/export/starexec/sandbox/solver/bin/starexec_run_standard /export/starexec/sandbox/benchmark/theBenchmark.pl /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- YES proof of /export/starexec/sandbox/benchmark/theBenchmark.pl # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern maxsort(g,a) w.r.t. the given Prolog program could successfully be proven: (0) Prolog (1) PrologToDTProblemTransformerProof [SOUND, 127 ms] (2) TRIPLES (3) TriplesToPiDPProof [SOUND, 21 ms] (4) PiDP (5) DependencyGraphProof [EQUIVALENT, 0 ms] (6) AND (7) PiDP (8) UsableRulesProof [EQUIVALENT, 0 ms] (9) PiDP (10) PiDPToQDPProof [SOUND, 0 ms] (11) QDP (12) QDPSizeChangeProof [EQUIVALENT, 0 ms] (13) YES (14) PiDP (15) UsableRulesProof [EQUIVALENT, 0 ms] (16) PiDP (17) PiDPToQDPProof [SOUND, 0 ms] (18) QDP (19) QDPSizeChangeProof [EQUIVALENT, 0 ms] (20) YES (21) PiDP (22) UsableRulesProof [EQUIVALENT, 0 ms] (23) PiDP (24) PiDPToQDPProof [SOUND, 0 ms] (25) QDP (26) QDPSizeChangeProof [EQUIVALENT, 0 ms] (27) YES (28) PiDP (29) UsableRulesProof [EQUIVALENT, 0 ms] (30) PiDP (31) PiDPToQDPProof [SOUND, 0 ms] (32) QDP (33) QDPSizeChangeProof [EQUIVALENT, 0 ms] (34) YES (35) PiDP (36) UsableRulesProof [EQUIVALENT, 0 ms] (37) PiDP (38) PiDPToQDPProof [SOUND, 0 ms] (39) QDP (40) TransformationProof [SOUND, 0 ms] (41) QDP (42) UsableRulesProof [EQUIVALENT, 0 ms] (43) QDP (44) QReductionProof [EQUIVALENT, 0 ms] (45) QDP (46) TransformationProof [EQUIVALENT, 0 ms] (47) QDP (48) DependencyGraphProof [EQUIVALENT, 0 ms] (49) QDP (50) QDPQMonotonicMRRProof [EQUIVALENT, 31 ms] (51) QDP (52) QReductionProof [EQUIVALENT, 0 ms] (53) QDP (54) TransformationProof [SOUND, 0 ms] (55) QDP (56) UsableRulesProof [EQUIVALENT, 0 ms] (57) QDP (58) QReductionProof [EQUIVALENT, 0 ms] (59) QDP (60) TransformationProof [EQUIVALENT, 0 ms] (61) QDP (62) QDPQMonotonicMRRProof [EQUIVALENT, 87 ms] (63) QDP (64) UsableRulesProof [EQUIVALENT, 0 ms] (65) QDP (66) QReductionProof [EQUIVALENT, 0 ms] (67) QDP (68) TransformationProof [SOUND, 0 ms] (69) QDP (70) UsableRulesProof [EQUIVALENT, 0 ms] (71) QDP (72) QReductionProof [EQUIVALENT, 1 ms] (73) QDP (74) TransformationProof [EQUIVALENT, 0 ms] (75) QDP (76) QDPQMonotonicMRRProof [EQUIVALENT, 41 ms] (77) QDP (78) DependencyGraphProof [EQUIVALENT, 0 ms] (79) QDP (80) UsableRulesProof [EQUIVALENT, 0 ms] (81) QDP (82) QDPQMonotonicMRRProof [EQUIVALENT, 25 ms] (83) QDP (84) QReductionProof [EQUIVALENT, 0 ms] (85) QDP (86) TransformationProof [SOUND, 1 ms] (87) QDP (88) UsableRulesProof [EQUIVALENT, 0 ms] (89) QDP (90) QReductionProof [EQUIVALENT, 0 ms] (91) QDP (92) TransformationProof [EQUIVALENT, 0 ms] (93) QDP (94) DependencyGraphProof [EQUIVALENT, 0 ms] (95) QDP (96) QDPQMonotonicMRRProof [EQUIVALENT, 17 ms] (97) QDP (98) QDPQMonotonicMRRProof [EQUIVALENT, 13 ms] (99) QDP (100) QDPQMonotonicMRRProof [EQUIVALENT, 10 ms] (101) QDP (102) PisEmptyProof [EQUIVALENT, 0 ms] (103) YES ---------------------------------------- (0) Obligation: Clauses: eq(t, t). eq(f, f). neq(t, f). neq(f, t). del(X1, [], []). del(X, .(Y, YS), YS) :- eq(X, Y). del(X, .(Y, YS), .(Y, ZS)) :- ','(neq(X, Y), del(X, YS, ZS)). ge(t, t). ge(t, f). ge(f, f). gt(t, f). max([], f). max(.(X, []), X). max(.(X, .(Y, XS)), Z) :- ','(ge(X, Y), max(.(X, XS), Z)). max(.(X, .(Y, XS)), Z) :- ','(gt(Y, X), max(.(Y, XS), Z)). maxsort([], []). maxsort(.(X, XS), .(Y, YS)) :- ','(max(.(X, XS), Y), ','(del(Y, .(X, XS), ZS), maxsort(ZS, YS))). Query: maxsort(g,a) ---------------------------------------- (1) PrologToDTProblemTransformerProof (SOUND) Built DT problem from termination graph DT10. { "root": 1, "program": { "directives": [], "clauses": [ [ "(eq (t) (t))", null ], [ "(eq (f) (f))", null ], [ "(neq (t) (f))", null ], [ "(neq (f) (t))", null ], [ "(del X1 ([]) ([]))", null ], [ "(del X (. Y YS) YS)", "(eq X Y)" ], [ "(del X (. Y YS) (. Y ZS))", "(',' (neq X Y) (del X YS ZS))" ], [ "(ge (t) (t))", null ], [ "(ge (t) (f))", null ], [ "(ge (f) (f))", null ], [ "(gt (t) (f))", null ], [ "(max ([]) (f))", null ], [ "(max (. X ([])) X)", null ], [ "(max (. X (. Y XS)) Z)", "(',' (ge X Y) (max (. X XS) Z))" ], [ "(max (. X (. Y XS)) Z)", "(',' (gt Y X) (max (. Y XS) Z))" ], [ "(maxsort ([]) ([]))", null ], [ "(maxsort (. X XS) (. Y YS))", "(',' (max (. X XS) Y) (',' (del Y (. X XS) ZS) (maxsort ZS YS)))" ] ] }, "graph": { "nodes": { "907": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "908": { "goal": [ { "clause": 4, "scope": 28, "term": "(del (t) (. (f) T214) X347)" }, { "clause": 5, "scope": 28, "term": "(del (t) (. (f) T214) X347)" }, { "clause": 6, "scope": 28, "term": "(del (t) (. (f) T214) X347)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T214"], "free": ["X347"], "exprvars": [] } }, "909": { "goal": [ { "clause": 5, "scope": 28, "term": "(del (t) (. (f) T214) X347)" }, { "clause": 6, "scope": 28, "term": "(del (t) (. 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T7 T8) T11) (',' (del T11 (. T7 T8) X14) (maxsort X14 T12)))" }, { "clause": 13, "scope": 2, "term": "(',' (max (. T7 T8) T11) (',' (del T11 (. T7 T8) X14) (maxsort X14 T12)))" }, { "clause": 14, "scope": 2, "term": "(',' (max (. T7 T8) T11) (',' (del T11 (. T7 T8) X14) (maxsort X14 T12)))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T7", "T8" ], "free": ["X14"], "exprvars": [] } }, "910": { "goal": [{ "clause": 5, "scope": 28, "term": "(del (t) (. (f) T214) X347)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T214"], "free": ["X347"], "exprvars": [] } }, "911": { "goal": [{ "clause": 6, "scope": 28, "term": "(del (t) (. 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(t) T72) T74)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T72"], "free": [], "exprvars": [] } }, "694": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "698": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "586": { "goal": [{ "clause": 5, "scope": 3, "term": "(del T17 (. T17 ([])) X14)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T17"], "free": ["X14"], "exprvars": [] } }, "587": { "goal": [{ "clause": 6, "scope": 3, "term": "(del T17 (. T17 ([])) X14)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T17"], "free": ["X14"], "exprvars": [] } }, "588": { "goal": [{ "clause": -1, "scope": -1, "term": "(eq T29 T29)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T29"], "free": [], "exprvars": [] } }, "106": { "goal": [ { "clause": 11, "scope": 2, "term": "(',' (max (. T7 T8) T11) (',' (del T11 (. T7 T8) X14) (maxsort X14 T12)))" }, { "clause": 12, "scope": 2, "term": "(',' (max (. T7 T8) T11) (',' (del T11 (. T7 T8) X14) (maxsort X14 T12)))" }, { "clause": 13, "scope": 2, "term": "(',' (max (. T7 T8) T11) (',' (del T11 (. T7 T8) X14) (maxsort X14 T12)))" }, { "clause": 14, "scope": 2, "term": "(',' (max (. T7 T8) T11) (',' (del T11 (. T7 T8) X14) (maxsort X14 T12)))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T7", "T8" ], "free": ["X14"], "exprvars": [] } }, "900": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "901": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "902": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (neq T213 (f)) (del T213 (. (f) T214) X347))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T213", "T214" ], "free": ["X347"], "exprvars": [] } }, "903": { "goal": [ { "clause": 2, "scope": 27, "term": "(',' (neq T213 (f)) (del T213 (. (f) T214) X347))" }, { "clause": 3, "scope": 27, "term": "(',' (neq T213 (f)) (del T213 (. (f) T214) X347))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T213", "T214" ], "free": ["X347"], "exprvars": [] } }, "904": { "goal": [{ "clause": 2, "scope": 27, "term": "(',' (neq T213 (f)) (del T213 (. (f) T214) X347))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T213", "T214" ], "free": ["X347"], "exprvars": [] } }, "905": { "goal": [{ "clause": 3, "scope": 27, "term": "(',' (neq T213 (f)) (del T213 (. (f) T214) X347))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T213", "T214" ], "free": ["X347"], "exprvars": [] } }, "906": { "goal": [{ "clause": -1, "scope": -1, "term": "(del (t) (. (f) T214) X347)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T214"], "free": ["X347"], "exprvars": [] } } }, "edges": [ { "from": 1, "to": 3, "label": "CASE" }, { "from": 3, "to": 37, "label": "EVAL with clause\nmaxsort([], []).\nand substitutionT1 -> [],\nT2 -> []" }, { "from": 3, "to": 38, "label": "EVAL-BACKTRACK" }, { "from": 37, "to": 39, "label": "SUCCESS" }, { "from": 38, "to": 75, "label": "EVAL with clause\nmaxsort(.(X10, X11), .(X12, X13)) :- ','(max(.(X10, X11), X12), ','(del(X12, .(X10, X11), X14), maxsort(X14, X13))).\nand substitutionX10 -> T7,\nX11 -> T8,\nT1 -> .(T7, T8),\nX12 -> T11,\nX13 -> T12,\nT2 -> .(T11, T12),\nT9 -> T11,\nT10 -> T12" }, { "from": 38, "to": 90, "label": "EVAL-BACKTRACK" }, { "from": 39, "to": 40, "label": "BACKTRACK\nfor clause: maxsort(.(X, XS), .(Y, YS)) :- ','(max(.(X, XS), Y), ','(del(Y, .(X, XS), ZS), maxsort(ZS, YS)))because of non-unification" }, { "from": 75, "to": 106, "label": "CASE" }, { "from": 106, "to": 115, "label": "BACKTRACK\nfor clause: max([], f)because of non-unification" }, { "from": 115, "to": 504, "label": "PARALLEL" }, { "from": 115, "to": 505, "label": "PARALLEL" }, { "from": 504, "to": 506, "label": "EVAL with clause\nmax(.(X19, []), X19).\nand substitutionT7 -> T17,\nX19 -> T17,\nT8 -> [],\nT11 -> T17,\nT12 -> T18" }, { "from": 504, "to": 507, "label": "EVAL-BACKTRACK" }, { "from": 505, "to": 624, "label": "PARALLEL" }, { "from": 505, "to": 625, "label": "PARALLEL" }, { "from": 506, "to": 508, "label": "SPLIT 1" }, { "from": 506, "to": 509, "label": "SPLIT 2\nnew knowledge:\nT17 is ground\nT21 is ground\nreplacements:X14 -> T21" }, { "from": 508, "to": 552, "label": "CASE" }, { "from": 509, "to": 1, "label": "INSTANCE with matching:\nT1 -> T21\nT2 -> T18" }, { "from": 552, "to": 553, "label": "BACKTRACK\nfor clause: del(X1, [], [])because of non-unification" }, { "from": 553, "to": 586, "label": "PARALLEL" }, { "from": 553, "to": 587, "label": "PARALLEL" }, { "from": 586, "to": 588, "label": "ONLY EVAL with clause\ndel(X37, .(X38, X39), X39) :- eq(X37, X38).\nand substitutionT17 -> T29,\nX37 -> T29,\nX38 -> T29,\nX39 -> [],\nX14 -> []" }, { "from": 587, "to": 612, "label": "ONLY EVAL with clause\ndel(X51, .(X52, X53), .(X52, X54)) :- ','(neq(X51, X52), del(X51, X53, X54)).\nand substitutionT17 -> T33,\nX51 -> T33,\nX52 -> T33,\nX53 -> [],\nX54 -> X55,\nX14 -> .(T33, X55)" }, { "from": 588, "to": 593, "label": "CASE" }, { "from": 593, "to": 595, "label": "PARALLEL" }, { "from": 593, "to": 596, "label": "PARALLEL" }, { "from": 595, "to": 600, "label": "EVAL with clause\neq(t, t).\nand substitutionT29 -> t" }, { "from": 595, "to": 602, "label": "EVAL-BACKTRACK" }, { "from": 596, "to": 605, "label": "EVAL with clause\neq(f, f).\nand substitutionT29 -> f" }, { "from": 596, "to": 606, "label": "EVAL-BACKTRACK" }, { "from": 600, "to": 604, "label": "SUCCESS" }, { "from": 605, "to": 607, "label": "SUCCESS" }, { "from": 612, "to": 613, "label": "CASE" }, { "from": 613, "to": 614, "label": "BACKTRACK\nfor clause: neq(t, f)because of non-unification" }, { "from": 614, "to": 616, "label": "BACKTRACK\nfor clause: neq(f, t)because of non-unification" }, { "from": 624, "to": 628, "label": "EVAL with clause\nmax(.(X73, .(X74, X75)), X76) :- ','(ge(X73, X74), max(.(X73, X75), X76)).\nand substitutionT7 -> T51,\nX73 -> T51,\nX74 -> T52,\nX75 -> T53,\nT8 -> .(T52, T53),\nT11 -> T55,\nX76 -> T55,\nT54 -> T55,\nT12 -> T56" }, { "from": 624, "to": 631, "label": "EVAL-BACKTRACK" }, { "from": 625, "to": 998, "label": "EVAL with clause\nmax(.(X440, .(X441, X442)), X443) :- ','(gt(X441, X440), max(.(X441, X442), X443)).\nand substitutionT7 -> T249,\nX440 -> T249,\nX441 -> T250,\nX442 -> T251,\nT8 -> .(T250, T251),\nT11 -> T253,\nX443 -> T253,\nT252 -> T253,\nT12 -> T254" }, { "from": 625, "to": 999, "label": "EVAL-BACKTRACK" }, { "from": 628, "to": 633, "label": "CASE" }, { "from": 633, "to": 636, "label": "PARALLEL" }, { "from": 633, "to": 637, "label": "PARALLEL" }, { "from": 636, "to": 641, "label": "EVAL with clause\nge(t, t).\nand substitutionT51 -> t,\nT52 -> t" }, { "from": 636, "to": 642, "label": "EVAL-BACKTRACK" }, { "from": 637, "to": 840, "label": "PARALLEL" }, { "from": 637, "to": 841, "label": "PARALLEL" }, { "from": 641, "to": 650, "label": "SPLIT 1" }, { "from": 641, "to": 651, "label": "SPLIT 2\nnew knowledge:\nT53 is ground\nT57 is ground\nreplacements:T55 -> T57,\nT56 -> T58" }, { "from": 650, "to": 655, "label": "CASE" }, { "from": 651, "to": 711, "label": "SPLIT 1" }, { "from": 651, "to": 712, "label": "SPLIT 2\nnew knowledge:\nT57 is ground\nT53 is ground\nT87 is ground\nreplacements:X14 -> T87" }, { "from": 655, "to": 657, "label": "BACKTRACK\nfor clause: max([], f)because of non-unification" }, { "from": 657, "to": 661, "label": "PARALLEL" }, { "from": 657, "to": 662, "label": "PARALLEL" }, { "from": 661, "to": 663, "label": "EVAL with clause\nmax(.(X81, []), X81).\nand substitutionX81 -> t,\nT53 -> [],\nT55 -> t" }, { "from": 661, "to": 665, "label": "EVAL-BACKTRACK" }, { "from": 662, "to": 673, "label": "PARALLEL" }, { "from": 662, "to": 674, "label": "PARALLEL" }, { "from": 663, "to": 666, "label": "SUCCESS" }, { "from": 673, "to": 677, "label": "EVAL with clause\nmax(.(X98, .(X99, X100)), X101) :- ','(ge(X98, X99), max(.(X98, X100), X101)).\nand substitutionX98 -> t,\nX99 -> T71,\nX100 -> T72,\nT53 -> .(T71, T72),\nT55 -> T74,\nX101 -> T74,\nT73 -> T74" }, { "from": 673, "to": 678, "label": "EVAL-BACKTRACK" }, { "from": 674, "to": 702, "label": "EVAL with clause\nmax(.(X110, .(X111, X112)), X113) :- ','(gt(X111, X110), max(.(X111, X112), X113)).\nand substitutionX110 -> t,\nX111 -> T81,\nX112 -> T82,\nT53 -> .(T81, T82),\nT55 -> T84,\nX113 -> T84,\nT83 -> T84" }, { "from": 674, "to": 703, "label": "EVAL-BACKTRACK" }, { "from": 677, "to": 681, "label": "CASE" }, { "from": 681, "to": 682, "label": "PARALLEL" }, { "from": 681, "to": 683, "label": "PARALLEL" }, { "from": 682, "to": 685, "label": "EVAL with clause\nge(t, t).\nand substitutionT71 -> t" }, { "from": 682, "to": 687, "label": "EVAL-BACKTRACK" }, { "from": 683, "to": 691, "label": "PARALLEL" }, { "from": 683, "to": 692, "label": "PARALLEL" }, { "from": 685, "to": 650, "label": "INSTANCE with matching:\nT53 -> T72\nT55 -> T74" }, { "from": 691, "to": 693, "label": "EVAL with clause\nge(t, f).\nand substitutionT71 -> f" }, { "from": 691, "to": 694, "label": "EVAL-BACKTRACK" }, { "from": 692, "to": 698, "label": "BACKTRACK\nfor clause: ge(f, f)because of non-unification" }, { "from": 693, "to": 650, "label": "INSTANCE with matching:\nT53 -> T72\nT55 -> T74" }, { "from": 702, "to": 704, "label": "CASE" }, { "from": 704, "to": 705, "label": "BACKTRACK\nfor clause: gt(t, f)because of non-unification" }, { "from": 711, "to": 764, "label": "CASE" }, { "from": 712, "to": 1, "label": "INSTANCE with matching:\nT1 -> T87\nT2 -> T58" }, { "from": 764, "to": 767, "label": "BACKTRACK\nfor clause: del(X1, [], [])because of non-unification" }, { "from": 767, "to": 768, "label": "PARALLEL" }, { "from": 767, "to": 769, "label": "PARALLEL" }, { "from": 768, "to": 770, "label": "ONLY EVAL with clause\ndel(X131, .(X132, X133), X133) :- eq(X131, X132).\nand substitutionT57 -> T99,\nX131 -> T99,\nX132 -> t,\nT53 -> T100,\nX133 -> .(t, T100),\nX14 -> .(t, T100)" }, { "from": 769, "to": 778, "label": "ONLY EVAL with clause\ndel(X146, .(X147, X148), .(X147, X149)) :- ','(neq(X146, X147), del(X146, X148, X149)).\nand substitutionT57 -> T107,\nX146 -> T107,\nX147 -> t,\nT53 -> T108,\nX148 -> .(t, T108),\nX149 -> X150,\nX14 -> .(t, X150)" }, { "from": 770, "to": 771, "label": "CASE" }, { "from": 771, "to": 772, "label": "PARALLEL" }, { "from": 771, "to": 773, "label": "PARALLEL" }, { "from": 772, "to": 774, "label": "EVAL with clause\neq(t, t).\nand substitutionT99 -> t" }, { "from": 772, "to": 775, "label": "EVAL-BACKTRACK" }, { "from": 773, "to": 777, "label": "BACKTRACK\nfor clause: eq(f, f)because of non-unification" }, { "from": 774, "to": 776, "label": "SUCCESS" }, { "from": 778, "to": 779, "label": "CASE" }, { "from": 779, "to": 780, "label": "BACKTRACK\nfor clause: neq(t, f)because of non-unification" }, { "from": 780, "to": 781, "label": "EVAL with clause\nneq(f, t).\nand substitutionT107 -> f" }, { "from": 780, "to": 782, "label": "EVAL-BACKTRACK" }, { "from": 781, "to": 783, "label": "CASE" }, { "from": 783, "to": 784, "label": "BACKTRACK\nfor clause: del(X1, [], [])because of non-unification" }, { "from": 784, "to": 785, "label": "PARALLEL" }, { "from": 784, "to": 786, "label": "PARALLEL" }, { "from": 785, "to": 787, "label": "ONLY EVAL with clause\ndel(X168, .(X169, X170), X170) :- eq(X168, X169).\nand substitutionX168 -> f,\nX169 -> t,\nT108 -> T115,\nX170 -> T115,\nX150 -> T115" }, { "from": 786, "to": 791, "label": "ONLY EVAL with clause\ndel(X183, .(X184, X185), .(X184, X186)) :- ','(neq(X183, X184), del(X183, X185, X186)).\nand substitutionX183 -> f,\nX184 -> t,\nT108 -> T118,\nX185 -> T118,\nX186 -> X187,\nX150 -> .(t, X187)" }, { "from": 787, "to": 788, "label": "CASE" }, { "from": 788, "to": 789, "label": "BACKTRACK\nfor clause: eq(t, t)because of non-unification" }, { "from": 789, "to": 790, "label": "BACKTRACK\nfor clause: eq(f, f)because of non-unification" }, { "from": 791, "to": 792, "label": "CASE" }, { "from": 792, "to": 793, "label": "BACKTRACK\nfor clause: neq(t, f)because of non-unification" }, { "from": 793, "to": 794, "label": "ONLY EVAL with clause\nneq(f, t).\nand substitution" }, { "from": 794, "to": 795, "label": "CASE" }, { "from": 795, "to": 796, "label": "PARALLEL" }, { "from": 795, "to": 797, "label": "PARALLEL" }, { "from": 796, "to": 822, "label": "EVAL with clause\ndel(X198, [], []).\nand substitutionX198 -> f,\nT118 -> [],\nX187 -> []" }, { "from": 796, "to": 823, "label": "EVAL-BACKTRACK" }, { "from": 797, "to": 825, "label": "PARALLEL" }, { "from": 797, "to": 826, "label": "PARALLEL" }, { "from": 822, "to": 824, "label": "SUCCESS" }, { "from": 825, "to": 827, "label": "EVAL with clause\ndel(X211, .(X212, X213), X213) :- eq(X211, X212).\nand substitutionX211 -> f,\nX212 -> T127,\nX213 -> T128,\nT118 -> .(T127, T128),\nX187 -> T128" }, { "from": 825, "to": 828, "label": "EVAL-BACKTRACK" }, { "from": 826, "to": 834, "label": "EVAL with clause\ndel(X224, .(X225, X226), .(X225, X227)) :- ','(neq(X224, X225), del(X224, X226, X227)).\nand substitutionX224 -> f,\nX225 -> T133,\nX226 -> T134,\nT118 -> .(T133, T134),\nX227 -> X228,\nX187 -> .(T133, X228)" }, { "from": 826, "to": 835, "label": "EVAL-BACKTRACK" }, { "from": 827, "to": 829, "label": "CASE" }, { "from": 829, "to": 830, "label": "BACKTRACK\nfor clause: eq(t, t)because of non-unification" }, { "from": 830, "to": 831, "label": "EVAL with clause\neq(f, f).\nand substitutionT127 -> f" }, { "from": 830, "to": 832, "label": "EVAL-BACKTRACK" }, { "from": 831, "to": 833, "label": "SUCCESS" }, { "from": 834, "to": 836, "label": "CASE" }, { "from": 836, "to": 837, "label": "BACKTRACK\nfor clause: neq(t, f)because of non-unification" }, { "from": 837, "to": 838, "label": "EVAL with clause\nneq(f, t).\nand substitutionT133 -> t" }, { "from": 837, "to": 839, "label": "EVAL-BACKTRACK" }, { "from": 838, "to": 794, "label": "INSTANCE with matching:\nT118 -> T134\nX187 -> X228" }, { "from": 840, "to": 842, "label": "EVAL with clause\nge(t, f).\nand substitutionT51 -> t,\nT52 -> f" }, { "from": 840, "to": 843, "label": "EVAL-BACKTRACK" }, { "from": 841, "to": 865, "label": "EVAL with clause\nge(f, f).\nand substitutionT51 -> f,\nT52 -> f" }, { "from": 841, "to": 866, "label": "EVAL-BACKTRACK" }, { "from": 842, "to": 844, "label": "SPLIT 1" }, { "from": 842, "to": 845, "label": "SPLIT 2\nnew knowledge:\nT53 is ground\nT135 is ground\nreplacements:T55 -> T135,\nT56 -> T136" }, { "from": 844, "to": 650, "label": "INSTANCE" }, { "from": 845, "to": 846, "label": "SPLIT 1" }, { "from": 845, "to": 847, "label": "SPLIT 2\nnew knowledge:\nT135 is ground\nT53 is ground\nT139 is ground\nreplacements:X14 -> T139" }, { "from": 846, "to": 848, "label": "CASE" }, { "from": 847, "to": 1, "label": "INSTANCE with matching:\nT1 -> T139\nT2 -> T136" }, { "from": 848, "to": 849, "label": "BACKTRACK\nfor clause: del(X1, [], [])because of non-unification" }, { "from": 849, "to": 850, "label": "PARALLEL" }, { "from": 849, "to": 851, "label": "PARALLEL" }, { "from": 850, "to": 852, "label": "ONLY EVAL with clause\ndel(X250, .(X251, X252), X252) :- eq(X250, X251).\nand substitutionT135 -> T151,\nX250 -> T151,\nX251 -> t,\nT53 -> T152,\nX252 -> .(f, T152),\nX14 -> .(f, T152)" }, { "from": 851, "to": 860, "label": "ONLY EVAL with clause\ndel(X265, .(X266, X267), .(X266, X268)) :- ','(neq(X265, X266), del(X265, X267, X268)).\nand substitutionT135 -> T159,\nX265 -> T159,\nX266 -> t,\nT53 -> T160,\nX267 -> .(f, T160),\nX268 -> X269,\nX14 -> .(t, X269)" }, { "from": 852, "to": 853, "label": "CASE" }, { "from": 853, "to": 854, "label": "PARALLEL" }, { "from": 853, "to": 855, "label": "PARALLEL" }, { "from": 854, "to": 856, "label": "EVAL with clause\neq(t, t).\nand substitutionT151 -> t" }, { "from": 854, "to": 857, "label": "EVAL-BACKTRACK" }, { "from": 855, "to": 859, "label": "BACKTRACK\nfor clause: eq(f, f)because of non-unification" }, { "from": 856, "to": 858, "label": "SUCCESS" }, { "from": 860, "to": 861, "label": "CASE" }, { "from": 861, "to": 862, "label": "BACKTRACK\nfor clause: neq(t, f)because of non-unification" }, { "from": 862, "to": 863, "label": "EVAL with clause\nneq(f, t).\nand substitutionT159 -> f" }, { "from": 862, "to": 864, "label": "EVAL-BACKTRACK" }, { "from": 863, "to": 794, "label": "INSTANCE with matching:\nT118 -> .(f, T160)\nX187 -> X269" }, { "from": 865, "to": 867, "label": "SPLIT 1" }, { "from": 865, "to": 868, "label": "SPLIT 2\nnew knowledge:\nT53 is ground\nT163 is ground\nreplacements:T55 -> T163,\nT56 -> T164" }, { "from": 867, "to": 869, "label": "CASE" }, { "from": 868, "to": 890, "label": "SPLIT 1" }, { "from": 868, "to": 891, "label": "SPLIT 2\nnew knowledge:\nT163 is ground\nT53 is ground\nT193 is ground\nreplacements:X14 -> T193" }, { "from": 869, "to": 870, "label": "BACKTRACK\nfor clause: max([], f)because of non-unification" }, { "from": 870, "to": 871, "label": "PARALLEL" }, { "from": 870, "to": 872, "label": "PARALLEL" }, { "from": 871, "to": 873, "label": "EVAL with clause\nmax(.