6.42/2.49 YES 6.56/2.51 proof of /export/starexec/sandbox2/benchmark/theBenchmark.pl 6.56/2.51 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 6.56/2.51 6.56/2.51 6.56/2.51 Left Termination of the query pattern 6.56/2.51 6.56/2.51 merge(g,g,a) 6.56/2.51 6.56/2.51 w.r.t. the given Prolog program could successfully be proven: 6.56/2.51 6.56/2.51 (0) Prolog 6.56/2.51 (1) PrologToDTProblemTransformerProof [SOUND, 46 ms] 6.56/2.51 (2) TRIPLES 6.56/2.51 (3) TriplesToPiDPProof [SOUND, 42 ms] 6.56/2.51 (4) PiDP 6.56/2.51 (5) DependencyGraphProof [EQUIVALENT, 0 ms] 6.56/2.51 (6) AND 6.56/2.51 (7) PiDP 6.56/2.51 (8) UsableRulesProof [EQUIVALENT, 0 ms] 6.56/2.51 (9) PiDP 6.56/2.51 (10) PiDPToQDPProof [EQUIVALENT, 0 ms] 6.56/2.51 (11) QDP 6.56/2.51 (12) QDPSizeChangeProof [EQUIVALENT, 0 ms] 6.56/2.51 (13) YES 6.56/2.51 (14) PiDP 6.56/2.51 (15) UsableRulesProof [EQUIVALENT, 0 ms] 6.56/2.51 (16) PiDP 6.56/2.51 (17) PiDPToQDPProof [EQUIVALENT, 0 ms] 6.56/2.51 (18) QDP 6.56/2.51 (19) QDPSizeChangeProof [EQUIVALENT, 0 ms] 6.56/2.51 (20) YES 6.56/2.51 (21) PiDP 6.56/2.51 (22) PiDPToQDPProof [SOUND, 0 ms] 6.56/2.51 (23) QDP 6.56/2.51 (24) MRRProof [EQUIVALENT, 35 ms] 6.56/2.51 (25) QDP 6.56/2.51 (26) DependencyGraphProof [EQUIVALENT, 0 ms] 6.56/2.51 (27) QDP 6.56/2.51 (28) UsableRulesProof [EQUIVALENT, 0 ms] 6.56/2.51 (29) QDP 6.56/2.51 (30) QReductionProof [EQUIVALENT, 0 ms] 6.56/2.51 (31) QDP 6.56/2.51 (32) QDPSizeChangeProof [EQUIVALENT, 0 ms] 6.56/2.51 (33) YES 6.56/2.51 6.56/2.51 6.56/2.51 ---------------------------------------- 6.56/2.51 6.56/2.51 (0) 6.56/2.51 Obligation: 6.56/2.51 Clauses: 6.56/2.51 6.56/2.51 merge([], X, X). 6.56/2.51 merge(X, [], X). 6.56/2.51 merge(.(X, Xs), .(Y, Ys), .(X, Zs)) :- ','(leq(X, Y), merge(Xs, .(Y, Ys), Zs)). 6.56/2.51 merge(.(X, Xs), .(Y, Ys), .(Y, Zs)) :- ','(less(Y, X), merge(.(X, Xs), Ys, Zs)). 6.56/2.51 less(0, s(0)). 6.56/2.51 less(s(X), s(Y)) :- less(X, Y). 6.56/2.51 leq(0, 0). 6.56/2.51 leq(0, s(0)). 6.56/2.51 leq(s(X), s(Y)) :- leq(X, Y). 6.56/2.51 6.56/2.51 6.56/2.51 Query: merge(g,g,a) 6.56/2.51 ---------------------------------------- 6.56/2.51 6.56/2.51 (1) PrologToDTProblemTransformerProof (SOUND) 6.56/2.51 Built DT problem from termination graph DT10. 6.56/2.51 6.56/2.51 { 6.56/2.51 "root": 5, 6.56/2.51 "program": { 6.56/2.51 "directives": [], 6.56/2.51 "clauses": [ 6.56/2.51 [ 6.56/2.51 "(merge ([]) X X)", 6.56/2.51 null 6.56/2.51 ], 6.56/2.51 [ 6.56/2.51 "(merge X ([]) X)", 6.56/2.51 null 6.56/2.51 ], 6.56/2.51 [ 6.56/2.51 "(merge (. X Xs) (. Y Ys) (. X Zs))", 6.56/2.51 "(',' (leq X Y) (merge Xs (. Y Ys) Zs))" 6.56/2.51 ], 6.56/2.51 [ 6.56/2.51 "(merge (. X Xs) (. Y Ys) (. Y Zs))", 6.56/2.51 "(',' (less Y X) (merge (. X Xs) Ys Zs))" 6.56/2.51 ], 6.56/2.51 [ 6.56/2.51 "(less (0) (s (0)))", 6.56/2.51 null 6.56/2.51 ], 6.56/2.51 [ 6.56/2.51 "(less (s X) (s Y))", 6.56/2.51 "(less X Y)" 6.56/2.51 ], 6.56/2.51 [ 6.56/2.51 "(leq (0) (0))", 6.56/2.51 null 6.56/2.51 ], 6.56/2.51 [ 6.56/2.51 "(leq (0) (s (0)))", 6.56/2.51 null 6.56/2.51 ], 6.56/2.51 [ 6.56/2.51 "(leq (s X) (s Y))", 6.56/2.51 "(leq X Y)" 6.56/2.51 ] 6.56/2.51 ] 6.56/2.51 }, 6.56/2.51 "graph": { 6.56/2.51 "nodes": { 6.56/2.51 "type": "Nodes", 6.56/2.51 "271": { 6.56/2.51 "goal": [ 6.56/2.51 { 6.56/2.51 "clause": -1, 6.56/2.51 "scope": -1, 6.56/2.51 "term": "(true)" 6.56/2.51 }, 6.56/2.51 { 6.56/2.51 "clause": 2, 6.56/2.51 "scope": 1, 6.56/2.51 "term": "(merge ([]) ([]) T3)" 6.56/2.51 }, 6.56/2.51 { 6.56/2.51 "clause": 3, 6.56/2.51 "scope": 1, 6.56/2.51 "term": "(merge ([]) ([]) T3)" 6.56/2.51 } 6.56/2.51 ], 6.56/2.51 "kb": { 6.56/2.51 "nonunifying": [], 6.56/2.51 "intvars": {}, 6.56/2.51 "arithmetic": { 6.56/2.51 "type": "PlainIntegerRelationState", 6.56/2.51 "relations": [] 6.56/2.51 }, 6.56/2.51 "ground": [], 6.56/2.51 "free": [], 6.56/2.51 "exprvars": [] 6.56/2.51 } 6.56/2.51 }, 6.56/2.51 "275": { 6.56/2.51 "goal": [ 6.56/2.51 { 6.56/2.51 "clause": 2, 6.56/2.51 "scope": 1, 6.56/2.51 "term": "(merge ([]) T5 T3)" 6.56/2.51 }, 6.56/2.51 { 6.56/2.51 "clause": 3, 6.56/2.51 "scope": 1, 6.56/2.51 "term": "(merge ([]) T5 T3)" 6.56/2.51 } 6.56/2.51 ], 6.56/2.51 "kb": { 6.56/2.51 "nonunifying": [[ 6.56/2.51 "(merge ([]) T5 T3)", 6.56/2.51 "(merge X4 ([]) X4)" 6.56/2.51 ]], 6.56/2.51 "intvars": {}, 6.56/2.51 "arithmetic": { 6.56/2.51 "type": "PlainIntegerRelationState", 6.56/2.51 "relations": [] 6.56/2.51 }, 6.56/2.51 "ground": ["T5"], 6.56/2.51 "free": ["X4"], 6.56/2.51 "exprvars": [] 6.56/2.51 } 6.56/2.51 }, 6.56/2.51 "276": { 6.56/2.51 "goal": [ 6.56/2.51 { 6.56/2.51 "clause": 2, 6.56/2.51 "scope": 1, 6.56/2.51 "term": "(merge ([]) ([]) T3)" 6.56/2.51 }, 6.56/2.51 { 6.56/2.51 "clause": 3, 6.56/2.51 "scope": 1, 6.56/2.51 "term": "(merge ([]) ([]) T3)" 6.56/2.51 } 6.56/2.51 ], 6.56/2.51 "kb": { 6.56/2.51 "nonunifying": [], 6.56/2.51 "intvars": {}, 6.56/2.51 "arithmetic": { 6.56/2.51 "type": "PlainIntegerRelationState", 6.56/2.51 "relations": [] 6.56/2.51 }, 6.56/2.51 "ground": [], 6.56/2.51 "free": [], 6.56/2.51 "exprvars": [] 6.56/2.51 } 6.56/2.51 }, 6.56/2.51 "430": { 6.56/2.51 "goal": [{ 6.56/2.51 "clause": -1, 6.56/2.51 "scope": -1, 6.56/2.51 "term": "(',' (less T76 T74) (merge (. 