/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.pl /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- YES proof of /export/starexec/sandbox2/benchmark/theBenchmark.pl # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern div(g,g,a) w.r.t. the given Prolog program could successfully be proven: (0) Prolog (1) PrologToDTProblemTransformerProof [SOUND, 97 ms] (2) TRIPLES (3) TriplesToPiDPProof [SOUND, 172 ms] (4) PiDP (5) DependencyGraphProof [EQUIVALENT, 0 ms] (6) AND (7) PiDP (8) PiDPToQDPProof [EQUIVALENT, 0 ms] (9) QDP (10) QDPSizeChangeProof [EQUIVALENT, 0 ms] (11) YES (12) PiDP (13) PiDPToQDPProof [EQUIVALENT, 0 ms] (14) QDP (15) QDPSizeChangeProof [EQUIVALENT, 0 ms] (16) YES (17) PiDP (18) PiDPToQDPProof [SOUND, 0 ms] (19) QDP (20) QDPSizeChangeProof [EQUIVALENT, 0 ms] (21) YES (22) PiDP (23) PiDPToQDPProof [SOUND, 0 ms] (24) QDP (25) QDPSizeChangeProof [EQUIVALENT, 0 ms] (26) YES ---------------------------------------- (0) Obligation: Clauses: div(X, Y, Z) :- quot(X, Y, Y, Z). quot(0, s(Y), s(Z), R) :- ','(!, eq(R, 0)). quot(X, 0, Z, U) :- ','(!, ','(eq(Z, s(X1)), ','(p(U, P), quot(X, Z, Z, P)))). quot(s(X), Y, Z, U) :- ','(p(Y, P), quot(X, P, Z, U)). p(0, 0). p(s(X), X). eq(X, X). Query: div(g,g,a) ---------------------------------------- (1) PrologToDTProblemTransformerProof (SOUND) Built DT problem from termination graph DT10. { "root": 1, "program": { "directives": [], "clauses": [ [ "(div X Y Z)", "(quot X Y Y Z)" ], [ "(quot (0) (s Y) (s Z) R)", "(',' (!) (eq R (0)))" ], [ "(quot X (0) Z U)", "(',' (!) (',' (eq Z (s X1)) (',' (p U P) (quot X Z Z P))))" ], [ "(quot (s X) Y Z U)", "(',' (p Y P) (quot X P Z U))" ], [ "(p (0) (0))", null ], [ "(p (s X) X)", null ], [ "(eq X X)", null ] ] }, "graph": { "nodes": { "590": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (!_11) (',' (eq (s (s (0))) (s X187)) (',' (p T112 X188) (quot T110 (s (s (0))) (s (s (0))) X188))))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T110"], "free": [ "X187", "X188" ], "exprvars": [] } }, "591": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "592": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (eq (s (s (0))) (s X187)) (',' (p T112 X188) (quot T110 (s (s (0))) (s (s (0))) X188)))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T110"], "free": [ "X187", "X188" ], "exprvars": [] } }, "593": { "goal": [{ "clause": 6, "scope": 13, "term": "(',' (eq (s (s (0))) (s X187)) (',' (p T112 X188) (quot T110 (s (s (0))) (s (s (0))) X188)))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T110"], "free": [ "X187", "X188" ], "exprvars": [] } }, "1215": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "597": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (p T112 X188) (quot T110 (s (s (0))) (s (s (0))) X188))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T110"], "free": ["X188"], "exprvars": [] } }, "1214": { "goal": [{ "clause": -1, "scope": -1, "term": "(quot T339 T347 (s (s (s (s (s (s (s (s 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"kb": { "nonunifying": [[ "(quot T81 T89 (s (s T89)) T84)", "(quot (0) (s X162) (s X163) X164)" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T81", "T89" ], "free": [ "X162", "X163", "X164" ], "exprvars": [] } }, "668": { "goal": [{ "clause": -1, "scope": -1, "term": "(quot T124 (0) (s (s (0))) T127)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T124"], "free": [], "exprvars": [] } }, "427": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "548": { "goal": [{ "clause": 3, "scope": 11, "term": "(quot T81 T89 (s (s T89)) T84)" }], "kb": { "nonunifying": [[ "(quot T81 T89 (s (s T89)) T84)", "(quot (0) (s X162) (s X163) X164)" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T81", "T89" ], "free": [ "X162", "X163", "X164" ], "exprvars": [] } }, "669": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "670": { "goal": [{ "clause": -1, "scope": -1, "term": "(quot T124 T132 (s (s (s T132))) T127)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T124", "T132" ], "free": [], "exprvars": [] } }, "671": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "676": { "goal": [ { "clause": 1, "scope": 16, "term": "(quot T124 T132 (s (s (s T132))) T127)" }, { "clause": 2, "scope": 16, "term": "(quot T124 T132 (s (s (s T132))) T127)" }, { "clause": 3, "scope": 16, "term": "(quot T124 T132 (s (s (s T132))) T127)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T124", "T132" ], "free": [], "exprvars": [] } }, "678": { "goal": [ { "clause": -1, "scope": -1, "term": "(',' (!_16) (eq T141 (0)))" }, { "clause": 2, "scope": 16, "term": "(quot (0) (s T139) (s (s (s (s T139)))) T127)" }, { "clause": 3, "scope": 16, "term": "(quot (0) (s T139) (s (s (s (s T139)))) T127)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T139"], "free": [], "exprvars": [] } }, "679": { "goal": [ { "clause": 2, "scope": 16, "term": "(quot T124 T132 (s (s (s T132))) T127)" }, { "clause": 3, "scope": 16, "term": "(quot T124 T132 (s (s (s T132))) T127)" } ], "kb": { "nonunifying": [[ "(quot T124 T132 (s (s (s T132))) T127)", "(quot (0) (s X246) (s X247) X248)" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T124", "T132" ], "free": [ "X246", "X247", "X248" ], "exprvars": [] } }, "680": { "goal": [{ "clause": -1, "scope": -1, "term": "(eq T141 (0))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "682": { "goal": [{ "clause": 6, "scope": 17, "term": "(eq T141 (0))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "683": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "684": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "685": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "690": { "goal": [{ "clause": 2, "scope": 16, "term": "(quot T124 T132 (s (s (s T132))) T127)" }], "kb": { "nonunifying": [[ "(quot T124 T132 (s (s (s T132))) T127)", "(quot (0) (s X246) (s X247) X248)" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T124", "T132" ], "free": [ "X246", "X247", "X248" ], "exprvars": [] } }, "691": { "goal": [{ "clause": 3, "scope": 16, "term": "(quot T124 T132 (s (s (s T132))) T127)" }], "kb": { "nonunifying": [[ "(quot T124 T132 (s (s (s T132))) T127)", "(quot (0) (s X246) (s X247) X248)" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T124", "T132" ], "free": [ "X246", "X247", "X248" ], "exprvars": [] } }, "1206": { "goal": [{ "clause": -1, "scope": -1, "term": "(quot T339 (0) (s (s (s (s (s (s (s (0)))))))) T342)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T339"], "free": [], "exprvars": [] } }, "1203": { "goal": [{ "clause": 5, "scope": 40, "term": "(',' (p T340 X646) (quot T339 X646 (s (s (s (s (s (s (s T340))))))) T342))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T339", "T340" ], "free": ["X646"], "exprvars": [] } }, "103": { "goal": [{ "clause": -1, "scope": -1, "term": "(quot T67 (s (0)) (s (0)) T73)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T67"], "free": [], "exprvars": [] } }, "1202": { "goal": [{ "clause": 4, "scope": 40, "term": "(',' (p T340 X646) (quot T339 X646 (s (s (s (s (s (s (s T340))))))) T342))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T339", "T340" ], "free": ["X646"], "exprvars": [] } }, "105": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1200": { "goal": [ { "clause": 4, "scope": 40, "term": "(',' (p T340 X646) (quot T339 X646 (s (s (s (s (s (s (s T340))))))) T342))" }, { "clause": 5, "scope": 40, "term": "(',' (p T340 X646) (quot T339 X646 (s (s (s (s (s (s (s T340))))))) T342))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T339", "T340" ], "free": ["X646"], "exprvars": [] } }, "1207": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } } }, "edges": [ { "from": 1, "to": 8, "label": "CASE" }, { "from": 8, "to": 9, "label": "ONLY EVAL with clause\ndiv(X5, X6, X7) :- quot(X5, X6, X6, X7).\nand substitutionT1 -> T7,\nX5 -> T7,\nT2 -> T8,\nX6 -> T8,\nT3 -> T10,\nX7 -> T10,\nT9 -> T10" }, { "from": 9, "to": 10, "label": "CASE" }, { "from": 10, "to": 11, "label": "EVAL with clause\nquot(0, s(X14), s(X15), X16) :- ','(!_2, eq(X16, 0)).\nand substitutionT7 -> 0,\nX14 -> T15,\nT8 -> s(T15),\nX15 -> T15,\nT10 -> T17,\nX16 -> T17,\nT16 -> T17" }, { "from": 10, "to": 12, "label": "EVAL-BACKTRACK" }, { "from": 11, "to": 13, "label": "CUT" }, { "from": 12, "to": 18, "label": "PARALLEL" }, { "from": 12, "to": 19, "label": "PARALLEL" }, { "from": 13, "to": 14, "label": "CASE" }, { "from": 14, "to": 15, "label": "EVAL with clause\neq(X19, X19).\nand substitutionT17 -> 0,\nX19 -> 0,\nT20 -> 0" }, { "from": 14, "to": 16, "label": "EVAL-BACKTRACK" }, { "from": 15, "to": 17, "label": "SUCCESS" }, { "from": 18, "to": 30, "label": "EVAL with clause\nquot(X36, 0, X37, X38) :- ','(!_2, ','(eq(X37, s(X39)), ','(p(X38, X40), quot(X36, X37, X37, X40)))).\nand substitutionT7 -> T29,\nX36 -> T29,\nT8 -> 0,\nX37 -> 0,\nT10 -> T31,\nX38 -> T31,\nT30 -> T31" }, { "from": 18, "to": 31, "label": "EVAL-BACKTRACK" }, { "from": 19, "to": 40, "label": "EVAL with clause\nquot(s(X54), X55, X56, X57) :- ','(p(X55, X58), quot(X54, X58, X56, X57)).\nand substitutionX54 -> T38,\nT7 -> s(T38),\nT8 -> T39,\nX55 -> T39,\nX56 -> T39,\nT10 -> T41,\nX57 -> T41,\nT40 -> T41" }, { "from": 19, "to": 43, "label": "EVAL-BACKTRACK" }, { "from": 30, "to": 32, "label": "CUT" }, { "from": 32, "to": 35, "label": "CASE" }, { "from": 35, "to": 37, "label": "BACKTRACK\nfor clause: eq(X, X)because of non-unification" }, { "from": 40, "to": 44, "label": "CASE" }, { "from": 44, "to": 46, "label": "PARALLEL" }, { "from": 44, "to": 48, "label": "PARALLEL" }, { "from": 46, "to": 50, "label": "EVAL with clause\np(0, 0).\nand substitutionT39 -> 0,\nX58 -> 0" }, { "from": 46, "to": 52, "label": "EVAL-BACKTRACK" }, { "from": 48, "to": 56, "label": "EVAL with clause\np(s(X68), X68).\nand substitutionX68 -> T46,\nT39 -> s(T46),\nX58 -> T46" }, { "from": 48, "to": 58, "label": "EVAL-BACKTRACK" }, { "from": 50, "to": 9, "label": "INSTANCE with matching:\nT7 -> T38\nT8 -> 0\nT10 -> T41" }, { "from": 56, "to": 60, "label": "CASE" }, { "from": 60, "to": 62, "label": "EVAL with clause\nquot(0, s(X78), s(X79), X80) :- ','(!_6, eq(X80, 0)).\nand substitutionT38 -> 0,\nX78 -> T53,\nT46 -> s(T53),\nX79 -> s(T53),\nT41 -> T55,\nX80 -> T55,\nT54 -> T55" }, { "from": 60, "to": 64, "label": "EVAL-BACKTRACK" }, { "from": 62, "to": 66, "label": "CUT" }, { "from": 64, "to": 76, "label": "PARALLEL" }, { "from": 64, "to": 78, "label": "PARALLEL" }, { "from": 66, "to": 68, "label": "CASE" }, { "from": 68, "to": 70, "label": "EVAL with clause\neq(X83, X83).