5.62/2.34 YES 5.91/2.38 proof of /export/starexec/sandbox/benchmark/theBenchmark.pl 5.91/2.38 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 5.91/2.38 5.91/2.38 5.91/2.38 Left Termination of the query pattern 5.91/2.38 5.91/2.38 ss(g,g) 5.91/2.38 5.91/2.38 w.r.t. the given Prolog program could successfully be proven: 5.91/2.38 5.91/2.38 (0) Prolog 5.91/2.38 (1) PrologToDTProblemTransformerProof [SOUND, 60 ms] 5.91/2.38 (2) TRIPLES 5.91/2.38 (3) TriplesToPiDPProof [SOUND, 36 ms] 5.91/2.38 (4) PiDP 5.91/2.38 (5) DependencyGraphProof [EQUIVALENT, 0 ms] 5.91/2.38 (6) AND 5.91/2.38 (7) PiDP 5.91/2.38 (8) UsableRulesProof [EQUIVALENT, 0 ms] 5.91/2.38 (9) PiDP 5.91/2.38 (10) PiDPToQDPProof [EQUIVALENT, 0 ms] 5.91/2.38 (11) QDP 5.91/2.38 (12) QDPSizeChangeProof [EQUIVALENT, 0 ms] 5.91/2.38 (13) YES 5.91/2.38 (14) PiDP 5.91/2.38 (15) UsableRulesProof [EQUIVALENT, 0 ms] 5.91/2.38 (16) PiDP 5.91/2.38 (17) PiDPToQDPProof [EQUIVALENT, 0 ms] 5.91/2.38 (18) QDP 5.91/2.38 (19) QDPSizeChangeProof [EQUIVALENT, 0 ms] 5.91/2.38 (20) YES 5.91/2.38 (21) PiDP 5.91/2.38 (22) UsableRulesProof [EQUIVALENT, 0 ms] 5.91/2.38 (23) PiDP 5.91/2.38 (24) PiDPToQDPProof [SOUND, 0 ms] 5.91/2.38 (25) QDP 5.91/2.38 (26) QDPSizeChangeProof [EQUIVALENT, 0 ms] 5.91/2.38 (27) YES 5.91/2.38 (28) PiDP 5.91/2.38 (29) UsableRulesProof [EQUIVALENT, 0 ms] 5.91/2.38 (30) PiDP 5.91/2.38 (31) PiDPToQDPProof [SOUND, 0 ms] 5.91/2.38 (32) QDP 5.91/2.38 (33) QDPSizeChangeProof [EQUIVALENT, 0 ms] 5.91/2.38 (34) YES 5.91/2.38 (35) PiDP 5.91/2.38 (36) UsableRulesProof [EQUIVALENT, 0 ms] 5.91/2.38 (37) PiDP 5.91/2.38 (38) PiDPToQDPProof [SOUND, 0 ms] 5.91/2.38 (39) QDP 5.91/2.38 (40) QDPSizeChangeProof [EQUIVALENT, 0 ms] 5.91/2.38 (41) YES 5.91/2.38 5.91/2.38 5.91/2.38 ---------------------------------------- 5.91/2.38 5.91/2.38 (0) 5.91/2.38 Obligation: 5.91/2.38 Clauses: 5.91/2.38 5.91/2.38 ss(Xs, Ys) :- ','(perm(Xs, Ys), ordered(Ys)). 5.91/2.38 perm([], []). 5.91/2.38 perm(Xs, .(X, Ys)) :- ','(app(X1s, .(X, X2s), Xs), ','(app(X1s, X2s, Zs), perm(Zs, Ys))). 5.91/2.38 app([], X, X). 5.91/2.38 app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs). 5.91/2.38 ordered([]). 5.91/2.38 ordered(.(X1, [])). 5.91/2.38 ordered(.(X, .(Y, Xs))) :- ','(less(X, s(Y)), ordered(.(Y, Xs))). 5.91/2.38 less(0, s(X2)). 5.91/2.38 less(s(X), s(Y)) :- less(X, Y). 