97/2.73 (11) QDP 6.97/2.73 (12) QDPSizeChangeProof [EQUIVALENT, 0 ms] 6.97/2.73 (13) YES 6.97/2.73 (14) PiDP 6.97/2.73 (15) UsableRulesProof [EQUIVALENT, 0 ms] 6.97/2.73 (16) PiDP 6.97/2.73 (17) PiDPToQDPProof [SOUND, 0 ms] 6.97/2.73 (18) QDP 6.97/2.73 (19) QDPSizeChangeProof [EQUIVALENT, 0 ms] 6.97/2.73 (20) YES 6.97/2.73 (21) PiDP 6.97/2.73 (22) PiDPToQDPProof [SOUND, 0 ms] 6.97/2.73 (23) QDP 6.97/2.73 (24) QDPOrderProof [EQUIVALENT, 33 ms] 6.97/2.73 (25) QDP 6.97/2.73 (26) PisEmptyProof [EQUIVALENT, 0 ms] 6.97/2.73 (27) YES 6.97/2.73 6.97/2.73 6.97/2.73 ---------------------------------------- 6.97/2.73 6.97/2.73 (0) 6.97/2.73 Obligation: 6.97/2.73 Clauses: 6.97/2.73 6.97/2.73 rem(X, Y, R) :- ','(notZero(Y), ','(sub(X, Y, Z), rem(Z, Y, R))). 6.97/2.73 rem(X, Y, X) :- ','(notZero(Y), geq(X, Y)). 6.97/2.73 sub(s(X), s(Y), Z) :- sub(X, Y, Z). 6.97/2.73 sub(X, 0, X). 6.97/2.73 notZero(s(X)). 6.97/2.73 geq(s(X), s(Y)) :- geq(X, Y). 6.97/2.73 geq(X, 0). 6.97/2.73 6.97/2.73 6.97/2.73 Query: rem(g,g,a) 6.97/2.73 ---------------------------------------- 6.97/2.73 6.97/2.73 (1) PrologToDTProblemTransformerProof (SOUND) 6.97/2.73 Built DT problem from termination graph DT10. 6.97/2.73 6.97/2.73 { 6.97/2.73 "root": 16, 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["X7"], 6.97/2.74 "exprvars": [] 6.97/2.74 } 6.97/2.74 }, 6.97/2.74 "165": { 6.97/2.74 "goal": [ 6.97/2.74 { 6.97/2.74 "clause": -1, 6.97/2.74 "scope": 2, 6.97/2.74 "term": null 6.97/2.74 }, 6.97/2.74 { 6.97/2.74 "clause": 1, 6.97/2.74 "scope": 1, 6.97/2.74 "term": "(rem T7 T8 T3)" 6.97/2.74 } 6.97/2.74 ], 6.97/2.74 "kb": { 6.97/2.74 "nonunifying": [], 6.97/2.74 "intvars": {}, 6.97/2.74 "arithmetic": { 6.97/2.74 "type": "PlainIntegerRelationState", 6.97/2.74 "relations": [] 6.97/2.74 }, 6.97/2.74 "ground": [ 6.97/2.74 "T7", 6.97/2.74 "T8" 6.97/2.74 ], 6.97/2.74 "free": [], 6.97/2.74 "exprvars": [] 6.97/2.74 } 6.97/2.74 }, 6.97/2.74 "363": { 6.97/2.74 "goal": [{ 6.97/2.74 "clause": -1, 6.97/2.74 "scope": -1, 6.97/2.74 "term": "(',' (notZero T57) (geq T56 T57))" 6.97/2.74 }], 6.97/2.74 "kb": { 6.97/2.74 "nonunifying": [], 6.97/2.74 "intvars": {}, 6.97/2.74 "arithmetic": { 6.97/2.74 "type": "PlainIntegerRelationState", 6.97/2.74 "relations": [] 6.97/2.74 }, 6.97/2.74 "ground": [ 6.97/2.74 "T56", 6.97/2.74 "T57" 6.97/2.74 ], 6.97/2.74 "free": [], 6.97/2.74 "exprvars": [] 6.97/2.74 } 6.97/2.74 }, 6.97/2.74 "364": { 6.97/2.74 "goal": [], 6.97/2.74 "kb": { 6.97/2.74 "nonunifying": [], 6.97/2.74 "intvars": {}, 6.97/2.74 "arithmetic": { 6.97/2.74 "type": "PlainIntegerRelationState", 6.97/2.74 "relations": [] 6.97/2.74 }, 6.97/2.74 "ground": [], 6.97/2.74 "free": [], 6.97/2.74 "exprvars": [] 6.97/2.74 } 6.97/2.74 }, 6.97/2.74 "365": { 6.97/2.74 "goal": [{ 6.97/2.74 "clause": 4, 6.97/2.74 "scope": 5, 6.97/2.74 "term": "(',' (notZero T57) (geq T56 T57))" 6.97/2.74 }], 6.97/2.74 "kb": { 6.97/2.74 "nonunifying": [], 6.97/2.74 "intvars": {}, 6.97/2.74 "arithmetic": { 6.97/2.74 "type": "PlainIntegerRelationState", 6.97/2.74 "relations": [] 6.97/2.74 }, 6.97/2.74 "ground": [ 6.97/2.74 "T56", 6.97/2.74 "T57" 6.97/2.74 ], 6.97/2.74 "free": [], 6.97/2.74 "exprvars": [] 6.97/2.74 } 6.97/2.74 }, 6.97/2.74 "147": { 6.97/2.74 "goal": [ 6.97/2.74 { 6.97/2.74 "clause": -1, 6.97/2.74 "scope": -1, 6.97/2.74 "term": "(',' (notZero T8) (',' (sub T7 T8 X7) (rem X7 T8 T10)))" 6.97/2.74 }, 6.97/2.74 { 6.97/2.74 "clause": 1, 6.97/2.74 "scope": 1, 6.97/2.74 "term": "(rem T7 T8 T3)" 6.97/2.74 } 6.97/2.74 ], 6.97/2.74 "kb": { 6.97/2.74 "nonunifying": [], 6.97/2.74 "intvars": {}, 6.97/2.74 "arithmetic": { 6.97/2.74 "type": "PlainIntegerRelationState", 6.97/2.74 "relations": [] 6.97/2.74 }, 6.97/2.74 "ground": [ 6.97/2.74 "T7", 6.97/2.74 "T8" 6.97/2.74 ], 6.97/2.74 "free": ["X7"], 6.97/2.74 "exprvars": [] 6.97/2.74 } 6.97/2.74 }, 6.97/2.74 "400": { 6.97/2.74 "goal": [{ 6.97/2.74 "clause": 6, 6.97/2.74 "scope": 7, 6.97/2.74 "term": "(geq T73 T74)" 6.97/2.74 }], 6.97/2.74 "kb": { 6.97/2.74 "nonunifying": [], 6.97/2.74 "intvars": {}, 6.97/2.74 "arithmetic": { 6.97/2.74 "type": "PlainIntegerRelationState", 6.97/2.74 "relations": [] 6.97/2.74 }, 6.97/2.74 "ground": [ 6.97/2.74 "T73", 6.97/2.74 "T74" 6.97/2.74 ], 6.97/2.74 "free": [], 6.97/2.74 "exprvars": [] 6.97/2.74 } 6.97/2.74 }, 6.97/2.74 "368": { 6.97/2.74 "goal": [{ 6.97/2.74 "clause": -1, 6.97/2.74 "scope": -1, 6.97/2.74 "term": "(geq T56 (s T62))" 6.97/2.74 }], 6.97/2.74 "kb": { 6.97/2.74 "nonunifying": [], 6.97/2.74 "intvars": {}, 6.97/2.74 "arithmetic": { 6.97/2.74 "type": "PlainIntegerRelationState", 6.97/2.74 "relations": [] 6.97/2.74 }, 6.97/2.74 "ground": [ 6.97/2.74 "T56", 6.97/2.74 "T62" 6.97/2.74 ], 6.97/2.74 "free": [], 6.97/2.74 "exprvars": [] 6.97/2.74 } 6.97/2.74 }, 6.97/2.74 "401": { 6.97/2.74 "goal": [{ 6.97/2.74 "clause": -1, 6.97/2.74 "scope": -1, 6.97/2.74 "term": "(geq T85 T86)" 6.97/2.74 }], 6.97/2.74 "kb": { 6.97/2.74 "nonunifying": [], 6.97/2.74 "intvars": {}, 6.97/2.74 "arithmetic": { 6.97/2.74 "type": "PlainIntegerRelationState", 6.97/2.74 "relations": [] 6.97/2.74 }, 6.97/2.74 "ground": [ 6.97/2.74 "T85", 6.97/2.74 "T86" 6.97/2.74 ], 6.97/2.74 "free": [], 6.97/2.74 "exprvars": [] 6.97/2.74 } 6.97/2.74 }, 6.97/2.74 "369": { 6.97/2.74 "goal": [], 6.97/2.74 "kb": { 6.97/2.74 "nonunifying": [], 6.97/2.74 "intvars": {}, 6.97/2.74 "arithmetic": { 6.97/2.74 "type": "PlainIntegerRelationState", 6.97/2.74 "relations": [] 6.97/2.74 }, 6.97/2.74 "ground": [], 6.97/2.74 "free": [], 6.97/2.74 "exprvars": [] 6.97/2.74 } 6.97/2.74 }, 6.97/2.74 "402": { 6.97/2.74 "goal": [], 6.97/2.74 "kb": { 6.97/2.74 "nonunifying": [], 6.97/2.74 "intvars": {}, 6.97/2.74 "arithmetic": { 6.97/2.74 "type": "PlainIntegerRelationState", 6.97/2.74 "relations": [] 6.97/2.74 }, 6.97/2.74 "ground": [], 6.97/2.74 "free": [], 6.97/2.74 "exprvars": [] 6.97/2.74 } 6.97/2.74 }, 6.97/2.74 "403": { 6.97/2.74 "goal": [{ 6.97/2.74 "clause": -1, 6.97/2.74 "scope": -1, 6.97/2.74 "term": "(true)" 6.97/2.74 }], 6.97/2.74 "kb": { 6.97/2.74 "nonunifying": [], 6.97/2.74 "intvars": {}, 6.97/2.74 "arithmetic": { 6.97/2.74 "type": "PlainIntegerRelationState", 6.97/2.74 "relations": [] 6.97/2.74 }, 6.97/2.74 "ground": [], 6.97/2.74 "free": [], 6.97/2.74 "exprvars": [] 6.97/2.74 } 6.97/2.74 }, 6.97/2.74 "327": { 6.97/2.74 "goal": [ 6.97/2.74 { 6.97/2.74 "clause": 2, 6.97/2.74 "scope": 3, 6.97/2.74 "term": "(sub T7 (s T15) X7)" 6.97/2.74 }, 6.97/2.74 { 6.97/2.74 "clause": 3, 6.97/2.74 "scope": 3, 6.97/2.74 "term": "(sub T7 (s T15) X7)" 6.97/2.74 } 6.97/2.74 ], 6.97/2.74 "kb": { 6.97/2.74 "nonunifying": [], 6.97/2.74 "intvars": {}, 6.97/2.74 "arithmetic": { 6.97/2.74 "type": "PlainIntegerRelationState", 6.97/2.74 "relations": [] 6.97/2.74 }, 6.97/2.74 "ground": [ 6.97/2.74 "T7", 6.97/2.74 "T15" 6.97/2.74 ], 6.97/2.74 "free": ["X7"], 6.97/2.74 "exprvars": [] 6.97/2.74 } 6.97/2.74 }, 6.97/2.74 "404": { 6.97/2.74 "goal": [], 6.97/2.74 "kb": { 6.97/2.74 "nonunifying": [], 6.97/2.74 "intvars": {}, 6.97/2.74 "arithmetic": { 6.97/2.74 "type": "PlainIntegerRelationState", 6.97/2.74 "relations": [] 6.97/2.74 }, 6.97/2.74 "ground": [], 6.97/2.74 "free": [], 6.97/2.74 "exprvars": [] 6.97/2.74 } 6.97/2.74 }, 6.97/2.74 "328": { 6.97/2.74 "goal": [{ 6.97/2.74 "clause": 2, 6.97/2.74 "scope": 3, 6.97/2.74 "term": "(sub T7 (s T15) X7)" 6.97/2.74 }], 6.97/2.74 "kb": { 6.97/2.74 "nonunifying": [], 6.97/2.74 "intvars": {}, 6.97/2.74 "arithmetic": { 6.97/2.74 "type": "PlainIntegerRelationState", 6.97/2.74 "relations": [] 6.97/2.74 }, 6.97/2.74 "ground": [ 6.97/2.74 "T7", 6.97/2.74 "T15" 6.97/2.74 ], 6.97/2.74 "free": ["X7"], 6.97/2.74 "exprvars": [] 6.97/2.74 } 6.97/2.74 }, 6.97/2.74 "405": { 6.97/2.74 "goal": [], 6.97/2.74 "kb": { 6.97/2.74 "nonunifying": [], 6.97/2.74 "intvars": {}, 6.97/2.74 "arithmetic": { 6.97/2.74 "type": "PlainIntegerRelationState", 6.97/2.74 "relations": [] 6.97/2.74 }, 6.97/2.74 "ground": [], 6.97/2.74 "free": [], 6.97/2.74 "exprvars": [] 6.97/2.74 } 6.97/2.74 }, 6.97/2.74 "329": { 6.97/2.74 "goal": [{ 6.97/2.74 "clause": 3, 6.97/2.74 "scope": 3, 6.97/2.74 "term": "(sub T7 (s T15) X7)" 6.97/2.74 }], 6.97/2.74 "kb": { 6.97/2.74 "nonunifying": [], 6.97/2.74 "intvars": {}, 6.97/2.74 "arithmetic": { 6.97/2.74 "type": "PlainIntegerRelationState", 6.97/2.74 "relations": [] 6.97/2.74 }, 6.97/2.74 "ground": [ 6.97/2.74 "T7", 6.97/2.74 "T15" 6.97/2.74 ], 6.97/2.74 "free": ["X7"], 6.97/2.74 "exprvars": [] 6.97/2.74 } 6.97/2.74 }, 6.97/2.74 "406": { 6.97/2.74 "goal": [], 6.97/2.74 "kb": { 6.97/2.74 "nonunifying": [], 6.97/2.74 "intvars": {}, 6.97/2.74 "arithmetic": { 6.97/2.74 "type": "PlainIntegerRelationState", 6.97/2.74 "relations": [] 6.97/2.74 }, 6.97/2.74 "ground": [], 6.97/2.74 "free": [], 6.97/2.74 "exprvars": [] 6.97/2.74 } 6.97/2.74 } 6.97/2.74 }, 6.97/2.74 "edges": [ 6.97/2.74 { 6.97/2.74 "from": 16, 6.97/2.74 "to": 19, 6.97/2.74 "label": "CASE" 6.97/2.74 }, 6.97/2.74 { 6.97/2.74 "from": 19, 6.97/2.74 "to": 147, 6.97/2.74 "label": "ONLY EVAL with clause\nrem(X4, X5, X6) :- ','(notZero(X5), ','(sub(X4, X5, X7), rem(X7, X5, X6))).