\nand substitutionT55 -> 0,\nX83 -> 0,\nT58 -> 0" }, { "from": 68, "to": 72, "label": "EVAL-BACKTRACK" }, { "from": 70, "to": 74, "label": "SUCCESS" }, { "from": 76, "to": 79, "label": "EVAL with clause\nquot(X100, 0, X101, X102) :- ','(!_6, ','(eq(X101, s(X103)), ','(p(X102, X104), quot(X100, X101, X101, X104)))).\nand substitutionT38 -> T67,\nX100 -> T67,\nT46 -> 0,\nX101 -> s(0),\nT41 -> T69,\nX102 -> T69,\nT68 -> T69" }, { "from": 76, "to": 81, "label": "EVAL-BACKTRACK" }, { "from": 78, "to": 283, "label": "EVAL with clause\nquot(s(X138), X139, X140, X141) :- ','(p(X139, X142), quot(X138, X142, X140, X141)).\nand substitutionX138 -> T81,\nT38 -> s(T81),\nT46 -> T82,\nX139 -> T82,\nX140 -> s(T82),\nT41 -> T84,\nX141 -> T84,\nT83 -> T84" }, { "from": 78, "to": 284, "label": "EVAL-BACKTRACK" }, { "from": 79, "to": 82, "label": "CUT" }, { "from": 82, "to": 84, "label": "CASE" }, { "from": 84, "to": 90, "label": "ONLY EVAL with clause\neq(X115, X115).\nand substitutionX115 -> s(0),\nX103 -> 0" }, { "from": 90, "to": 92, "label": "CASE" }, { "from": 92, "to": 94, "label": "PARALLEL" }, { "from": 92, "to": 95, "label": "PARALLEL" }, { "from": 94, "to": 97, "label": "EVAL with clause\np(0, 0).\nand substitutionT69 -> 0,\nX104 -> 0" }, { "from": 94, "to": 98, "label": "EVAL-BACKTRACK" }, { "from": 95, "to": 103, "label": "EVAL with clause\np(s(X125), X125).\nand substitutionX125 -> T73,\nT69 -> s(T73),\nX104 -> T73,\nT72 -> T73" }, { "from": 95, "to": 105, "label": "EVAL-BACKTRACK" }, { "from": 97, "to": 9, "label": "INSTANCE with matching:\nT7 -> T67\nT8 -> s(0)\nT10 -> 0" }, { "from": 103, "to": 9, "label": "INSTANCE with matching:\nT7 -> T67\nT8 -> s(0)\nT10 -> T73" }, { "from": 283, "to": 300, "label": "CASE" }, { "from": 300, "to": 301, "label": "PARALLEL" }, { "from": 300, "to": 302, "label": "PARALLEL" }, { "from": 301, "to": 303, "label": "EVAL with clause\np(0, 0).\nand substitutionT82 -> 0,\nX142 -> 0" }, { "from": 301, "to": 304, "label": "EVAL-BACKTRACK" }, { "from": 302, "to": 362, "label": "EVAL with clause\np(s(X152), X152).\nand substitutionX152 -> T89,\nT82 -> s(T89),\nX142 -> T89" }, { "from": 302, "to": 369, "label": "EVAL-BACKTRACK" }, { "from": 303, "to": 56, "label": "INSTANCE with matching:\nT38 -> T81\nT46 -> 0\nT41 -> T84" }, { "from": 362, "to": 414, "label": "CASE" }, { "from": 414, "to": 417, "label": "EVAL with clause\nquot(0, s(X162), s(X163), X164) :- ','(!_11, eq(X164, 0)).\nand substitutionT81 -> 0,\nX162 -> T96,\nT89 -> s(T96),\nX163 -> s(s(T96)),\nT84 -> T98,\nX164 -> T98,\nT97 -> T98" }, { "from": 414, "to": 418, "label": "EVAL-BACKTRACK" }, { "from": 417, "to": 419, "label": "CUT" }, { "from": 418, "to": 547, "label": "PARALLEL" }, { "from": 418, "to": 548, "label": "PARALLEL" }, { "from": 419, "to": 420, "label": "CASE" }, { "from": 420, "to": 424, "label": "EVAL with clause\neq(X167, X167).\nand substitutionT98 -> 0,\nX167 -> 0,\nT101 -> 0" }, { "from": 420, "to": 425, "label": "EVAL-BACKTRACK" }, { "from": 424, "to": 427, "label": "SUCCESS" }, { "from": 547, "to": 590, "label": "EVAL with clause\nquot(X184, 0, X185, X186) :- ','(!_11, ','(eq(X185, s(X187)), ','(p(X186, X188), quot(X184, X185, X185, X188)))).\nand substitutionT81 -> T110,\nX184 -> T110,\nT89 -> 0,\nX185 -> s(s(0)),\nT84 -> T112,\nX186 -> T112,\nT111 -> T112" }, { "from": 547, "to": 591, "label": "EVAL-BACKTRACK" }, { "from": 548, "to": 623, "label": "EVAL with clause\nquot(s(X222), X223, X224, X225) :- ','(p(X223, X226), quot(X222, X226, X224, X225)).\nand substitutionX222 -> T124,\nT81 -> s(T124),\nT89 -> T125,\nX223 -> T125,\nX224 -> s(s(T125)),\nT84 -> T127,\nX225 -> T127,\nT126 -> T127" }, { "from": 548, "to": 664, "label": "EVAL-BACKTRACK" }, { "from": 590, "to": 592, "label": "CUT" }, { "from": 592, "to": 593, "label": "CASE" }, { "from": 593, "to": 597, "label": "ONLY EVAL with clause\neq(X199, X199).\nand substitutionX199 -> s(s(0)),\nX187 -> s(0)" }, { "from": 597, "to": 602, "label": "CASE" }, { "from": 602, "to": 604, "label": "PARALLEL" }, { "from": 602, "to": 605, "label": "PARALLEL" }, { "from": 604, "to": 606, "label": "EVAL with clause\np(0, 0).\nand substitutionT112 -> 0,\nX188 -> 0" }, { "from": 604, "to": 607, "label": "EVAL-BACKTRACK" }, { "from": 605, "to": 611, "label": "EVAL with clause\np(s(X209), X209).\nand substitutionX209 -> T116,\nT112 -> s(T116),\nX188 -> T116,\nT115 -> T116" }, { "from": 605, "to": 613, "label": "EVAL-BACKTRACK" }, { "from": 606, "to": 9, "label": "INSTANCE with matching:\nT7 -> T110\nT8 -> s(s(0))\nT10 -> 0" }, { "from": 611, "to": 9, "label": "INSTANCE with matching:\nT7 -> T110\nT8 -> s(s(0))\nT10 -> T116" }, { "from": 623, "to": 665, "label": "CASE" }, { "from": 665, "to": 666, "label": "PARALLEL" }, { "from": 665, "to": 667, "label": "PARALLEL" }, { "from": 666, "to": 668, "label": "EVAL with clause\np(0, 0).\nand substitutionT125 -> 0,\nX226 -> 0" }, { "from": 666, "to": 669, "label": "EVAL-BACKTRACK" }, { "from": 667, "to": 670, "label": "EVAL with clause\np(s(X236), X236).\nand substitutionX236 -> T132,\nT125 -> s(T132),\nX226 -> T132" }, { "from": 667, "to": 671, "label": "EVAL-BACKTRACK" }, { "from": 668, "to": 362, "label": "INSTANCE with matching:\nT81 -> T124\nT89 -> 0\nT84 -> T127" }, { "from": 670, "to": 676, "label": "CASE" }, { "from": 676, "to": 678, "label": "EVAL with clause\nquot(0, s(X246), s(X247), X248) :- ','(!_16, eq(X248, 0)).\nand substitutionT124 -> 0,\nX246 -> T139,\nT132 -> s(T139),\nX247 -> s(s(s(T139))),\nT127 -> T141,\nX248 -> T141,\nT140 -> T141" }, { "from": 676, "to": 679, "label": "EVAL-BACKTRACK" }, { "from": 678, "to": 680, "label": "CUT" }, { "from": 679, "to": 690, "label": "PARALLEL" }, { "from": 679, "to": 691, "label": "PARALLEL" }, { "from": 680, "to": 682, "label": "CASE" }, { "from": 682, "to": 683, "label": "EVAL with clause\neq(X251, X251).\nand substitutionT141 -> 0,\nX251 -> 0,\nT144 -> 0" }, { "from": 682, "to": 684, "label": "EVAL-BACKTRACK" }, { "from": 683, "to": 685, "label": "SUCCESS" }, { "from": 690, "to": 765, "label": "EVAL with clause\nquot(X268, 0, X269, X270) :- ','(!_16, ','(eq(X269, s(X271)), ','(p(X270, X272), quot(X268, X269, X269, X272)))).\nand substitutionT124 -> T153,\nX268 -> T153,\nT132 -> 0,\nX269 -> s(s(s(0))),\nT127 -> T155,\nX270 -> T155,\nT154 -> T155" }, { "from": 690, "to": 766, "label": "EVAL-BACKTRACK" }, { "from": 691, "to": 834, "label": "EVAL with clause\nquot(s(X306), X307, X308, X309) :- ','(p(X307, X310), quot(X306, X310, X308, X309)).\nand substitutionX306 -> T167,\nT124 -> s(T167),\nT132 -> T168,\nX307 -> T168,\nX308 -> s(s(s(T168))),\nT127 -> T170,\nX309 -> T170,\nT169 -> T170" }, { "from": 691, "to": 835, "label": "EVAL-BACKTRACK" }, { "from": 765, "to": 767, "label": "CUT" }, { "from": 767, "to": 768, "label": "CASE" }, { "from": 768, "to": 816, "label": "ONLY EVAL with clause\neq(X283, X283).\nand substitutionX283 -> s(s(s(0))),\nX271 -> s(s(0))" }, { "from": 816, "to": 819, "label": "CASE" }, { "from": 819, "to": 820, "label": "PARALLEL" }, { "from": 819, "to": 821, "label": "PARALLEL" }, { "from": 820, "to": 822, "label": "EVAL with clause\np(0, 0).\nand substitutionT155 -> 0,\nX272 -> 0" }, { "from": 820, "to": 823, "label": "EVAL-BACKTRACK" }, { "from": 821, "to": 829, "label": "EVAL with clause\np(s(X293), X293).\nand substitutionX293 -> T159,\nT155 -> s(T159),\nX272 -> T159,\nT158 -> T159" }, { "from": 821, "to": 830, "label": "EVAL-BACKTRACK" }, { "from": 822, "to": 9, "label": "INSTANCE with matching:\nT7 -> T153\nT8 -> s(s(s(0)))\nT10 -> 0" }, { "from": 829, "to": 9, "label": "INSTANCE with matching:\nT7 -> T153\nT8 -> s(s(s(0)))\nT10 -> T159" }, { "from": 834, "to": 838, "label": "CASE" }, { "from": 838, "to": 839, "label": "PARALLEL" }, { "from": 838, "to": 840, "label": "PARALLEL" }, { "from": 839, "to": 841, "label": "EVAL with clause\np(0, 0).\nand substitutionT168 -> 0,\nX310 -> 0" }, { "from": 839, "to": 843, "label": "EVAL-BACKTRACK" }, { "from": 840, "to": 846, "label": "EVAL with clause\np(s(X320), X320).\nand substitutionX320 -> T175,\nT168 -> s(T175),\nX310 -> T175" }, { "from": 840, "to": 847, "label": "EVAL-BACKTRACK" }, { "from": 841, "to": 670, "label": "INSTANCE with matching:\nT124 -> T167\nT132 -> 0\nT127 -> T170" }, { "from": 846, "to": 849, "label": "CASE" }, { "from": 849, "to": 854, "label": "EVAL with clause\nquot(0, s(X330), s(X331), X332) :- ','(!_21, eq(X332, 0)).\nand substitutionT167 -> 0,\nX330 -> T182,\nT175 -> s(T182),\nX331 -> s(s(s(s(T182)))),\nT170 -> T184,\nX332 -> T184,\nT183 -> T184" }, { "from": 849, "to": 855, "label": "EVAL-BACKTRACK" }, { "from": 854, "to": 856, "label": "CUT" }, { "from": 855, "to": 864, "label": "PARALLEL" }, { "from": 855, "to": 865, "label": "PARALLEL" }, { "from": 856, "to": 859, "label": "CASE" }, { "from": 859, "to": 860, "label": "EVAL with clause\neq(X335, X335).\nand substitutionT184 -> 0,\nX335 -> 0,\nT187 -> 0" }, { "from": 859, "to": 861, "label": "EVAL-BACKTRACK" }, { "from": 860, "to": 862, "label": "SUCCESS" }, { "from": 864, "to": 867, "label": "EVAL with clause\nquot(X352, 0, X353, X354) :- ','(!_21, ','(eq(X353, s(X355)), ','(p(X354, X356), quot(X352, X353, X353, X356)))).\nand substitutionT167 -> T196,\nX352 -> T196,\nT175 -> 0,\nX353 -> s(s(s(s(0)))),\nT170 -> T198,\nX354 -> T198,\nT197 -> T198" }, { "from": 864, "to": 869, "label": "EVAL-BACKTRACK" }, { "from": 865, "to": 913, "label": "EVAL with clause\nquot(s(X390), X391, X392, X393) :- ','(p(X391, X394), quot(X390, X394, X392, X393)).\nand substitutionX390 -> T210,\nT167 -> s(T210),\nT175 -> T211,\nX391 -> T211,\nX392 -> s(s(s(s(T211)))),\nT170 -> T213,\nX393 -> T213,\nT212 -> T213" }, { "from": 865, "to": 916, "label": "EVAL-BACKTRACK" }, { "from": 867, "to": 870, "label": "CUT" }, { "from": 870, "to": 871, "label": "CASE" }, { "from": 871, "to": 878, "label": "ONLY EVAL with clause\neq(X367, X367).\nand substitutionX367 -> s(s(s(s(0)))),\nX355 -> s(s(s(0)))" }, { "from": 878, "to": 880, "label": "CASE" }, { "from": 880, "to": 881, "label": "PARALLEL" }, { "from": 880, "to": 882, "label": "PARALLEL" }, { "from": 881, "to": 883, "label": "EVAL with clause\np(0, 0).\nand substitutionT198 -> 0,\nX356 -> 0" }, { "from": 881, "to": 884, "label": "EVAL-BACKTRACK" }, { "from": 882, "to": 888, "label": "EVAL with clause\np(s(X377), X377).\nand substitutionX377 -> T202,\nT198 -> s(T202),\nX356 -> T202,\nT201 -> T202" }, { "from": 882, "to": 889, "label": "EVAL-BACKTRACK" }, { "from": 883, "to": 9, "label": "INSTANCE with matching:\nT7 -> T196\nT8 -> s(s(s(s(0))))\nT10 -> 0" }, { "from": 888, "to": 9, "label": "INSTANCE with matching:\nT7 -> T196\nT8 -> s(s(s(s(0))))\nT10 -> T202" }, { "from": 913, "to": 917, "label": "CASE" }, { "from": 917, "to": 918, "label": "PARALLEL" }, { "from": 917, "to": 919, "label": "PARALLEL" }, { "from": 918, "to": 921, "label": "EVAL with clause\np(0, 0).