10.77/3.64 YES 10.90/3.69 proof of /export/starexec/sandbox/benchmark/theBenchmark.pl 10.90/3.69 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 10.90/3.69 10.90/3.69 10.90/3.69 Left Termination of the query pattern 10.90/3.69 10.90/3.69 p(g) 10.90/3.69 10.90/3.69 w.r.t. the given Prolog program could successfully be proven: 10.90/3.69 10.90/3.69 (0) Prolog 10.90/3.69 (1) PrologToDTProblemTransformerProof [SOUND, 0 ms] 10.90/3.69 (2) TRIPLES 10.90/3.69 (3) TriplesToPiDPProof [SOUND, 0 ms] 10.90/3.69 (4) PiDP 10.90/3.69 (5) DependencyGraphProof [EQUIVALENT, 0 ms] 10.90/3.69 (6) PiDP 10.90/3.69 (7) PiDPToQDPProof [EQUIVALENT, 24 ms] 10.90/3.69 (8) QDP 10.90/3.69 (9) QDPQMonotonicMRRProof [EQUIVALENT, 217 ms] 10.90/3.69 (10) QDP 10.90/3.69 (11) DependencyGraphProof [EQUIVALENT, 0 ms] 10.90/3.69 (12) QDP 10.90/3.69 (13) UsableRulesProof [EQUIVALENT, 0 ms] 10.90/3.69 (14) QDP 10.90/3.69 (15) QReductionProof [EQUIVALENT, 1 ms] 10.90/3.69 (16) QDP 10.90/3.69 (17) UsableRulesReductionPairsProof [EQUIVALENT, 6 ms] 10.90/3.69 (18) QDP 10.90/3.69 (19) PisEmptyProof [EQUIVALENT, 0 ms] 10.90/3.69 (20) YES 10.90/3.69 10.90/3.69 10.90/3.69 ---------------------------------------- 10.90/3.69 10.90/3.69 (0) 10.90/3.69 Obligation: 10.90/3.69 Clauses: 10.90/3.69 10.90/3.69 p(.(X, [])). 10.90/3.69 p(.(s(s(X)), .(Y, Xs))) :- ','(p(.(X, .(Y, Xs))), p(.(s(s(s(s(Y)))), Xs))). 10.90/3.69 p(.(0, Xs)) :- p(Xs). 10.90/3.69 10.90/3.69 10.90/3.69 Query: p(g) 10.90/3.69 ---------------------------------------- 10.90/3.69 10.90/3.69 (1) PrologToDTProblemTransformerProof (SOUND) 10.90/3.69 Built DT problem from termination graph DT10. 10.90/3.69 10.90/3.69 { 10.90/3.69 "root": 1, 10.90/3.69 "program": { 10.90/3.69 "directives": [], 10.90/3.69 "clauses": [ 10.90/3.69 [ 10.90/3.69 "(p (. X ([])))", 10.90/3.69 null 10.90/3.69 ], 10.90/3.69 [ 10.90/3.69 "(p (. (s (s X)) (. Y Xs)))", 10.90/3.69 "(',' (p (. X (. Y Xs))) (p (. (s (s (s (s Y)))) Xs)))" 10.90/3.69 ], 10.90/3.69 [ 10.90/3.69 "(p (. (0) Xs))", 10.90/3.69 "(p Xs)" 10.90/3.69 ] 10.90/3.69 ] 10.90/3.69 }, 10.90/3.69 "graph": { 10.90/3.69 "nodes": { 10.90/3.69 "type": "Nodes", 10.90/3.69 "150": { 10.90/3.69 "goal": [{ 10.90/3.69 "clause": -1, 10.90/3.69 "scope": -1, 10.90/3.69 "term": "(',' (p (. (s (s (s (s T25)))) T26)) (p (. (s (s (s (s T25)))) T26)))" 10.90/3.69 }], 10.90/3.69 "kb": { 10.90/3.69 "nonunifying": [], 10.90/3.69 "intvars": {}, 10.90/3.69 "arithmetic": { 10.90/3.69 "type": "PlainIntegerRelationState", 10.90/3.69 "relations": [] 10.90/3.69 }, 10.90/3.69 "ground": [ 10.90/3.69 "T25", 10.90/3.69 "T26" 10.90/3.69 ], 10.90/3.69 "free": [], 10.90/3.69 "exprvars": [] 10.90/3.69 } 10.90/3.69 }, 10.90/3.69 "151": { 10.90/3.69 "goal": [{ 10.90/3.69 "clause": -1, 10.90/3.69 "scope": -1, 10.90/3.69 "term": "(p (. (s (s (s (s T25)))) T26))" 10.90/3.69 }], 10.90/3.69 "kb": { 10.90/3.69 "nonunifying": [], 10.90/3.69 "intvars": {}, 10.90/3.69 "arithmetic": { 10.90/3.69 "type": "PlainIntegerRelationState", 10.90/3.69 "relations": [] 10.90/3.69 }, 10.90/3.69 "ground": [ 10.90/3.69 "T25", 10.90/3.69 "T26" 10.90/3.69 ], 10.90/3.69 "free": [], 10.90/3.69 "exprvars": [] 10.90/3.69 } 10.90/3.69 }, 10.90/3.69 "152": { 10.90/3.69 "goal": [{ 10.90/3.69 "clause": -1, 10.90/3.69 "scope": -1, 10.90/3.69 "term": "(p (. (s (s (s (s T25)))) T26))" 10.90/3.69 }], 10.90/3.69 "kb": { 10.90/3.69 "nonunifying": [], 10.90/3.69 "intvars": {}, 10.90/3.69 "arithmetic": { 10.90/3.69 "type": "PlainIntegerRelationState", 10.90/3.69 "relations": [] 10.90/3.69 }, 10.90/3.69 "ground": [ 10.90/3.69 "T25", 10.90/3.69 "T26" 10.90/3.69 ], 10.90/3.69 "free": [], 10.90/3.69 "exprvars": [] 10.90/3.69 } 10.90/3.69 }, 10.90/3.69 "153": { 10.90/3.69 "goal": [{ 10.90/3.69 "clause": 2, 10.90/3.69 "scope": 3, 10.90/3.69 "term": "(',' (p (. T8 (. T9 T10))) (p (. (s (s (s (s T9)))) T10)))" 10.90/3.69 }], 10.90/3.69 "kb": { 10.90/3.69 "nonunifying": [], 10.90/3.69 "intvars": {}, 10.90/3.69 "arithmetic": { 10.90/3.69 "type": "PlainIntegerRelationState", 10.90/3.69 "relations": [] 10.90/3.69 }, 10.90/3.69 "ground": [ 10.90/3.69 "T8", 10.90/3.69 "T9", 10.90/3.69 "T10" 10.90/3.69 ], 10.90/3.69 "free": [], 10.90/3.69 "exprvars": [] 10.90/3.69 } 10.90/3.69 }, 10.90/3.69 "154": { 10.90/3.69 "goal": [ 10.90/3.69 { 10.90/3.69 "clause": -1, 10.90/3.69 "scope": 3, 10.90/3.69 "term": null 10.90/3.69 }, 10.90/3.69 { 10.90/3.69 "clause": 2, 10.90/3.69 "scope": 1, 10.90/3.69 "term": "(p (. (s (s T8)) (. T9 T10)))" 10.90/3.69 } 10.90/3.69 ], 10.90/3.69 "kb": { 10.90/3.69 "nonunifying": [], 10.90/3.69 "intvars": {}, 10.90/3.69 "arithmetic": { 10.90/3.69 "type": "PlainIntegerRelationState", 10.90/3.69 "relations": [] 10.90/3.69 }, 10.90/3.69 "ground": [ 10.90/3.69 "T8", 10.90/3.69 "T9", 10.90/3.69 "T10" 10.90/3.69 ], 10.90/3.69 "free": [], 10.90/3.69 "exprvars": [] 10.90/3.69 } 10.90/3.69 }, 10.90/3.69 "155": { 10.90/3.69 "goal": [{ 10.90/3.69 "clause": -1, 10.90/3.69 "scope": -1, 10.90/3.69 "term": "(',' (p (. T45 T46)) (p (. (s (s (s (s T45)))) T46)))" 10.90/3.69 }], 10.90/3.69 "kb": { 10.90/3.69 "nonunifying": [], 10.90/3.69 "intvars": {}, 10.90/3.69 "arithmetic": { 10.90/3.69 "type": "PlainIntegerRelationState", 10.90/3.69 "relations": [] 10.90/3.69 }, 10.90/3.69 "ground": [ 10.90/3.69 "T45", 10.90/3.69 "T46" 10.90/3.69 ], 10.90/3.69 "free": [], 10.90/3.69 "exprvars": [] 10.90/3.69 } 10.90/3.69 }, 10.90/3.69 "156": { 10.90/3.69 "goal": [], 10.90/3.69 "kb": { 10.90/3.69 "nonunifying": [], 10.90/3.69 "intvars": {}, 10.90/3.69 "arithmetic": { 10.90/3.69 "type": "PlainIntegerRelationState", 10.90/3.69 "relations": [] 10.90/3.69 }, 10.90/3.69 "ground": [], 10.90/3.69 "free": [], 10.90/3.69 "exprvars": [] 10.90/3.69 } 10.90/3.69 }, 10.90/3.69 "157": { 10.90/3.69 "goal": [{ 10.90/3.69 "clause": -1, 10.90/3.69 "scope": -1, 10.90/3.69 "term": "(p (. 