\nand substitutionT1 -> T7,\nX4 -> T7,\nT2 -> T8,\nX5 -> T8,\nT3 -> T10,\nX6 -> T10,\nT9 -> T10" 6.97/2.74 }, 6.97/2.74 { 6.97/2.74 "from": 147, 6.97/2.74 "to": 155, 6.97/2.74 "label": "CASE" 6.97/2.74 }, 6.97/2.74 { 6.97/2.74 "from": 155, 6.97/2.74 "to": 164, 6.97/2.74 "label": "PARALLEL" 6.97/2.74 }, 6.97/2.74 { 6.97/2.74 "from": 155, 6.97/2.74 "to": 165, 6.97/2.74 "label": "PARALLEL" 6.97/2.74 }, 6.97/2.74 { 6.97/2.74 "from": 164, 6.97/2.74 "to": 173, 6.97/2.74 "label": "EVAL with clause\nnotZero(s(X12)).\nand substitutionX12 -> T15,\nT8 -> s(T15)" 6.97/2.74 }, 6.97/2.74 { 6.97/2.74 "from": 164, 6.97/2.74 "to": 177, 6.97/2.74 "label": "EVAL-BACKTRACK" 6.97/2.74 }, 6.97/2.74 { 6.97/2.74 "from": 165, 6.97/2.74 "to": 357, 6.97/2.74 "label": "FAILURE" 6.97/2.74 }, 6.97/2.74 { 6.97/2.74 "from": 173, 6.97/2.74 "to": 179, 6.97/2.74 "label": "SPLIT 1" 6.97/2.74 }, 6.97/2.74 { 6.97/2.74 "from": 173, 6.97/2.74 "to": 181, 6.97/2.74 "label": "SPLIT 2\nnew knowledge:\nT7 is ground\nT15 is ground\nT18 is ground\nreplacements:X7 -> T18" 6.97/2.74 }, 6.97/2.74 { 6.97/2.74 "from": 179, 6.97/2.74 "to": 327, 6.97/2.74 "label": "CASE" 6.97/2.74 }, 6.97/2.74 { 6.97/2.74 "from": 181, 6.97/2.74 "to": 16, 6.97/2.74 "label": "INSTANCE with matching:\nT1 -> T18\nT2 -> s(T15)\nT3 -> T10" 6.97/2.74 }, 6.97/2.74 { 6.97/2.74 "from": 327, 6.97/2.74 "to": 328, 6.97/2.74 "label": "PARALLEL" 6.97/2.74 }, 6.97/2.74 { 6.97/2.74 "from": 327, 6.97/2.74 "to": 329, 6.97/2.74 "label": "PARALLEL" 6.97/2.74 }, 6.97/2.74 { 6.97/2.74 "from": 328, 6.97/2.74 "to": 330, 6.97/2.74 "label": "EVAL with clause\nsub(s(X37), s(X38), X39) :- sub(X37, X38, X39).\nand substitutionX37 -> T29,\nT7 -> s(T29),\nT15 -> T30,\nX38 -> T30,\nX7 -> X40,\nX39 -> X40" 6.97/2.74 }, 6.97/2.74 { 6.97/2.74 "from": 328, 6.97/2.74 "to": 331, 6.97/2.74 "label": "EVAL-BACKTRACK" 6.97/2.74 }, 6.97/2.74 { 6.97/2.74 "from": 329, 6.97/2.74 "to": 353, 6.97/2.74 "label": "BACKTRACK\nfor clause: sub(X, 0, X)because of non-unification" 6.97/2.74 }, 6.97/2.74 { 6.97/2.74 "from": 330, 6.97/2.74 "to": 332, 6.97/2.74 "label": "CASE" 6.97/2.74 }, 6.97/2.74 { 6.97/2.74 "from": 332, 6.97/2.74 "to": 333, 6.97/2.74 "label": "PARALLEL" 6.97/2.74 }, 6.97/2.74 { 6.97/2.74 "from": 332, 6.97/2.74 "to": 334, 6.97/2.74 "label": "PARALLEL" 6.97/2.74 }, 6.97/2.74 { 6.97/2.74 "from": 333, 6.97/2.74 "to": 335, 6.97/2.74 "label": "EVAL with clause\nsub(s(X61), s(X62), X63) :- sub(X61, X62, X63).\nand substitutionX61 -> T41,\nT29 -> s(T41),\nX62 -> T42,\nT30 -> s(T42),\nX40 -> X64,\nX63 -> X64" 6.97/2.74 }, 6.97/2.74 { 6.97/2.74 "from": 333, 6.97/2.74 "to": 336, 6.97/2.74 "label": "EVAL-BACKTRACK" 6.97/2.74 }, 6.97/2.74 { 6.97/2.74 "from": 334, 6.97/2.74 "to": 337, 6.97/2.74 "label": "EVAL with clause\nsub(X71, 0, X71).\nand substitutionT29 -> T47,\nX71 -> T47,\nT30 -> 0,\nX40 -> T47" 6.97/2.74 }, 6.97/2.74 { 6.97/2.74 "from": 334, 6.97/2.74 "to": 338, 6.97/2.74 "label": "EVAL-BACKTRACK" 6.97/2.74 }, 6.97/2.74 { 6.97/2.74 "from": 335, 6.97/2.74 "to": 330, 6.97/2.74 "label": "INSTANCE with matching:\nT29 -> T41\nT30 -> T42\nX40 -> X64" 6.97/2.74 }, 6.97/2.74 { 6.97/2.74 "from": 337, 6.97/2.74 "to": 339, 6.97/2.74 "label": "SUCCESS" 6.97/2.74 }, 6.97/2.74 { 6.97/2.74 "from": 357, 6.97/2.74 "to": 363, 6.97/2.74 "label": "EVAL with clause\nrem(X81, X82, X81) :- ','(notZero(X82), geq(X81, X82)).\nand substitutionT7 -> T56,\nX81 -> T56,\nT8 -> T57,\nX82 -> T57,\nT3 -> T56" 6.97/2.74 }, 6.97/2.74 { 6.97/2.74 "from": 357, 6.97/2.74 "to": 364, 6.97/2.74 "label": "EVAL-BACKTRACK" 6.97/2.74 }, 6.97/2.74 { 6.97/2.74 "from": 363, 6.97/2.74 "to": 365, 6.97/2.74 "label": "CASE" 6.97/2.74 }, 6.97/2.74 { 6.97/2.74 "from": 365, 6.97/2.74 "to": 368, 6.97/2.74 "label": "EVAL with clause\nnotZero(s(X87)).