/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern ordered(g) w.r.t. the given Prolog program could successfully be proven: (0) Prolog (1) PrologToDTProblemTransformerProof [SOUND, 86 ms] (2) TRIPLES (3) TriplesToPiDPProof [SOUND, 0 ms] (4) PiDP (5) DependencyGraphProof [EQUIVALENT, 0 ms] (6) AND (7) PiDP (8) UsableRulesProof [EQUIVALENT, 0 ms] (9) PiDP (10) PiDPToQDPProof [EQUIVALENT, 0 ms] (11) QDP (12) QDPSizeChangeProof [EQUIVALENT, 0 ms] (13) YES (14) PiDP (15) UsableRulesProof [EQUIVALENT, 0 ms] (16) PiDP (17) PiDPToQDPProof [EQUIVALENT, 0 ms] (18) QDP (19) QDPSizeChangeProof [EQUIVALENT, 0 ms] (20) YES (21) PiDP (22) PiDPToQDPProof [EQUIVALENT, 5 ms] (23) QDP (24) MRRProof [EQUIVALENT, 67 ms] (25) QDP (26) DependencyGraphProof [EQUIVALENT, 0 ms] (27) TRUE ---------------------------------------- (0) Obligation: Clauses: ordered([]). ordered(.(X1, [])). ordered(Xs) :- ','(no(max1el_list(Xs)), ','(head(Xs, X), ','(tail(Xs, Ys), ','(head(Ys, Y), ','(tail(Ys, Zs), ','(less(X, s(Y)), ordered(.(Y, Zs)))))))). head([], X2). head(.(X, X3), X). tail([], []). tail(.(X4, Xs), Xs). less(0, s(X5)). less(X, Y) :- ','(no(zero(X)), ','(p(X, Px), ','(p(Y, Py), less(Px, Py)))). p(0, 0). p(s(X), X). max1el_list([]). max1el_list(.(X6, [])). zero(0). no(X) :- ','(X, ','(!, failure(a))). no(X7). failure(b). Query: ordered(g) ---------------------------------------- (1) PrologToDTProblemTransformerProof (SOUND) Built DT problem from termination graph DT10. { "root": 1, "program": { "directives": [], "clauses": [ [ "(ordered ([]))", null ], [ "(ordered (. X1 ([])))", null ], [ "(ordered Xs)", "(',' (no (max1el_list Xs)) (',' (head Xs X) (',' (tail Xs Ys) (',' (head Ys Y) (',' (tail Ys Zs) (',' (less X (s Y)) (ordered (. Y Zs))))))))" ], [ "(head ([]) X2)", null ], [ "(head (. X X3) X)", null ], [ "(tail ([]) ([]))", null ], [ "(tail (. X4 Xs) Xs)", null ], [ "(less (0) (s X5))", null ], [ "(less X Y)", "(',' (no (zero X)) (',' (p X Px) (',' (p Y Py) (less Px Py))))" ], [ "(p (0) (0))", null ], [ "(p (s X) X)", null ], [ "(max1el_list ([]))", null ], [ "(max1el_list (. X6 ([])))", null ], [ "(zero (0))", null ], [ "(no X)", "(',' X (',' (!) (failure (a))))" ], [ "(no X7)", null ], [ "(failure (b))", null ] ] }, "graph": { "nodes": { "88": { "goal": [ { "clause": -1, "scope": -1, "term": "(',' (',' (!_6) (failure (a))) (',' (head (. T11 ([])) X24) (',' (tail (. T11 ([])) X25) (',' (head X25 X26) (',' (tail X25 X27) (',' (less X24 (s X26)) (ordered (. X26 X27))))))))" }, { "clause": -1, "scope": 8, "term": null }, { "clause": -1, "scope": 7, "term": null }, { "clause": 15, "scope": 6, "term": "(',' (no (max1el_list (. T11 ([])))) (',' (head (. T11 ([])) X24) (',' (tail (. T11 ([])) X25) (',' (head X25 X26) (',' (tail X25 X27) (',' (less X24 (s X26)) (ordered (. X26 X27))))))))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T11"], "free": [ "X24", "X25", "X26", "X27" ], "exprvars": [] } }, "89": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (failure (a)) (',' (head (. T11 ([])) X24) (',' (tail (. 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T8 ([]))) (',' (',' (!