(X278, []), X278).\nand substitutionX278 -> f,\nT53 -> [],\nT55 -> f" }, { "from": 871, "to": 874, "label": "EVAL-BACKTRACK" }, { "from": 872, "to": 876, "label": "PARALLEL" }, { "from": 872, "to": 877, "label": "PARALLEL" }, { "from": 873, "to": 875, "label": "SUCCESS" }, { "from": 876, "to": 878, "label": "EVAL with clause\nmax(.(X295, .(X296, X297)), X298) :- ','(ge(X295, X296), max(.(X295, X297), X298)).\nand substitutionX295 -> f,\nX296 -> T177,\nX297 -> T178,\nT53 -> .(T177, T178),\nT55 -> T180,\nX298 -> T180,\nT179 -> T180" }, { "from": 876, "to": 879, "label": "EVAL-BACKTRACK" }, { "from": 877, "to": 885, "label": "EVAL with clause\nmax(.(X307, .(X308, X309)), X310) :- ','(gt(X308, X307), max(.(X308, X309), X310)).\nand substitutionX307 -> f,\nX308 -> T187,\nX309 -> T188,\nT53 -> .(T187, T188),\nT55 -> T190,\nX310 -> T190,\nT189 -> T190" }, { "from": 877, "to": 886, "label": "EVAL-BACKTRACK" }, { "from": 878, "to": 880, "label": "CASE" }, { "from": 880, "to": 881, "label": "BACKTRACK\nfor clause: ge(t, t)because of non-unification" }, { "from": 881, "to": 882, "label": "BACKTRACK\nfor clause: ge(t, f)because of non-unification" }, { "from": 882, "to": 883, "label": "EVAL with clause\nge(f, f).\nand substitutionT177 -> f" }, { "from": 882, "to": 884, "label": "EVAL-BACKTRACK" }, { "from": 883, "to": 867, "label": "INSTANCE with matching:\nT53 -> T178\nT55 -> T180" }, { "from": 885, "to": 887, "label": "CASE" }, { "from": 887, "to": 888, "label": "EVAL with clause\ngt(t, f).\nand substitutionT187 -> t" }, { "from": 887, "to": 889, "label": "EVAL-BACKTRACK" }, { "from": 888, "to": 650, "label": "INSTANCE with matching:\nT53 -> T188\nT55 -> T190" }, { "from": 890, "to": 892, "label": "CASE" }, { "from": 891, "to": 1, "label": "INSTANCE with matching:\nT1 -> T193\nT2 -> T164" }, { "from": 892, "to": 893, "label": "BACKTRACK\nfor clause: del(X1, [], [])because of non-unification" }, { "from": 893, "to": 894, "label": "PARALLEL" }, { "from": 893, "to": 895, "label": "PARALLEL" }, { "from": 894, "to": 896, "label": "ONLY EVAL with clause\ndel(X328, .(X329, X330), X330) :- eq(X328, X329).\nand substitutionT163 -> T205,\nX328 -> T205,\nX329 -> f,\nT53 -> T206,\nX330 -> .(f, T206),\nX14 -> .(f, T206)" }, { "from": 895, "to": 902, "label": "ONLY EVAL with clause\ndel(X343, .(X344, X345), .(X344, X346)) :- ','(neq(X343, X344), del(X343, X345, X346)).\nand substitutionT163 -> T213,\nX343 -> T213,\nX344 -> f,\nT53 -> T214,\nX345 -> .(f, T214),\nX346 -> X347,\nX14 -> .(f, X347)" }, { "from": 896, "to": 897, "label": "CASE" }, { "from": 897, "to": 898, "label": "BACKTRACK\nfor clause: eq(t, t)because of non-unification" }, { "from": 898, "to": 899, "label": "EVAL with clause\neq(f, f).\nand substitutionT205 -> f" }, { "from": 898, "to": 900, "label": "EVAL-BACKTRACK" }, { "from": 899, "to": 901, "label": "SUCCESS" }, { "from": 902, "to": 903, "label": "CASE" }, { "from": 903, "to": 904, "label": "PARALLEL" }, { "from": 903, "to": 905, "label": "PARALLEL" }, { "from": 904, "to": 906, "label": "EVAL with clause\nneq(t, f).\nand substitutionT213 -> t" }, { "from": 904, "to": 907, "label": "EVAL-BACKTRACK" }, { "from": 905, "to": 997, "label": "BACKTRACK\nfor clause: neq(f, t)because of non-unification" }, { "from": 906, "to": 908, "label": "CASE" }, { "from": 908, "to": 909, "label": "BACKTRACK\nfor clause: del(X1, [], [])because of non-unification" }, { "from": 909, "to": 910, "label": "PARALLEL" }, { "from": 909, "to": 911, "label": "PARALLEL" }, { "from": 910, "to": 912, "label": "ONLY EVAL with clause\ndel(X365, .(X366, X367), X367) :- eq(X365, X366).\nand substitutionX365 -> t,\nX366 -> f,\nT214 -> T221,\nX367 -> T221,\nX347 -> T221" }, { "from": 911, "to": 916, "label": "ONLY EVAL with clause\ndel(X380, .(X381, X382), .(X381, X383)) :- ','(neq(X380, X381), del(X380, X382, X383)).\nand substitutionX380 -> t,\nX381 -> f,\nT214 -> T224,\nX382 -> T224,\nX383 -> X384,\nX347 -> .(f, X384)" }, { "from": 912, "to": 913, "label": "CASE" }, { "from": 913, "to": 914, "label": "BACKTRACK\nfor clause: eq(t, t)because of non-unification" }, { "from": 914, "to": 915, "label": "BACKTRACK\nfor clause: eq(f, f)because of non-unification" }, { "from": 916, "to": 917, "label": "CASE" }, { "from": 917, "to": 918, "label": "PARALLEL" }, { "from": 917, "to": 919, "label": "PARALLEL" }, { "from": 918, "to": 920, "label": "ONLY EVAL with clause\nneq(t, f).\nand substitution" }, { "from": 919, "to": 996, "label": "BACKTRACK\nfor clause: neq(f, t)because of non-unification" }, { "from": 920, "to": 921, "label": "CASE" }, { "from": 921, "to": 922, "label": "PARALLEL" }, { "from": 921, "to": 923, "label": "PARALLEL" }, { "from": 922, "to": 924, "label": "EVAL with clause\ndel(X397, [], []).\nand substitutionX397 -> t,\nT224 -> [],\nX384 -> []" }, { "from": 922, "to": 925, "label": "EVAL-BACKTRACK" }, { "from": 923, "to": 927, "label": "PARALLEL" }, { "from": 923, "to": 928, "label": "PARALLEL" }, { "from": 924, "to": 926, "label": "SUCCESS" }, { "from": 927, "to": 929, "label": "EVAL with clause\ndel(X410, .(X411, X412), X412) :- eq(X410, X411).\nand substitutionX410 -> t,\nX411 -> T233,\nX412 -> T234,\nT224 -> .(T233, T234),\nX384 -> T234" }, { "from": 927, "to": 930, "label": "EVAL-BACKTRACK" }, { "from": 928, "to": 988, "label": "EVAL with clause\ndel(X423, .(X424, X425), .(X424, X426)) :- ','(neq(X423, X424), del(X423, X425, X426)).\nand substitutionX423 -> t,\nX424 -> T239,\nX425 -> T240,\nT224 -> .(T239, T240),\nX426 -> X427,\nX384 -> .(T239, X427)" }, { "from": 928, "to": 989, "label": "EVAL-BACKTRACK" }, { "from": 929, "to": 931, "label": "CASE" }, { "from": 931, "to": 932, "label": "PARALLEL" }, { "from": 931, "to": 933, "label": "PARALLEL" }, { "from": 932, "to": 934, "label": "EVAL with clause\neq(t, t).\nand substitutionT233 -> t" }, { "from": 932, "to": 935, "label": "EVAL-BACKTRACK" }, { "from": 933, "to": 937, "label": "BACKTRACK\nfor clause: eq(f, f)because of non-unification" }, { "from": 934, "to": 936, "label": "SUCCESS" }, { "from": 988, "to": 990, "label": "CASE" }, { "from": 990, "to": 991, "label": "PARALLEL" }, { "from": 990, "to": 992, "label": "PARALLEL" }, { "from": 991, "to": 993, "label": "EVAL with clause\nneq(t, f).\nand substitutionT239 -> f" }, { "from": 991, "to": 994, "label": "EVAL-BACKTRACK" }, { "from": 992, "to": 995, "label": "BACKTRACK\nfor clause: neq(f, t)because of non-unification" }, { "from": 993, "to": 920, "label": "INSTANCE with matching:\nT224 -> T240\nX384 -> X427" }, { "from": 998, "to": 1000, "label": "CASE" }, { "from": 1000, "to": 1001, "label": "EVAL with clause\ngt(t, f).\nand substitutionT250 -> t,\nT249 -> f" }, { "from": 1000, "to": 1002, "label": "EVAL-BACKTRACK" }, { "from": 1001, "to": 1003, "label": "SPLIT 1" }, { "from": 1001, "to": 1004, "label": "SPLIT 2\nnew knowledge:\nT251 is ground\nT255 is ground\nreplacements:T253 -> T255,\nT254 -> T256" }, { "from": 1003, "to": 650, "label": "INSTANCE with matching:\nT53 -> T251\nT55 -> T253" }, { "from": 1004, "to": 1005, "label": "SPLIT 1" }, { "from": 1004, "to": 1006, "label": "SPLIT 2\nnew knowledge:\nT255 is ground\nT251 is ground\nT259 is ground\nreplacements:X14 -> T259" }, { "from": 1005, "to": 1007, "label": "CASE" }, { "from": 1006, "to": 1, "label": "INSTANCE with matching:\nT1 -> T259\nT2 -> T256" }, { "from": 1007, "to": 1008, "label": "BACKTRACK\nfor clause: del(X1, [], [])because of non-unification" }, { "from": 1008, "to": 1009, "label": "PARALLEL" }, { "from": 1008, "to": 1010, "label": "PARALLEL" }, { "from": 1009, "to": 1011, "label": "ONLY EVAL with clause\ndel(X461, .(X462, X463), X463) :- eq(X461, X462).\nand substitutionT255 -> T271,\nX461 -> T271,\nX462 -> f,\nT251 -> T272,\nX463 -> .(t, T272),\nX14 -> .(t, T272)" }, { "from": 1010, "to": 1017, "label": "ONLY EVAL with clause\ndel(X476, .(X477, X478), .(X477, X479)) :- ','(neq(X476, X477), del(X476, X478, X479)).\nand substitutionT255 -> T279,\nX476 -> T279,\nX477 -> f,\nT251 -> T280,\nX478 -> .(t, T280),\nX479 -> X480,\nX14 -> .(f, X480)" }, { "from": 1011, "to": 1012, "label": "CASE" }, { "from": 1012, "to": 1013, "label": "BACKTRACK\nfor clause: eq(t, t)because of non-unification" }, { "from": 1013, "to": 1014, "label": "EVAL with clause\neq(f, f).\nand substitutionT271 -> f" }, { "from": 1013, "to": 1015, "label": "EVAL-BACKTRACK" }, { "from": 1014, "to": 1016, "label": "SUCCESS" }, { "from": 1017, "to": 1018, "label": "CASE" }, { "from": 1018, "to": 1019, "label": "PARALLEL" }, { "from": 1018, "to": 1020, "label": "PARALLEL" }, { "from": 1019, "to": 1021, "label": "EVAL with clause\nneq(t, f).\nand substitutionT279 -> t" }, { "from": 1019, "to": 1022, "label": "EVAL-BACKTRACK" }, { "from": 1020, "to": 1023, "label": "BACKTRACK\nfor clause: neq(f, t)because of non-unification" }, { "from": 1021, "to": 920, "label": "INSTANCE with matching:\nT224 -> .(t, T280)\nX384 -> X480" } ], "type": "Graph" } } ---------------------------------------- (2) Obligation: Triples: maxC(.(t, X1), X2) :- maxC(X1, X2). maxC(.(f, X1), X2) :- maxC(X1, X2). delI(.(t, X1), .(t, X2)) :- delI(X1, X2). maxF(.(f, X1), X2) :- maxF(X1, X2). maxF(.(t, X1), X2) :- maxC(X1, X2). delJ(.(f, X1), .(f, X2)) :- delJ(X1, X2). maxsortA(.(X1, []), .(X1, X2)) :- ','(delcB(X1, X3), maxsortA(X3, X2)). maxsortA(.(t, .(t, X1)), .(X2, X3)) :- maxC(X1, X2). maxsortA(.(t, .(t, X1)), .(f, X2)) :- ','(maxcC(X1, f), delI(X1, X3)). maxsortA(.(t, .(t, X1)), .(X2, X3)) :- ','(maxcC(X1, X2), ','(delcD(X2, X1, X4), maxsortA(X4, X3))). maxsortA(.(t, .(f, X1)), .(X2, X3)) :- maxC(X1, X2). maxsortA(.(t, .(f, X1)), .(f, X2)) :- ','(maxcC(X1, f), delI(.(f, X1), X3)). maxsortA(.(t, .(f, X1)), .(X2, X3)) :- ','(maxcC(X1, X2), ','(delcE(X2, X1, X4), maxsortA(X4, X3))). maxsortA(.(f, .(f, X1)), .(X2, X3)) :- maxF(X1, X2). maxsortA(.(f, .(f, X1)), .(t, X2)) :- ','(maxcF(X1, t), delJ(X1, X3)). maxsortA(.(f, .(f, X1)), .(X2, X3)) :- ','(maxcF(X1, X2), ','(delcG(X2, X1, X4), maxsortA(X4, X3))). maxsortA(.(f, .(t, X1)), .(X2, X3)) :- maxC(X1, X2). maxsortA(.(f, .(t, X1)), .(t, X2)) :- ','(maxcC(X1, t), delJ(.(t, X1), X3)). maxsortA(.(f, .(t, X1)), .(X2, X3)) :- ','(maxcC(X1, X2), ','(delcH(X2, X1, X4), maxsortA(X4, X3))). Clauses: maxsortcA([], []). maxsortcA(.(X1, []), .(X1, X2)) :- ','(delcB(X1, X3), maxsortcA(X3, X2)). maxsortcA(.(t, .(t, X1)), .(X2, X3)) :- ','(maxcC(X1, X2), ','(delcD(X2, X1, X4), maxsortcA(X4, X3))). maxsortcA(.(t, .(f, X1)), .(X2, X3)) :- ','(maxcC(X1, X2), ','(delcE(X2, X1, X4), maxsortcA(X4, X3))). maxsortcA(.(f, .(f, X1)), .(X2, X3)) :- ','(maxcF(X1, X2), ','(delcG(X2, X1, X4), maxsortcA(X4, X3))). maxsortcA(.(f, .(t, X1)), .(X2, X3)) :- ','(maxcC(X1, X2), ','(delcH(X2, X1, X4), maxsortcA(X4, X3))). maxcC([], t). maxcC(.(t, X1), X2) :- maxcC(X1, X2). maxcC(.(f, X1), X2) :- maxcC(X1, X2). delcI([], []). delcI(.(f, X1), X1). delcI(.(t, X1), .(t, X2)) :- delcI(X1, X2). maxcF([], f). maxcF(.(f, X1), X2) :- maxcF(X1, X2). maxcF(.(t, X1), X2) :- maxcC(X1, X2). delcJ([], []). delcJ(.(t, X1), X1). delcJ(.(f, X1), .(f, X2)) :- delcJ(X1, X2). delcB(t, []). delcB(f, []). delcD(t, X1, .(t, X1)). delcD(f, X1, .(t, .(t, X2))) :- delcI(X1, X2). delcE(t, X1, .(f, X1)). delcE(f, X1, .(t, X2)) :- delcI(.(f, X1), X2). delcG(f, X1, .(f, X1)). delcG(t, X1, .(f, .(f, X2))) :- delcJ(X1, X2). delcH(f, X1, .(t, X1)). delcH(t, X1, .(f, X2)) :- delcJ(.(t, X1), X2). Afs: maxsortA(x1, x2) = maxsortA(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: maxsortA_in_2: (b,f) maxC_in_2: (b,f) maxcC_in_2: (b,b) (b,f) delI_in_2: (b,f) delcD_in_3: (b,b,f) delcI_in_2: (b,f) delcE_in_3: (b,b,f) maxF_in_2: (b,f) maxcF_in_2: (b,b) (b,f) delJ_in_2: (b,f) delcG_in_3: (b,b,f) delcJ_in_2: (b,f) delcH_in_3: (b,b,f) Transforming TRIPLES into the following Term Rewriting System: Pi DP problem: The TRS P consists of the following rules: MAXSORTA_IN_GA(.(X1, []), .(X1, X2)) -> U7_GA(X1, X2, delcB_in_ga(X1, X3)) U7_GA(X1, X2, delcB_out_ga(X1, X3)) -> U8_GA(X1, X2, maxsortA_in_ga(X3, X2)) U7_GA(X1, X2, delcB_out_ga(X1, X3)) -> MAXSORTA_IN_GA(X3, X2) MAXSORTA_IN_GA(.(t, .(t, X1)), .(X2, X3)) -> U9_GA(X1, X2, X3, maxC_in_ga(X1, X2)) MAXSORTA_IN_GA(.(t, .(t, X1)), .(X2, X3)) -> MAXC_IN_GA(X1, X2) MAXC_IN_GA(.(t, X1), X2) -> U1_GA(X1, X2, maxC_in_ga(X1, X2)) MAXC_IN_GA(.(t, X1), X2) -> MAXC_IN_GA(X1, X2) MAXC_IN_GA(.(f, X1), X2) -> U2_GA(X1, X2, maxC_in_ga(X1, X2)) MAXC_IN_GA(.(f, X1), X2) -> MAXC_IN_GA(X1, X2) MAXSORTA_IN_GA(.(t, .(t, X1)), .(f, X2)) -> U10_GA(X1, X2, maxcC_in_gg(X1, f)) U10_GA(X1, X2, maxcC_out_gg(X1, f)) -> U11_GA(X1, X2, delI_in_ga(X1, X3)) U10_GA(X1, X2, maxcC_out_gg(X1, f)) -> DELI_IN_GA(X1, X3) DELI_IN_GA(.(t, X1), .(t, X2)) -> U3_GA(X1, X2, delI_in_ga(X1, X2)) DELI_IN_GA(.(t, X1), .(t, X2)) -> DELI_IN_GA(X1, X2) MAXSORTA_IN_GA(.(t, .(t, X1)), .(X2, X3)) -> U12_GA(X1, X2, X3, maxcC_in_ga(X1, X2)) U12_GA(X1, X2, X3, maxcC_out_ga(X1, X2)) -> U13_GA(X1, X2, X3, delcD_in_gga(X2, X1, X4)) U13_GA(X1, X2, X3, delcD_out_gga(X2, X1, X4)) -> U14_GA(X1, X2, X3, maxsortA_in_ga(X4, X3)) U13_GA(X1, X2, X3, delcD_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4, X3) MAXSORTA_IN_GA(.(t, .(f, X1)), .(X2, X3)) -> U15_GA(X1, X2, X3, maxC_in_ga(X1, X2)) MAXSORTA_IN_GA(.(t, .(f, X1)), .(X2, X3)) -> MAXC_IN_GA(X1, X2) MAXSORTA_IN_GA(.(t, .(f, X1)), .(f, X2)) -> U16_GA(X1, X2, maxcC_in_gg(X1, f)) U16_GA(X1, X2, maxcC_out_gg(X1, f)) -> U17_GA(X1, X2, delI_in_ga(.(f, X1), X3)) U16_GA(X1, X2, maxcC_out_gg(X1, f)) -> DELI_IN_GA(.(f, X1), X3) MAXSORTA_IN_GA(.(t, .(f, X1)), .(X2, X3)) -> U18_GA(X1, X2, X3, maxcC_in_ga(X1, X2)) U18_GA(X1, X2, X3, maxcC_out_ga(X1, X2)) -> U19_GA(X1, X2, X3, delcE_in_gga(X2, X1, X4)) U19_GA(X1, X2, X3, delcE_out_gga(X2, X1, X4)) -> U20_GA(X1, X2, X3, maxsortA_in_ga(X4, X3)) U19_GA(X1, X2, X3, delcE_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4, X3) MAXSORTA_IN_GA(.(f, .(f, X1)), .(X2, X3)) -> U21_GA(X1, X2, X3, maxF_in_ga(X1, X2)) MAXSORTA_IN_GA(.(f, .(f, X1)), .(X2, X3)) -> MAXF_IN_GA(X1, X2) MAXF_IN_GA(.(f, X1), X2) -> U4_GA(X1, X2, maxF_in_ga(X1, X2)) MAXF_IN_GA(.(f, X1), X2) -> MAXF_IN_GA(X1, X2) MAXF_IN_GA(.(t, X1), X2) -> U5_GA(X1, X2, maxC_in_ga(X1, X2)) MAXF_IN_GA(.(t, X1), X2) -> MAXC_IN_GA(X1, X2) MAXSORTA_IN_GA(.(f, .(f, X1)), .(t, X2)) -> U22_GA(X1, X2, maxcF_in_gg(X1, t)) U22_GA(X1, X2, maxcF_out_gg(X1, t)) -> U23_GA(X1, X2, delJ_in_ga(X1, X3)) U22_GA(X1, X2, maxcF_out_gg(X1, t)) -> DELJ_IN_GA(X1, X3) DELJ_IN_GA(.(f, X1), .(f, X2)) -> U6_GA(X1, X2, delJ_in_ga(X1, X2)) DELJ_IN_GA(.(f, X1), .(f, X2)) -> DELJ_IN_GA(X1, X2) MAXSORTA_IN_GA(.(f, .(f, X1)), .(X2, X3)) -> U24_GA(X1, X2, X3, maxcF_in_ga(X1, X2)) U24_GA(X1, X2, X3, maxcF_out_ga(X1, X2)) -> U25_GA(X1, X2, X3, delcG_in_gga(X2, X1, X4)) U25_GA(X1, X2, X3, delcG_out_gga(X2, X1, X4)) -> U26_GA(X1, X2, X3, maxsortA_in_ga(X4, X3)) U25_GA(X1, X2, X3, delcG_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4, X3) MAXSORTA_IN_GA(.(f, .(t, X1)), .(X2, X3)) -> U27_GA(X1, X2, X3, maxC_in_ga(X1, X2)) MAXSORTA_IN_GA(.(f, .(t, X1)), .(X2, X3)) -> MAXC_IN_GA(X1, X2) MAXSORTA_IN_GA(.(f, .(t, X1)), .(t, X2)) -> U28_GA(X1, X2, maxcC_in_gg(X1, t)) U28_GA(X1, X2, maxcC_out_gg(X1, t)) -> U29_GA(X1, X2, delJ_in_ga(.(t, X1), X3)) U28_GA(X1, X2, maxcC_out_gg(X1, t)) -> DELJ_IN_GA(.(t, X1), X3) MAXSORTA_IN_GA(.(f, .(t, X1)), .(X2, X3)) -> U30_GA(X1, X2, X3, maxcC_in_ga(X1, X2)) U30_GA(X1, X2, X3, maxcC_out_ga(X1, X2)) -> U31_GA(X1, X2, X3, delcH_in_gga(X2, X1, X4)) U31_GA(X1, X2, X3, delcH_out_gga(X2, X1, X4)) -> U32_GA(X1, X2, X3, maxsortA_in_ga(X4, X3)) U31_GA(X1, X2, X3, delcH_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4, X3) The TRS R consists of the following rules: delcB_in_ga(t, []) -> delcB_out_ga(t, []) delcB_in_ga(f, []) -> delcB_out_ga(f, []) maxcC_in_gg([], t) -> maxcC_out_gg([], t) maxcC_in_gg(.(t, X1), X2) -> U48_gg(X1, X2, maxcC_in_gg(X1, X2)) maxcC_in_gg(.(f, X1), X2) -> U49_gg(X1, X2, maxcC_in_gg(X1, X2)) U49_gg(X1, X2, maxcC_out_gg(X1, X2)) -> maxcC_out_gg(.(f, X1), X2) U48_gg(X1, X2, maxcC_out_gg(X1, X2)) -> maxcC_out_gg(.(t, X1), X2) maxcC_in_ga([], t) -> maxcC_out_ga([], t) maxcC_in_ga(.(t, X1), X2) -> U48_ga(X1, X2, maxcC_in_ga(X1, X2)) maxcC_in_ga(.(f, X1), X2) -> U49_ga(X1, X2, maxcC_in_ga(X1, X2)) U49_ga(X1, X2, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(f, X1), X2) U48_ga(X1, X2, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(t, X1), X2) delcD_in_gga(t, X1, .(t, X1)) -> delcD_out_gga(t, X1, .(t, X1)) delcD_in_gga(f, X1, .(t, .(t, X2))) -> U54_gga(X1, X2, delcI_in_ga(X1, X2)) delcI_in_ga([], []) -> delcI_out_ga([], []) delcI_in_ga(.(f, X1), X1) -> delcI_out_ga(.(f, X1), X1) delcI_in_ga(.(t, X1), .(t, X2)) -> U50_ga(X1, X2, delcI_in_ga(X1, X2)) U50_ga(X1, X2, delcI_out_ga(X1, X2)) -> delcI_out_ga(.(t, X1), .(t, X2)) U54_gga(X1, X2, delcI_out_ga(X1, X2)) -> delcD_out_gga(f, X1, .(t, .(t, X2))) delcE_in_gga(t, X1, .(f, X1)) -> delcE_out_gga(t, X1, .(f, X1)) delcE_in_gga(f, X1, .(t, X2)) -> U55_gga(X1, X2, delcI_in_ga(.(f, X1), X2)) U55_gga(X1, X2, delcI_out_ga(.(f, X1), X2)) -> delcE_out_gga(f, X1, .(t, X2)) maxcF_in_gg([], f) -> maxcF_out_gg([], f) maxcF_in_gg(.(f, X1), X2) -> U51_gg(X1, X2, maxcF_in_gg(X1, X2)) maxcF_in_gg(.(t, X1), X2) -> U52_gg(X1, X2, maxcC_in_gg(X1, X2)) U52_gg(X1, X2, maxcC_out_gg(X1, X2)) -> maxcF_out_gg(.(t, X1), X2) U51_gg(X1, X2, maxcF_out_gg(X1, X2)) -> maxcF_out_gg(.(f, X1), X2) maxcF_in_ga([], f) -> maxcF_out_ga([], f) maxcF_in_ga(.(f, X1), X2) -> U51_ga(X1, X2, maxcF_in_ga(X1, X2)) maxcF_in_ga(.(t, X1), X2) -> U52_ga(X1, X2, maxcC_in_ga(X1, X2)) U52_ga(X1, X2, maxcC_out_ga(X1, X2)) -> maxcF_out_ga(.(t, X1), X2) U51_ga(X1, X2, maxcF_out_ga(X1, X2)) -> maxcF_out_ga(.(f, X1), X2) delcG_in_gga(f, X1, .(f, X1)) -> delcG_out_gga(f, X1, .(f, X1)) delcG_in_gga(t, X1, .(f, .(f, X2))) -> U56_gga(X1, X2, delcJ_in_ga(X1, X2)) delcJ_in_ga([], []) -> delcJ_out_ga([], []) delcJ_in_ga(.(t, X1), X1) -> delcJ_out_ga(.(t, X1), X1) delcJ_in_ga(.(f, X1), .(f, X2)) -> U53_ga(X1, X2, delcJ_in_ga(X1, X2)) U53_ga(X1, X2, delcJ_out_ga(X1, X2)) -> delcJ_out_ga(.(f, X1), .(f, X2)) U56_gga(X1, X2, delcJ_out_ga(X1, X2)) -> delcG_out_gga(t, X1, .(f, .(f, X2))) delcH_in_gga(f, X1, .(t, X1)) -> delcH_out_gga(f, X1, .(t, X1)) delcH_in_gga(t, X1, .(f, X2)) -> U57_gga(X1, X2, delcJ_in_ga(.(t, X1), X2)) U57_gga(X1, X2, delcJ_out_ga(.(t, X1), X2)) -> delcH_out_gga(t, X1, .(f, X2)) The argument filtering Pi contains the following mapping: maxsortA_in_ga(x1, x2) = maxsortA_in_ga(x1) .(x1, x2) = .(x1, x2) [] = [] delcB_in_ga(x1, x2) = delcB_in_ga(x1) t = t delcB_out_ga(x1, x2) = delcB_out_ga(x1, x2) f = f maxC_in_ga(x1, x2) = maxC_in_ga(x1) maxcC_in_gg(x1, x2) = maxcC_in_gg(x1, x2) maxcC_out_gg(x1, x2) = maxcC_out_gg(x1, x2) U48_gg(x1, x2, x3) = U48_gg(x1, x2, x3) U49_gg(x1, x2, x3) = U49_gg(x1, x2, x3) delI_in_ga(x1, x2) = delI_in_ga(x1) maxcC_in_ga(x1, x2) = maxcC_in_ga(x1) maxcC_out_ga(x1, x2) = maxcC_out_ga(x1, x2) U48_ga(x1, x2, x3) = U48_ga(x1, x3) U49_ga(x1, x2, x3) = U49_ga(x1, x3) delcD_in_gga(x1, x2, x3) = delcD_in_gga(x1, x2) delcD_out_gga(x1, x2, x3) = delcD_out_gga(x1, x2, x3) U54_gga(x1, x2, x3) = U54_gga(x1, x3) delcI_in_ga(x1, x2) = delcI_in_ga(x1) delcI_out_ga(x1, x2) = delcI_out_ga(x1, x2) U50_ga(x1, x2, x3) = U50_ga(x1, x3) delcE_in_gga(x1, x2, x3) = delcE_in_gga(x1, x2) delcE_out_gga(x1, x2, x3) = delcE_out_gga(x1, x2, x3) U55_gga(x1, x2, x3) = U55_gga(x1, x3) maxF_in_ga(x1, x2) = maxF_in_ga(x1) maxcF_in_gg(x1, x2) = maxcF_in_gg(x1, x2) maxcF_out_gg(x1, x2) = maxcF_out_gg(x1, x2) U51_gg(x1, x2, x3) = U51_gg(x1, x2, x3) U52_gg(x1, x2, x3) = U52_gg(x1, x2, x3) delJ_in_ga(x1, x2) = delJ_in_ga(x1) maxcF_in_ga(x1, x2) = maxcF_in_ga(x1) maxcF_out_ga(x1, x2) = maxcF_out_ga(x1, x2) U51_ga(x1, x2, x3) = U51_ga(x1, x3) U52_ga(x1, x2, x3) = U52_ga(x1, x3) delcG_in_gga(x1, x2, x3) = delcG_in_gga(x1, x2) delcG_out_gga(x1, x2, x3) = delcG_out_gga(x1, x2, x3) U56_gga(x1, x2, x3) = U56_gga(x1, x3) delcJ_in_ga(x1, x2) = delcJ_in_ga(x1) delcJ_out_ga(x1, x2) = delcJ_out_ga(x1, x2) U53_ga(x1, x2, x3) = U53_ga(x1, x3) delcH_in_gga(x1, x2, x3) = delcH_in_gga(x1, x2) delcH_out_gga(x1, x2, x3) = delcH_out_gga(x1, x2, x3) U57_gga(x1, x2, x3) = U57_gga(x1, x3) MAXSORTA_IN_GA(x1, x2) = MAXSORTA_IN_GA(x1) U7_GA(x1, x2, x3) = U7_GA(x1, x3) U8_GA(x1, x2, x3) = U8_GA(x1, x3) U9_GA(x1, x2, x3, x4) = U9_GA(x1, x4) MAXC_IN_GA(x1, x2) = MAXC_IN_GA(x1) U1_GA(x1, x2, x3) = U1_GA(x1, x3) U2_GA(x1, x2, x3) = U2_GA(x1, x3) U10_GA(x1, x2, x3) = U10_GA(x1, x3) U11_GA(x1, x2, x3) = U11_GA(x1, x3) DELI_IN_GA(x1, x2) = DELI_IN_GA(x1) U3_GA(x1, x2, x3) = U3_GA(x1, x3) U12_GA(x1, x2, x3, x4) = U12_GA(x1, x4) U13_GA(x1, x2, x3, x4) = U13_GA(x1, x4) U14_GA(x1, x2, x3, x4) = U14_GA(x1, x4) U15_GA(x1, x2, x3, x4) = U15_GA(x1, x4) U16_GA(x1, x2, x3) = U16_GA(x1, x3) U17_GA(x1, x2, x3) = U17_GA(x1, x3) U18_GA(x1, x2, x3, x4) = U18_GA(x1, x4) U19_GA(x1, x2, x3, x4) = U19_GA(x1, x4) U20_GA(x1, x2, x3, x4) = U20_GA(x1, x4) U21_GA(x1, x2, x3, x4) = U21_GA(x1, x4) MAXF_IN_GA(x1, x2) = MAXF_IN_GA(x1) U4_GA(x1, x2, x3) = U4_GA(x1, x3) U5_GA(x1, x2, x3) = U5_GA(x1, x3) U22_GA(x1, x2, x3) = U22_GA(x1, x3) U23_GA(x1, x2, x3) = U23_GA(x1, x3) DELJ_IN_GA(x1, x2) = DELJ_IN_GA(x1) U6_GA(x1, x2, x3) = U6_GA(x1, x3) U24_GA(x1, x2, x3, x4) = U24_GA(x1, x4) U25_GA(x1, x2, x3, x4) = U25_GA(x1, x4) U26_GA(x1, x2, x3, x4) = U26_GA(x1, x4) U27_GA(x1, x2, x3, x4) = U27_GA(x1, x4) U28_GA(x1, x2, x3) = U28_GA(x1, x3) U29_GA(x1, x2, x3) = U29_GA(x1, x3) U30_GA(x1, x2, x3, x4) = U30_GA(x1, x4) U31_GA(x1, x2, x3, x4) = U31_GA(x1, x4) U32_GA(x1, x2, x3, x4) = U32_GA(x1, x4) 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: MAXSORTA_IN_GA(.(X1, []), .(X1, X2)) -> U7_GA(X1, X2, delcB_in_ga(X1, X3)) U7_GA(X1, X2, delcB_out_ga(X1, X3)) -> U8_GA(X1, X2, maxsortA_in_ga(X3, X2)) U7_GA(X1, X2, delcB_out_ga(X1, X3)) -> MAXSORTA_IN_GA(X3, X2) MAXSORTA_IN_GA(.