\nand substitutionX87 -> T62,\nT57 -> s(T62)" 6.97/2.74 }, 6.97/2.74 { 6.97/2.74 "from": 365, 6.97/2.74 "to": 369, 6.97/2.74 "label": "EVAL-BACKTRACK" 6.97/2.74 }, 6.97/2.74 { 6.97/2.74 "from": 368, 6.97/2.74 "to": 380, 6.97/2.74 "label": "CASE" 6.97/2.74 }, 6.97/2.74 { 6.97/2.74 "from": 380, 6.97/2.74 "to": 381, 6.97/2.74 "label": "PARALLEL" 6.97/2.74 }, 6.97/2.74 { 6.97/2.74 "from": 380, 6.97/2.74 "to": 382, 6.97/2.74 "label": "PARALLEL" 6.97/2.74 }, 6.97/2.74 { 6.97/2.74 "from": 381, 6.97/2.74 "to": 396, 6.97/2.74 "label": "EVAL with clause\ngeq(s(X98), s(X99)) :- geq(X98, X99).\nand substitutionX98 -> T73,\nT56 -> s(T73),\nT62 -> T74,\nX99 -> T74" 6.97/2.74 }, 6.97/2.74 { 6.97/2.74 "from": 381, 6.97/2.74 "to": 397, 6.97/2.74 "label": "EVAL-BACKTRACK" 6.97/2.74 }, 6.97/2.74 { 6.97/2.74 "from": 382, 6.97/2.74 "to": 406, 6.97/2.74 "label": "BACKTRACK\nfor clause: geq(X, 0)because of non-unification" 6.97/2.74 }, 6.97/2.74 { 6.97/2.74 "from": 396, 6.97/2.74 "to": 398, 6.97/2.74 "label": "CASE" 6.97/2.74 }, 6.97/2.74 { 6.97/2.74 "from": 398, 6.97/2.74 "to": 399, 6.97/2.74 "label": "PARALLEL" 6.97/2.74 }, 6.97/2.74 { 6.97/2.74 "from": 398, 6.97/2.74 "to": 400, 6.97/2.74 "label": "PARALLEL" 6.97/2.74 }, 6.97/2.74 { 6.97/2.74 "from": 399, 6.97/2.74 "to": 401, 6.97/2.74 "label": "EVAL with clause\ngeq(s(X110), s(X111)) :- geq(X110, X111).\nand substitutionX110 -> T85,\nT73 -> s(T85),\nX111 -> T86,\nT74 -> s(T86)" 6.97/2.74 }, 6.97/2.74 { 6.97/2.74 "from": 399, 6.97/2.74 "to": 402, 6.97/2.74 "label": "EVAL-BACKTRACK" 6.97/2.74 }, 6.97/2.74 { 6.97/2.74 "from": 400, 6.97/2.74 "to": 403, 6.97/2.74 "label": "EVAL with clause\ngeq(X116, 0).\nand substitutionT73 -> T91,\nX116 -> T91,\nT74 -> 0" 6.97/2.74 }, 6.97/2.74 { 6.97/2.74 "from": 400, 6.97/2.74 "to": 404, 6.97/2.74 "label": "EVAL-BACKTRACK" 6.97/2.74 }, 6.97/2.74 { 6.97/2.74 "from": 401, 6.97/2.74 "to": 396, 6.97/2.74 "label": "INSTANCE with matching:\nT73 -> T85\nT74 -> T86" 6.97/2.74 }, 6.97/2.74 { 6.97/2.74 "from": 403, 6.97/2.74 "to": 405, 6.97/2.74 "label": "SUCCESS" 6.97/2.74 } 6.97/2.74 ], 6.97/2.74 "type": "Graph" 6.97/2.74 } 6.97/2.74 } 6.97/2.74 6.97/2.74 ---------------------------------------- 6.97/2.74 6.97/2.74 (2) 6.97/2.74 Obligation: 6.97/2.74 Triples: 6.97/2.74 6.97/2.74 subD(s(X1), s(X2), X3) :- subD(X1, X2, X3). 6.97/2.74 geqC(s(X1), s(X2)) :- geqC(X1, X2). 6.97/2.74 remA(s(X1), s(X2), X3) :- subD(X1, X2, X4). 6.97/2.74 remA(X1, s(X2), X3) :- ','(subcB(X1, X2, X4), remA(X4, s(X2), X3)). 6.97/2.74 remA(s(X1), s(X2), s(X1)) :- geqC(X1, X2). 6.97/2.74 6.97/2.74 Clauses: 6.97/2.74 6.97/2.74 remcA(X1, s(X2), X3) :- ','(subcB(X1, X2, X4), remcA(X4, s(X2), X3)). 6.97/2.74 remcA(s(X1), s(X2), s(X1)) :- geqcC(X1, X2). 6.97/2.74 subcD(s(X1), s(X2), X3) :- subcD(X1, X2, X3). 6.97/2.74 subcD(X1, 0, X1). 6.97/2.74 geqcC(s(X1), s(X2)) :- geqcC(X1, X2). 6.97/2.74 geqcC(X1, 0). 6.97/2.74 subcB(s(X1), X2, X3) :- subcD(X1, X2, X3). 6.97/2.74 6.97/2.74 Afs: 6.97/2.74 6.97/2.74 remA(x1, x2, x3) = remA(x1, x2) 6.97/2.74 6.97/2.74 6.97/2.74 ---------------------------------------- 6.97/2.74 6.97/2.74 (3) TriplesToPiDPProof (SOUND) 6.97/2.74 We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes: 6.97/2.74 6.97/2.74 remA_in_3: (b,b,f) 6.97/2.74 6.97/2.74 subD_in_3: (b,b,f) 6.97/2.74 6.97/2.74 subcB_in_3: (b,b,f) 6.97/2.74 6.97/2.74 subcD_in_3: (b,b,f) 6.97/2.74 6.97/2.74 geqC_in_2: (b,b) 6.97/2.74 6.97/2.74 Transforming TRIPLES into the following Term Rewriting System: 6.97/2.74 6.97/2.74 Pi DP problem: 6.97/2.74 The TRS P consists of the following rules: 6.97/2.74 6.97/2.74 REMA_IN_GGA(s(X1), s(X2), X3) -> U3_GGA(X1, X2, X3, subD_in_gga(X1, X2, X4)) 6.97/2.74 REMA_IN_GGA(s(X1), s(X2), X3) -> SUBD_IN_GGA(X1, X2, X4) 6.97/2.74 SUBD_IN_GGA(s(X1), s(X2), X3) -> U1_GGA(X1, X2, X3, subD_in_gga(X1, X2, X3)) 6.97/2.74 SUBD_IN_GGA(s(X1), s(X2), X3) -> SUBD_IN_GGA(X1, X2, X3) 6.97/2.74 REMA_IN_GGA(X1, s(X2), X3) -> U4_GGA(X1, X2, X3, subcB_in_gga(X1, X2, X4)) 6.97/2.74 U4_GGA(X1, X2, X3, subcB_out_gga(X1, X2, X4)) -> U5_GGA(X1, X2, X3, remA_in_gga(X4, s(X2), X3)) 6.97/2.74 U4_GGA(X1, X2, X3, subcB_out_gga(X1, X2, X4)) -> REMA_IN_GGA(X4, s(X2), X3) 6.97/2.74 