_6) (failure (a))) (',' (head (. T8 ([])) X24) (',' (tail (. T8 ([])) X25) (',' (head X25 X26) (',' (tail X25 X27) (',' (less X24 (s X26)) (ordered (. X26 X27)))))))))" }, { "clause": -1, "scope": 7, "term": null }, { "clause": 15, "scope": 6, "term": "(',' (no (max1el_list (. T8 ([])))) (',' (head (. T8 ([])) X24) (',' (tail (. T8 ([])) X25) (',' (head X25 X26) (',' (tail X25 X27) (',' (less X24 (s X26)) (ordered (. X26 X27))))))))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T8"], "free": [ "X24", "X25", "X26", "X27" ], "exprvars": [] } }, "84": { "goal": [ { "clause": 11, "scope": 8, "term": "(',' (max1el_list (. T8 ([]))) (',' (',' (!_6) (failure (a))) (',' (head (. T8 ([])) X24) (',' (tail (. T8 ([])) X25) (',' (head X25 X26) (',' (tail X25 X27) (',' (less X24 (s X26)) (ordered (. X26 X27)))))))))" }, { "clause": 12, "scope": 8, "term": "(',' (max1el_list (. T8 ([]))) (',' (',' (!_6) (failure (a))) (',' (head (. T8 ([])) X24) (',' (tail (. T8 ([])) X25) (',' (head X25 X26) (',' (tail X25 X27) (',' (less X24 (s X26)) (ordered (. X26 X27)))))))))" }, { "clause": -1, "scope": 8, "term": null }, { "clause": -1, "scope": 7, "term": null }, { "clause": 15, "scope": 6, "term": "(',' (no (max1el_list (. T8 ([])))) (',' (head (. T8 ([])) X24) (',' (tail (. T8 ([])) X25) (',' (head X25 X26) (',' (tail X25 X27) (',' (less X24 (s X26)) (ordered (. X26 X27))))))))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T8"], "free": [ "X24", "X25", "X26", "X27" ], "exprvars": [] } }, "85": { "goal": [ { "clause": 12, "scope": 8, "term": "(',' (max1el_list (. T8 ([]))) (',' (',' (!_6) (failure (a))) (',' (head (. T8 ([])) X24) (',' (tail (. T8 ([])) X25) (',' (head X25 X26) (',' (tail X25 X27) (',' (less X24 (s X26)) (ordered (. X26 X27)))))))))" }, { "clause": -1, "scope": 8, "term": null }, { "clause": -1, "scope": 7, "term": null }, { "clause": 15, "scope": 6, "term": "(',' (no (max1el_list (. T8 ([])))) (',' (head (. T8 ([])) X24) (',' (tail (. T8 ([])) X25) (',' (head X25 X26) (',' (tail X25 X27) (',' (less X24 (s X26)) (ordered (. X26 X27))))))))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T8"], "free": [ "X24", "X25", "X26", "X27" ], "exprvars": [] } } }, "edges": [ { "from": 1, "to": 4, "label": "CASE" }, { "from": 4, "to": 21, "label": "EVAL with clause\nordered([]).\nand substitutionT1 -> []" }, { "from": 4, "to": 22, "label": "EVAL-BACKTRACK" }, { "from": 21, "to": 25, "label": "SUCCESS" }, { "from": 22, "to": 64, "label": "EVAL with clause\nordered(.(X21, [])).\nand substitutionX21 -> T3,\nT1 -> .(T3, [])" }, { "from": 22, "to": 66, "label": "EVAL-BACKTRACK" }, { "from": 25, "to": 32, "label": "BACKTRACK\nfor clause: ordered(.(X1, []))because of non-unification" }, { "from": 32, "to": 33, "label": "ONLY EVAL with clause\nordered(X10) :- ','(no(max1el_list(X10)), ','(head(X10, X11), ','(tail(X10, X12), ','(head(X12, X13), ','(tail(X12, X14), ','(less(X11, s(X13)), ordered(.(X13, X14)))))))).\nand substitutionX10 -> []" }, { "from": 33, "to": 35, "label": "CASE" }, { "from": 35, "to": 36, "label": "ONLY EVAL with clause\nno(X17) :- ','(call(X17), ','(!_2, failure(a))).