(t, .(t, X1)), .(X2, X3)) -> U9_GA(X1, X2, X3, maxC_in_ga(X1, X2)) MAXSORTA_IN_GA(.(t, .(t, X1)), .(X2, X3)) -> MAXC_IN_GA(X1, X2) MAXC_IN_GA(.(t, X1), X2) -> U1_GA(X1, X2, maxC_in_ga(X1, X2)) MAXC_IN_GA(.(t, X1), X2) -> MAXC_IN_GA(X1, X2) MAXC_IN_GA(.(f, X1), X2) -> U2_GA(X1, X2, maxC_in_ga(X1, X2)) MAXC_IN_GA(.(f, X1), X2) -> MAXC_IN_GA(X1, X2) MAXSORTA_IN_GA(.(t, .(t, X1)), .(f, X2)) -> U10_GA(X1, X2, maxcC_in_gg(X1, f)) U10_GA(X1, X2, maxcC_out_gg(X1, f)) -> U11_GA(X1, X2, delI_in_ga(X1, X3)) U10_GA(X1, X2, maxcC_out_gg(X1, f)) -> DELI_IN_GA(X1, X3) DELI_IN_GA(.(t, X1), .(t, X2)) -> U3_GA(X1, X2, delI_in_ga(X1, X2)) DELI_IN_GA(.(t, X1), .(t, X2)) -> DELI_IN_GA(X1, X2) MAXSORTA_IN_GA(.(t, .(t, X1)), .(X2, X3)) -> U12_GA(X1, X2, X3, maxcC_in_ga(X1, X2)) U12_GA(X1, X2, X3, maxcC_out_ga(X1, X2)) -> U13_GA(X1, X2, X3, delcD_in_gga(X2, X1, X4)) U13_GA(X1, X2, X3, delcD_out_gga(X2, X1, X4)) -> U14_GA(X1, X2, X3, maxsortA_in_ga(X4, X3)) U13_GA(X1, X2, X3, delcD_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4, X3) MAXSORTA_IN_GA(.(t, .(f, X1)), .(X2, X3)) -> U15_GA(X1, X2, X3, maxC_in_ga(X1, X2)) MAXSORTA_IN_GA(.(t, .(f, X1)), .(X2, X3)) -> MAXC_IN_GA(X1, X2) MAXSORTA_IN_GA(.(t, .(f, X1)), .(f, X2)) -> U16_GA(X1, X2, maxcC_in_gg(X1, f)) U16_GA(X1, X2, maxcC_out_gg(X1, f)) -> U17_GA(X1, X2, delI_in_ga(.(f, X1), X3)) U16_GA(X1, X2, maxcC_out_gg(X1, f)) -> DELI_IN_GA(.(f, X1), X3) MAXSORTA_IN_GA(.(t, .(f, X1)), .(X2, X3)) -> U18_GA(X1, X2, X3, maxcC_in_ga(X1, X2)) U18_GA(X1, X2, X3, maxcC_out_ga(X1, X2)) -> U19_GA(X1, X2, X3, delcE_in_gga(X2, X1, X4)) U19_GA(X1, X2, X3, delcE_out_gga(X2, X1, X4)) -> U20_GA(X1, X2, X3, maxsortA_in_ga(X4, X3)) U19_GA(X1, X2, X3, delcE_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4, X3) MAXSORTA_IN_GA(.(f, .(f, X1)), .(X2, X3)) -> U21_GA(X1, X2, X3, maxF_in_ga(X1, X2)) MAXSORTA_IN_GA(.(f, .(f, X1)), .(X2, X3)) -> MAXF_IN_GA(X1, X2) MAXF_IN_GA(.(f, X1), X2) -> U4_GA(X1, X2, maxF_in_ga(X1, X2)) MAXF_IN_GA(.(f, X1), X2) -> MAXF_IN_GA(X1, X2) MAXF_IN_GA(.(t, X1), X2) -> U5_GA(X1, X2, maxC_in_ga(X1, X2)) MAXF_IN_GA(.(t, X1), X2) -> MAXC_IN_GA(X1, X2) MAXSORTA_IN_GA(.(f, .(f, X1)), .(t, X2)) -> U22_GA(X1, X2, maxcF_in_gg(X1, t)) U22_GA(X1, X2, maxcF_out_gg(X1, t)) -> U23_GA(X1, X2, delJ_in_ga(X1, X3)) U22_GA(X1, X2, maxcF_out_gg(X1, t)) -> DELJ_IN_GA(X1, X3) DELJ_IN_GA(.(f, X1), .(f, X2)) -> U6_GA(X1, X2, delJ_in_ga(X1, X2)) DELJ_IN_GA(.(f, X1), .(f, X2)) -> DELJ_IN_GA(X1, X2) MAXSORTA_IN_GA(.(f, .(f, X1)), .(X2, X3)) -> U24_GA(X1, X2, X3, maxcF_in_ga(X1, X2)) U24_GA(X1, X2, X3, maxcF_out_ga(X1, X2)) -> U25_GA(X1, X2, X3, delcG_in_gga(X2, X1, X4)) U25_GA(X1, X2, X3, delcG_out_gga(X2, X1, X4)) -> U26_GA(X1, X2, X3, maxsortA_in_ga(X4, X3)) U25_GA(X1, X2, X3, delcG_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4, X3) MAXSORTA_IN_GA(.(f, .(t, X1)), .(X2, X3)) -> U27_GA(X1, X2, X3, maxC_in_ga(X1, X2)) MAXSORTA_IN_GA(.(f, .(t, X1)), .(X2, X3)) -> MAXC_IN_GA(X1, X2) MAXSORTA_IN_GA(.(f, .(t, X1)), .(t, X2)) -> U28_GA(X1, X2, maxcC_in_gg(X1, t)) U28_GA(X1, X2, maxcC_out_gg(X1, t)) -> U29_GA(X1, X2, delJ_in_ga(.(t, X1), X3)) U28_GA(X1, X2, maxcC_out_gg(X1, t)) -> DELJ_IN_GA(.(t, X1), X3) MAXSORTA_IN_GA(.(f, .(t, X1)), .(X2, X3)) -> U30_GA(X1, X2, X3, maxcC_in_ga(X1, X2)) U30_GA(X1, X2, X3, maxcC_out_ga(X1, X2)) -> U31_GA(X1, X2, X3, delcH_in_gga(X2, X1, X4)) U31_GA(X1, X2, X3, delcH_out_gga(X2, X1, X4)) -> U32_GA(X1, X2, X3, maxsortA_in_ga(X4, X3)) U31_GA(X1, X2, X3, delcH_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4, X3) The TRS R consists of the following rules: delcB_in_ga(t, []) -> delcB_out_ga(t, []) delcB_in_ga(f, []) -> delcB_out_ga(f, []) maxcC_in_gg([], t) -> maxcC_out_gg([], t) maxcC_in_gg(.(t, X1), X2) -> U48_gg(X1, X2, maxcC_in_gg(X1, X2)) maxcC_in_gg(.(f, X1), X2) -> U49_gg(X1, X2, maxcC_in_gg(X1, X2)) U49_gg(X1, X2, maxcC_out_gg(X1, X2)) -> maxcC_out_gg(.(f, X1), X2) U48_gg(X1, X2, maxcC_out_gg(X1, X2)) -> maxcC_out_gg(.(t, X1), X2) maxcC_in_ga([], t) -> maxcC_out_ga([], t) maxcC_in_ga(.(t, X1), X2) -> U48_ga(X1, X2, maxcC_in_ga(X1, X2)) maxcC_in_ga(.(f, X1), X2) -> U49_ga(X1, X2, maxcC_in_ga(X1, X2)) U49_ga(X1, X2, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(f, X1), X2) U48_ga(X1, X2, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(t, X1), X2) delcD_in_gga(t, X1, .(t, X1)) -> delcD_out_gga(t, X1, .(t, X1)) delcD_in_gga(f, X1, .(t, .(t, X2))) -> U54_gga(X1, X2, delcI_in_ga(X1, X2)) delcI_in_ga([], []) -> delcI_out_ga([], []) delcI_in_ga(.(f, X1), X1) -> delcI_out_ga(.(f, X1), X1) delcI_in_ga(.(t, X1), .(t, X2)) -> U50_ga(X1, X2, delcI_in_ga(X1, X2)) U50_ga(X1, X2, delcI_out_ga(X1, X2)) -> delcI_out_ga(.(t, X1), .(t, X2)) U54_gga(X1, X2, delcI_out_ga(X1, X2)) -> delcD_out_gga(f, X1, .(t, .(t, X2))) delcE_in_gga(t, X1, .(f, X1)) -> delcE_out_gga(t, X1, .(f, X1)) delcE_in_gga(f, X1, .(t, X2)) -> U55_gga(X1, X2, delcI_in_ga(.(f, X1), X2)) U55_gga(X1, X2, delcI_out_ga(.(f, X1), X2)) -> delcE_out_gga(f, X1, .(t, X2)) maxcF_in_gg([], f) -> maxcF_out_gg([], f) maxcF_in_gg(.(f, X1), X2) -> U51_gg(X1, X2, maxcF_in_gg(X1, X2)) maxcF_in_gg(.(t, X1), X2) -> U52_gg(X1, X2, maxcC_in_gg(X1, X2)) U52_gg(X1, X2, maxcC_out_gg(X1, X2)) -> maxcF_out_gg(.(t, X1), X2) U51_gg(X1, X2, maxcF_out_gg(X1, X2)) -> maxcF_out_gg(.(f, X1), X2) maxcF_in_ga([], f) -> maxcF_out_ga([], f) maxcF_in_ga(.(f, X1), X2) -> U51_ga(X1, X2, maxcF_in_ga(X1, X2)) maxcF_in_ga(.(t, X1), X2) -> U52_ga(X1, X2, maxcC_in_ga(X1, X2)) U52_ga(X1, X2, maxcC_out_ga(X1, X2)) -> maxcF_out_ga(.(t, X1), X2) U51_ga(X1, X2, maxcF_out_ga(X1, X2)) -> maxcF_out_ga(.(f, X1), X2) delcG_in_gga(f, X1, .(f, X1)) -> delcG_out_gga(f, X1, .(f, X1)) delcG_in_gga(t, X1, .(f, .(f, X2))) -> U56_gga(X1, X2, delcJ_in_ga(X1, X2)) delcJ_in_ga([], []) -> delcJ_out_ga([], []) delcJ_in_ga(.(t, X1), X1) -> delcJ_out_ga(.(t, X1), X1) delcJ_in_ga(.(f, X1), .(f, X2)) -> U53_ga(X1, X2, delcJ_in_ga(X1, X2)) U53_ga(X1, X2, delcJ_out_ga(X1, X2)) -> delcJ_out_ga(.(f, X1), .(f, X2)) U56_gga(X1, X2, delcJ_out_ga(X1, X2)) -> delcG_out_gga(t, X1, .(f, .(f, X2))) delcH_in_gga(f, X1, .(t, X1)) -> delcH_out_gga(f, X1, .(t, X1)) delcH_in_gga(t, X1, .(f, X2)) -> U57_gga(X1, X2, delcJ_in_ga(.(t, X1), X2)) U57_gga(X1, X2, delcJ_out_ga(.(t, X1), X2)) -> delcH_out_gga(t, X1, .(f, X2)) The argument filtering Pi contains the following mapping: maxsortA_in_ga(x1, x2) = maxsortA_in_ga(x1) .(x1, x2) = .(x1, x2) [] = [] delcB_in_ga(x1, x2) = delcB_in_ga(x1) t = t delcB_out_ga(x1, x2) = delcB_out_ga(x1, x2) f = f maxC_in_ga(x1, x2) = maxC_in_ga(x1) maxcC_in_gg(x1, x2) = maxcC_in_gg(x1, x2) maxcC_out_gg(x1, x2) = maxcC_out_gg(x1, x2) U48_gg(x1, x2, x3) = U48_gg(x1, x2, x3) U49_gg(x1, x2, x3) = U49_gg(x1, x2, x3) delI_in_ga(x1, x2) = delI_in_ga(x1) maxcC_in_ga(x1, x2) = maxcC_in_ga(x1) maxcC_out_ga(x1, x2) = maxcC_out_ga(x1, x2) U48_ga(x1, x2, x3) = U48_ga(x1, x3) U49_ga(x1, x2, x3) = U49_ga(x1, x3) delcD_in_gga(x1, x2, x3) = delcD_in_gga(x1, x2) delcD_out_gga(x1, x2, x3) = delcD_out_gga(x1, x2, x3) U54_gga(x1, x2, x3) = U54_gga(x1, x3) delcI_in_ga(x1, x2) = delcI_in_ga(x1) delcI_out_ga(x1, x2) = delcI_out_ga(x1, x2) U50_ga(x1, x2, x3) = U50_ga(x1, x3) delcE_in_gga(x1, x2, x3) = delcE_in_gga(x1, x2) delcE_out_gga(x1, x2, x3) = delcE_out_gga(x1, x2, x3) U55_gga(x1, x2, x3) = U55_gga(x1, x3) maxF_in_ga(x1, x2) = maxF_in_ga(x1) maxcF_in_gg(x1, x2) = maxcF_in_gg(x1, x2) maxcF_out_gg(x1, x2) = maxcF_out_gg(x1, x2) U51_gg(x1, x2, x3) = U51_gg(x1, x2, x3) U52_gg(x1, x2, x3) = U52_gg(x1, x2, x3) delJ_in_ga(x1, x2) = delJ_in_ga(x1) maxcF_in_ga(x1, x2) = maxcF_in_ga(x1) maxcF_out_ga(x1, x2) = maxcF_out_ga(x1, x2) U51_ga(x1, x2, x3) = U51_ga(x1, x3) U52_ga(x1, x2, x3) = U52_ga(x1, x3) delcG_in_gga(x1, x2, x3) = delcG_in_gga(x1, x2) delcG_out_gga(x1, x2, x3) = delcG_out_gga(x1, x2, x3) U56_gga(x1, x2, x3) = U56_gga(x1, x3) delcJ_in_ga(x1, x2) = delcJ_in_ga(x1) delcJ_out_ga(x1, x2) = delcJ_out_ga(x1, x2) U53_ga(x1, x2, x3) = U53_ga(x1, x3) delcH_in_gga(x1, x2, x3) = delcH_in_gga(x1, x2) delcH_out_gga(x1, x2, x3) = delcH_out_gga(x1, x2, x3) U57_gga(x1, x2, x3) = U57_gga(x1, x3) MAXSORTA_IN_GA(x1, x2) = MAXSORTA_IN_GA(x1) U7_GA(x1, x2, x3) = U7_GA(x1, x3) U8_GA(x1, x2, x3) = U8_GA(x1, x3) U9_GA(x1, x2, x3, x4) = U9_GA(x1, x4) MAXC_IN_GA(x1, x2) = MAXC_IN_GA(x1) U1_GA(x1, x2, x3) = U1_GA(x1, x3) U2_GA(x1, x2, x3) = U2_GA(x1, x3) U10_GA(x1, x2, x3) = U10_GA(x1, x3) U11_GA(x1, x2, x3) = U11_GA(x1, x3) DELI_IN_GA(x1, x2) = DELI_IN_GA(x1) U3_GA(x1, x2, x3) = U3_GA(x1, x3) U12_GA(x1, x2, x3, x4) = U12_GA(x1, x4) U13_GA(x1, x2, x3, x4) = U13_GA(x1, x4) U14_GA(x1, x2, x3, x4) = U14_GA(x1, x4) U15_GA(x1, x2, x3, x4) = U15_GA(x1, x4) U16_GA(x1, x2, x3) = U16_GA(x1, x3) U17_GA(x1, x2, x3) = U17_GA(x1, x3) U18_GA(x1, x2, x3, x4) = U18_GA(x1, x4) U19_GA(x1, x2, x3, x4) = U19_GA(x1, x4) U20_GA(x1, x2, x3, x4) = U20_GA(x1, x4) U21_GA(x1, x2, x3, x4) = U21_GA(x1, x4) MAXF_IN_GA(x1, x2) = MAXF_IN_GA(x1) U4_GA(x1, x2, x3) = U4_GA(x1, x3) U5_GA(x1, x2, x3) = U5_GA(x1, x3) U22_GA(x1, x2, x3) = U22_GA(x1, x3) U23_GA(x1, x2, x3) = U23_GA(x1, x3) DELJ_IN_GA(x1, x2) = DELJ_IN_GA(x1) U6_GA(x1, x2, x3) = U6_GA(x1, x3) U24_GA(x1, x2, x3, x4) = U24_GA(x1, x4) U25_GA(x1, x2, x3, x4) = U25_GA(x1, x4) U26_GA(x1, x2, x3, x4) = U26_GA(x1, x4) U27_GA(x1, x2, x3, x4) = U27_GA(x1, x4) U28_GA(x1, x2, x3) = U28_GA(x1, x3) U29_GA(x1, x2, x3) = U29_GA(x1, x3) U30_GA(x1, x2, x3, x4) = U30_GA(x1, x4) U31_GA(x1, x2, x3, x4) = U31_GA(x1, x4) U32_GA(x1, x2, x3, x4) = U32_GA(x1, x4) We have to consider all (P,R,Pi)-chains ---------------------------------------- (5) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 5 SCCs with 32 less nodes. ---------------------------------------- (6) Complex Obligation (AND) ---------------------------------------- (7) Obligation: Pi DP problem: The TRS P consists of the following rules: DELJ_IN_GA(.(f, X1), .(f, X2)) -> DELJ_IN_GA(X1, X2) The TRS R consists of the following rules: delcB_in_ga(t, []) -> delcB_out_ga(t, []) delcB_in_ga(f, []) -> delcB_out_ga(f, []) maxcC_in_gg([], t) -> maxcC_out_gg([], t) maxcC_in_gg(.(t, X1), X2) -> U48_gg(X1, X2, maxcC_in_gg(X1, X2)) maxcC_in_gg(.(f, X1), X2) -> U49_gg(X1, X2, maxcC_in_gg(X1, X2)) U49_gg(X1, X2, maxcC_out_gg(X1, X2)) -> maxcC_out_gg(.(f, X1), X2) U48_gg(X1, X2, maxcC_out_gg(X1, X2)) -> maxcC_out_gg(.(t, X1), X2) maxcC_in_ga([], t) -> maxcC_out_ga([], t) maxcC_in_ga(.(t, X1), X2) -> U48_ga(X1, X2, maxcC_in_ga(X1, X2)) maxcC_in_ga(.(f, X1), X2) -> U49_ga(X1, X2, maxcC_in_ga(X1, X2)) U49_ga(X1, X2, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(f, X1), X2) U48_ga(X1, X2, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(t, X1), X2) delcD_in_gga(t, X1, .(t, X1)) -> delcD_out_gga(t, X1, .(t, X1)) delcD_in_gga(f, X1, .(t, .(t, X2))) -> U54_gga(X1, X2, delcI_in_ga(X1, X2)) delcI_in_ga([], []) -> delcI_out_ga([], []) delcI_in_ga(.(f, X1), X1) -> delcI_out_ga(.(f, X1), X1) delcI_in_ga(.(t, X1), .(t, X2)) -> U50_ga(X1, X2, delcI_in_ga(X1, X2)) U50_ga(X1, X2, delcI_out_ga(X1, X2)) -> delcI_out_ga(.(t, X1), .(t, X2)) U54_gga(X1, X2, delcI_out_ga(X1, X2)) -> delcD_out_gga(f, X1, .(t, .(t, X2))) delcE_in_gga(t, X1, .(f, X1)) -> delcE_out_gga(t, X1, .(f, X1)) delcE_in_gga(f, X1, .(t, X2)) -> U55_gga(X1, X2, delcI_in_ga(.(f, X1), X2)) U55_gga(X1, X2, delcI_out_ga(.(f, X1), X2)) -> delcE_out_gga(f, X1, .(t, X2)) maxcF_in_gg([], f) -> maxcF_out_gg([], f) maxcF_in_gg(.(f, X1), X2) -> U51_gg(X1, X2, maxcF_in_gg(X1, X2)) maxcF_in_gg(.(t, X1), X2) -> U52_gg(X1, X2, maxcC_in_gg(X1, X2)) U52_gg(X1, X2, maxcC_out_gg(X1, X2)) -> maxcF_out_gg(.(t, X1), X2) U51_gg(X1, X2, maxcF_out_gg(X1, X2)) -> maxcF_out_gg(.(f, X1), X2) maxcF_in_ga([], f) -> maxcF_out_ga([], f) maxcF_in_ga(.(f, X1), X2) -> U51_ga(X1, X2, maxcF_in_ga(X1, X2)) maxcF_in_ga(.(t, X1), X2) -> U52_ga(X1, X2, maxcC_in_ga(X1, X2)) U52_ga(X1, X2, maxcC_out_ga(X1, X2)) -> maxcF_out_ga(.(t, X1), X2) U51_ga(X1, X2, maxcF_out_ga(X1, X2)) -> maxcF_out_ga(.(f, X1), X2) delcG_in_gga(f, X1, .(f, X1)) -> delcG_out_gga(f, X1, .(f, X1)) delcG_in_gga(t, X1, .(f, .(f, X2))) -> U56_gga(X1, X2, delcJ_in_ga(X1, X2)) delcJ_in_ga([], []) -> delcJ_out_ga([], []) delcJ_in_ga(.(t, X1), X1) -> delcJ_out_ga(.(t, X1), X1) delcJ_in_ga(.(f, X1), .(f, X2)) -> U53_ga(X1, X2, delcJ_in_ga(X1, X2)) U53_ga(X1, X2, delcJ_out_ga(X1, X2)) -> delcJ_out_ga(.(f, X1), .(f, X2)) U56_gga(X1, X2, delcJ_out_ga(X1, X2)) -> delcG_out_gga(t, X1, .(f, .(f, X2))) delcH_in_gga(f, X1, .(t, X1)) -> delcH_out_gga(f, X1, .(t, X1)) delcH_in_gga(t, X1, .(f, X2)) -> U57_gga(X1, X2, delcJ_in_ga(.(t, X1), X2)) U57_gga(X1, X2, delcJ_out_ga(.(t, X1), X2)) -> delcH_out_gga(t, X1, .(f, X2)) The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) [] = [] delcB_in_ga(x1, x2) = delcB_in_ga(x1) t = t delcB_out_ga(x1, x2) = delcB_out_ga(x1, x2) f = f maxcC_in_gg(x1, x2) = maxcC_in_gg(x1, x2) maxcC_out_gg(x1, x2) = maxcC_out_gg(x1, x2) U48_gg(x1, x2, x3) = U48_gg(x1, x2, x3) U49_gg(x1, x2, x3) = U49_gg(x1, x2, x3) maxcC_in_ga(x1, x2) = maxcC_in_ga(x1) maxcC_out_ga(x1, x2) = maxcC_out_ga(x1, x2) U48_ga(x1, x2, x3) = U48_ga(x1, x3) U49_ga(x1, x2, x3) = U49_ga(x1, x3) delcD_in_gga(x1, x2, x3) = delcD_in_gga(x1, x2) delcD_out_gga(x1, x2, x3) = delcD_out_gga(x1, x2, x3) U54_gga(x1, x2, x3) = U54_gga(x1, x3) delcI_in_ga(x1, x2) = delcI_in_ga(x1) delcI_out_ga(x1, x2) = delcI_out_ga(x1, x2) U50_ga(x1, x2, x3) = U50_ga(x1, x3) delcE_in_gga(x1, x2, x3) = delcE_in_gga(x1, x2) delcE_out_gga(x1, x2, x3) = delcE_out_gga(x1, x2, x3) U55_gga(x1, x2, x3) = U55_gga(x1, x3) maxcF_in_gg(x1, x2) = maxcF_in_gg(x1, x2) maxcF_out_gg(x1, x2) = maxcF_out_gg(x1, x2) U51_gg(x1, x2, x3) = U51_gg(x1, x2, x3) U52_gg(x1, x2, x3) = U52_gg(x1, x2, x3) maxcF_in_ga(x1, x2) = maxcF_in_ga(x1) maxcF_out_ga(x1, x2) = maxcF_out_ga(x1, x2) U51_ga(x1, x2, x3) = U51_ga(x1, x3) U52_ga(x1, x2, x3) = U52_ga(x1, x3) delcG_in_gga(x1, x2, x3) = delcG_in_gga(x1, x2) delcG_out_gga(x1, x2, x3) = delcG_out_gga(x1, x2, x3) U56_gga(x1, x2, x3) = U56_gga(x1, x3) delcJ_in_ga(x1, x2) = delcJ_in_ga(x1) delcJ_out_ga(x1, x2) = delcJ_out_ga(x1, x2) U53_ga(x1, x2, x3) = U53_ga(x1, x3) delcH_in_gga(x1, x2, x3) = delcH_in_gga(x1, x2) delcH_out_gga(x1, x2, x3) = delcH_out_gga(x1, x2, x3) U57_gga(x1, x2, x3) = U57_gga(x1, x3) DELJ_IN_GA(x1, x2) = DELJ_IN_GA(x1) 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: DELJ_IN_GA(.(f, X1), .(f, X2)) -> DELJ_IN_GA(X1, X2) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) f = f DELJ_IN_GA(x1, x2) = DELJ_IN_GA(x1) We have to consider all (P,R,Pi)-chains ---------------------------------------- (10) PiDPToQDPProof (SOUND) 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: DELJ_IN_GA(.(f, X1)) -> DELJ_IN_GA(X1) 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: *DELJ_IN_GA(.(f, X1)) -> DELJ_IN_GA(X1) The graph contains the following edges 1 > 1 ---------------------------------------- (13) YES ---------------------------------------- (14) Obligation: Pi DP problem: The TRS P consists of the following rules: DELI_IN_GA(.(t, X1), .(t, X2)) -> DELI_IN_GA(X1, X2) The TRS R consists of the following rules: delcB_in_ga(t, []) -> delcB_out_ga(t, []) delcB_in_ga(f, []) -> delcB_out_ga(f, []) maxcC_in_gg([], t) -> maxcC_out_gg([], t) maxcC_in_gg(.(t, X1), X2) -> U48_gg(X1, X2, maxcC_in_gg(X1, X2)) maxcC_in_gg(.(f, X1), X2) -> U49_gg(X1, X2, maxcC_in_gg(X1, X2)) U49_gg(X1, X2, maxcC_out_gg(X1, X2)) -> maxcC_out_gg(.(f, X1), X2) U48_gg(X1, X2, maxcC_out_gg(X1, X2)) -> maxcC_out_gg(.(t, X1), X2) maxcC_in_ga([], t) -> maxcC_out_ga([], t) maxcC_in_ga(.(t, X1), X2) -> U48_ga(X1, X2, maxcC_in_ga(X1, X2)) maxcC_in_ga(.(f, X1), X2) -> U49_ga(X1, X2, maxcC_in_ga(X1, X2)) U49_ga(X1, X2, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(f, X1), X2) U48_ga(X1, X2, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(t, X1), X2) delcD_in_gga(t, X1, .(t, X1)) -> delcD_out_gga(t, X1, .(t, X1)) delcD_in_gga(f, X1, .(t, .(t, X2))) -> U54_gga(X1, X2, delcI_in_ga(X1, X2)) delcI_in_ga([], []) -> delcI_out_ga([], []) delcI_in_ga(.(f, X1), X1) -> delcI_out_ga(.(f, X1), X1) delcI_in_ga(.(t, X1), .(t, X2)) -> U50_ga(X1, X2, delcI_in_ga(X1, X2)) U50_ga(X1, X2, delcI_out_ga(X1, X2)) -> delcI_out_ga(.(t, X1), .(t, X2)) U54_gga(X1, X2, delcI_out_ga(X1, X2)) -> delcD_out_gga(f, X1, .(t, .(t, X2))) delcE_in_gga(t, X1, .(f, X1)) -> delcE_out_gga(t, X1, .(f, X1)) delcE_in_gga(f, X1, .(t, X2)) -> U55_gga(X1, X2, delcI_in_ga(.(f, X1), X2)) U55_gga(X1, X2, delcI_out_ga(.(f, X1), X2)) -> delcE_out_gga(f, X1, .(t, X2)) maxcF_in_gg([], f) -> maxcF_out_gg([], f) maxcF_in_gg(.(f, X1), X2) -> U51_gg(X1, X2, maxcF_in_gg(X1, X2)) maxcF_in_gg(.(t, X1), X2) -> U52_gg(X1, X2, maxcC_in_gg(X1, X2)) U52_gg(X1, X2, maxcC_out_gg(X1, X2)) -> maxcF_out_gg(.(t, X1), X2) U51_gg(X1, X2, maxcF_out_gg(X1, X2)) -> maxcF_out_gg(.(f, X1), X2) maxcF_in_ga([], f) -> maxcF_out_ga([], f) maxcF_in_ga(.(f, X1), X2) -> U51_ga(X1, X2, maxcF_in_ga(X1, X2)) maxcF_in_ga(.(t, X1), X2) -> U52_ga(X1, X2, maxcC_in_ga(X1, X2)) U52_ga(X1, X2, maxcC_out_ga(X1, X2)) -> maxcF_out_ga(.(t, X1), X2) U51_ga(X1, X2, maxcF_out_ga(X1, X2)) -> maxcF_out_ga(.(f, X1), X2) delcG_in_gga(f, X1, .(f, X1)) -> delcG_out_gga(f, X1, .(f, X1)) delcG_in_gga(t, X1, .(f, .(f, X2))) -> U56_gga(X1, X2, delcJ_in_ga(X1, X2)) delcJ_in_ga([], []) -> delcJ_out_ga([], []) delcJ_in_ga(.(t, X1), X1) -> delcJ_out_ga(.(t, X1), X1) delcJ_in_ga(.(f, X1), .(f, X2)) -> U53_ga(X1, X2, delcJ_in_ga(X1, X2)) U53_ga(X1, X2, delcJ_out_ga(X1, X2)) -> delcJ_out_ga(.(f, X1), .(f, X2)) U56_gga(X1, X2, delcJ_out_ga(X1, X2)) -> delcG_out_gga(t, X1, .(f, .(f, X2))) delcH_in_gga(f, X1, .(t, X1)) -> delcH_out_gga(f, X1, .(t, X1)) delcH_in_gga(t, X1, .(f, X2)) -> U57_gga(X1, X2, delcJ_in_ga(.(t, X1), X2)) U57_gga(X1, X2, delcJ_out_ga(.(t, X1), X2)) -> delcH_out_gga(t, X1, .(f, X2)) The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) [] = [] delcB_in_ga(x1, x2) = delcB_in_ga(x1) t = t delcB_out_ga(x1, x2) = delcB_out_ga(x1, x2) f = f maxcC_in_gg(x1, x2) = maxcC_in_gg(x1, x2) maxcC_out_gg(x1, x2) = maxcC_out_gg(x1, x2) U48_gg(x1, x2, x3) = U48_gg(x1, x2, x3) U49_gg(x1, x2, x3) = U49_gg(x1, x2, x3) maxcC_in_ga(x1, x2) = maxcC_in_ga(x1) maxcC_out_ga(x1, x2) = maxcC_out_ga(x1, x2) U48_ga(x1, x2, x3) = U48_ga(x1, x3) U49_ga(x1, x2, x3) = U49_ga(x1, x3) delcD_in_gga(x1, x2, x3) = delcD_in_gga(x1, x2) delcD_out_gga(x1, x2, x3) = delcD_out_gga(x1, x2, x3) U54_gga(x1, x2, x3) = U54_gga(x1, x3) delcI_in_ga(x1, x2) = delcI_in_ga(x1) delcI_out_ga(x1, x2) = delcI_out_ga(x1, x2) U50_ga(x1, x2, x3) = U50_ga(x1, x3) delcE_in_gga(x1, x2, x3) = delcE_in_gga(x1, x2) delcE_out_gga(x1, x2, x3) = delcE_out_gga(x1, x2, x3) U55_gga(x1, x2, x3) = U55_gga(x1, x3) maxcF_in_gg(x1, x2) = maxcF_in_gg(x1, x2) maxcF_out_gg(x1, x2) = maxcF_out_gg(x1, x2) U51_gg(x1, x2, x3) = U51_gg(x1, x2, x3) U52_gg(x1, x2, x3) = U52_gg(x1, x2, x3) maxcF_in_ga(x1, x2) = maxcF_in_ga(x1) maxcF_out_ga(x1, x2) = maxcF_out_ga(x1, x2) U51_ga(x1, x2, x3) = U51_ga(x1, x3) U52_ga(x1, x2, x3) = U52_ga(x1, x3) delcG_in_gga(x1, x2, x3) = delcG_in_gga(x1, x2) delcG_out_gga(x1, x2, x3) = delcG_out_gga(x1, x2, x3) U56_gga(x1, x2, x3) = U56_gga(x1, x3) delcJ_in_ga(x1, x2) = delcJ_in_ga(x1) delcJ_out_ga(x1, x2) = delcJ_out_ga(x1, x2) U53_ga(x1, x2, x3) = U53_ga(x1, x3) delcH_in_gga(x1, x2, x3) = delcH_in_gga(x1, x2) delcH_out_gga(x1, x2, x3) = delcH_out_gga(x1, x2, x3) U57_gga(x1, x2, x3) = U57_gga(x1, x3) DELI_IN_GA(x1, x2) = DELI_IN_GA(x1) 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: DELI_IN_GA(.(t, X1), .(t, X2)) -> DELI_IN_GA(X1, X2) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) t = t DELI_IN_GA(x1, x2) = DELI_IN_GA(x1) We have to consider all (P,R,Pi)-chains ---------------------------------------- (17) PiDPToQDPProof (SOUND) 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: DELI_IN_GA(.(t, X1)) -> DELI_IN_GA(X1) 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: *DELI_IN_GA(.(t, X1)) -> DELI_IN_GA(X1) The graph contains the following edges 1 > 1 ---------------------------------------- (20) YES ---------------------------------------- (21) Obligation: Pi DP problem: The TRS P consists of the following rules: MAXC_IN_GA(.(f, X1), X2) -> MAXC_IN_GA(X1, X2) MAXC_IN_GA(.(t, X1), X2) -> MAXC_IN_GA(X1, X2) The TRS R consists of the following rules: delcB_in_ga(t, []) -> delcB_out_ga(t, []) delcB_in_ga(f, []) -> delcB_out_ga(f, []) maxcC_in_gg([], t) -> maxcC_out_gg([], t) maxcC_in_gg(.(t, X1), X2) -> U48_gg(X1, X2, maxcC_in_gg(X1, X2)) maxcC_in_gg(.(f, X1), X2) -> U49_gg(X1, X2, maxcC_in_gg(X1, X2)) U49_gg(X1, X2, maxcC_out_gg(X1, X2)) -> maxcC_out_gg(.(f, X1), X2) U48_gg(X1, X2, maxcC_out_gg(X1, X2)) -> maxcC_out_gg(.(t, X1), X2) maxcC_in_ga([], t) -> maxcC_out_ga([], t) maxcC_in_ga(.(t, X1), X2) -> U48_ga(X1, X2, maxcC_in_ga(X1, X2)) maxcC_in_ga(.(f, X1), X2) -> U49_ga(X1, X2, maxcC_in_ga(X1, X2)) U49_ga(X1, X2, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(f, X1), X2) U48_ga(X1, X2, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(t, X1), X2) delcD_in_gga(t, X1, .(t, X1)) -> delcD_out_gga(t, X1, .(t, X1)) delcD_in_gga(f, X1, .(t, .(t, X2))) -> U54_gga(X1, X2, delcI_in_ga(X1, X2)) delcI_in_ga([], []) -> delcI_out_ga([], []) delcI_in_ga(.(f, X1), X1) -> delcI_out_ga(.(f, X1), X1) delcI_in_ga(.(t, X1), .(t, X2)) -> U50_ga(X1, X2, delcI_in_ga(X1, X2)) U50_ga(X1, X2, delcI_out_ga(X1, X2)) -> delcI_out_ga(.(t, X1), .(t, X2)) U54_gga(X1, X2, delcI_out_ga(X1, X2)) -> delcD_out_gga(f, X1, .(t, .(t, X2))) delcE_in_gga(t, X1, .(f, X1)) -> delcE_out_gga(t, X1, .(f, X1)) delcE_in_gga(f, X1, .(t, X2)) -> U55_gga(X1, X2, delcI_in_ga(.(f, X1), X2)) U55_gga(X1, X2, delcI_out_ga(.(f, X1), X2)) -> delcE_out_gga(f, X1, .(t, X2)) maxcF_in_gg([], f) -> maxcF_out_gg([], f) maxcF_in_gg(.(f, X1), X2) -> U51_gg(X1, X2, maxcF_in_gg(X1, X2)) maxcF_in_gg(.(t, X1), X2) -> U52_gg(X1, X2, maxcC_in_gg(X1, X2)) U52_gg(X1, X2, maxcC_out_gg(X1, X2)) -> maxcF_out_gg(.(t, X1), X2) U51_gg(X1, X2, maxcF_out_gg(X1, X2)) -> maxcF_out_gg(.(f, X1), X2) maxcF_in_ga([], f) -> maxcF_out_ga([], f) maxcF_in_ga(.(f, X1), X2) -> U51_ga(X1, X2, maxcF_in_ga(X1, X2)) maxcF_in_ga(.