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6.56/2.52 }, 6.56/2.52 "ground": [], 6.56/2.52 "free": [], 6.56/2.52 "exprvars": [] 6.56/2.52 } 6.56/2.52 }, 6.56/2.52 "427": { 6.56/2.52 "goal": [], 6.56/2.52 "kb": { 6.56/2.52 "nonunifying": [], 6.56/2.52 "intvars": {}, 6.56/2.52 "arithmetic": { 6.56/2.52 "type": "PlainIntegerRelationState", 6.56/2.52 "relations": [] 6.56/2.52 }, 6.56/2.52 "ground": [], 6.56/2.52 "free": [], 6.56/2.52 "exprvars": [] 6.56/2.52 } 6.56/2.52 }, 6.56/2.52 "428": { 6.56/2.52 "goal": [{ 6.56/2.52 "clause": -1, 6.56/2.52 "scope": -1, 6.56/2.52 "term": "(less T67 T68)" 6.56/2.52 }], 6.56/2.52 "kb": { 6.56/2.52 "nonunifying": [], 6.56/2.52 "intvars": {}, 6.56/2.52 "arithmetic": { 6.56/2.52 "type": "PlainIntegerRelationState", 6.56/2.52 "relations": [] 6.56/2.52 }, 6.56/2.52 "ground": [ 6.56/2.52 "T67", 6.56/2.52 "T68" 6.56/2.52 ], 6.56/2.52 "free": [], 6.56/2.52 "exprvars": [] 6.56/2.52 } 6.56/2.52 }, 6.56/2.52 "429": { 6.56/2.52 "goal": [], 6.56/2.52 "kb": { 6.56/2.52 "nonunifying": [], 6.56/2.52 "intvars": {}, 6.56/2.52 "arithmetic": { 6.56/2.52 "type": "PlainIntegerRelationState", 6.56/2.52 "relations": [] 6.56/2.52 }, 6.56/2.52 "ground": [], 6.56/2.52 "free": [], 6.56/2.52 "exprvars": [] 6.56/2.52 } 6.56/2.52 }, 6.56/2.52 "309": { 6.56/2.52 "goal": [], 6.56/2.52 "kb": { 6.56/2.52 "nonunifying": [], 6.56/2.52 "intvars": {}, 6.56/2.52 "arithmetic": { 6.56/2.52 "type": "PlainIntegerRelationState", 6.56/2.52 "relations": [] 6.56/2.52 }, 6.56/2.52 "ground": [], 6.56/2.52 "free": [], 6.56/2.52 "exprvars": [] 6.56/2.52 } 6.56/2.52 } 6.56/2.52 }, 6.56/2.52 "edges": [ 6.56/2.52 { 6.56/2.52 "from": 5, 6.56/2.52 "to": 6, 6.56/2.52 "label": "CASE" 6.56/2.52 }, 6.56/2.52 { 6.56/2.52 "from": 6, 6.56/2.52 "to": 235, 6.56/2.52 "label": "EVAL with clause\nmerge([], X2, X2).\nand substitutionT1 -> [],\nT2 -> T5,\nX2 -> T5,\nT3 -> T5" 6.56/2.52 }, 6.56/2.52 { 6.56/2.52 "from": 6, 6.56/2.52 "to": 255, 6.56/2.52 "label": "EVAL-BACKTRACK" 6.56/2.52 }, 6.56/2.52 { 6.56/2.52 "from": 235, 6.56/2.52 "to": 259, 6.56/2.52 "label": "SUCCESS" 6.56/2.52 }, 6.56/2.52 { 6.56/2.52 "from": 255, 6.56/2.52 "to": 287, 6.56/2.52 "label": "EVAL with clause\nmerge(X26, [], X26).\nand substitutionT1 -> T7,\nX26 -> T7,\nT2 -> [],\nT3 -> T7" 6.56/2.52 }, 6.56/2.52 { 6.56/2.52 "from": 255, 6.56/2.52 "to": 297, 6.56/2.52 "label": "EVAL-BACKTRACK" 6.56/2.52 }, 6.56/2.52 { 6.56/2.52 "from": 259, 6.56/2.52 "to": 271, 6.56/2.52 "label": "EVAL with clause\nmerge(X4, [], X4).\nand substitutionX4 -> [],\nT5 -> [],\nT3 -> []" 6.56/2.52 }, 6.56/2.52 { 6.56/2.52 "from": 259, 6.56/2.52 "to": 275, 6.56/2.52 "label": "EVAL-BACKTRACK" 6.56/2.52 }, 6.56/2.52 { 6.56/2.52 "from": 271, 6.56/2.52 "to": 276, 6.56/2.52 "label": "SUCCESS" 6.56/2.52 }, 6.56/2.52 { 6.56/2.52 "from": 275, 6.56/2.52 "to": 281, 6.56/2.52 "label": "BACKTRACK\nfor clause: merge(.(X, Xs), .(Y, Ys), .(X, Zs)) :- ','(leq(X, Y), merge(Xs, .(Y, Ys), Zs))because of non-unification" 6.56/2.52 }, 6.56/2.52 { 6.56/2.52 "from": 276, 6.56/2.52 "to": 277, 6.56/2.52 "label": "BACKTRACK\nfor clause: merge(.(X, Xs), .(Y, Ys), .(X, Zs)) :- ','(leq(X, Y), merge(Xs, .(Y, Ys), Zs))because of non-unification" 6.56/2.52 }, 6.56/2.52 { 6.56/2.52 "from": 277, 6.56/2.52 "to": 278, 6.56/2.52 "label": "BACKTRACK\nfor clause: merge(.(X, Xs), .(Y, Ys), .(Y, Zs)) :- ','(less(Y, X), merge(.(X, Xs), Ys, Zs))because of non-unification" 6.56/2.52 }, 6.56/2.52 { 6.56/2.52 "from": 281, 6.56/2.52 "to": 283, 6.56/2.52 "label": "BACKTRACK\nfor clause: merge(.(X, Xs), .(Y, Ys), .(Y, Zs)) :- ','(less(Y, X), merge(.(X, Xs), Ys, Zs))because of non-unification" 6.56/2.52 }, 6.56/2.52 { 6.56/2.52 "from": 287, 6.56/2.52 "to": 300, 6.56/2.52 "label": "SUCCESS" 6.56/2.52 }, 6.56/2.52 { 6.56/2.52 "from": 297, 6.56/2.52 "to": 322, 6.56/2.52 "label": "EVAL with clause\nmerge(.(X42, X43), .(X44, X45), .(X42, X46)) :- ','(leq(X42, X44), merge(X43, .(X44, X45), X46)).\nand substitutionX42 -> T17,\nX43 -> T18,\nT1 -> .(T17, T18),\nX44 -> T19,\nX45 -> T20,\nT2 -> .(T19, T20),\nX46 -> T22,\nT3 -> .(T17, T22),\nT21 -> T22" 6.56/2.52 }, 6.56/2.52 { 6.56/2.52 "from": 297, 6.56/2.52 "to": 327, 6.56/2.52 "label": "EVAL-BACKTRACK" 6.56/2.52 }, 6.56/2.52 { 6.56/2.52 "from": 300, 6.56/2.52 "to": 305, 6.56/2.52 "label": "BACKTRACK\nfor clause: merge(.(X, Xs), .(Y, Ys), .(X, Zs)) :- ','(leq(X, Y), merge(Xs, .(Y, Ys), Zs))because of non-unification" 6.56/2.52 }, 6.56/2.52 { 6.56/2.52 "from": 305, 6.56/2.52 "to": 309, 6.56/2.52 "label": "BACKTRACK\nfor clause: merge(.(X, Xs), .(Y, Ys), .(Y, Zs)) :- ','(less(Y, X), merge(.(X, Xs), Ys, Zs))because of non-unification" 6.56/2.52 }, 6.56/2.52 { 6.56/2.52 "from": 322, 6.56/2.52 "to": 328, 6.56/2.52 "label": "CASE" 6.56/2.52 }, 6.56/2.52 { 6.56/2.52 "from": 327, 6.56/2.52 "to": 430, 6.56/2.52 "label": "EVAL with clause\nmerge(.(X97, X98), .(X99, X100), .(X99, X101)) :- ','(less(X99, X97), merge(.(X97, X98), X100, X101)).\nand substitutionX97 -> T74,\nX98 -> T75,\nT1 -> .(T74, T75),\nX99 -> T76,\nX100 -> T77,\nT2 -> .(T76, T77),\nX101 -> T79,\nT3 -> .(T76, T79),\nT78 -> T79" 6.56/2.52 }, 6.56/2.52 { 6.56/2.52 "from": 327, 6.56/2.52 "to": 431, 6.56/2.52 "label": "EVAL-BACKTRACK" 6.56/2.52 }, 6.56/2.52 { 6.56/2.52 "from": 328, 6.56/2.52 "to": 329, 6.56/2.52 "label": "PARALLEL" 6.56/2.52 }, 6.56/2.52 { 6.56/2.52 "from": 328, 6.56/2.52 "to": 330, 6.56/2.52 "label": "PARALLEL" 6.56/2.52 }, 6.56/2.52 { 6.56/2.52 "from": 329, 6.56/2.52 "to": 331, 6.56/2.52 "label": "EVAL with clause\nleq(0, 0).\nand substitutionT17 -> 0,\nT19 -> 0" 6.56/2.52 }, 6.56/2.52 { 6.56/2.52 "from": 329, 6.56/2.52 "to": 332, 6.56/2.52 "label": "EVAL-BACKTRACK" 6.56/2.52 }, 6.56/2.52 { 6.56/2.52 "from": 330, 6.56/2.52 "to": 372, 6.56/2.52 "label": "PARALLEL" 6.56/2.52 }, 6.56/2.52 { 6.56/2.52 "from": 330, 6.56/2.52 "to": 373, 6.56/2.52 "label": "PARALLEL" 6.56/2.52 }, 6.56/2.52 { 6.56/2.52 "from": 331, 6.56/2.52 "to": 5, 6.56/2.52 "label": "INSTANCE with matching:\nT1 -> T18\nT2 -> .(0, T20)\nT3 -> T22" 6.56/2.52 }, 6.56/2.52 { 6.56/2.52 "from": 372, 6.56/2.52 "to": 377, 6.56/2.52 "label": "EVAL with clause\nleq(0, s(0)).\nand substitutionT17 -> 0,\nT19 -> s(0)" 6.56/2.52 }, 6.56/2.52 { 6.56/2.52 "from": 372, 6.56/2.52 "to": 378, 6.56/2.52 "label": "EVAL-BACKTRACK" 6.56/2.52 }, 6.56/2.52 { 6.56/2.52 "from": 373, 6.56/2.52 "to": 398, 6.56/2.52 "label": "PARALLEL" 6.56/2.52 }, 6.56/2.52 { 6.56/2.52 "from": 373, 6.56/2.52 "to": 399, 6.56/2.52 "label": "PARALLEL" 6.56/2.52 }, 6.56/2.52 { 6.56/2.52 "from": 377, 6.56/2.52 "to": 5, 6.56/2.52 "label": "INSTANCE with matching:\nT1 -> T18\nT2 -> .(s(0), T20)\nT3 -> T22" 6.56/2.52 }, 6.56/2.52 { 6.56/2.52 "from": 398, 6.56/2.52 "to": 400, 6.56/2.52 "label": "EVAL with clause\nleq(s(X59), s(X60)) :- leq(X59, X60).