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T3 ([])))" 10.90/3.69 }, 10.90/3.69 { 10.90/3.69 "clause": 2, 10.90/3.69 "scope": 1, 10.90/3.69 "term": "(p (. T3 ([])))" 10.90/3.69 } 10.90/3.69 ], 10.90/3.69 "kb": { 10.90/3.69 "nonunifying": [], 10.90/3.69 "intvars": {}, 10.90/3.69 "arithmetic": { 10.90/3.69 "type": "PlainIntegerRelationState", 10.90/3.69 "relations": [] 10.90/3.69 }, 10.90/3.69 "ground": ["T3"], 10.90/3.69 "free": [], 10.90/3.69 "exprvars": [] 10.90/3.69 } 10.90/3.69 }, 10.90/3.69 "171": { 10.90/3.69 "goal": [{ 10.90/3.69 "clause": 2, 10.90/3.69 "scope": 1, 10.90/3.69 "term": "(p (. (s (s T8)) (. T9 T10)))" 10.90/3.69 }], 10.90/3.69 "kb": { 10.90/3.69 "nonunifying": [], 10.90/3.69 "intvars": {}, 10.90/3.69 "arithmetic": { 10.90/3.69 "type": "PlainIntegerRelationState", 10.90/3.69 "relations": [] 10.90/3.69 }, 10.90/3.69 "ground": [ 10.90/3.69 "T8", 10.90/3.69 "T9", 10.90/3.69 "T10" 10.90/3.69 ], 10.90/3.69 "free": [], 10.90/3.69 "exprvars": [] 10.90/3.69 } 10.90/3.69 }, 10.90/3.69 "172": { 10.90/3.69 "goal": [], 10.90/3.69 "kb": { 10.90/3.69 "nonunifying": [], 10.90/3.69 "intvars": {}, 10.90/3.69 "arithmetic": { 10.90/3.69 "type": "PlainIntegerRelationState", 10.90/3.69 "relations": [] 10.90/3.69 }, 10.90/3.69 "ground": [], 10.90/3.69 "free": [], 10.90/3.69 "exprvars": [] 10.90/3.69 } 10.90/3.69 }, 10.90/3.69 "173": { 10.90/3.69 "goal": [{ 10.90/3.69 "clause": -1, 10.90/3.69 "scope": -1, 10.90/3.69 "term": "(p T54)" 10.90/3.69 }], 10.90/3.69 "kb": { 10.90/3.69 "nonunifying": [[ 10.90/3.69 "(p (. (0) T54))", 10.90/3.69 "(p (. 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T3 ([])))" 10.90/3.69 } 10.90/3.69 ], 10.90/3.69 "kb": { 10.90/3.69 "nonunifying": [], 10.90/3.69 "intvars": {}, 10.90/3.69 "arithmetic": { 10.90/3.69 "type": "PlainIntegerRelationState", 10.90/3.69 "relations": [] 10.90/3.69 }, 10.90/3.69 "ground": ["T3"], 10.90/3.69 "free": [], 10.90/3.69 "exprvars": [] 10.90/3.69 } 10.90/3.69 }, 10.90/3.69 "175": { 10.90/3.69 "goal": [ 10.90/3.69 { 10.90/3.69 "clause": 0, 10.90/3.69 "scope": 4, 10.90/3.69 "term": "(p T54)" 10.90/3.69 }, 10.90/3.69 { 10.90/3.69 "clause": 1, 10.90/3.69 "scope": 4, 10.90/3.69 "term": "(p T54)" 10.90/3.69 }, 10.90/3.69 { 10.90/3.69 "clause": 2, 10.90/3.69 "scope": 4, 10.90/3.69 "term": "(p T54)" 10.90/3.69 } 10.90/3.69 ], 10.90/3.69 "kb": { 10.90/3.69 "nonunifying": [[ 10.90/3.69 "(p (. (0) T54))", 10.90/3.69 "(p (. 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(s (s (s (s T25)))) T26)))" 10.90/3.69 }], 10.90/3.69 "kb": { 10.90/3.69 "nonunifying": [], 10.90/3.69 "intvars": {}, 10.90/3.69 "arithmetic": { 10.90/3.69 "type": "PlainIntegerRelationState", 10.90/3.69 "relations": [] 10.90/3.69 }, 10.90/3.69 "ground": [ 10.90/3.69 "T24", 10.90/3.69 "T25", 10.90/3.69 "T26" 10.90/3.69 ], 10.90/3.69 "free": [], 10.90/3.69 "exprvars": [] 10.90/3.69 } 10.90/3.69 }, 10.90/3.69 "148": { 10.90/3.69 "goal": [], 10.90/3.69 "kb": { 10.90/3.69 "nonunifying": [], 10.90/3.69 "intvars": {}, 10.90/3.69 "arithmetic": { 10.90/3.69 "type": "PlainIntegerRelationState", 10.90/3.69 "relations": [] 10.90/3.69 }, 10.90/3.69 "ground": [], 10.90/3.69 "free": [], 10.90/3.69 "exprvars": [] 10.90/3.69 } 10.90/3.69 }, 10.90/3.69 "149": { 10.90/3.69 "goal": [{ 10.90/3.69 "clause": -1, 10.90/3.69 "scope": -1, 10.90/3.69 "term": "(p (. T24 (. T25 T26)))" 10.90/3.69 }], 10.90/3.69 "kb": { 10.90/3.69 "nonunifying": [], 10.90/3.69 "intvars": {}, 10.90/3.69 "arithmetic": { 10.90/3.69 "type": "PlainIntegerRelationState", 10.90/3.69 "relations": [] 10.90/3.69 }, 10.90/3.69 "ground": [ 10.90/3.69 "T24", 10.90/3.69 "T25", 10.90/3.69 "T26" 10.90/3.69 ], 10.90/3.69 "free": [], 10.90/3.69 "exprvars": [] 10.90/3.69 } 10.90/3.69 } 10.90/3.69 }, 10.90/3.69 "edges": [ 10.90/3.69 { 10.90/3.69 "from": 1, 10.90/3.69 "to": 128, 10.90/3.69 "label": "CASE" 10.90/3.69 }, 10.90/3.69 { 10.90/3.69 "from": 128, 10.90/3.69 "to": 129, 10.90/3.69 "label": "EVAL with clause\np(.(X2, [])).\nand substitutionX2 -> T3,\nT1 -> .(T3, [])" 10.90/3.69 }, 10.90/3.69 { 10.90/3.69 "from": 128, 10.90/3.69 "to": 130, 10.90/3.69 "label": "EVAL-BACKTRACK" 10.90/3.69 }, 10.90/3.69 { 10.90/3.69 "from": 129, 10.90/3.69 "to": 131, 10.90/3.69 "label": "SUCCESS" 10.90/3.69 }, 10.90/3.69 { 10.90/3.69 "from": 130, 10.90/3.69 "to": 141, 10.90/3.69 "label": "EVAL with clause\np(.(s(s(X16)), .(X17, X18))) :- ','(p(.(X16, .(X17, X18))), p(.(s(s(s(s(X17)))), X18))).\nand substitutionX16 -> T8,\nX17 -> T9,\nX18 -> T10,\nT1 -> .(s(s(T8)), .