\nand substitutionX17 -> max1el_list([])" }, { "from": 36, "to": 37, "label": "CALL" }, { "from": 37, "to": 38, "label": "CASE" }, { "from": 38, "to": 39, "label": "ONLY EVAL with clause\nmax1el_list([]).\nand substitution" }, { "from": 39, "to": 40, "label": "CUT" }, { "from": 40, "to": 41, "label": "CASE" }, { "from": 41, "to": 42, "label": "BACKTRACK\nfor clause: failure(b)because of non-unification" }, { "from": 64, "to": 68, "label": "SUCCESS" }, { "from": 66, "to": 95, "label": "ONLY EVAL with clause\nordered(X37) :- ','(no(max1el_list(X37)), ','(head(X37, X38), ','(tail(X37, X39), ','(head(X39, X40), ','(tail(X39, X41), ','(less(X38, s(X40)), ordered(.(X40, X41)))))))).\nand substitutionT1 -> T13,\nX37 -> T13" }, { "from": 68, "to": 71, "label": "ONLY EVAL with clause\nordered(X23) :- ','(no(max1el_list(X23)), ','(head(X23, X24), ','(tail(X23, X25), ','(head(X25, X26), ','(tail(X25, X27), ','(less(X24, s(X26)), ordered(.(X26, X27)))))))).\nand substitutionT3 -> T5,\nX23 -> .(T5, [])" }, { "from": 71, "to": 74, "label": "CASE" }, { "from": 74, "to": 78, "label": "ONLY EVAL with clause\nno(X30) :- ','(call(X30), ','(!_6, failure(a))).\nand substitutionT5 -> T8,\nX30 -> max1el_list(.(T8, []))" }, { "from": 78, "to": 82, "label": "CALL" }, { "from": 82, "to": 84, "label": "CASE" }, { "from": 84, "to": 85, "label": "BACKTRACK\nfor clause: max1el_list([])because of non-unification" }, { "from": 85, "to": 88, "label": "ONLY EVAL with clause\nmax1el_list(.(X33, [])).\nand substitutionT8 -> T11,\nX33 -> T11" }, { "from": 88, "to": 89, "label": "CUT" }, { "from": 89, "to": 92, "label": "CASE" }, { "from": 92, "to": 93, "label": "BACKTRACK\nfor clause: failure(b)because of non-unification" }, { "from": 95, "to": 98, "label": "CASE" }, { "from": 98, "to": 102, "label": "ONLY EVAL with clause\nno(X44) :- ','(call(X44), ','(!_10, failure(a))).\nand substitutionT13 -> T16,\nX44 -> max1el_list(T16)" }, { "from": 102, "to": 112, "label": "CALL" }, { "from": 112, "to": 113, "label": "CASE" }, { "from": 113, "to": 114, "label": "BACKTRACK\nfor clause: max1el_list([])\nwith clash: (ordered(T16), ordered([]))" }, { "from": 114, "to": 115, "label": "BACKTRACK\nfor clause: max1el_list(.(X6, []))\nwith clash: (ordered(T16), ordered(.(X21, [])))" }, { "from": 115, "to": 116, "label": "FAILURE" }, { "from": 116, "to": 117, "label": "FAILURE" }, { "from": 117, "to": 118, "label": "ONLY EVAL with clause\nno(X50).\nand substitutionT16 -> T20,\nX50 -> max1el_list(T20)" }, { "from": 118, "to": 119, "label": "CASE" }, { "from": 119, "to": 120, "label": "BACKTRACK\nfor clause: head([], X2)\nwith clash: (ordered(T20), ordered([]))" }, { "from": 120, "to": 121, "label": "EVAL with clause\nhead(.(X59, X60), X59).\nand substitutionX59 -> T25,\nX60 -> T26,\nT20 -> .(T25, T26),\nX38 -> T25" }, { "from": 120, "to": 122, "label": "EVAL-BACKTRACK" }, { "from": 121, "to": 123, "label": "CASE" }, { "from": 123, "to": 124, "label": "BACKTRACK\nfor clause: tail([], [])because of non-unification" }, { "from": 124, "to": 127, "label": "ONLY EVAL with clause\ntail(.(X73, X74), X74).