5.91/2.38 5.91/2.38 5.91/2.38 Query: ss(g,g) 5.91/2.38 ---------------------------------------- 5.91/2.38 5.91/2.38 (1) PrologToDTProblemTransformerProof (SOUND) 5.91/2.38 Built DT problem from termination graph DT10. 5.91/2.38 5.91/2.38 { 5.91/2.38 "root": 3, 5.91/2.38 "program": { 5.91/2.38 "directives": [], 5.91/2.38 "clauses": [ 5.91/2.38 [ 5.91/2.38 "(ss Xs Ys)", 5.91/2.38 "(',' (perm Xs Ys) (ordered Ys))" 5.91/2.38 ], 5.91/2.38 [ 5.91/2.38 "(perm ([]) ([]))", 5.91/2.38 null 5.91/2.38 ], 5.91/2.38 [ 5.91/2.38 "(perm Xs (. X Ys))", 5.91/2.38 "(',' (app X1s (. X X2s) Xs) (',' (app X1s X2s Zs) (perm Zs Ys)))" 5.91/2.38 ], 5.91/2.38 [ 5.91/2.38 "(app ([]) X X)", 5.91/2.38 null 5.91/2.38 ], 5.91/2.38 [ 5.91/2.38 "(app (. X Xs) Ys (. X Zs))", 5.91/2.38 "(app Xs Ys Zs)" 5.91/2.38 ], 5.91/2.38 [ 5.91/2.38 "(ordered ([]))", 5.91/2.38 null 5.91/2.38 ], 5.91/2.38 [ 5.91/2.38 "(ordered (. X1 ([])))", 5.91/2.38 null 5.91/2.38 ], 5.91/2.38 [ 5.91/2.38 "(ordered (. X (. Y Xs)))", 5.91/2.38 "(',' (less X (s Y)) (ordered (. Y Xs)))" 5.91/2.38 ], 5.91/2.38 [ 5.91/2.38 "(less (0) (s X2))", 5.91/2.38 null 5.91/2.38 ], 5.91/2.38 [ 5.91/2.38 "(less (s X) (s Y))", 5.91/2.38 "(less X Y)" 5.91/2.38 ] 5.91/2.38 ] 5.91/2.38 }, 5.91/2.38 "graph": { 5.91/2.38 "nodes": { 5.91/2.38 "type": "Nodes", 5.91/2.38 "590": { 5.91/2.38 "goal": [], 5.91/2.38 "kb": { 5.91/2.38 "nonunifying": [], 5.91/2.38 "intvars": {}, 5.91/2.38 "arithmetic": { 5.91/2.38 "type": "PlainIntegerRelationState", 5.91/2.38 "relations": [] 5.91/2.38 }, 5.91/2.38 "ground": [], 5.91/2.38 "free": [], 5.91/2.38 "exprvars": [] 5.91/2.38 } 5.91/2.38 }, 5.91/2.38 "393": { 5.91/2.38 "goal": [], 5.91/2.38 "kb": { 5.91/2.38 "nonunifying": [], 5.91/2.38 "intvars": {}, 5.91/2.38 "arithmetic": { 5.91/2.38 "type": "PlainIntegerRelationState", 5.91/2.38 "relations": [] 5.91/2.38 }, 5.91/2.38 "ground": [], 5.91/2.38 "free": [], 5.91/2.38 "exprvars": [] 5.91/2.38 } 5.91/2.38 }, 5.91/2.38 "591": { 5.91/2.38 "goal": [{ 5.91/2.38 "clause": -1, 5.91/2.38 "scope": -1, 5.91/2.38 "term": "(less T134 T135)" 5.91/2.38 }], 5.91/2.38 "kb": { 5.91/2.38 "nonunifying": [], 5.91/2.38 "intvars": {}, 5.91/2.38 "arithmetic": { 5.91/2.38 "type": "PlainIntegerRelationState", 5.91/2.38 "relations": [] 5.91/2.38 }, 5.91/2.38 "ground": [ 5.91/2.38 "T134", 5.91/2.38 "T135" 5.91/2.38 ], 5.91/2.38 "free": [], 5.91/2.38 "exprvars": [] 5.91/2.38 } 5.91/2.38 }, 5.91/2.38 "394": { 5.91/2.38 "goal": [{ 5.91/2.38 "clause": -1, 5.91/2.38 "scope": -1, 5.91/2.38 "term": "(',' (',' (app X23 (. 