\nand substitutionX59 -> T35,\nT17 -> s(T35),\nX60 -> T36,\nT19 -> s(T36)" 6.56/2.52 }, 6.56/2.52 { 6.56/2.52 "from": 398, 6.56/2.52 "to": 401, 6.56/2.52 "label": "EVAL-BACKTRACK" 6.56/2.52 }, 6.56/2.52 { 6.56/2.52 "from": 399, 6.56/2.52 "to": 417, 6.56/2.52 "label": "FAILURE" 6.56/2.52 }, 6.56/2.52 { 6.56/2.52 "from": 400, 6.56/2.52 "to": 402, 6.56/2.52 "label": "SPLIT 1" 6.56/2.52 }, 6.56/2.52 { 6.56/2.52 "from": 400, 6.56/2.52 "to": 403, 6.56/2.52 "label": "SPLIT 2\nnew knowledge:\nT35 is ground\nT36 is ground" 6.56/2.52 }, 6.56/2.52 { 6.56/2.52 "from": 402, 6.56/2.52 "to": 404, 6.56/2.52 "label": "CASE" 6.56/2.52 }, 6.56/2.52 { 6.56/2.52 "from": 403, 6.56/2.52 "to": 5, 6.56/2.52 "label": "INSTANCE with matching:\nT1 -> T18\nT2 -> .(s(T36), T20)\nT3 -> T22" 6.56/2.52 }, 6.56/2.52 { 6.56/2.52 "from": 404, 6.56/2.52 "to": 405, 6.56/2.52 "label": "PARALLEL" 6.56/2.52 }, 6.56/2.52 { 6.56/2.52 "from": 404, 6.56/2.52 "to": 406, 6.56/2.52 "label": "PARALLEL" 6.56/2.52 }, 6.56/2.52 { 6.56/2.52 "from": 405, 6.56/2.52 "to": 407, 6.56/2.52 "label": "EVAL with clause\nleq(0, 0).\nand substitutionT35 -> 0,\nT36 -> 0" 6.56/2.52 }, 6.56/2.52 { 6.56/2.52 "from": 405, 6.56/2.52 "to": 408, 6.56/2.52 "label": "EVAL-BACKTRACK" 6.56/2.52 }, 6.56/2.52 { 6.56/2.52 "from": 406, 6.56/2.52 "to": 410, 6.56/2.52 "label": "PARALLEL" 6.56/2.52 }, 6.56/2.52 { 6.56/2.52 "from": 406, 6.56/2.52 "to": 411, 6.56/2.52 "label": "PARALLEL" 6.56/2.52 }, 6.56/2.52 { 6.56/2.52 "from": 407, 6.56/2.52 "to": 409, 6.56/2.52 "label": "SUCCESS" 6.56/2.52 }, 6.56/2.52 { 6.56/2.52 "from": 410, 6.56/2.52 "to": 412, 6.56/2.52 "label": "EVAL with clause\nleq(0, s(0)).\nand substitutionT35 -> 0,\nT36 -> s(0)" 6.56/2.52 }, 6.56/2.52 { 6.56/2.52 "from": 410, 6.56/2.52 "to": 413, 6.56/2.52 "label": "EVAL-BACKTRACK" 6.56/2.52 }, 6.56/2.52 { 6.56/2.52 "from": 411, 6.56/2.52 "to": 415, 6.56/2.52 "label": "EVAL with clause\nleq(s(X65), s(X66)) :- leq(X65, X66).\nand substitutionX65 -> T41,\nT35 -> s(T41),\nX66 -> T42,\nT36 -> s(T42)" 6.56/2.52 }, 6.56/2.52 { 6.56/2.52 "from": 411, 6.56/2.52 "to": 416, 6.56/2.52 "label": "EVAL-BACKTRACK" 6.56/2.52 }, 6.56/2.52 { 6.56/2.52 "from": 412, 6.56/2.52 "to": 414, 6.56/2.52 "label": "SUCCESS" 6.56/2.52 }, 6.56/2.52 { 6.56/2.52 "from": 415, 6.56/2.52 "to": 402, 6.56/2.52 "label": "INSTANCE with matching:\nT35 -> T41\nT36 -> T42" 6.56/2.52 }, 6.56/2.52 { 6.56/2.52 "from": 417, 6.56/2.52 "to": 418, 6.56/2.52 "label": "EVAL with clause\nmerge(.(X79, X80), .(X81, X82), .(X81, X83)) :- ','(less(X81, X79), merge(.(X79, X80), X82, X83)).\nand substitutionT17 -> T57,\nX79 -> T57,\nT18 -> T58,\nX80 -> T58,\nT19 -> T59,\nX81 -> T59,\nT20 -> T60,\nX82 -> T60,\nX83 -> T62,\nT3 -> .(T59, T62),\nT61 -> T62" 6.56/2.52 }, 6.56/2.52 { 6.56/2.52 "from": 417, 6.56/2.52 "to": 419, 6.56/2.52 "label": "EVAL-BACKTRACK" 6.56/2.52 }, 6.56/2.52 { 6.56/2.52 "from": 418, 6.56/2.52 "to": 420, 6.56/2.52 "label": "SPLIT 1" 6.56/2.52 }, 6.56/2.52 { 6.56/2.52 "from": 418, 6.56/2.52 "to": 421, 6.56/2.52 "label": "SPLIT 2\nnew knowledge:\nT59 is ground\nT57 is ground" 6.56/2.52 }, 6.56/2.52 { 6.56/2.52 "from": 420, 6.56/2.52 "to": 422, 6.56/2.52 "label": "CASE" 6.56/2.52 }, 6.56/2.52 { 6.56/2.52 "from": 421, 6.56/2.52 "to": 5, 6.56/2.52 "label": "INSTANCE with matching:\nT1 -> .(T57, T58)\nT2 -> T60\nT3 -> T62" 6.56/2.52 }, 6.56/2.52 { 6.56/2.52 "from": 422, 6.56/2.52 "to": 423, 6.56/2.52 "label": "PARALLEL" 6.56/2.52 }, 6.56/2.52 { 6.56/2.52 "from": 422, 6.56/2.52 "to": 424, 6.56/2.52 "label": "PARALLEL" 6.56/2.52 }, 6.56/2.52 { 6.56/2.52 "from": 423, 6.56/2.52 "to": 425, 6.56/2.52 "label": "EVAL with clause\nless(0, s(0)).\nand substitutionT59 -> 0,\nT57 -> s(0)" 6.56/2.52 }, 6.56/2.52 { 6.56/2.52 "from": 423, 6.56/2.52 "to": 426, 6.56/2.52 "label": "EVAL-BACKTRACK" 6.56/2.52 }, 6.56/2.52 { 6.56/2.52 "from": 424, 6.56/2.52 "to": 428, 6.56/2.52 "label": "EVAL with clause\nless(s(X88), s(X89)) :- less(X88, X89).\nand substitutionX88 -> T67,\nT59 -> s(T67),\nX89 -> T68,\nT57 -> s(T68)" 6.56/2.52 }, 6.56/2.52 { 6.56/2.52 "from": 424, 6.56/2.52 "to": 429, 6.56/2.52 "label": "EVAL-BACKTRACK" 6.56/2.52 }, 6.56/2.52 { 6.56/2.52 "from": 425, 6.56/2.52 "to": 427, 6.56/2.52 "label": "SUCCESS" 6.56/2.52 }, 6.56/2.52 { 6.56/2.52 "from": 428, 6.56/2.52 "to": 420, 6.56/2.52 "label": "INSTANCE with matching:\nT59 -> T67\nT57 -> T68" 6.56/2.52 }, 6.56/2.52 { 6.56/2.52 "from": 430, 6.56/2.52 "to": 432, 6.56/2.52 "label": "CASE" 6.56/2.52 }, 6.56/2.52 { 6.56/2.52 "from": 432, 6.56/2.52 "to": 433, 6.56/2.52 "label": "PARALLEL" 6.56/2.52 }, 6.56/2.52 { 6.56/2.52 "from": 432, 6.56/2.52 "to": 434, 6.56/2.52 "label": "PARALLEL" 6.56/2.52 }, 6.56/2.52 { 6.56/2.52 "from": 433, 6.56/2.52 "to": 435, 6.56/2.52 "label": "EVAL with clause\nless(0, s(0)).\nand substitutionT76 -> 0,\nT74 -> s(0)" 6.56/2.52 }, 6.56/2.52 { 6.56/2.52 "from": 433, 6.56/2.52 "to": 436, 6.56/2.52 "label": "EVAL-BACKTRACK" 6.56/2.52 }, 6.56/2.52 { 6.56/2.52 "from": 434, 6.56/2.52 "to": 437, 6.56/2.52 "label": "EVAL with clause\nless(s(X108), s(X109)) :- less(X108, X109).\nand substitutionX108 -> T84,\nT76 -> s(T84),\nX109 -> T85,\nT74 -> s(T85)" 6.56/2.52 }, 6.56/2.52 { 6.56/2.52 "from": 434, 6.56/2.52 "to": 438, 6.56/2.52 "label": "EVAL-BACKTRACK" 6.56/2.52 }, 6.56/2.52 { 6.56/2.52 "from": 435, 6.56/2.52 "to": 5, 6.56/2.52 "label": "INSTANCE with matching:\nT1 -> .(s(0), T75)\nT2 -> T77\nT3 -> T79" 6.56/2.52 }, 6.56/2.52 { 6.56/2.52 "from": 437, 6.56/2.52 "to": 439, 6.56/2.52 "label": "SPLIT 1" 6.56/2.52 }, 6.56/2.52 { 6.56/2.52 "from": 437, 6.56/2.52 "to": 440, 6.56/2.52 "label": "SPLIT 2\nnew knowledge:\nT84 is ground\nT85 is ground" 6.56/2.52 }, 6.56/2.52 { 6.56/2.52 "from": 439, 6.56/2.52 "to": 420, 6.56/2.52 "label": "INSTANCE with matching:\nT59 -> T84\nT57 -> T85" 6.56/2.52 }, 6.56/2.52 { 6.56/2.52 "from": 440, 6.56/2.52 "to": 5, 6.56/2.52 "label": "INSTANCE with matching:\nT1 -> .(s(T85), T75)\nT2 -> T77\nT3 -> T79" 6.56/2.52 } 6.56/2.52 ], 6.56/2.52 "type": "Graph" 6.56/2.52 } 6.56/2.52 } 6.56/2.52 6.56/2.52 ---------------------------------------- 6.56/2.52 6.56/2.52 (2) 6.56/2.52 Obligation: 6.56/2.52 Triples: 6.56/2.52 6.56/2.52 leqB(s(X1), s(X2)) :- leqB(X1, X2). 6.56/2.52 lessC(s(X1), s(X2)) :- lessC(X1, X2). 6.56/2.52 mergeA(.(0, X1), .(0, X2), .