(T9, T10))" 10.90/3.69 }, 10.90/3.69 { 10.90/3.69 "from": 130, 10.90/3.69 "to": 142, 10.90/3.69 "label": "EVAL-BACKTRACK" 10.90/3.69 }, 10.90/3.69 { 10.90/3.69 "from": 131, 10.90/3.69 "to": 132, 10.90/3.69 "label": "BACKTRACK\nfor clause: p(.(s(s(X)), .(Y, Xs))) :- ','(p(.(X, .(Y, Xs))), p(.(s(s(s(s(Y)))), Xs)))because of non-unification" 10.90/3.69 }, 10.90/3.69 { 10.90/3.69 "from": 132, 10.90/3.69 "to": 135, 10.90/3.69 "label": "EVAL with clause\np(.(0, X7)) :- p(X7).\nand substitutionT3 -> 0,\nX7 -> []" 10.90/3.69 }, 10.90/3.69 { 10.90/3.69 "from": 132, 10.90/3.69 "to": 136, 10.90/3.69 "label": "EVAL-BACKTRACK" 10.90/3.69 }, 10.90/3.69 { 10.90/3.69 "from": 135, 10.90/3.69 "to": 137, 10.90/3.69 "label": "CASE" 10.90/3.69 }, 10.90/3.69 { 10.90/3.69 "from": 137, 10.90/3.69 "to": 138, 10.90/3.69 "label": "BACKTRACK\nfor clause: p(.(X, []))because of non-unification" 10.90/3.69 }, 10.90/3.69 { 10.90/3.69 "from": 138, 10.90/3.69 "to": 139, 10.90/3.69 "label": "BACKTRACK\nfor clause: p(.(s(s(X)), .(Y, Xs))) :- ','(p(.(X, .(Y, Xs))), p(.(s(s(s(s(Y)))), Xs)))because of non-unification" 10.90/3.69 }, 10.90/3.69 { 10.90/3.69 "from": 139, 10.90/3.69 "to": 140, 10.90/3.69 "label": "BACKTRACK\nfor clause: p(.(0, Xs)) :- p(Xs)because of non-unification" 10.90/3.69 }, 10.90/3.69 { 10.90/3.69 "from": 141, 10.90/3.69 "to": 143, 10.90/3.69 "label": "CASE" 10.90/3.69 }, 10.90/3.69 { 10.90/3.69 "from": 142, 10.90/3.69 "to": 173, 10.90/3.69 "label": "EVAL with clause\np(.(0, X58)) :- p(X58).\nand substitutionX58 -> T54,\nT1 -> .(0, T54)" 10.90/3.69 }, 10.90/3.69 { 10.90/3.69 "from": 142, 10.90/3.69 "to": 174, 10.90/3.69 "label": "EVAL-BACKTRACK" 10.90/3.69 }, 10.90/3.69 { 10.90/3.69 "from": 143, 10.90/3.69 "to": 144, 10.90/3.69 "label": "BACKTRACK\nfor clause: p(.(X, []))because of non-unification" 10.90/3.70 }, 10.90/3.70 { 10.90/3.70 "from": 144, 10.90/3.70 "to": 145, 10.90/3.70 "label": "PARALLEL" 10.90/3.70 }, 10.90/3.70 { 10.90/3.70 "from": 144, 10.90/3.70 "to": 146, 10.90/3.70 "label": "PARALLEL" 10.90/3.70 }, 10.90/3.70 { 10.90/3.70 "from": 145, 10.90/3.70 "to": 147, 10.90/3.70 "label": "EVAL with clause\np(.(s(s(X32)), .(X33, X34))) :- ','(p(.(X32, .(X33, X34))), p(.(s(s(s(s(X33)))), X34))).\nand substitutionX32 -> T24,\nT8 -> s(s(T24)),\nT9 -> T25,\nX33 -> T25,\nT10 -> T26,\nX34 -> T26" 10.90/3.70 }, 10.90/3.70 { 10.90/3.70 "from": 145, 10.90/3.70 "to": 148, 10.90/3.70 "label": "EVAL-BACKTRACK" 10.90/3.70 }, 10.90/3.70 { 10.90/3.70 "from": 146, 10.90/3.70 "to": 153, 10.90/3.70 "label": "PARALLEL" 10.90/3.70 }, 10.90/3.70 { 10.90/3.70 "from": 146, 10.90/3.70 "to": 154, 10.90/3.70 "label": "PARALLEL" 10.90/3.70 }, 10.90/3.70 { 10.90/3.70 "from": 147, 10.90/3.70 "to": 149, 10.90/3.70 "label": "SPLIT 1" 10.90/3.70 }, 10.90/3.70 { 10.90/3.70 "from": 147, 10.90/3.70 "to": 150, 10.90/3.70 "label": "SPLIT 2\nnew knowledge:\nT24 is ground\nT25 is ground\nT26 is ground" 10.90/3.70 }, 10.90/3.70 { 10.90/3.70 "from": 149, 10.90/3.70 "to": 1, 10.90/3.70 "label": "INSTANCE with matching:\nT1 -> .(T24, .(T25, T26))" 10.90/3.70 }, 10.90/3.70 { 10.90/3.70 "from": 150, 10.90/3.70 "to": 151, 10.90/3.70 "label": "SPLIT 1" 10.90/3.70 }, 10.90/3.70 { 10.90/3.70 "from": 150, 10.90/3.70 "to": 152, 10.90/3.70 "label": "SPLIT 2\nnew knowledge:\nT25 is ground\nT26 is ground" 10.90/3.70 }, 10.90/3.70 { 10.90/3.70 "from": 151, 10.90/3.70 "to": 1, 10.90/3.70 "label": "INSTANCE with matching:\nT1 -> .(s(s(s(s(T25)))), T26)" 10.90/3.70 }, 10.90/3.70 { 10.90/3.70 "from": 152, 10.90/3.70 "to": 1, 10.90/3.70 "label": "INSTANCE with matching:\nT1 -> .(s(s(s(s(T25)))), T26)" 10.90/3.70 }, 10.90/3.70 { 10.90/3.70 "from": 153, 10.90/3.70 "to": 155, 10.90/3.70 "label": "EVAL with clause\np(.(0, X49)) :- p(X49).\nand substitutionT8 -> 0,\nT9 -> T45,\nT10 -> T46,\nX49 -> .(T45, T46)" 10.90/3.70 }, 10.90/3.70 { 10.90/3.70 "from": 153, 10.90/3.70 "to": 156, 10.90/3.70 "label": "EVAL-BACKTRACK" 10.90/3.70 }, 10.90/3.70 { 10.90/3.70 "from": 154, 10.90/3.70 "to": 171, 10.90/3.70 "label": "FAILURE" 10.90/3.70 }, 10.90/3.70 { 10.90/3.70 "from": 155, 10.90/3.70 "to": 157, 10.90/3.70 "label": "SPLIT 1" 10.90/3.70 }, 10.90/3.70 { 10.90/3.70 "from": 155, 10.90/3.70 "to": 158, 10.90/3.70 "label": "SPLIT 2\nnew knowledge:\nT45 is ground\nT46 is ground" 10.90/3.70 }, 10.90/3.70 { 10.90/3.70 "from": 157, 10.90/3.70 "to": 1, 10.90/3.70 "label": "INSTANCE with matching:\nT1 -> .(T45, T46)" 10.90/3.70 }, 10.90/3.70 { 10.90/3.70 "from": 158, 10.90/3.70 "to": 1, 10.90/3.70 "label": "INSTANCE with matching:\nT1 -> .(s(s(s(s(T45)))), T46)" 10.90/3.70 }, 10.90/3.70 { 10.90/3.70 "from": 171, 10.90/3.70 "to": 172, 10.90/3.70 "label": "BACKTRACK\nfor clause: p(.(0, Xs)) :- p(Xs)because of non-unification" 10.90/3.70 }, 10.90/3.70 { 10.90/3.70 "from": 173, 10.90/3.70 "to": 175, 10.90/3.70 "label": "CASE" 10.90/3.70 }, 10.90/3.70 { 10.90/3.70 "from": 175, 10.90/3.70 "to": 176, 10.90/3.70 "label": "PARALLEL" 10.90/3.70 }, 10.90/3.70 { 10.90/3.70 "from": 175, 10.90/3.70 "to": 177, 10.90/3.70 "label": "PARALLEL" 10.90/3.70 }, 10.90/3.70 { 10.90/3.70 "from": 176, 10.90/3.70 "to": 178, 10.90/3.70 "label": "EVAL with clause\np(.(X63, [])).