(t, X1), X2) -> U52_ga(X1, X2, maxcC_in_ga(X1, X2)) U52_ga(X1, X2, maxcC_out_ga(X1, X2)) -> maxcF_out_ga(.(t, X1), X2) U51_ga(X1, X2, maxcF_out_ga(X1, X2)) -> maxcF_out_ga(.(f, X1), X2) delcG_in_gga(f, X1, .(f, X1)) -> delcG_out_gga(f, X1, .(f, X1)) delcG_in_gga(t, X1, .(f, .(f, X2))) -> U56_gga(X1, X2, delcJ_in_ga(X1, X2)) delcJ_in_ga([], []) -> delcJ_out_ga([], []) delcJ_in_ga(.(t, X1), X1) -> delcJ_out_ga(.(t, X1), X1) delcJ_in_ga(.(f, X1), .(f, X2)) -> U53_ga(X1, X2, delcJ_in_ga(X1, X2)) U53_ga(X1, X2, delcJ_out_ga(X1, X2)) -> delcJ_out_ga(.(f, X1), .(f, X2)) U56_gga(X1, X2, delcJ_out_ga(X1, X2)) -> delcG_out_gga(t, X1, .(f, .(f, X2))) delcH_in_gga(f, X1, .(t, X1)) -> delcH_out_gga(f, X1, .(t, X1)) delcH_in_gga(t, X1, .(f, X2)) -> U57_gga(X1, X2, delcJ_in_ga(.(t, X1), X2)) U57_gga(X1, X2, delcJ_out_ga(.(t, X1), X2)) -> delcH_out_gga(t, X1, .(f, X2)) The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) [] = [] delcB_in_ga(x1, x2) = delcB_in_ga(x1) t = t delcB_out_ga(x1, x2) = delcB_out_ga(x1, x2) f = f maxcC_in_gg(x1, x2) = maxcC_in_gg(x1, x2) maxcC_out_gg(x1, x2) = maxcC_out_gg(x1, x2) U48_gg(x1, x2, x3) = U48_gg(x1, x2, x3) U49_gg(x1, x2, x3) = U49_gg(x1, x2, x3) maxcC_in_ga(x1, x2) = maxcC_in_ga(x1) maxcC_out_ga(x1, x2) = maxcC_out_ga(x1, x2) U48_ga(x1, x2, x3) = U48_ga(x1, x3) U49_ga(x1, x2, x3) = U49_ga(x1, x3) delcD_in_gga(x1, x2, x3) = delcD_in_gga(x1, x2) delcD_out_gga(x1, x2, x3) = delcD_out_gga(x1, x2, x3) U54_gga(x1, x2, x3) = U54_gga(x1, x3) delcI_in_ga(x1, x2) = delcI_in_ga(x1) delcI_out_ga(x1, x2) = delcI_out_ga(x1, x2) U50_ga(x1, x2, x3) = U50_ga(x1, x3) delcE_in_gga(x1, x2, x3) = delcE_in_gga(x1, x2) delcE_out_gga(x1, x2, x3) = delcE_out_gga(x1, x2, x3) U55_gga(x1, x2, x3) = U55_gga(x1, x3) maxcF_in_gg(x1, x2) = maxcF_in_gg(x1, x2) maxcF_out_gg(x1, x2) = maxcF_out_gg(x1, x2) U51_gg(x1, x2, x3) = U51_gg(x1, x2, x3) U52_gg(x1, x2, x3) = U52_gg(x1, x2, x3) maxcF_in_ga(x1, x2) = maxcF_in_ga(x1) maxcF_out_ga(x1, x2) = maxcF_out_ga(x1, x2) U51_ga(x1, x2, x3) = U51_ga(x1, x3) U52_ga(x1, x2, x3) = U52_ga(x1, x3) delcG_in_gga(x1, x2, x3) = delcG_in_gga(x1, x2) delcG_out_gga(x1, x2, x3) = delcG_out_gga(x1, x2, x3) U56_gga(x1, x2, x3) = U56_gga(x1, x3) delcJ_in_ga(x1, x2) = delcJ_in_ga(x1) delcJ_out_ga(x1, x2) = delcJ_out_ga(x1, x2) U53_ga(x1, x2, x3) = U53_ga(x1, x3) delcH_in_gga(x1, x2, x3) = delcH_in_gga(x1, x2) delcH_out_gga(x1, x2, x3) = delcH_out_gga(x1, x2, x3) U57_gga(x1, x2, x3) = U57_gga(x1, x3) MAXC_IN_GA(x1, x2) = MAXC_IN_GA(x1) We have to consider all (P,R,Pi)-chains ---------------------------------------- (22) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (23) Obligation: Pi DP problem: The TRS P consists of the following rules: MAXC_IN_GA(.(f, X1), X2) -> MAXC_IN_GA(X1, X2) MAXC_IN_GA(.(t, X1), X2) -> MAXC_IN_GA(X1, X2) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) t = t f = f MAXC_IN_GA(x1, x2) = MAXC_IN_GA(x1) We have to consider all (P,R,Pi)-chains ---------------------------------------- (24) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (25) Obligation: Q DP problem: The TRS P consists of the following rules: MAXC_IN_GA(.(f, X1)) -> MAXC_IN_GA(X1) MAXC_IN_GA(.(t, X1)) -> MAXC_IN_GA(X1) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (26) 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: *MAXC_IN_GA(.(f, X1)) -> MAXC_IN_GA(X1) The graph contains the following edges 1 > 1 *MAXC_IN_GA(.(t, X1)) -> MAXC_IN_GA(X1) The graph contains the following edges 1 > 1 ---------------------------------------- (27) YES ---------------------------------------- (28) Obligation: Pi DP problem: The TRS P consists of the following rules: MAXF_IN_GA(.(f, X1), X2) -> MAXF_IN_GA(X1, X2) The TRS R consists of the following rules: delcB_in_ga(t, []) -> delcB_out_ga(t, []) delcB_in_ga(f, []) -> delcB_out_ga(f, []) maxcC_in_gg([], t) -> maxcC_out_gg([], t) maxcC_in_gg(.(t, X1), X2) -> U48_gg(X1, X2, maxcC_in_gg(X1, X2)) maxcC_in_gg(.(f, X1), X2) -> U49_gg(X1, X2, maxcC_in_gg(X1, X2)) U49_gg(X1, X2, maxcC_out_gg(X1, X2)) -> maxcC_out_gg(.(f, X1), X2) U48_gg(X1, X2, maxcC_out_gg(X1, X2)) -> maxcC_out_gg(.(t, X1), X2) maxcC_in_ga([], t) -> maxcC_out_ga([], t) maxcC_in_ga(.(t, X1), X2) -> U48_ga(X1, X2, maxcC_in_ga(X1, X2)) maxcC_in_ga(.(f, X1), X2) -> U49_ga(X1, X2, maxcC_in_ga(X1, X2)) U49_ga(X1, X2, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(f, X1), X2) U48_ga(X1, X2, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(t, X1), X2) delcD_in_gga(t, X1, .(t, X1)) -> delcD_out_gga(t, X1, .(t, X1)) delcD_in_gga(f, X1, .(t, .(t, X2))) -> U54_gga(X1, X2, delcI_in_ga(X1, X2)) delcI_in_ga([], []) -> delcI_out_ga([], []) delcI_in_ga(.(f, X1), X1) -> delcI_out_ga(.(f, X1), X1) delcI_in_ga(.(t, X1), .(t, X2)) -> U50_ga(X1, X2, delcI_in_ga(X1, X2)) U50_ga(X1, X2, delcI_out_ga(X1, X2)) -> delcI_out_ga(.(t, X1), .(t, X2)) U54_gga(X1, X2, delcI_out_ga(X1, X2)) -> delcD_out_gga(f, X1, .(t, .(t, X2))) delcE_in_gga(t, X1, .(f, X1)) -> delcE_out_gga(t, X1, .(f, X1)) delcE_in_gga(f, X1, .(t, X2)) -> U55_gga(X1, X2, delcI_in_ga(.(f, X1), X2)) U55_gga(X1, X2, delcI_out_ga(.(f, X1), X2)) -> delcE_out_gga(f, X1, .(t, X2)) maxcF_in_gg([], f) -> maxcF_out_gg([], f) maxcF_in_gg(.(f, X1), X2) -> U51_gg(X1, X2, maxcF_in_gg(X1, X2)) maxcF_in_gg(.(t, X1), X2) -> U52_gg(X1, X2, maxcC_in_gg(X1, X2)) U52_gg(X1, X2, maxcC_out_gg(X1, X2)) -> maxcF_out_gg(.(t, X1), X2) U51_gg(X1, X2, maxcF_out_gg(X1, X2)) -> maxcF_out_gg(.(f, X1), X2) maxcF_in_ga([], f) -> maxcF_out_ga([], f) maxcF_in_ga(.(f, X1), X2) -> U51_ga(X1, X2, maxcF_in_ga(X1, X2)) maxcF_in_ga(.(t, X1), X2) -> U52_ga(X1, X2, maxcC_in_ga(X1, X2)) U52_ga(X1, X2, maxcC_out_ga(X1, X2)) -> maxcF_out_ga(.(t, X1), X2) U51_ga(X1, X2, maxcF_out_ga(X1, X2)) -> maxcF_out_ga(.(f, X1), X2) delcG_in_gga(f, X1, .(f, X1)) -> delcG_out_gga(f, X1, .(f, X1)) delcG_in_gga(t, X1, .(f, .(f, X2))) -> U56_gga(X1, X2, delcJ_in_ga(X1, X2)) delcJ_in_ga([], []) -> delcJ_out_ga([], []) delcJ_in_ga(.(t, X1), X1) -> delcJ_out_ga(.(t, X1), X1) delcJ_in_ga(.(f, X1), .(f, X2)) -> U53_ga(X1, X2, delcJ_in_ga(X1, X2)) U53_ga(X1, X2, delcJ_out_ga(X1, X2)) -> delcJ_out_ga(.(f, X1), .(f, X2)) U56_gga(X1, X2, delcJ_out_ga(X1, X2)) -> delcG_out_gga(t, X1, .(f, .(f, X2))) delcH_in_gga(f, X1, .(t, X1)) -> delcH_out_gga(f, X1, .(t, X1)) delcH_in_gga(t, X1, .(f, X2)) -> U57_gga(X1, X2, delcJ_in_ga(.(t, X1), X2)) U57_gga(X1, X2, delcJ_out_ga(.(t, X1), X2)) -> delcH_out_gga(t, X1, .(f, X2)) The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) [] = [] delcB_in_ga(x1, x2) = delcB_in_ga(x1) t = t delcB_out_ga(x1, x2) = delcB_out_ga(x1, x2) f = f maxcC_in_gg(x1, x2) = maxcC_in_gg(x1, x2) maxcC_out_gg(x1, x2) = maxcC_out_gg(x1, x2) U48_gg(x1, x2, x3) = U48_gg(x1, x2, x3) U49_gg(x1, x2, x3) = U49_gg(x1, x2, x3) maxcC_in_ga(x1, x2) = maxcC_in_ga(x1) maxcC_out_ga(x1, x2) = maxcC_out_ga(x1, x2) U48_ga(x1, x2, x3) = U48_ga(x1, x3) U49_ga(x1, x2, x3) = U49_ga(x1, x3) delcD_in_gga(x1, x2, x3) = delcD_in_gga(x1, x2) delcD_out_gga(x1, x2, x3) = delcD_out_gga(x1, x2, x3) U54_gga(x1, x2, x3) = U54_gga(x1, x3) delcI_in_ga(x1, x2) = delcI_in_ga(x1) delcI_out_ga(x1, x2) = delcI_out_ga(x1, x2) U50_ga(x1, x2, x3) = U50_ga(x1, x3) delcE_in_gga(x1, x2, x3) = delcE_in_gga(x1, x2) delcE_out_gga(x1, x2, x3) = delcE_out_gga(x1, x2, x3) U55_gga(x1, x2, x3) = U55_gga(x1, x3) maxcF_in_gg(x1, x2) = maxcF_in_gg(x1, x2) maxcF_out_gg(x1, x2) = maxcF_out_gg(x1, x2) U51_gg(x1, x2, x3) = U51_gg(x1, x2, x3) U52_gg(x1, x2, x3) = U52_gg(x1, x2, x3) maxcF_in_ga(x1, x2) = maxcF_in_ga(x1) maxcF_out_ga(x1, x2) = maxcF_out_ga(x1, x2) U51_ga(x1, x2, x3) = U51_ga(x1, x3) U52_ga(x1, x2, x3) = U52_ga(x1, x3) delcG_in_gga(x1, x2, x3) = delcG_in_gga(x1, x2) delcG_out_gga(x1, x2, x3) = delcG_out_gga(x1, x2, x3) U56_gga(x1, x2, x3) = U56_gga(x1, x3) delcJ_in_ga(x1, x2) = delcJ_in_ga(x1) delcJ_out_ga(x1, x2) = delcJ_out_ga(x1, x2) U53_ga(x1, x2, x3) = U53_ga(x1, x3) delcH_in_gga(x1, x2, x3) = delcH_in_gga(x1, x2) delcH_out_gga(x1, x2, x3) = delcH_out_gga(x1, x2, x3) U57_gga(x1, x2, x3) = U57_gga(x1, x3) MAXF_IN_GA(x1, x2) = MAXF_IN_GA(x1) We have to consider all (P,R,Pi)-chains ---------------------------------------- (29) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (30) Obligation: Pi DP problem: The TRS P consists of the following rules: MAXF_IN_GA(.(f, X1), X2) -> MAXF_IN_GA(X1, X2) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) f = f MAXF_IN_GA(x1, x2) = MAXF_IN_GA(x1) We have to consider all (P,R,Pi)-chains ---------------------------------------- (31) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (32) Obligation: Q DP problem: The TRS P consists of the following rules: MAXF_IN_GA(.(f, X1)) -> MAXF_IN_GA(X1) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (33) 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: *MAXF_IN_GA(.(f, X1)) -> MAXF_IN_GA(X1) The graph contains the following edges 1 > 1 ---------------------------------------- (34) YES ---------------------------------------- (35) Obligation: Pi DP problem: The TRS P consists of the following rules: U7_GA(X1, X2, delcB_out_ga(X1, X3)) -> MAXSORTA_IN_GA(X3, X2) MAXSORTA_IN_GA(.(X1, []), .(X1, X2)) -> U7_GA(X1, X2, delcB_in_ga(X1, X3)) MAXSORTA_IN_GA(.(t, .(t, X1)), .(X2, X3)) -> U12_GA(X1, X2, X3, maxcC_in_ga(X1, X2)) U12_GA(X1, X2, X3, maxcC_out_ga(X1, X2)) -> U13_GA(X1, X2, X3, delcD_in_gga(X2, X1, X4)) U13_GA(X1, X2, X3, delcD_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4, X3) MAXSORTA_IN_GA(.(t, .(f, X1)), .(X2, X3)) -> U18_GA(X1, X2, X3, maxcC_in_ga(X1, X2)) U18_GA(X1, X2, X3, maxcC_out_ga(X1, X2)) -> U19_GA(X1, X2, X3, delcE_in_gga(X2, X1, X4)) U19_GA(X1, X2, X3, delcE_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4, X3) MAXSORTA_IN_GA(.(f, .(f, X1)), .(X2, X3)) -> U24_GA(X1, X2, X3, maxcF_in_ga(X1, X2)) U24_GA(X1, X2, X3, maxcF_out_ga(X1, X2)) -> U25_GA(X1, X2, X3, delcG_in_gga(X2, X1, X4)) U25_GA(X1, X2, X3, delcG_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4, X3) MAXSORTA_IN_GA(.(f, .(t, X1)), .(X2, X3)) -> U30_GA(X1, X2, X3, maxcC_in_ga(X1, X2)) U30_GA(X1, X2, X3, maxcC_out_ga(X1, X2)) -> U31_GA(X1, X2, X3, delcH_in_gga(X2, X1, X4)) U31_GA(X1, X2, X3, delcH_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4, X3) The TRS R consists of the following rules: delcB_in_ga(t, []) -> delcB_out_ga(t, []) delcB_in_ga(f, []) -> delcB_out_ga(f, []) maxcC_in_gg([], t) -> maxcC_out_gg([], t) maxcC_in_gg(.(t, X1), X2) -> U48_gg(X1, X2, maxcC_in_gg(X1, X2)) maxcC_in_gg(.(f, X1), X2) -> U49_gg(X1, X2, maxcC_in_gg(X1, X2)) U49_gg(X1, X2, maxcC_out_gg(X1, X2)) -> maxcC_out_gg(.(f, X1), X2) U48_gg(X1, X2, maxcC_out_gg(X1, X2)) -> maxcC_out_gg(.(t, X1), X2) maxcC_in_ga([], t) -> maxcC_out_ga([], t) maxcC_in_ga(.(t, X1), X2) -> U48_ga(X1, X2, maxcC_in_ga(X1, X2)) maxcC_in_ga(.(f, X1), X2) -> U49_ga(X1, X2, maxcC_in_ga(X1, X2)) U49_ga(X1, X2, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(f, X1), X2) U48_ga(X1, X2, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(t, X1), X2) delcD_in_gga(t, X1, .(t, X1)) -> delcD_out_gga(t, X1, .(t, X1)) delcD_in_gga(f, X1, .(t, .(t, X2))) -> U54_gga(X1, X2, delcI_in_ga(X1, X2)) delcI_in_ga([], []) -> delcI_out_ga([], []) delcI_in_ga(.(f, X1), X1) -> delcI_out_ga(.(f, X1), X1) delcI_in_ga(.(t, X1), .(t, X2)) -> U50_ga(X1, X2, delcI_in_ga(X1, X2)) U50_ga(X1, X2, delcI_out_ga(X1, X2)) -> delcI_out_ga(.(t, X1), .(t, X2)) U54_gga(X1, X2, delcI_out_ga(X1, X2)) -> delcD_out_gga(f, X1, .(t, .(t, X2))) delcE_in_gga(t, X1, .(f, X1)) -> delcE_out_gga(t, X1, .(f, X1)) delcE_in_gga(f, X1, .(t, X2)) -> U55_gga(X1, X2, delcI_in_ga(.(f, X1), X2)) U55_gga(X1, X2, delcI_out_ga(.(f, X1), X2)) -> delcE_out_gga(f, X1, .(t, X2)) maxcF_in_gg([], f) -> maxcF_out_gg([], f) maxcF_in_gg(.(f, X1), X2) -> U51_gg(X1, X2, maxcF_in_gg(X1, X2)) maxcF_in_gg(.(t, X1), X2) -> U52_gg(X1, X2, maxcC_in_gg(X1, X2)) U52_gg(X1, X2, maxcC_out_gg(X1, X2)) -> maxcF_out_gg(.(t, X1), X2) U51_gg(X1, X2, maxcF_out_gg(X1, X2)) -> maxcF_out_gg(.(f, X1), X2) maxcF_in_ga([], f) -> maxcF_out_ga([], f) maxcF_in_ga(.(f, X1), X2) -> U51_ga(X1, X2, maxcF_in_ga(X1, X2)) maxcF_in_ga(.(t, X1), X2) -> U52_ga(X1, X2, maxcC_in_ga(X1, X2)) U52_ga(X1, X2, maxcC_out_ga(X1, X2)) -> maxcF_out_ga(.(t, X1), X2) U51_ga(X1, X2, maxcF_out_ga(X1, X2)) -> maxcF_out_ga(.(f, X1), X2) delcG_in_gga(f, X1, .(f, X1)) -> delcG_out_gga(f, X1, .(f, X1)) delcG_in_gga(t, X1, .(f, .(f, X2))) -> U56_gga(X1, X2, delcJ_in_ga(X1, X2)) delcJ_in_ga([], []) -> delcJ_out_ga([], []) delcJ_in_ga(.(t, X1), X1) -> delcJ_out_ga(.(t, X1), X1) delcJ_in_ga(.(f, X1), .(f, X2)) -> U53_ga(X1, X2, delcJ_in_ga(X1, X2)) U53_ga(X1, X2, delcJ_out_ga(X1, X2)) -> delcJ_out_ga(.(f, X1), .(f, X2)) U56_gga(X1, X2, delcJ_out_ga(X1, X2)) -> delcG_out_gga(t, X1, .(f, .(f, X2))) delcH_in_gga(f, X1, .(t, X1)) -> delcH_out_gga(f, X1, .(t, X1)) delcH_in_gga(t, X1, .(f, X2)) -> U57_gga(X1, X2, delcJ_in_ga(.(t, X1), X2)) U57_gga(X1, X2, delcJ_out_ga(.(t, X1), X2)) -> delcH_out_gga(t, X1, .(f, X2)) The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) [] = [] delcB_in_ga(x1, x2) = delcB_in_ga(x1) t = t delcB_out_ga(x1, x2) = delcB_out_ga(x1, x2) f = f maxcC_in_gg(x1, x2) = maxcC_in_gg(x1, x2) maxcC_out_gg(x1, x2) = maxcC_out_gg(x1, x2) U48_gg(x1, x2, x3) = U48_gg(x1, x2, x3) U49_gg(x1, x2, x3) = U49_gg(x1, x2, x3) maxcC_in_ga(x1, x2) = maxcC_in_ga(x1) maxcC_out_ga(x1, x2) = maxcC_out_ga(x1, x2) U48_ga(x1, x2, x3) = U48_ga(x1, x3) U49_ga(x1, x2, x3) = U49_ga(x1, x3) delcD_in_gga(x1, x2, x3) = delcD_in_gga(x1, x2) delcD_out_gga(x1, x2, x3) = delcD_out_gga(x1, x2, x3) U54_gga(x1, x2, x3) = U54_gga(x1, x3) delcI_in_ga(x1, x2) = delcI_in_ga(x1) delcI_out_ga(x1, x2) = delcI_out_ga(x1, x2) U50_ga(x1, x2, x3) = U50_ga(x1, x3) delcE_in_gga(x1, x2, x3) = delcE_in_gga(x1, x2) delcE_out_gga(x1, x2, x3) = delcE_out_gga(x1, x2, x3) U55_gga(x1, x2, x3) = U55_gga(x1, x3) maxcF_in_gg(x1, x2) = maxcF_in_gg(x1, x2) maxcF_out_gg(x1, x2) = maxcF_out_gg(x1, x2) U51_gg(x1, x2, x3) = U51_gg(x1, x2, x3) U52_gg(x1, x2, x3) = U52_gg(x1, x2, x3) maxcF_in_ga(x1, x2) = maxcF_in_ga(x1) maxcF_out_ga(x1, x2) = maxcF_out_ga(x1, x2) U51_ga(x1, x2, x3) = U51_ga(x1, x3) U52_ga(x1, x2, x3) = U52_ga(x1, x3) delcG_in_gga(x1, x2, x3) = delcG_in_gga(x1, x2) delcG_out_gga(x1, x2, x3) = delcG_out_gga(x1, x2, x3) U56_gga(x1, x2, x3) = U56_gga(x1, x3) delcJ_in_ga(x1, x2) = delcJ_in_ga(x1) delcJ_out_ga(x1, x2) = delcJ_out_ga(x1, x2) U53_ga(x1, x2, x3) = U53_ga(x1, x3) delcH_in_gga(x1, x2, x3) = delcH_in_gga(x1, x2) delcH_out_gga(x1, x2, x3) = delcH_out_gga(x1, x2, x3) U57_gga(x1, x2, x3) = U57_gga(x1, x3) MAXSORTA_IN_GA(x1, x2) = MAXSORTA_IN_GA(x1) U7_GA(x1, x2, x3) = U7_GA(x1, x3) U12_GA(x1, x2, x3, x4) = U12_GA(x1, x4) U13_GA(x1, x2, x3, x4) = U13_GA(x1, x4) U18_GA(x1, x2, x3, x4) = U18_GA(x1, x4) U19_GA(x1, x2, x3, x4) = U19_GA(x1, x4) U24_GA(x1, x2, x3, x4) = U24_GA(x1, x4) U25_GA(x1, x2, x3, x4) = U25_GA(x1, x4) U30_GA(x1, x2, x3, x4) = U30_GA(x1, x4) U31_GA(x1, x2, x3, x4) = U31_GA(x1, x4) We have to consider all (P,R,Pi)-chains ---------------------------------------- (36) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (37) Obligation: Pi DP problem: The TRS P consists of the following rules: U7_GA(X1, X2, delcB_out_ga(X1, X3)) -> MAXSORTA_IN_GA(X3, X2) MAXSORTA_IN_GA(.(X1, []), .(X1, X2)) -> U7_GA(X1, X2, delcB_in_ga(X1, X3)) MAXSORTA_IN_GA(.(t, .(t, X1)), .(X2, X3)) -> U12_GA(X1, X2, X3, maxcC_in_ga(X1, X2)) U12_GA(X1, X2, X3, maxcC_out_ga(X1, X2)) -> U13_GA(X1, X2, X3, delcD_in_gga(X2, X1, X4)) U13_GA(X1, X2, X3, delcD_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4, X3) MAXSORTA_IN_GA(.(t, .(f, X1)), .(X2, X3)) -> U18_GA(X1, X2, X3, maxcC_in_ga(X1, X2)) U18_GA(X1, X2, X3, maxcC_out_ga(X1, X2)) -> U19_GA(X1, X2, X3, delcE_in_gga(X2, X1, X4)) U19_GA(X1, X2, X3, delcE_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4, X3) MAXSORTA_IN_GA(.(f, .(f, X1)), .(X2, X3)) -> U24_GA(X1, X2, X3, maxcF_in_ga(X1, X2)) U24_GA(X1, X2, X3, maxcF_out_ga(X1, X2)) -> U25_GA(X1, X2, X3, delcG_in_gga(X2, X1, X4)) U25_GA(X1, X2, X3, delcG_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4, X3) MAXSORTA_IN_GA(.(f, .(t, X1)), .(X2, X3)) -> U30_GA(X1, X2, X3, maxcC_in_ga(X1, X2)) U30_GA(X1, X2, X3, maxcC_out_ga(X1, X2)) -> U31_GA(X1, X2, X3, delcH_in_gga(X2, X1, X4)) U31_GA(X1, X2, X3, delcH_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4, X3) The TRS R consists of the following rules: delcB_in_ga(t, []) -> delcB_out_ga(t, []) delcB_in_ga(f, []) -> delcB_out_ga(f, []) maxcC_in_ga([], t) -> maxcC_out_ga([], t) maxcC_in_ga(.(t, X1), X2) -> U48_ga(X1, X2, maxcC_in_ga(X1, X2)) maxcC_in_ga(.(f, X1), X2) -> U49_ga(X1, X2, maxcC_in_ga(X1, X2)) delcD_in_gga(t, X1, .(t, X1)) -> delcD_out_gga(t, X1, .(t, X1)) delcD_in_gga(f, X1, .(t, .(t, X2))) -> U54_gga(X1, X2, delcI_in_ga(X1, X2)) delcE_in_gga(t, X1, .(f, X1)) -> delcE_out_gga(t, X1, .(f, X1)) delcE_in_gga(f, X1, .(t, X2)) -> U55_gga(X1, X2, delcI_in_ga(.(f, X1), X2)) maxcF_in_ga([], f) -> maxcF_out_ga([], f) maxcF_in_ga(.(f, X1), X2) -> U51_ga(X1, X2, maxcF_in_ga(X1, X2)) maxcF_in_ga(.(t, X1), X2) -> U52_ga(X1, X2, maxcC_in_ga(X1, X2)) delcG_in_gga(f, X1, .(f, X1)) -> delcG_out_gga(f, X1, .(f, X1)) delcG_in_gga(t, X1, .(f, .(f, X2))) -> U56_gga(X1, X2, delcJ_in_ga(X1, X2)) delcH_in_gga(f, X1, .(t, X1)) -> delcH_out_gga(f, X1, .(t, X1)) delcH_in_gga(t, X1, .(f, X2)) -> U57_gga(X1, X2, delcJ_in_ga(.(t, X1), X2)) U48_ga(X1, X2, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(t, X1), X2) U49_ga(X1, X2, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(f, X1), X2) U54_gga(X1, X2, delcI_out_ga(X1, X2)) -> delcD_out_gga(f, X1, .(t, .(t, X2))) U55_gga(X1, X2, delcI_out_ga(.(f, X1), X2)) -> delcE_out_gga(f, X1, .(t, X2)) U51_ga(X1, X2, maxcF_out_ga(X1, X2)) -> maxcF_out_ga(.(f, X1), X2) U52_ga(X1, X2, maxcC_out_ga(X1, X2)) -> maxcF_out_ga(.(t, X1), X2) U56_gga(X1, X2, delcJ_out_ga(X1, X2)) -> delcG_out_gga(t, X1, .(f, .(f, X2))) U57_gga(X1, X2, delcJ_out_ga(.(t, X1), X2)) -> delcH_out_gga(t, X1, .(f, X2)) delcI_in_ga([], []) -> delcI_out_ga([], []) delcI_in_ga(.(f, X1), X1) -> delcI_out_ga(.(f, X1), X1) delcI_in_ga(.(t, X1), .(t, X2)) -> U50_ga(X1, X2, delcI_in_ga(X1, X2)) delcJ_in_ga([], []) -> delcJ_out_ga([], []) delcJ_in_ga(.(t, X1), X1) -> delcJ_out_ga(.(t, X1), X1) delcJ_in_ga(.(f, X1), .(f, X2)) -> U53_ga(X1, X2, delcJ_in_ga(X1, X2)) U50_ga(X1, X2, delcI_out_ga(X1, X2)) -> delcI_out_ga(.(t, X1), .(t, X2)) U53_ga(X1, X2, delcJ_out_ga(X1, X2)) -> delcJ_out_ga(.(f, X1), .(f, X2)) The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) [] = [] delcB_in_ga(x1, x2) = delcB_in_ga(x1) t = t delcB_out_ga(x1, x2) = delcB_out_ga(x1, x2) f = f maxcC_in_ga(x1, x2) = maxcC_in_ga(x1) maxcC_out_ga(x1, x2) = maxcC_out_ga(x1, x2) U48_ga(x1, x2, x3) = U48_ga(x1, x3) U49_ga(x1, x2, x3) = U49_ga(x1, x3) delcD_in_gga(x1, x2, x3) = delcD_in_gga(x1, x2) delcD_out_gga(x1, x2, x3) = delcD_out_gga(x1, x2, x3) U54_gga(x1, x2, x3) = U54_gga(x1, x3) delcI_in_ga(x1, x2) = delcI_in_ga(x1) delcI_out_ga(x1, x2) = delcI_out_ga(x1, x2) U50_ga(x1, x2, x3) = U50_ga(x1, x3) delcE_in_gga(x1, x2, x3) = delcE_in_gga(x1, x2) delcE_out_gga(x1, x2, x3) = delcE_out_gga(x1, x2, x3) U55_gga(x1, x2, x3) = U55_gga(x1, x3) maxcF_in_ga(x1, x2) = maxcF_in_ga(x1) maxcF_out_ga(x1, x2) = maxcF_out_ga(x1, x2) U51_ga(x1, x2, x3) = U51_ga(x1, x3) U52_ga(x1, x2, x3) = U52_ga(x1, x3) delcG_in_gga(x1, x2, x3) = delcG_in_gga(x1, x2) delcG_out_gga(x1, x2, x3) = delcG_out_gga(x1, x2, x3) U56_gga(x1, x2, x3) = U56_gga(x1, x3) delcJ_in_ga(x1, x2) = delcJ_in_ga(x1) delcJ_out_ga(x1, x2) = delcJ_out_ga(x1, x2) U53_ga(x1, x2, x3) = U53_ga(x1, x3) delcH_in_gga(x1, x2, x3) = delcH_in_gga(x1, x2) delcH_out_gga(x1, x2, x3) = delcH_out_gga(x1, x2, x3) U57_gga(x1, x2, x3) = U57_gga(x1, x3) MAXSORTA_IN_GA(x1, x2) = MAXSORTA_IN_GA(x1) U7_GA(x1, x2, x3) = U7_GA(x1, x3) U12_GA(x1, x2, x3, x4) = U12_GA(x1, x4) U13_GA(x1, x2, x3, x4) = U13_GA(x1, x4) U18_GA(x1, x2, x3, x4) = U18_GA(x1, x4) U19_GA(x1, x2, x3, x4) = U19_GA(x1, x4) U24_GA(x1, x2, x3, x4) = U24_GA(x1, x4) U25_GA(x1, x2, x3, x4) = U25_GA(x1, x4) U30_GA(x1, x2, x3, x4) = U30_GA(x1, x4) U31_GA(x1, x2, x3, x4) = U31_GA(x1, x4) We have to consider all (P,R,Pi)-chains ---------------------------------------- (38) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (39) Obligation: Q DP problem: The TRS P consists of the following rules: U7_GA(X1, delcB_out_ga(X1, X3)) -> MAXSORTA_IN_GA(X3) MAXSORTA_IN_GA(.(X1, [])) -> U7_GA(X1, delcB_in_ga(X1)) MAXSORTA_IN_GA(.(t, .(t, X1))) -> U12_GA(X1, maxcC_in_ga(X1)) U12_GA(X1, maxcC_out_ga(X1, X2)) -> U13_GA(X1, delcD_in_gga(X2, X1)) U13_GA(X1, delcD_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) MAXSORTA_IN_GA(.(t, .(f, X1))) -> U18_GA(X1, maxcC_in_ga(X1)) U18_GA(X1, maxcC_out_ga(X1, X2)) -> U19_GA(X1, delcE_in_gga(X2, X1)) U19_GA(X1, delcE_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) MAXSORTA_IN_GA(.(f, .(f, X1))) -> U24_GA(X1, maxcF_in_ga(X1)) U24_GA(X1, maxcF_out_ga(X1, X2)) -> U25_GA(X1, delcG_in_gga(X2, X1)) U25_GA(X1, delcG_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) MAXSORTA_IN_GA(.(f, .(t, X1))) -> U30_GA(X1, maxcC_in_ga(X1)) U30_GA(X1, maxcC_out_ga(X1, X2)) -> U31_GA(X1, delcH_in_gga(X2, X1)) U31_GA(X1, delcH_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) The TRS R consists of the following rules: delcB_in_ga(t) -> delcB_out_ga(t, []) delcB_in_ga(f) -> delcB_out_ga(f, []) maxcC_in_ga([]) -> maxcC_out_ga([], t) maxcC_in_ga(.(t, X1)) -> U48_ga(X1, maxcC_in_ga(X1)) maxcC_in_ga(.(f, X1)) -> U49_ga(X1, maxcC_in_ga(X1)) delcD_in_gga(t, X1) -> delcD_out_gga(t, X1, .(t, X1)) delcD_in_gga(f, X1) -> U54_gga(X1, delcI_in_ga(X1)) delcE_in_gga(t, X1) -> delcE_out_gga(t, X1, .(f, X1)) delcE_in_gga(f, X1) -> U55_gga(X1, delcI_in_ga(.(f, X1))) maxcF_in_ga([]) -> maxcF_out_ga([], f) maxcF_in_ga(.(f, X1)) -> U51_ga(X1, maxcF_in_ga(X1)) maxcF_in_ga(.(t, X1)) -> U52_ga(X1, maxcC_in_ga(X1)) delcG_in_gga(f, X1) -> delcG_out_gga(f, X1, .(f, X1)) delcG_in_gga(t, X1) -> U56_gga(X1, delcJ_in_ga(X1)) delcH_in_gga(f, X1) -> delcH_out_gga(f, X1, .(t, X1)) delcH_in_gga(t, X1) -> U57_gga(X1, delcJ_in_ga(.