\nand substitutionT211 -> 0,\nX394 -> 0" }, { "from": 918, "to": 922, "label": "EVAL-BACKTRACK" }, { "from": 919, "to": 928, "label": "EVAL with clause\np(s(X404), X404).\nand substitutionX404 -> T218,\nT211 -> s(T218),\nX394 -> T218" }, { "from": 919, "to": 929, "label": "EVAL-BACKTRACK" }, { "from": 921, "to": 846, "label": "INSTANCE with matching:\nT167 -> T210\nT175 -> 0\nT170 -> T213" }, { "from": 928, "to": 932, "label": "CASE" }, { "from": 932, "to": 935, "label": "EVAL with clause\nquot(0, s(X414), s(X415), X416) :- ','(!_26, eq(X416, 0)).\nand substitutionT210 -> 0,\nX414 -> T225,\nT218 -> s(T225),\nX415 -> s(s(s(s(s(T225))))),\nT213 -> T227,\nX416 -> T227,\nT226 -> T227" }, { "from": 932, "to": 937, "label": "EVAL-BACKTRACK" }, { "from": 935, "to": 938, "label": "CUT" }, { "from": 937, "to": 945, "label": "PARALLEL" }, { "from": 937, "to": 946, "label": "PARALLEL" }, { "from": 938, "to": 939, "label": "CASE" }, { "from": 939, "to": 940, "label": "EVAL with clause\neq(X419, X419).\nand substitutionT227 -> 0,\nX419 -> 0,\nT230 -> 0" }, { "from": 939, "to": 941, "label": "EVAL-BACKTRACK" }, { "from": 940, "to": 942, "label": "SUCCESS" }, { "from": 945, "to": 951, "label": "EVAL with clause\nquot(X436, 0, X437, X438) :- ','(!_26, ','(eq(X437, s(X439)), ','(p(X438, X440), quot(X436, X437, X437, X440)))).\nand substitutionT210 -> T239,\nX436 -> T239,\nT218 -> 0,\nX437 -> s(s(s(s(s(0))))),\nT213 -> T241,\nX438 -> T241,\nT240 -> T241" }, { "from": 945, "to": 954, "label": "EVAL-BACKTRACK" }, { "from": 946, "to": 992, "label": "EVAL with clause\nquot(s(X474), X475, X476, X477) :- ','(p(X475, X478), quot(X474, X478, X476, X477)).\nand substitutionX474 -> T253,\nT210 -> s(T253),\nT218 -> T254,\nX475 -> T254,\nX476 -> s(s(s(s(s(T254))))),\nT213 -> T256,\nX477 -> T256,\nT255 -> T256" }, { "from": 946, "to": 994, "label": "EVAL-BACKTRACK" }, { "from": 951, "to": 956, "label": "CUT" }, { "from": 956, "to": 960, "label": "CASE" }, { "from": 960, "to": 965, "label": "ONLY EVAL with clause\neq(X451, X451).\nand substitutionX451 -> s(s(s(s(s(0))))),\nX439 -> s(s(s(s(0))))" }, { "from": 965, "to": 966, "label": "CASE" }, { "from": 966, "to": 967, "label": "PARALLEL" }, { "from": 966, "to": 968, "label": "PARALLEL" }, { "from": 967, "to": 969, "label": "EVAL with clause\np(0, 0).\nand substitutionT241 -> 0,\nX440 -> 0" }, { "from": 967, "to": 970, "label": "EVAL-BACKTRACK" }, { "from": 968, "to": 973, "label": "EVAL with clause\np(s(X461), X461).\nand substitutionX461 -> T245,\nT241 -> s(T245),\nX440 -> T245,\nT244 -> T245" }, { "from": 968, "to": 974, "label": "EVAL-BACKTRACK" }, { "from": 969, "to": 9, "label": "INSTANCE with matching:\nT7 -> T239\nT8 -> s(s(s(s(s(0)))))\nT10 -> 0" }, { "from": 973, "to": 9, "label": "INSTANCE with matching:\nT7 -> T239\nT8 -> s(s(s(s(s(0)))))\nT10 -> T245" }, { "from": 992, "to": 996, "label": "CASE" }, { "from": 996, "to": 997, "label": "PARALLEL" }, { "from": 996, "to": 998, "label": "PARALLEL" }, { "from": 997, "to": 999, "label": "EVAL with clause\np(0, 0).\nand substitutionT254 -> 0,\nX478 -> 0" }, { "from": 997, "to": 1000, "label": "EVAL-BACKTRACK" }, { "from": 998, "to": 1012, "label": "EVAL with clause\np(s(X488), X488).\nand substitutionX488 -> T261,\nT254 -> s(T261),\nX478 -> T261" }, { "from": 998, "to": 1014, "label": "EVAL-BACKTRACK" }, { "from": 999, "to": 928, "label": "INSTANCE with matching:\nT210 -> T253\nT218 -> 0\nT213 -> T256" }, { "from": 1012, "to": 1018, "label": "CASE" }, { "from": 1018, "to": 1025, "label": "EVAL with clause\nquot(0, s(X498), s(X499), X500) :- ','(!_31, eq(X500, 0)).\nand substitutionT253 -> 0,\nX498 -> T268,\nT261 -> s(T268),\nX499 -> s(s(s(s(s(s(T268)))))),\nT256 -> T270,\nX500 -> T270,\nT269 -> T270" }, { "from": 1018, "to": 1027, "label": "EVAL-BACKTRACK" }, { "from": 1025, "to": 1029, "label": "CUT" }, { "from": 1027, "to": 1039, "label": "PARALLEL" }, { "from": 1027, "to": 1040, "label": "PARALLEL" }, { "from": 1029, "to": 1030, "label": "CASE" }, { "from": 1030, "to": 1033, "label": "EVAL with clause\neq(X503, X503).\nand substitutionT270 -> 0,\nX503 -> 0,\nT273 -> 0" }, { "from": 1030, "to": 1035, "label": "EVAL-BACKTRACK" }, { "from": 1033, "to": 1036, "label": "SUCCESS" }, { "from": 1039, "to": 1044, "label": "EVAL with clause\nquot(X520, 0, X521, X522) :- ','(!_31, ','(eq(X521, s(X523)), ','(p(X522, X524), quot(X520, X521, X521, X524)))).\nand substitutionT253 -> T282,\nX520 -> T282,\nT261 -> 0,\nX521 -> s(s(s(s(s(s(0)))))),\nT256 -> T284,\nX522 -> T284,\nT283 -> T284" }, { "from": 1039, "to": 1045, "label": "EVAL-BACKTRACK" }, { "from": 1040, "to": 1082, "label": "EVAL with clause\nquot(s(X558), X559, X560, X561) :- ','(p(X559, X562), quot(X558, X562, X560, X561)).\nand substitutionX558 -> T296,\nT253 -> s(T296),\nT261 -> T297,\nX559 -> T297,\nX560 -> s(s(s(s(s(s(T297)))))),\nT256 -> T299,\nX561 -> T299,\nT298 -> T299" }, { "from": 1040, "to": 1083, "label": "EVAL-BACKTRACK" }, { "from": 1044, "to": 1046, "label": "CUT" }, { "from": 1046, "to": 1052, "label": "CASE" }, { "from": 1052, "to": 1057, "label": "ONLY EVAL with clause\neq(X535, X535).\nand substitutionX535 -> s(s(s(s(s(s(0)))))),\nX523 -> s(s(s(s(s(0)))))" }, { "from": 1057, "to": 1061, "label": "CASE" }, { "from": 1061, "to": 1063, "label": "PARALLEL" }, { "from": 1061, "to": 1064, "label": "PARALLEL" }, { "from": 1063, "to": 1065, "label": "EVAL with clause\np(0, 0).\nand substitutionT284 -> 0,\nX524 -> 0" }, { "from": 1063, "to": 1066, "label": "EVAL-BACKTRACK" }, { "from": 1064, "to": 1073, "label": "EVAL with clause\np(s(X545), X545).\nand substitutionX545 -> T288,\nT284 -> s(T288),\nX524 -> T288,\nT287 -> T288" }, { "from": 1064, "to": 1074, "label": "EVAL-BACKTRACK" }, { "from": 1065, "to": 9, "label": "INSTANCE with matching:\nT7 -> T282\nT8 -> s(s(s(s(s(s(0))))))\nT10 -> 0" }, { "from": 1073, "to": 9, "label": "INSTANCE with matching:\nT7 -> T282\nT8 -> s(s(s(s(s(s(0))))))\nT10 -> T288" }, { "from": 1082, "to": 1085, "label": "CASE" }, { "from": 1085, "to": 1087, "label": "PARALLEL" }, { "from": 1085, "to": 1088, "label": "PARALLEL" }, { "from": 1087, "to": 1089, "label": "EVAL with clause\np(0, 0).\nand substitutionT297 -> 0,\nX562 -> 0" }, { "from": 1087, "to": 1108, "label": "EVAL-BACKTRACK" }, { "from": 1088, "to": 1161, "label": "EVAL with clause\np(s(X572), X572).\nand substitutionX572 -> T304,\nT297 -> s(T304),\nX562 -> T304" }, { "from": 1088, "to": 1162, "label": "EVAL-BACKTRACK" }, { "from": 1089, "to": 1012, "label": "INSTANCE with matching:\nT253 -> T296\nT261 -> 0\nT256 -> T299" }, { "from": 1161, "to": 1163, "label": "CASE" }, { "from": 1163, "to": 1168, "label": "EVAL with clause\nquot(0, s(X582), s(X583), X584) :- ','(!_36, eq(X584, 0)).\nand substitutionT296 -> 0,\nX582 -> T311,\nT304 -> s(T311),\nX583 -> s(s(s(s(s(s(s(T311))))))),\nT299 -> T313,\nX584 -> T313,\nT312 -> T313" }, { "from": 1163, "to": 1169, "label": "EVAL-BACKTRACK" }, { "from": 1168, "to": 1170, "label": "CUT" }, { "from": 1169, "to": 1175, "label": "PARALLEL" }, { "from": 1169, "to": 1176, "label": "PARALLEL" }, { "from": 1170, "to": 1171, "label": "CASE" }, { "from": 1171, "to": 1172, "label": "EVAL with clause\neq(X587, X587).\nand substitutionT313 -> 0,\nX587 -> 0,\nT316 -> 0" }, { "from": 1171, "to": 1173, "label": "EVAL-BACKTRACK" }, { "from": 1172, "to": 1174, "label": "SUCCESS" }, { "from": 1175, "to": 1177, "label": "EVAL with clause\nquot(X604, 0, X605, X606) :- ','(!_36, ','(eq(X605, s(X607)), ','(p(X606, X608), quot(X604, X605, X605, X608)))).\nand substitutionT296 -> T325,\nX604 -> T325,\nT304 -> 0,\nX605 -> s(s(s(s(s(s(s(0))))))),\nT299 -> T327,\nX606 -> T327,\nT326 -> T327" }, { "from": 1175, "to": 1178, "label": "EVAL-BACKTRACK" }, { "from": 1176, "to": 1197, "label": "EVAL with clause\nquot(s(X642), X643, X644, X645) :- ','(p(X643, X646), quot(X642, X646, X644, X645)).\nand substitutionX642 -> T339,\nT296 -> s(T339),\nT304 -> T340,\nX643 -> T340,\nX644 -> s(s(s(s(s(s(s(T340))))))),\nT299 -> T342,\nX645 -> T342,\nT341 -> T342" }, { "from": 1176, "to": 1198, "label": "EVAL-BACKTRACK" }, { "from": 1177, "to": 1179, "label": "CUT" }, { "from": 1179, "to": 1183, "label": "CASE" }, { "from": 1183, "to": 1184, "label": "ONLY EVAL with clause\neq(X619, X619).\nand substitutionX619 -> s(s(s(s(s(s(s(0))))))),\nX607 -> s(s(s(s(s(s(0))))))" }, { "from": 1184, "to": 1187, "label": "CASE" }, { "from": 1187, "to": 1188, "label": "PARALLEL" }, { "from": 1187, "to": 1189, "label": "PARALLEL" }, { "from": 1188, "to": 1190, "label": "EVAL with clause\np(0, 0).\nand substitutionT327 -> 0,\nX608 -> 0" }, { "from": 1188, "to": 1191, "label": "EVAL-BACKTRACK" }, { "from": 1189, "to": 1192, "label": "EVAL with clause\np(s(X629), X629).\nand substitutionX629 -> T331,\nT327 -> s(T331),\nX608 -> T331,\nT330 -> T331" }, { "from": 1189, "to": 1193, "label": "EVAL-BACKTRACK" }, { "from": 1190, "to": 9, "label": "INSTANCE with matching:\nT7 -> T325\nT8 -> s(s(s(s(s(s(s(0)))))))\nT10 -> 0" }, { "from": 1192, "to": 9, "label": "INSTANCE with matching:\nT7 -> T325\nT8 -> s(s(s(s(s(s(s(0)))))))\nT10 -> T331" }, { "from": 1197, "to": 1200, "label": "CASE" }, { "from": 1200, "to": 1202, "label": "PARALLEL" }, { "from": 1200, "to": 1203, "label": "PARALLEL" }, { "from": 1202, "to": 1206, "label": "EVAL with clause\np(0, 0).\nand substitutionT340 -> 0,\nX646 -> 0" }, { "from": 1202, "to": 1207, "label": "EVAL-BACKTRACK" }, { "from": 1203, "to": 1214, "label": "EVAL with clause\np(s(X656), X656).\nand substitutionX656 -> T347,\nT340 -> s(T347),\nX646 -> T347" }, { "from": 1203, "to": 1215, "label": "EVAL-BACKTRACK" }, { "from": 1206, "to": 1161, "label": "INSTANCE with matching:\nT296 -> T339\nT304 -> 0\nT299 -> T342" }, { "from": 1214, "to": 1218, "label": "GENERALIZATION\nT350 <-- s(s(s(s(s(s(s(T347)))))))\n\nNew Knowledge:\nT350 is ground" }, { "from": 1218, "to": 1219, "label": "CASE" }, { "from": 1219, "to": 1222, "label": "EVAL with clause\nquot(0, s(X669), s(X670), X671) :- ','(!_41, eq(X671, 0)).\nand substitutionT339 -> 0,\nX669 -> T360,\nT347 -> s(T360),\nT350 -> T361,\nX670 -> T361,\nT342 -> T363,\nX671 -> T363,\nT362 -> T363" }, { "from": 1219, "to": 1223, "label": "EVAL-BACKTRACK" }, { "from": 1222, "to": 1224, "label": "CUT" }, { "from": 1223, "to": 1238, "label": "PARALLEL" }, { "from": 1223, "to": 1239, "label": "PARALLEL" }, { "from": 1224, "to": 1225, "label": "CASE" }, { "from": 1225, "to": 1226, "label": "EVAL with clause\neq(X674, X674).