\nand substitutionT25 -> T31,\nX73 -> T31,\nT26 -> T32,\nX74 -> T32,\nX39 -> T32" }, { "from": 127, "to": 131, "label": "CASE" }, { "from": 131, "to": 132, "label": "BACKTRACK\nfor clause: head([], X2)\nwith clash: (ordered(.(T31, T32)), ordered(.(X21, [])))" }, { "from": 132, "to": 136, "label": "EVAL with clause\nhead(.(X85, X86), X85).\nand substitutionX85 -> T37,\nX86 -> T38,\nT32 -> .(T37, T38),\nX40 -> T37" }, { "from": 132, "to": 137, "label": "EVAL-BACKTRACK" }, { "from": 136, "to": 138, "label": "CASE" }, { "from": 138, "to": 139, "label": "BACKTRACK\nfor clause: tail([], [])because of non-unification" }, { "from": 139, "to": 141, "label": "ONLY EVAL with clause\ntail(.(X95, X96), X96).\nand substitutionT37 -> T47,\nX95 -> T47,\nT38 -> T48,\nX96 -> T48,\nX41 -> T48" }, { "from": 141, "to": 149, "label": "SPLIT 1" }, { "from": 141, "to": 150, "label": "SPLIT 2\nnew knowledge:\nT31 is ground\nT47 is ground" }, { "from": 149, "to": 155, "label": "CASE" }, { "from": 150, "to": 1, "label": "INSTANCE with matching:\nT1 -> .(T47, T48)" }, { "from": 155, "to": 156, "label": "PARALLEL" }, { "from": 155, "to": 157, "label": "PARALLEL" }, { "from": 156, "to": 159, "label": "EVAL with clause\nless(0, s(X105)).\nand substitutionT31 -> 0,\nT47 -> T57,\nX105 -> T57" }, { "from": 156, "to": 160, "label": "EVAL-BACKTRACK" }, { "from": 157, "to": 166, "label": "ONLY EVAL with clause\nless(X114, X115) :- ','(no(zero(X114)), ','(p(X114, X116), ','(p(X115, X117), less(X116, X117)))).\nand substitutionT31 -> T62,\nX114 -> T62,\nT47 -> T63,\nX115 -> s(T63)" }, { "from": 159, "to": 161, "label": "SUCCESS" }, { "from": 166, "to": 169, "label": "CASE" }, { "from": 169, "to": 170, "label": "ONLY EVAL with clause\nno(X122) :- ','(call(X122), ','(!_18, failure(a))).\nand substitutionT62 -> T66,\nX122 -> zero(T66)" }, { "from": 170, "to": 171, "label": "CALL" }, { "from": 171, "to": 172, "label": "CASE" }, { "from": 172, "to": 174, "label": "EVAL with clause\nzero(0).\nand substitutionT66 -> 0" }, { "from": 172, "to": 175, "label": "EVAL-BACKTRACK" }, { "from": 174, "to": 176, "label": "CUT" }, { "from": 175, "to": 179, "label": "FAILURE" }, { "from": 176, "to": 177, "label": "CASE" }, { "from": 177, "to": 178, "label": "BACKTRACK\nfor clause: failure(b)because of non-unification" }, { "from": 179, "to": 180, "label": "FAILURE" }, { "from": 180, "to": 183, "label": "ONLY EVAL with clause\nno(X129).\nand substitutionT66 -> T69,\nX129 -> zero(T69)" }, { "from": 183, "to": 186, "label": "CASE" }, { "from": 186, "to": 188, "label": "BACKTRACK\nfor clause: p(0, 0)\nwith clash: (zero(T69), zero(0))" }, { "from": 188, "to": 192, "label": "EVAL with clause\np(s(X134), X134).\nand substitutionX134 -> T72,\nT69 -> s(T72),\nX116 -> T72" }, { "from": 188, "to": 193, "label": "EVAL-BACKTRACK" }, { "from": 192, "to": 194, "label": "CASE" }, { "from": 194, "to": 195, "label": "BACKTRACK\nfor clause: p(0, 0)because of non-unification" }, { "from": 195, "to": 199, "label": "ONLY EVAL with clause\np(s(X143), X143).