(0, X3)) :- mergeA(X1, .(0, X2), X3). 6.56/2.52 mergeA(.(0, X1), .(s(0), X2), .(0, X3)) :- mergeA(X1, .(s(0), X2), X3). 6.56/2.52 mergeA(.(s(X1), X2), .(s(X3), X4), .(s(X1), X5)) :- leqB(X1, X3). 6.56/2.52 mergeA(.(s(X1), X2), .(s(X3), X4), .(s(X1), X5)) :- ','(leqcB(X1, X3), mergeA(X2, .(s(X3), X4), X5)). 6.56/2.52 mergeA(.(X1, X2), .(X3, X4), .(X3, X5)) :- lessC(X3, X1). 6.56/2.52 mergeA(.(X1, X2), .(X3, X4), .(X3, X5)) :- ','(lesscC(X3, X1), mergeA(.(X1, X2), X4, X5)). 6.56/2.52 mergeA(.(s(0), X1), .(0, X2), .(0, X3)) :- mergeA(.(s(0), X1), X2, X3). 6.56/2.52 mergeA(.(s(X1), X2), .(s(X3), X4), .(s(X3), X5)) :- lessC(X3, X1). 6.56/2.52 mergeA(.(s(X1), X2), .(s(X3), X4), .(s(X3), X5)) :- ','(lesscC(X3, X1), mergeA(.(s(X1), X2), X4, X5)). 6.56/2.52 6.56/2.52 Clauses: 6.56/2.52 6.56/2.52 mergecA([], X1, X1). 6.56/2.52 mergecA([], [], []). 6.56/2.52 mergecA(X1, [], X1). 6.56/2.52 mergecA(.(0, X1), .(0, X2), .(0, X3)) :- mergecA(X1, .(0, X2), X3). 6.56/2.52 mergecA(.(0, X1), .(s(0), X2), .(0, X3)) :- mergecA(X1, .(s(0), X2), X3). 6.56/2.52 mergecA(.(s(X1), X2), .(s(X3), X4), .(s(X1), X5)) :- ','(leqcB(X1, X3), mergecA(X2, .(s(X3), X4), X5)). 6.56/2.52 mergecA(.(X1, X2), .(X3, X4), .(X3, X5)) :- ','(lesscC(X3, X1), mergecA(.(X1, X2), X4, X5)). 6.56/2.52 mergecA(.(s(0), X1), .(0, X2), .(0, X3)) :- mergecA(.(s(0), X1), X2, X3). 6.56/2.52 mergecA(.(s(X1), X2), .(s(X3), X4), .(s(X3), X5)) :- ','(lesscC(X3, X1), mergecA(.(s(X1), X2), X4, X5)). 6.56/2.52 leqcB(0, 0). 6.56/2.52 leqcB(0, s(0)). 6.56/2.52 leqcB(s(X1), s(X2)) :- leqcB(X1, X2). 6.56/2.52 lesscC(0, s(0)). 6.56/2.52 lesscC(s(X1), s(X2)) :- lesscC(X1, X2). 6.56/2.52 6.56/2.52 Afs: 6.56/2.52 6.56/2.52 mergeA(x1, x2, x3) = mergeA(x1, x2) 6.56/2.52 6.56/2.52 6.56/2.52 ---------------------------------------- 6.56/2.52 6.56/2.52 (3) TriplesToPiDPProof (SOUND) 6.56/2.52 We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes: 6.56/2.52 6.56/2.52 mergeA_in_3: (b,b,f) 6.56/2.52 6.56/2.52 leqB_in_2: (b,b) 6.56/2.52 6.56/2.52 leqcB_in_2: (b,b) 6.56/2.52 6.56/2.52 lessC_in_2: (b,b) 6.56/2.52 6.56/2.52 lesscC_in_2: (b,b) 6.56/2.52 6.56/2.52 Transforming TRIPLES into the following Term Rewriting System: 6.56/2.52 6.56/2.52 Pi DP problem: 6.56/2.52 The TRS P consists of the following rules: 6.56/2.52 6.56/2.52 MERGEA_IN_GGA(.(0, X1), .(0, X2), .(0, X3)) -> U3_GGA(X1, X2, X3, mergeA_in_gga(X1, .(0, X2), X3)) 6.56/2.52 MERGEA_IN_GGA(.(0, X1), .(0, X2), .(0, X3)) -> MERGEA_IN_GGA(X1, .(0, X2), X3) 6.56/2.52 MERGEA_IN_GGA(.(0, X1), .(s(0), X2), .(0, X3)) -> U4_GGA(X1, X2, X3, mergeA_in_gga(X1, .(s(0), X2), X3)) 6.56/2.52 MERGEA_IN_GGA(.(0, X1), .(s(0), X2), .(0, X3)) -> MERGEA_IN_GGA(X1, .(s(0), X2), X3) 6.56/2.52 MERGEA_IN_GGA(.(s(X1), X2), .(s(X3), X4), .(s(X1), X5)) -> U5_GGA(X1, X2, X3, X4, X5, leqB_in_gg(X1, X3)) 6.56/2.52 MERGEA_IN_GGA(.(s(X1), X2), .(s(X3), X4), .(s(X1), X5)) -> LEQB_IN_GG(X1, X3) 6.56/2.52 LEQB_IN_GG(s(X1), s(X2)) -> U1_GG(X1, X2, leqB_in_gg(X1, X2)) 6.56/2.52 LEQB_IN_GG(s(X1), s(X2)) -> LEQB_IN_GG(X1, X2) 6.56/2.52 MERGEA_IN_GGA(.(s(X1), X2), .(s(X3), X4), .(s(X1), X5)) -> U6_GGA(X1, X2, X3, X4, X5, leqcB_in_gg(X1, X3)) 6.56/2.52 U6_GGA(X1, X2, X3, X4, X5, leqcB_out_gg(X1, X3)) -> U7_GGA(X1, X2, X3, X4, X5, mergeA_in_gga(X2, .(s(X3), X4), X5)) 6.56/2.52 U6_GGA(X1, X2, X3, X4, X5, leqcB_out_gg(X1, X3)) -> MERGEA_IN_GGA(X2, .(s(X3), X4), X5) 6.56/2.52 MERGEA_IN_GGA(.(X1, X2), .(X3, X4), .(X3, X5)) -> U8_GGA(X1, X2, X3, X4, X5, lessC_in_gg(X3, X1)) 6.56/2.52 MERGEA_IN_GGA(.(X1, X2), .(X3, X4), .(X3, X5)) -> LESSC_IN_GG(X3, X1) 6.56/2.52 LESSC_IN_GG(s(X1), s(X2)) -> U2_GG(X1, X2, lessC_in_gg(X1, X2)) 6.56/2.52 LESSC_IN_GG(s(X1), s(X2)) -> LESSC_IN_GG(X1, X2) 6.56/2.52 MERGEA_IN_GGA(.(X1, X2), .(X3, X4), .(X3, X5)) -> U9_GGA(X1, X2, X3, X4, X5, lesscC_in_gg(X3, X1)) 6.56/2.52 U9_GGA(X1, X2, X3, X4, X5, lesscC_out_gg(X3, X1)) -> U10_GGA(X1, X2, X3, X4, X5, mergeA_in_gga(.(X1, X2), X4, X5)) 6.56/2.52 U9_GGA(X1, X2, X3, X4, X5, lesscC_out_gg(X3, X1)) -> MERGEA_IN_GGA(.(X1, X2), X4, X5) 6.56/2.52 MERGEA_IN_GGA(.(s(0), X1), .(0, X2), .(0, X3)) -> U11_GGA(X1, X2, X3, mergeA_in_gga(.(s(0), X1), X2, X3)) 6.56/2.52 MERGEA_IN_GGA(.(s(0), X1), .(0, X2), .(0, X3)) -> MERGEA_IN_GGA(.(s(0), X1), X2, X3) 6.56/2.52 MERGEA_IN_GGA(.(s(X1), X2), .(s(X3), X4), .(s(X3), X5)) -> U12_GGA(X1, X2, X3, X4, X5, lessC_in_gg(X3, X1)) 6.56/2.52 MERGEA_IN_GGA(.(s(X1), X2), .(s(X3), X4), .(s(X3), X5)) -> LESSC_IN_GG(X3, X1) 6.56/2.52 MERGEA_IN_GGA(.(s(X1), X2), .(s(X3), X4), .(s(X3), X5)) -> U13_GGA(X1, X2, X3, X4, X5, lesscC_in_gg(X3, X1)) 6.56/2.52 U13_GGA(X1, X2, X3, X4, X5, lesscC_out_gg(X3, X1)) -> U14_GGA(X1, X2, X3, X4, X5, mergeA_in_gga(.(s(X1), X2), X4, X5)) 6.56/2.52 U13_GGA(X1, X2, X3, X4, X5, lesscC_out_gg(X3, X1)) -> MERGEA_IN_GGA(.(s(X1), X2), X4, X5) 6.56/2.52 6.56/2.52 The TRS R consists of the following rules: 6.56/2.52 6.56/2.52 leqcB_in_gg(0, 0) -> leqcB_out_gg(0, 0) 6.56/2.52 leqcB_in_gg(0, s(0)) -> leqcB_out_gg(0, s(0)) 6.56/2.52 leqcB_in_gg(s(X1), s(X2)) -> U25_gg(X1, X2, leqcB_in_gg(X1, X2)) 6.56/2.52 U25_gg(X1, X2, leqcB_out_gg(X1, X2)) -> leqcB_out_gg(s(X1), s(X2)) 6.56/2.52 lesscC_in_gg(0, s(0)) -> lesscC_out_gg(0, s(0)) 6.56/2.52 lesscC_in_gg(s(X1), s(X2)) -> U26_gg(X1, X2, lesscC_in_gg(X1, X2)) 6.56/2.52 U26_gg(X1, X2, lesscC_out_gg(X1, X2)) -> lesscC_out_gg(s(X1), s(X2)) 6.56/2.52 6.56/2.52 The argument filtering Pi contains the following mapping: 6.56/2.52 mergeA_in_gga(x1, x2, x3) = mergeA_in_gga(x1, x2) 6.56/2.52 6.56/2.52 .(x1, x2) = .