(t, X1))) U48_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(t, X1), X2) U49_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(f, X1), X2) U54_gga(X1, delcI_out_ga(X1, X2)) -> delcD_out_gga(f, X1, .(t, .(t, X2))) U55_gga(X1, delcI_out_ga(.(f, X1), X2)) -> delcE_out_gga(f, X1, .(t, X2)) U51_ga(X1, maxcF_out_ga(X1, X2)) -> maxcF_out_ga(.(f, X1), X2) U52_ga(X1, maxcC_out_ga(X1, X2)) -> maxcF_out_ga(.(t, X1), X2) U56_gga(X1, delcJ_out_ga(X1, X2)) -> delcG_out_gga(t, X1, .(f, .(f, X2))) U57_gga(X1, delcJ_out_ga(.(t, X1), X2)) -> delcH_out_gga(t, X1, .(f, X2)) delcI_in_ga([]) -> delcI_out_ga([], []) delcI_in_ga(.(f, X1)) -> delcI_out_ga(.(f, X1), X1) delcI_in_ga(.(t, X1)) -> U50_ga(X1, delcI_in_ga(X1)) delcJ_in_ga([]) -> delcJ_out_ga([], []) delcJ_in_ga(.(t, X1)) -> delcJ_out_ga(.(t, X1), X1) delcJ_in_ga(.(f, X1)) -> U53_ga(X1, delcJ_in_ga(X1)) U50_ga(X1, delcI_out_ga(X1, X2)) -> delcI_out_ga(.(t, X1), .(t, X2)) U53_ga(X1, delcJ_out_ga(X1, X2)) -> delcJ_out_ga(.(f, X1), .(f, X2)) The set Q consists of the following terms: delcB_in_ga(x0) maxcC_in_ga(x0) delcD_in_gga(x0, x1) delcE_in_gga(x0, x1) maxcF_in_ga(x0) delcG_in_gga(x0, x1) delcH_in_gga(x0, x1) U48_ga(x0, x1) U49_ga(x0, x1) U54_gga(x0, x1) U55_gga(x0, x1) U51_ga(x0, x1) U52_ga(x0, x1) U56_gga(x0, x1) U57_gga(x0, x1) delcI_in_ga(x0) delcJ_in_ga(x0) U50_ga(x0, x1) U53_ga(x0, x1) We have to consider all (P,Q,R)-chains. ---------------------------------------- (40) TransformationProof (SOUND) By narrowing [LPAR04] the rule MAXSORTA_IN_GA(.(X1, [])) -> U7_GA(X1, delcB_in_ga(X1)) at position [1] we obtained the following new rules [LPAR04]: (MAXSORTA_IN_GA(.(t, [])) -> U7_GA(t, delcB_out_ga(t, [])),MAXSORTA_IN_GA(.(t, [])) -> U7_GA(t, delcB_out_ga(t, []))) (MAXSORTA_IN_GA(.(f, [])) -> U7_GA(f, delcB_out_ga(f, [])),MAXSORTA_IN_GA(.(f, [])) -> U7_GA(f, delcB_out_ga(f, []))) ---------------------------------------- (41) Obligation: Q DP problem: The TRS P consists of the following rules: U7_GA(X1, delcB_out_ga(X1, X3)) -> MAXSORTA_IN_GA(X3) MAXSORTA_IN_GA(.(t, .(t, X1))) -> U12_GA(X1, maxcC_in_ga(X1)) U12_GA(X1, maxcC_out_ga(X1, X2)) -> U13_GA(X1, delcD_in_gga(X2, X1)) U13_GA(X1, delcD_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) MAXSORTA_IN_GA(.(t, .(f, X1))) -> U18_GA(X1, maxcC_in_ga(X1)) U18_GA(X1, maxcC_out_ga(X1, X2)) -> U19_GA(X1, delcE_in_gga(X2, X1)) U19_GA(X1, delcE_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) MAXSORTA_IN_GA(.(f, .(f, X1))) -> U24_GA(X1, maxcF_in_ga(X1)) U24_GA(X1, maxcF_out_ga(X1, X2)) -> U25_GA(X1, delcG_in_gga(X2, X1)) U25_GA(X1, delcG_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) MAXSORTA_IN_GA(.(f, .(t, X1))) -> U30_GA(X1, maxcC_in_ga(X1)) U30_GA(X1, maxcC_out_ga(X1, X2)) -> U31_GA(X1, delcH_in_gga(X2, X1)) U31_GA(X1, delcH_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) MAXSORTA_IN_GA(.(t, [])) -> U7_GA(t, delcB_out_ga(t, [])) MAXSORTA_IN_GA(.(f, [])) -> U7_GA(f, delcB_out_ga(f, [])) The TRS R consists of the following rules: delcB_in_ga(t) -> delcB_out_ga(t, []) delcB_in_ga(f) -> delcB_out_ga(f, []) maxcC_in_ga([]) -> maxcC_out_ga([], t) maxcC_in_ga(.(t, X1)) -> U48_ga(X1, maxcC_in_ga(X1)) maxcC_in_ga(.(f, X1)) -> U49_ga(X1, maxcC_in_ga(X1)) delcD_in_gga(t, X1) -> delcD_out_gga(t, X1, .(t, X1)) delcD_in_gga(f, X1) -> U54_gga(X1, delcI_in_ga(X1)) delcE_in_gga(t, X1) -> delcE_out_gga(t, X1, .(f, X1)) delcE_in_gga(f, X1) -> U55_gga(X1, delcI_in_ga(.(f, X1))) maxcF_in_ga([]) -> maxcF_out_ga([], f) maxcF_in_ga(.(f, X1)) -> U51_ga(X1, maxcF_in_ga(X1)) maxcF_in_ga(.(t, X1)) -> U52_ga(X1, maxcC_in_ga(X1)) delcG_in_gga(f, X1) -> delcG_out_gga(f, X1, .(f, X1)) delcG_in_gga(t, X1) -> U56_gga(X1, delcJ_in_ga(X1)) delcH_in_gga(f, X1) -> delcH_out_gga(f, X1, .(t, X1)) delcH_in_gga(t, X1) -> U57_gga(X1, delcJ_in_ga(.(t, X1))) U48_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(t, X1), X2) U49_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(f, X1), X2) U54_gga(X1, delcI_out_ga(X1, X2)) -> delcD_out_gga(f, X1, .(t, .(t, X2))) U55_gga(X1, delcI_out_ga(.(f, X1), X2)) -> delcE_out_gga(f, X1, .(t, X2)) U51_ga(X1, maxcF_out_ga(X1, X2)) -> maxcF_out_ga(.(f, X1), X2) U52_ga(X1, maxcC_out_ga(X1, X2)) -> maxcF_out_ga(.(t, X1), X2) U56_gga(X1, delcJ_out_ga(X1, X2)) -> delcG_out_gga(t, X1, .(f, .(f, X2))) U57_gga(X1, delcJ_out_ga(.(t, X1), X2)) -> delcH_out_gga(t, X1, .(f, X2)) delcI_in_ga([]) -> delcI_out_ga([], []) delcI_in_ga(.(f, X1)) -> delcI_out_ga(.(f, X1), X1) delcI_in_ga(.(t, X1)) -> U50_ga(X1, delcI_in_ga(X1)) delcJ_in_ga([]) -> delcJ_out_ga([], []) delcJ_in_ga(.(t, X1)) -> delcJ_out_ga(.(t, X1), X1) delcJ_in_ga(.(f, X1)) -> U53_ga(X1, delcJ_in_ga(X1)) U50_ga(X1, delcI_out_ga(X1, X2)) -> delcI_out_ga(.(t, X1), .(t, X2)) U53_ga(X1, delcJ_out_ga(X1, X2)) -> delcJ_out_ga(.(f, X1), .(f, X2)) The set Q consists of the following terms: delcB_in_ga(x0) maxcC_in_ga(x0) delcD_in_gga(x0, x1) delcE_in_gga(x0, x1) maxcF_in_ga(x0) delcG_in_gga(x0, x1) delcH_in_gga(x0, x1) U48_ga(x0, x1) U49_ga(x0, x1) U54_gga(x0, x1) U55_gga(x0, x1) U51_ga(x0, x1) U52_ga(x0, x1) U56_gga(x0, x1) U57_gga(x0, x1) delcI_in_ga(x0) delcJ_in_ga(x0) U50_ga(x0, x1) U53_ga(x0, x1) We have to consider all (P,Q,R)-chains. ---------------------------------------- (42) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (43) Obligation: Q DP problem: The TRS P consists of the following rules: U7_GA(X1, delcB_out_ga(X1, X3)) -> MAXSORTA_IN_GA(X3) MAXSORTA_IN_GA(.(t, .(t, X1))) -> U12_GA(X1, maxcC_in_ga(X1)) U12_GA(X1, maxcC_out_ga(X1, X2)) -> U13_GA(X1, delcD_in_gga(X2, X1)) U13_GA(X1, delcD_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) MAXSORTA_IN_GA(.(t, .(f, X1))) -> U18_GA(X1, maxcC_in_ga(X1)) U18_GA(X1, maxcC_out_ga(X1, X2)) -> U19_GA(X1, delcE_in_gga(X2, X1)) U19_GA(X1, delcE_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) MAXSORTA_IN_GA(.(f, .(f, X1))) -> U24_GA(X1, maxcF_in_ga(X1)) U24_GA(X1, maxcF_out_ga(X1, X2)) -> U25_GA(X1, delcG_in_gga(X2, X1)) U25_GA(X1, delcG_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) MAXSORTA_IN_GA(.(f, .(t, X1))) -> U30_GA(X1, maxcC_in_ga(X1)) U30_GA(X1, maxcC_out_ga(X1, X2)) -> U31_GA(X1, delcH_in_gga(X2, X1)) U31_GA(X1, delcH_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) MAXSORTA_IN_GA(.(t, [])) -> U7_GA(t, delcB_out_ga(t, [])) MAXSORTA_IN_GA(.(f, [])) -> U7_GA(f, delcB_out_ga(f, [])) The TRS R consists of the following rules: delcH_in_gga(f, X1) -> delcH_out_gga(f, X1, .(t, X1)) delcH_in_gga(t, X1) -> U57_gga(X1, delcJ_in_ga(.(t, X1))) delcJ_in_ga(.(t, X1)) -> delcJ_out_ga(.(t, X1), X1) U57_gga(X1, delcJ_out_ga(.(t, X1), X2)) -> delcH_out_gga(t, X1, .(f, X2)) maxcC_in_ga([]) -> maxcC_out_ga([], t) maxcC_in_ga(.(t, X1)) -> U48_ga(X1, maxcC_in_ga(X1)) maxcC_in_ga(.(f, X1)) -> U49_ga(X1, maxcC_in_ga(X1)) U49_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(f, X1), X2) U48_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(t, X1), X2) delcG_in_gga(f, X1) -> delcG_out_gga(f, X1, .(f, X1)) delcG_in_gga(t, X1) -> U56_gga(X1, delcJ_in_ga(X1)) delcJ_in_ga([]) -> delcJ_out_ga([], []) delcJ_in_ga(.(f, X1)) -> U53_ga(X1, delcJ_in_ga(X1)) U56_gga(X1, delcJ_out_ga(X1, X2)) -> delcG_out_gga(t, X1, .(f, .(f, X2))) U53_ga(X1, delcJ_out_ga(X1, X2)) -> delcJ_out_ga(.(f, X1), .(f, X2)) maxcF_in_ga([]) -> maxcF_out_ga([], f) maxcF_in_ga(.(f, X1)) -> U51_ga(X1, maxcF_in_ga(X1)) maxcF_in_ga(.(t, X1)) -> U52_ga(X1, maxcC_in_ga(X1)) U52_ga(X1, maxcC_out_ga(X1, X2)) -> maxcF_out_ga(.(t, X1), X2) U51_ga(X1, maxcF_out_ga(X1, X2)) -> maxcF_out_ga(.(f, X1), X2) delcE_in_gga(t, X1) -> delcE_out_gga(t, X1, .(f, X1)) delcE_in_gga(f, X1) -> U55_gga(X1, delcI_in_ga(.(f, X1))) delcI_in_ga(.(f, X1)) -> delcI_out_ga(.(f, X1), X1) U55_gga(X1, delcI_out_ga(.(f, X1), X2)) -> delcE_out_gga(f, X1, .(t, X2)) delcD_in_gga(t, X1) -> delcD_out_gga(t, X1, .(t, X1)) delcD_in_gga(f, X1) -> U54_gga(X1, delcI_in_ga(X1)) delcI_in_ga([]) -> delcI_out_ga([], []) delcI_in_ga(.(t, X1)) -> U50_ga(X1, delcI_in_ga(X1)) U54_gga(X1, delcI_out_ga(X1, X2)) -> delcD_out_gga(f, X1, .(t, .(t, X2))) U50_ga(X1, delcI_out_ga(X1, X2)) -> delcI_out_ga(.(t, X1), .(t, X2)) The set Q consists of the following terms: delcB_in_ga(x0) maxcC_in_ga(x0) delcD_in_gga(x0, x1) delcE_in_gga(x0, x1) maxcF_in_ga(x0) delcG_in_gga(x0, x1) delcH_in_gga(x0, x1) U48_ga(x0, x1) U49_ga(x0, x1) U54_gga(x0, x1) U55_gga(x0, x1) U51_ga(x0, x1) U52_ga(x0, x1) U56_gga(x0, x1) U57_gga(x0, x1) delcI_in_ga(x0) delcJ_in_ga(x0) U50_ga(x0, x1) U53_ga(x0, x1) We have to consider all (P,Q,R)-chains. ---------------------------------------- (44) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. delcB_in_ga(x0) ---------------------------------------- (45) Obligation: Q DP problem: The TRS P consists of the following rules: U7_GA(X1, delcB_out_ga(X1, X3)) -> MAXSORTA_IN_GA(X3) MAXSORTA_IN_GA(.(t, .(t, X1))) -> U12_GA(X1, maxcC_in_ga(X1)) U12_GA(X1, maxcC_out_ga(X1, X2)) -> U13_GA(X1, delcD_in_gga(X2, X1)) U13_GA(X1, delcD_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) MAXSORTA_IN_GA(.(t, .(f, X1))) -> U18_GA(X1, maxcC_in_ga(X1)) U18_GA(X1, maxcC_out_ga(X1, X2)) -> U19_GA(X1, delcE_in_gga(X2, X1)) U19_GA(X1, delcE_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) MAXSORTA_IN_GA(.(f, .(f, X1))) -> U24_GA(X1, maxcF_in_ga(X1)) U24_GA(X1, maxcF_out_ga(X1, X2)) -> U25_GA(X1, delcG_in_gga(X2, X1)) U25_GA(X1, delcG_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) MAXSORTA_IN_GA(.(f, .(t, X1))) -> U30_GA(X1, maxcC_in_ga(X1)) U30_GA(X1, maxcC_out_ga(X1, X2)) -> U31_GA(X1, delcH_in_gga(X2, X1)) U31_GA(X1, delcH_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) MAXSORTA_IN_GA(.(t, [])) -> U7_GA(t, delcB_out_ga(t, [])) MAXSORTA_IN_GA(.(f, [])) -> U7_GA(f, delcB_out_ga(f, [])) The TRS R consists of the following rules: delcH_in_gga(f, X1) -> delcH_out_gga(f, X1, .(t, X1)) delcH_in_gga(t, X1) -> U57_gga(X1, delcJ_in_ga(.(t, X1))) delcJ_in_ga(.(t, X1)) -> delcJ_out_ga(.(t, X1), X1) U57_gga(X1, delcJ_out_ga(.(t, X1), X2)) -> delcH_out_gga(t, X1, .(f, X2)) maxcC_in_ga([]) -> maxcC_out_ga([], t) maxcC_in_ga(.(t, X1)) -> U48_ga(X1, maxcC_in_ga(X1)) maxcC_in_ga(.(f, X1)) -> U49_ga(X1, maxcC_in_ga(X1)) U49_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(f, X1), X2) U48_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(t, X1), X2) delcG_in_gga(f, X1) -> delcG_out_gga(f, X1, .(f, X1)) delcG_in_gga(t, X1) -> U56_gga(X1, delcJ_in_ga(X1)) delcJ_in_ga([]) -> delcJ_out_ga([], []) delcJ_in_ga(.(f, X1)) -> U53_ga(X1, delcJ_in_ga(X1)) U56_gga(X1, delcJ_out_ga(X1, X2)) -> delcG_out_gga(t, X1, .(f, .(f, X2))) U53_ga(X1, delcJ_out_ga(X1, X2)) -> delcJ_out_ga(.(f, X1), .(f, X2)) maxcF_in_ga([]) -> maxcF_out_ga([], f) maxcF_in_ga(.(f, X1)) -> U51_ga(X1, maxcF_in_ga(X1)) maxcF_in_ga(.(t, X1)) -> U52_ga(X1, maxcC_in_ga(X1)) U52_ga(X1, maxcC_out_ga(X1, X2)) -> maxcF_out_ga(.(t, X1), X2) U51_ga(X1, maxcF_out_ga(X1, X2)) -> maxcF_out_ga(.(f, X1), X2) delcE_in_gga(t, X1) -> delcE_out_gga(t, X1, .(f, X1)) delcE_in_gga(f, X1) -> U55_gga(X1, delcI_in_ga(.(f, X1))) delcI_in_ga(.(f, X1)) -> delcI_out_ga(.(f, X1), X1) U55_gga(X1, delcI_out_ga(.(f, X1), X2)) -> delcE_out_gga(f, X1, .(t, X2)) delcD_in_gga(t, X1) -> delcD_out_gga(t, X1, .(t, X1)) delcD_in_gga(f, X1) -> U54_gga(X1, delcI_in_ga(X1)) delcI_in_ga([]) -> delcI_out_ga([], []) delcI_in_ga(.(t, X1)) -> U50_ga(X1, delcI_in_ga(X1)) U54_gga(X1, delcI_out_ga(X1, X2)) -> delcD_out_gga(f, X1, .(t, .(t, X2))) U50_ga(X1, delcI_out_ga(X1, X2)) -> delcI_out_ga(.(t, X1), .(t, X2)) The set Q consists of the following terms: maxcC_in_ga(x0) delcD_in_gga(x0, x1) delcE_in_gga(x0, x1) maxcF_in_ga(x0) delcG_in_gga(x0, x1) delcH_in_gga(x0, x1) U48_ga(x0, x1) U49_ga(x0, x1) U54_gga(x0, x1) U55_gga(x0, x1) U51_ga(x0, x1) U52_ga(x0, x1) U56_gga(x0, x1) U57_gga(x0, x1) delcI_in_ga(x0) delcJ_in_ga(x0) U50_ga(x0, x1) U53_ga(x0, x1) We have to consider all (P,Q,R)-chains. ---------------------------------------- (46) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule U7_GA(X1, delcB_out_ga(X1, X3)) -> MAXSORTA_IN_GA(X3) we obtained the following new rules [LPAR04]: (U7_GA(t, delcB_out_ga(t, [])) -> MAXSORTA_IN_GA([]),U7_GA(t, delcB_out_ga(t, [])) -> MAXSORTA_IN_GA([])) (U7_GA(f, delcB_out_ga(f, [])) -> MAXSORTA_IN_GA([]),U7_GA(f, delcB_out_ga(f, [])) -> MAXSORTA_IN_GA([])) ---------------------------------------- (47) Obligation: Q DP problem: The TRS P consists of the following rules: MAXSORTA_IN_GA(.(t, .(t, X1))) -> U12_GA(X1, maxcC_in_ga(X1)) U12_GA(X1, maxcC_out_ga(X1, X2)) -> U13_GA(X1, delcD_in_gga(X2, X1)) U13_GA(X1, delcD_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) MAXSORTA_IN_GA(.(t, .(f, X1))) -> U18_GA(X1, maxcC_in_ga(X1)) U18_GA(X1, maxcC_out_ga(X1, X2)) -> U19_GA(X1, delcE_in_gga(X2, X1)) U19_GA(X1, delcE_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) MAXSORTA_IN_GA(.(f, .(f, X1))) -> U24_GA(X1, maxcF_in_ga(X1)) U24_GA(X1, maxcF_out_ga(X1, X2)) -> U25_GA(X1, delcG_in_gga(X2, X1)) U25_GA(X1, delcG_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) MAXSORTA_IN_GA(.(f, .(t, X1))) -> U30_GA(X1, maxcC_in_ga(X1)) U30_GA(X1, maxcC_out_ga(X1, X2)) -> U31_GA(X1, delcH_in_gga(X2, X1)) U31_GA(X1, delcH_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) MAXSORTA_IN_GA(.(t, [])) -> U7_GA(t, delcB_out_ga(t, [])) MAXSORTA_IN_GA(.(f, [])) -> U7_GA(f, delcB_out_ga(f, [])) U7_GA(t, delcB_out_ga(t, [])) -> MAXSORTA_IN_GA([]) U7_GA(f, delcB_out_ga(f, [])) -> MAXSORTA_IN_GA([]) The TRS R consists of the following rules: delcH_in_gga(f, X1) -> delcH_out_gga(f, X1, .(t, X1)) delcH_in_gga(t, X1) -> U57_gga(X1, delcJ_in_ga(.(t, X1))) delcJ_in_ga(.(t, X1)) -> delcJ_out_ga(.(t, X1), X1) U57_gga(X1, delcJ_out_ga(.(t, X1), X2)) -> delcH_out_gga(t, X1, .(f, X2)) maxcC_in_ga([]) -> maxcC_out_ga([], t) maxcC_in_ga(.(t, X1)) -> U48_ga(X1, maxcC_in_ga(X1)) maxcC_in_ga(.(f, X1)) -> U49_ga(X1, maxcC_in_ga(X1)) U49_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(f, X1), X2) U48_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(t, X1), X2) delcG_in_gga(f, X1) -> delcG_out_gga(f, X1, .(f, X1)) delcG_in_gga(t, X1) -> U56_gga(X1, delcJ_in_ga(X1)) delcJ_in_ga([]) -> delcJ_out_ga([], []) delcJ_in_ga(.(f, X1)) -> U53_ga(X1, delcJ_in_ga(X1)) U56_gga(X1, delcJ_out_ga(X1, X2)) -> delcG_out_gga(t, X1, .(f, .(f, X2))) U53_ga(X1, delcJ_out_ga(X1, X2)) -> delcJ_out_ga(.(f, X1), .(f, X2)) maxcF_in_ga([]) -> maxcF_out_ga([], f) maxcF_in_ga(.(f, X1)) -> U51_ga(X1, maxcF_in_ga(X1)) maxcF_in_ga(.(t, X1)) -> U52_ga(X1, maxcC_in_ga(X1)) U52_ga(X1, maxcC_out_ga(X1, X2)) -> maxcF_out_ga(.(t, X1), X2) U51_ga(X1, maxcF_out_ga(X1, X2)) -> maxcF_out_ga(.(f, X1), X2) delcE_in_gga(t, X1) -> delcE_out_gga(t, X1, .(f, X1)) delcE_in_gga(f, X1) -> U55_gga(X1, delcI_in_ga(.(f, X1))) delcI_in_ga(.(f, X1)) -> delcI_out_ga(.(f, X1), X1) U55_gga(X1, delcI_out_ga(.(f, X1), X2)) -> delcE_out_gga(f, X1, .(t, X2)) delcD_in_gga(t, X1) -> delcD_out_gga(t, X1, .(t, X1)) delcD_in_gga(f, X1) -> U54_gga(X1, delcI_in_ga(X1)) delcI_in_ga([]) -> delcI_out_ga([], []) delcI_in_ga(.(t, X1)) -> U50_ga(X1, delcI_in_ga(X1)) U54_gga(X1, delcI_out_ga(X1, X2)) -> delcD_out_gga(f, X1, .(t, .(t, X2))) U50_ga(X1, delcI_out_ga(X1, X2)) -> delcI_out_ga(.(t, X1), .(t, X2)) The set Q consists of the following terms: maxcC_in_ga(x0) delcD_in_gga(x0, x1) delcE_in_gga(x0, x1) maxcF_in_ga(x0) delcG_in_gga(x0, x1) delcH_in_gga(x0, x1) U48_ga(x0, x1) U49_ga(x0, x1) U54_gga(x0, x1) U55_gga(x0, x1) U51_ga(x0, x1) U52_ga(x0, x1) U56_gga(x0, x1) U57_gga(x0, x1) delcI_in_ga(x0) delcJ_in_ga(x0) U50_ga(x0, x1) U53_ga(x0, x1) We have to consider all (P,Q,R)-chains. ---------------------------------------- (48) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 4 less nodes. ---------------------------------------- (49) Obligation: Q DP problem: The TRS P consists of the following rules: U12_GA(X1, maxcC_out_ga(X1, X2)) -> U13_GA(X1, delcD_in_gga(X2, X1)) U13_GA(X1, delcD_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) MAXSORTA_IN_GA(.(t, .(t, X1))) -> U12_GA(X1, maxcC_in_ga(X1)) MAXSORTA_IN_GA(.(t, .(f, X1))) -> U18_GA(X1, maxcC_in_ga(X1)) U18_GA(X1, maxcC_out_ga(X1, X2)) -> U19_GA(X1, delcE_in_gga(X2, X1)) U19_GA(X1, delcE_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) MAXSORTA_IN_GA(.(f, .(f, X1))) -> U24_GA(X1, maxcF_in_ga(X1)) U24_GA(X1, maxcF_out_ga(X1, X2)) -> U25_GA(X1, delcG_in_gga(X2, X1)) U25_GA(X1, delcG_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) MAXSORTA_IN_GA(.(f, .(t, X1))) -> U30_GA(X1, maxcC_in_ga(X1)) U30_GA(X1, maxcC_out_ga(X1, X2)) -> U31_GA(X1, delcH_in_gga(X2, X1)) U31_GA(X1, delcH_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) The TRS R consists of the following rules: delcH_in_gga(f, X1) -> delcH_out_gga(f, X1, .(t, X1)) delcH_in_gga(t, X1) -> U57_gga(X1, delcJ_in_ga(.(t, X1))) delcJ_in_ga(.(t, X1)) -> delcJ_out_ga(.(t, X1), X1) U57_gga(X1, delcJ_out_ga(.(t, X1), X2)) -> delcH_out_gga(t, X1, .(f, X2)) maxcC_in_ga([]) -> maxcC_out_ga([], t) maxcC_in_ga(.(t, X1)) -> U48_ga(X1, maxcC_in_ga(X1)) maxcC_in_ga(.(f, X1)) -> U49_ga(X1, maxcC_in_ga(X1)) U49_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(f, X1), X2) U48_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(t, X1), X2) delcG_in_gga(f, X1) -> delcG_out_gga(f, X1, .(f, X1)) delcG_in_gga(t, X1) -> U56_gga(X1, delcJ_in_ga(X1)) delcJ_in_ga([]) -> delcJ_out_ga([], []) delcJ_in_ga(.(f, X1)) -> U53_ga(X1, delcJ_in_ga(X1)) U56_gga(X1, delcJ_out_ga(X1, X2)) -> delcG_out_gga(t, X1, .(f, .(f, X2))) U53_ga(X1, delcJ_out_ga(X1, X2)) -> delcJ_out_ga(.(f, X1), .(f, X2)) maxcF_in_ga([]) -> maxcF_out_ga([], f) maxcF_in_ga(.(f, X1)) -> U51_ga(X1, maxcF_in_ga(X1)) maxcF_in_ga(.(t, X1)) -> U52_ga(X1, maxcC_in_ga(X1)) U52_ga(X1, maxcC_out_ga(X1, X2)) -> maxcF_out_ga(.(t, X1), X2) U51_ga(X1, maxcF_out_ga(X1, X2)) -> maxcF_out_ga(.(f, X1), X2) delcE_in_gga(t, X1) -> delcE_out_gga(t, X1, .(f, X1)) delcE_in_gga(f, X1) -> U55_gga(X1, delcI_in_ga(.(f, X1))) delcI_in_ga(.(f, X1)) -> delcI_out_ga(.(f, X1), X1) U55_gga(X1, delcI_out_ga(.(f, X1), X2)) -> delcE_out_gga(f, X1, .(t, X2)) delcD_in_gga(t, X1) -> delcD_out_gga(t, X1, .(t, X1)) delcD_in_gga(f, X1) -> U54_gga(X1, delcI_in_ga(X1)) delcI_in_ga([]) -> delcI_out_ga([], []) delcI_in_ga(.(t, X1)) -> U50_ga(X1, delcI_in_ga(X1)) U54_gga(X1, delcI_out_ga(X1, X2)) -> delcD_out_gga(f, X1, .(t, .(t, X2))) U50_ga(X1, delcI_out_ga(X1, X2)) -> delcI_out_ga(.(t, X1), .(t, X2)) The set Q consists of the following terms: maxcC_in_ga(x0) delcD_in_gga(x0, x1) delcE_in_gga(x0, x1) maxcF_in_ga(x0) delcG_in_gga(x0, x1) delcH_in_gga(x0, x1) U48_ga(x0, x1) U49_ga(x0, x1) U54_gga(x0, x1) U55_gga(x0, x1) U51_ga(x0, x1) U52_ga(x0, x1) U56_gga(x0, x1) U57_gga(x0, x1) delcI_in_ga(x0) delcJ_in_ga(x0) U50_ga(x0, x1) U53_ga(x0, x1) We have to consider all (P,Q,R)-chains. ---------------------------------------- (50) QDPQMonotonicMRRProof (EQUIVALENT) By using the Q-monotonic rule removal processor with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented such that it always occurs at a strongly monotonic position in a (P,Q,R)-chain. Strictly oriented rules of the TRS R: delcE_in_gga(f, X1) -> U55_gga(X1, delcI_in_ga(.(f, X1))) U55_gga(X1, delcI_out_ga(.(f, X1), X2)) -> delcE_out_gga(f, X1, .(t, X2)) Used ordering: Polynomial interpretation [POLO]: POL(.(x_1, x_2)) = 0 POL(MAXSORTA_IN_GA(x_1)) = 0 POL(U12_GA(x_1, x_2)) = 0 POL(U13_GA(x_1, x_2)) = 0 POL(U18_GA(x_1, x_2)) = x_2 POL(U19_GA(x_1, x_2)) = 2*x_2 POL(U24_GA(x_1, x_2)) = 0 POL(U25_GA(x_1, x_2)) = 0 POL(U30_GA(x_1, x_2)) = 0 POL(U31_GA(x_1, x_2)) = 0 POL(U48_ga(x_1, x_2)) = 2*x_2 POL(U49_ga(x_1, x_2)) = x_2 POL(U50_ga(x_1, x_2)) = 0 POL(U51_ga(x_1, x_2)) = 0 POL(U52_ga(x_1, x_2)) = 0 POL(U53_ga(x_1, x_2)) = 0 POL(U54_gga(x_1, x_2)) = 0 POL(U55_gga(x_1, x_2)) = 1 POL(U56_gga(x_1, x_2)) = 0 POL(U57_gga(x_1, x_2)) = 0 POL([]) = 0 POL(delcD_in_gga(x_1, x_2)) = 0 POL(delcD_out_gga(x_1, x_2, x_3)) = 0 POL(delcE_in_gga(x_1, x_2)) = x_1 POL(delcE_out_gga(x_1, x_2, x_3)) = 0 POL(delcG_in_gga(x_1, x_2)) = 2*x_1 POL(delcG_out_gga(x_1, x_2, x_3)) = x_1 POL(delcH_in_gga(x_1, x_2)) = 0 POL(delcH_out_gga(x_1, x_2, x_3)) = 0 POL(delcI_in_ga(x_1)) = 0 POL(delcI_out_ga(x_1, x_2)) = 0 POL(delcJ_in_ga(x_1)) = 0 POL(delcJ_out_ga(x_1, x_2)) = 0 POL(f) = 2 POL(maxcC_in_ga(x_1)) = 0 POL(maxcC_out_ga(x_1, x_2)) = 2*x_2 POL(maxcF_in_ga(x_1)) = 1 POL(maxcF_out_ga(x_1, x_2)) = 0 POL(t) = 0 ---------------------------------------- (51) Obligation: Q DP problem: The TRS P consists of the following rules: U12_GA(X1, maxcC_out_ga(X1, X2)) -> U13_GA(X1, delcD_in_gga(X2, X1)) U13_GA(X1, delcD_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) MAXSORTA_IN_GA(.(t, .(t, X1))) -> U12_GA(X1, maxcC_in_ga(X1)) MAXSORTA_IN_GA(.(t, .(f, X1))) -> U18_GA(X1, maxcC_in_ga(X1)) U18_GA(X1, maxcC_out_ga(X1, X2)) -> U19_GA(X1, delcE_in_gga(X2, X1)) U19_GA(X1, delcE_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) MAXSORTA_IN_GA(.(f, .(f, X1))) -> U24_GA(X1, maxcF_in_ga(X1)) U24_GA(X1, maxcF_out_ga(X1, X2)) -> U25_GA(X1, delcG_in_gga(X2, X1)) U25_GA(X1, delcG_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) MAXSORTA_IN_GA(.(f, .(t, X1))) -> U30_GA(X1, maxcC_in_ga(X1)) U30_GA(X1, maxcC_out_ga(X1, X2)) -> U31_GA(X1, delcH_in_gga(X2, X1)) U31_GA(X1, delcH_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) The TRS R consists of the following rules: delcH_in_gga(f, X1) -> delcH_out_gga(f, X1, .(t, X1)) delcH_in_gga(t, X1) -> U57_gga(X1, delcJ_in_ga(.(t, X1))) delcJ_in_ga(.(t, X1)) -> delcJ_out_ga(.(t, X1), X1) U57_gga(X1, delcJ_out_ga(.(t, X1), X2)) -> delcH_out_gga(t, X1, .(f, X2)) maxcC_in_ga([]) -> maxcC_out_ga([], t) maxcC_in_ga(.(t, X1)) -> U48_ga(X1, maxcC_in_ga(X1)) maxcC_in_ga(.(f, X1)) -> U49_ga(X1, maxcC_in_ga(X1)) U49_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(f, X1), X2) U48_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(t, X1), X2) delcG_in_gga(f, X1) -> delcG_out_gga(f, X1, .(f, X1)) delcG_in_gga(t, X1) -> U56_gga(X1, delcJ_in_ga(X1)) delcJ_in_ga([]) -> delcJ_out_ga([], []) delcJ_in_ga(.(f, X1)) -> U53_ga(X1, delcJ_in_ga(X1)) U56_gga(X1, delcJ_out_ga(X1, X2)) -> delcG_out_gga(t, X1, .(f, .