\nand substitutionT63 -> T77,\nX143 -> T77,\nX117 -> T77" }, { "from": 199, "to": 206, "label": "CASE" }, { "from": 206, "to": 210, "label": "PARALLEL" }, { "from": 206, "to": 211, "label": "PARALLEL" }, { "from": 210, "to": 212, "label": "EVAL with clause\nless(0, s(X150)).\nand substitutionT72 -> 0,\nX150 -> T84,\nT77 -> s(T84)" }, { "from": 210, "to": 213, "label": "EVAL-BACKTRACK" }, { "from": 211, "to": 218, "label": "ONLY EVAL with clause\nless(X159, X160) :- ','(no(zero(X159)), ','(p(X159, X161), ','(p(X160, X162), less(X161, X162)))).\nand substitutionT72 -> T89,\nX159 -> T89,\nT77 -> T90,\nX160 -> T90" }, { "from": 212, "to": 214, "label": "SUCCESS" }, { "from": 218, "to": 219, "label": "CASE" }, { "from": 219, "to": 223, "label": "ONLY EVAL with clause\nno(X167) :- ','(call(X167), ','(!_25, failure(a))).\nand substitutionT89 -> T93,\nX167 -> zero(T93)" }, { "from": 223, "to": 224, "label": "CALL" }, { "from": 224, "to": 225, "label": "CASE" }, { "from": 225, "to": 228, "label": "EVAL with clause\nzero(0).\nand substitutionT93 -> 0" }, { "from": 225, "to": 229, "label": "EVAL-BACKTRACK" }, { "from": 228, "to": 230, "label": "CUT" }, { "from": 229, "to": 235, "label": "FAILURE" }, { "from": 230, "to": 233, "label": "CASE" }, { "from": 233, "to": 234, "label": "BACKTRACK\nfor clause: failure(b)because of non-unification" }, { "from": 235, "to": 236, "label": "FAILURE" }, { "from": 236, "to": 238, "label": "ONLY EVAL with clause\nno(X174).\nand substitutionT93 -> T96,\nX174 -> zero(T96)" }, { "from": 238, "to": 239, "label": "CASE" }, { "from": 239, "to": 241, "label": "BACKTRACK\nfor clause: p(0, 0)\nwith clash: (zero(T96), zero(0))" }, { "from": 241, "to": 244, "label": "EVAL with clause\np(s(X179), X179).\nand substitutionX179 -> T99,\nT96 -> s(T99),\nX161 -> T99" }, { "from": 241, "to": 245, "label": "EVAL-BACKTRACK" }, { "from": 244, "to": 249, "label": "CASE" }, { "from": 249, "to": 250, "label": "PARALLEL" }, { "from": 249, "to": 251, "label": "PARALLEL" }, { "from": 250, "to": 252, "label": "EVAL with clause\np(0, 0).\nand substitutionT90 -> 0,\nX162 -> 0" }, { "from": 250, "to": 253, "label": "EVAL-BACKTRACK" }, { "from": 251, "to": 276, "label": "EVAL with clause\np(s(X188), X188).\nand substitutionX188 -> T102,\nT90 -> s(T102),\nX162 -> T102" }, { "from": 251, "to": 277, "label": "EVAL-BACKTRACK" }, { "from": 252, "to": 199, "label": "INSTANCE with matching:\nT72 -> T99\nT77 -> 0" }, { "from": 276, "to": 199, "label": "INSTANCE with matching:\nT72 -> T99\nT77 -> T102" } ], "type": "Graph" } } ---------------------------------------- (2) Obligation: Triples: lessC(s(X1), 0) :- lessC(X1, 0). lessC(s(X1), s(X2)) :- lessC(X1, X2). orderedA(.(s(X1), .(X2, X3))) :- lessC(X1, X2). orderedA(.(X1, .(X2, X3))) :- ','(lesscB(X1, X2), orderedA(.(X2, X3))). Clauses: orderedcA([]). orderedcA(.(X1, [])). orderedcA(.(X1, .(X2, X3))) :- ','(lesscB(X1, X2), orderedcA(.(X2, X3))). lesscC(0, s(X1)). lesscC(s(X1), 0) :- lesscC(X1, 0). lesscC(s(X1), s(X2)) :- lesscC(X1, X2). lesscB(0, X1). lesscB(s(X1), X2) :- lesscC(X1, X2). Afs: orderedA(x1) = orderedA(x1) ---------------------------------------- (3) TriplesToPiDPProof (SOUND) We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes: orderedA_in_1: (b) lessC_in_2: (b,b) lesscB_in_2: (b,b) lesscC_in_2: (b,b) Transforming TRIPLES into the following Term Rewriting System: Pi DP problem: The TRS P consists of the following rules: ORDEREDA_IN_G(.