(x1, x2) 6.56/2.52 6.56/2.52 0 = 0 6.56/2.52 6.56/2.52 s(x1) = s(x1) 6.56/2.52 6.56/2.52 leqB_in_gg(x1, x2) = leqB_in_gg(x1, x2) 6.56/2.52 6.56/2.52 leqcB_in_gg(x1, x2) = leqcB_in_gg(x1, x2) 6.56/2.52 6.56/2.52 leqcB_out_gg(x1, x2) = leqcB_out_gg(x1, x2) 6.56/2.52 6.56/2.52 U25_gg(x1, x2, x3) = U25_gg(x1, x2, x3) 6.56/2.52 6.56/2.52 lessC_in_gg(x1, x2) = lessC_in_gg(x1, x2) 6.56/2.52 6.56/2.52 lesscC_in_gg(x1, x2) = lesscC_in_gg(x1, x2) 6.56/2.52 6.56/2.52 lesscC_out_gg(x1, x2) = lesscC_out_gg(x1, x2) 6.56/2.52 6.56/2.52 U26_gg(x1, x2, x3) = U26_gg(x1, x2, x3) 6.56/2.52 6.56/2.52 MERGEA_IN_GGA(x1, x2, x3) = MERGEA_IN_GGA(x1, x2) 6.56/2.52 6.56/2.52 U3_GGA(x1, x2, x3, x4) = U3_GGA(x1, x2, x4) 6.56/2.52 6.56/2.52 U4_GGA(x1, x2, x3, x4) = U4_GGA(x1, x2, x4) 6.56/2.52 6.56/2.52 U5_GGA(x1, x2, x3, x4, x5, x6) = U5_GGA(x1, x2, x3, x4, x6) 6.56/2.52 6.56/2.52 LEQB_IN_GG(x1, x2) = LEQB_IN_GG(x1, x2) 6.56/2.52 6.56/2.52 U1_GG(x1, x2, x3) = U1_GG(x1, x2, x3) 6.56/2.52 6.56/2.52 U6_GGA(x1, x2, x3, x4, x5, x6) = U6_GGA(x1, x2, x3, x4, x6) 6.56/2.52 6.56/2.52 U7_GGA(x1, x2, x3, x4, x5, x6) = U7_GGA(x1, x2, x3, x4, x6) 6.56/2.52 6.56/2.52 U8_GGA(x1, x2, x3, x4, x5, x6) = U8_GGA(x1, x2, x3, x4, x6) 6.56/2.52 6.56/2.52 LESSC_IN_GG(x1, x2) = LESSC_IN_GG(x1, x2) 6.56/2.52 6.56/2.52 U2_GG(x1, x2, x3) = U2_GG(x1, x2, x3) 6.56/2.52 6.56/2.52 U9_GGA(x1, x2, x3, x4, x5, x6) = U9_GGA(x1, x2, x3, x4, x6) 6.56/2.52 6.56/2.52 U10_GGA(x1, x2, x3, x4, x5, x6) = U10_GGA(x1, x2, x3, x4, x6) 6.56/2.52 6.56/2.52 U11_GGA(x1, x2, x3, x4) = U11_GGA(x1, x2, x4) 6.56/2.52 6.56/2.52 U12_GGA(x1, x2, x3, x4, x5, x6) = U12_GGA(x1, x2, x3, x4, x6) 6.56/2.52 6.56/2.52 U13_GGA(x1, x2, x3, x4, x5, x6) = U13_GGA(x1, x2, x3, x4, x6) 6.56/2.52 6.56/2.52 U14_GGA(x1, x2, x3, x4, x5, x6) = U14_GGA(x1, x2, x3, x4, x6) 6.56/2.52 6.56/2.52 6.56/2.52 We have to consider all (P,R,Pi)-chains 6.56/2.52 6.56/2.52 6.56/2.52 Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES 6.56/2.52 6.56/2.52 6.56/2.52 6.56/2.52 ---------------------------------------- 6.56/2.52 6.56/2.52 (4) 6.56/2.52 Obligation: 6.56/2.52 Pi DP problem: 6.56/2.52 The TRS P consists of the following rules: 6.56/2.52 6.56/2.52 MERGEA_IN_GGA(.(0, X1), .(0, X2), .(0, X3)) -> U3_GGA(X1, X2, X3, mergeA_in_gga(X1, .(0, X2), X3)) 6.56/2.52 MERGEA_IN_GGA(.(0, X1), .(0, X2), .(0, X3)) -> MERGEA_IN_GGA(X1, .(0, X2), X3) 6.56/2.52 MERGEA_IN_GGA(.(0, X1), .(s(0), X2), .(0, X3)) -> U4_GGA(X1, X2, X3, mergeA_in_gga(X1, .(s(0), X2), X3)) 6.56/2.52 MERGEA_IN_GGA(.(0, X1), .(s(0), X2), .(0, X3)) -> MERGEA_IN_GGA(X1, .(s(0), X2), X3) 6.56/2.52 MERGEA_IN_GGA(.(s(X1), X2), .(s(X3), X4), .(s(X1), X5)) -> U5_GGA(X1, X2, X3, X4, X5, leqB_in_gg(X1, X3)) 6.56/2.52 MERGEA_IN_GGA(.(s(X1), X2), .(s(X3), X4), .(s(X1), X5)) -> LEQB_IN_GG(X1, X3) 6.56/2.52 LEQB_IN_GG(s(X1), s(X2)) -> U1_GG(X1, X2, leqB_in_gg(X1, X2)) 6.56/2.52 LEQB_IN_GG(s(X1), s(X2)) -> LEQB_IN_GG(X1, X2) 6.56/2.52 MERGEA_IN_GGA(.(s(X1), X2), .(s(X3), X4), .(s(X1), X5)) -> U6_GGA(X1, X2, X3, X4, X5, leqcB_in_gg(X1, X3)) 6.56/2.52 U6_GGA(X1, X2, X3, X4, X5, leqcB_out_gg(X1, X3)) -> U7_GGA(X1, X2, X3, X4, X5, mergeA_in_gga(X2, .(s(X3), X4), X5)) 6.56/2.52 U6_GGA(X1, X2, X3, X4, X5, leqcB_out_gg(X1, X3)) -> MERGEA_IN_GGA(X2, .(s(X3), X4), X5) 6.56/2.52 MERGEA_IN_GGA(.(X1, X2), .(X3, X4), .(X3, X5)) -> U8_GGA(X1, X2, X3, X4, X5, lessC_in_gg(X3, X1)) 6.56/2.52 MERGEA_IN_GGA(.(X1, X2), .(X3, X4), .(X3, X5)) -> LESSC_IN_GG(X3, X1) 6.56/2.52 LESSC_IN_GG(s(X1), s(X2)) -> U2_GG(X1, X2, lessC_in_gg(X1, X2)) 6.56/2.52 LESSC_IN_GG(s(X1), s(X2)) -> LESSC_IN_GG(X1, X2) 6.56/2.52 MERGEA_IN_GGA(.(X1, X2), .(X3, X4), .(X3, X5)) -> U9_GGA(X1, X2, X3, X4, X5, lesscC_in_gg(X3, X1)) 6.56/2.52 U9_GGA(X1, X2, X3, X4, X5, lesscC_out_gg(X3, X1)) -> U10_GGA(X1, X2, X3, X4, X5, mergeA_in_gga(.(X1, X2), X4, X5)) 6.56/2.52 U9_GGA(X1, X2, X3, X4, X5, lesscC_out_gg(X3, X1)) -> MERGEA_IN_GGA(.(X1, X2), X4, X5) 6.56/2.52 MERGEA_IN_GGA(.(s(0), X1), .(0, X2), .(0, X3)) -> U11_GGA(X1, X2, X3, mergeA_in_gga(.(s(0), X1), X2, X3)) 6.56/2.52 MERGEA_IN_GGA(.(s(0), X1), .(0, X2), .(0, X3)) -> MERGEA_IN_GGA(.(s(0), X1), X2, X3) 6.56/2.52 MERGEA_IN_GGA(.(s(X1), X2), .(s(X3), X4), .(s(X3), X5)) -> U12_GGA(X1, X2, X3, X4, X5, lessC_in_gg(X3, X1)) 6.56/2.52 MERGEA_IN_GGA(.(s(X1), X2), .(s(X3), X4), .(s(X3), X5)) -> LESSC_IN_GG(X3, X1) 6.56/2.52 MERGEA_IN_GGA(.(s(X1), X2), .(s(X3), X4), .(s(X3), X5)) -> U13_GGA(X1, X2, X3, X4, X5, lesscC_in_gg(X3, X1)) 6.56/2.52 U13_GGA(X1, X2, X3, X4, X5, lesscC_out_gg(X3, X1)) -> U14_GGA(X1, X2, X3, X4, X5, mergeA_in_gga(.(s(X1), X2), X4, X5)) 6.56/2.52 U13_GGA(X1, X2, X3, X4, X5, lesscC_out_gg(X3, X1)) -> MERGEA_IN_GGA(.(s(X1), X2), X4, X5) 6.56/2.52 6.56/2.52 The TRS R consists of the following rules: 6.56/2.52 6.56/2.52 leqcB_in_gg(0, 0) -> leqcB_out_gg(0, 0) 6.56/2.52 leqcB_in_gg(0, s(0)) -> leqcB_out_gg(0, s(0)) 6.56/2.52 leqcB_in_gg(s(X1), s(X2)) -> U25_gg(X1, X2, leqcB_in_gg(X1, X2)) 6.56/2.52 U25_gg(X1, X2, leqcB_out_gg(X1, X2)) -> leqcB_out_gg(s(X1), s(X2)) 6.56/2.52 lesscC_in_gg(0, s(0)) -> lesscC_out_gg(0, s(0)) 6.56/2.52 lesscC_in_gg(s(X1), s(X2)) -> U26_gg(X1, X2, lesscC_in_gg(X1, X2)) 6.56/2.52 U26_gg(X1, X2, lesscC_out_gg(X1, X2)) -> lesscC_out_gg(s(X1), s(X2)) 6.56/2.52 6.56/2.52 The argument filtering Pi contains the following mapping: 6.56/2.52 mergeA_in_gga(x1, x2, x3) = mergeA_in_gga(x1, x2) 6.56/2.52 6.56/2.52 .(x1, x2) = .(x1, x2) 6.56/2.52 6.56/2.52 0 = 0 6.56/2.52 6.56/2.52 s(x1) = s(x1) 6.56/2.52 6.56/2.52 leqB_in_gg(x1, x2) = leqB_in_gg(x1, x2) 6.56/2.52 6.56/2.52 leqcB_in_gg(x1, x2) = leqcB_in_gg(x1, x2) 6.56/2.52 6.56/2.52 leqcB_out_gg(x1, x2) = leqcB_out_gg(x1, x2) 6.56/2.52 6.56/2.52 U25_gg(x1, x2, x3) = U25_gg(x1, x2, x3) 6.56/2.52 6.56/2.52 lessC_in_gg(x1, x2) = lessC_in_gg(x1, x2) 6.56/2.52 6.56/2.52 lesscC_in_gg(x1, x2) = lesscC_in_gg(x1, x2) 6.56/2.52 6.56/2.52 lesscC_out_gg(x1, x2) = lesscC_out_gg(x1, x2) 6.56/2.52 6.56/2.52 U26_gg(x1, x2, x3) = U26_gg(x1, x2, x3) 6.56/2.52 6.56/2.52 MERGEA_IN_GGA(x1, x2, x3) = MERGEA_IN_GGA(x1, x2) 6.56/2.52 6.56/2.52 U3_GGA(x1, x2, x3, x4) = U3_GGA(x1, x2, x4) 6.56/2.52 6.56/2.52 U4_GGA(x1, x2, x3, x4) = U4_GGA(x1, x2, x4) 6.56/2.52 6.56/2.52 U5_GGA(x1, x2, x3, x4, x5, x6) = U5_GGA(x1, x2, x3, x4, x6) 6.56/2.52 6.56/2.52 LEQB_IN_GG(x1, x2) = LEQB_IN_GG(x1, x2) 6.56/2.52 6.56/2.52 U1_GG(x1, x2, x3) = U1_GG(x1, x2, x3) 6.56/2.52 6.56/2.52 U6_GGA(x1, x2, x3, x4, x5, x6) = U6_GGA(x1, x2, x3, x4, x6) 6.56/2.52 6.56/2.52 U7_GGA(x1, x2, x3, x4, x5, x6) = U7_GGA(x1, x2, x3, x4, x6) 6.56/2.52 6.56/2.52 U8_GGA(x1, x2, x3, x4, x5, x6) = U8_GGA(x1, x2, x3, x4, x6) 6.56/2.52 6.56/2.52 LESSC_IN_GG(x1, x2) = LESSC_IN_GG(x1, x2) 6.56/2.52 6.56/2.52 U2_GG(x1, x2, x3) = U2_GG(x1, x2, x3) 6.56/2.52 6.56/2.52 U9_GGA(x1, x2, x3, x4, x5, x6) = U9_GGA(x1, x2, x3, x4, x6) 6.56/2.53 6.56/2.53 U10_GGA(x1, x2, x3, x4, x5, x6) = U10_GGA(x1, x2, x3, x4, x6) 6.56/2.53 6.56/2.53 U11_GGA(x1, x2, x3, x4) = U11_GGA(x1, x2, x4) 6.56/2.53 6.56/2.53 U12_GGA(x1, x2, x3, x4, x5, x6) = U12_GGA(x1, x2, x3, x4, x6) 6.56/2.53 6.56/2.53 U13_GGA(x1, x2, x3, x4, x5, x6) = U13_GGA(x1, x2, x3, x4, x6) 6.56/2.53 6.56/2.53 U14_GGA(x1, x2, x3, x4, x5, x6) = U14_GGA(x1, x2, x3, x4, x6) 6.56/2.53 6.56/2.53 6.56/2.53 We have to consider all (P,R,Pi)-chains 6.56/2.53 ---------------------------------------- 6.56/2.53 6.56/2.53 (5) DependencyGraphProof (EQUIVALENT) 6.56/2.53 The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 14 less nodes. 