\nand substitutionT363 -> 0,\nX674 -> 0,\nT366 -> 0" }, { "from": 1225, "to": 1227, "label": "EVAL-BACKTRACK" }, { "from": 1226, "to": 1229, "label": "SUCCESS" }, { "from": 1238, "to": 1242, "label": "EVAL with clause\nquot(X691, 0, X692, X693) :- ','(!_41, ','(eq(X692, s(X694)), ','(p(X693, X695), quot(X691, X692, X692, X695)))).\nand substitutionT339 -> T379,\nX691 -> T379,\nT347 -> 0,\nT350 -> T380,\nX692 -> s(T380),\nT342 -> T382,\nX693 -> T382,\nT381 -> T382" }, { "from": 1238, "to": 1243, "label": "EVAL-BACKTRACK" }, { "from": 1239, "to": 1367, "label": "EVAL with clause\nquot(s(X729), X730, X731, X732) :- ','(p(X730, X733), quot(X729, X733, X731, X732)).\nand substitutionX729 -> T404,\nT339 -> s(T404),\nT347 -> T405,\nX730 -> T405,\nT350 -> T406,\nX731 -> s(T406),\nT342 -> T408,\nX732 -> T408,\nT407 -> T408" }, { "from": 1239, "to": 1369, "label": "EVAL-BACKTRACK" }, { "from": 1242, "to": 1244, "label": "CUT" }, { "from": 1244, "to": 1251, "label": "CASE" }, { "from": 1251, "to": 1349, "label": "ONLY EVAL with clause\neq(X706, X706).\nand substitutionT380 -> T387,\nX706 -> s(T387),\nX694 -> T387" }, { "from": 1349, "to": 1350, "label": "CASE" }, { "from": 1350, "to": 1351, "label": "PARALLEL" }, { "from": 1350, "to": 1352, "label": "PARALLEL" }, { "from": 1351, "to": 1353, "label": "EVAL with clause\np(0, 0).\nand substitutionT382 -> 0,\nX695 -> 0" }, { "from": 1351, "to": 1354, "label": "EVAL-BACKTRACK" }, { "from": 1352, "to": 1356, "label": "EVAL with clause\np(s(X716), X716).\nand substitutionX716 -> T393,\nT382 -> s(T393),\nX695 -> T393,\nT392 -> T393" }, { "from": 1352, "to": 1357, "label": "EVAL-BACKTRACK" }, { "from": 1353, "to": 9, "label": "INSTANCE with matching:\nT7 -> T379\nT8 -> s(T387)\nT10 -> 0" }, { "from": 1356, "to": 9, "label": "INSTANCE with matching:\nT7 -> T379\nT8 -> s(T387)\nT10 -> T393" }, { "from": 1367, "to": 1373, "label": "CASE" }, { "from": 1373, "to": 1376, "label": "PARALLEL" }, { "from": 1373, "to": 1377, "label": "PARALLEL" }, { "from": 1376, "to": 1378, "label": "EVAL with clause\np(0, 0).\nand substitutionT405 -> 0,\nX733 -> 0" }, { "from": 1376, "to": 1379, "label": "EVAL-BACKTRACK" }, { "from": 1377, "to": 1382, "label": "EVAL with clause\np(s(X743), X743).\nand substitutionX743 -> T413,\nT405 -> s(T413),\nX733 -> T413" }, { "from": 1377, "to": 1383, "label": "EVAL-BACKTRACK" }, { "from": 1378, "to": 1218, "label": "INSTANCE with matching:\nT339 -> T404\nT347 -> 0\nT350 -> T406\nT342 -> T408" }, { "from": 1382, "to": 1218, "label": "INSTANCE with matching:\nT339 -> T404\nT347 -> T413\nT350 -> T406\nT342 -> T408" } ], "type": "Graph" } } ---------------------------------------- (2) Obligation: Triples: quotA(s(X1), 0, X2) :- quotA(X1, 0, X2). quotA(s(X1), s(X2), X3) :- quotB(X1, X2, X3). quotB(X1, 0, 0) :- quotA(X1, s(0), 0). quotB(X1, 0, s(X2)) :- quotA(X1, s(0), X2). quotB(s(X1), 0, X2) :- quotB(X1, 0, X2). quotB(s(X1), s(X2), X3) :- quotC(X1, X2, X3). quotC(X1, 0, 0) :- quotA(X1, s(s(0)), 0). quotC(X1, 0, s(X2)) :- quotA(X1, s(s(0)), X2). quotC(s(X1), 0, X2) :- quotC(X1, 0, X2). quotC(s(X1), s(X2), X3) :- quotD(X1, X2, X3). quotD(X1, 0, 0) :- quotA(X1, s(s(s(0))), 0). quotD(X1, 0, s(X2)) :- quotA(X1, s(s(s(0))), X2). quotD(s(X1), 0, X2) :- quotD(X1, 0, X2). quotD(s(X1), s(X2), X3) :- quotE(X1, X2, X3). quotE(X1, 0, 0) :- quotA(X1, s(s(s(s(0)))), 0). quotE(X1, 0, s(X2)) :- quotA(X1, s(s(s(s(0)))), X2). quotE(s(X1), 0, X2) :- quotE(X1, 0, X2). quotE(s(X1), s(X2), X3) :- quotF(X1, X2, X3). quotF(X1, 0, 0) :- quotA(X1, s(s(s(s(s(0))))), 0). quotF(X1, 0, s(X2)) :- quotA(X1, s(s(s(s(s(0))))), X2). quotF(s(X1), 0, X2) :- quotF(X1, 0, X2). quotF(s(X1), s(X2), X3) :- quotG(X1, X2, X3). quotG(X1, 0, 0) :- quotA(X1, s(s(s(s(s(s(0)))))), 0). quotG(X1, 0, s(X2)) :- quotA(X1, s(s(s(s(s(s(0)))))), X2). quotG(s(X1), 0, X2) :- quotG(X1, 0, X2). quotG(s(X1), s(X2), X3) :- quotH(X1, X2, X3). quotH(X1, 0, 0) :- quotA(X1, s(s(s(s(s(s(s(0))))))), 0). quotH(X1, 0, s(X2)) :- quotA(X1, s(s(s(s(s(s(s(0))))))), X2). quotH(s(X1), 0, X2) :- quotH(X1, 0, X2). quotH(s(X1), s(X2), X3) :- quotI(X1, X2, s(s(s(s(s(s(s(X2))))))), X3). quotI(X1, 0, X2, 0) :- quotA(X1, s(X2), 0). quotI(X1, 0, X2, s(X3)) :- quotA(X1, s(X2), X3). quotI(s(X1), 0, X2, X3) :- quotI(X1, 0, X2, X3). quotI(s(X1), s(X2), X3, X4) :- quotI(X1, X2, X3, X4). divJ(X1, X2, X3) :- quotA(X1, X2, X3). Clauses: quotcA(0, s(X1), 0). quotcA(s(X1), 0, X2) :- quotcA(X1, 0, X2). quotcA(s(X1), s(X2), X3) :- quotcB(X1, X2, X3). quotcB(0, s(X1), 0). quotcB(X1, 0, 0) :- quotcA(X1, s(0), 0). quotcB(X1, 0, s(X2)) :- quotcA(X1, s(0), X2). quotcB(s(X1), 0, X2) :- quotcB(X1, 0, X2). quotcB(s(X1), s(X2), X3) :- quotcC(X1, X2, X3). quotcC(0, s(X1), 0). quotcC(X1, 0, 0) :- quotcA(X1, s(s(0)), 0). quotcC(X1, 0, s(X2)) :- quotcA(X1, s(s(0)), X2). quotcC(s(X1), 0, X2) :- quotcC(X1, 0, X2). quotcC(s(X1), s(X2), X3) :- quotcD(X1, X2, X3). quotcD(0, s(X1), 0). quotcD(X1, 0, 0) :- quotcA(X1, s(s(s(0))), 0). quotcD(X1, 0, s(X2)) :- quotcA(X1, s(s(s(0))), X2). quotcD(s(X1), 0, X2) :- quotcD(X1, 0, X2). quotcD(s(X1), s(X2), X3) :- quotcE(X1, X2, X3). quotcE(0, s(X1), 0). quotcE(X1, 0, 0) :- quotcA(X1, s(s(s(s(0)))), 0). quotcE(X1, 0, s(X2)) :- quotcA(X1, s(s(s(s(0)))), X2). quotcE(s(X1), 0, X2) :- quotcE(X1, 0, X2). quotcE(s(X1), s(X2), X3) :- quotcF(X1, X2, X3). quotcF(0, s(X1), 0). quotcF(X1, 0, 0) :- quotcA(X1, s(s(s(s(s(0))))), 0). quotcF(X1, 0, s(X2)) :- quotcA(X1, s(s(s(s(s(0))))), X2). quotcF(s(X1), 0, X2) :- quotcF(X1, 0, X2). quotcF(s(X1), s(X2), X3) :- quotcG(X1, X2, X3). quotcG(0, s(X1), 0). quotcG(X1, 0, 0) :- quotcA(X1, s(s(s(s(s(s(0)))))), 0). quotcG(X1, 0, s(X2)) :- quotcA(X1, s(s(s(s(s(s(0)))))), X2). quotcG(s(X1), 0, X2) :- quotcG(X1, 0, X2). quotcG(s(X1), s(X2), X3) :- quotcH(X1, X2, X3). quotcH(0, s(X1), 0). quotcH(X1, 0, 0) :- quotcA(X1, s(s(s(s(s(s(s(0))))))), 0). quotcH(X1, 0, s(X2)) :- quotcA(X1, s(s(s(s(s(s(s(0))))))), X2). quotcH(s(X1), 0, X2) :- quotcH(X1, 0, X2). quotcH(s(X1), s(X2), X3) :- quotcI(X1, X2, s(s(s(s(s(s(s(X2))))))), X3). quotcI(0, s(X1), X2, 0). quotcI(X1, 0, X2, 0) :- quotcA(X1, s(X2), 0). quotcI(X1, 0, X2, s(X3)) :- quotcA(X1, s(X2), X3). quotcI(s(X1), 0, X2, X3) :- quotcI(X1, 0, X2, X3). quotcI(s(X1), s(X2), X3, X4) :- quotcI(X1, X2, X3, X4). Afs: divJ(x1, x2, x3) = divJ(x1, x2) ---------------------------------------- (3) TriplesToPiDPProof (SOUND) We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes: divJ_in_3: (b,b,f) quotA_in_3: (b,b,f) (b,b,b) quotB_in_3: (b,b,f) (b,b,b) quotC_in_3: (b,b,b) (b,b,f) quotD_in_3: (b,b,b) (b,b,f) quotE_in_3: (b,b,b) (b,b,f) quotF_in_3: (b,b,b) (b,b,f) quotG_in_3: (b,b,b) (b,b,f) quotH_in_3: (b,b,b) (b,b,f) quotI_in_4: (b,b,b,b) (b,b,b,f) Transforming TRIPLES into the following Term Rewriting System: Pi DP problem: The TRS P consists of the following rules: DIVJ_IN_GGA(X1, X2, X3) -> U35_GGA(X1, X2, X3, quotA_in_gga(X1, X2, X3)) DIVJ_IN_GGA(X1, X2, X3) -> QUOTA_IN_GGA(X1, X2, X3) QUOTA_IN_GGA(s(X1), 0, X2) -> U1_GGA(X1, X2, quotA_in_gga(X1, 0, X2)) QUOTA_IN_GGA(s(X1), 0, X2) -> QUOTA_IN_GGA(X1, 0, X2) QUOTA_IN_GGA(s(X1), s(X2), X3) -> U2_GGA(X1, X2, X3, quotB_in_gga(X1, X2, X3)) QUOTA_IN_GGA(s(X1), s(X2), X3) -> QUOTB_IN_GGA(X1, X2, X3) QUOTB_IN_GGA(X1, 0, 0) -> U3_GGA(X1, quotA_in_ggg(X1, s(0), 0)) QUOTB_IN_GGA(X1, 0, 0) -> QUOTA_IN_GGG(X1, s(0), 0) QUOTA_IN_GGG(s(X1), 0, X2) -> U1_GGG(X1, X2, quotA_in_ggg(X1, 0, X2)) QUOTA_IN_GGG(s(X1), 0, X2) -> QUOTA_IN_GGG(X1, 0, X2) QUOTA_IN_GGG(s(X1), s(X2), X3) -> U2_GGG(X1, X2, X3, quotB_in_ggg(X1, X2, X3)) QUOTA_IN_GGG(s(X1), s(X2), X3) -> QUOTB_IN_GGG(X1, X2, X3) QUOTB_IN_GGG(X1, 0, 0) -> U3_GGG(X1, quotA_in_ggg(X1, s(0), 0)) QUOTB_IN_GGG(X1, 0, 0) -> QUOTA_IN_GGG(X1, s(0), 0) QUOTB_IN_GGG(X1, 0, s(X2)) -> U4_GGG(X1, X2, quotA_in_ggg(X1, s(0), X2)) QUOTB_IN_GGG(X1, 0, s(X2)) -> QUOTA_IN_GGG(X1, s(0), X2) QUOTB_IN_GGG(s(X1), 0, X2) -> U5_GGG(X1, X2, quotB_in_ggg(X1, 0, X2)) QUOTB_IN_GGG(s(X1), 0, X2) -> QUOTB_IN_GGG(X1, 0, X2) QUOTB_IN_GGG(s(X1), s(X2), X3) -> U6_GGG(X1, X2, X3, quotC_in_ggg(X1, X2, X3)) QUOTB_IN_GGG(s(X1), s(X2), X3) -> QUOTC_IN_GGG(X1, X2, X3) QUOTC_IN_GGG(X1, 0, 0) -> U7_GGG(X1, quotA_in_ggg(X1, s(s(0)), 0)) QUOTC_IN_GGG(X1, 0, 0) -> QUOTA_IN_GGG(X1, s(s(0)), 0) QUOTC_IN_GGG(X1, 0, s(X2)) -> U8_GGG(X1, X2, quotA_in_ggg(X1, s(s(0)), X2)) QUOTC_IN_GGG(X1, 0, s(X2)) -> QUOTA_IN_GGG(X1, s(s(0)), X2) QUOTC_IN_GGG(s(X1), 0, X2) -> U9_GGG(X1, X2, quotC_in_ggg(X1, 0, X2)) QUOTC_IN_GGG(s(X1), 0, X2) -> QUOTC_IN_GGG(X1, 0, X2) QUOTC_IN_GGG(s(X1), s(X2), X3) -> U10_GGG(X1, X2, X3, quotD_in_ggg(X1, X2, X3)) QUOTC_IN_GGG(s(X1), s(X2), X3) -> QUOTD_IN_GGG(X1, X2, X3) QUOTD_IN_GGG(X1, 0, 0) -> U11_GGG(X1, quotA_in_ggg(X1, s(s(s(0))), 0)) QUOTD_IN_GGG(X1, 0, 0) -> QUOTA_IN_GGG(X1, s(s(s(0))), 0) QUOTD_IN_GGG(X1, 0, s(X2)) -> U12_GGG(X1, X2, quotA_in_ggg(X1, s(s(s(0))), X2)) QUOTD_IN_GGG(X1, 0, s(X2)) -> QUOTA_IN_GGG(X1, s(s(s(0))), X2) QUOTD_IN_GGG(s(X1), 0, X2) -> U13_GGG(X1, X2, quotD_in_ggg(X1, 0, X2)) QUOTD_IN_GGG(s(X1), 0, X2) -> QUOTD_IN_GGG(X1, 0, X2) QUOTD_IN_GGG(s(X1), s(X2), X3) -> U14_GGG(X1, X2, X3, quotE_in_ggg(X1, X2, X3)) QUOTD_IN_GGG(s(X1), s(X2), X3) -> QUOTE_IN_GGG(X1, X2, X3) QUOTE_IN_GGG(X1, 0, 0) -> U15_GGG(X1, quotA_in_ggg(X1, s(s(s(s(0)))), 0)) QUOTE_IN_GGG(X1, 0, 0) -> QUOTA_IN_GGG(X1, s(s(s(s(0)))), 0) QUOTE_IN_GGG(X1, 0, s(X2)) -> U16_GGG(X1, X2, quotA_in_ggg(X1, s(s(s(s(0)))), X2)) QUOTE_IN_GGG(X1, 0, s(X2)) -> QUOTA_IN_GGG(X1, s(s(s(s(0)))), X2) QUOTE_IN_GGG(s(X1), 0, X2) -> U17_GGG(X1, X2, quotE_in_ggg(X1, 0, X2)) QUOTE_IN_GGG(s(X1), 0, X2) -> QUOTE_IN_GGG(X1, 0, X2) QUOTE_IN_GGG(s(X1), s(X2), X3) -> U18_GGG(X1, X2, X3, quotF_in_ggg(X1, X2, X3)) QUOTE_IN_GGG(s(X1), s(X2), X3) -> QUOTF_IN_GGG(X1, X2, X3) QUOTF_IN_GGG(X1, 0, 0) -> U19_GGG(X1, quotA_in_ggg(X1, s(s(s(s(s(0))))), 0)) QUOTF_IN_GGG(X1, 0, 0) -> QUOTA_IN_GGG(X1, s(s(s(s(s(0))))), 0) QUOTF_IN_GGG(X1, 0, s(X2)) -> U20_GGG(X1, X2, quotA_in_ggg(X1, s(s(s(s(s(0))))), X2)) QUOTF_IN_GGG(X1, 0, s(X2)) -> QUOTA_IN_GGG(X1, s(s(s(s(s(0))))), X2) QUOTF_IN_GGG(s(X1), 0, X2) -> U21_GGG(X1, X2, quotF_in_ggg(X1, 0, X2)) QUOTF_IN_GGG(s(X1), 0, X2) -> QUOTF_IN_GGG(X1, 0, X2) QUOTF_IN_GGG(s(X1), s(X2), X3) -> U22_GGG(X1, X2, X3, quotG_in_ggg(X1, X2, X3)) QUOTF_IN_GGG(s(X1), s(X2), X3) -> QUOTG_IN_GGG(X1, X2, X3) QUOTG_IN_GGG(X1, 0, 0) -> U23_GGG(X1, quotA_in_ggg(X1, s(s(s(s(s(s(0)))))), 0)) QUOTG_IN_GGG(X1, 0, 0) -> QUOTA_IN_GGG(X1, s(s(s(s(s(s(0)))))), 0) QUOTG_IN_GGG(X1, 0, s(X2)) -> U24_GGG(X1, X2, quotA_in_ggg(X1, s(s(s(s(s(s(0)))))), X2)) QUOTG_IN_GGG(X1, 0, s(X2)) -> QUOTA_IN_GGG(X1, s(s(s(s(s(s(0)))))), X2) QUOTG_IN_GGG(s(X1), 0, X2) -> U25_GGG(X1, X2, quotG_in_ggg(X1, 