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"goal": [], 5.91/2.39 "kb": { 5.91/2.39 "nonunifying": [], 5.91/2.39 "intvars": {}, 5.91/2.39 "arithmetic": { 5.91/2.39 "type": "PlainIntegerRelationState", 5.91/2.39 "relations": [] 5.91/2.39 }, 5.91/2.39 "ground": [], 5.91/2.39 "free": [], 5.91/2.39 "exprvars": [] 5.91/2.39 } 5.91/2.39 }, 5.91/2.39 "587": { 5.91/2.39 "goal": [{ 5.91/2.39 "clause": 9, 5.91/2.39 "scope": 9, 5.91/2.39 "term": "(less T121 T122)" 5.91/2.39 }], 5.91/2.39 "kb": { 5.91/2.39 "nonunifying": [], 5.91/2.39 "intvars": {}, 5.91/2.39 "arithmetic": { 5.91/2.39 "type": "PlainIntegerRelationState", 5.91/2.39 "relations": [] 5.91/2.39 }, 5.91/2.39 "ground": [ 5.91/2.39 "T121", 5.91/2.39 "T122" 5.91/2.39 ], 5.91/2.39 "free": [], 5.91/2.39 "exprvars": [] 5.91/2.39 } 5.91/2.39 }, 5.91/2.39 "588": { 5.91/2.39 "goal": [{ 5.91/2.39 "clause": -1, 5.91/2.39 "scope": -1, 5.91/2.39 "term": "(true)" 5.91/2.39 }], 5.91/2.39 "kb": { 5.91/2.39 "nonunifying": [], 5.91/2.39 "intvars": {}, 5.91/2.39 "arithmetic": { 5.91/2.39 "type": "PlainIntegerRelationState", 5.91/2.39 "relations": [] 5.91/2.39 }, 5.91/2.39 "ground": [], 5.91/2.39 "free": [], 5.91/2.39 "exprvars": [] 5.91/2.39 } 5.91/2.39 }, 5.91/2.39 "589": { 5.91/2.39 "goal": [], 5.91/2.39 "kb": { 5.91/2.39 "nonunifying": [], 5.91/2.39 "intvars": {}, 5.91/2.39 "arithmetic": { 5.91/2.39 "type": "PlainIntegerRelationState", 5.91/2.39 "relations": [] 5.91/2.39 }, 5.91/2.39 "ground": [], 5.91/2.39 "free": [], 5.91/2.39 "exprvars": [] 5.91/2.39 } 5.91/2.39 }, 5.91/2.39 "40": { 5.91/2.39 "goal": [{ 5.91/2.39 "clause": -1, 5.91/2.39 "scope": -1, 5.91/2.39 "term": "(',' (perm T5 T6) (ordered T6))" 5.91/2.39 }], 5.91/2.39 "kb": { 5.91/2.39 "nonunifying": [], 5.91/2.39 "intvars": {}, 5.91/2.39 "arithmetic": { 5.91/2.39 "type": "PlainIntegerRelationState", 5.91/2.39 "relations": [] 5.91/2.39 }, 5.91/2.39 "ground": [ 5.91/2.39 "T5", 5.91/2.39 "T6" 5.91/2.39 ], 5.91/2.39 "free": [], 5.91/2.39 "exprvars": [] 5.91/2.39 } 5.91/2.39 } 5.91/2.39 }, 5.91/2.39 "edges": [ 5.91/2.39 { 5.91/2.39 "from": 3, 5.91/2.39 "to": 23, 5.91/2.39 "label": "CASE" 5.91/2.39 }, 5.91/2.39 { 5.91/2.39 "from": 23, 5.91/2.39 "to": 40, 5.91/2.39 "label": "ONLY EVAL with clause\nss(X5, X6) :- ','(perm(X5, X6), ordered(X6)).