6.56/2.53 ---------------------------------------- 6.56/2.53 6.56/2.53 (6) 6.56/2.53 Complex Obligation (AND) 6.56/2.53 6.56/2.53 ---------------------------------------- 6.56/2.53 6.56/2.53 (7) 6.56/2.53 Obligation: 6.56/2.53 Pi DP problem: 6.56/2.53 The TRS P consists of the following rules: 6.56/2.53 6.56/2.53 LESSC_IN_GG(s(X1), s(X2)) -> LESSC_IN_GG(X1, X2) 6.56/2.53 6.56/2.53 The TRS R consists of the following rules: 6.56/2.53 6.56/2.53 leqcB_in_gg(0, 0) -> leqcB_out_gg(0, 0) 6.56/2.53 leqcB_in_gg(0, s(0)) -> leqcB_out_gg(0, s(0)) 6.56/2.53 leqcB_in_gg(s(X1), s(X2)) -> U25_gg(X1, X2, leqcB_in_gg(X1, X2)) 6.56/2.53 U25_gg(X1, X2, leqcB_out_gg(X1, X2)) -> leqcB_out_gg(s(X1), s(X2)) 6.56/2.53 lesscC_in_gg(0, s(0)) -> lesscC_out_gg(0, s(0)) 6.56/2.53 lesscC_in_gg(s(X1), s(X2)) -> U26_gg(X1, X2, lesscC_in_gg(X1, X2)) 6.56/2.53 U26_gg(X1, X2, lesscC_out_gg(X1, X2)) -> lesscC_out_gg(s(X1), s(X2)) 6.56/2.53 6.56/2.53 Pi is empty. 6.56/2.53 We have to consider all (P,R,Pi)-chains 6.56/2.53 ---------------------------------------- 6.56/2.53 6.56/2.53 (8) UsableRulesProof (EQUIVALENT) 6.56/2.53 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 6.56/2.53 ---------------------------------------- 6.56/2.53 6.56/2.53 (9) 6.56/2.53 Obligation: 6.56/2.53 Pi DP problem: 6.56/2.53 The TRS P consists of the following rules: 6.56/2.53 6.56/2.53 LESSC_IN_GG(s(X1), s(X2)) -> LESSC_IN_GG(X1, X2) 6.56/2.53 6.56/2.53 R is empty. 6.56/2.53 Pi is empty. 6.56/2.53 We have to consider all (P,R,Pi)-chains 6.56/2.53 ---------------------------------------- 6.56/2.53 6.56/2.53 (10) PiDPToQDPProof (EQUIVALENT) 6.56/2.53 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 6.56/2.53 ---------------------------------------- 6.56/2.53 6.56/2.53 (11) 6.56/2.53 Obligation: 6.56/2.53 Q DP problem: 6.56/2.53 The TRS P consists of the following rules: 6.56/2.53 6.56/2.53 LESSC_IN_GG(s(X1), s(X2)) -> LESSC_IN_GG(X1, X2) 6.56/2.53 6.56/2.53 R is empty. 6.56/2.53 Q is empty. 6.56/2.53 We have to consider all (P,Q,R)-chains. 6.56/2.53 ---------------------------------------- 6.56/2.53 6.56/2.53 (12) QDPSizeChangeProof (EQUIVALENT) 6.56/2.53 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. 6.56/2.53 6.56/2.53 From the DPs we obtained the following set of size-change graphs: 6.56/2.53 *LESSC_IN_GG(s(X1), s(X2)) -> LESSC_IN_GG(X1, X2) 6.56/2.53 The graph contains the following edges 1 > 1, 2 > 2 6.56/2.53 6.56/2.53 6.56/2.53 ---------------------------------------- 6.56/2.53 6.56/2.53 (13) 6.56/2.53 YES 6.56/2.53 6.56/2.53 ---------------------------------------- 6.56/2.53 6.56/2.53 (14) 6.56/2.53 Obligation: 6.56/2.53 Pi DP problem: 6.56/2.53 The TRS P consists of the following rules: 6.56/2.53 6.56/2.53 LEQB_IN_GG(s(X1), s(X2)) -> LEQB_IN_GG(X1, X2) 6.56/2.53 6.56/2.53 The TRS R consists of the following rules: 6.56/2.53 6.56/2.53 leqcB_in_gg(0, 0) -> leqcB_out_gg(0, 0) 6.56/2.53 leqcB_in_gg(0, s(0)) -> leqcB_out_gg(0, s(0)) 6.56/2.53 leqcB_in_gg(s(X1), s(X2)) -> U25_gg(X1, X2, leqcB_in_gg(X1, X2)) 6.56/2.53 U25_gg(X1, X2, leqcB_out_gg(X1, X2)) -> leqcB_out_gg(s(X1), s(X2)) 6.56/2.53 lesscC_in_gg(0, s(0)) -> lesscC_out_gg(0, s(0)) 6.56/2.53 lesscC_in_gg(s(X1), s(X2)) -> U26_gg(X1, X2, lesscC_in_gg(X1, X2)) 6.56/2.53 U26_gg(X1, X2, lesscC_out_gg(X1, X2)) -> lesscC_out_gg(s(X1), s(X2)) 6.56/2.53 6.56/2.53 Pi is empty. 6.56/2.53 We have to consider all (P,R,Pi)-chains 6.56/2.53 ---------------------------------------- 6.56/2.53 6.56/2.53 (15) UsableRulesProof (EQUIVALENT) 6.56/2.53 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 6.56/2.53 ---------------------------------------- 6.56/2.53 6.56/2.53 (16) 6.56/2.53 Obligation: 6.56/2.53 Pi DP problem: 6.56/2.53 The TRS P consists of the following rules: 6.56/2.53 6.56/2.53 LEQB_IN_GG(s(X1), s(X2)) -> LEQB_IN_GG(X1, X2) 6.56/2.53 6.56/2.53 R is empty. 6.56/2.53 Pi is empty. 6.56/2.53 We have to consider all (P,R,Pi)-chains 6.56/2.53 ---------------------------------------- 6.56/2.53 6.56/2.53 (17) PiDPToQDPProof (EQUIVALENT) 6.56/2.53 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 6.56/2.53 ---------------------------------------- 6.56/2.53 6.56/2.53 (18) 6.56/2.53 Obligation: 6.56/2.53 Q DP problem: 6.56/2.53 The TRS P consists of the following rules: 6.56/2.53 6.56/2.53 LEQB_IN_GG(s(X1), s(X2)) -> LEQB_IN_GG(X1, X2) 6.56/2.53 6.56/2.53 R is empty. 6.56/2.53 Q is empty. 6.56/2.53 We have to consider all (P,Q,R)-chains. 6.56/2.53 ---------------------------------------- 6.56/2.53 6.56/2.53 (19) QDPSizeChangeProof (EQUIVALENT) 6.56/2.53 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. 6.56/2.53 6.56/2.53 From the DPs we obtained the following set of size-change graphs: 6.56/2.53 *LEQB_IN_GG(s(X1), s(X2)) -> LEQB_IN_GG(X1, X2) 6.56/2.53 The graph contains the following edges 1 > 1, 2 > 2 6.56/2.53 6.56/2.53 6.56/2.53 ---------------------------------------- 6.56/2.53 6.56/2.53 (20) 6.56/2.53 YES 6.56/2.53 6.56/2.53 ---------------------------------------- 6.56/2.53 6.56/2.53 (21) 6.56/2.53 Obligation: 6.56/2.53 Pi DP problem: 6.56/2.53 The TRS P consists of the following rules: 6.56/2.53 6.56/2.53 MERGEA_IN_GGA(.(X1, X2), .(X3, X4), .(X3, X5)) -> U9_GGA(X1, X2, X3, X4, X5, lesscC_in_gg(X3, X1)) 6.56/2.53 U9_GGA(X1, X2, X3, X4, X5, lesscC_out_gg(X3, X1)) -> MERGEA_IN_GGA(.