0, X2)) QUOTG_IN_GGG(s(X1), 0, X2) -> QUOTG_IN_GGG(X1, 0, X2) QUOTG_IN_GGG(s(X1), s(X2), X3) -> U26_GGG(X1, X2, X3, quotH_in_ggg(X1, X2, X3)) QUOTG_IN_GGG(s(X1), s(X2), X3) -> QUOTH_IN_GGG(X1, X2, X3) QUOTH_IN_GGG(X1, 0, 0) -> U27_GGG(X1, quotA_in_ggg(X1, s(s(s(s(s(s(s(0))))))), 0)) QUOTH_IN_GGG(X1, 0, 0) -> QUOTA_IN_GGG(X1, s(s(s(s(s(s(s(0))))))), 0) QUOTH_IN_GGG(X1, 0, s(X2)) -> U28_GGG(X1, X2, quotA_in_ggg(X1, s(s(s(s(s(s(s(0))))))), X2)) QUOTH_IN_GGG(X1, 0, s(X2)) -> QUOTA_IN_GGG(X1, s(s(s(s(s(s(s(0))))))), X2) QUOTH_IN_GGG(s(X1), 0, X2) -> U29_GGG(X1, X2, quotH_in_ggg(X1, 0, X2)) QUOTH_IN_GGG(s(X1), 0, X2) -> QUOTH_IN_GGG(X1, 0, X2) QUOTH_IN_GGG(s(X1), s(X2), X3) -> U30_GGG(X1, X2, X3, quotI_in_gggg(X1, X2, s(s(s(s(s(s(s(X2))))))), X3)) QUOTH_IN_GGG(s(X1), s(X2), X3) -> QUOTI_IN_GGGG(X1, X2, s(s(s(s(s(s(s(X2))))))), X3) QUOTI_IN_GGGG(X1, 0, X2, 0) -> U31_GGGG(X1, X2, quotA_in_ggg(X1, s(X2), 0)) QUOTI_IN_GGGG(X1, 0, X2, 0) -> QUOTA_IN_GGG(X1, s(X2), 0) QUOTI_IN_GGGG(X1, 0, X2, s(X3)) -> U32_GGGG(X1, X2, X3, quotA_in_ggg(X1, s(X2), X3)) QUOTI_IN_GGGG(X1, 0, X2, s(X3)) -> QUOTA_IN_GGG(X1, s(X2), X3) QUOTI_IN_GGGG(s(X1), 0, X2, X3) -> U33_GGGG(X1, X2, X3, quotI_in_gggg(X1, 0, X2, X3)) QUOTI_IN_GGGG(s(X1), 0, X2, X3) -> QUOTI_IN_GGGG(X1, 0, X2, X3) QUOTI_IN_GGGG(s(X1), s(X2), X3, X4) -> U34_GGGG(X1, X2, X3, X4, quotI_in_gggg(X1, X2, X3, X4)) QUOTI_IN_GGGG(s(X1), s(X2), X3, X4) -> QUOTI_IN_GGGG(X1, X2, X3, X4) QUOTB_IN_GGA(X1, 0, s(X2)) -> U4_GGA(X1, X2, quotA_in_gga(X1, s(0), X2)) QUOTB_IN_GGA(X1, 0, s(X2)) -> QUOTA_IN_GGA(X1, s(0), X2) QUOTB_IN_GGA(s(X1), 0, X2) -> U5_GGA(X1, X2, quotB_in_gga(X1, 0, X2)) QUOTB_IN_GGA(s(X1), 0, X2) -> QUOTB_IN_GGA(X1, 0, X2) QUOTB_IN_GGA(s(X1), s(X2), X3) -> U6_GGA(X1, X2, X3, quotC_in_gga(X1, X2, X3)) QUOTB_IN_GGA(s(X1), s(X2), X3) -> QUOTC_IN_GGA(X1, X2, X3) QUOTC_IN_GGA(X1, 0, 0) -> U7_GGA(X1, quotA_in_ggg(X1, s(s(0)), 0)) QUOTC_IN_GGA(X1, 0, 0) -> QUOTA_IN_GGG(X1, s(s(0)), 0) QUOTC_IN_GGA(X1, 0, s(X2)) -> U8_GGA(X1, X2, quotA_in_gga(X1, s(s(0)), X2)) QUOTC_IN_GGA(X1, 0, s(X2)) -> QUOTA_IN_GGA(X1, s(s(0)), X2) QUOTC_IN_GGA(s(X1), 0, X2) -> U9_GGA(X1, X2, quotC_in_gga(X1, 0, X2)) QUOTC_IN_GGA(s(X1), 0, X2) -> QUOTC_IN_GGA(X1, 0, X2) QUOTC_IN_GGA(s(X1), s(X2), X3) -> U10_GGA(X1, X2, X3, quotD_in_gga(X1, X2, X3)) QUOTC_IN_GGA(s(X1), s(X2), X3) -> QUOTD_IN_GGA(X1, X2, X3) QUOTD_IN_GGA(X1, 0, 0) -> U11_GGA(X1, quotA_in_ggg(X1, s(s(s(0))), 0)) QUOTD_IN_GGA(X1, 0, 0) -> QUOTA_IN_GGG(X1, s(s(s(0))), 0) QUOTD_IN_GGA(X1, 0, s(X2)) -> U12_GGA(X1, X2, quotA_in_gga(X1, s(s(s(0))), X2)) QUOTD_IN_GGA(X1, 0, s(X2)) -> QUOTA_IN_GGA(X1, s(s(s(0))), X2) QUOTD_IN_GGA(s(X1), 0, X2) -> U13_GGA(X1, X2, quotD_in_gga(X1, 0, X2)) QUOTD_IN_GGA(s(X1), 0, X2) -> QUOTD_IN_GGA(X1, 0, X2) QUOTD_IN_GGA(s(X1), s(X2), X3) -> U14_GGA(X1, X2, X3, quotE_in_gga(X1, X2, X3)) QUOTD_IN_GGA(s(X1), s(X2), X3) -> QUOTE_IN_GGA(X1, X2, X3) QUOTE_IN_GGA(X1, 0, 0) -> U15_GGA(X1, quotA_in_ggg(X1, s(s(s(s(0)))), 0)) QUOTE_IN_GGA(X1, 0, 0) -> QUOTA_IN_GGG(X1, s(s(s(s(0)))), 0) QUOTE_IN_GGA(X1, 0, s(X2)) -> U16_GGA(X1, X2, quotA_in_gga(X1, s(s(s(s(0)))), X2)) QUOTE_IN_GGA(X1, 0, s(X2)) -> QUOTA_IN_GGA(X1, s(s(s(s(0)))), X2) QUOTE_IN_GGA(s(X1), 0, X2) -> U17_GGA(X1, X2, quotE_in_gga(X1, 0, X2)) QUOTE_IN_GGA(s(X1), 0, X2) -> QUOTE_IN_GGA(X1, 0, X2) QUOTE_IN_GGA(s(X1), s(X2), X3) -> U18_GGA(X1, X2, X3, quotF_in_gga(X1, X2, X3)) QUOTE_IN_GGA(s(X1), s(X2), X3) -> QUOTF_IN_GGA(X1, X2, X3) QUOTF_IN_GGA(X1, 0, 0) -> U19_GGA(X1, quotA_in_ggg(X1, s(s(s(s(s(0))))), 0)) QUOTF_IN_GGA(X1, 0, 0) -> QUOTA_IN_GGG(X1, s(s(s(s(s(0))))), 0) QUOTF_IN_GGA(X1, 0, s(X2)) -> U20_GGA(X1, X2, quotA_in_gga(X1, s(s(s(s(s(0))))), X2)) QUOTF_IN_GGA(X1, 0, s(X2)) -> QUOTA_IN_GGA(X1, s(s(s(s(s(0))))), X2) QUOTF_IN_GGA(s(X1), 0, X2) -> U21_GGA(X1, X2, quotF_in_gga(X1, 0, X2)) QUOTF_IN_GGA(s(X1), 0, X2) -> QUOTF_IN_GGA(X1, 0, X2) QUOTF_IN_GGA(s(X1), s(X2), X3) -> U22_GGA(X1, X2, X3, quotG_in_gga(X1, X2, X3)) QUOTF_IN_GGA(s(X1), s(X2), X3) -> QUOTG_IN_GGA(X1, X2, X3) QUOTG_IN_GGA(X1, 0, 0) -> U23_GGA(X1, quotA_in_ggg(X1, s(s(s(s(s(s(0)))))), 0)) QUOTG_IN_GGA(X1, 0, 0) -> QUOTA_IN_GGG(X1, s(s(s(s(s(s(0)))))), 0) QUOTG_IN_GGA(X1, 0, s(X2)) -> U24_GGA(X1, X2, quotA_in_gga(X1, s(s(s(s(s(s(0)))))), X2)) QUOTG_IN_GGA(X1, 0, s(X2)) -> QUOTA_IN_GGA(X1, s(s(s(s(s(s(0)))))), X2) QUOTG_IN_GGA(s(X1), 0, X2) -> U25_GGA(X1, X2, quotG_in_gga(X1, 0, X2)) QUOTG_IN_GGA(s(X1), 0, X2) -> QUOTG_IN_GGA(X1, 0, X2) QUOTG_IN_GGA(s(X1), s(X2), X3) -> U26_GGA(X1, X2, X3, quotH_in_gga(X1, X2, X3)) QUOTG_IN_GGA(s(X1), s(X2), X3) -> QUOTH_IN_GGA(X1, X2, X3) QUOTH_IN_GGA(X1, 0, 0) -> U27_GGA(X1, quotA_in_ggg(X1, s(s(s(s(s(s(s(0))))))), 0)) QUOTH_IN_GGA(X1, 0, 0) -> QUOTA_IN_GGG(X1, s(s(s(s(s(s(s(0))))))), 0) QUOTH_IN_GGA(X1, 0, s(X2)) -> U28_GGA(X1, X2, quotA_in_gga(X1, s(s(s(s(s(s(s(0))))))), X2)) QUOTH_IN_GGA(X1, 0, s(X2)) -> QUOTA_IN_GGA(X1, s(s(s(s(s(s(s(0))))))), X2) QUOTH_IN_GGA(s(X1), 0, X2) -> U29_GGA(X1, X2, quotH_in_gga(X1, 0, X2)) QUOTH_IN_GGA(s(X1), 0, X2) -> QUOTH_IN_GGA(X1, 0, X2) QUOTH_IN_GGA(s(X1), s(X2), X3) -> U30_GGA(X1, X2, X3, quotI_in_ggga(X1, X2, s(s(s(s(s(s(s(X2))))))), X3)) QUOTH_IN_GGA(s(X1), s(X2), X3) -> QUOTI_IN_GGGA(X1, X2, s(s(s(s(s(s(s(X2))))))), X3) QUOTI_IN_GGGA(X1, 0, X2, 0) -> U31_GGGA(X1, X2, quotA_in_ggg(X1, s(X2), 0)) QUOTI_IN_GGGA(X1, 0, X2, 0) -> QUOTA_IN_GGG(X1, s(X2), 0) QUOTI_IN_GGGA(X1, 0, X2, s(X3)) -> U32_GGGA(X1, X2, X3, quotA_in_gga(X1, s(X2), X3)) QUOTI_IN_GGGA(X1, 0, X2, s(X3)) -> QUOTA_IN_GGA(X1, s(X2), X3) QUOTI_IN_GGGA(s(X1), 0, X2, X3) -> U33_GGGA(X1, X2, X3, quotI_in_ggga(X1, 0, X2, X3)) QUOTI_IN_GGGA(s(X1), 0, X2, X3) -> QUOTI_IN_GGGA(X1, 0, X2, X3) QUOTI_IN_GGGA(s(X1), s(X2), X3, X4) -> U34_GGGA(X1, X2, X3, X4, quotI_in_ggga(X1, X2, X3, X4)) QUOTI_IN_GGGA(s(X1), s(X2), X3, X4) -> QUOTI_IN_GGGA(X1, X2, X3, X4) R is empty. The argument filtering Pi contains the following mapping: quotA_in_gga(x1, x2, x3) = quotA_in_gga(x1, x2) s(x1) = s(x1) 0 = 0 quotB_in_gga(x1, x2, x3) = quotB_in_gga(x1, x2) quotA_in_ggg(x1, x2, x3) = quotA_in_ggg(x1, x2, x3) quotB_in_ggg(x1, x2, x3) = quotB_in_ggg(x1, x2, x3) quotC_in_ggg(x1, x2, x3) = quotC_in_ggg(x1, x2, x3) quotD_in_ggg(x1, x2, x3) = quotD_in_ggg(x1, x2, x3) quotE_in_ggg(x1, x2, x3) = quotE_in_ggg(x1, x2, x3) quotF_in_ggg(x1, x2, x3) = quotF_in_ggg(x1, x2, x3) quotG_in_ggg(x1, x2, x3) = quotG_in_ggg(x1, x2, x3) quotH_in_ggg(x1, x2, x3) = quotH_in_ggg(x1, x2, x3) quotI_in_gggg(x1, x2, x3, x4) = quotI_in_gggg(x1, x2, x3, x4) quotC_in_gga(x1, x2, x3) = quotC_in_gga(x1, x2) quotD_in_gga(x1, x2, x3) = quotD_in_gga(x1, x2) quotE_in_gga(x1, x2, x3) = quotE_in_gga(x1, x2) quotF_in_gga(x1, x2, x3) = quotF_in_gga(x1, x2) quotG_in_gga(x1, x2, x3) = quotG_in_gga(x1, x2) quotH_in_gga(x1, x2, x3) = quotH_in_gga(x1, x2) quotI_in_ggga(x1, x2, x3, x4) = quotI_in_ggga(x1, x2, x3) DIVJ_IN_GGA(x1, x2, x3) = DIVJ_IN_GGA(x1, x2) U35_GGA(x1, x2, x3, x4) = U35_GGA(x1, x2, x4) QUOTA_IN_GGA(x1, x2, x3) = QUOTA_IN_GGA(x1, x2) U1_GGA(x1, x2, x3) = U1_GGA(x1, x3) U2_GGA(x1, x2, x3, x4) = U2_GGA(x1, x2, x4) QUOTB_IN_GGA(x1, x2, x3) = QUOTB_IN_GGA(x1, x2) U3_GGA(x1, x2) = U3_GGA(x1, x2) QUOTA_IN_GGG(x1, x2, x3) = QUOTA_IN_GGG(x1, x2, x3) U1_GGG(x1, x2, x3) = U1_GGG(x1, x2, x3) U2_GGG(x1, x2, x3, x4) = U2_GGG(x1, x2, x3, x4) QUOTB_IN_GGG(x1, x2, x3) = QUOTB_IN_GGG(x1, x2, x3) U3_GGG(x1, x2) = U3_GGG(x1, x2) U4_GGG(x1, x2, x3) = U4_GGG(x1, x2, x3) U5_GGG(x1, x2, x3) = U5_GGG(x1, x2, x3) U6_GGG(x1, x2, x3, x4) = U6_GGG(x1, x2, x3, x4) QUOTC_IN_GGG(x1, x2, x3) = QUOTC_IN_GGG(x1, x2, x3) U7_GGG(x1, x2) = U7_GGG(x1, x2) U8_GGG(x1, x2, x3) = U8_GGG(x1, x2, x3) U9_GGG(x1, x2, x3) = U9_GGG(x1, x2, x3) U10_GGG(x1, x2, x3, x4) = U10_GGG(x1, x2, x3, x4) QUOTD_IN_GGG(x1, x2, x3) = QUOTD_IN_GGG(x1, x2, x3) U11_GGG(x1, x2) = U11_GGG(x1, x2) U12_GGG(x1, x2, x3) = U12_GGG(x1, x2, x3) U13_GGG(x1, x2, x3) = U13_GGG(x1, x2, x3) U14_GGG(x1, x2, x3, x4) = U14_GGG(x1, x2, x3, x4) QUOTE_IN_GGG(x1, x2, x3) = QUOTE_IN_GGG(x1, x2, x3) U15_GGG(x1, x2) = U15_GGG(x1, x2) U16_GGG(x1, x2, x3) = U16_GGG(x1, x2, x3) U17_GGG(x1, x2, x3) = U17_GGG(x1, x2, x3) U18_GGG(x1, x2, x3, x4) = U18_GGG(x1, x2, x3, x4) QUOTF_IN_GGG(x1, x2, x3) = QUOTF_IN_GGG'(x1, x2, x3) U19_GGG(x1, x2) = U19_GGG(x1, x2) U20_GGG(x1, x2, x3) = U20_GGG(x1, x2, x3) U21_GGG(x1, x2, x3) = U21_GGG(x1, x2, x3) U22_GGG(x1, x2, x3, x4) = U22_GGG(x1, x2, x3, x4) QUOTG_IN_GGG(x1, x2, x3) = QUOTG_IN_GGG(x1, x2, x3) U23_GGG(x1, x2) = U23_GGG(x1, x2) U24_GGG(x1, x2, x3) = U24_GGG(x1, x2, x3) U25_GGG(x1, x2, x3) = U25_GGG(x1, x2, x3) U26_GGG(x1, x2, x3, x4) = U26_GGG(x1, x2, x3, x4) QUOTH_IN_GGG(x1, x2, x3) = QUOTH_IN_GGG(x1, x2, x3) U27_GGG(x1, x2) = U27_GGG(x1, x2) U28_GGG(x1, x2, x3) = U28_GGG(x1, x2, x3) U29_GGG(x1, x2, x3) = U29_GGG(x1, x2, x3) U30_GGG(x1, x2, x3, x4) = U30_GGG(x1, x2, x3, x4) QUOTI_IN_GGGG(x1, x2, x3, x4) = QUOTI_IN_GGGG(x1, x2, x3, x4) U31_GGGG(x1, x2, x3) = U31_GGGG(x1, x2, x3) U32_GGGG(x1, x2, x3, x4) = U32_GGGG(x1, x2, x3, x4) U33_GGGG(x1, x2, x3, x4) = U33_GGGG(x1, x2, x3, x4) U34_GGGG(x1, x2, x3, x4, x5) = U34_GGGG(x1, x2, x3, x4, x5) U4_GGA(x1, x2, x3) = U4_GGA(x1, x3) U5_GGA(x1, x2, x3) = U5_GGA(x1, x3) U6_GGA(x1, x2, x3, x4) = U6_GGA(x1, x2, x4) QUOTC_IN_GGA(x1, x2, x3) = QUOTC_IN_GGA(x1, x2) U7_GGA(x1, x2) = U7_GGA(x1, x2) U8_GGA(x1, x2, x3) = U8_GGA(x1, x3) U9_GGA(x1, x2, x3) = U9_GGA(x1, x3) U10_GGA(x1, x2, x3, x4) = U10_GGA(x1, x2, x4) QUOTD_IN_GGA(x1, x2, x3) = QUOTD_IN_GGA(x1, x2) U11_GGA(x1, x2) = U11_GGA(x1, x2) U12_GGA(x1, x2, x3) = U12_GGA(x1, x3) U13_GGA(x1, x2, x3) = U13_GGA(x1, x3) U14_GGA(x1, x2, x3, x4) = U14_GGA(x1, x2, x4) QUOTE_IN_GGA(x1, x2, x3) = QUOTE_IN_GGA(x1, x2) U15_GGA(x1, x2) = U15_GGA(x1, x2) U16_GGA(x1, x2, x3) = U16_GGA(x1, x3) U17_GGA(x1, x2, x3) = U17_GGA(x1, x3) U18_GGA(x1, x2, x3, x4) = U18_GGA(x1, x2, x4) QUOTF_IN_GGA(x1, x2, x3) = QUOTF_IN_GGA(x1, x2) U19_GGA(x1, x2) = U19_GGA(x1, x2) U20_GGA(x1, x2, x3) = U20_GGA(x1, x3) U21_GGA(x1, x2, x3) = U21_GGA(x1, x3) U22_GGA(x1, x2, x3, x4) = U22_GGA(x1, x2, x4) QUOTG_IN_GGA(x1, x2, x3) = QUOTG_IN_GGA(x1, x2) U23_GGA(x1, x2) = U23_GGA(x1, x2) U24_GGA(x1, x2, x3) = U24_GGA(x1, x3) U25_GGA(x1, x2, x3) = U25_GGA(x1, x3) U26_GGA(x1, x2, x3, x4) = U26_GGA(x1, x2, x4) QUOTH_IN_GGA(x1, x2, x3) = QUOTH_IN_GGA(x1, x2) U27_GGA(x1, x2) = U27_GGA(x1, x2) U28_GGA(x1, x2, x3) = U28_GGA(x1, x3) U29_GGA(x1, x2, x3) = U29_GGA(x1, x3) U30_GGA(x1, x2, x3, x4) = U30_GGA(x1, x2, x4) QUOTI_IN_GGGA(x1, x2, x3, x4) = QUOTI_IN_GGGA(x1, x2, x3) U31_GGGA(x1, x2, x3) = U31_GGGA(x1, x2, x3) U32_GGGA(x1, x2, x3, x4) = U32_GGGA(x1, x2, x4) U33_GGGA(x1, x2, x3, x4) = U33_GGGA(x1, x2, x4) U34_GGGA(x1, x2, x3, x4, x5) = U34_GGGA(x1, x2, x3, x5) We