REMA_IN_GGA(s(X1), s(X2), s(X1)) -> U6_GGA(X1, X2, geqC_in_gg(X1, X2)) 6.97/2.74 REMA_IN_GGA(s(X1), s(X2), s(X1)) -> GEQC_IN_GG(X1, X2) 6.97/2.74 GEQC_IN_GG(s(X1), s(X2)) -> U2_GG(X1, X2, geqC_in_gg(X1, X2)) 6.97/2.74 GEQC_IN_GG(s(X1), s(X2)) -> GEQC_IN_GG(X1, X2) 6.97/2.74 6.97/2.74 The TRS R consists of the following rules: 6.97/2.74 6.97/2.74 subcB_in_gga(s(X1), X2, X3) -> U13_gga(X1, X2, X3, subcD_in_gga(X1, X2, X3)) 6.97/2.74 subcD_in_gga(s(X1), s(X2), X3) -> U11_gga(X1, X2, X3, subcD_in_gga(X1, X2, X3)) 6.97/2.74 subcD_in_gga(X1, 0, X1) -> subcD_out_gga(X1, 0, X1) 6.97/2.74 U11_gga(X1, X2, X3, subcD_out_gga(X1, X2, X3)) -> subcD_out_gga(s(X1), s(X2), X3) 6.97/2.74 U13_gga(X1, X2, X3, subcD_out_gga(X1, X2, X3)) -> subcB_out_gga(s(X1), X2, X3) 6.97/2.74 6.97/2.74 The argument filtering Pi contains the following mapping: 6.97/2.74 remA_in_gga(x1, x2, x3) = remA_in_gga(x1, x2) 6.97/2.74 6.97/2.74 s(x1) = s(x1) 6.97/2.74 6.97/2.74 subD_in_gga(x1, x2, x3) = subD_in_gga(x1, x2) 6.97/2.74 6.97/2.74 subcB_in_gga(x1, x2, x3) = subcB_in_gga(x1, x2) 6.97/2.74 6.97/2.74 U13_gga(x1, x2, x3, x4) = U13_gga(x1, x2, x4) 6.97/2.74 6.97/2.74 subcD_in_gga(x1, x2, x3) = subcD_in_gga(x1, x2) 6.97/2.74 6.97/2.74 U11_gga(x1, x2, x3, x4) = U11_gga(x1, x2, x4) 6.97/2.74 6.97/2.74 0 = 0 6.97/2.74 6.97/2.74 subcD_out_gga(x1, x2, x3) = subcD_out_gga(x1, x2, x3) 6.97/2.74 6.97/2.74 subcB_out_gga(x1, x2, x3) = subcB_out_gga(x1, x2, x3) 6.97/2.74 6.97/2.74 geqC_in_gg(x1, x2) = geqC_in_gg(x1, x2) 6.97/2.74 6.97/2.74 REMA_IN_GGA(x1, x2, x3) = REMA_IN_GGA(x1, x2) 6.97/2.74 6.97/2.74 U3_GGA(x1, x2, x3, x4) = U3_GGA(x1, x2, x4) 6.97/2.74 6.97/2.74 SUBD_IN_GGA(x1, x2, x3) = SUBD_IN_GGA(x1, x2) 6.97/2.74 6.97/2.74 U1_GGA(x1, x2, x3, x4) = U1_GGA(x1, x2, x4) 6.97/2.74 6.97/2.74 U4_GGA(x1, x2, x3, x4) = U4_GGA(x1, x2, x4) 6.97/2.74 6.97/2.74 U5_GGA(x1, x2, x3, x4) = U5_GGA(x1, x2, x4) 6.97/2.74 6.97/2.74 U6_GGA(x1, x2, x3) = U6_GGA(x1, x2, x3) 6.97/2.74 6.97/2.74 GEQC_IN_GG(x1, x2) = GEQC_IN_GG(x1, x2) 6.97/2.74 6.97/2.74 U2_GG(x1, x2, x3) = U2_GG(x1, x2, x3) 6.97/2.74 6.97/2.74 6.97/2.74 We have to consider all (P,R,Pi)-chains 6.97/2.74 6.97/2.74 6.97/2.74 Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES 6.97/2.74 6.97/2.74 6.97/2.74 6.97/2.74 ---------------------------------------- 6.97/2.74 6.97/2.74 (4) 6.97/2.74 Obligation: 6.97/2.74 Pi DP problem: 6.97/2.74 The TRS P consists of the following rules: 6.97/2.74 6.97/2.74 REMA_IN_GGA(s(X1), s(X2), X3) -> U3_GGA(X1, X2, X3, subD_in_gga(X1, X2, X4)) 6.97/2.74 REMA_IN_GGA(s(X1), s(X2), X3) -> SUBD_IN_GGA(X1, X2, X4) 6.97/2.74 SUBD_IN_GGA(s(X1), s(X2), X3) -> U1_GGA(X1, X2, X3, subD_in_gga(X1, X2, X3)) 6.97/2.74 SUBD_IN_GGA(s(X1), s(X2), X3) -> SUBD_IN_GGA(X1, X2, X3) 6.97/2.74 REMA_IN_GGA(X1, s(X2), X3) -> U4_GGA(X1, X2, X3, subcB_in_gga(X1, X2, X4)) 6.97/2.74 U4_GGA(X1, X2, X3, subcB_out_gga(X1, X2, X4)) -> U5_GGA(X1, X2, X3, remA_in_gga(X4, s(X2), X3)) 6.97/2.74 U4_GGA(X1, X2, X3, subcB_out_gga(X1, X2, X4)) -> REMA_IN_GGA(X4, s(X2), X3) 6.97/2.74 REMA_IN_GGA(s(X1), s(X2), s(X1)) -> U6_GGA(X1, X2, geqC_in_gg(X1, X2)) 6.97/2.74 REMA_IN_GGA(s(X1), s(X2), s(X1)) -> GEQC_IN_GG(X1, X2) 6.97/2.74 GEQC_IN_GG(s(X1), s(X2)) -> U2_GG(X1, X2, geqC_in_gg(X1, X2)) 6.97/2.74 GEQC_IN_GG(s(X1), s(X2)) -> GEQC_IN_GG(X1, X2) 6.97/2.74 6.97/2.74 The TRS R consists of the following rules: 6.97/2.74 6.97/2.74 subcB_in_gga(s(X1), X2, X3) -> U13_gga(X1, X2, X3, subcD_in_gga(X1, X2, X3)) 6.97/2.74 subcD_in_gga(s(X1), s(X2), X3) -> U11_gga(X1, X2, X3, subcD_in_gga(X1, X2, X3)) 6.97/2.74 subcD_in_gga(X1, 0, X1) -> subcD_out_gga(X1, 0, X1) 6.97/2.74 U11_gga(X1, X2, X3, subcD_out_gga(X1, X2, X3)) -> subcD_out_gga(s(X1), s(X2), X3) 6.97/2.74 U13_gga(X1, X2, X3, subcD_out_gga(X1, X2, X3)) -> subcB_out_gga(s(X1), X2, X3) 6.97/2.74 6.97/2.74 The argument filtering Pi contains the following mapping: 6.97/2.74 remA_in_gga(x1, x2, x3) = remA_in_gga(x1, x2) 6.97/2.74 6.97/2.74 s(x1) = s(x1) 6.97/2.74 6.97/2.74 subD_in_gga(x1, x2, x3) = subD_in_gga(x1, x2) 6.97/2.74 6.97/2.74 subcB_in_gga(x1, x2, x3) = subcB_in_gga(x1, x2) 6.97/2.74 6.97/2.74 U13_gga(x1, x2, x3, x4) = U13_gga(x1, x2, x4) 6.97/2.74 6.97/2.74 subcD_in_gga(x1, x2, x3) = subcD_in_gga(x1, x2) 6.97/2.74 6.97/2.74 