(X1, X2), X4, X5) 6.56/2.53 MERGEA_IN_GGA(.(0, X1), .(0, X2), .(0, X3)) -> MERGEA_IN_GGA(X1, .(0, X2), X3) 6.56/2.53 MERGEA_IN_GGA(.(s(0), X1), .(0, X2), .(0, X3)) -> MERGEA_IN_GGA(.(s(0), X1), X2, X3) 6.56/2.53 MERGEA_IN_GGA(.(s(X1), X2), .(s(X3), X4), .(s(X1), X5)) -> U6_GGA(X1, X2, X3, X4, X5, leqcB_in_gg(X1, X3)) 6.56/2.53 U6_GGA(X1, X2, X3, X4, X5, leqcB_out_gg(X1, X3)) -> MERGEA_IN_GGA(X2, .(s(X3), X4), X5) 6.56/2.53 MERGEA_IN_GGA(.(0, X1), .(s(0), X2), .(0, X3)) -> MERGEA_IN_GGA(X1, .(s(0), X2), X3) 6.56/2.53 MERGEA_IN_GGA(.(s(X1), X2), .(s(X3), X4), .(s(X3), X5)) -> U13_GGA(X1, X2, X3, X4, X5, lesscC_in_gg(X3, X1)) 6.56/2.53 U13_GGA(X1, X2, X3, X4, X5, lesscC_out_gg(X3, X1)) -> MERGEA_IN_GGA(.(s(X1), X2), X4, X5) 6.56/2.53 6.56/2.53 The TRS R consists of the following rules: 6.56/2.53 6.56/2.53 leqcB_in_gg(0, 0) -> leqcB_out_gg(0, 0) 6.56/2.53 leqcB_in_gg(0, s(0)) -> leqcB_out_gg(0, s(0)) 6.56/2.53 leqcB_in_gg(s(X1), s(X2)) -> U25_gg(X1, X2, leqcB_in_gg(X1, X2)) 6.56/2.53 U25_gg(X1, X2, leqcB_out_gg(X1, X2)) -> leqcB_out_gg(s(X1), s(X2)) 6.56/2.53 lesscC_in_gg(0, s(0)) -> lesscC_out_gg(0, s(0)) 6.56/2.53 lesscC_in_gg(s(X1), s(X2)) -> U26_gg(X1, X2, lesscC_in_gg(X1, X2)) 6.56/2.53 U26_gg(X1, X2, lesscC_out_gg(X1, X2)) -> lesscC_out_gg(s(X1), s(X2)) 6.56/2.53 6.56/2.53 The argument filtering Pi contains the following mapping: 6.56/2.53 .(x1, x2) = .(x1, x2) 6.56/2.53 6.56/2.53 0 = 0 6.56/2.53 6.56/2.53 s(x1) = s(x1) 6.56/2.53 6.56/2.53 leqcB_in_gg(x1, x2) = leqcB_in_gg(x1, x2) 6.56/2.53 6.56/2.53 leqcB_out_gg(x1, x2) = leqcB_out_gg(x1, x2) 6.56/2.53 6.56/2.53 U25_gg(x1, x2, x3) = U25_gg(x1, x2, x3) 6.56/2.53 6.56/2.53 lesscC_in_gg(x1, x2) = lesscC_in_gg(x1, x2) 6.56/2.53 6.56/2.53 lesscC_out_gg(x1, x2) = lesscC_out_gg(x1, x2) 6.56/2.53 6.56/2.53 U26_gg(x1, x2, x3) = U26_gg(x1, x2, x3) 6.56/2.53 6.56/2.53 MERGEA_IN_GGA(x1, x2, x3) = MERGEA_IN_GGA(x1, x2) 6.56/2.53 6.56/2.53 U6_GGA(x1, x2, x3, x4, x5, x6) = U6_GGA(x1, x2, x3, x4, x6) 6.56/2.53 6.56/2.53 U9_GGA(x1, x2, x3, x4, x5, x6) = U9_GGA(x1, x2, x3, x4, x6) 6.56/2.53 6.56/2.53 U13_GGA(x1, x2, x3, x4, x5, x6) = U13_GGA(x1, x2, x3, x4, x6) 6.56/2.53 6.56/2.53 6.56/2.53 We have to consider all (P,R,Pi)-chains 6.56/2.53 ---------------------------------------- 6.56/2.53 6.56/2.53 (22) PiDPToQDPProof (SOUND) 6.56/2.53 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 6.56/2.53 ---------------------------------------- 6.56/2.53 6.56/2.53 (23) 6.56/2.53 Obligation: 6.56/2.53 Q DP problem: 6.56/2.53 The TRS P consists of the following rules: 6.56/2.53 6.56/2.53 MERGEA_IN_GGA(.(X1, X2), .(X3, X4)) -> U9_GGA(X1, X2, X3, X4, lesscC_in_gg(X3, X1)) 6.56/2.53 U9_GGA(X1, X2, X3, X4, lesscC_out_gg(X3, X1)) -> MERGEA_IN_GGA(.(X1, X2), X4) 6.56/2.53 MERGEA_IN_GGA(.(0, X1), .(0, X2)) -> MERGEA_IN_GGA(X1, .(0, X2)) 6.56/2.53 MERGEA_IN_GGA(.(s(0), X1), .(0, X2)) -> MERGEA_IN_GGA(.(s(0), X1), X2) 6.56/2.53 MERGEA_IN_GGA(.(s(X1), X2), .(s(X3), X4)) -> U6_GGA(X1, X2, X3, X4, leqcB_in_gg(X1, X3)) 6.56/2.53 U6_GGA(X1, X2, X3, X4, leqcB_out_gg(X1, X3)) -> MERGEA_IN_GGA(X2, .(s(X3), X4)) 6.56/2.53 MERGEA_IN_GGA(.(0, X1), .(s(0), X2)) -> MERGEA_IN_GGA(X1, .(s(0), X2)) 6.56/2.53 MERGEA_IN_GGA(.(s(X1), X2), .(s(X3), X4)) -> U13_GGA(X1, X2, X3, X4, lesscC_in_gg(X3, X1)) 6.56/2.53 U13_GGA(X1, X2, X3, X4, lesscC_out_gg(X3, X1)) -> MERGEA_IN_GGA(.(s(X1), X2), X4) 6.56/2.53 6.56/2.53 The TRS R consists of the following rules: 6.56/2.53 6.56/2.53 leqcB_in_gg(0, 0) -> leqcB_out_gg(0, 0) 6.56/2.53 leqcB_in_gg(0, s(0)) -> leqcB_out_gg(0, s(0)) 6.56/2.53 leqcB_in_gg(s(X1), s(X2)) -> U25_gg(X1, X2, leqcB_in_gg(X1, X2)) 6.56/2.53 U25_gg(X1, X2, leqcB_out_gg(X1, X2)) -> leqcB_out_gg(s(X1), s(X2)) 6.56/2.53 lesscC_in_gg(0, s(0)) -> lesscC_out_gg(0, s(0)) 6.56/2.53 lesscC_in_gg(s(X1), s(X2)) -> U26_gg(X1, X2, lesscC_in_gg(X1, X2)) 6.56/2.53 U26_gg(X1, X2, lesscC_out_gg(X1, X2)) -> lesscC_out_gg(s(X1), s(X2)) 6.56/2.53 6.56/2.53 The set Q consists of the following terms: 6.56/2.53 6.56/2.53 leqcB_in_gg(x0, x1) 6.56/2.53 U25_gg(x0, x1, x2) 6.56/2.53 lesscC_in_gg(x0, x1) 6.56/2.53 U26_gg(x0, x1, x2) 6.56/2.53 6.56/2.53 We have to consider all (P,Q,R)-chains. 6.56/2.53 ---------------------------------------- 6.56/2.53 6.56/2.53 (24) MRRProof (EQUIVALENT) 6.56/2.53 By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented. 6.56/2.53 6.56/2.53 Strictly oriented dependency pairs: 6.56/2.53 6.56/2.53 MERGEA_IN_GGA(.(0, X1), .(0, X2)) -> MERGEA_IN_GGA(X1, .(0, X2)) 6.56/2.53 MERGEA_IN_GGA(.(s(0), X1), .(0, X2)) -> MERGEA_IN_GGA(.(s(0), X1), X2) 6.56/2.53 U6_GGA(X1, X2, X3, X4, leqcB_out_gg(X1, X3)) -> MERGEA_IN_GGA(X2, .(s(X3), X4)) 6.56/2.53 MERGEA_IN_GGA(.(0, X1), .(s(0), X2)) -> MERGEA_IN_GGA(X1, .(s(0), X2)) 6.56/2.53 6.56/2.53 Strictly oriented rules of the TRS R: 6.56/2.53 6.56/2.53 leqcB_in_gg(0, 0) -> leqcB_out_gg(0, 0) 6.56/2.53 leqcB_in_gg(0, s(0)) -> leqcB_out_gg(0, s(0)) 6.56/2.53 6.56/2.53 Used ordering: Polynomial interpretation [POLO]: 6.56/2.53 6.56/2.53 POL(.(x_1, x_2)) = x_1 + x_2 6.56/2.53 POL(0) = 2 6.56/2.53 POL(MERGEA_IN_GGA(x_1, x_2)) = 2*x_1 + 2*x_2 6.56/2.53 POL(U13_GGA(x_1, x_2, x_3, x_4, x_5)) = 2*x_1 + 2*x_2 + 2*x_3 + 2*x_4 + 2*x_5 6.56/2.53 POL(U25_gg(x_1, x_2, x_3)) = x_1 + 2*x_2 + x_3 6.56/2.53 POL(U26_gg(x_1, x_2, x_3)) = x_1 + x_2 + x_3 6.56/2.53 POL(U6_GGA(x_1, x_2, x_3, x_4, x_5)) = x_1 + 2*x_2 + 2*x_3 + 2*x_4 + x_5 6.56/2.53 POL(U9_GGA(x_1, x_2, x_3, x_4, x_5)) = x_1 + 2*x_2 + x_3 + 2*x_4 + x_5 6.56/2.53 POL(leqcB_in_gg(x_1, x_2)) = 2*x_1 + 2*x_2 6.56/2.53 POL(leqcB_out_gg(x_1, x_2)) = 1 + x_1 + 2*x_2 6.56/2.53 POL(lesscC_in_gg(x_1, x_2)) = x_1 + x_2 6.56/2.53 POL(lesscC_out_gg(x_1, x_2)) = x_1 + x_2 6.56/2.53 POL(s(x_1)) = 2*x_1 6.56/2.53 6.56/2.53 6.56/2.53 ---------------------------------------- 6.56/2.53 6.56/2.53 (25) 6.56/2.53 Obligation: 6.56/2.53 Q DP problem: 6.56/2.53 The TRS P consists of the following rules: 6.56/2.53 6.56/2.53 MERGEA_IN_GGA(.(X1, X2), .(X3, X4)) -> U9_GGA(X1, X2, X3, X4, lesscC_in_gg(X3, X1)) 6.56/2.53 U9_GGA(X1, X2, X3, X4, lesscC_out_gg(X3, X1)) -> MERGEA_IN_GGA(.