(s(X1), .(X2, X3))) -> U3_G(X1, X2, X3, lessC_in_gg(X1, X2)) ORDEREDA_IN_G(.(s(X1), .(X2, X3))) -> LESSC_IN_GG(X1, X2) LESSC_IN_GG(s(X1), 0) -> U1_GG(X1, lessC_in_gg(X1, 0)) LESSC_IN_GG(s(X1), 0) -> LESSC_IN_GG(X1, 0) LESSC_IN_GG(s(X1), s(X2)) -> U2_GG(X1, X2, lessC_in_gg(X1, X2)) LESSC_IN_GG(s(X1), s(X2)) -> LESSC_IN_GG(X1, X2) ORDEREDA_IN_G(.(X1, .(X2, X3))) -> U4_G(X1, X2, X3, lesscB_in_gg(X1, X2)) U4_G(X1, X2, X3, lesscB_out_gg(X1, X2)) -> U5_G(X1, X2, X3, orderedA_in_g(.(X2, X3))) U4_G(X1, X2, X3, lesscB_out_gg(X1, X2)) -> ORDEREDA_IN_G(.(X2, X3)) The TRS R consists of the following rules: lesscB_in_gg(0, X1) -> lesscB_out_gg(0, X1) lesscB_in_gg(s(X1), X2) -> U11_gg(X1, X2, lesscC_in_gg(X1, X2)) lesscC_in_gg(0, s(X1)) -> lesscC_out_gg(0, s(X1)) lesscC_in_gg(s(X1), 0) -> U9_gg(X1, lesscC_in_gg(X1, 0)) lesscC_in_gg(s(X1), s(X2)) -> U10_gg(X1, X2, lesscC_in_gg(X1, X2)) U10_gg(X1, X2, lesscC_out_gg(X1, X2)) -> lesscC_out_gg(s(X1), s(X2)) U9_gg(X1, lesscC_out_gg(X1, 0)) -> lesscC_out_gg(s(X1), 0) U11_gg(X1, X2, lesscC_out_gg(X1, X2)) -> lesscB_out_gg(s(X1), X2) Pi is empty. 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: ORDEREDA_IN_G(.(s(X1), .(X2, X3))) -> U3_G(X1, X2, X3, lessC_in_gg(X1, X2)) ORDEREDA_IN_G(.(s(X1), .(X2, X3))) -> LESSC_IN_GG(X1, X2) LESSC_IN_GG(s(X1), 0) -> U1_GG(X1, lessC_in_gg(X1, 0)) LESSC_IN_GG(s(X1), 0) -> LESSC_IN_GG(X1, 0) LESSC_IN_GG(s(X1), s(X2)) -> U2_GG(X1, X2, lessC_in_gg(X1, X2)) LESSC_IN_GG(s(X1), s(X2)) -> LESSC_IN_GG(X1, X2) ORDEREDA_IN_G(.(X1, .(X2, X3))) -> U4_G(X1, X2, X3, lesscB_in_gg(X1, X2)) U4_G(X1, X2, X3, lesscB_out_gg(X1, X2)) -> U5_G(X1, X2, X3, orderedA_in_g(.(X2, X3))) U4_G(X1, X2, X3, lesscB_out_gg(X1, X2)) -> ORDEREDA_IN_G(.(X2, X3)) The TRS R consists of the following rules: lesscB_in_gg(0, X1) -> lesscB_out_gg(0, X1) lesscB_in_gg(s(X1), X2) -> U11_gg(X1, X2, lesscC_in_gg(X1, X2)) lesscC_in_gg(0, s(X1)) -> lesscC_out_gg(0, s(X1)) lesscC_in_gg(s(X1), 0) -> U9_gg(X1, lesscC_in_gg(X1, 0)) lesscC_in_gg(s(X1), s(X2)) -> U10_gg(X1, X2, lesscC_in_gg(X1, X2)) U10_gg(X1, X2, lesscC_out_gg(X1, X2)) -> lesscC_out_gg(s(X1), s(X2)) U9_gg(X1, lesscC_out_gg(X1, 0)) -> lesscC_out_gg(s(X1), 0) U11_gg(X1, X2, lesscC_out_gg(X1, X2)) -> lesscB_out_gg(s(X1), X2) Pi is empty. We have to consider all (P,R,Pi)-chains ---------------------------------------- (5) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 5 less nodes. ---------------------------------------- (6) Complex Obligation (AND) ---------------------------------------- (7) Obligation: Pi DP problem: The TRS P consists of the following rules: LESSC_IN_GG(s(X1), 0) -> LESSC_IN_GG(X1, 0) The TRS R consists of the following rules: lesscB_in_gg(0, X1) -> lesscB_out_gg(0, X1) lesscB_in_gg(s(X1), X2) -> U11_gg(X1, X2, lesscC_in_gg(X1, X2)) lesscC_in_gg(0, s(X1)) -> lesscC_out_gg(0, s(X1)) lesscC_in_gg(s(X1), 0) -> U9_gg(X1, lesscC_in_gg(X1, 