\nand substitutionX63 -> T59,\nT54 -> .(T59, [])" 10.90/3.70 }, 10.90/3.70 { 10.90/3.70 "from": 176, 10.90/3.70 "to": 179, 10.90/3.70 "label": "EVAL-BACKTRACK" 10.90/3.70 }, 10.90/3.70 { 10.90/3.70 "from": 177, 10.90/3.70 "to": 181, 10.90/3.70 "label": "PARALLEL" 10.90/3.70 }, 10.90/3.70 { 10.90/3.70 "from": 177, 10.90/3.70 "to": 182, 10.90/3.70 "label": "PARALLEL" 10.90/3.70 }, 10.90/3.70 { 10.90/3.70 "from": 178, 10.90/3.70 "to": 180, 10.90/3.70 "label": "SUCCESS" 10.90/3.70 }, 10.90/3.70 { 10.90/3.70 "from": 181, 10.90/3.70 "to": 183, 10.90/3.70 "label": "EVAL with clause\np(.(s(s(X76)), .(X77, X78))) :- ','(p(.(X76, .(X77, X78))), p(.(s(s(s(s(X77)))), X78))).\nand substitutionX76 -> T72,\nX77 -> T73,\nX78 -> T74,\nT54 -> .(s(s(T72)), .(T73, T74))" 10.90/3.70 }, 10.90/3.70 { 10.90/3.70 "from": 181, 10.90/3.70 "to": 184, 10.90/3.70 "label": "EVAL-BACKTRACK" 10.90/3.70 }, 10.90/3.70 { 10.90/3.70 "from": 182, 10.90/3.70 "to": 187, 10.90/3.70 "label": "EVAL with clause\np(.(0, X87)) :- p(X87).\nand substitutionX87 -> T83,\nT54 -> .(0, T83)" 10.90/3.70 }, 10.90/3.70 { 10.90/3.70 "from": 182, 10.90/3.70 "to": 188, 10.90/3.70 "label": "EVAL-BACKTRACK" 10.90/3.70 }, 10.90/3.70 { 10.90/3.70 "from": 183, 10.90/3.70 "to": 185, 10.90/3.70 "label": "SPLIT 1" 10.90/3.70 }, 10.90/3.70 { 10.90/3.70 "from": 183, 10.90/3.70 "to": 186, 10.90/3.70 "label": "SPLIT 2\nnew knowledge:\nT72 is ground\nT73 is ground\nT74 is ground" 10.90/3.70 }, 10.90/3.70 { 10.90/3.70 "from": 185, 10.90/3.70 "to": 1, 10.90/3.70 "label": "INSTANCE with matching:\nT1 -> .(T72, .(T73, T74))" 10.90/3.70 }, 10.90/3.70 { 10.90/3.70 "from": 186, 10.90/3.70 "to": 1, 10.90/3.70 "label": "INSTANCE with matching:\nT1 -> .(s(s(s(s(T73)))), T74)" 10.90/3.70 }, 10.90/3.70 { 10.90/3.70 "from": 187, 10.90/3.70 "to": 1, 10.90/3.70 "label": "INSTANCE with matching:\nT1 -> T83" 10.90/3.70 } 10.90/3.70 ], 10.90/3.70 "type": "Graph" 10.90/3.70 } 10.90/3.70 } 10.90/3.70 10.90/3.70 ---------------------------------------- 10.90/3.70 10.90/3.70 (2) 10.90/3.70 Obligation: 10.90/3.70 Triples: 10.90/3.70 10.90/3.70 pA(.(s(s(s(s(X1)))), .(X2, X3))) :- pA(.(X1, .(X2, X3))). 10.90/3.70 pA(.(s(s(s(s(X1)))), .(X2, X3))) :- ','(pcA(.(X1, .(X2, X3))), pA(.(s(s(s(s(X2)))), X3))). 10.90/3.70 pA(.(s(s(s(s(X1)))), .(X2, X3))) :- ','(pcA(.(X1, .(X2, X3))), ','(pcA(.(s(s(s(s(X2)))), X3)), pA(.(s(s(s(s(X2)))), X3)))). 10.90/3.70 pA(.(s(s(0)), .(X1, X2))) :- pA(.(X1, X2)). 10.90/3.70 pA(.(s(s(0)), .(X1, X2))) :- ','(pcA(.(X1, X2)), pA(.(s(s(s(s(X1)))), X2))). 10.90/3.70 pA(.(0, .(s(s(X1)), .(X2, X3)))) :- pA(.(X1, .(X2, X3))). 10.90/3.70 pA(.(0, .(s(s(X1)), .(X2, X3)))) :- ','(pcA(.(X1, .(X2, X3))), pA(.(s(s(s(s(X2)))), X3))). 10.90/3.70 pA(.(0, .(0, X1))) :- pA(X1). 10.90/3.70 10.90/3.70 Clauses: 10.90/3.70 10.90/3.70 pcA(.(X1, [])). 10.90/3.70 pcA(.(s(s(s(s(X1)))), .(X2, X3))) :- ','(pcA(.(X1, .(X2, X3))), ','(pcA(.(s(s(s(s(X2)))), X3)), pcA(.(s(s(s(s(X2)))), X3)))). 10.90/3.70 pcA(.(s(s(0)), .(X1, X2))) :- ','(pcA(.(X1, X2)), pcA(.(s(s(s(s(X1)))), X2))). 10.90/3.70 pcA(.(0, .(X1, []))). 10.90/3.70 pcA(.(0, .(s(s(X1)), .(X2, X3)))) :- ','(pcA(.(X1, .(X2, X3))), pcA(.(s(s(s(s(X2)))), X3))). 10.90/3.70 pcA(.(0, .(0, X1))) :- pcA(X1). 10.90/3.70 10.90/3.70 Afs: 10.90/3.70 10.90/3.70 pA(x1) = pA(x1) 10.90/3.70 10.90/3.70 10.90/3.70 ---------------------------------------- 10.90/3.70 10.90/3.70 (3) TriplesToPiDPProof (SOUND) 10.90/3.70 We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes: 10.90/3.70 10.90/3.70 pA_in_1: (b) 10.90/3.70 10.90/3.70 pcA_in_1: (b) 10.90/3.70 10.90/3.70 Transforming TRIPLES into the following Term Rewriting System: 10.90/3.70 10.90/3.70 Pi DP problem: 10.90/3.70 The TRS P consists of the following rules: 10.90/3.70 10.90/3.70 PA_IN_G(.(s(s(s(s(X1)))), .(X2, X3))) -> U1_G(X1, X2, X3, pA_in_g(.(X1, .(X2, X3)))) 10.90/3.70 PA_IN_G(.(s(s(s(s(X1)))), .(X2, X3))) -> PA_IN_G(.(X1, .(X2, X3))) 10.90/3.70 PA_IN_G(.(s(s(s(s(X1)))), .(X2, X3))) -> U2_G(X1, X2, X3, pcA_in_g(.(X1, .(X2, X3)))) 10.90/3.70 U2_G(X1, X2, X3, pcA_out_g(.(X1, .(X2, X3)))) -> U3_G(X1, X2, X3, pA_in_g(.(s(s(s(s(X2)))), X3))) 10.90/3.70 U2_G(X1, X2, X3, pcA_out_g(.(X1, .(X2, X3)))) -> PA_IN_G(.(s(s(s(s(X2)))), X3)) 10.90/3.70 PA_IN_G(.(s(s(0)), .(X1, X2))) -> U6_G(X1, X2, pA_in_g(.(X1, X2))) 10.90/3.70 PA_IN_G(.(s(s(0)), .(X1, X2))) -> PA_IN_G(.(X1, X2)) 10.90/3.70 PA_IN_G(.(s(s(0)), .(X1, X2))) -> U7_G(X1, X2, pcA_in_g(.(X1, X2))) 10.90/3.70 U7_G(X1, X2, pcA_out_g(.(X1, X2))) -> U8_G(X1, X2, pA_in_g(.(s(s(s(s(X1)))), X2))) 10.90/3.70 U7_G(X1, X2, pcA_out_g(.(X1, X2))) -> PA_IN_G(.(s(s(s(s(X1)))), X2)) 10.90/3.70 PA_IN_G(.(0, .(s(s(X1)), .(X2, X3)))) -> U9_G(X1, X2, X3, pA_in_g(.(X1, .(X2, X3)))) 10.90/3.70 PA_IN_G(.(0, .(s(s(X1)), .(X2, X3)))) -> PA_IN_G(.(X1, .(X2, X3))) 10.90/3.70 PA_IN_G(.(0, .(s(s(X1)), .(X2, X3)))) -> U10_G(X1, X2, X3, pcA_in_g(.(X1, .(X2, X3)))) 10.90/3.70 U10_G(X1, X2, X3, pcA_out_g(.(X1, .(X2, X3)))) -> U11_G(X1, X2, X3, pA_in_g(.(s(s(s(s(X2)))), X3))) 10.90/3.70 U10_G(X1, X2, X3, pcA_out_g(.(X1, .(X2, X3)))) -> PA_IN_G(.(s(s(s(s(X2)))), X3)) 10.90/3.70 PA_IN_G(.(0, .(0, X1))) -> U12_G(X1, pA_in_g(X1)) 10.90/3.70 PA_IN_G(.