(f, X2))) U53_ga(X1, delcJ_out_ga(X1, X2)) -> delcJ_out_ga(.(f, X1), .(f, X2)) maxcF_in_ga([]) -> maxcF_out_ga([], f) maxcF_in_ga(.(f, X1)) -> U51_ga(X1, maxcF_in_ga(X1)) maxcF_in_ga(.(t, X1)) -> U52_ga(X1, maxcC_in_ga(X1)) U52_ga(X1, maxcC_out_ga(X1, X2)) -> maxcF_out_ga(.(t, X1), X2) U51_ga(X1, maxcF_out_ga(X1, X2)) -> maxcF_out_ga(.(f, X1), X2) delcE_in_gga(t, X1) -> delcE_out_gga(t, X1, .(f, X1)) delcI_in_ga(.(f, X1)) -> delcI_out_ga(.(f, X1), X1) delcD_in_gga(t, X1) -> delcD_out_gga(t, X1, .(t, X1)) delcD_in_gga(f, X1) -> U54_gga(X1, delcI_in_ga(X1)) delcI_in_ga([]) -> delcI_out_ga([], []) delcI_in_ga(.(t, X1)) -> U50_ga(X1, delcI_in_ga(X1)) U54_gga(X1, delcI_out_ga(X1, X2)) -> delcD_out_gga(f, X1, .(t, .(t, X2))) U50_ga(X1, delcI_out_ga(X1, X2)) -> delcI_out_ga(.(t, X1), .(t, X2)) The set Q consists of the following terms: maxcC_in_ga(x0) delcD_in_gga(x0, x1) delcE_in_gga(x0, x1) maxcF_in_ga(x0) delcG_in_gga(x0, x1) delcH_in_gga(x0, x1) U48_ga(x0, x1) U49_ga(x0, x1) U54_gga(x0, x1) U55_gga(x0, x1) U51_ga(x0, x1) U52_ga(x0, x1) U56_gga(x0, x1) U57_gga(x0, x1) delcI_in_ga(x0) delcJ_in_ga(x0) U50_ga(x0, x1) U53_ga(x0, x1) We have to consider all (P,Q,R)-chains. ---------------------------------------- (52) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. U55_gga(x0, x1) ---------------------------------------- (53) Obligation: Q DP problem: The TRS P consists of the following rules: U12_GA(X1, maxcC_out_ga(X1, X2)) -> U13_GA(X1, delcD_in_gga(X2, X1)) U13_GA(X1, delcD_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) MAXSORTA_IN_GA(.(t, .(t, X1))) -> U12_GA(X1, maxcC_in_ga(X1)) MAXSORTA_IN_GA(.(t, .(f, X1))) -> U18_GA(X1, maxcC_in_ga(X1)) U18_GA(X1, maxcC_out_ga(X1, X2)) -> U19_GA(X1, delcE_in_gga(X2, X1)) U19_GA(X1, delcE_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) MAXSORTA_IN_GA(.(f, .(f, X1))) -> U24_GA(X1, maxcF_in_ga(X1)) U24_GA(X1, maxcF_out_ga(X1, X2)) -> U25_GA(X1, delcG_in_gga(X2, X1)) U25_GA(X1, delcG_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) MAXSORTA_IN_GA(.(f, .(t, X1))) -> U30_GA(X1, maxcC_in_ga(X1)) U30_GA(X1, maxcC_out_ga(X1, X2)) -> U31_GA(X1, delcH_in_gga(X2, X1)) U31_GA(X1, delcH_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) The TRS R consists of the following rules: delcH_in_gga(f, X1) -> delcH_out_gga(f, X1, .(t, X1)) delcH_in_gga(t, X1) -> U57_gga(X1, delcJ_in_ga(.(t, X1))) delcJ_in_ga(.(t, X1)) -> delcJ_out_ga(.(t, X1), X1) U57_gga(X1, delcJ_out_ga(.(t, X1), X2)) -> delcH_out_gga(t, X1, .(f, X2)) maxcC_in_ga([]) -> maxcC_out_ga([], t) maxcC_in_ga(.(t, X1)) -> U48_ga(X1, maxcC_in_ga(X1)) maxcC_in_ga(.(f, X1)) -> U49_ga(X1, maxcC_in_ga(X1)) U49_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(f, X1), X2) U48_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(t, X1), X2) delcG_in_gga(f, X1) -> delcG_out_gga(f, X1, .(f, X1)) delcG_in_gga(t, X1) -> U56_gga(X1, delcJ_in_ga(X1)) delcJ_in_ga([]) -> delcJ_out_ga([], []) delcJ_in_ga(.(f, X1)) -> U53_ga(X1, delcJ_in_ga(X1)) U56_gga(X1, delcJ_out_ga(X1, X2)) -> delcG_out_gga(t, X1, .(f, .(f, X2))) U53_ga(X1, delcJ_out_ga(X1, X2)) -> delcJ_out_ga(.(f, X1), .(f, X2)) maxcF_in_ga([]) -> maxcF_out_ga([], f) maxcF_in_ga(.(f, X1)) -> U51_ga(X1, maxcF_in_ga(X1)) maxcF_in_ga(.(t, X1)) -> U52_ga(X1, maxcC_in_ga(X1)) U52_ga(X1, maxcC_out_ga(X1, X2)) -> maxcF_out_ga(.(t, X1), X2) U51_ga(X1, maxcF_out_ga(X1, X2)) -> maxcF_out_ga(.(f, X1), X2) delcE_in_gga(t, X1) -> delcE_out_gga(t, X1, .(f, X1)) delcI_in_ga(.(f, X1)) -> delcI_out_ga(.(f, X1), X1) delcD_in_gga(t, X1) -> delcD_out_gga(t, X1, .(t, X1)) delcD_in_gga(f, X1) -> U54_gga(X1, delcI_in_ga(X1)) delcI_in_ga([]) -> delcI_out_ga([], []) delcI_in_ga(.(t, X1)) -> U50_ga(X1, delcI_in_ga(X1)) U54_gga(X1, delcI_out_ga(X1, X2)) -> delcD_out_gga(f, X1, .(t, .(t, X2))) U50_ga(X1, delcI_out_ga(X1, X2)) -> delcI_out_ga(.(t, X1), .(t, X2)) The set Q consists of the following terms: maxcC_in_ga(x0) delcD_in_gga(x0, x1) delcE_in_gga(x0, x1) maxcF_in_ga(x0) delcG_in_gga(x0, x1) delcH_in_gga(x0, x1) U48_ga(x0, x1) U49_ga(x0, x1) U54_gga(x0, x1) U51_ga(x0, x1) U52_ga(x0, x1) U56_gga(x0, x1) U57_gga(x0, x1) delcI_in_ga(x0) delcJ_in_ga(x0) U50_ga(x0, x1) U53_ga(x0, x1) We have to consider all (P,Q,R)-chains. ---------------------------------------- (54) TransformationProof (SOUND) By narrowing [LPAR04] the rule U18_GA(X1, maxcC_out_ga(X1, X2)) -> U19_GA(X1, delcE_in_gga(X2, X1)) at position [1] we obtained the following new rules [LPAR04]: (U18_GA(x0, maxcC_out_ga(x0, t)) -> U19_GA(x0, delcE_out_gga(t, x0, .(f, x0))),U18_GA(x0, maxcC_out_ga(x0, t)) -> U19_GA(x0, delcE_out_gga(t, x0, .(f, x0)))) ---------------------------------------- (55) Obligation: Q DP problem: The TRS P consists of the following rules: U12_GA(X1, maxcC_out_ga(X1, X2)) -> U13_GA(X1, delcD_in_gga(X2, X1)) U13_GA(X1, delcD_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) MAXSORTA_IN_GA(.(t, .(t, X1))) -> U12_GA(X1, maxcC_in_ga(X1)) MAXSORTA_IN_GA(.(t, .(f, X1))) -> U18_GA(X1, maxcC_in_ga(X1)) U19_GA(X1, delcE_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) MAXSORTA_IN_GA(.(f, .(f, X1))) -> U24_GA(X1, maxcF_in_ga(X1)) U24_GA(X1, maxcF_out_ga(X1, X2)) -> U25_GA(X1, delcG_in_gga(X2, X1)) U25_GA(X1, delcG_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) MAXSORTA_IN_GA(.(f, .(t, X1))) -> U30_GA(X1, maxcC_in_ga(X1)) U30_GA(X1, maxcC_out_ga(X1, X2)) -> U31_GA(X1, delcH_in_gga(X2, X1)) U31_GA(X1, delcH_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) U18_GA(x0, maxcC_out_ga(x0, t)) -> U19_GA(x0, delcE_out_gga(t, x0, .(f, x0))) The TRS R consists of the following rules: delcH_in_gga(f, X1) -> delcH_out_gga(f, X1, .(t, X1)) delcH_in_gga(t, X1) -> U57_gga(X1, delcJ_in_ga(.(t, X1))) delcJ_in_ga(.(t, X1)) -> delcJ_out_ga(.(t, X1), X1) U57_gga(X1, delcJ_out_ga(.(t, X1), X2)) -> delcH_out_gga(t, X1, .(f, X2)) maxcC_in_ga([]) -> maxcC_out_ga([], t) maxcC_in_ga(.(t, X1)) -> U48_ga(X1, maxcC_in_ga(X1)) maxcC_in_ga(.(f, X1)) -> U49_ga(X1, maxcC_in_ga(X1)) U49_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(f, X1), X2) U48_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(t, X1), X2) delcG_in_gga(f, X1) -> delcG_out_gga(f, X1, .(f, X1)) delcG_in_gga(t, X1) -> U56_gga(X1, delcJ_in_ga(X1)) delcJ_in_ga([]) -> delcJ_out_ga([], []) delcJ_in_ga(.(f, X1)) -> U53_ga(X1, delcJ_in_ga(X1)) U56_gga(X1, delcJ_out_ga(X1, X2)) -> delcG_out_gga(t, X1, .(f, .(f, X2))) U53_ga(X1, delcJ_out_ga(X1, X2)) -> delcJ_out_ga(.(f, X1), .(f, X2)) maxcF_in_ga([]) -> maxcF_out_ga([], f) maxcF_in_ga(.(f, X1)) -> U51_ga(X1, maxcF_in_ga(X1)) maxcF_in_ga(.(t, X1)) -> U52_ga(X1, maxcC_in_ga(X1)) U52_ga(X1, maxcC_out_ga(X1, X2)) -> maxcF_out_ga(.(t, X1), X2) U51_ga(X1, maxcF_out_ga(X1, X2)) -> maxcF_out_ga(.(f, X1), X2) delcE_in_gga(t, X1) -> delcE_out_gga(t, X1, .(f, X1)) delcI_in_ga(.(f, X1)) -> delcI_out_ga(.(f, X1), X1) delcD_in_gga(t, X1) -> delcD_out_gga(t, X1, .(t, X1)) delcD_in_gga(f, X1) -> U54_gga(X1, delcI_in_ga(X1)) delcI_in_ga([]) -> delcI_out_ga([], []) delcI_in_ga(.(t, X1)) -> U50_ga(X1, delcI_in_ga(X1)) U54_gga(X1, delcI_out_ga(X1, X2)) -> delcD_out_gga(f, X1, .(t, .(t, X2))) U50_ga(X1, delcI_out_ga(X1, X2)) -> delcI_out_ga(.(t, X1), .(t, X2)) The set Q consists of the following terms: maxcC_in_ga(x0) delcD_in_gga(x0, x1) delcE_in_gga(x0, x1) maxcF_in_ga(x0) delcG_in_gga(x0, x1) delcH_in_gga(x0, x1) U48_ga(x0, x1) U49_ga(x0, x1) U54_gga(x0, x1) U51_ga(x0, x1) U52_ga(x0, x1) U56_gga(x0, x1) U57_gga(x0, x1) delcI_in_ga(x0) delcJ_in_ga(x0) U50_ga(x0, x1) U53_ga(x0, x1) We have to consider all (P,Q,R)-chains. ---------------------------------------- (56) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (57) Obligation: Q DP problem: The TRS P consists of the following rules: U12_GA(X1, maxcC_out_ga(X1, X2)) -> U13_GA(X1, delcD_in_gga(X2, X1)) U13_GA(X1, delcD_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) MAXSORTA_IN_GA(.(t, .(t, X1))) -> U12_GA(X1, maxcC_in_ga(X1)) MAXSORTA_IN_GA(.(t, .(f, X1))) -> U18_GA(X1, maxcC_in_ga(X1)) U19_GA(X1, delcE_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) MAXSORTA_IN_GA(.(f, .(f, X1))) -> U24_GA(X1, maxcF_in_ga(X1)) U24_GA(X1, maxcF_out_ga(X1, X2)) -> U25_GA(X1, delcG_in_gga(X2, X1)) U25_GA(X1, delcG_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) MAXSORTA_IN_GA(.(f, .(t, X1))) -> U30_GA(X1, maxcC_in_ga(X1)) U30_GA(X1, maxcC_out_ga(X1, X2)) -> U31_GA(X1, delcH_in_gga(X2, X1)) U31_GA(X1, delcH_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) U18_GA(x0, maxcC_out_ga(x0, t)) -> U19_GA(x0, delcE_out_gga(t, x0, .(f, x0))) The TRS R consists of the following rules: delcH_in_gga(f, X1) -> delcH_out_gga(f, X1, .(t, X1)) delcH_in_gga(t, X1) -> U57_gga(X1, delcJ_in_ga(.(t, X1))) delcJ_in_ga(.(t, X1)) -> delcJ_out_ga(.(t, X1), X1) U57_gga(X1, delcJ_out_ga(.(t, X1), X2)) -> delcH_out_gga(t, X1, .(f, X2)) maxcC_in_ga([]) -> maxcC_out_ga([], t) maxcC_in_ga(.(t, X1)) -> U48_ga(X1, maxcC_in_ga(X1)) maxcC_in_ga(.(f, X1)) -> U49_ga(X1, maxcC_in_ga(X1)) U49_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(f, X1), X2) U48_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(t, X1), X2) delcG_in_gga(f, X1) -> delcG_out_gga(f, X1, .(f, X1)) delcG_in_gga(t, X1) -> U56_gga(X1, delcJ_in_ga(X1)) delcJ_in_ga([]) -> delcJ_out_ga([], []) delcJ_in_ga(.(f, X1)) -> U53_ga(X1, delcJ_in_ga(X1)) U56_gga(X1, delcJ_out_ga(X1, X2)) -> delcG_out_gga(t, X1, .(f, .(f, X2))) U53_ga(X1, delcJ_out_ga(X1, X2)) -> delcJ_out_ga(.(f, X1), .(f, X2)) maxcF_in_ga([]) -> maxcF_out_ga([], f) maxcF_in_ga(.(f, X1)) -> U51_ga(X1, maxcF_in_ga(X1)) maxcF_in_ga(.(t, X1)) -> U52_ga(X1, maxcC_in_ga(X1)) U52_ga(X1, maxcC_out_ga(X1, X2)) -> maxcF_out_ga(.(t, X1), X2) U51_ga(X1, maxcF_out_ga(X1, X2)) -> maxcF_out_ga(.(f, X1), X2) delcD_in_gga(t, X1) -> delcD_out_gga(t, X1, .(t, X1)) delcD_in_gga(f, X1) -> U54_gga(X1, delcI_in_ga(X1)) delcI_in_ga(.(f, X1)) -> delcI_out_ga(.(f, X1), X1) delcI_in_ga([]) -> delcI_out_ga([], []) delcI_in_ga(.(t, X1)) -> U50_ga(X1, delcI_in_ga(X1)) U54_gga(X1, delcI_out_ga(X1, X2)) -> delcD_out_gga(f, X1, .(t, .(t, X2))) U50_ga(X1, delcI_out_ga(X1, X2)) -> delcI_out_ga(.(t, X1), .(t, X2)) The set Q consists of the following terms: maxcC_in_ga(x0) delcD_in_gga(x0, x1) delcE_in_gga(x0, x1) maxcF_in_ga(x0) delcG_in_gga(x0, x1) delcH_in_gga(x0, x1) U48_ga(x0, x1) U49_ga(x0, x1) U54_gga(x0, x1) U51_ga(x0, x1) U52_ga(x0, x1) U56_gga(x0, x1) U57_gga(x0, x1) delcI_in_ga(x0) delcJ_in_ga(x0) U50_ga(x0, x1) U53_ga(x0, x1) We have to consider all (P,Q,R)-chains. ---------------------------------------- (58) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. delcE_in_gga(x0, x1) ---------------------------------------- (59) Obligation: Q DP problem: The TRS P consists of the following rules: U12_GA(X1, maxcC_out_ga(X1, X2)) -> U13_GA(X1, delcD_in_gga(X2, X1)) U13_GA(X1, delcD_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) MAXSORTA_IN_GA(.(t, .(t, X1))) -> U12_GA(X1, maxcC_in_ga(X1)) MAXSORTA_IN_GA(.(t, .(f, X1))) -> U18_GA(X1, maxcC_in_ga(X1)) U19_GA(X1, delcE_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) MAXSORTA_IN_GA(.(f, .(f, X1))) -> U24_GA(X1, maxcF_in_ga(X1)) U24_GA(X1, maxcF_out_ga(X1, X2)) -> U25_GA(X1, delcG_in_gga(X2, X1)) U25_GA(X1, delcG_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) MAXSORTA_IN_GA(.(f, .(t, X1))) -> U30_GA(X1, maxcC_in_ga(X1)) U30_GA(X1, maxcC_out_ga(X1, X2)) -> U31_GA(X1, delcH_in_gga(X2, X1)) U31_GA(X1, delcH_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) U18_GA(x0, maxcC_out_ga(x0, t)) -> U19_GA(x0, delcE_out_gga(t, x0, .(f, x0))) The TRS R consists of the following rules: delcH_in_gga(f, X1) -> delcH_out_gga(f, X1, .(t, X1)) delcH_in_gga(t, X1) -> U57_gga(X1, delcJ_in_ga(.(t, X1))) delcJ_in_ga(.(t, X1)) -> delcJ_out_ga(.(t, X1), X1) U57_gga(X1, delcJ_out_ga(.(t, X1), X2)) -> delcH_out_gga(t, X1, .(f, X2)) maxcC_in_ga([]) -> maxcC_out_ga([], t) maxcC_in_ga(.(t, X1)) -> U48_ga(X1, maxcC_in_ga(X1)) maxcC_in_ga(.(f, X1)) -> U49_ga(X1, maxcC_in_ga(X1)) U49_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(f, X1), X2) U48_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(t, X1), X2) delcG_in_gga(f, X1) -> delcG_out_gga(f, X1, .(f, X1)) delcG_in_gga(t, X1) -> U56_gga(X1, delcJ_in_ga(X1)) delcJ_in_ga([]) -> delcJ_out_ga([], []) delcJ_in_ga(.(f, X1)) -> U53_ga(X1, delcJ_in_ga(X1)) U56_gga(X1, delcJ_out_ga(X1, X2)) -> delcG_out_gga(t, X1, .(f, .(f, X2))) U53_ga(X1, delcJ_out_ga(X1, X2)) -> delcJ_out_ga(.(f, X1), .(f, X2)) maxcF_in_ga([]) -> maxcF_out_ga([], f) maxcF_in_ga(.(f, X1)) -> U51_ga(X1, maxcF_in_ga(X1)) maxcF_in_ga(.(t, X1)) -> U52_ga(X1, maxcC_in_ga(X1)) U52_ga(X1, maxcC_out_ga(X1, X2)) -> maxcF_out_ga(.(t, X1), X2) U51_ga(X1, maxcF_out_ga(X1, X2)) -> maxcF_out_ga(.(f, X1), X2) delcD_in_gga(t, X1) -> delcD_out_gga(t, X1, .(t, X1)) delcD_in_gga(f, X1) -> U54_gga(X1, delcI_in_ga(X1)) delcI_in_ga(.(f, X1)) -> delcI_out_ga(.(f, X1), X1) delcI_in_ga([]) -> delcI_out_ga([], []) delcI_in_ga(.(t, X1)) -> U50_ga(X1, delcI_in_ga(X1)) U54_gga(X1, delcI_out_ga(X1, X2)) -> delcD_out_gga(f, X1, .(t, .(t, X2))) U50_ga(X1, delcI_out_ga(X1, X2)) -> delcI_out_ga(.(t, X1), .(t, X2)) The set Q consists of the following terms: maxcC_in_ga(x0) delcD_in_gga(x0, x1) maxcF_in_ga(x0) delcG_in_gga(x0, x1) delcH_in_gga(x0, x1) U48_ga(x0, x1) U49_ga(x0, x1) U54_gga(x0, x1) U51_ga(x0, x1) U52_ga(x0, x1) U56_gga(x0, x1) U57_gga(x0, x1) delcI_in_ga(x0) delcJ_in_ga(x0) U50_ga(x0, x1) U53_ga(x0, x1) We have to consider all (P,Q,R)-chains. ---------------------------------------- (60) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule U19_GA(X1, delcE_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) we obtained the following new rules [LPAR04]: (U19_GA(z0, delcE_out_gga(t, z0, .(f, z0))) -> MAXSORTA_IN_GA(.(f, z0)),U19_GA(z0, delcE_out_gga(t, z0, .(f, z0))) -> MAXSORTA_IN_GA(.(f, z0))) ---------------------------------------- (61) Obligation: Q DP problem: The TRS P consists of the following rules: U12_GA(X1, maxcC_out_ga(X1, X2)) -> U13_GA(X1, delcD_in_gga(X2, X1)) U13_GA(X1, delcD_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) MAXSORTA_IN_GA(.(t, .(t, X1))) -> U12_GA(X1, maxcC_in_ga(X1)) MAXSORTA_IN_GA(.(t, .(f, X1))) -> U18_GA(X1, maxcC_in_ga(X1)) MAXSORTA_IN_GA(.(f, .(f, X1))) -> U24_GA(X1, maxcF_in_ga(X1)) U24_GA(X1, maxcF_out_ga(X1, X2)) -> U25_GA(X1, delcG_in_gga(X2, X1)) U25_GA(X1, delcG_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) MAXSORTA_IN_GA(.(f, .(t, X1))) -> U30_GA(X1, maxcC_in_ga(X1)) U30_GA(X1, maxcC_out_ga(X1, X2)) -> U31_GA(X1, delcH_in_gga(X2, X1)) U31_GA(X1, delcH_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) U18_GA(x0, maxcC_out_ga(x0, t)) -> U19_GA(x0, delcE_out_gga(t, x0, .(f, x0))) U19_GA(z0, delcE_out_gga(t, z0, .(f, z0))) -> MAXSORTA_IN_GA(.(f, z0)) The TRS R consists of the following rules: delcH_in_gga(f, X1) -> delcH_out_gga(f, X1, .(t, X1)) delcH_in_gga(t, X1) -> U57_gga(X1, delcJ_in_ga(.(t, X1))) delcJ_in_ga(.(t, X1)) -> delcJ_out_ga(.(t, X1), X1) U57_gga(X1, delcJ_out_ga(.(t, X1), X2)) -> delcH_out_gga(t, X1, .(f, X2)) maxcC_in_ga([]) -> maxcC_out_ga([], t) maxcC_in_ga(.(t, X1)) -> U48_ga(X1, maxcC_in_ga(X1)) maxcC_in_ga(.(f, X1)) -> U49_ga(X1, maxcC_in_ga(X1)) U49_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(f, X1), X2) U48_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(t, X1), X2) delcG_in_gga(f, X1) -> delcG_out_gga(f, X1, .(f, X1)) delcG_in_gga(t, X1) -> U56_gga(X1, delcJ_in_ga(X1)) delcJ_in_ga([]) -> delcJ_out_ga([], []) delcJ_in_ga(.(f, X1)) -> U53_ga(X1, delcJ_in_ga(X1)) U56_gga(X1, delcJ_out_ga(X1, X2)) -> delcG_out_gga(t, X1, .(f, .(f, X2))) U53_ga(X1, delcJ_out_ga(X1, X2)) -> delcJ_out_ga(.(f, X1), .(f, X2)) maxcF_in_ga([]) -> maxcF_out_ga([], f) maxcF_in_ga(.(f, X1)) -> U51_ga(X1, maxcF_in_ga(X1)) maxcF_in_ga(.(t, X1)) -> U52_ga(X1, maxcC_in_ga(X1)) U52_ga(X1, maxcC_out_ga(X1, X2)) -> maxcF_out_ga(.(t, X1), X2) U51_ga(X1, maxcF_out_ga(X1, X2)) -> maxcF_out_ga(.(f, X1), X2) delcD_in_gga(t, X1) -> delcD_out_gga(t, X1, .(t, X1)) delcD_in_gga(f, X1) -> U54_gga(X1, delcI_in_ga(X1)) delcI_in_ga(.(f, X1)) -> delcI_out_ga(.(f, X1), X1) delcI_in_ga([]) -> delcI_out_ga([], []) delcI_in_ga(.(t, X1)) -> U50_ga(X1, delcI_in_ga(X1)) U54_gga(X1, delcI_out_ga(X1, X2)) -> delcD_out_gga(f, X1, .(t, .(t, X2))) U50_ga(X1, delcI_out_ga(X1, X2)) -> delcI_out_ga(.(t, X1), .(t, X2)) The set Q consists of the following terms: maxcC_in_ga(x0) delcD_in_gga(x0, x1) maxcF_in_ga(x0) delcG_in_gga(x0, x1) delcH_in_gga(x0, x1) U48_ga(x0, x1) U49_ga(x0, x1) U54_gga(x0, x1) U51_ga(x0, x1) U52_ga(x0, x1) U56_gga(x0, x1) U57_gga(x0, x1) delcI_in_ga(x0) delcJ_in_ga(x0) U50_ga(x0, x1) U53_ga(x0, x1) We have to consider all (P,Q,R)-chains. ---------------------------------------- (62) QDPQMonotonicMRRProof (EQUIVALENT) By using the Q-monotonic rule removal processor with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented such that it always occurs at a strongly monotonic position in a (P,Q,R)-chain. Strictly oriented rules of the TRS R: delcD_in_gga(f, X1) -> U54_gga(X1, delcI_in_ga(X1)) Used ordering: Polynomial interpretation [POLO]: POL(.(x_1, x_2)) = 0 POL(MAXSORTA_IN_GA(x_1)) = 0 POL(U12_GA(x_1, x_2)) = x_2 POL(U13_GA(x_1, x_2)) = x_2 POL(U18_GA(x_1, x_2)) = 0 POL(U19_GA(x_1, x_2)) = 0 POL(U24_GA(x_1, x_2)) = 0 POL(U25_GA(x_1, x_2)) = 0 POL(U30_GA(x_1, x_2)) = 0 POL(U31_GA(x_1, x_2)) = 0 POL(U48_ga(x_1, x_2)) = 2*x_2 POL(U49_ga(x_1, x_2)) = 2*x_2 POL(U50_ga(x_1, x_2)) = 0 POL(U51_ga(x_1, x_2)) = 0 POL(U52_ga(x_1, x_2)) = 0 POL(U53_ga(x_1, x_2)) = 0 POL(U54_gga(x_1, x_2)) = 0 POL(U56_gga(x_1, x_2)) = 0 POL(U57_gga(x_1, x_2)) = 0 POL([]) = 0 POL(delcD_in_gga(x_1, x_2)) = x_1 POL(delcD_out_gga(x_1, x_2, x_3)) = 0 POL(delcE_out_gga(x_1, x_2, x_3)) = 2 + 2*x_1 + x_2 POL(delcG_in_gga(x_1, x_2)) = 0 POL(delcG_out_gga(x_1, x_2, x_3)) = 0 POL(delcH_in_gga(x_1, x_2)) = 0 POL(delcH_out_gga(x_1, x_2, x_3)) = 0 POL(delcI_in_ga(x_1)) = 0 POL(delcI_out_ga(x_1, x_2)) = 0 POL(delcJ_in_ga(x_1)) = 0 POL(delcJ_out_ga(x_1, x_2)) = 0 POL(f) = 2 POL(maxcC_in_ga(x_1)) = 0 POL(maxcC_out_ga(x_1, x_2)) = x_2 POL(maxcF_in_ga(x_1)) = 0 POL(maxcF_out_ga(x_1, x_2)) = 0 POL(t) = 0 ---------------------------------------- (63) Obligation: Q DP problem: The TRS P consists of the following rules: U12_GA(X1, maxcC_out_ga(X1, X2)) -> U13_GA(X1, delcD_in_gga(X2, X1)) U13_GA(X1, delcD_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) MAXSORTA_IN_GA(.(t, .(t, X1))) -> U12_GA(X1, maxcC_in_ga(X1)) MAXSORTA_IN_GA(.(t, .(f, X1))) -> U18_GA(X1, maxcC_in_ga(X1)) MAXSORTA_IN_GA(.(f, .(f, X1))) -> U24_GA(X1, maxcF_in_ga(X1)) U24_GA(X1, maxcF_out_ga(X1, X2)) -> U25_GA(X1, delcG_in_gga(X2, X1)) U25_GA(X1, delcG_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) MAXSORTA_IN_GA(.(f, .(t, X1))) -> U30_GA(X1, maxcC_in_ga(X1)) U30_GA(X1, maxcC_out_ga(X1, X2)) -> U31_GA(X1, delcH_in_gga(X2, X1)) U31_GA(X1, delcH_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) U18_GA(x0, maxcC_out_ga(x0, t)) -> U19_GA(x0, delcE_out_gga(t, x0, .(f, x0))) U19_GA(z0, delcE_out_gga(t, z0, .(f, z0))) -> MAXSORTA_IN_GA(.(f, z0)) The TRS R consists of the following rules: delcH_in_gga(f, X1) -> delcH_out_gga(f, X1, .(t, X1)) delcH_in_gga(t, X1) -> U57_gga(X1, delcJ_in_ga(.(t, X1))) delcJ_in_ga(.(t, X1)) -> delcJ_out_ga(.(t, X1), X1) U57_gga(X1, delcJ_out_ga(.(t, X1), X2)) -> delcH_out_gga(t, X1, .(f, X2)) maxcC_in_ga([]) -> maxcC_out_ga([], t) maxcC_in_ga(.(t, X1)) -> U48_ga(X1, maxcC_in_ga(X1)) maxcC_in_ga(.(f, X1)) -> U49_ga(X1, maxcC_in_ga(X1)) U49_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(f, X1), X2) U48_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(t, X1), X2) delcG_in_gga(f, X1) -> delcG_out_gga(f, X1, .(f, X1)) delcG_in_gga(t, X1) -> U56_gga(X1, delcJ_in_ga(X1)) delcJ_in_ga([]) -> delcJ_out_ga([], []) delcJ_in_ga(.(f, X1)) -> U53_ga(X1, delcJ_in_ga(X1)) U56_gga(X1, delcJ_out_ga(X1, X2)) -> delcG_out_gga(t, X1, .(f, .(f, X2))) U53_ga(X1, delcJ_out_ga(X1, X2)) -> delcJ_out_ga(.(f, X1), .(f, X2)) maxcF_in_ga([]) -> maxcF_out_ga([], f) maxcF_in_ga(.(f, X1)) -> U51_ga(X1, maxcF_in_ga(X1)) maxcF_in_ga(.(t, X1)) -> U52_ga(X1, maxcC_in_ga(X1)) U52_ga(X1, maxcC_out_ga(X1, X2)) -> maxcF_out_ga(.(t, X1), X2) U51_ga(X1, maxcF_out_ga(X1, X2)) -> maxcF_out_ga(.(f, X1), X2) delcD_in_gga(t, X1) -> delcD_out_gga(t, X1, .(t, X1)) delcI_in_ga(.(f, X1)) -> delcI_out_ga(.(f, X1), X1) delcI_in_ga([]) -> delcI_out_ga([], []) delcI_in_ga(.(t, X1)) -> U50_ga(X1, delcI_in_ga(X1)) U54_gga(X1, delcI_out_ga(X1, X2)) -> delcD_out_gga(f, X1, .(t, .(t, X2))) U50_ga(X1, delcI_out_ga(X1, X2)) -> delcI_out_ga(.(t, X1), .(t, X2)) The set Q consists of the following terms: maxcC_in_ga(x0) delcD_in_gga(x0, x1) maxcF_in_ga(x0) delcG_in_gga(x0, x1) delcH_in_gga(x0, x1) U48_ga(x0, x1) U49_ga(x0, x1) U54_gga(x0, x1) U51_ga(x0, x1) U52_ga(x0, x1) U56_gga(x0, x1) U57_gga(x0, x1) delcI_in_ga(x0) delcJ_in_ga(x0) U50_ga(x0, x1) U53_ga(x0, x1) We have to consider all (P,Q,R)-chains. ---------------------------------------- (64) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (65) Obligation: Q DP problem: The TRS P consists of the following rules: U12_GA(X1, maxcC_out_ga(X1, X2)) -> U13_GA(X1, delcD_in_gga(X2, X1)) U13_GA(X1, delcD_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) MAXSORTA_IN_GA(.(t, .(t, X1))) -> U12_GA(X1, maxcC_in_ga(X1)) MAXSORTA_IN_GA(.(t, .(f, X1))) -> U18_GA(X1, maxcC_in_ga(X1)) MAXSORTA_IN_GA(.(f, .(f, X1))) -> U24_GA(X1, maxcF_in_ga(X1)) U24_GA(X1, maxcF_out_ga(X1, X2)) -> U25_GA(X1, delcG_in_gga(X2, X1)) U25_GA(X1, delcG_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) MAXSORTA_IN_GA(.(f, .(t, X1))) -> U30_GA(X1, maxcC_in_ga(X1)) U30_GA(X1, maxcC_out_ga(X1, X2)) -> U31_GA(X1, delcH_in_gga(X2, X1)) U31_GA(X1, delcH_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) U18_GA(x0, maxcC_out_ga(x0, t)) -> U19_GA(x0, delcE_out_gga(t, x0, .(f, x0))) U19_GA(z0, delcE_out_gga(t, z0, .(f, z0))) -> MAXSORTA_IN_GA(.(f, z0)) The TRS R consists of the following rules: delcH_in_gga(f, X1) -> delcH_out_gga(f, X1, .(t, X1)) delcH_in_gga(t, X1) -> U57_gga(X1, delcJ_in_ga(.(t, X1))) delcJ_in_ga(.