0)) lesscC_in_gg(s(X1), s(X2)) -> U10_gg(X1, X2, lesscC_in_gg(X1, X2)) U10_gg(X1, X2, lesscC_out_gg(X1, X2)) -> lesscC_out_gg(s(X1), s(X2)) U9_gg(X1, lesscC_out_gg(X1, 0)) -> lesscC_out_gg(s(X1), 0) U11_gg(X1, X2, lesscC_out_gg(X1, X2)) -> lesscB_out_gg(s(X1), X2) Pi is empty. We have to consider all (P,R,Pi)-chains ---------------------------------------- (8) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (9) Obligation: Pi DP problem: The TRS P consists of the following rules: LESSC_IN_GG(s(X1), 0) -> LESSC_IN_GG(X1, 0) R is empty. Pi is empty. We have to consider all (P,R,Pi)-chains ---------------------------------------- (10) PiDPToQDPProof (EQUIVALENT) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (11) Obligation: Q DP problem: The TRS P consists of the following rules: LESSC_IN_GG(s(X1), 0) -> LESSC_IN_GG(X1, 0) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (12) 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: *LESSC_IN_GG(s(X1), 0) -> LESSC_IN_GG(X1, 0) The graph contains the following edges 1 > 1, 2 >= 2 ---------------------------------------- (13) YES ---------------------------------------- (14) Obligation: Pi DP problem: The TRS P consists of the following rules: LESSC_IN_GG(s(X1), s(X2)) -> LESSC_IN_GG(X1, X2) The TRS R consists of the following rules: lesscB_in_gg(0, X1) -> lesscB_out_gg(0, X1) lesscB_in_gg(s(X1), X2) -> U11_gg(X1, X2, lesscC_in_gg(X1, X2)) lesscC_in_gg(0, s(X1)) -> lesscC_out_gg(0, s(X1)) lesscC_in_gg(s(X1), 0) -> U9_gg(X1, lesscC_in_gg(X1, 0)) lesscC_in_gg(s(X1), s(X2)) -> U10_gg(X1, X2, lesscC_in_gg(X1, X2)) U10_gg(X1, X2, lesscC_out_gg(X1, X2)) -> lesscC_out_gg(s(X1), s(X2)) U9_gg(X1, lesscC_out_gg(X1, 0)) -> lesscC_out_gg(s(X1), 0) U11_gg(X1, X2, lesscC_out_gg(X1, X2)) -> lesscB_out_gg(s(X1), X2) Pi is empty. We have to consider all (P,R,Pi)-chains ---------------------------------------- (15) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (16) Obligation: Pi DP problem: The TRS P consists of the following rules: LESSC_IN_GG(s(X1), s(X2)) -> LESSC_IN_GG(X1, X2) R is empty. Pi is empty. We have to consider all (P,R,Pi)-chains ---------------------------------------- (17) PiDPToQDPProof (EQUIVALENT) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (18) Obligation: Q DP problem: The TRS P consists of the following rules: LESSC_IN_GG(s(X1), s(X2)) -> LESSC_IN_GG(X1, X2) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (19) 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: *LESSC_IN_GG(s(X1), s(X2)) -> LESSC_IN_GG(X1, X2) The graph contains the following edges 1 > 1, 2 > 2 ---------------------------------------- (20) YES ---------------------------------------- (21) Obligation: Pi DP problem: The TRS P consists of the following rules: ORDEREDA_IN_G(.(X1, .(X2, X3))) -> U4_G(X1, X2, X3, lesscB_in_gg(X1, X2)) U4_G(X1, X2, X3, lesscB_out_gg(X1, X2)) -> ORDEREDA_IN_G(.