have to consider all (P,R,Pi)-chains Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES ---------------------------------------- (4) Obligation: Pi DP problem: The TRS P consists of the following rules: DIVJ_IN_GGA(X1, X2, X3) -> U35_GGA(X1, X2, X3, quotA_in_gga(X1, X2, X3)) DIVJ_IN_GGA(X1, X2, X3) -> QUOTA_IN_GGA(X1, X2, X3) QUOTA_IN_GGA(s(X1), 0, X2) -> U1_GGA(X1, X2, quotA_in_gga(X1, 0, X2)) QUOTA_IN_GGA(s(X1), 0, X2) -> QUOTA_IN_GGA(X1, 0, X2) QUOTA_IN_GGA(s(X1), s(X2), X3) -> U2_GGA(X1, X2, X3, quotB_in_gga(X1, X2, X3)) QUOTA_IN_GGA(s(X1), s(X2), X3) -> QUOTB_IN_GGA(X1, X2, X3) QUOTB_IN_GGA(X1, 0, 0) -> U3_GGA(X1, quotA_in_ggg(X1, s(0), 0)) QUOTB_IN_GGA(X1, 0, 0) -> QUOTA_IN_GGG(X1, s(0), 0) QUOTA_IN_GGG(s(X1), 0, X2) -> U1_GGG(X1, X2, quotA_in_ggg(X1, 0, X2)) QUOTA_IN_GGG(s(X1), 0, X2) -> QUOTA_IN_GGG(X1, 0, X2) QUOTA_IN_GGG(s(X1), s(X2), X3) -> U2_GGG(X1, X2, X3, quotB_in_ggg(X1, X2, X3)) QUOTA_IN_GGG(s(X1), s(X2), X3) -> QUOTB_IN_GGG(X1, X2, X3) QUOTB_IN_GGG(X1, 0, 0) -> U3_GGG(X1, quotA_in_ggg(X1, s(0), 0)) QUOTB_IN_GGG(X1, 0, 0) -> QUOTA_IN_GGG(X1, s(0), 0) QUOTB_IN_GGG(X1, 0, s(X2)) -> U4_GGG(X1, X2, quotA_in_ggg(X1, s(0), X2)) QUOTB_IN_GGG(X1, 0, s(X2)) -> QUOTA_IN_GGG(X1, s(0), X2) QUOTB_IN_GGG(s(X1), 0, X2) -> U5_GGG(X1, X2, quotB_in_ggg(X1, 0, X2)) QUOTB_IN_GGG(s(X1), 0, X2) -> QUOTB_IN_GGG(X1, 0, X2) QUOTB_IN_GGG(s(X1), s(X2), X3) -> U6_GGG(X1, X2, X3, quotC_in_ggg(X1, X2, X3)) QUOTB_IN_GGG(s(X1), s(X2), X3) -> QUOTC_IN_GGG(X1, X2, X3) QUOTC_IN_GGG(X1, 0, 0) -> U7_GGG(X1, quotA_in_ggg(X1, s(s(0)), 0)) QUOTC_IN_GGG(X1, 0, 0) -> QUOTA_IN_GGG(X1, s(s(0)), 0) QUOTC_IN_GGG(X1, 0, s(X2)) -> U8_GGG(X1, X2, quotA_in_ggg(X1, s(s(0)), X2)) QUOTC_IN_GGG(X1, 0, s(X2)) -> QUOTA_IN_GGG(X1, s(s(0)), X2) QUOTC_IN_GGG(s(X1), 0, X2) -> U9_GGG(X1, X2, quotC_in_ggg(X1, 0, X2)) QUOTC_IN_GGG(s(X1), 0, X2) -> QUOTC_IN_GGG(X1, 0, X2) QUOTC_IN_GGG(s(X1), s(X2), X3) -> U10_GGG(X1, X2, X3, quotD_in_ggg(X1, X2, X3)) QUOTC_IN_GGG(s(X1), s(X2), X3) -> QUOTD_IN_GGG(X1, X2, X3) QUOTD_IN_GGG(X1, 0, 0) -> U11_GGG(X1, quotA_in_ggg(X1, s(s(s(0))), 0)) QUOTD_IN_GGG(X1, 0, 0) -> QUOTA_IN_GGG(X1, s(s(s(0))), 0) QUOTD_IN_GGG(X1, 0, s(X2)) -> U12_GGG(X1, X2, quotA_in_ggg(X1, s(s(s(0))), X2)) QUOTD_IN_GGG(X1, 0, s(X2)) -> QUOTA_IN_GGG(X1, s(s(s(0))), X2) QUOTD_IN_GGG(s(X1), 0, X2) -> U13_GGG(X1, X2, quotD_in_ggg(X1, 0, X2)) QUOTD_IN_GGG(s(X1), 0, X2) -> QUOTD_IN_GGG(X1, 0, X2) QUOTD_IN_GGG(s(X1), s(X2), X3) -> U14_GGG(X1, X2, X3, quotE_in_ggg(X1, X2, X3)) QUOTD_IN_GGG(s(X1), s(X2), X3) -> QUOTE_IN_GGG(X1, X2, X3) QUOTE_IN_GGG(X1, 0, 0) -> U15_GGG(X1, quotA_in_ggg(X1, s(s(s(s(0)))), 0)) QUOTE_IN_GGG(X1, 0, 0) -> QUOTA_IN_GGG(X1, s(s(s(s(0)))), 0) QUOTE_IN_GGG(X1, 0, s(X2)) -> U16_GGG(X1, X2, quotA_in_ggg(X1, s(s(s(s(0)))), X2)) QUOTE_IN_GGG(X1, 0, s(X2)) -> QUOTA_IN_GGG(X1, s(s(s(s(0)))), X2) QUOTE_IN_GGG(s(X1), 0, X2) -> U17_GGG(X1, X2, quotE_in_ggg(X1, 0, X2)) QUOTE_IN_GGG(s(X1), 0, X2) -> QUOTE_IN_GGG(X1, 0, X2) QUOTE_IN_GGG(s(X1), s(X2), X3) -> U18_GGG(X1, X2, X3, quotF_in_ggg(X1, X2, X3)) QUOTE_IN_GGG(s(X1), s(X2), X3) -> QUOTF_IN_GGG(X1, X2, X3) QUOTF_IN_GGG(X1, 0, 0) -> U19_GGG(X1, quotA_in_ggg(X1, s(s(s(s(s(0))))), 0)) QUOTF_IN_GGG(X1, 0, 0) -> QUOTA_IN_GGG(X1, s(s(s(s(s(0))))), 0) QUOTF_IN_GGG(X1, 0, s(X2)) -> U20_GGG(X1, X2, quotA_in_ggg(X1, s(s(s(s(s(0))))), X2)) QUOTF_IN_GGG(X1, 0, s(X2)) -> QUOTA_IN_GGG(X1, s(s(s(s(s(0))))), X2) QUOTF_IN_GGG(s(X1), 0, X2) -> U21_GGG(X1, X2, quotF_in_ggg(X1, 0, X2)) QUOTF_IN_GGG(s(X1), 0, X2) -> QUOTF_IN_GGG(X1, 0, X2) QUOTF_IN_GGG(s(X1), s(X2), X3) -> U22_GGG(X1, X2, X3, quotG_in_ggg(X1, X2, X3)) QUOTF_IN_GGG(s(X1), s(X2), X3) -> QUOTG_IN_GGG(X1, X2, X3) QUOTG_IN_GGG(X1, 0, 0) -> U23_GGG(X1, quotA_in_ggg(X1, s(s(s(s(s(s(0)))))), 0)) QUOTG_IN_GGG(X1, 0, 0) -> QUOTA_IN_GGG(X1, s(s(s(s(s(s(0)))))), 0) QUOTG_IN_GGG(X1, 0, s(X2)) -> U24_GGG(X1, X2, quotA_in_ggg(X1, s(s(s(s(s(s(0)))))), X2)) QUOTG_IN_GGG(X1, 0, s(X2)) -> QUOTA_IN_GGG(X1, s(s(s(s(s(s(0)))))), X2) QUOTG_IN_GGG(s(X1), 0, X2) -> U25_GGG(X1, X2, quotG_in_ggg(X1, 0, X2)) QUOTG_IN_GGG(s(X1), 0, X2) -> QUOTG_IN_GGG(X1, 0, X2) QUOTG_IN_GGG(s(X1), s(X2), X3) -> U26_GGG(X1, X2, X3, quotH_in_ggg(X1, X2, X3)) QUOTG_IN_GGG(s(X1), s(X2), X3) -> QUOTH_IN_GGG(X1, X2, X3) QUOTH_IN_GGG(X1, 0, 0) -> U27_GGG(X1, quotA_in_ggg(X1, s(s(s(s(s(s(s(0))))))), 0)) QUOTH_IN_GGG(X1, 0, 0) -> QUOTA_IN_GGG(X1, s(s(s(s(s(s(s(0))))))), 0) QUOTH_IN_GGG(X1, 0, s(X2)) -> U28_GGG(X1, X2, quotA_in_ggg(X1, s(s(s(s(s(s(s(0))))))), X2)) QUOTH_IN_GGG(X1, 0, s(X2)) -> QUOTA_IN_GGG(X1, s(s(s(s(s(s(s(0))))))), X2) QUOTH_IN_GGG(s(X1), 0, X2) -> U29_GGG(X1, X2, quotH_in_ggg(X1, 0, X2)) QUOTH_IN_GGG(s(X1), 0, X2) -> QUOTH_IN_GGG(X1, 0, X2) QUOTH_IN_GGG(s(X1), s(X2), X3) -> U30_GGG(X1, X2, X3, quotI_in_gggg(X1, X2, s(s(s(s(s(s(s(X2))))))), X3)) QUOTH_IN_GGG(s(X1), s(X2), X3) -> QUOTI_IN_GGGG(X1, X2, s(s(s(s(s(s(s(X2))))))), X3) QUOTI_IN_GGGG(X1, 0, X2, 0) -> U31_GGGG(X1, X2, quotA_in_ggg(X1, s(X2), 0)) QUOTI_IN_GGGG(X1, 0, X2, 0) -> QUOTA_IN_GGG(X1, s(X2), 0) QUOTI_IN_GGGG(X1, 0, X2, s(X3)) -> U32_GGGG(X1, X2, X3, quotA_in_ggg(X1, s(X2), X3)) QUOTI_IN_GGGG(X1, 0, X2, s(X3)) -> QUOTA_IN_GGG(X1, s(X2), X3) QUOTI_IN_GGGG(s(X1), 0, X2, X3) -> U33_GGGG(X1, X2, X3, quotI_in_gggg(X1, 0, X2, X3)) QUOTI_IN_GGGG(s(X1), 0, X2, X3) -> QUOTI_IN_GGGG(X1, 0, X2, X3) QUOTI_IN_GGGG(s(X1), s(X2), X3, X4) -> U34_GGGG(X1, X2, X3, X4, quotI_in_gggg(X1, X2, X3, X4)) QUOTI_IN_GGGG(s(X1), s(X2), X3, X4) -> QUOTI_IN_GGGG(X1, X2, X3, X4) QUOTB_IN_GGA(X1, 0, s(X2)) -> U4_GGA(X1, X2, quotA_in_gga(X1, s(0), X2)) QUOTB_IN_GGA(X1, 0, s(X2)) -> QUOTA_IN_GGA(X1, s(0), X2) QUOTB_IN_GGA(s(X1), 0, X2) -> U5_GGA(X1, X2, quotB_in_gga(X1, 0, X2)) QUOTB_IN_GGA(s(X1), 0, X2) -> QUOTB_IN_GGA(X1, 0, X2) QUOTB_IN_GGA(s(X1), s(X2), X3) -> U6_GGA(X1, X2, X3, quotC_in_gga(X1, X2, X3)) QUOTB_IN_GGA(s(X1), s(X2), X3) -> QUOTC_IN_GGA(X1, X2, X3) QUOTC_IN_GGA(X1, 0, 0) -> U7_GGA(X1, quotA_in_ggg(X1, s(s(0)), 0)) QUOTC_IN_GGA(X1, 0, 0) -> QUOTA_IN_GGG(X1, s(s(0)), 0) QUOTC_IN_GGA(X1, 0, s(X2)) -> U8_GGA(X1, X2, quotA_in_gga(X1, s(s(0)), X2)) QUOTC_IN_GGA(X1, 0, s(X2)) -> QUOTA_IN_GGA(X1, s(s(0)), X2) QUOTC_IN_GGA(s(X1), 0, X2) -> U9_GGA(X1, X2, quotC_in_gga(X1, 0, X2)) QUOTC_IN_GGA(s(X1), 0, X2) -> QUOTC_IN_GGA(X1, 0, X2) QUOTC_IN_GGA(s(X1), s(X2), X3) -> U10_GGA(X1, X2, X3, quotD_in_gga(X1, X2, X3)) QUOTC_IN_GGA(s(X1), s(X2), X3) -> QUOTD_IN_GGA(X1, X2, X3) QUOTD_IN_GGA(X1, 0, 0) -> U11_GGA(X1, quotA_in_ggg(X1, s(s(s(0))), 0)) QUOTD_IN_GGA(X1, 0, 0) -> QUOTA_IN_GGG(X1, s(s(s(0))), 0) QUOTD_IN_GGA(X1, 0, s(X2)) -> U12_GGA(X1, X2, quotA_in_gga(X1, s(s(s(0))), X2)) QUOTD_IN_GGA(X1, 0, s(X2)) -> QUOTA_IN_GGA(X1, s(s(s(0))), X2) QUOTD_IN_GGA(s(X1), 0, X2) -> U13_GGA(X1, X2, quotD_in_gga(X1, 0, X2)) QUOTD_IN_GGA(s(X1), 0, X2) -> QUOTD_IN_GGA(X1, 0, X2) QUOTD_IN_GGA(s(X1), s(X2), X3) -> U14_GGA(X1, X2, X3, quotE_in_gga(X1, X2, X3)) QUOTD_IN_GGA(s(X1), s(X2), X3) -> QUOTE_IN_GGA(X1, X2, X3) QUOTE_IN_GGA(X1, 0, 0) -> U15_GGA(X1, quotA_in_ggg(X1, s(s(s(s(0)))), 0)) QUOTE_IN_GGA(X1, 0, 0) -> QUOTA_IN_GGG(X1, s(s(s(s(0)))), 0) QUOTE_IN_GGA(X1, 0, s(X2)) -> U16_GGA(X1, X2, quotA_in_gga(X1, s(s(s(s(0)))), X2)) QUOTE_IN_GGA(X1, 0, s(X2)) -> QUOTA_IN_GGA(X1, s(s(s(s(0)))), X2) QUOTE_IN_GGA(s(X1), 0, X2) -> U17_GGA(X1, X2, quotE_in_gga(X1, 0, X2)) QUOTE_IN_GGA(s(X1), 0, X2) -> QUOTE_IN_GGA(X1, 0, X2) QUOTE_IN_GGA(s(X1), s(X2), X3) -> U18_GGA(X1, X2, X3, quotF_in_gga(X1, X2, X3)) QUOTE_IN_GGA(s(X1), s(X2), X3) -> QUOTF_IN_GGA(X1, X2, X3) QUOTF_IN_GGA(X1, 0, 0) -> U19_GGA(X1, quotA_in_ggg(X1, s(s(s(s(s(0))))), 0)) QUOTF_IN_GGA(X1, 0, 0) -> QUOTA_IN_GGG(X1, s(s(s(s(s(0))))), 0) QUOTF_IN_GGA(X1, 0, s(X2)) -> U20_GGA(X1, X2, quotA_in_gga(X1, s(s(s(s(s(0))))), X2)) QUOTF_IN_GGA(X1, 0, s(X2)) -> QUOTA_IN_GGA(X1, s(s(s(s(s(0))))), X2) QUOTF_IN_GGA(s(X1), 0, X2) -> U21_GGA(X1, X2, quotF_in_gga(X1, 0, X2)) QUOTF_IN_GGA(s(X1), 0, X2) -> QUOTF_IN_GGA(X1, 0, X2) QUOTF_IN_GGA(s(X1), s(X2), X3) -> U22_GGA(X1, X2, X3, quotG_in_gga(X1, X2, X3)) QUOTF_IN_GGA(s(X1), s(X2), X3) -> QUOTG_IN_GGA(X1, X2, X3) QUOTG_IN_GGA(X1, 0, 0) -> U23_GGA(X1, quotA_in_ggg(X1, s(s(s(s(s(s(0)))))), 0)) QUOTG_IN_GGA(X1, 0, 0) -> QUOTA_IN_GGG(X1, s(s(s(s(s(s(0)))))), 0) QUOTG_IN_GGA(X1, 0, s(X2)) -> U24_GGA(X1, X2, quotA_in_gga(X1, s(s(s(s(s(s(0)))))), X2)) QUOTG_IN_GGA(X1, 0, s(X2)) -> QUOTA_IN_GGA(X1, s(s(s(s(s(s(0)))))), X2) QUOTG_IN_GGA(s(X1), 0, X2) -> U25_GGA(X1, X2, quotG_in_gga(X1, 0, X2)) QUOTG_IN_GGA(s(X1), 0, X2) -> QUOTG_IN_GGA(X1, 0, X2) QUOTG_IN_GGA(s(X1), s(X2), X3) -> U26_GGA(X1, X2, X3, quotH_in_gga(X1, X2, X3)) QUOTG_IN_GGA(s(X1), s(X2), X3) -> QUOTH_IN_GGA(X1, X2, X3) QUOTH_IN_GGA(X1, 0, 0) -> U27_GGA(X1, quotA_in_ggg(X1, s(s(s(s(s(s(s(0))))))), 0)) QUOTH_IN_GGA(X1, 0, 0) -> QUOTA_IN_GGG(X1, s(s(s(s(s(s(s(0))))))), 0) QUOTH_IN_GGA(X1, 0, s(X2)) -> U28_GGA(X1, X2, quotA_in_gga(X1, s(s(s(s(s(s(s(0))))))), X2)) QUOTH_IN_GGA(X1, 0, s(X2)) -> QUOTA_IN_GGA(X1, s(s(s(s(s(s(s(0))))))), X2) QUOTH_IN_GGA(s(X1), 0, X2) -> U29_GGA(X1, X2, quotH_in_gga(X1, 0, X2)) QUOTH_IN_GGA(s(X1), 0, X2) -> QUOTH_IN_GGA(X1, 0, X2) QUOTH_IN_GGA(s(X1), s(X2), X3) -> U30_GGA(X1, X2, X3, quotI_in_ggga(X1, X2, s(s(s(s(s(s(s(X2))))))), X3)) QUOTH_IN_GGA(s(X1), s(X2), X3) -> QUOTI_IN_GGGA(X1, X2, s(s(s(s(s(s(s(X2))))))), X3) QUOTI_IN_GGGA(X1, 0, X2, 0) -> U31_GGGA(X1, X2, quotA_in_ggg(X1, s(X2), 0)) QUOTI_IN_GGGA(X1, 0, X2, 0) -> QUOTA_IN_GGG(X1, s(X2), 0) QUOTI_IN_GGGA(X1, 0, X2, s(X3)) -> U32_GGGA(X1, X2, X3, quotA_in_gga(X1, s(X2), X3)) QUOTI_IN_GGGA(X1, 0, X2, s(X3)) -> QUOTA_IN_GGA(X1, s(X2), X3) QUOTI_IN_GGGA(s(X1), 0, X2, X3) -> U33_GGGA(X1, X2, X3, quotI_in_ggga(X1, 0, X2, X3)) QUOTI_IN_GGGA(s(X1), 0, X2, X3) -> QUOTI_IN_GGGA(X1, 0, X2, X3) QUOTI_IN_GGGA(s(X1), s(X2), X3, X4) -> U34_GGGA(X1, X2, X3, X4, quotI_in_ggga(X1, X2, X3, X4)) QUOTI_IN_GGGA(s(X1), s(X2), X3, X4) -> QUOTI_IN_GGGA(X1, X2, X3, X4) R is empty. The argument filtering Pi contains the following mapping: quotA_in_gga(x1, x2, x3) = quotA_in_gga(x1, x2) s(x1) = s(x1) 0 = 0 