(t, X1)) -> delcJ_out_ga(.(t, X1), X1) U57_gga(X1, delcJ_out_ga(.(t, X1), X2)) -> delcH_out_gga(t, X1, .(f, X2)) maxcC_in_ga([]) -> maxcC_out_ga([], t) maxcC_in_ga(.(t, X1)) -> U48_ga(X1, maxcC_in_ga(X1)) maxcC_in_ga(.(f, X1)) -> U49_ga(X1, maxcC_in_ga(X1)) U49_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(f, X1), X2) U48_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(t, X1), X2) delcG_in_gga(f, X1) -> delcG_out_gga(f, X1, .(f, X1)) delcG_in_gga(t, X1) -> U56_gga(X1, delcJ_in_ga(X1)) delcJ_in_ga([]) -> delcJ_out_ga([], []) delcJ_in_ga(.(f, X1)) -> U53_ga(X1, delcJ_in_ga(X1)) U56_gga(X1, delcJ_out_ga(X1, X2)) -> delcG_out_gga(t, X1, .(f, .(f, X2))) U53_ga(X1, delcJ_out_ga(X1, X2)) -> delcJ_out_ga(.(f, X1), .(f, X2)) maxcF_in_ga([]) -> maxcF_out_ga([], f) maxcF_in_ga(.(f, X1)) -> U51_ga(X1, maxcF_in_ga(X1)) maxcF_in_ga(.(t, X1)) -> U52_ga(X1, maxcC_in_ga(X1)) U52_ga(X1, maxcC_out_ga(X1, X2)) -> maxcF_out_ga(.(t, X1), X2) U51_ga(X1, maxcF_out_ga(X1, X2)) -> maxcF_out_ga(.(f, X1), X2) delcD_in_gga(t, X1) -> delcD_out_gga(t, X1, .(t, X1)) The set Q consists of the following terms: maxcC_in_ga(x0) delcD_in_gga(x0, x1) maxcF_in_ga(x0) delcG_in_gga(x0, x1) delcH_in_gga(x0, x1) U48_ga(x0, x1) U49_ga(x0, x1) U54_gga(x0, x1) U51_ga(x0, x1) U52_ga(x0, x1) U56_gga(x0, x1) U57_gga(x0, x1) delcI_in_ga(x0) delcJ_in_ga(x0) U50_ga(x0, x1) U53_ga(x0, x1) We have to consider all (P,Q,R)-chains. ---------------------------------------- (66) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. U54_gga(x0, x1) delcI_in_ga(x0) U50_ga(x0, x1) ---------------------------------------- (67) Obligation: Q DP problem: The TRS P consists of the following rules: U12_GA(X1, maxcC_out_ga(X1, X2)) -> U13_GA(X1, delcD_in_gga(X2, X1)) U13_GA(X1, delcD_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) MAXSORTA_IN_GA(.(t, .(t, X1))) -> U12_GA(X1, maxcC_in_ga(X1)) MAXSORTA_IN_GA(.(t, .(f, X1))) -> U18_GA(X1, maxcC_in_ga(X1)) MAXSORTA_IN_GA(.(f, .(f, X1))) -> U24_GA(X1, maxcF_in_ga(X1)) U24_GA(X1, maxcF_out_ga(X1, X2)) -> U25_GA(X1, delcG_in_gga(X2, X1)) U25_GA(X1, delcG_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) MAXSORTA_IN_GA(.(f, .(t, X1))) -> U30_GA(X1, maxcC_in_ga(X1)) U30_GA(X1, maxcC_out_ga(X1, X2)) -> U31_GA(X1, delcH_in_gga(X2, X1)) U31_GA(X1, delcH_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) U18_GA(x0, maxcC_out_ga(x0, t)) -> U19_GA(x0, delcE_out_gga(t, x0, .(f, x0))) U19_GA(z0, delcE_out_gga(t, z0, .(f, z0))) -> MAXSORTA_IN_GA(.(f, z0)) The TRS R consists of the following rules: delcH_in_gga(f, X1) -> delcH_out_gga(f, X1, .(t, X1)) delcH_in_gga(t, X1) -> U57_gga(X1, delcJ_in_ga(.(t, X1))) delcJ_in_ga(.(t, X1)) -> delcJ_out_ga(.(t, X1), X1) U57_gga(X1, delcJ_out_ga(.(t, X1), X2)) -> delcH_out_gga(t, X1, .(f, X2)) maxcC_in_ga([]) -> maxcC_out_ga([], t) maxcC_in_ga(.(t, X1)) -> U48_ga(X1, maxcC_in_ga(X1)) maxcC_in_ga(.(f, X1)) -> U49_ga(X1, maxcC_in_ga(X1)) U49_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(f, X1), X2) U48_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(t, X1), X2) delcG_in_gga(f, X1) -> delcG_out_gga(f, X1, .(f, X1)) delcG_in_gga(t, X1) -> U56_gga(X1, delcJ_in_ga(X1)) delcJ_in_ga([]) -> delcJ_out_ga([], []) delcJ_in_ga(.(f, X1)) -> U53_ga(X1, delcJ_in_ga(X1)) U56_gga(X1, delcJ_out_ga(X1, X2)) -> delcG_out_gga(t, X1, .(f, .(f, X2))) U53_ga(X1, delcJ_out_ga(X1, X2)) -> delcJ_out_ga(.(f, X1), .(f, X2)) maxcF_in_ga([]) -> maxcF_out_ga([], f) maxcF_in_ga(.(f, X1)) -> U51_ga(X1, maxcF_in_ga(X1)) maxcF_in_ga(.(t, X1)) -> U52_ga(X1, maxcC_in_ga(X1)) U52_ga(X1, maxcC_out_ga(X1, X2)) -> maxcF_out_ga(.(t, X1), X2) U51_ga(X1, maxcF_out_ga(X1, X2)) -> maxcF_out_ga(.(f, X1), X2) delcD_in_gga(t, X1) -> delcD_out_gga(t, X1, .(t, X1)) The set Q consists of the following terms: maxcC_in_ga(x0) delcD_in_gga(x0, x1) maxcF_in_ga(x0) delcG_in_gga(x0, x1) delcH_in_gga(x0, x1) U48_ga(x0, x1) U49_ga(x0, x1) U51_ga(x0, x1) U52_ga(x0, x1) U56_gga(x0, x1) U57_gga(x0, x1) delcJ_in_ga(x0) U53_ga(x0, x1) We have to consider all (P,Q,R)-chains. ---------------------------------------- (68) TransformationProof (SOUND) By narrowing [LPAR04] the rule U12_GA(X1, maxcC_out_ga(X1, X2)) -> U13_GA(X1, delcD_in_gga(X2, X1)) at position [1] we obtained the following new rules [LPAR04]: (U12_GA(x0, maxcC_out_ga(x0, t)) -> U13_GA(x0, delcD_out_gga(t, x0, .(t, x0))),U12_GA(x0, maxcC_out_ga(x0, t)) -> U13_GA(x0, delcD_out_gga(t, x0, .(t, x0)))) ---------------------------------------- (69) Obligation: Q DP problem: The TRS P consists of the following rules: U13_GA(X1, delcD_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) MAXSORTA_IN_GA(.(t, .(t, X1))) -> U12_GA(X1, maxcC_in_ga(X1)) MAXSORTA_IN_GA(.(t, .(f, X1))) -> U18_GA(X1, maxcC_in_ga(X1)) MAXSORTA_IN_GA(.(f, .(f, X1))) -> U24_GA(X1, maxcF_in_ga(X1)) U24_GA(X1, maxcF_out_ga(X1, X2)) -> U25_GA(X1, delcG_in_gga(X2, X1)) U25_GA(X1, delcG_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) MAXSORTA_IN_GA(.(f, .(t, X1))) -> U30_GA(X1, maxcC_in_ga(X1)) U30_GA(X1, maxcC_out_ga(X1, X2)) -> U31_GA(X1, delcH_in_gga(X2, X1)) U31_GA(X1, delcH_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) U18_GA(x0, maxcC_out_ga(x0, t)) -> U19_GA(x0, delcE_out_gga(t, x0, .(f, x0))) U19_GA(z0, delcE_out_gga(t, z0, .(f, z0))) -> MAXSORTA_IN_GA(.(f, z0)) U12_GA(x0, maxcC_out_ga(x0, t)) -> U13_GA(x0, delcD_out_gga(t, x0, .(t, x0))) The TRS R consists of the following rules: delcH_in_gga(f, X1) -> delcH_out_gga(f, X1, .(t, X1)) delcH_in_gga(t, X1) -> U57_gga(X1, delcJ_in_ga(.(t, X1))) delcJ_in_ga(.(t, X1)) -> delcJ_out_ga(.(t, X1), X1) U57_gga(X1, delcJ_out_ga(.(t, X1), X2)) -> delcH_out_gga(t, X1, .(f, X2)) maxcC_in_ga([]) -> maxcC_out_ga([], t) maxcC_in_ga(.(t, X1)) -> U48_ga(X1, maxcC_in_ga(X1)) maxcC_in_ga(.(f, X1)) -> U49_ga(X1, maxcC_in_ga(X1)) U49_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(f, X1), X2) U48_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(t, X1), X2) delcG_in_gga(f, X1) -> delcG_out_gga(f, X1, .(f, X1)) delcG_in_gga(t, X1) -> U56_gga(X1, delcJ_in_ga(X1)) delcJ_in_ga([]) -> delcJ_out_ga([], []) delcJ_in_ga(.(f, X1)) -> U53_ga(X1, delcJ_in_ga(X1)) U56_gga(X1, delcJ_out_ga(X1, X2)) -> delcG_out_gga(t, X1, .(f, .(f, X2))) U53_ga(X1, delcJ_out_ga(X1, X2)) -> delcJ_out_ga(.(f, X1), .(f, X2)) maxcF_in_ga([]) -> maxcF_out_ga([], f) maxcF_in_ga(.(f, X1)) -> U51_ga(X1, maxcF_in_ga(X1)) maxcF_in_ga(.(t, X1)) -> U52_ga(X1, maxcC_in_ga(X1)) U52_ga(X1, maxcC_out_ga(X1, X2)) -> maxcF_out_ga(.(t, X1), X2) U51_ga(X1, maxcF_out_ga(X1, X2)) -> maxcF_out_ga(.(f, X1), X2) delcD_in_gga(t, X1) -> delcD_out_gga(t, X1, .(t, X1)) The set Q consists of the following terms: maxcC_in_ga(x0) delcD_in_gga(x0, x1) maxcF_in_ga(x0) delcG_in_gga(x0, x1) delcH_in_gga(x0, x1) U48_ga(x0, x1) U49_ga(x0, x1) U51_ga(x0, x1) U52_ga(x0, x1) U56_gga(x0, x1) U57_gga(x0, x1) delcJ_in_ga(x0) U53_ga(x0, x1) We have to consider all (P,Q,R)-chains. ---------------------------------------- (70) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (71) Obligation: Q DP problem: The TRS P consists of the following rules: U13_GA(X1, delcD_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) MAXSORTA_IN_GA(.(t, .(t, X1))) -> U12_GA(X1, maxcC_in_ga(X1)) MAXSORTA_IN_GA(.(t, .(f, X1))) -> U18_GA(X1, maxcC_in_ga(X1)) MAXSORTA_IN_GA(.(f, .(f, X1))) -> U24_GA(X1, maxcF_in_ga(X1)) U24_GA(X1, maxcF_out_ga(X1, X2)) -> U25_GA(X1, delcG_in_gga(X2, X1)) U25_GA(X1, delcG_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) MAXSORTA_IN_GA(.(f, .(t, X1))) -> U30_GA(X1, maxcC_in_ga(X1)) U30_GA(X1, maxcC_out_ga(X1, X2)) -> U31_GA(X1, delcH_in_gga(X2, X1)) U31_GA(X1, delcH_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) U18_GA(x0, maxcC_out_ga(x0, t)) -> U19_GA(x0, delcE_out_gga(t, x0, .(f, x0))) U19_GA(z0, delcE_out_gga(t, z0, .(f, z0))) -> MAXSORTA_IN_GA(.(f, z0)) U12_GA(x0, maxcC_out_ga(x0, t)) -> U13_GA(x0, delcD_out_gga(t, x0, .(t, x0))) The TRS R consists of the following rules: delcH_in_gga(f, X1) -> delcH_out_gga(f, X1, .(t, X1)) delcH_in_gga(t, X1) -> U57_gga(X1, delcJ_in_ga(.(t, X1))) delcJ_in_ga(.(t, X1)) -> delcJ_out_ga(.(t, X1), X1) U57_gga(X1, delcJ_out_ga(.(t, X1), X2)) -> delcH_out_gga(t, X1, .(f, X2)) maxcC_in_ga([]) -> maxcC_out_ga([], t) maxcC_in_ga(.(t, X1)) -> U48_ga(X1, maxcC_in_ga(X1)) maxcC_in_ga(.(f, X1)) -> U49_ga(X1, maxcC_in_ga(X1)) U49_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(f, X1), X2) U48_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(t, X1), X2) delcG_in_gga(f, X1) -> delcG_out_gga(f, X1, .(f, X1)) delcG_in_gga(t, X1) -> U56_gga(X1, delcJ_in_ga(X1)) delcJ_in_ga([]) -> delcJ_out_ga([], []) delcJ_in_ga(.(f, X1)) -> U53_ga(X1, delcJ_in_ga(X1)) U56_gga(X1, delcJ_out_ga(X1, X2)) -> delcG_out_gga(t, X1, .(f, .(f, X2))) U53_ga(X1, delcJ_out_ga(X1, X2)) -> delcJ_out_ga(.(f, X1), .(f, X2)) maxcF_in_ga([]) -> maxcF_out_ga([], f) maxcF_in_ga(.(f, X1)) -> U51_ga(X1, maxcF_in_ga(X1)) maxcF_in_ga(.(t, X1)) -> U52_ga(X1, maxcC_in_ga(X1)) U52_ga(X1, maxcC_out_ga(X1, X2)) -> maxcF_out_ga(.(t, X1), X2) U51_ga(X1, maxcF_out_ga(X1, X2)) -> maxcF_out_ga(.(f, X1), X2) The set Q consists of the following terms: maxcC_in_ga(x0) delcD_in_gga(x0, x1) maxcF_in_ga(x0) delcG_in_gga(x0, x1) delcH_in_gga(x0, x1) U48_ga(x0, x1) U49_ga(x0, x1) U51_ga(x0, x1) U52_ga(x0, x1) U56_gga(x0, x1) U57_gga(x0, x1) delcJ_in_ga(x0) U53_ga(x0, x1) We have to consider all (P,Q,R)-chains. ---------------------------------------- (72) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. delcD_in_gga(x0, x1) ---------------------------------------- (73) Obligation: Q DP problem: The TRS P consists of the following rules: U13_GA(X1, delcD_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) MAXSORTA_IN_GA(.(t, .(t, X1))) -> U12_GA(X1, maxcC_in_ga(X1)) MAXSORTA_IN_GA(.(t, .(f, X1))) -> U18_GA(X1, maxcC_in_ga(X1)) MAXSORTA_IN_GA(.(f, .(f, X1))) -> U24_GA(X1, maxcF_in_ga(X1)) U24_GA(X1, maxcF_out_ga(X1, X2)) -> U25_GA(X1, delcG_in_gga(X2, X1)) U25_GA(X1, delcG_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) MAXSORTA_IN_GA(.(f, .(t, X1))) -> U30_GA(X1, maxcC_in_ga(X1)) U30_GA(X1, maxcC_out_ga(X1, X2)) -> U31_GA(X1, delcH_in_gga(X2, X1)) U31_GA(X1, delcH_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) U18_GA(x0, maxcC_out_ga(x0, t)) -> U19_GA(x0, delcE_out_gga(t, x0, .(f, x0))) U19_GA(z0, delcE_out_gga(t, z0, .(f, z0))) -> MAXSORTA_IN_GA(.(f, z0)) U12_GA(x0, maxcC_out_ga(x0, t)) -> U13_GA(x0, delcD_out_gga(t, x0, .(t, x0))) The TRS R consists of the following rules: delcH_in_gga(f, X1) -> delcH_out_gga(f, X1, .(t, X1)) delcH_in_gga(t, X1) -> U57_gga(X1, delcJ_in_ga(.(t, X1))) delcJ_in_ga(.(t, X1)) -> delcJ_out_ga(.(t, X1), X1) U57_gga(X1, delcJ_out_ga(.(t, X1), X2)) -> delcH_out_gga(t, X1, .(f, X2)) maxcC_in_ga([]) -> maxcC_out_ga([], t) maxcC_in_ga(.(t, X1)) -> U48_ga(X1, maxcC_in_ga(X1)) maxcC_in_ga(.(f, X1)) -> U49_ga(X1, maxcC_in_ga(X1)) U49_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(f, X1), X2) U48_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(t, X1), X2) delcG_in_gga(f, X1) -> delcG_out_gga(f, X1, .(f, X1)) delcG_in_gga(t, X1) -> U56_gga(X1, delcJ_in_ga(X1)) delcJ_in_ga([]) -> delcJ_out_ga([], []) delcJ_in_ga(.(f, X1)) -> U53_ga(X1, delcJ_in_ga(X1)) U56_gga(X1, delcJ_out_ga(X1, X2)) -> delcG_out_gga(t, X1, .(f, .(f, X2))) U53_ga(X1, delcJ_out_ga(X1, X2)) -> delcJ_out_ga(.(f, X1), .(f, X2)) maxcF_in_ga([]) -> maxcF_out_ga([], f) maxcF_in_ga(.(f, X1)) -> U51_ga(X1, maxcF_in_ga(X1)) maxcF_in_ga(.(t, X1)) -> U52_ga(X1, maxcC_in_ga(X1)) U52_ga(X1, maxcC_out_ga(X1, X2)) -> maxcF_out_ga(.(t, X1), X2) U51_ga(X1, maxcF_out_ga(X1, X2)) -> maxcF_out_ga(.(f, X1), X2) The set Q consists of the following terms: maxcC_in_ga(x0) maxcF_in_ga(x0) delcG_in_gga(x0, x1) delcH_in_gga(x0, x1) U48_ga(x0, x1) U49_ga(x0, x1) U51_ga(x0, x1) U52_ga(x0, x1) U56_gga(x0, x1) U57_gga(x0, x1) delcJ_in_ga(x0) U53_ga(x0, x1) We have to consider all (P,Q,R)-chains. ---------------------------------------- (74) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule U13_GA(X1, delcD_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) we obtained the following new rules [LPAR04]: (U13_GA(z0, delcD_out_gga(t, z0, .(t, z0))) -> MAXSORTA_IN_GA(.(t, z0)),U13_GA(z0, delcD_out_gga(t, z0, .(t, z0))) -> MAXSORTA_IN_GA(.(t, z0))) ---------------------------------------- (75) Obligation: Q DP problem: The TRS P consists of the following rules: MAXSORTA_IN_GA(.(t, .(t, X1))) -> U12_GA(X1, maxcC_in_ga(X1)) MAXSORTA_IN_GA(.(t, .(f, X1))) -> U18_GA(X1, maxcC_in_ga(X1)) MAXSORTA_IN_GA(.(f, .(f, X1))) -> U24_GA(X1, maxcF_in_ga(X1)) U24_GA(X1, maxcF_out_ga(X1, X2)) -> U25_GA(X1, delcG_in_gga(X2, X1)) U25_GA(X1, delcG_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) MAXSORTA_IN_GA(.(f, .(t, X1))) -> U30_GA(X1, maxcC_in_ga(X1)) U30_GA(X1, maxcC_out_ga(X1, X2)) -> U31_GA(X1, delcH_in_gga(X2, X1)) U31_GA(X1, delcH_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) U18_GA(x0, maxcC_out_ga(x0, t)) -> U19_GA(x0, delcE_out_gga(t, x0, .(f, x0))) U19_GA(z0, delcE_out_gga(t, z0, .(f, z0))) -> MAXSORTA_IN_GA(.(f, z0)) U12_GA(x0, maxcC_out_ga(x0, t)) -> U13_GA(x0, delcD_out_gga(t, x0, .(t, x0))) U13_GA(z0, delcD_out_gga(t, z0, .(t, z0))) -> MAXSORTA_IN_GA(.(t, z0)) The TRS R consists of the following rules: delcH_in_gga(f, X1) -> delcH_out_gga(f, X1, .(t, X1)) delcH_in_gga(t, X1) -> U57_gga(X1, delcJ_in_ga(.(t, X1))) delcJ_in_ga(.(t, X1)) -> delcJ_out_ga(.(t, X1), X1) U57_gga(X1, delcJ_out_ga(.(t, X1), X2)) -> delcH_out_gga(t, X1, .(f, X2)) maxcC_in_ga([]) -> maxcC_out_ga([], t) maxcC_in_ga(.(t, X1)) -> U48_ga(X1, maxcC_in_ga(X1)) maxcC_in_ga(.(f, X1)) -> U49_ga(X1, maxcC_in_ga(X1)) U49_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(f, X1), X2) U48_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(t, X1), X2) delcG_in_gga(f, X1) -> delcG_out_gga(f, X1, .(f, X1)) delcG_in_gga(t, X1) -> U56_gga(X1, delcJ_in_ga(X1)) delcJ_in_ga([]) -> delcJ_out_ga([], []) delcJ_in_ga(.(f, X1)) -> U53_ga(X1, delcJ_in_ga(X1)) U56_gga(X1, delcJ_out_ga(X1, X2)) -> delcG_out_gga(t, X1, .(f, .(f, X2))) U53_ga(X1, delcJ_out_ga(X1, X2)) -> delcJ_out_ga(.(f, X1), .(f, X2)) maxcF_in_ga([]) -> maxcF_out_ga([], f) maxcF_in_ga(.(f, X1)) -> U51_ga(X1, maxcF_in_ga(X1)) maxcF_in_ga(.(t, X1)) -> U52_ga(X1, maxcC_in_ga(X1)) U52_ga(X1, maxcC_out_ga(X1, X2)) -> maxcF_out_ga(.(t, X1), X2) U51_ga(X1, maxcF_out_ga(X1, X2)) -> maxcF_out_ga(.(f, X1), X2) The set Q consists of the following terms: maxcC_in_ga(x0) maxcF_in_ga(x0) delcG_in_gga(x0, x1) delcH_in_gga(x0, x1) U48_ga(x0, x1) U49_ga(x0, x1) U51_ga(x0, x1) U52_ga(x0, x1) U56_gga(x0, x1) U57_gga(x0, x1) delcJ_in_ga(x0) U53_ga(x0, x1) We have to consider all (P,Q,R)-chains. ---------------------------------------- (76) QDPQMonotonicMRRProof (EQUIVALENT) By using the Q-monotonic rule removal processor with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented such that it always occurs at a strongly monotonic position in a (P,Q,R)-chain. Strictly oriented dependency pairs: MAXSORTA_IN_GA(.(t, .(t, X1))) -> U12_GA(X1, maxcC_in_ga(X1)) MAXSORTA_IN_GA(.(t, .(f, X1))) -> U18_GA(X1, maxcC_in_ga(X1)) U18_GA(x0, maxcC_out_ga(x0, t)) -> U19_GA(x0, delcE_out_gga(t, x0, .(f, x0))) U12_GA(x0, maxcC_out_ga(x0, t)) -> U13_GA(x0, delcD_out_gga(t, x0, .(t, x0))) Strictly oriented rules of the TRS R: delcJ_in_ga(.(t, X1)) -> delcJ_out_ga(.(t, X1), X1) Used ordering: Polynomial interpretation [POLO]: POL(.(x_1, x_2)) = x_1 + x_2 POL(MAXSORTA_IN_GA(x_1)) = 2 + x_1 POL(U12_GA(x_1, x_2)) = 1 + x_1 + 2*x_2 POL(U13_GA(x_1, x_2)) = x_2 POL(U18_GA(x_1, x_2)) = 1 + x_1 + x_2 POL(U19_GA(x_1, x_2)) = 2 + x_1 POL(U24_GA(x_1, x_2)) = 2 + x_1 POL(U25_GA(x_1, x_2)) = 1 + x_2 POL(U30_GA(x_1, x_2)) = x_1 + 2*x_2 POL(U31_GA(x_1, x_2)) = 2 + x_2 POL(U48_ga(x_1, x_2)) = 2 POL(U49_ga(x_1, x_2)) = 2 POL(U51_ga(x_1, x_2)) = 0 POL(U52_ga(x_1, x_2)) = 0 POL(U53_ga(x_1, x_2)) = x_2 POL(U56_gga(x_1, x_2)) = 1 + x_2 POL(U57_gga(x_1, x_2)) = x_2 POL([]) = 0 POL(delcD_out_gga(x_1, x_2, x_3)) = 2*x_1 + x_2 POL(delcE_out_gga(x_1, x_2, x_3)) = 2 + x_2 POL(delcG_in_gga(x_1, x_2)) = 1 + x_2 POL(delcG_out_gga(x_1, x_2, x_3)) = 1 + x_3 POL(delcH_in_gga(x_1, x_2)) = 2 + x_2 POL(delcH_out_gga(x_1, x_2, x_3)) = x_3 POL(delcJ_in_ga(x_1)) = x_1 POL(delcJ_out_ga(x_1, x_2)) = x_2 POL(f) = 0 POL(maxcC_in_ga(x_1)) = 2 POL(maxcC_out_ga(x_1, x_2)) = 2 POL(maxcF_in_ga(x_1)) = 0 POL(maxcF_out_ga(x_1, x_2)) = 0 POL(t) = 2 ---------------------------------------- (77) Obligation: Q DP problem: The TRS P consists of the following rules: MAXSORTA_IN_GA(.(f, .(f, X1))) -> U24_GA(X1, maxcF_in_ga(X1)) U24_GA(X1, maxcF_out_ga(X1, X2)) -> U25_GA(X1, delcG_in_gga(X2, X1)) U25_GA(X1, delcG_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) MAXSORTA_IN_GA(.(f, .(t, X1))) -> U30_GA(X1, maxcC_in_ga(X1)) U30_GA(X1, maxcC_out_ga(X1, X2)) -> U31_GA(X1, delcH_in_gga(X2, X1)) U31_GA(X1, delcH_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) U19_GA(z0, delcE_out_gga(t, z0, .(f, z0))) -> MAXSORTA_IN_GA(.(f, z0)) U13_GA(z0, delcD_out_gga(t, z0, .(t, z0))) -> MAXSORTA_IN_GA(.(t, z0)) The TRS R consists of the following rules: delcH_in_gga(f, X1) -> delcH_out_gga(f, X1, .(t, X1)) delcH_in_gga(t, X1) -> U57_gga(X1, delcJ_in_ga(.(t, X1))) U57_gga(X1, delcJ_out_ga(.(t, X1), X2)) -> delcH_out_gga(t, X1, .(f, X2)) maxcC_in_ga([]) -> maxcC_out_ga([], t) maxcC_in_ga(.(t, X1)) -> U48_ga(X1, maxcC_in_ga(X1)) maxcC_in_ga(.(f, X1)) -> U49_ga(X1, maxcC_in_ga(X1)) U49_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(f, X1), X2) U48_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(t, X1), X2) delcG_in_gga(f, X1) -> delcG_out_gga(f, X1, .(f, X1)) delcG_in_gga(t, X1) -> U56_gga(X1, delcJ_in_ga(X1)) delcJ_in_ga([]) -> delcJ_out_ga([], []) delcJ_in_ga(.(f, X1)) -> U53_ga(X1, delcJ_in_ga(X1)) U56_gga(X1, delcJ_out_ga(X1, X2)) -> delcG_out_gga(t, X1, .(f, .(f, X2))) U53_ga(X1, delcJ_out_ga(X1, X2)) -> delcJ_out_ga(.(f, X1), .(f, X2)) maxcF_in_ga([]) -> maxcF_out_ga([], f) maxcF_in_ga(.(f, X1)) -> U51_ga(X1, maxcF_in_ga(X1)) maxcF_in_ga(.(t, X1)) -> U52_ga(X1, maxcC_in_ga(X1)) U52_ga(X1, maxcC_out_ga(X1, X2)) -> maxcF_out_ga(.(t, X1), X2) U51_ga(X1, maxcF_out_ga(X1, X2)) -> maxcF_out_ga(.(f, X1), X2) The set Q consists of the following terms: maxcC_in_ga(x0) maxcF_in_ga(x0) delcG_in_gga(x0, x1) delcH_in_gga(x0, x1) U48_ga(x0, x1) U49_ga(x0, x1) U51_ga(x0, x1) U52_ga(x0, x1) U56_gga(x0, x1) U57_gga(x0, x1) delcJ_in_ga(x0) U53_ga(x0, x1) We have to consider all (P,Q,R)-chains. ---------------------------------------- (78) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes. ---------------------------------------- (79) Obligation: Q DP problem: The TRS P consists of the following rules: U24_GA(X1, maxcF_out_ga(X1, X2)) -> U25_GA(X1, delcG_in_gga(X2, X1)) U25_GA(X1, delcG_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) MAXSORTA_IN_GA(.(f, .(f, X1))) -> U24_GA(X1, maxcF_in_ga(X1)) MAXSORTA_IN_GA(.(f, .(t, X1))) -> U30_GA(X1, maxcC_in_ga(X1)) U30_GA(X1, maxcC_out_ga(X1, X2)) -> U31_GA(X1, delcH_in_gga(X2, X1)) U31_GA(X1, delcH_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) The TRS R consists of the following rules: delcH_in_gga(f, X1) -> delcH_out_gga(f, X1, .(t, X1)) delcH_in_gga(t, X1) -> U57_gga(X1, delcJ_in_ga(.(t, X1))) U57_gga(X1, delcJ_out_ga(.(t, X1), X2)) -> delcH_out_gga(t, X1, .(f, X2)) maxcC_in_ga([]) -> maxcC_out_ga([], t) maxcC_in_ga(.(t, X1)) -> U48_ga(X1, maxcC_in_ga(X1)) maxcC_in_ga(.(f, X1)) -> U49_ga(X1, maxcC_in_ga(X1)) U49_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(f, X1), X2) U48_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(t, X1), X2) delcG_in_gga(f, X1) -> delcG_out_gga(f, X1, .(f, X1)) delcG_in_gga(t, X1) -> U56_gga(X1, delcJ_in_ga(X1)) delcJ_in_ga([]) -> delcJ_out_ga([], []) delcJ_in_ga(.(f, X1)) -> U53_ga(X1, delcJ_in_ga(X1)) U56_gga(X1, delcJ_out_ga(X1, X2)) -> delcG_out_gga(t, X1, .(f, .(f, X2))) U53_ga(X1, delcJ_out_ga(X1, X2)) -> delcJ_out_ga(.(f, X1), .(f, X2)) maxcF_in_ga([]) -> maxcF_out_ga([], f) maxcF_in_ga(.(f, X1)) -> U51_ga(X1, maxcF_in_ga(X1)) maxcF_in_ga(.(t, X1)) -> U52_ga(X1, maxcC_in_ga(X1)) U52_ga(X1, maxcC_out_ga(X1, X2)) -> maxcF_out_ga(.(t, X1), X2) U51_ga(X1, maxcF_out_ga(X1, X2)) -> maxcF_out_ga(.(f, X1), X2) The set Q consists of the following terms: maxcC_in_ga(x0) maxcF_in_ga(x0) delcG_in_gga(x0, x1) delcH_in_gga(x0, x1) U48_ga(x0, x1) U49_ga(x0, x1) U51_ga(x0, x1) U52_ga(x0, x1) U56_gga(x0, x1) U57_gga(x0, x1) delcJ_in_ga(x0) U53_ga(x0, x1) We have to consider all (P,Q,R)-chains. ---------------------------------------- (80) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (81) Obligation: Q DP problem: The TRS P consists of the following rules: U24_GA(X1, maxcF_out_ga(X1, X2)) -> U25_GA(X1, delcG_in_gga(X2, X1)) U25_GA(X1, delcG_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) MAXSORTA_IN_GA(.(f, .(f, X1))) -> U24_GA(X1, maxcF_in_ga(X1)) MAXSORTA_IN_GA(.(f, .(t, X1))) -> U30_GA(X1, maxcC_in_ga(X1)) U30_GA(X1, maxcC_out_ga(X1, X2)) -> U31_GA(X1, delcH_in_gga(X2, X1)) U31_GA(X1, delcH_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) The TRS R consists of the following rules: delcH_in_gga(f, X1) -> delcH_out_gga(f, X1, .(t, X1)) delcH_in_gga(t, X1) -> U57_gga(X1, delcJ_in_ga(.(t, X1))) maxcC_in_ga([]) -> maxcC_out_ga([], t) maxcC_in_ga(.(t, X1)) -> U48_ga(X1, maxcC_in_ga(X1)) maxcC_in_ga(.(f, X1)) -> U49_ga(X1, maxcC_in_ga(X1)) U49_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(f, X1), X2) U48_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(t, X1), X2) maxcF_in_ga([]) -> maxcF_out_ga([], f) maxcF_in_ga(.(f, X1)) -> U51_ga(X1, maxcF_in_ga(X1)) maxcF_in_ga(.(t, X1)) -> U52_ga(X1, maxcC_in_ga(X1)) U52_ga(X1, maxcC_out_ga(X1, X2)) -> maxcF_out_ga(.(t, X1), X2) U51_ga(X1, maxcF_out_ga(X1, X2)) -> maxcF_out_ga(.(f, X1), X2) delcG_in_gga(f, X1) -> delcG_out_gga(f, X1, .(f, X1)) delcG_in_gga(t, X1) -> U56_gga(X1, delcJ_in_ga(X1)) delcJ_in_ga([]) -> delcJ_out_ga([], []) delcJ_in_ga(.(f, X1)) -> U53_ga(X1, delcJ_in_ga(X1)) U56_gga(X1, delcJ_out_ga(X1, X2)) -> delcG_out_gga(t, X1, .(f, .(f, X2))) U53_ga(X1, delcJ_out_ga(X1, X2)) -> delcJ_out_ga(.