(X2, X3)) The TRS R consists of the following rules: lesscB_in_gg(0, X1) -> lesscB_out_gg(0, X1) lesscB_in_gg(s(X1), X2) -> U11_gg(X1, X2, lesscC_in_gg(X1, X2)) lesscC_in_gg(0, s(X1)) -> lesscC_out_gg(0, s(X1)) lesscC_in_gg(s(X1), 0) -> U9_gg(X1, lesscC_in_gg(X1, 0)) lesscC_in_gg(s(X1), s(X2)) -> U10_gg(X1, X2, lesscC_in_gg(X1, X2)) U10_gg(X1, X2, lesscC_out_gg(X1, X2)) -> lesscC_out_gg(s(X1), s(X2)) U9_gg(X1, lesscC_out_gg(X1, 0)) -> lesscC_out_gg(s(X1), 0) U11_gg(X1, X2, lesscC_out_gg(X1, X2)) -> lesscB_out_gg(s(X1), X2) Pi is empty. We have to consider all (P,R,Pi)-chains ---------------------------------------- (22) PiDPToQDPProof (EQUIVALENT) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (23) Obligation: Q DP problem: The TRS P consists of the following rules: ORDEREDA_IN_G(.(X1, .(X2, X3))) -> U4_G(X1, X2, X3, lesscB_in_gg(X1, X2)) U4_G(X1, X2, X3, lesscB_out_gg(X1, X2)) -> ORDEREDA_IN_G(.(X2, X3)) The TRS R consists of the following rules: lesscB_in_gg(0, X1) -> lesscB_out_gg(0, X1) lesscB_in_gg(s(X1), X2) -> U11_gg(X1, X2, lesscC_in_gg(X1, X2)) lesscC_in_gg(0, s(X1)) -> lesscC_out_gg(0, s(X1)) lesscC_in_gg(s(X1), 0) -> U9_gg(X1, lesscC_in_gg(X1, 0)) lesscC_in_gg(s(X1), s(X2)) -> U10_gg(X1, X2, lesscC_in_gg(X1, X2)) U10_gg(X1, X2, lesscC_out_gg(X1, X2)) -> lesscC_out_gg(s(X1), s(X2)) U9_gg(X1, lesscC_out_gg(X1, 0)) -> lesscC_out_gg(s(X1), 0) U11_gg(X1, X2, lesscC_out_gg(X1, X2)) -> lesscB_out_gg(s(X1), X2) The set Q consists of the following terms: lesscB_in_gg(x0, x1) lesscC_in_gg(x0, x1) U10_gg(x0, x1, x2) U9_gg(x0, x1) U11_gg(x0, x1, x2) We have to consider all (P,Q,R)-chains. ---------------------------------------- (24) MRRProof (EQUIVALENT) 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. Strictly oriented rules of the TRS R: lesscB_in_gg(0, X1) -> lesscB_out_gg(0, X1) lesscC_in_gg(0, s(X1)) -> lesscC_out_gg(0, s(X1)) lesscC_in_gg(s(X1), 0) -> U9_gg(X1, lesscC_in_gg(X1, 0)) lesscC_in_gg(s(X1), s(X2)) -> U10_gg(X1, X2, lesscC_in_gg(X1, X2)) U11_gg(X1, X2, lesscC_out_gg(X1, X2)) -> lesscB_out_gg(s(X1), X2) Used ordering: Polynomial interpretation [POLO]: POL(.(x_1, x_2)) = 2*x_1 + x_2 POL(0) = 2 POL(ORDEREDA_IN_G(x_1)) = 2*x_1 POL(U10_gg(x_1, x_2, x_3)) = 2 + x_1 + x_2 + x_3 POL(U11_gg(x_1, x_2, x_3)) = 2 + 2*x_1 + x_2 + x_3 POL(U4_G(x_1, x_2, x_3, x_4)) = 2*x_1 + 2*x_2 + 2*x_3 + x_4 POL(U9_gg(x_1, x_2)) = 1 + 2*x_1 + x_2 POL(lesscB_in_gg(x_1, x_2)) = 2*x_1 + 2*x_2 POL(lesscB_out_gg(x_1, x_2)) = x_1 + 2*x_2 POL(lesscC_in_gg(x_1, x_2)) = 2*x_1 + x_2 POL(lesscC_out_gg(x_1, x_2)) = x_1 + x_2 POL(s(x_1)) = 1 + 2*x_1 ---------------------------------------- (25) Obligation: Q DP problem: The TRS P consists of the following rules: ORDEREDA_IN_G(.(X1, .(X2, X3))) -> U4_G(X1, X2, X3, lesscB_in_gg(X1, X2)) U4_G(X1, X2, X3, lesscB_out_gg(X1, X2)) -> ORDEREDA_IN_G(.(X2, X3)) The TRS R consists of the following rules: lesscB_in_gg(s(X1), X2) -> U11_gg(X1, X2, lesscC_in_gg(X1, X2)) U10_gg(X1, X2, lesscC_out_gg(X1, X2)) -> lesscC_out_gg(s(X1), s(X2)) U9_gg(X1, lesscC_out_gg(X1, 0)) -> lesscC_out_gg(s(X1), 0) The set Q consists of the following terms: lesscB_in_gg(x0, x1) lesscC_in_gg(x0, x1) U10_gg(x0, x1, x2) U9_gg(x0, x1) U11_gg(x0, x1, x2) We have to consider all (P,Q,R)-chains. ---------------------------------------- (26) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes. ---------------------------------------- (27) TRUE