U11_gga(x1, x2, x3, x4) = U11_gga(x1, x2, x4) 6.97/2.74 6.97/2.74 0 = 0 6.97/2.74 6.97/2.74 subcD_out_gga(x1, x2, x3) = subcD_out_gga(x1, x2, x3) 6.97/2.74 6.97/2.74 subcB_out_gga(x1, x2, x3) = subcB_out_gga(x1, x2, x3) 6.97/2.74 6.97/2.74 geqC_in_gg(x1, x2) = geqC_in_gg(x1, x2) 6.97/2.74 6.97/2.74 REMA_IN_GGA(x1, x2, x3) = REMA_IN_GGA(x1, x2) 6.97/2.74 6.97/2.74 U3_GGA(x1, x2, x3, x4) = U3_GGA(x1, x2, x4) 6.97/2.74 6.97/2.74 SUBD_IN_GGA(x1, x2, x3) = SUBD_IN_GGA(x1, x2) 6.97/2.74 6.97/2.74 U1_GGA(x1, x2, x3, x4) = U1_GGA(x1, x2, x4) 6.97/2.74 6.97/2.74 U4_GGA(x1, x2, x3, x4) = U4_GGA(x1, x2, x4) 6.97/2.74 6.97/2.74 U5_GGA(x1, x2, x3, x4) = U5_GGA(x1, x2, x4) 6.97/2.74 6.97/2.74 U6_GGA(x1, x2, x3) = U6_GGA(x1, x2, x3) 6.97/2.74 6.97/2.74 GEQC_IN_GG(x1, x2) = GEQC_IN_GG(x1, x2) 6.97/2.74 6.97/2.74 U2_GG(x1, x2, x3) = U2_GG(x1, x2, x3) 6.97/2.74 6.97/2.74 6.97/2.74 We have to consider all (P,R,Pi)-chains 6.97/2.74 ---------------------------------------- 6.97/2.74 6.97/2.74 (5) DependencyGraphProof (EQUIVALENT) 6.97/2.74 The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 7 less nodes. 6.97/2.74 ---------------------------------------- 6.97/2.74 6.97/2.74 (6) 6.97/2.74 Complex Obligation (AND) 6.97/2.74 6.97/2.74 ---------------------------------------- 6.97/2.74 6.97/2.74 (7) 6.97/2.74 Obligation: 6.97/2.74 Pi DP problem: 6.97/2.74 The TRS P consists of the following rules: 6.97/2.74 6.97/2.74 GEQC_IN_GG(s(X1), s(X2)) -> GEQC_IN_GG(X1, X2) 6.97/2.74 6.97/2.74 The TRS R consists of the following rules: 6.97/2.74 6.97/2.74 subcB_in_gga(s(X1), X2, X3) -> U13_gga(X1, X2, X3, subcD_in_gga(X1, X2, X3)) 6.97/2.74 subcD_in_gga(s(X1), s(X2), X3) -> U11_gga(X1, X2, X3, subcD_in_gga(X1, X2, X3)) 6.97/2.74 subcD_in_gga(X1, 0, X1) -> subcD_out_gga(X1, 0, X1) 6.97/2.74 U11_gga(X1, X2, X3, subcD_out_gga(X1, X2, X3)) -> subcD_out_gga(s(X1), s(X2), X3) 6.97/2.74 U13_gga(X1, X2, X3, subcD_out_gga(X1, X2, X3)) -> subcB_out_gga(s(X1), X2, X3) 6.97/2.74 6.97/2.74 The argument filtering Pi contains the following mapping: 6.97/2.74 s(x1) = s(x1) 6.97/2.74 6.97/2.74 subcB_in_gga(x1, x2, x3) = subcB_in_gga(x1, x2) 6.97/2.74 6.97/2.74 U13_gga(x1, x2, x3, x4) = U13_gga(x1, x2, x4) 6.97/2.74 6.97/2.74 subcD_in_gga(x1, x2, x3) = subcD_in_gga(x1, x2) 6.97/2.74 6.97/2.74 U11_gga(x1, x2, x3, x4) = U11_gga(x1, x2, x4) 6.97/2.74 6.97/2.74 0 = 0 6.97/2.74 6.97/2.74 subcD_out_gga(x1, x2, x3) = subcD_out_gga(x1, x2, x3) 6.97/2.74 6.97/2.74 subcB_out_gga(x1, x2, x3) = subcB_out_gga(x1, x2, x3) 6.97/2.74 6.97/2.74 GEQC_IN_GG(x1, x2) = GEQC_IN_GG(x1, x2) 6.97/2.74 6.97/2.74 6.97/2.74 We have to consider all (P,R,Pi)-chains 6.97/2.74 ---------------------------------------- 6.97/2.74 6.97/2.74 (8) UsableRulesProof (EQUIVALENT) 6.97/2.74 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 6.97/2.74 ---------------------------------------- 6.97/2.74 6.97/2.74 (9) 6.97/2.74 Obligation: 6.97/2.74 Pi DP problem: 6.97/2.74 The TRS P consists of the following rules: 6.97/2.74 6.97/2.74 GEQC_IN_GG(s(X1), s(X2)) -> GEQC_IN_GG(X1, X2) 6.97/2.74 6.97/2.74 R is empty. 6.97/2.74 Pi is empty. 6.97/2.74 We have to consider all (P,R,Pi)-chains 6.97/2.74 ---------------------------------------- 6.97/2.74 6.97/2.74 (10) PiDPToQDPProof (EQUIVALENT) 6.97/2.74 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 6.97/2.74 ---------------------------------------- 6.97/2.74 6.97/2.74 (11) 6.97/2.74 Obligation: 6.97/2.74 Q DP problem: 6.97/2.74 The TRS P consists of the following rules: 6.97/2.74 6.97/2.74 GEQC_IN_GG(s(X1), s(X2)) -> GEQC_IN_GG(X1, X2) 6.97/2.74 6.97/2.74 R is empty. 6.97/2.74 Q is empty. 6.97/2.74 We have to consider all (P,Q,R)-chains. 