(0, .(0, X1))) -> PA_IN_G(X1) 10.90/3.70 U2_G(X1, X2, X3, pcA_out_g(.(X1, .(X2, X3)))) -> U4_G(X1, X2, X3, pcA_in_g(.(s(s(s(s(X2)))), X3))) 10.90/3.70 U4_G(X1, X2, X3, pcA_out_g(.(s(s(s(s(X2)))), X3))) -> U5_G(X1, X2, X3, pA_in_g(.(s(s(s(s(X2)))), X3))) 10.90/3.70 U4_G(X1, X2, X3, pcA_out_g(.(s(s(s(s(X2)))), X3))) -> PA_IN_G(.(s(s(s(s(X2)))), X3)) 10.90/3.70 10.90/3.70 The TRS R consists of the following rules: 10.90/3.70 10.90/3.70 pcA_in_g(.(X1, [])) -> pcA_out_g(.(X1, [])) 10.90/3.70 pcA_in_g(.(s(s(s(s(X1)))), .(X2, X3))) -> U14_g(X1, X2, X3, pcA_in_g(.(X1, .(X2, X3)))) 10.90/3.70 pcA_in_g(.(s(s(0)), .(X1, X2))) -> U17_g(X1, X2, pcA_in_g(.(X1, X2))) 10.90/3.70 pcA_in_g(.(0, .(X1, []))) -> pcA_out_g(.(0, .(X1, []))) 10.90/3.70 pcA_in_g(.(0, .(s(s(X1)), .(X2, X3)))) -> U19_g(X1, X2, X3, pcA_in_g(.(X1, .(X2, X3)))) 10.90/3.70 pcA_in_g(.(0, .(0, X1))) -> U21_g(X1, pcA_in_g(X1)) 10.90/3.70 U21_g(X1, pcA_out_g(X1)) -> pcA_out_g(.(0, .(0, X1))) 10.90/3.70 U19_g(X1, X2, X3, pcA_out_g(.(X1, .(X2, X3)))) -> U20_g(X1, X2, X3, pcA_in_g(.(s(s(s(s(X2)))), X3))) 10.90/3.70 U20_g(X1, X2, X3, pcA_out_g(.(s(s(s(s(X2)))), X3))) -> pcA_out_g(.(0, .(s(s(X1)), .(X2, X3)))) 10.90/3.70 U17_g(X1, X2, pcA_out_g(.(X1, X2))) -> U18_g(X1, X2, pcA_in_g(.(s(s(s(s(X1)))), X2))) 10.90/3.70 U18_g(X1, X2, pcA_out_g(.(s(s(s(s(X1)))), X2))) -> pcA_out_g(.(s(s(0)), .(X1, X2))) 10.90/3.70 U14_g(X1, X2, X3, pcA_out_g(.(X1, .(X2, X3)))) -> U15_g(X1, X2, X3, pcA_in_g(.(s(s(s(s(X2)))), X3))) 10.90/3.70 U15_g(X1, X2, X3, pcA_out_g(.(s(s(s(s(X2)))), X3))) -> U16_g(X1, X2, X3, pcA_in_g(.(s(s(s(s(X2)))), X3))) 10.90/3.70 U16_g(X1, X2, X3, pcA_out_g(.(s(s(s(s(X2)))), X3))) -> pcA_out_g(.(s(s(s(s(X1)))), .(X2, X3))) 10.90/3.70 10.90/3.70 Pi is empty. 10.90/3.70 We have to consider all (P,R,Pi)-chains 10.90/3.70 10.90/3.70 10.90/3.70 Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES 10.90/3.70 10.90/3.70 10.90/3.70 10.90/3.70 ---------------------------------------- 10.90/3.70 10.90/3.70 (4) 10.90/3.70 Obligation: 10.90/3.70 Pi DP problem: 10.90/3.70 The TRS P consists of the following rules: 10.90/3.70 10.90/3.70 PA_IN_G(.(s(s(s(s(X1)))), .(X2, X3))) -> U1_G(X1, X2, X3, pA_in_g(.(X1, .(X2, X3)))) 10.90/3.70 PA_IN_G(.(s(s(s(s(X1)))), .(X2, X3))) -> PA_IN_G(.(X1, .(X2, X3))) 10.90/3.70 PA_IN_G(.(s(s(s(s(X1)))), .(X2, X3))) -> U2_G(X1, X2, X3, pcA_in_g(.(X1, .(X2, X3)))) 10.90/3.70 U2_G(X1, X2, X3, pcA_out_g(.(X1, .(X2, X3)))) -> U3_G(X1, X2, X3, pA_in_g(.(s(s(s(s(X2)))), X3))) 10.90/3.70 U2_G(X1, X2, X3, pcA_out_g(.(X1, .(X2, X3)))) -> PA_IN_G(.(s(s(s(s(X2)))), X3)) 10.90/3.70 PA_IN_G(.(s(s(0)), .(X1, X2))) -> U6_G(X1, X2, pA_in_g(.(X1, X2))) 10.90/3.70 PA_IN_G(.(s(s(0)), .(X1, X2))) -> PA_IN_G(.(X1, X2)) 10.90/3.70 PA_IN_G(.(s(s(0)), .(X1, X2))) -> U7_G(X1, X2, pcA_in_g(.(X1, X2))) 10.90/3.70 U7_G(X1, X2, pcA_out_g(.(X1, X2))) -> U8_G(X1, X2, pA_in_g(.(s(s(s(s(X1)))), X2))) 10.90/3.70 U7_G(X1, X2, pcA_out_g(.(X1, X2))) -> PA_IN_G(.(s(s(s(s(X1)))), X2)) 10.90/3.70 PA_IN_G(.(0, .(s(s(X1)), .(X2, X3)))) -> U9_G(X1, X2, X3, pA_in_g(.(X1, .(X2, X3)))) 10.90/3.70 PA_IN_G(.(0, .(s(s(X1)), .(X2, X3)))) -> PA_IN_G(.(X1, .(X2, X3))) 10.90/3.70 PA_IN_G(.(0, .(s(s(X1)), .(X2, X3)))) -> U10_G(X1, X2, X3, pcA_in_g(.(X1, .(X2, X3)))) 10.90/3.70 U10_G(X1, X2, X3, pcA_out_g(.(X1, .(X2, X3)))) -> U11_G(X1, X2, X3, pA_in_g(.(s(s(s(s(X2)))), X3))) 10.90/3.70 U10_G(X1, X2, X3, pcA_out_g(.(X1, .(X2, X3)))) -> PA_IN_G(.(s(s(s(s(X2)))), X3)) 10.90/3.70 PA_IN_G(.(0, .(0, X1))) -> U12_G(X1, pA_in_g(X1)) 10.90/3.70 PA_IN_G(.(0, .(0, X1))) -> PA_IN_G(X1) 10.90/3.70 U2_G(X1, X2, X3, pcA_out_g(.(X1, .(X2, X3)))) -> U4_G(X1, X2, X3, pcA_in_g(.(s(s(s(s(X2)))), X3))) 10.90/3.70 U4_G(X1, X2, X3, pcA_out_g(.(s(s(s(s(X2)))), X3))) -> U5_G(X1, X2, X3, pA_in_g(.(s(s(s(s(X2)))), X3))) 10.90/3.70 U4_G(X1, X2, X3, pcA_out_g(.(s(s(s(s(X2)))), X3))) -> PA_IN_G(.(s(s(s(s(X2)))), X3)) 10.90/3.70 10.90/3.70 The TRS R consists of the following rules: 10.90/3.70 10.90/3.70 pcA_in_g(.(X1, [])) -> pcA_out_g(.(X1, [])) 10.90/3.70 pcA_in_g(.(s(s(s(s(X1)))), .(X2, X3))) -> U14_g(X1, X2, X3, pcA_in_g(.(X1, .(X2, X3)))) 10.90/3.70 pcA_in_g(.(s(s(0)), .(X1, X2))) -> U17_g(X1, X2, pcA_in_g(.(X1, X2))) 10.90/3.70 pcA_in_g(.(0, .(X1, []))) -> pcA_out_g(.(0, .(X1, []))) 10.90/3.70 pcA_in_g(.(0, .(s(s(X1)), .(X2, X3)))) -> U19_g(X1, X2, X3, pcA_in_g(.(X1, .(X2, X3)))) 10.90/3.70 pcA_in_g(.(0, .(0, X1))) -> U21_g(X1, pcA_in_g(X1)) 10.90/3.70 U21_g(X1, pcA_out_g(X1)) -> pcA_out_g(.(0, .(0, X1))) 10.90/3.70 U19_g(X1, X2, X3, pcA_out_g(.(X1, .(X2, X3)))) -> U20_g(X1, X2, X3, pcA_in_g(.(s(s(s(s(X2)))), X3))) 10.90/3.70 U20_g(X1, X2, X3, pcA_out_g(.(s(s(s(s(X2)))), X3))) -> pcA_out_g(.(0, .(s(s(X1)), .(X2, X3)))) 10.90/3.70 U17_g(X1, X2, pcA_out_g(.(X1, X2))) -> U18_g(X1, X2, pcA_in_g(.(s(s(s(s(X1)))), X2))) 10.90/3.70 U18_g(X1, X2, pcA_out_g(.(s(s(s(s(X1)))), X2))) -> pcA_out_g(.(s(s(0)), .(X1, X2))) 10.90/3.70 U14_g(X1, X2, X3, pcA_out_g(.(X1, .(X2, X3)))) -> U15_g(X1, X2, X3, pcA_in_g(.(s(s(s(s(X2)))), X3))) 10.90/3.70 U15_g(X1, X2, X3, pcA_out_g(.(s(s(s(s(X2)))), X3))) -> U16_g(X1, X2, X3, pcA_in_g(.(s(s(s(s(X2)))), X3))) 10.90/3.70 U16_g(X1, X2, X3, pcA_out_g(.