quotB_in_gga(x1, x2, x3) = quotB_in_gga(x1, x2) quotA_in_ggg(x1, x2, x3) = quotA_in_ggg(x1, x2, x3) quotB_in_ggg(x1, x2, x3) = quotB_in_ggg(x1, x2, x3) quotC_in_ggg(x1, x2, x3) = quotC_in_ggg(x1, x2, x3) quotD_in_ggg(x1, x2, x3) = quotD_in_ggg(x1, x2, x3) quotE_in_ggg(x1, x2, x3) = quotE_in_ggg(x1, x2, x3) quotF_in_ggg(x1, x2, x3) = quotF_in_ggg(x1, x2, x3) quotG_in_ggg(x1, x2, x3) = quotG_in_ggg(x1, x2, x3) quotH_in_ggg(x1, x2, x3) = quotH_in_ggg(x1, x2, x3) quotI_in_gggg(x1, x2, x3, x4) = quotI_in_gggg(x1, x2, x3, x4) quotC_in_gga(x1, x2, x3) = quotC_in_gga(x1, x2) quotD_in_gga(x1, x2, x3) = quotD_in_gga(x1, x2) quotE_in_gga(x1, x2, x3) = quotE_in_gga(x1, x2) quotF_in_gga(x1, x2, x3) = quotF_in_gga(x1, x2) quotG_in_gga(x1, x2, x3) = quotG_in_gga(x1, x2) quotH_in_gga(x1, x2, x3) = quotH_in_gga(x1, x2) quotI_in_ggga(x1, x2, x3, x4) = quotI_in_ggga(x1, x2, x3) DIVJ_IN_GGA(x1, x2, x3) = DIVJ_IN_GGA(x1, x2) U35_GGA(x1, x2, x3, x4) = U35_GGA(x1, x2, x4) QUOTA_IN_GGA(x1, x2, x3) = QUOTA_IN_GGA(x1, x2) U1_GGA(x1, x2, x3) = U1_GGA(x1, x3) U2_GGA(x1, x2, x3, x4) = U2_GGA(x1, x2, x4) QUOTB_IN_GGA(x1, x2, x3) = QUOTB_IN_GGA(x1, x2) U3_GGA(x1, x2) = U3_GGA(x1, x2) QUOTA_IN_GGG(x1, x2, x3) = QUOTA_IN_GGG(x1, x2, x3) U1_GGG(x1, x2, x3) = U1_GGG(x1, x2, x3) U2_GGG(x1, x2, x3, x4) = U2_GGG(x1, x2, x3, x4) QUOTB_IN_GGG(x1, x2, x3) = QUOTB_IN_GGG(x1, x2, x3) U3_GGG(x1, x2) = U3_GGG(x1, x2) U4_GGG(x1, x2, x3) = U4_GGG(x1, x2, x3) U5_GGG(x1, x2, x3) = U5_GGG(x1, x2, x3) U6_GGG(x1, x2, x3, x4) = U6_GGG(x1, x2, x3, x4) QUOTC_IN_GGG(x1, x2, x3) = QUOTC_IN_GGG(x1, x2, x3) U7_GGG(x1, x2) = U7_GGG(x1, x2) U8_GGG(x1, x2, x3) = U8_GGG(x1, x2, x3) U9_GGG(x1, x2, x3) = U9_GGG(x1, x2, x3) U10_GGG(x1, x2, x3, x4) = U10_GGG(x1, x2, x3, x4) QUOTD_IN_GGG(x1, x2, x3) = QUOTD_IN_GGG(x1, x2, x3) U11_GGG(x1, x2) = U11_GGG(x1, x2) U12_GGG(x1, x2, x3) = U12_GGG(x1, x2, x3) U13_GGG(x1, x2, x3) = U13_GGG(x1, x2, x3) U14_GGG(x1, x2, x3, x4) = U14_GGG(x1, x2, x3, x4) QUOTE_IN_GGG(x1, x2, x3) = QUOTE_IN_GGG(x1, x2, x3) U15_GGG(x1, x2) = U15_GGG(x1, x2) U16_GGG(x1, x2, x3) = U16_GGG(x1, x2, x3) U17_GGG(x1, x2, x3) = U17_GGG(x1, x2, x3) U18_GGG(x1, x2, x3, x4) = U18_GGG(x1, x2, x3, x4) QUOTF_IN_GGG(x1, x2, x3) = QUOTF_IN_GGG(x1, x2, x3) U19_GGG(x1, x2) = U19_GGG(x1, x2) U20_GGG(x1, x2, x3) = U20_GGG(x1, x2, x3) U21_GGG(x1, x2, x3) = U21_GGG(x1, x2, x3) U22_GGG(x1, x2, x3, x4) = U22_GGG(x1, x2, x3, x4) QUOTG_IN_GGG(x1, x2, x3) = QUOTG_IN_GGG(x1, x2, x3) U23_GGG(x1, x2) = U23_GGG(x1, x2) U24_GGG(x1, x2, x3) = U24_GGG(x1, x2, x3) U25_GGG(x1, x2, x3) = U25_GGG(x1, x2, x3) U26_GGG(x1, x2, x3, x4) = U26_GGG(x1, x2, x3, x4) QUOTH_IN_GGG(x1, x2, x3) = QUOTH_IN_GGG(x1, x2, x3) U27_GGG(x1, x2) = U27_GGG(x1, x2) U28_GGG(x1, x2, x3) = U28_GGG(x1, x2, x3) U29_GGG(x1, x2, x3) = U29_GGG(x1, x2, x3) U30_GGG(x1, x2, x3, x4) = U30_GGG(x1, x2, x3, x4) QUOTI_IN_GGGG(x1, x2, x3, x4) = QUOTI_IN_GGGG(x1, x2, x3, x4) U31_GGGG(x1, x2, x3) = U31_GGGG(x1, x2, x3) U32_GGGG(x1, x2, x3, x4) = U32_GGGG(x1, x2, x3, x4) U33_GGGG(x1, x2, x3, x4) = U33_GGGG(x1, x2, x3, x4) U34_GGGG(x1, x2, x3, x4, x5) = U34_GGGG(x1, x2, x3, x4, x5) U4_GGA(x1, x2, x3) = U4_GGA(x1, x3) U5_GGA(x1, x2, x3) = U5_GGA(x1, x3) U6_GGA(x1, x2, x3, x4) = U6_GGA(x1, x2, x4) QUOTC_IN_GGA(x1, x2, x3) = QUOTC_IN_GGA(x1, x2) U7_GGA(x1, x2) = U7_GGA(x1, x2) U8_GGA(x1, x2, x3) = U8_GGA(x1, x3) U9_GGA(x1, x2, x3) = U9_GGA(x1, x3) U10_GGA(x1, x2, x3, x4) = U10_GGA(x1, x2, x4) QUOTD_IN_GGA(x1, x2, x3) = QUOTD_IN_GGA(x1, x2) U11_GGA(x1, x2) = U11_GGA(x1, x2) U12_GGA(x1, x2, x3) = U12_GGA(x1, x3) U13_GGA(x1, x2, x3) = U13_GGA(x1, x3) U14_GGA(x1, x2, x3, x4) = U14_GGA(x1, x2, x4) QUOTE_IN_GGA(x1, x2, x3) = QUOTE_IN_GGA(x1, x2) U15_GGA(x1, x2) = U15_GGA(x1, x2) U16_GGA(x1, x2, x3) = U16_GGA(x1, x3) U17_GGA(x1, x2, x3) = U17_GGA(x1, x3) U18_GGA(x1, x2, x3, x4) = U18_GGA(x1, x2, x4) QUOTF_IN_GGA(x1, x2, x3) = QUOTF_IN_GGA(x1, x2) U19_GGA(x1, x2) = U19_GGA(x1, x2) U20_GGA(x1, x2, x3) = U20_GGA(x1, x3) U21_GGA(x1, x2, x3) = U21_GGA(x1, x3) U22_GGA(x1, x2, x3, x4) = U22_GGA(x1, x2, x4) QUOTG_IN_GGA(x1, x2, x3) = QUOTG_IN_GGA(x1, x2) U23_GGA(x1, x2) = U23_GGA(x1, x2) U24_GGA(x1, x2, x3) = U24_GGA(x1, x3) U25_GGA(x1, x2, x3) = U25_GGA(x1, x3) U26_GGA(x1, x2, x3, x4) = U26_GGA(x1, x2, x4) QUOTH_IN_GGA(x1, x2, x3) = QUOTH_IN_GGA(x1, x2) U27_GGA(x1, x2) = U27_GGA(x1, x2) U28_GGA(x1, x2, x3) = U28_GGA(x1, x3) U29_GGA(x1, x2, x3) = U29_GGA(x1, x3) U30_GGA(x1, x2, x3, x4) = U30_GGA(x1, x2, x4) QUOTI_IN_GGGA(x1, x2, x3, x4) = QUOTI_IN_GGGA(x1, x2, x3) U31_GGGA(x1, x2, x3) = U31_GGGA(x1, x2, x3) U32_GGGA(x1, x2, x3, x4) = U32_GGGA(x1, x2, x4) U33_GGGA(x1, x2, x3, x4) = U33_GGGA(x1, x2, x4) U34_GGGA(x1, x2, x3, x4, x5) = U34_GGGA(x1, x2, x3, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (5) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 4 SCCs with 78 less nodes. ---------------------------------------- (6) Complex Obligation (AND) ---------------------------------------- (7) Obligation: Pi DP problem: The TRS P consists of the following rules: QUOTA_IN_GGG(s(X1), s(X2), X3) -> QUOTB_IN_GGG(X1, X2, X3) QUOTB_IN_GGG(X1, 0, 0) -> QUOTA_IN_GGG(X1, s(0), 0) QUOTB_IN_GGG(X1, 0, s(X2)) -> QUOTA_IN_GGG(X1, s(0), X2) QUOTB_IN_GGG(s(X1), 0, X2) -> QUOTB_IN_GGG(X1, 0, X2) QUOTB_IN_GGG(s(X1), s(X2), X3) -> QUOTC_IN_GGG(X1, X2, X3) QUOTC_IN_GGG(X1, 0, 0) -> QUOTA_IN_GGG(X1, s(s(0)), 0) QUOTC_IN_GGG(X1, 0, s(X2)) -> QUOTA_IN_GGG(X1, s(s(0)), X2) QUOTC_IN_GGG(s(X1), 0, X2) -> QUOTC_IN_GGG(X1, 0, X2) QUOTC_IN_GGG(s(X1), s(X2), X3) -> QUOTD_IN_GGG(X1, X2, X3) QUOTD_IN_GGG(X1, 0, 0) -> QUOTA_IN_GGG(X1, s(s(s(0))), 0) QUOTD_IN_GGG(X1, 0, s(X2)) -> QUOTA_IN_GGG(X1, s(s(s(0))), X2) QUOTD_IN_GGG(s(X1), 0, X2) -> QUOTD_IN_GGG(X1, 0, X2) QUOTD_IN_GGG(s(X1), s(X2), X3) -> QUOTE_IN_GGG(X1, X2, X3) QUOTE_IN_GGG(X1, 0, 0) -> QUOTA_IN_GGG(X1, s(s(s(s(0)))), 0) QUOTE_IN_GGG(X1, 0, s(X2)) -> QUOTA_IN_GGG(X1, s(s(s(s(0)))), X2) QUOTE_IN_GGG(s(X1), 0, X2) -> QUOTE_IN_GGG(X1, 0, X2) QUOTE_IN_GGG(s(X1), s(X2), X3) -> QUOTF_IN_GGG(X1, X2, X3) QUOTF_IN_GGG(X1, 0, 0) -> QUOTA_IN_GGG(X1, s(s(s(s(s(0))))), 0) QUOTF_IN_GGG(X1, 0, s(X2)) -> QUOTA_IN_GGG(X1, s(s(s(s(s(0))))), X2) QUOTF_IN_GGG(s(X1), 0, X2) -> QUOTF_IN_GGG(X1, 0, X2) QUOTF_IN_GGG(s(X1), s(X2), X3) -> QUOTG_IN_GGG(X1, X2, X3) QUOTG_IN_GGG(X1, 0, 0) -> QUOTA_IN_GGG(X1, s(s(s(s(s(s(0)))))), 0) QUOTG_IN_GGG(X1, 0, s(X2)) -> QUOTA_IN_GGG(X1, s(s(s(s(s(s(0)))))), X2) QUOTG_IN_GGG(s(X1), 0, X2) -> QUOTG_IN_GGG(X1, 0, X2) QUOTG_IN_GGG(s(X1), s(X2), X3) -> QUOTH_IN_GGG(X1, X2, X3) QUOTH_IN_GGG(X1, 0, 0) -> QUOTA_IN_GGG(X1, s(s(s(s(s(s(s(0))))))), 0) QUOTH_IN_GGG(X1, 0, s(X2)) -> QUOTA_IN_GGG(X1, s(s(s(s(s(s(s(0))))))), X2) QUOTH_IN_GGG(s(X1), 0, X2) -> QUOTH_IN_GGG(X1, 0, X2) QUOTH_IN_GGG(s(X1), s(X2), X3) -> QUOTI_IN_GGGG(X1, X2, s(s(s(s(s(s(s(X2))))))), X3) QUOTI_IN_GGGG(X1, 0, X2, 0) -> QUOTA_IN_GGG(X1, s(X2), 0) QUOTI_IN_GGGG(X1, 0, X2, s(X3)) -> QUOTA_IN_GGG(X1, s(X2), X3) QUOTI_IN_GGGG(s(X1), 0, X2, X3) -> QUOTI_IN_GGGG(X1, 0, X2, X3) QUOTI_IN_GGGG(s(X1), s(X2), X3, X4) -> QUOTI_IN_GGGG(X1, X2, X3, X4) R is empty. Pi is empty. We have to consider all (P,R,Pi)-chains ---------------------------------------- (8) PiDPToQDPProof (EQUIVALENT) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (9) Obligation: Q DP problem: The TRS P consists of the following rules: QUOTA_IN_GGG(s(X1), s(X2), X3) -> QUOTB_IN_GGG(X1, X2, X3) QUOTB_IN_GGG(X1, 0, 0) -> QUOTA_IN_GGG(X1, s(0), 0) QUOTB_IN_GGG(X1, 0, s(X2)) -> QUOTA_IN_GGG(X1, s(0), X2) QUOTB_IN_GGG(s(X1), 0, X2) -> QUOTB_IN_GGG(X1, 0, X2) QUOTB_IN_GGG(s(X1), s(X2), X3) -> QUOTC_IN_GGG(X1, X2, X3) QUOTC_IN_GGG(X1, 0, 0) -> QUOTA_IN_GGG(X1, s(s(0)), 0) QUOTC_IN_GGG(X1, 0, s(X2)) -> QUOTA_IN_GGG(X1, s(s(0)), X2) QUOTC_IN_GGG(s(X1), 0, X2) -> QUOTC_IN_GGG(X1, 0, X2) QUOTC_IN_GGG(s(X1), s(X2), X3) -> QUOTD_IN_GGG(X1, X2, X3) QUOTD_IN_GGG(X1, 0, 0) -> QUOTA_IN_GGG(X1, s(s(s(0))), 0) QUOTD_IN_GGG(X1, 0, s(X2)) -> QUOTA_IN_GGG(X1, s(s(s(0))), X2) QUOTD_IN_GGG(s(X1), 0, X2) -> QUOTD_IN_GGG(X1, 0, X2) QUOTD_IN_GGG(s(X1), s(X2), X3) -> QUOTE_IN_GGG(X1, X2, X3) QUOTE_IN_GGG(X1, 0, 0) -> QUOTA_IN_GGG(X1, s(s(s(s(0)))), 0) QUOTE_IN_GGG(X1, 0, s(X2)) -> QUOTA_IN_GGG(X1, s(s(s(s(0)))), X2) QUOTE_IN_GGG(s(X1), 0, X2) -> QUOTE_IN_GGG(X1, 0, X2) QUOTE_IN_GGG(s(X1), s(X2), X3) -> QUOTF_IN_GGG(X1, X2, X3) QUOTF_IN_GGG(X1, 0, 0) -> QUOTA_IN_GGG(X1, s(s(s(s(s(0))))), 0) QUOTF_IN_GGG(X1, 0, s(X2)) -> QUOTA_IN_GGG(X1, s(s(s(s(s(0))))), X2) QUOTF_IN_GGG(s(X1), 0, X2) -> QUOTF_IN_GGG(X1, 0, X2) QUOTF_IN_GGG(s(X1), s(X2), X3) -> QUOTG_IN_GGG(X1, X2, X3) QUOTG_IN_GGG(X1, 0, 0) -> QUOTA_IN_GGG(X1, s(s(s(s(s(s(0)))))), 0) QUOTG_IN_GGG(X1, 0, s(X2)) -> QUOTA_IN_GGG(X1, s(s(s(s(s(s(0)))))), X2) QUOTG_IN_GGG(s(X1), 0, X2) -> QUOTG_IN_GGG(X1, 0, X2) QUOTG_IN_GGG(s(X1), s(X2), X3) -> QUOTH_IN_GGG(X1, X2, X3) QUOTH_IN_GGG(X1, 0, 0) -> QUOTA_IN_GGG(X1, s(s(s(s(s(s(s(0))))))), 0) QUOTH_IN_GGG(X1, 0, s(X2)) -> QUOTA_IN_GGG(X1, s(s(s(s(s(s(s(0))))))), X2) QUOTH_IN_GGG(s(X1), 0, X2) -> QUOTH_IN_GGG(X1, 0, X2) QUOTH_IN_GGG(s(X1), s(X2), X3) -> QUOTI_IN_GGGG(X1, X2, s(s(s(s(s(s(s(X2))))))), X3) QUOTI_IN_GGGG(X1, 0, X2, 0) -> QUOTA_IN_GGG(X1, s(X2), 0) QUOTI_IN_GGGG(X1, 0, X2, s(X3)) -> QUOTA_IN_GGG(X1, s(X2), X3) QUOTI_IN_GGGG(s(X1), 0, X2, X3) -> QUOTI_IN_GGGG(X1, 0, X2, X3) QUOTI_IN_GGGG(s(X1), s(X2), X3, X4) -> QUOTI_IN_GGGG(X1, X2, X3, X4) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (10) QDPSizeChangeProof (EQUIVALENT) 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. From the DPs we obtained the following set of size-change graphs: *QUOTB_IN_GGG(s(X1), 0, X2) -> QUOTB_IN_GGG(X1, 0, X2) The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3 *QUOTB_IN_GGG(s(X1), s(X2), X3) -> QUOTC_IN_GGG(X1, X2, X3) The graph contains the following edges 1 > 1, 2 > 2, 3 >= 3 *QUOTA_IN_GGG(s(X1), s(X2), X3) -> QUOTB_IN_GGG(X1, X2, X3) The graph contains the following edges 1 > 1, 2 > 2, 3 >= 3 *QUOTC_IN_GGG(s(X1), 0, X2) -> QUOTC_IN_GGG(X1, 0, X2) The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3 *QUOTC_IN_GGG(s(X1), s(X2), X3) -> QUOTD_IN_GGG(X1, X2, X3) The graph contains the following edges 1 > 1, 2 > 2, 3 >= 3 *QUOTD_IN_GGG(s(X1), 0, X2) -> QUOTD_IN_GGG(X1, 0, X2) The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3 *QUOTD_IN_GGG(s(X1), s(X2), X3) -> QUOTE_IN_GGG(X1, X2, X3) The graph contains the following edges 1 > 1, 2 > 2, 3 >= 3 *QUOTE_IN_GGG(s(X1), 0, X2) -> QUOTE_IN_GGG(X1, 0, X2) The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3 *QUOTE_IN_GGG(s(X1), s(X2), X3) -> QUOTF_IN_GGG(X1, X2, X3) The graph contains the following edges 1 > 1, 2 > 2, 3 >= 3 *QUOTF_IN_GGG(s(X1), 0, X2) -> QUOTF_IN_GGG(X1, 0, X2) The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3 *QUOTF_IN_GGG(s(X1), s(X2), X3) -> QUOTG_IN_GGG(X1, X2, X3) The graph contains the following edges 1 > 1, 2 > 2, 3 >= 3 *QUOTG_IN_GGG(s(X1), 0, X2) -> QUOTG_IN_GGG(X1, 0, X2) The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3 *QUOTG_IN_GGG(s(X1), s(X2), X3) -> QUOTH_IN_GGG(X1, X2, X3) The graph contains the following edges 1 > 1, 2 > 2, 3 >= 3 *QUOTH_IN_GGG(s(X1), 0, X2) -> QUOTH_IN_GGG(X1, 0, X2) The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3 *QUOTH_IN_GGG(s(X1), s(X2), X3) -> QUOTI_IN_GGGG(X1, X2, s(s(s(s(s(s(s(X2))))))), X3) The graph contains the following