(f, X1), .(f, X2)) The set Q consists of the following terms: maxcC_in_ga(x0) maxcF_in_ga(x0) delcG_in_gga(x0, x1) delcH_in_gga(x0, x1) U48_ga(x0, x1) U49_ga(x0, x1) U51_ga(x0, x1) U52_ga(x0, x1) U56_gga(x0, x1) U57_gga(x0, x1) delcJ_in_ga(x0) U53_ga(x0, x1) We have to consider all (P,Q,R)-chains. ---------------------------------------- (82) QDPQMonotonicMRRProof (EQUIVALENT) By using the Q-monotonic rule removal processor with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented such that it always occurs at a strongly monotonic position in a (P,Q,R)-chain. Strictly oriented rules of the TRS R: delcH_in_gga(t, X1) -> U57_gga(X1, delcJ_in_ga(.(t, X1))) Used ordering: Polynomial interpretation [POLO]: POL(.(x_1, x_2)) = 2 + 2*x_2 POL(MAXSORTA_IN_GA(x_1)) = 2 POL(U24_GA(x_1, x_2)) = 2 POL(U25_GA(x_1, x_2)) = 2 POL(U30_GA(x_1, x_2)) = 2 POL(U31_GA(x_1, x_2)) = x_2 POL(U48_ga(x_1, x_2)) = 0 POL(U49_ga(x_1, x_2)) = 2 POL(U51_ga(x_1, x_2)) = 2 + 2*x_1 POL(U52_ga(x_1, x_2)) = 2 + 2*x_1 POL(U53_ga(x_1, x_2)) = 0 POL(U56_gga(x_1, x_2)) = 2 POL(U57_gga(x_1, x_2)) = 0 POL([]) = 0 POL(delcG_in_gga(x_1, x_2)) = 2*x_1 POL(delcG_out_gga(x_1, x_2, x_3)) = 0 POL(delcH_in_gga(x_1, x_2)) = 2 POL(delcH_out_gga(x_1, x_2, x_3)) = 2 + 2*x_1 POL(delcJ_in_ga(x_1)) = 0 POL(delcJ_out_ga(x_1, x_2)) = 0 POL(f) = 0 POL(maxcC_in_ga(x_1)) = 2 + x_1 POL(maxcC_out_ga(x_1, x_2)) = 0 POL(maxcF_in_ga(x_1)) = 2*x_1 POL(maxcF_out_ga(x_1, x_2)) = 0 POL(t) = 2 ---------------------------------------- (83) Obligation: Q DP problem: The TRS P consists of the following rules: U24_GA(X1, maxcF_out_ga(X1, X2)) -> U25_GA(X1, delcG_in_gga(X2, X1)) U25_GA(X1, delcG_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) MAXSORTA_IN_GA(.(f, .(f, X1))) -> U24_GA(X1, maxcF_in_ga(X1)) MAXSORTA_IN_GA(.(f, .(t, X1))) -> U30_GA(X1, maxcC_in_ga(X1)) U30_GA(X1, maxcC_out_ga(X1, X2)) -> U31_GA(X1, delcH_in_gga(X2, X1)) U31_GA(X1, delcH_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) The TRS R consists of the following rules: delcH_in_gga(f, X1) -> delcH_out_gga(f, X1, .(t, X1)) maxcC_in_ga([]) -> maxcC_out_ga([], t) maxcC_in_ga(.(t, X1)) -> U48_ga(X1, maxcC_in_ga(X1)) maxcC_in_ga(.(f, X1)) -> U49_ga(X1, maxcC_in_ga(X1)) U49_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(f, X1), X2) U48_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(t, X1), X2) maxcF_in_ga([]) -> maxcF_out_ga([], f) maxcF_in_ga(.(f, X1)) -> U51_ga(X1, maxcF_in_ga(X1)) maxcF_in_ga(.(t, X1)) -> U52_ga(X1, maxcC_in_ga(X1)) U52_ga(X1, maxcC_out_ga(X1, X2)) -> maxcF_out_ga(.(t, X1), X2) U51_ga(X1, maxcF_out_ga(X1, X2)) -> maxcF_out_ga(.(f, X1), X2) delcG_in_gga(f, X1) -> delcG_out_gga(f, X1, .(f, X1)) delcG_in_gga(t, X1) -> U56_gga(X1, delcJ_in_ga(X1)) delcJ_in_ga([]) -> delcJ_out_ga([], []) delcJ_in_ga(.(f, X1)) -> U53_ga(X1, delcJ_in_ga(X1)) U56_gga(X1, delcJ_out_ga(X1, X2)) -> delcG_out_gga(t, X1, .(f, .(f, X2))) U53_ga(X1, delcJ_out_ga(X1, X2)) -> delcJ_out_ga(.(f, X1), .(f, X2)) The set Q consists of the following terms: maxcC_in_ga(x0) maxcF_in_ga(x0) delcG_in_gga(x0, x1) delcH_in_gga(x0, x1) U48_ga(x0, x1) U49_ga(x0, x1) U51_ga(x0, x1) U52_ga(x0, x1) U56_gga(x0, x1) U57_gga(x0, x1) delcJ_in_ga(x0) U53_ga(x0, x1) We have to consider all (P,Q,R)-chains. ---------------------------------------- (84) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. U57_gga(x0, x1) ---------------------------------------- (85) Obligation: Q DP problem: The TRS P consists of the following rules: U24_GA(X1, maxcF_out_ga(X1, X2)) -> U25_GA(X1, delcG_in_gga(X2, X1)) U25_GA(X1, delcG_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) MAXSORTA_IN_GA(.(f, .(f, X1))) -> U24_GA(X1, maxcF_in_ga(X1)) MAXSORTA_IN_GA(.(f, .(t, X1))) -> U30_GA(X1, maxcC_in_ga(X1)) U30_GA(X1, maxcC_out_ga(X1, X2)) -> U31_GA(X1, delcH_in_gga(X2, X1)) U31_GA(X1, delcH_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) The TRS R consists of the following rules: delcH_in_gga(f, X1) -> delcH_out_gga(f, X1, .(t, X1)) maxcC_in_ga([]) -> maxcC_out_ga([], t) maxcC_in_ga(.(t, X1)) -> U48_ga(X1, maxcC_in_ga(X1)) maxcC_in_ga(.(f, X1)) -> U49_ga(X1, maxcC_in_ga(X1)) U49_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(f, X1), X2) U48_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(t, X1), X2) maxcF_in_ga([]) -> maxcF_out_ga([], f) maxcF_in_ga(.(f, X1)) -> U51_ga(X1, maxcF_in_ga(X1)) maxcF_in_ga(.(t, X1)) -> U52_ga(X1, maxcC_in_ga(X1)) U52_ga(X1, maxcC_out_ga(X1, X2)) -> maxcF_out_ga(.(t, X1), X2) U51_ga(X1, maxcF_out_ga(X1, X2)) -> maxcF_out_ga(.(f, X1), X2) delcG_in_gga(f, X1) -> delcG_out_gga(f, X1, .(f, X1)) delcG_in_gga(t, X1) -> U56_gga(X1, delcJ_in_ga(X1)) delcJ_in_ga([]) -> delcJ_out_ga([], []) delcJ_in_ga(.(f, X1)) -> U53_ga(X1, delcJ_in_ga(X1)) U56_gga(X1, delcJ_out_ga(X1, X2)) -> delcG_out_gga(t, X1, .(f, .(f, X2))) U53_ga(X1, delcJ_out_ga(X1, X2)) -> delcJ_out_ga(.(f, X1), .(f, X2)) The set Q consists of the following terms: maxcC_in_ga(x0) maxcF_in_ga(x0) delcG_in_gga(x0, x1) delcH_in_gga(x0, x1) U48_ga(x0, x1) U49_ga(x0, x1) U51_ga(x0, x1) U52_ga(x0, x1) U56_gga(x0, x1) delcJ_in_ga(x0) U53_ga(x0, x1) We have to consider all (P,Q,R)-chains. ---------------------------------------- (86) TransformationProof (SOUND) By narrowing [LPAR04] the rule U30_GA(X1, maxcC_out_ga(X1, X2)) -> U31_GA(X1, delcH_in_gga(X2, X1)) at position [1] we obtained the following new rules [LPAR04]: (U30_GA(x0, maxcC_out_ga(x0, f)) -> U31_GA(x0, delcH_out_gga(f, x0, .(t, x0))),U30_GA(x0, maxcC_out_ga(x0, f)) -> U31_GA(x0, delcH_out_gga(f, x0, .(t, x0)))) ---------------------------------------- (87) Obligation: Q DP problem: The TRS P consists of the following rules: U24_GA(X1, maxcF_out_ga(X1, X2)) -> U25_GA(X1, delcG_in_gga(X2, X1)) U25_GA(X1, delcG_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) MAXSORTA_IN_GA(.(f, .(f, X1))) -> U24_GA(X1, maxcF_in_ga(X1)) MAXSORTA_IN_GA(.(f, .(t, X1))) -> U30_GA(X1, maxcC_in_ga(X1)) U31_GA(X1, delcH_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) U30_GA(x0, maxcC_out_ga(x0, f)) -> U31_GA(x0, delcH_out_gga(f, x0, .(t, x0))) The TRS R consists of the following rules: delcH_in_gga(f, X1) -> delcH_out_gga(f, X1, .(t, X1)) maxcC_in_ga([]) -> maxcC_out_ga([], t) maxcC_in_ga(.(t, X1)) -> U48_ga(X1, maxcC_in_ga(X1)) maxcC_in_ga(.(f, X1)) -> U49_ga(X1, maxcC_in_ga(X1)) U49_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(f, X1), X2) U48_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(t, X1), X2) maxcF_in_ga([]) -> maxcF_out_ga([], f) maxcF_in_ga(.(f, X1)) -> U51_ga(X1, maxcF_in_ga(X1)) maxcF_in_ga(.(t, X1)) -> U52_ga(X1, maxcC_in_ga(X1)) U52_ga(X1, maxcC_out_ga(X1, X2)) -> maxcF_out_ga(.(t, X1), X2) U51_ga(X1, maxcF_out_ga(X1, X2)) -> maxcF_out_ga(.(f, X1), X2) delcG_in_gga(f, X1) -> delcG_out_gga(f, X1, .(f, X1)) delcG_in_gga(t, X1) -> U56_gga(X1, delcJ_in_ga(X1)) delcJ_in_ga([]) -> delcJ_out_ga([], []) delcJ_in_ga(.(f, X1)) -> U53_ga(X1, delcJ_in_ga(X1)) U56_gga(X1, delcJ_out_ga(X1, X2)) -> delcG_out_gga(t, X1, .(f, .(f, X2))) U53_ga(X1, delcJ_out_ga(X1, X2)) -> delcJ_out_ga(.(f, X1), .(f, X2)) The set Q consists of the following terms: maxcC_in_ga(x0) maxcF_in_ga(x0) delcG_in_gga(x0, x1) delcH_in_gga(x0, x1) U48_ga(x0, x1) U49_ga(x0, x1) U51_ga(x0, x1) U52_ga(x0, x1) U56_gga(x0, x1) delcJ_in_ga(x0) U53_ga(x0, x1) We have to consider all (P,Q,R)-chains. ---------------------------------------- (88) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (89) Obligation: Q DP problem: The TRS P consists of the following rules: U24_GA(X1, maxcF_out_ga(X1, X2)) -> U25_GA(X1, delcG_in_gga(X2, X1)) U25_GA(X1, delcG_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) MAXSORTA_IN_GA(.(f, .(f, X1))) -> U24_GA(X1, maxcF_in_ga(X1)) MAXSORTA_IN_GA(.(f, .(t, X1))) -> U30_GA(X1, maxcC_in_ga(X1)) U31_GA(X1, delcH_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) U30_GA(x0, maxcC_out_ga(x0, f)) -> U31_GA(x0, delcH_out_gga(f, x0, .(t, x0))) The TRS R consists of the following rules: maxcC_in_ga([]) -> maxcC_out_ga([], t) maxcC_in_ga(.(t, X1)) -> U48_ga(X1, maxcC_in_ga(X1)) maxcC_in_ga(.(f, X1)) -> U49_ga(X1, maxcC_in_ga(X1)) U49_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(f, X1), X2) U48_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(t, X1), X2) maxcF_in_ga([]) -> maxcF_out_ga([], f) maxcF_in_ga(.(f, X1)) -> U51_ga(X1, maxcF_in_ga(X1)) maxcF_in_ga(.(t, X1)) -> U52_ga(X1, maxcC_in_ga(X1)) U52_ga(X1, maxcC_out_ga(X1, X2)) -> maxcF_out_ga(.(t, X1), X2) U51_ga(X1, maxcF_out_ga(X1, X2)) -> maxcF_out_ga(.(f, X1), X2) delcG_in_gga(f, X1) -> delcG_out_gga(f, X1, .(f, X1)) delcG_in_gga(t, X1) -> U56_gga(X1, delcJ_in_ga(X1)) delcJ_in_ga([]) -> delcJ_out_ga([], []) delcJ_in_ga(.(f, X1)) -> U53_ga(X1, delcJ_in_ga(X1)) U56_gga(X1, delcJ_out_ga(X1, X2)) -> delcG_out_gga(t, X1, .(f, .(f, X2))) U53_ga(X1, delcJ_out_ga(X1, X2)) -> delcJ_out_ga(.(f, X1), .(f, X2)) The set Q consists of the following terms: maxcC_in_ga(x0) maxcF_in_ga(x0) delcG_in_gga(x0, x1) delcH_in_gga(x0, x1) U48_ga(x0, x1) U49_ga(x0, x1) U51_ga(x0, x1) U52_ga(x0, x1) U56_gga(x0, x1) delcJ_in_ga(x0) U53_ga(x0, x1) We have to consider all (P,Q,R)-chains. ---------------------------------------- (90) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. delcH_in_gga(x0, x1) ---------------------------------------- (91) Obligation: Q DP problem: The TRS P consists of the following rules: U24_GA(X1, maxcF_out_ga(X1, X2)) -> U25_GA(X1, delcG_in_gga(X2, X1)) U25_GA(X1, delcG_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) MAXSORTA_IN_GA(.(f, .(f, X1))) -> U24_GA(X1, maxcF_in_ga(X1)) MAXSORTA_IN_GA(.(f, .(t, X1))) -> U30_GA(X1, maxcC_in_ga(X1)) U31_GA(X1, delcH_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) U30_GA(x0, maxcC_out_ga(x0, f)) -> U31_GA(x0, delcH_out_gga(f, x0, .(t, x0))) The TRS R consists of the following rules: maxcC_in_ga([]) -> maxcC_out_ga([], t) maxcC_in_ga(.(t, X1)) -> U48_ga(X1, maxcC_in_ga(X1)) maxcC_in_ga(.(f, X1)) -> U49_ga(X1, maxcC_in_ga(X1)) U49_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(f, X1), X2) U48_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(t, X1), X2) maxcF_in_ga([]) -> maxcF_out_ga([], f) maxcF_in_ga(.(f, X1)) -> U51_ga(X1, maxcF_in_ga(X1)) maxcF_in_ga(.(t, X1)) -> U52_ga(X1, maxcC_in_ga(X1)) U52_ga(X1, maxcC_out_ga(X1, X2)) -> maxcF_out_ga(.(t, X1), X2) U51_ga(X1, maxcF_out_ga(X1, X2)) -> maxcF_out_ga(.(f, X1), X2) delcG_in_gga(f, X1) -> delcG_out_gga(f, X1, .(f, X1)) delcG_in_gga(t, X1) -> U56_gga(X1, delcJ_in_ga(X1)) delcJ_in_ga([]) -> delcJ_out_ga([], []) delcJ_in_ga(.(f, X1)) -> U53_ga(X1, delcJ_in_ga(X1)) U56_gga(X1, delcJ_out_ga(X1, X2)) -> delcG_out_gga(t, X1, .(f, .(f, X2))) U53_ga(X1, delcJ_out_ga(X1, X2)) -> delcJ_out_ga(.(f, X1), .(f, X2)) The set Q consists of the following terms: maxcC_in_ga(x0) maxcF_in_ga(x0) delcG_in_gga(x0, x1) U48_ga(x0, x1) U49_ga(x0, x1) U51_ga(x0, x1) U52_ga(x0, x1) U56_gga(x0, x1) delcJ_in_ga(x0) U53_ga(x0, x1) We have to consider all (P,Q,R)-chains. ---------------------------------------- (92) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule U31_GA(X1, delcH_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) we obtained the following new rules [LPAR04]: (U31_GA(z0, delcH_out_gga(f, z0, .(t, z0))) -> MAXSORTA_IN_GA(.(t, z0)),U31_GA(z0, delcH_out_gga(f, z0, .(t, z0))) -> MAXSORTA_IN_GA(.(t, z0))) ---------------------------------------- (93) Obligation: Q DP problem: The TRS P consists of the following rules: U24_GA(X1, maxcF_out_ga(X1, X2)) -> U25_GA(X1, delcG_in_gga(X2, X1)) U25_GA(X1, delcG_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) MAXSORTA_IN_GA(.(f, .(f, X1))) -> U24_GA(X1, maxcF_in_ga(X1)) MAXSORTA_IN_GA(.(f, .(t, X1))) -> U30_GA(X1, maxcC_in_ga(X1)) U30_GA(x0, maxcC_out_ga(x0, f)) -> U31_GA(x0, delcH_out_gga(f, x0, .(t, x0))) U31_GA(z0, delcH_out_gga(f, z0, .(t, z0))) -> MAXSORTA_IN_GA(.(t, z0)) The TRS R consists of the following rules: maxcC_in_ga([]) -> maxcC_out_ga([], t) maxcC_in_ga(.(t, X1)) -> U48_ga(X1, maxcC_in_ga(X1)) maxcC_in_ga(.(f, X1)) -> U49_ga(X1, maxcC_in_ga(X1)) U49_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(f, X1), X2) U48_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(t, X1), X2) maxcF_in_ga([]) -> maxcF_out_ga([], f) maxcF_in_ga(.(f, X1)) -> U51_ga(X1, maxcF_in_ga(X1)) maxcF_in_ga(.(t, X1)) -> U52_ga(X1, maxcC_in_ga(X1)) U52_ga(X1, maxcC_out_ga(X1, X2)) -> maxcF_out_ga(.(t, X1), X2) U51_ga(X1, maxcF_out_ga(X1, X2)) -> maxcF_out_ga(.(f, X1), X2) delcG_in_gga(f, X1) -> delcG_out_gga(f, X1, .(f, X1)) delcG_in_gga(t, X1) -> U56_gga(X1, delcJ_in_ga(X1)) delcJ_in_ga([]) -> delcJ_out_ga([], []) delcJ_in_ga(.(f, X1)) -> U53_ga(X1, delcJ_in_ga(X1)) U56_gga(X1, delcJ_out_ga(X1, X2)) -> delcG_out_gga(t, X1, .(f, .(f, X2))) U53_ga(X1, delcJ_out_ga(X1, X2)) -> delcJ_out_ga(.(f, X1), .(f, X2)) The set Q consists of the following terms: maxcC_in_ga(x0) maxcF_in_ga(x0) delcG_in_gga(x0, x1) U48_ga(x0, x1) U49_ga(x0, x1) U51_ga(x0, x1) U52_ga(x0, x1) U56_gga(x0, x1) delcJ_in_ga(x0) U53_ga(x0, x1) We have to consider all (P,Q,R)-chains. ---------------------------------------- (94) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes. ---------------------------------------- (95) Obligation: Q DP problem: The TRS P consists of the following rules: U25_GA(X1, delcG_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) MAXSORTA_IN_GA(.(f, .(f, X1))) -> U24_GA(X1, maxcF_in_ga(X1)) U24_GA(X1, maxcF_out_ga(X1, X2)) -> U25_GA(X1, delcG_in_gga(X2, X1)) The TRS R consists of the following rules: maxcC_in_ga([]) -> maxcC_out_ga([], t) maxcC_in_ga(.(t, X1)) -> U48_ga(X1, maxcC_in_ga(X1)) maxcC_in_ga(.(f, X1)) -> U49_ga(X1, maxcC_in_ga(X1)) U49_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(f, X1), X2) U48_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(t, X1), X2) maxcF_in_ga([]) -> maxcF_out_ga([], f) maxcF_in_ga(.(f, X1)) -> U51_ga(X1, maxcF_in_ga(X1)) maxcF_in_ga(.(t, X1)) -> U52_ga(X1, maxcC_in_ga(X1)) U52_ga(X1, maxcC_out_ga(X1, X2)) -> maxcF_out_ga(.(t, X1), X2) U51_ga(X1, maxcF_out_ga(X1, X2)) -> maxcF_out_ga(.(f, X1), X2) delcG_in_gga(f, X1) -> delcG_out_gga(f, X1, .(f, X1)) delcG_in_gga(t, X1) -> U56_gga(X1, delcJ_in_ga(X1)) delcJ_in_ga([]) -> delcJ_out_ga([], []) delcJ_in_ga(.(f, X1)) -> U53_ga(X1, delcJ_in_ga(X1)) U56_gga(X1, delcJ_out_ga(X1, X2)) -> delcG_out_gga(t, X1, .(f, .(f, X2))) U53_ga(X1, delcJ_out_ga(X1, X2)) -> delcJ_out_ga(.(f, X1), .(f, X2)) The set Q consists of the following terms: maxcC_in_ga(x0) maxcF_in_ga(x0) delcG_in_gga(x0, x1) U48_ga(x0, x1) U49_ga(x0, x1) U51_ga(x0, x1) U52_ga(x0, x1) U56_gga(x0, x1) delcJ_in_ga(x0) U53_ga(x0, x1) We have to consider all (P,Q,R)-chains. ---------------------------------------- (96) QDPQMonotonicMRRProof (EQUIVALENT) By using the Q-monotonic rule removal processor with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented such that it always occurs at a strongly monotonic position in a (P,Q,R)-chain. Strictly oriented rules of the TRS R: maxcC_in_ga([]) -> maxcC_out_ga([], t) Used ordering: Polynomial interpretation [POLO]: POL(.(x_1, x_2)) = x_1 + 2*x_2 POL(MAXSORTA_IN_GA(x_1)) = 2*x_1 POL(U24_GA(x_1, x_2)) = 2*x_1 + 2*x_2 POL(U25_GA(x_1, x_2)) = 2*x_2 POL(U48_ga(x_1, x_2)) = 2 + x_1 + x_2 POL(U49_ga(x_1, x_2)) = x_1 + x_2 POL(U51_ga(x_1, x_2)) = 2*x_1 + x_2 POL(U52_ga(x_1, x_2)) = 2*x_1 + 2*x_2 POL(U53_ga(x_1, x_2)) = 2*x_2 POL(U56_gga(x_1, x_2)) = 2*x_2 POL([]) = 0 POL(delcG_in_gga(x_1, x_2)) = 2*x_2 POL(delcG_out_gga(x_1, x_2, x_3)) = x_3 POL(delcJ_in_ga(x_1)) = 0 POL(delcJ_out_ga(x_1, x_2)) = 2*x_2 POL(f) = 0 POL(maxcC_in_ga(x_1)) = 2 + x_1 POL(maxcC_out_ga(x_1, x_2)) = 1 + x_1 POL(maxcF_in_ga(x_1)) = 2*x_1 POL(maxcF_out_ga(x_1, x_2)) = x_1 POL(t) = 2 ---------------------------------------- (97) Obligation: Q DP problem: The TRS P consists of the following rules: U25_GA(X1, delcG_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) MAXSORTA_IN_GA(.(f, .(f, X1))) -> U24_GA(X1, maxcF_in_ga(X1)) U24_GA(X1, maxcF_out_ga(X1, X2)) -> U25_GA(X1, delcG_in_gga(X2, X1)) The TRS R consists of the following rules: maxcC_in_ga(.(t, X1)) -> U48_ga(X1, maxcC_in_ga(X1)) maxcC_in_ga(.(f, X1)) -> U49_ga(X1, maxcC_in_ga(X1)) U49_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(f, X1), X2) U48_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(t, X1), X2) maxcF_in_ga([]) -> maxcF_out_ga([], f) maxcF_in_ga(.(f, X1)) -> U51_ga(X1, maxcF_in_ga(X1)) maxcF_in_ga(.(t, X1)) -> U52_ga(X1, maxcC_in_ga(X1)) U52_ga(X1, maxcC_out_ga(X1, X2)) -> maxcF_out_ga(.(t, X1), X2) U51_ga(X1, maxcF_out_ga(X1, X2)) -> maxcF_out_ga(.(f, X1), X2) delcG_in_gga(f, X1) -> delcG_out_gga(f, X1, .(f, X1)) delcG_in_gga(t, X1) -> U56_gga(X1, delcJ_in_ga(X1)) delcJ_in_ga([]) -> delcJ_out_ga([], []) delcJ_in_ga(.(f, X1)) -> U53_ga(X1, delcJ_in_ga(X1)) U56_gga(X1, delcJ_out_ga(X1, X2)) -> delcG_out_gga(t, X1, .(f, .(f, X2))) U53_ga(X1, delcJ_out_ga(X1, X2)) -> delcJ_out_ga(.(f, X1), .(f, X2)) The set Q consists of the following terms: maxcC_in_ga(x0) maxcF_in_ga(x0) delcG_in_gga(x0, x1) U48_ga(x0, x1) U49_ga(x0, x1) U51_ga(x0, x1) U52_ga(x0, x1) U56_gga(x0, x1) delcJ_in_ga(x0) U53_ga(x0, x1) We have to consider all (P,Q,R)-chains. ---------------------------------------- (98) QDPQMonotonicMRRProof (EQUIVALENT) By using the Q-monotonic rule removal processor with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented such that it always occurs at a strongly monotonic position in a (P,Q,R)-chain. Strictly oriented rules of the TRS R: U52_ga(X1, maxcC_out_ga(X1, X2)) -> maxcF_out_ga(.(t, X1), X2) Used ordering: Polynomial interpretation [POLO]: POL(.(x_1, x_2)) = 2*x_2 POL(MAXSORTA_IN_GA(x_1)) = 0 POL(U24_GA(x_1, x_2)) = 2*x_2 POL(U25_GA(x_1, x_2)) = 0 POL(U48_ga(x_1, x_2)) = x_2 POL(U49_ga(x_1, x_2)) = x_2 POL(U51_ga(x_1, x_2)) = x_2 POL(U52_ga(x_1, x_2)) = 2*x_2 POL(U53_ga(x_1, x_2)) = 1 + 2*x_1 POL(U56_gga(x_1, x_2)) = 0 POL([]) = 0 POL(delcG_in_gga(x_1, x_2)) = 0 POL(delcG_out_gga(x_1, x_2, x_3)) = 0 POL(delcJ_in_ga(x_1)) = 2 + 2*x_1 POL(delcJ_out_ga(x_1, x_2)) = 0 POL(f) = 0 POL(maxcC_in_ga(x_1)) = 0 POL(maxcC_out_ga(x_1, x_2)) = 1 POL(maxcF_in_ga(x_1)) = 0 POL(maxcF_out_ga(x_1, x_2)) = 0 POL(t) = 0 ---------------------------------------- (99) Obligation: Q DP problem: The TRS P consists of the following rules: U25_GA(X1, delcG_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) MAXSORTA_IN_GA(.(f, .(f, X1))) -> U24_GA(X1, maxcF_in_ga(X1)) U24_GA(X1, maxcF_out_ga(X1, X2)) -> U25_GA(X1, delcG_in_gga(X2, X1)) The TRS R consists of the following rules: maxcC_in_ga(.(t, X1)) -> U48_ga(X1, maxcC_in_ga(X1)) maxcC_in_ga(.(f, X1)) -> U49_ga(X1, maxcC_in_ga(X1)) U49_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(f, X1), X2) U48_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(t, X1), X2) maxcF_in_ga([]) -> maxcF_out_ga([], f) maxcF_in_ga(.(f, X1)) -> U51_ga(X1, maxcF_in_ga(X1)) maxcF_in_ga(.(t, X1)) -> U52_ga(X1, maxcC_in_ga(X1)) U51_ga(X1, maxcF_out_ga(X1, X2)) -> maxcF_out_ga(.(f, X1), X2) delcG_in_gga(f, X1) -> delcG_out_gga(f, X1, .(f, X1)) delcG_in_gga(t, X1) -> U56_gga(X1, delcJ_in_ga(X1)) delcJ_in_ga([]) -> delcJ_out_ga([], []) delcJ_in_ga(.(f, X1)) -> U53_ga(X1, delcJ_in_ga(X1)) U56_gga(X1, delcJ_out_ga(X1, X2)) -> delcG_out_gga(t, X1, .(f, .(f, X2))) U53_ga(X1, delcJ_out_ga(X1, X2)) -> delcJ_out_ga(.(f, X1), .(f, X2)) The set Q consists of the following terms: maxcC_in_ga(x0) maxcF_in_ga(x0) delcG_in_gga(x0, x1) U48_ga(x0, x1) U49_ga(x0, x1) U51_ga(x0, x1) U52_ga(x0, x1) U56_gga(x0, x1) delcJ_in_ga(x0) U53_ga(x0, x1) We have to consider all (P,Q,R)-chains. ---------------------------------------- (100) QDPQMonotonicMRRProof (EQUIVALENT) By using the Q-monotonic rule removal processor with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented such that it always occurs at a strongly monotonic position in a (P,Q,R)-chain. Strictly oriented dependency pairs: U25_GA(X1, delcG_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) MAXSORTA_IN_GA(.(f, .(f, X1))) -> U24_GA(X1, maxcF_in_ga(X1)) U24_GA(X1, maxcF_out_ga(X1, X2)) -> U25_GA(X1, delcG_in_gga(X2, X1)) Used ordering: Polynomial interpretation [POLO]: POL(.(x_1, x_2)) = 2 + x_2 POL(MAXSORTA_IN_GA(x_1)) = 2*x_1 POL(U24_GA(x_1, x_2)) = 2 + 2*x_1 + 2*x_2 POL(U25_GA(x_1, x_2)) = 1 + 2*x_2 POL(U48_ga(x_1, x_2)) = 0 POL(U49_ga(x_1, x_2)) = 2 POL(U51_ga(x_1, x_2)) = x_2 POL(U52_ga(x_1, x_2)) = 2 POL(U53_ga(x_1, x_2)) = 2 + x_2 POL(U56_gga(x_1, x_2)) = 2 + x_2 POL([]) = 0 POL(delcG_in_gga(x_1, x_2)) = 2 + x_1 + x_2 POL(delcG_out_gga(x_1, x_2, x_3)) = x_3 POL(delcJ_in_ga(x_1)) = 2 + x_1 POL(delcJ_out_ga(x_1, x_2)) = 2 + x_2 POL(f) = 0 POL(maxcC_in_ga(x_1)) = 2*x_1 POL(maxcC_out_ga(x_1, x_2)) = 0 POL(maxcF_in_ga(x_1)) = 2 POL(maxcF_out_ga(x_1, x_2)) = 2 + x_2 POL(t) = 2 ---------------------------------------- (101) Obligation: Q DP problem: P is empty. The TRS R consists of the following rules: maxcC_in_ga(.(t, X1)) -> U48_ga(X1, maxcC_in_ga(X1)) maxcC_in_ga(.(f, X1)) -> U49_ga(X1, maxcC_in_ga(X1)) U49_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(f, X1), X2) U48_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(t, X1), X2) maxcF_in_ga([]) -> maxcF_out_ga([], f) maxcF_in_ga(.(f, X1)) -> U51_ga(X1, maxcF_in_ga(X1)) maxcF_in_ga(.(t, X1)) -> U52_ga(X1, maxcC_in_ga(X1)) U51_ga(X1, maxcF_out_ga(X1, X2)) -> maxcF_out_ga(.(f, X1), X2) delcG_in_gga(f, X1) -> delcG_out_gga(f, X1, .(f, X1)) delcG_in_gga(t, X1) -> U56_gga(X1, delcJ_in_ga(X1)) delcJ_in_ga([]) -> delcJ_out_ga([], []) delcJ_in_ga(.(f, X1)) -> U53_ga(X1, delcJ_in_ga(X1)) U56_gga(X1, delcJ_out_ga(X1, X2)) -> delcG_out_gga(t, X1, .(f, .(f, X2))) U53_ga(X1, delcJ_out_ga(X1, X2)) -> delcJ_out_ga(.(f, X1), .(f, X2)) The set Q consists of the following terms: maxcC_in_ga(x0) maxcF_in_ga(x0) delcG_in_gga(x0, x1) U48_ga(x0, x1) U49_ga(x0, x1) U51_ga(x0, x1) U52_ga(x0, x1) U56_gga(x0, x1) delcJ_in_ga(x0) U53_ga(x0, x1) We have to consider all (P,Q,R)-chains. ---------------------------------------- (102) PisEmptyProof (EQUIVALENT) The TRS P is empty. Hence, there is no (P,Q,R) chain. ---------------------------------------- (103) YES