(s(s(s(s(X2)))), X3))) -> pcA_out_g(.(s(s(s(s(X1)))), .(X2, X3))) 10.90/3.70 10.90/3.70 Pi is empty. 10.90/3.70 We have to consider all (P,R,Pi)-chains 10.90/3.70 ---------------------------------------- 10.90/3.70 10.90/3.70 (5) DependencyGraphProof (EQUIVALENT) 10.90/3.70 The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 8 less nodes. 10.90/3.70 ---------------------------------------- 10.90/3.70 10.90/3.70 (6) 10.90/3.70 Obligation: 10.90/3.70 Pi DP problem: 10.90/3.70 The TRS P consists of the following rules: 10.90/3.70 10.90/3.70 PA_IN_G(.(s(s(s(s(X1)))), .(X2, X3))) -> U2_G(X1, X2, X3, pcA_in_g(.(X1, .(X2, X3)))) 10.90/3.70 U2_G(X1, X2, X3, pcA_out_g(.(X1, .(X2, X3)))) -> PA_IN_G(.(s(s(s(s(X2)))), X3)) 10.90/3.70 PA_IN_G(.(s(s(s(s(X1)))), .(X2, X3))) -> PA_IN_G(.(X1, .(X2, X3))) 10.90/3.70 PA_IN_G(.(s(s(0)), .(X1, X2))) -> PA_IN_G(.(X1, X2)) 10.90/3.70 PA_IN_G(.(s(s(0)), .(X1, X2))) -> U7_G(X1, X2, pcA_in_g(.(X1, X2))) 10.90/3.70 U7_G(X1, X2, pcA_out_g(.(X1, X2))) -> PA_IN_G(.(s(s(s(s(X1)))), X2)) 10.90/3.70 PA_IN_G(.(0, .(s(s(X1)), .(X2, X3)))) -> PA_IN_G(.(X1, .(X2, X3))) 10.90/3.70 PA_IN_G(.(0, .(s(s(X1)), .(X2, X3)))) -> U10_G(X1, X2, X3, pcA_in_g(.(X1, .(X2, X3)))) 10.90/3.70 U10_G(X1, X2, X3, pcA_out_g(.(X1, .(X2, X3)))) -> PA_IN_G(.(s(s(s(s(X2)))), X3)) 10.90/3.70 PA_IN_G(.(0, .(0, X1))) -> PA_IN_G(X1) 10.90/3.70 U2_G(X1, X2, X3, pcA_out_g(.(X1, .(X2, X3)))) -> U4_G(X1, X2, X3, pcA_in_g(.(s(s(s(s(X2)))), X3))) 10.90/3.70 U4_G(X1, X2, X3, pcA_out_g(.(s(s(s(s(X2)))), X3))) -> PA_IN_G(.(s(s(s(s(X2)))), X3)) 10.90/3.70 10.90/3.70 The TRS R consists of the following rules: 10.90/3.70 10.90/3.70 pcA_in_g(.(X1, [])) -> pcA_out_g(.(X1, [])) 10.90/3.70 pcA_in_g(.(s(s(s(s(X1)))), .(X2, X3))) -> U14_g(X1, X2, X3, pcA_in_g(.(X1, .(X2, X3)))) 10.90/3.70 pcA_in_g(.(s(s(0)), .(X1, X2))) -> U17_g(X1, X2, pcA_in_g(.(X1, X2))) 10.90/3.70 pcA_in_g(.(0, .(X1, []))) -> pcA_out_g(.(0, .(X1, []))) 10.90/3.70 pcA_in_g(.(0, .(s(s(X1)), .(X2, X3)))) -> U19_g(X1, X2, X3, pcA_in_g(.(X1, .(X2, X3)))) 10.90/3.70 pcA_in_g(.(0, .(0, X1))) -> U21_g(X1, pcA_in_g(X1)) 10.90/3.70 U21_g(X1, pcA_out_g(X1)) -> pcA_out_g(.(0, .(0, X1))) 10.90/3.70 U19_g(X1, X2, X3, pcA_out_g(.(X1, .(X2, X3)))) -> U20_g(X1, X2, X3, pcA_in_g(.(s(s(s(s(X2)))), X3))) 10.90/3.70 U20_g(X1, X2, X3, pcA_out_g(.(s(s(s(s(X2)))), X3))) -> pcA_out_g(.(0, .(s(s(X1)), .(X2, X3)))) 10.90/3.70 U17_g(X1, X2, pcA_out_g(.(X1, X2))) -> U18_g(X1, X2, pcA_in_g(.(s(s(s(s(X1)))), X2))) 10.90/3.70 U18_g(X1, X2, pcA_out_g(.(s(s(s(s(X1)))), X2))) -> pcA_out_g(.(s(s(0)), .(X1, X2))) 10.90/3.70 U14_g(X1, X2, X3, pcA_out_g(.(X1, .(X2, X3)))) -> U15_g(X1, X2, X3, pcA_in_g(.(s(s(s(s(X2)))), X3))) 10.90/3.70 U15_g(X1, X2, X3, pcA_out_g(.(s(s(s(s(X2)))), X3))) -> U16_g(X1, X2, X3, pcA_in_g(.(s(s(s(s(X2)))), X3))) 10.90/3.70 U16_g(X1, X2, X3, pcA_out_g(.(s(s(s(s(X2)))), X3))) -> pcA_out_g(.(s(s(s(s(X1)))), .(X2, X3))) 10.90/3.70 10.90/3.70 Pi is empty. 10.90/3.70 We have to consider all (P,R,Pi)-chains 10.90/3.70 ---------------------------------------- 10.90/3.70 10.90/3.70 (7) PiDPToQDPProof (EQUIVALENT) 10.90/3.70 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 10.90/3.70 ---------------------------------------- 10.90/3.70 10.90/3.70 (8) 10.90/3.70 Obligation: 10.90/3.70 Q DP problem: 10.90/3.70 The TRS P consists of the following rules: 10.90/3.70 10.90/3.70 PA_IN_G(.(s(s(s(s(X1)))), .(X2, X3))) -> U2_G(X1, X2, X3, pcA_in_g(.(X1, .(X2, X3)))) 10.90/3.70 U2_G(X1, X2, X3, pcA_out_g(.(X1, .(X2, X3)))) -> PA_IN_G(.(s(s(s(s(X2)))), X3)) 10.90/3.70 PA_IN_G(.(s(s(s(s(X1)))), .(X2, X3))) -> PA_IN_G(.(X1, .(X2, X3))) 10.90/3.70 PA_IN_G(.(s(s(0)), .(X1, X2))) -> PA_IN_G(.(X1, X2)) 10.90/3.70 PA_IN_G(.(s(s(0)), .(X1, X2))) -> U7_G(X1, X2, pcA_in_g(.(X1, X2))) 10.90/3.70 U7_G(X1, X2, pcA_out_g(.(X1, X2))) -> PA_IN_G(.(s(s(s(s(X1)))), X2)) 10.90/3.70 PA_IN_G(.(0, .(s(s(X1)), .(X2, X3)))) -> PA_IN_G(.(X1, .(X2, X3))) 10.90/3.70 PA_IN_G(.(0, .(s(s(X1)), .(X2, X3)))) -> U10_G(X1, X2, X3, pcA_in_g(.(X1, .(X2, X3)))) 10.90/3.70 U10_G(X1, X2, X3, pcA_out_g(.(X1, .(X2, X3)))) -> PA_IN_G(.(s(s(s(s(X2)))), X3)) 10.90/3.70 PA_IN_G(.(0, .(0, X1))) -> PA_IN_G(X1) 10.90/3.70 U2_G(X1, X2, X3, pcA_out_g(.(X1, .(X2, X3)))) -> U4_G(X1, X2, X3, pcA_in_g(.(s(s(s(s(X2)))), X3))) 10.90/3.70 U4_G(X1, X2, X3, pcA_out_g(.(s(s(s(s(X2)))), X3))) -> PA_IN_G(.(s(s(s(s(X2)))), X3)) 10.90/3.70 10.90/3.70 The TRS R consists of the following rules: 10.90/3.70 10.90/3.70 pcA_in_g(.(X1, [])) -> pcA_out_g(.(X1, [])) 10.90/3.70 pcA_in_g(.(s(s(s(s(X1)))), .(X2, X3))) -> U14_g(X1, X2, X3, pcA_in_g(.(X1, .(X2, X3)))) 10.90/3.70 pcA_in_g(.(s(s(0)), .(X1, X2))) -> U17_g(X1, X2, pcA_in_g(.(X1, X2))) 10.90/3.70 pcA_in_g(.(0, .(X1, []))) -> pcA_out_g(.(0, .(X1, []))) 10.90/3.70 pcA_in_g(.(0, .(s(s(X1)), .(X2, X3)))) -> U19_g(X1, X2, X3, pcA_in_g(.(X1, .(X2, X3)))) 10.90/3.70 pcA_in_g(.(0, .(0, X1))) -> U21_g(X1, pcA_in_g(X1)) 10.90/3.70 U21_g(X1, pcA_out_g(X1)) -> pcA_out_g(.(0, .(0, X1))) 10.90/3.70 U19_g(X1, X2, X3, pcA_out_g(.(X1, .(X2, X3)))) -> U20_g(X1, X2, X3, pcA_in_g(.(s(s(s(s(X2)))), X3))) 10.90/3.70 U20_g(X1, X2, X3, pcA_out_g(.(s(s(s(s(X2)))), X3))) -> pcA_out_g(.