6.97/2.74 ---------------------------------------- 6.97/2.74 6.97/2.74 (12) QDPSizeChangeProof (EQUIVALENT) 6.97/2.74 By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 6.97/2.74 6.97/2.74 From the DPs we obtained the following set of size-change graphs: 6.97/2.74 *GEQC_IN_GG(s(X1), s(X2)) -> GEQC_IN_GG(X1, X2) 6.97/2.74 The graph contains the following edges 1 > 1, 2 > 2 6.97/2.74 6.97/2.74 6.97/2.74 ---------------------------------------- 6.97/2.74 6.97/2.74 (13) 6.97/2.74 YES 6.97/2.74 6.97/2.74 ---------------------------------------- 6.97/2.74 6.97/2.74 (14) 6.97/2.74 Obligation: 6.97/2.74 Pi DP problem: 6.97/2.74 The TRS P consists of the following rules: 6.97/2.74 6.97/2.74 SUBD_IN_GGA(s(X1), s(X2), X3) -> SUBD_IN_GGA(X1, X2, X3) 6.97/2.74 6.97/2.74 The TRS R consists of the following rules: 6.97/2.74 6.97/2.74 subcB_in_gga(s(X1), X2, X3) -> U13_gga(X1, X2, X3, subcD_in_gga(X1, X2, X3)) 6.97/2.74 subcD_in_gga(s(X1), s(X2), X3) -> U11_gga(X1, X2, X3, subcD_in_gga(X1, X2, X3)) 6.97/2.74 subcD_in_gga(X1, 0, X1) -> subcD_out_gga(X1, 0, X1) 6.97/2.74 U11_gga(X1, X2, X3, subcD_out_gga(X1, X2, X3)) -> subcD_out_gga(s(X1), s(X2), X3) 6.97/2.74 U13_gga(X1, X2, X3, subcD_out_gga(X1, X2, X3)) -> subcB_out_gga(s(X1), X2, X3) 6.97/2.74 6.97/2.74 The argument filtering Pi contains the following mapping: 6.97/2.74 s(x1) = s(x1) 6.97/2.74 6.97/2.74 subcB_in_gga(x1, x2, x3) = subcB_in_gga(x1, x2) 6.97/2.74 6.97/2.74 U13_gga(x1, x2, x3, x4) = U13_gga(x1, x2, x4) 6.97/2.74 6.97/2.74 subcD_in_gga(x1, x2, x3) = subcD_in_gga(x1, x2) 6.97/2.74 6.97/2.74 U11_gga(x1, x2, x3, x4) = U11_gga(x1, x2, x4) 6.97/2.74 6.97/2.74 0 = 0 6.97/2.74 6.97/2.74 subcD_out_gga(x1, x2, x3) = subcD_out_gga(x1, x2, x3) 6.97/2.74 6.97/2.74 subcB_out_gga(x1, x2, x3) = subcB_out_gga(x1, x2, x3) 6.97/2.74 6.97/2.74 SUBD_IN_GGA(x1, x2, x3) = SUBD_IN_GGA(x1, x2) 6.97/2.74 6.97/2.74 6.97/2.74 We have to consider all (P,R,Pi)-chains 6.97/2.74 ---------------------------------------- 6.97/2.74 6.97/2.74 (15) UsableRulesProof (EQUIVALENT) 6.97/2.74 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 6.97/2.74 ---------------------------------------- 6.97/2.74 6.97/2.74 (16) 6.97/2.74 Obligation: 6.97/2.74 Pi DP problem: 6.97/2.74 The TRS P consists of the following rules: 6.97/2.74 6.97/2.74 SUBD_IN_GGA(s(X1), s(X2), X3) -> SUBD_IN_GGA(X1, X2, X3) 6.97/2.74 6.97/2.74 R is empty. 6.97/2.74 The argument filtering Pi contains the following mapping: 6.97/2.74 s(x1) = s(x1) 6.97/2.74 6.97/2.74 SUBD_IN_GGA(x1, x2, x3) = SUBD_IN_GGA(x1, x2) 6.97/2.74 6.97/2.74 6.97/2.74 We have to consider all (P,R,Pi)-chains 6.97/2.74 ---------------------------------------- 6.97/2.74 6.97/2.74 (17) PiDPToQDPProof (SOUND) 6.97/2.74 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 6.97/2.74 ---------------------------------------- 6.97/2.74 6.97/2.74 (18) 6.97/2.74 Obligation: 6.97/2.74 Q DP problem: 6.97/2.74 The TRS P consists of the following rules: 6.97/2.74 6.97/2.74 SUBD_IN_GGA(s(X1), s(X2)) -> SUBD_IN_GGA(X1, X2) 6.97/2.74 6.97/2.74 R is empty. 6.97/2.74 Q is empty. 6.97/2.74 We have to consider all (P,Q,R)-chains. 6.97/2.74 ---------------------------------------- 6.97/2.74 6.97/2.74 (19) QDPSizeChangeProof (EQUIVALENT) 6.97/2.74 By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 6.97/2.74 6.97/2.74 From the DPs we obtained the following set of size-change graphs: 6.97/2.74 *SUBD_IN_GGA(s(X1), s(X2)) -> SUBD_IN_GGA(X1, X2) 6.97/2.74 The graph contains the following edges 1 > 1, 2 > 2 6.97/2.74 6.97/2.74 6.97/2.74 ---------------------------------------- 6.97/2.74 6.97/2.74 (20) 6.97/2.74 YES 6.97/2.74 6.97/2.74 ---------------------------------------- 6.97/2.74 6.97/2.74 (21) 6.97/2.74 Obligation: 6.97/2.74 Pi DP problem: 6.97/2.74 The TRS P consists of the following rules: 6.97/2.74 6.97/2.74 REMA_IN_GGA(X1, s(X2), X3) -> U4_GGA(X1, X2, X3, subcB_in_gga(X1, X2, X4)) 6.97/2.74 U4_GGA(X1, X2, X3, subcB_out_gga(X1, X2, X4)) -> REMA_IN_GGA(X4, s(X2), X3) 6.97/2.74 6.97/2.74 The TRS R consists of the following rules: 6.97/2.74 6.97/2.74 subcB_in_gga(s(X1), X2, X3) -> U13_gga(X1, X2, X3, subcD_in_gga(X1, X2, X3)) 6.97/2.74 subcD_in_gga(s(X1), s(X2), X3) -> U11_gga(X1, X2, X3, subcD_in_gga(X1, X2, X3)) 6.97/2.74 subcD_in_gga(X1, 0, X1) -> subcD_out_gga(X1, 0, X1) 6.97/2.74 U11_gga(X1, X2, X3, subcD_out_gga(X1, X2, X3)) -> subcD_out_gga(s(X1), s(X2), X3) 6.97/2.74 U13_gga(X1, X2, X3, subcD_out_gga(X1, X2, X3)) -> subcB_out_gga(s(X1), X2, X3) 6.97/2.74 6.97/2.74 The argument filtering Pi contains the following mapping: 6.97/2.74 s(x1) = s(x1) 6.97/2.74 6.97/2.74 subcB_in_gga(x1, x2, x3) = subcB_in_gga(x1, x2) 6.97/2.74 6.97/2.74 U13_gga(x1, x2, x3, x4) = U13_gga(x1, x2, x4) 6.97/2.74 6.97/2.74 subcD_in_gga(x1, x2, x3) = subcD_in_gga(x1, x2) 6.97/2.74 6.97/2.74 U11_gga(x1, x2, x3, x4) = U11_gga(x1, x2, x4) 