edges 1 > 1, 2 > 2, 3 >= 4 *QUOTI_IN_GGGG(s(X1), 0, X2, X3) -> QUOTI_IN_GGGG(X1, 0, X2, X3) The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3, 4 >= 4 *QUOTI_IN_GGGG(s(X1), s(X2), X3, X4) -> QUOTI_IN_GGGG(X1, X2, X3, X4) The graph contains the following edges 1 > 1, 2 > 2, 3 >= 3, 4 >= 4 *QUOTB_IN_GGG(X1, 0, 0) -> QUOTA_IN_GGG(X1, s(0), 0) The graph contains the following edges 1 >= 1, 2 >= 3, 3 >= 3 *QUOTB_IN_GGG(X1, 0, s(X2)) -> QUOTA_IN_GGG(X1, s(0), X2) The graph contains the following edges 1 >= 1, 3 > 3 *QUOTC_IN_GGG(X1, 0, 0) -> QUOTA_IN_GGG(X1, s(s(0)), 0) The graph contains the following edges 1 >= 1, 2 >= 3, 3 >= 3 *QUOTC_IN_GGG(X1, 0, s(X2)) -> QUOTA_IN_GGG(X1, s(s(0)), X2) The graph contains the following edges 1 >= 1, 3 > 3 *QUOTD_IN_GGG(X1, 0, 0) -> QUOTA_IN_GGG(X1, s(s(s(0))), 0) The graph contains the following edges 1 >= 1, 2 >= 3, 3 >= 3 *QUOTD_IN_GGG(X1, 0, s(X2)) -> QUOTA_IN_GGG(X1, s(s(s(0))), X2) The graph contains the following edges 1 >= 1, 3 > 3 *QUOTE_IN_GGG(X1, 0, 0) -> QUOTA_IN_GGG(X1, s(s(s(s(0)))), 0) The graph contains the following edges 1 >= 1, 2 >= 3, 3 >= 3 *QUOTE_IN_GGG(X1, 0, s(X2)) -> QUOTA_IN_GGG(X1, s(s(s(s(0)))), X2) The graph contains the following edges 1 >= 1, 3 > 3 *QUOTF_IN_GGG(X1, 0, 0) -> QUOTA_IN_GGG(X1, s(s(s(s(s(0))))), 0) The graph contains the following edges 1 >= 1, 2 >= 3, 3 >= 3 *QUOTF_IN_GGG(X1, 0, s(X2)) -> QUOTA_IN_GGG(X1, s(s(s(s(s(0))))), X2) The graph contains the following edges 1 >= 1, 3 > 3 *QUOTG_IN_GGG(X1, 0, 0) -> QUOTA_IN_GGG(X1, s(s(s(s(s(s(0)))))), 0) The graph contains the following edges 1 >= 1, 2 >= 3, 3 >= 3 *QUOTG_IN_GGG(X1, 0, s(X2)) -> QUOTA_IN_GGG(X1, s(s(s(s(s(s(0)))))), X2) The graph contains the following edges 1 >= 1, 3 > 3 *QUOTH_IN_GGG(X1, 0, 0) -> QUOTA_IN_GGG(X1, s(s(s(s(s(s(s(0))))))), 0) The graph contains the following edges 1 >= 1, 2 >= 3, 3 >= 3 *QUOTH_IN_GGG(X1, 0, s(X2)) -> QUOTA_IN_GGG(X1, s(s(s(s(s(s(s(0))))))), X2) The graph contains the following edges 1 >= 1, 3 > 3 *QUOTI_IN_GGGG(X1, 0, X2, 0) -> QUOTA_IN_GGG(X1, s(X2), 0) The graph contains the following edges 1 >= 1, 2 >= 3, 4 >= 3 *QUOTI_IN_GGGG(X1, 0, X2, s(X3)) -> QUOTA_IN_GGG(X1, s(X2), X3) The graph contains the following edges 1 >= 1, 4 > 3 ---------------------------------------- (11) YES ---------------------------------------- (12) Obligation: Pi DP problem: The TRS P consists of the following rules: QUOTA_IN_GGG(s(X1), 0, X2) -> QUOTA_IN_GGG(X1, 0, X2) R is empty. Pi is empty. We have to consider all (P,R,Pi)-chains ---------------------------------------- (13) PiDPToQDPProof (EQUIVALENT) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (14) Obligation: Q DP problem: The TRS P consists of the following rules: QUOTA_IN_GGG(s(X1), 0, X2) -> QUOTA_IN_GGG(X1, 0, X2) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (15) QDPSizeChangeProof (EQUIVALENT) 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. From the DPs we obtained the following set of size-change graphs: *QUOTA_IN_GGG(s(X1), 0, X2) -> QUOTA_IN_GGG(X1, 0, X2) The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3 ---------------------------------------- (16) YES ---------------------------------------- (17) Obligation: Pi DP problem: The TRS P consists of the following rules: QUOTA_IN_GGA(s(X1), s(X2), X3) -> QUOTB_IN_GGA(X1, X2, X3) QUOTB_IN_GGA(X1, 0, s(X2)) -> QUOTA_IN_GGA(X1, s(0), X2) QUOTB_IN_GGA(s(X1), 0, X2) -> QUOTB_IN_GGA(X1, 0, X2) QUOTB_IN_GGA(s(X1), s(X2), X3) -> QUOTC_IN_GGA(X1, X2, X3) QUOTC_IN_GGA(X1, 0, s(X2)) -> QUOTA_IN_GGA(X1, s(s(0)), X2) QUOTC_IN_GGA(s(X1), 0, X2) -> QUOTC_IN_GGA(X1, 0, X2) QUOTC_IN_GGA(s(X1), s(X2), X3) -> QUOTD_IN_GGA(X1, X2, X3) QUOTD_IN_GGA(X1, 0, s(X2)) -> QUOTA_IN_GGA(X1, s(s(s(0))), X2) QUOTD_IN_GGA(s(X1), 0, X2) -> QUOTD_IN_GGA(X1, 0, X2) QUOTD_IN_GGA(s(X1), s(X2), X3) -> QUOTE_IN_GGA(X1, X2, X3) QUOTE_IN_GGA(X1, 0, s(X2)) -> QUOTA_IN_GGA(X1, s(s(s(s(0)))), X2) QUOTE_IN_GGA(s(X1), 0, X2) -> QUOTE_IN_GGA(X1, 0, X2) QUOTE_IN_GGA(s(X1), s(X2), X3) -> QUOTF_IN_GGA(X1, X2, X3) QUOTF_IN_GGA(X1, 0, s(X2)) -> QUOTA_IN_GGA(X1, s(s(s(s(s(0))))), X2) QUOTF_IN_GGA(s(X1), 0, X2) -> QUOTF_IN_GGA(X1, 0, X2) QUOTF_IN_GGA(s(X1), s(X2), X3) -> QUOTG_IN_GGA(X1, X2, X3) QUOTG_IN_GGA(X1, 0, s(X2)) -> QUOTA_IN_GGA(X1, s(s(s(s(s(s(0)))))), X2) QUOTG_IN_GGA(s(X1), 0, X2) -> QUOTG_IN_GGA(X1, 0, X2) QUOTG_IN_GGA(s(X1), s(X2), X3) -> QUOTH_IN_GGA(X1, X2, X3) QUOTH_IN_GGA(X1, 0, s(X2)) -> QUOTA_IN_GGA(X1, s(s(s(s(s(s(s(0))))))), X2) QUOTH_IN_GGA(s(X1), 0, X2) -> QUOTH_IN_GGA(X1, 0, X2) QUOTH_IN_GGA(s(X1), s(X2), X3) -> QUOTI_IN_GGGA(X1, X2, s(s(s(s(s(s(s(X2))))))), X3) QUOTI_IN_GGGA(X1, 0, X2, s(X3)) -> QUOTA_IN_GGA(X1, s(X2), X3) QUOTI_IN_GGGA(s(X1), 0, X2, X3) -> QUOTI_IN_GGGA(X1, 0, X2, X3) QUOTI_IN_GGGA(s(X1), s(X2), X3, X4) -> QUOTI_IN_GGGA(X1, X2, X3, X4) R is empty. The argument filtering Pi contains the following mapping: s(x1) = s(x1) 0 = 0 QUOTA_IN_GGA(x1, x2, x3) = QUOTA_IN_GGA(x1, x2) QUOTB_IN_GGA(x1, x2, x3) = QUOTB_IN_GGA(x1, x2) QUOTC_IN_GGA(x1, x2, x3) = QUOTC_IN_GGA(x1, x2) QUOTD_IN_GGA(x1, x2, x3) = QUOTD_IN_GGA(x1, x2) QUOTE_IN_GGA(x1, x2, x3) = QUOTE_IN_GGA(x1, x2) QUOTF_IN_GGA(x1, x2, x3) = QUOTF_IN_GGA(x1, x2) QUOTG_IN_GGA(x1, x2, x3) = QUOTG_IN_GGA(x1, x2) QUOTH_IN_GGA(x1, x2, x3) = QUOTH_IN_GGA(x1, x2) QUOTI_IN_GGGA(x1, x2, x3, x4) = QUOTI_IN_GGGA(x1, x2, x3) We have to consider all (P,R,Pi)-chains ---------------------------------------- (18) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (19) Obligation: Q DP problem: The TRS P consists of the following rules: QUOTA_IN_GGA(s(X1), s(X2)) -> QUOTB_IN_GGA(X1, X2) QUOTB_IN_GGA(X1, 0) -> QUOTA_IN_GGA(X1, s(0)) QUOTB_IN_GGA(s(X1), 0) -> QUOTB_IN_GGA(X1, 0) QUOTB_IN_GGA(s(X1), s(X2)) -> QUOTC_IN_GGA(X1, X2) QUOTC_IN_GGA(X1, 0) -> QUOTA_IN_GGA(X1, s(s(0))) QUOTC_IN_GGA(s(X1), 0) -> QUOTC_IN_GGA(X1, 0) QUOTC_IN_GGA(s(X1), s(X2)) -> QUOTD_IN_GGA(X1, X2) QUOTD_IN_GGA(X1, 0) -> QUOTA_IN_GGA(X1, s(s(s(0)))) QUOTD_IN_GGA(s(X1), 0) -> QUOTD_IN_GGA(X1, 0) QUOTD_IN_GGA(s(X1), s(X2)) -> QUOTE_IN_GGA(X1, X2) QUOTE_IN_GGA(X1, 0) -> QUOTA_IN_GGA(X1, s(s(s(s(0))))) QUOTE_IN_GGA(s(X1), 0) -> QUOTE_IN_GGA(X1, 0) QUOTE_IN_GGA(s(X1), s(X2)) -> QUOTF_IN_GGA(X1, X2) QUOTF_IN_GGA(X1, 0) -> QUOTA_IN_GGA(X1, s(s(s(s(s(0)))))) QUOTF_IN_GGA(s(X1), 0) -> QUOTF_IN_GGA(X1, 0) QUOTF_IN_GGA(s(X1), s(X2)) -> QUOTG_IN_GGA(X1, X2) QUOTG_IN_GGA(X1, 0) -> QUOTA_IN_GGA(X1, s(s(s(s(s(s(0))))))) QUOTG_IN_GGA(s(X1), 0) -> QUOTG_IN_GGA(X1, 0) QUOTG_IN_GGA(s(X1), s(X2)) -> QUOTH_IN_GGA(X1, X2) QUOTH_IN_GGA(X1, 0) -> QUOTA_IN_GGA(X1, s(s(s(s(s(s(s(0)))))))) QUOTH_IN_GGA(s(X1), 0) -> QUOTH_IN_GGA(X1, 0) QUOTH_IN_GGA(s(X1), s(X2)) -> QUOTI_IN_GGGA(X1, X2, s(s(s(s(s(s(s(X2)))))))) QUOTI_IN_GGGA(X1, 0, X2) -> QUOTA_IN_GGA(X1, s(X2)) QUOTI_IN_GGGA(s(X1), 0, X2) -> QUOTI_IN_GGGA(X1, 0, X2) QUOTI_IN_GGGA(s(X1), s(X2), X3) -> QUOTI_IN_GGGA(X1, X2, X3) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (20) QDPSizeChangeProof (EQUIVALENT) 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. From the DPs we obtained the following set of size-change graphs: *QUOTB_IN_GGA(X1, 0) -> QUOTA_IN_GGA(X1, s(0)) The graph contains the following edges 1 >= 1 *QUOTB_IN_GGA(s(X1), 0) -> QUOTB_IN_GGA(X1, 0) The graph contains the following edges 1 > 1, 2 >= 2 *QUOTB_IN_GGA(s(X1), s(X2)) -> QUOTC_IN_GGA(X1, X2) The graph contains the following edges 1 > 1, 2 > 2 *QUOTA_IN_GGA(s(X1), s(X2)) -> QUOTB_IN_GGA(X1, X2) The graph contains the following edges 1 > 1, 2 > 2 *QUOTC_IN_GGA(X1, 0) -> QUOTA_IN_GGA(X1, s(s(0))) The graph contains the following edges 1 >= 1 *QUOTC_IN_GGA(s(X1), 0) -> QUOTC_IN_GGA(X1, 0) The graph contains the following edges 1 > 1, 2 >= 2 *QUOTC_IN_GGA(s(X1), s(X2)) -> QUOTD_IN_GGA(X1, X2) The graph contains the following edges 1 > 1, 2 > 2 *QUOTD_IN_GGA(X1, 0) -> QUOTA_IN_GGA(X1, s(s(s(0)))) The graph contains the following edges 1 >= 1 *QUOTD_IN_GGA(s(X1), 0) -> QUOTD_IN_GGA(X1, 0) The graph contains the following edges 1 > 1, 2 >= 2 *QUOTD_IN_GGA(s(X1), s(X2)) -> QUOTE_IN_GGA(X1, X2) The graph contains the following edges 1 > 1, 2 > 2 *QUOTE_IN_GGA(X1, 0) -> QUOTA_IN_GGA(X1, s(s(s(s(0))))) The graph contains the following edges 1 >= 1 *QUOTE_IN_GGA(s(X1), 0) -> QUOTE_IN_GGA(X1, 0) The graph contains the following edges 1 > 1, 2 >= 2 *QUOTE_IN_GGA(s(X1), s(X2)) -> QUOTF_IN_GGA(X1, X2) The graph contains the following edges 1 > 1, 2 > 2 *QUOTF_IN_GGA(X1, 0) -> QUOTA_IN_GGA(X1, s(s(s(s(s(0)))))) The graph contains the following edges 1 >= 1 *QUOTF_IN_GGA(s(X1), 0) -> QUOTF_IN_GGA(X1, 0) The graph contains the following edges 1 > 1, 2 >= 2 *QUOTF_IN_GGA(s(X1), s(X2)) -> QUOTG_IN_GGA(X1, X2) The graph contains the following edges 1 > 1, 2 > 2 *QUOTG_IN_GGA(X1, 0) -> QUOTA_IN_GGA(X1, s(s(s(s(s(s(0))))))) The graph contains the following edges 1 >= 1 *QUOTG_IN_GGA(s(X1), 0) -> QUOTG_IN_GGA(X1, 0) The graph contains the following edges 1 > 1, 2 >= 2 *QUOTG_IN_GGA(s(X1), s(X2)) -> QUOTH_IN_GGA(X1, X2) The graph contains the following edges 1 > 1, 2 > 2 *QUOTH_IN_GGA(X1, 0) -> QUOTA_IN_GGA(X1, s(s(s(s(s(s(s(0)))))))) The graph contains the following edges 1 >= 1 *QUOTI_IN_GGGA(X1, 0, X2) -> QUOTA_IN_GGA(X1, s(X2)) The graph contains the following edges 1 >= 1 *QUOTH_IN_GGA(s(X1), 0) -> QUOTH_IN_GGA(X1, 0) The graph contains the following edges 1 > 1, 2 >= 2 *QUOTH_IN_GGA(s(X1), s(X2)) -> QUOTI_IN_GGGA(X1, X2, s(s(s(s(s(s(s(X2)))))))) The graph contains the following edges 1 > 1, 2 > 2 *QUOTI_IN_GGGA(s(X1), 0, X2) -> QUOTI_IN_GGGA(X1, 0, X2) The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3 *QUOTI_IN_GGGA(s(X1), s(X2), X3) -> QUOTI_IN_GGGA(X1, X2, X3) The graph contains the following edges 1 > 1, 2 > 2, 3 >= 3 ---------------------------------------- (21) YES ---------------------------------------- (22) Obligation: Pi DP problem: The TRS P consists of the following rules: QUOTA_IN_GGA(s(X1), 0, X2) -> QUOTA_IN_GGA(X1, 0, X2) R is empty. The argument filtering Pi contains the following mapping: s(x1) = s(x1) 0 = 0 QUOTA_IN_GGA(x1, x2, x3) = QUOTA_IN_GGA(x1, x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (23) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (24) Obligation: Q DP problem: The TRS P consists of the following rules: QUOTA_IN_GGA(s(X1), 0) -> QUOTA_IN_GGA(X1, 0) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (25) QDPSizeChangeProof (EQUIVALENT) 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. From the DPs we obtained the following set of size-change graphs: *QUOTA_IN_GGA(s(X1), 0) -> QUOTA_IN_GGA(X1, 0) The graph contains the following edges 1 > 1, 2 >= 2 ---------------------------------------- (26) YES