(0, .(s(s(X1)), .(X2, X3)))) 10.90/3.70 U17_g(X1, X2, pcA_out_g(.(X1, X2))) -> U18_g(X1, X2, pcA_in_g(.(s(s(s(s(X1)))), X2))) 10.90/3.70 U18_g(X1, X2, pcA_out_g(.(s(s(s(s(X1)))), X2))) -> pcA_out_g(.(s(s(0)), .(X1, X2))) 10.90/3.70 U14_g(X1, X2, X3, pcA_out_g(.(X1, .(X2, X3)))) -> U15_g(X1, X2, X3, pcA_in_g(.(s(s(s(s(X2)))), X3))) 10.90/3.70 U15_g(X1, X2, X3, pcA_out_g(.(s(s(s(s(X2)))), X3))) -> U16_g(X1, X2, X3, pcA_in_g(.(s(s(s(s(X2)))), X3))) 10.90/3.70 U16_g(X1, X2, X3, pcA_out_g(.(s(s(s(s(X2)))), X3))) -> pcA_out_g(.(s(s(s(s(X1)))), .(X2, X3))) 10.90/3.70 10.90/3.70 The set Q consists of the following terms: 10.90/3.70 10.90/3.70 pcA_in_g(x0) 10.90/3.70 U21_g(x0, x1) 10.90/3.70 U19_g(x0, x1, x2, x3) 10.90/3.70 U20_g(x0, x1, x2, x3) 10.90/3.70 U17_g(x0, x1, x2) 10.90/3.70 U18_g(x0, x1, x2) 10.90/3.70 U14_g(x0, x1, x2, x3) 10.90/3.70 U15_g(x0, x1, x2, x3) 10.90/3.70 U16_g(x0, x1, x2, x3) 10.90/3.70 10.90/3.70 We have to consider all (P,Q,R)-chains. 10.90/3.70 ---------------------------------------- 10.90/3.70 10.90/3.70 (9) QDPQMonotonicMRRProof (EQUIVALENT) 10.90/3.70 By using the Q-monotonic rule removal processor with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented such that it always occurs at a strongly monotonic position in a (P,Q,R)-chain. 10.90/3.70 10.90/3.70 Strictly oriented dependency pairs: 10.90/3.70 10.90/3.70 PA_IN_G(.(s(s(s(s(X1)))), .(X2, X3))) -> U2_G(X1, X2, X3, pcA_in_g(.(X1, .(X2, X3)))) 10.90/3.70 PA_IN_G(.(s(s(0)), .(X1, X2))) -> PA_IN_G(.(X1, X2)) 10.90/3.70 PA_IN_G(.(s(s(0)), .(X1, X2))) -> U7_G(X1, X2, pcA_in_g(.(X1, X2))) 10.90/3.70 PA_IN_G(.(0, .(s(s(X1)), .(X2, X3)))) -> PA_IN_G(.(X1, .(X2, X3))) 10.90/3.70 PA_IN_G(.(0, .(s(s(X1)), .(X2, X3)))) -> U10_G(X1, X2, X3, pcA_in_g(.(X1, .(X2, X3)))) 10.90/3.70 PA_IN_G(.(0, .(0, X1))) -> PA_IN_G(X1) 10.90/3.70 10.90/3.70 10.90/3.70 Used ordering: Polynomial interpretation [POLO]: 10.90/3.70 10.90/3.70 POL(.(x_1, x_2)) = 1 + x_2 10.90/3.70 POL(0) = 0 10.90/3.70 POL(PA_IN_G(x_1)) = 2*x_1 10.90/3.70 POL(U10_G(x_1, x_2, x_3, x_4)) = 2 + 2*x_3 10.90/3.70 POL(U14_g(x_1, x_2, x_3, x_4)) = 0 10.90/3.70 POL(U15_g(x_1, x_2, x_3, x_4)) = 0 10.90/3.70 POL(U16_g(x_1, x_2, x_3, x_4)) = 0 10.90/3.70 POL(U17_g(x_1, x_2, x_3)) = 0 10.90/3.70 POL(U18_g(x_1, x_2, x_3)) = 0 10.90/3.70 POL(U19_g(x_1, x_2, x_3, x_4)) = 0 10.90/3.70 POL(U20_g(x_1, x_2, x_3, x_4)) = 0 10.90/3.70 POL(U21_g(x_1, x_2)) = 0 10.90/3.70 POL(U2_G(x_1, x_2, x_3, x_4)) = 2 + 2*x_3 10.90/3.70 POL(U4_G(x_1, x_2, x_3, x_4)) = 2 + 2*x_3 10.90/3.70 POL(U7_G(x_1, x_2, x_3)) = 2 + 2*x_2 10.90/3.70 POL([]) = 0 10.90/3.70 POL(pcA_in_g(x_1)) = 0 10.90/3.70 POL(pcA_out_g(x_1)) = 0 10.90/3.70 POL(s(x_1)) = 0 10.90/3.70 10.90/3.70 10.90/3.70 ---------------------------------------- 10.90/3.70 10.90/3.70 (10) 10.90/3.70 Obligation: 10.90/3.70 Q DP problem: 10.90/3.70 The TRS P consists of the following rules: 10.90/3.70 10.90/3.70 U2_G(X1, X2, X3, pcA_out_g(.(X1, .(X2, X3)))) -> PA_IN_G(.(s(s(s(s(X2)))), X3)) 10.90/3.70 PA_IN_G(.(s(s(s(s(X1)))), .(X2, X3))) -> PA_IN_G(.(X1, .(X2, X3))) 10.90/3.70 U7_G(X1, X2, pcA_out_g(.(X1, X2))) -> PA_IN_G(.(s(s(s(s(X1)))), X2)) 10.90/3.70 U10_G(X1, X2, X3, pcA_out_g(.(X1, .(X2, X3)))) -> PA_IN_G(.(s(s(s(s(X2)))), X3)) 10.90/3.70 U2_G(X1, X2, X3, pcA_out_g(.(X1, .(X2, X3)))) -> U4_G(X1, X2, X3, pcA_in_g(.(s(s(s(s(X2)))), X3))) 10.90/3.70 U4_G(X1, X2, X3, pcA_out_g(.(s(s(s(s(X2)))), X3))) -> PA_IN_G(.(s(s(s(s(X2)))), X3)) 10.90/3.70 10.90/3.70 The TRS R consists of the following rules: 10.90/3.70 10.90/3.70 pcA_in_g(.(X1, [])) -> pcA_out_g(.(X1, [])) 10.90/3.70 pcA_in_g(.(s(s(s(s(X1)))), .(X2, X3))) -> U14_g(X1, X2, X3, pcA_in_g(.(X1, .(X2, X3)))) 10.90/3.70 pcA_in_g(.(s(s(0)), .(X1, X2))) -> U17_g(X1, X2, pcA_in_g(.(X1, X2))) 10.90/3.70 pcA_in_g(.(0, .(X1, []))) -> pcA_out_g(.(0, .(X1, []))) 10.90/3.70 pcA_in_g(.(0, .(s(s(X1)), .(X2, X3)))) -> U19_g(X1, X2, X3, pcA_in_g(.(X1, .(X2, X3)))) 10.90/3.70 pcA_in_g(.(0, .(0, X1))) -> U21_g(X1, pcA_in_g(X1)) 10.90/3.70 U21_g(X1, pcA_out_g(X1)) -> pcA_out_g(.(0, .(0, X1))) 10.90/3.70 U19_g(X1, X2, X3, pcA_out_g(.(X1, .(X2, X3)))) -> U20_g(X1, X2, X3, pcA_in_g(.(s(s(s(s(X2)))), X3))) 10.90/3.70 U20_g(X1, X2, X3, pcA_out_g(.(s(s(s(s(X2)))), X3))) -> pcA_out_g(.(0, .(s(s(X1)), .(X2, X3)))) 10.90/3.70 U17_g(X1, X2, pcA_out_g(.(X1, X2))) -> U18_g(X1, X2, pcA_in_g(.(s(s(s(s(X1)))), X2))) 10.90/3.70 U18_g(X1, X2, pcA_out_g(.(s(s(s(s(X1)))), X2))) -> pcA_out_g(.(s(s(0)), .(X1, X2))) 10.90/3.70 U14_g(X1, X2, X3, pcA_out_g(.(X1, .(X2, X3)))) -> U15_g(X1, X2, X3, pcA_in_g(.(s(s(s(s(X2)))), X3))) 10.90/3.70 U15_g(X1, X2, X3, pcA_out_g(.(s(s(s(s(X2)))), X3))) -> U16_g(X1, X2, X3, pcA_in_g(.(s(s(s(s(X2)))), X3))) 10.90/3.70 U16_g(X1, X2, X3, pcA_out_g(.(s(s(s(s(X2)))), X3))) -> pcA_out_g(.(s(s(s(s(X1)))), .(X2, X3))) 10.90/3.70 10.90/3.70 The set Q consists of the following terms: 10.90/3.70 10.90/3.70 pcA_in_g(x0) 10.90/3.70 U21_g(x0, x1) 10.90/3.70 U19_g(x0, x1, x2, x3) 10.90/3.70 U20_g(x0, x1, x2, x3) 10.90/3.70 U17_g(x0, x1, x2) 10.90/3.70 U18_g(x0, x1, x2) 10.90/3.70 U14_g(x0, x1, x2, x3) 10.90/3.70 U15_g(x0, x1, x2, x3) 10.90/3.70 U16_g(x0, x1, x2, x3) 10.90/3.70 10.90/3.70 We have to consider all (P,Q,R)-chains. 