6.97/2.74 6.97/2.74 0 = 0 6.97/2.74 6.97/2.74 subcD_out_gga(x1, x2, x3) = subcD_out_gga(x1, x2, x3) 6.97/2.74 6.97/2.74 subcB_out_gga(x1, x2, x3) = subcB_out_gga(x1, x2, x3) 6.97/2.74 6.97/2.74 REMA_IN_GGA(x1, x2, x3) = REMA_IN_GGA(x1, x2) 6.97/2.74 6.97/2.74 U4_GGA(x1, x2, x3, x4) = U4_GGA(x1, x2, x4) 6.97/2.74 6.97/2.74 6.97/2.74 We have to consider all (P,R,Pi)-chains 6.97/2.74 ---------------------------------------- 6.97/2.74 6.97/2.74 (22) PiDPToQDPProof (SOUND) 6.97/2.74 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 6.97/2.74 ---------------------------------------- 6.97/2.74 6.97/2.74 (23) 6.97/2.74 Obligation: 6.97/2.74 Q DP problem: 6.97/2.74 The TRS P consists of the following rules: 6.97/2.74 6.97/2.74 REMA_IN_GGA(X1, s(X2)) -> U4_GGA(X1, X2, subcB_in_gga(X1, X2)) 6.97/2.74 U4_GGA(X1, X2, subcB_out_gga(X1, X2, X4)) -> REMA_IN_GGA(X4, s(X2)) 6.97/2.74 6.97/2.74 The TRS R consists of the following rules: 6.97/2.74 6.97/2.74 subcB_in_gga(s(X1), X2) -> U13_gga(X1, X2, subcD_in_gga(X1, X2)) 6.97/2.74 subcD_in_gga(s(X1), s(X2)) -> U11_gga(X1, X2, subcD_in_gga(X1, X2)) 6.97/2.74 subcD_in_gga(X1, 0) -> subcD_out_gga(X1, 0, X1) 6.97/2.74 U11_gga(X1, X2, subcD_out_gga(X1, X2, X3)) -> subcD_out_gga(s(X1), s(X2), X3) 6.97/2.74 U13_gga(X1, X2, subcD_out_gga(X1, X2, X3)) -> subcB_out_gga(s(X1), X2, X3) 6.97/2.74 6.97/2.74 The set Q consists of the following terms: 6.97/2.74 6.97/2.74 subcB_in_gga(x0, x1) 6.97/2.74 subcD_in_gga(x0, x1) 6.97/2.74 U11_gga(x0, x1, x2) 6.97/2.74 U13_gga(x0, x1, x2) 6.97/2.74 6.97/2.74 We have to consider all (P,Q,R)-chains. 6.97/2.74 ---------------------------------------- 6.97/2.74 6.97/2.74 (24) QDPOrderProof (EQUIVALENT) 6.97/2.74 We use the reduction pair processor [LPAR04,JAR06]. 6.97/2.74 6.97/2.74 6.97/2.74 The following pairs can be oriented strictly and are deleted. 6.97/2.74 6.97/2.74 REMA_IN_GGA(X1, s(X2)) -> U4_GGA(X1, X2, subcB_in_gga(X1, X2)) 6.97/2.74 U4_GGA(X1, X2, subcB_out_gga(X1, X2, X4)) -> REMA_IN_GGA(X4, s(X2)) 6.97/2.74 The remaining pairs can at least be oriented weakly. 6.97/2.74 Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: 6.97/2.74 6.97/2.74 POL( U4_GGA_3(x_1, ..., x_3) ) = x_3 6.97/2.74 POL( subcB_in_gga_2(x_1, x_2) ) = 2x_1 6.97/2.74 POL( s_1(x_1) ) = 2x_1 + 1 6.97/2.74 POL( U13_gga_3(x_1, ..., x_3) ) = x_3 + 2 6.97/2.74 POL( subcD_in_gga_2(x_1, x_2) ) = 2x_1 6.97/2.74 POL( U11_gga_3(x_1, ..., x_3) ) = x_1 + x_3 + 1 6.97/2.74 POL( 0 ) = 0 6.97/2.74 POL( subcD_out_gga_3(x_1, ..., x_3) ) = 2x_3 6.97/2.74 POL( subcB_out_gga_3(x_1, ..., x_3) ) = 2x_3 + 2 6.97/2.74 POL( REMA_IN_GGA_2(x_1, x_2) ) = 2x_1 + 1 6.97/2.74 6.97/2.74 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 6.97/2.74 6.97/2.74 subcB_in_gga(s(X1), X2) -> U13_gga(X1, X2, subcD_in_gga(X1, X2)) 6.97/2.74 subcD_in_gga(s(X1), s(X2)) -> U11_gga(X1, X2, subcD_in_gga(X1, X2)) 6.97/2.74 subcD_in_gga(X1, 0) -> subcD_out_gga(X1, 0, X1) 6.97/2.74 U13_gga(X1, X2, subcD_out_gga(X1, X2, X3)) -> subcB_out_gga(s(X1), X2, X3) 6.97/2.74 U11_gga(X1, X2, subcD_out_gga(X1, X2, X3)) -> subcD_out_gga(s(X1), s(X2), X3) 6.97/2.74 6.97/2.74 6.97/2.74 ---------------------------------------- 6.97/2.74 6.97/2.74 (25) 6.97/2.74 Obligation: 6.97/2.74 Q DP problem: 6.97/2.74 P is empty. 6.97/2.74 The TRS R consists of the following rules: 6.97/2.74 6.97/2.74 subcB_in_gga(s(X1), X2) -> U13_gga(X1, X2, subcD_in_gga(X1, X2)) 6.97/2.74 subcD_in_gga(s(X1), s(X2)) -> U11_gga(X1, X2, subcD_in_gga(X1, X2)) 6.97/2.74 subcD_in_gga(X1, 0) -> subcD_out_gga(X1, 0, X1) 6.97/2.74 U11_gga(X1, X2, subcD_out_gga(X1, X2, X3)) -> subcD_out_gga(s(X1), s(X2), X3) 6.97/2.74 U13_gga(X1, X2, subcD_out_gga(X1, X2, X3)) -> subcB_out_gga(s(X1), X2, X3) 6.97/2.74 6.97/2.74 The set Q consists of the following terms: 6.97/2.74 6.97/2.74 subcB_in_gga(x0, x1) 6.97/2.74 subcD_in_gga(x0, x1) 6.97/2.74 U11_gga(x0, x1, x2) 6.97/2.74 U13_gga(x0, x1, x2) 6.97/2.74 6.97/2.74 We have to consider all (P,Q,R)-chains. 6.97/2.74 ---------------------------------------- 6.97/2.74 6.97/2.74 (26) PisEmptyProof (EQUIVALENT) 6.97/2.74 The TRS P is empty. Hence, there is no (P,Q,R) chain. 6.97/2.74 ---------------------------------------- 6.97/2.74 6.97/2.74 (27) 6.97/2.74 YES 6.97/2.77 EOF