(X1, X2), X4) 6.56/2.53 MERGEA_IN_GGA(.(s(X1), X2), .(s(X3), X4)) -> U6_GGA(X1, X2, X3, X4, leqcB_in_gg(X1, X3)) 6.56/2.53 MERGEA_IN_GGA(.(s(X1), X2), .(s(X3), X4)) -> U13_GGA(X1, X2, X3, X4, lesscC_in_gg(X3, X1)) 6.56/2.53 U13_GGA(X1, X2, X3, X4, lesscC_out_gg(X3, X1)) -> MERGEA_IN_GGA(.(s(X1), X2), X4) 6.56/2.53 6.56/2.53 The TRS R consists of the following rules: 6.56/2.53 6.56/2.53 leqcB_in_gg(s(X1), s(X2)) -> U25_gg(X1, X2, leqcB_in_gg(X1, X2)) 6.56/2.53 U25_gg(X1, X2, leqcB_out_gg(X1, X2)) -> leqcB_out_gg(s(X1), s(X2)) 6.56/2.53 lesscC_in_gg(0, s(0)) -> lesscC_out_gg(0, s(0)) 6.56/2.53 lesscC_in_gg(s(X1), s(X2)) -> U26_gg(X1, X2, lesscC_in_gg(X1, X2)) 6.56/2.53 U26_gg(X1, X2, lesscC_out_gg(X1, X2)) -> lesscC_out_gg(s(X1), s(X2)) 6.56/2.53 6.56/2.53 The set Q consists of the following terms: 6.56/2.53 6.56/2.53 leqcB_in_gg(x0, x1) 6.56/2.53 U25_gg(x0, x1, x2) 6.56/2.53 lesscC_in_gg(x0, x1) 6.56/2.53 U26_gg(x0, x1, x2) 6.56/2.53 6.56/2.53 We have to consider all (P,Q,R)-chains. 6.56/2.53 ---------------------------------------- 6.56/2.53 6.56/2.53 (26) DependencyGraphProof (EQUIVALENT) 6.56/2.53 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 6.56/2.53 ---------------------------------------- 6.56/2.53 6.56/2.53 (27) 6.56/2.53 Obligation: 6.56/2.53 Q DP problem: 6.56/2.53 The TRS P consists of the following rules: 6.56/2.53 6.56/2.53 U9_GGA(X1, X2, X3, X4, lesscC_out_gg(X3, X1)) -> MERGEA_IN_GGA(.(X1, X2), X4) 6.56/2.53 MERGEA_IN_GGA(.(X1, X2), .(X3, X4)) -> U9_GGA(X1, X2, X3, X4, lesscC_in_gg(X3, X1)) 6.56/2.53 MERGEA_IN_GGA(.(s(X1), X2), .(s(X3), X4)) -> U13_GGA(X1, X2, X3, X4, lesscC_in_gg(X3, X1)) 6.56/2.53 U13_GGA(X1, X2, X3, X4, lesscC_out_gg(X3, X1)) -> MERGEA_IN_GGA(.(s(X1), X2), X4) 6.56/2.53 6.56/2.53 The TRS R consists of the following rules: 6.56/2.53 6.56/2.53 leqcB_in_gg(s(X1), s(X2)) -> U25_gg(X1, X2, leqcB_in_gg(X1, X2)) 6.56/2.53 U25_gg(X1, X2, leqcB_out_gg(X1, X2)) -> leqcB_out_gg(s(X1), s(X2)) 6.56/2.53 lesscC_in_gg(0, s(0)) -> lesscC_out_gg(0, s(0)) 6.56/2.53 lesscC_in_gg(s(X1), s(X2)) -> U26_gg(X1, X2, lesscC_in_gg(X1, X2)) 6.56/2.53 U26_gg(X1, X2, lesscC_out_gg(X1, X2)) -> lesscC_out_gg(s(X1), s(X2)) 6.56/2.53 6.56/2.53 The set Q consists of the following terms: 6.56/2.53 6.56/2.53 leqcB_in_gg(x0, x1) 6.56/2.53 U25_gg(x0, x1, x2) 6.56/2.53 lesscC_in_gg(x0, x1) 6.56/2.53 U26_gg(x0, x1, x2) 6.56/2.53 6.56/2.53 We have to consider all (P,Q,R)-chains. 6.56/2.53 ---------------------------------------- 6.56/2.53 6.56/2.53 (28) UsableRulesProof (EQUIVALENT) 6.56/2.53 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. 6.56/2.53 ---------------------------------------- 6.56/2.53 6.56/2.53 (29) 6.56/2.53 Obligation: 6.56/2.53 Q DP problem: 6.56/2.53 The TRS P consists of the following rules: 6.56/2.53 6.56/2.53 U9_GGA(X1, X2, X3, X4, lesscC_out_gg(X3, X1)) -> MERGEA_IN_GGA(.(X1, X2), X4) 6.56/2.53 MERGEA_IN_GGA(.(X1, X2), .(X3, X4)) -> U9_GGA(X1, X2, X3, X4, lesscC_in_gg(X3, X1)) 6.56/2.53 MERGEA_IN_GGA(.(s(X1), X2), .(s(X3), X4)) -> U13_GGA(X1, X2, X3, X4, lesscC_in_gg(X3, X1)) 6.56/2.53 U13_GGA(X1, X2, X3, X4, lesscC_out_gg(X3, X1)) -> MERGEA_IN_GGA(.(s(X1), X2), X4) 6.56/2.53 6.56/2.53 The TRS R consists of the following rules: 6.56/2.53 6.56/2.53 lesscC_in_gg(0, s(0)) -> lesscC_out_gg(0, s(0)) 6.56/2.53 lesscC_in_gg(s(X1), s(X2)) -> U26_gg(X1, X2, lesscC_in_gg(X1, X2)) 6.56/2.53 U26_gg(X1, X2, lesscC_out_gg(X1, X2)) -> lesscC_out_gg(s(X1), s(X2)) 6.56/2.53 6.56/2.53 The set Q consists of the following terms: 6.56/2.53 6.56/2.53 leqcB_in_gg(x0, x1) 6.56/2.53 U25_gg(x0, x1, x2) 6.56/2.53 lesscC_in_gg(x0, x1) 6.56/2.53 U26_gg(x0, x1, x2) 6.56/2.53 6.56/2.53 We have to consider all (P,Q,R)-chains. 6.56/2.53 ---------------------------------------- 6.56/2.53 6.56/2.53 (30) QReductionProof (EQUIVALENT) 6.56/2.53 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 6.56/2.53 6.56/2.53 leqcB_in_gg(x0, x1) 6.56/2.53 U25_gg(x0, x1, x2) 6.56/2.53 6.56/2.53 6.56/2.53 ---------------------------------------- 6.56/2.53 6.56/2.53 (31) 6.56/2.53 Obligation: 6.56/2.53 Q DP problem: 6.56/2.53 The TRS P consists of the following rules: 6.56/2.53 6.56/2.53 U9_GGA(X1, X2, X3, X4, lesscC_out_gg(X3, X1)) -> MERGEA_IN_GGA(.(X1, X2), X4) 6.56/2.53 MERGEA_IN_GGA(.(X1, X2), .(X3, X4)) -> U9_GGA(X1, X2, X3, X4, lesscC_in_gg(X3, X1)) 6.56/2.53 MERGEA_IN_GGA(.(s(X1), X2), .(s(X3), X4)) -> U13_GGA(X1, X2, X3, X4, lesscC_in_gg(X3, X1)) 6.56/2.53 U13_GGA(X1, X2, X3, X4, lesscC_out_gg(X3, X1)) -> MERGEA_IN_GGA(.(s(X1), X2), X4) 6.56/2.53 6.56/2.53 The TRS R consists of the following rules: 6.56/2.53 6.56/2.53 lesscC_in_gg(0, s(0)) -> lesscC_out_gg(0, s(0)) 6.56/2.53 lesscC_in_gg(s(X1), s(X2)) -> U26_gg(X1, X2, lesscC_in_gg(X1, X2)) 6.56/2.53 U26_gg(X1, X2, lesscC_out_gg(X1, X2)) -> lesscC_out_gg(s(X1), s(X2)) 6.56/2.53 6.56/2.53 The set Q consists of the following terms: 6.56/2.53 6.56/2.53 lesscC_in_gg(x0, x1) 6.56/2.53 U26_gg(x0, x1, x2) 6.56/2.53 6.56/2.53 We have to consider all (P,Q,R)-chains. 6.56/2.53 ---------------------------------------- 6.56/2.53 6.56/2.53 (32) QDPSizeChangeProof (EQUIVALENT) 6.56/2.53 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. 6.56/2.53 6.56/2.53 From the DPs we obtained the following set of size-change graphs: 6.56/2.53 *MERGEA_IN_GGA(.(X1, X2), .(X3, X4)) -> U9_GGA(X1, X2, X3, X4, lesscC_in_gg(X3, X1)) 6.56/2.53 The graph contains the following edges 1 > 1, 1 > 2, 2 > 3, 2 > 4 6.56/2.53 6.56/2.53 6.56/2.53 *MERGEA_IN_GGA(.(s(X1), X2), .(s(X3), X4)) -> U13_GGA(X1, X2, X3, X4, lesscC_in_gg(X3, X1)) 6.56/2.53 The graph contains the following edges 1 > 1, 1 > 2, 2 > 3, 2 > 4 6.56/2.53 6.56/2.53 6.56/2.53 *U9_GGA(X1, X2, X3, X4, lesscC_out_gg(X3, X1)) -> MERGEA_IN_GGA(.(X1, X2), X4) 6.56/2.53 The graph contains the following edges 4 >= 2 6.56/2.53 6.56/2.53 6.56/2.53 *U13_GGA(X1, X2, X3, X4, lesscC_out_gg(X3, X1)) -> MERGEA_IN_GGA(.(s(X1), X2), X4) 6.56/2.53 The graph contains the following edges 4 >= 2 6.56/2.53 6.56/2.53 6.56/2.53 ---------------------------------------- 6.56/2.53 6.56/2.53 (33) 6.56/2.53 YES 6.65/2.55 EOF