10.90/3.70 ---------------------------------------- 10.90/3.70 10.90/3.70 (11) DependencyGraphProof (EQUIVALENT) 10.90/3.70 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 5 less nodes. 10.90/3.70 ---------------------------------------- 10.90/3.70 10.90/3.70 (12) 10.90/3.70 Obligation: 10.90/3.70 Q DP problem: 10.90/3.70 The TRS P consists of the following rules: 10.90/3.70 10.90/3.70 PA_IN_G(.(s(s(s(s(X1)))), .(X2, X3))) -> PA_IN_G(.(X1, .(X2, X3))) 10.90/3.70 10.90/3.70 The TRS R consists of the following rules: 10.90/3.70 10.90/3.70 pcA_in_g(.(X1, [])) -> pcA_out_g(.(X1, [])) 10.90/3.70 pcA_in_g(.(s(s(s(s(X1)))), .(X2, X3))) -> U14_g(X1, X2, X3, pcA_in_g(.(X1, .(X2, X3)))) 10.90/3.70 pcA_in_g(.(s(s(0)), .(X1, X2))) -> U17_g(X1, X2, pcA_in_g(.(X1, X2))) 10.90/3.70 pcA_in_g(.(0, .(X1, []))) -> pcA_out_g(.(0, .(X1, []))) 10.90/3.70 pcA_in_g(.(0, .(s(s(X1)), .(X2, X3)))) -> U19_g(X1, X2, X3, pcA_in_g(.(X1, .(X2, X3)))) 10.90/3.70 pcA_in_g(.(0, .(0, X1))) -> U21_g(X1, pcA_in_g(X1)) 10.90/3.70 U21_g(X1, pcA_out_g(X1)) -> pcA_out_g(.(0, .(0, X1))) 10.90/3.70 U19_g(X1, X2, X3, pcA_out_g(.(X1, .(X2, X3)))) -> U20_g(X1, X2, X3, pcA_in_g(.(s(s(s(s(X2)))), X3))) 10.90/3.70 U20_g(X1, X2, X3, pcA_out_g(.(s(s(s(s(X2)))), X3))) -> pcA_out_g(.(0, .(s(s(X1)), .(X2, X3)))) 10.90/3.70 U17_g(X1, X2, pcA_out_g(.(X1, X2))) -> U18_g(X1, X2, pcA_in_g(.(s(s(s(s(X1)))), X2))) 10.90/3.70 U18_g(X1, X2, pcA_out_g(.(s(s(s(s(X1)))), X2))) -> pcA_out_g(.(s(s(0)), .(X1, X2))) 10.90/3.70 U14_g(X1, X2, X3, pcA_out_g(.(X1, .(X2, X3)))) -> U15_g(X1, X2, X3, pcA_in_g(.(s(s(s(s(X2)))), X3))) 10.90/3.70 U15_g(X1, X2, X3, pcA_out_g(.(s(s(s(s(X2)))), X3))) -> U16_g(X1, X2, X3, pcA_in_g(.(s(s(s(s(X2)))), X3))) 10.90/3.70 U16_g(X1, X2, X3, pcA_out_g(.(s(s(s(s(X2)))), X3))) -> pcA_out_g(.(s(s(s(s(X1)))), .(X2, X3))) 10.90/3.70 10.90/3.70 The set Q consists of the following terms: 10.90/3.70 10.90/3.70 pcA_in_g(x0) 10.90/3.70 U21_g(x0, x1) 10.90/3.70 U19_g(x0, x1, x2, x3) 10.90/3.70 U20_g(x0, x1, x2, x3) 10.90/3.70 U17_g(x0, x1, x2) 10.90/3.70 U18_g(x0, x1, x2) 10.90/3.70 U14_g(x0, x1, x2, x3) 10.90/3.70 U15_g(x0, x1, x2, x3) 10.90/3.70 U16_g(x0, x1, x2, x3) 10.90/3.70 10.90/3.70 We have to consider all (P,Q,R)-chains. 10.90/3.70 ---------------------------------------- 10.90/3.70 10.90/3.70 (13) UsableRulesProof (EQUIVALENT) 10.90/3.70 As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. 10.90/3.70 ---------------------------------------- 10.90/3.70 10.90/3.70 (14) 10.90/3.70 Obligation: 10.90/3.70 Q DP problem: 10.90/3.70 The TRS P consists of the following rules: 10.90/3.70 10.90/3.70 PA_IN_G(.(s(s(s(s(X1)))), .(X2, X3))) -> PA_IN_G(.(X1, .(X2, X3))) 10.90/3.70 10.90/3.70 R is empty. 10.90/3.70 The set Q consists of the following terms: 10.90/3.70 10.90/3.70 pcA_in_g(x0) 10.90/3.70 U21_g(x0, x1) 10.90/3.70 U19_g(x0, x1, x2, x3) 10.90/3.70 U20_g(x0, x1, x2, x3) 10.90/3.70 U17_g(x0, x1, x2) 10.90/3.70 U18_g(x0, x1, x2) 10.90/3.70 U14_g(x0, x1, x2, x3) 10.90/3.70 U15_g(x0, x1, x2, x3) 10.90/3.70 U16_g(x0, x1, x2, x3) 10.90/3.70 10.90/3.70 We have to consider all (P,Q,R)-chains. 10.90/3.70 ---------------------------------------- 10.90/3.70 10.90/3.70 (15) QReductionProof (EQUIVALENT) 10.90/3.70 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 10.90/3.70 10.90/3.70 pcA_in_g(x0) 10.90/3.70 U21_g(x0, x1) 10.90/3.70 U19_g(x0, x1, x2, x3) 10.90/3.70 U20_g(x0, x1, x2, x3) 10.90/3.70 U17_g(x0, x1, x2) 10.90/3.70 U18_g(x0, x1, x2) 10.90/3.70 U14_g(x0, x1, x2, x3) 10.90/3.70 U15_g(x0, x1, x2, x3) 10.90/3.70 U16_g(x0, x1, x2, x3) 10.90/3.70 10.90/3.70 10.90/3.70 ---------------------------------------- 10.90/3.70 10.90/3.70 (16) 10.90/3.70 Obligation: 10.90/3.70 Q DP problem: 10.90/3.70 The TRS P consists of the following rules: 10.90/3.70 10.90/3.70 PA_IN_G(.(s(s(s(s(X1)))), .(X2, X3))) -> PA_IN_G(.(X1, .(X2, X3))) 10.90/3.70 10.90/3.70 R is empty. 10.90/3.70 Q is empty. 10.90/3.70 We have to consider all (P,Q,R)-chains. 10.90/3.70 ---------------------------------------- 10.90/3.70 10.90/3.70 (17) UsableRulesReductionPairsProof (EQUIVALENT) 10.90/3.70 By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well. 10.90/3.70 10.90/3.70 The following dependency pairs can be deleted: 10.90/3.70 10.90/3.70 PA_IN_G(.(s(s(s(s(X1)))), .(X2, X3))) -> PA_IN_G(.(X1, .(X2, X3))) 10.90/3.70 No rules are removed from R. 10.90/3.70 10.90/3.70 Used ordering: POLO with Polynomial interpretation [POLO]: 10.90/3.70 10.90/3.70 POL(.(x_1, x_2)) = x_1 + x_2 10.90/3.70 POL(PA_IN_G(x_1)) = 2*x_1 10.90/3.70 POL(s(x_1)) = 2*x_1 10.90/3.70 10.90/3.70 10.90/3.70 ---------------------------------------- 10.90/3.70 10.90/3.70 (18) 10.90/3.70 Obligation: 10.90/3.70 Q DP problem: 10.90/3.70 P is empty. 10.90/3.70 R is empty. 10.90/3.70 Q is empty. 10.90/3.70 We have to consider all (P,Q,R)-chains. 10.90/3.70 ---------------------------------------- 10.90/3.70 10.90/3.70 (19) PisEmptyProof (EQUIVALENT) 10.90/3.70 The TRS P is empty. Hence, there is no (P,Q,R) chain. 10.90/3.70 ---------------------------------------- 10.90/3.70 10.90/3.70 (20) 10.90/3.70 YES 11.15/3.75 EOF