\nand substitutionT1 -> T5,\nX5 -> T5,\nT2 -> T6,\nX6 -> T6" 5.91/2.39 }, 5.91/2.39 { 5.91/2.39 "from": 40, 5.91/2.39 "to": 290, 5.91/2.39 "label": "CASE" 5.91/2.39 }, 5.91/2.39 { 5.91/2.39 "from": 290, 5.91/2.39 "to": 324, 5.91/2.39 "label": "PARALLEL" 5.91/2.39 }, 5.91/2.39 { 5.91/2.39 "from": 290, 5.91/2.39 "to": 325, 5.91/2.39 "label": "PARALLEL" 5.91/2.39 }, 5.91/2.39 { 5.91/2.39 "from": 324, 5.91/2.39 "to": 326, 5.91/2.39 "label": "EVAL with clause\nperm([], []).\nand substitutionT5 -> [],\nT6 -> []" 5.91/2.39 }, 5.91/2.39 { 5.91/2.39 "from": 324, 5.91/2.39 "to": 327, 5.91/2.39 "label": "EVAL-BACKTRACK" 5.91/2.39 }, 5.91/2.39 { 5.91/2.39 "from": 325, 5.91/2.39 "to": 394, 5.91/2.39 "label": "EVAL with clause\nperm(X20, .(X21, X22)) :- ','(app(X23, .(X21, X24), X20), ','(app(X23, X24, X25), perm(X25, X22))).\nand substitutionT5 -> T13,\nX20 -> T13,\nX21 -> T14,\nX22 -> T15,\nT6 -> .(T14, T15)" 5.91/2.39 }, 5.91/2.39 { 5.91/2.39 "from": 325, 5.91/2.39 "to": 395, 5.91/2.39 "label": "EVAL-BACKTRACK" 5.91/2.39 }, 5.91/2.39 { 5.91/2.39 "from": 326, 5.91/2.39 "to": 328, 5.91/2.39 "label": "CASE" 5.91/2.39 }, 5.91/2.39 { 5.91/2.39 "from": 328, 5.91/2.39 "to": 330, 5.91/2.39 "label": "PARALLEL" 5.91/2.39 }, 5.91/2.39 { 5.91/2.39 "from": 328, 5.91/2.39 "to": 332, 5.91/2.39 "label": "PARALLEL" 5.91/2.39 }, 5.91/2.39 { 5.91/2.39 "from": 330, 5.91/2.39 "to": 333, 5.91/2.39 "label": "ONLY EVAL with clause\nordered([]).\nand substitution" 5.91/2.39 }, 5.91/2.39 { 5.91/2.39 "from": 332, 5.91/2.39 "to": 338, 5.91/2.39 "label": "BACKTRACK\nfor clause: ordered(.(X1, []))because of non-unification" 5.91/2.39 }, 5.91/2.39 { 5.91/2.39 "from": 333, 5.91/2.39 "to": 334, 5.91/2.39 "label": "SUCCESS" 5.91/2.39 }, 5.91/2.39 { 5.91/2.39 "from": 338, 5.91/2.39 "to": 393, 5.91/2.39 "label": "BACKTRACK\nfor clause: ordered(.(X, .(Y, Xs))) :- ','(less(X, s(Y)), ordered(.(Y, Xs)))because of non-unification" 5.91/2.39 }, 5.91/2.39 { 5.91/2.39 "from": 394, 5.91/2.39 "to": 398, 5.91/2.39 "label": "SPLIT 1" 5.91/2.39 }, 5.91/2.39 { 5.91/2.39 "from": 394, 5.91/2.39 "to": 399, 5.91/2.39 "label": "SPLIT 2\nnew knowledge:\nT20 is ground\nT14 is ground\nT21 is ground\nT13 is ground\nreplacements:X23 -> T20,\nX24 -> T21" 5.91/2.39 }, 5.91/2.39 { 5.91/2.39 "from": 398, 5.91/2.39 "to": 402, 5.91/2.39 "label": "CASE" 5.91/2.39 }, 5.91/2.39 { 5.91/2.39 "from": 399, 5.91/2.39 "to": 440, 5.91/2.39 "label": "SPLIT 1" 5.91/2.39 }, 5.91/2.39 { 5.91/2.39 "from": 399, 5.91/2.39 "to": 441, 5.91/2.39 "label": "SPLIT 2\nnew knowledge:\nT20 is ground\nT21 is ground\nT51 is ground\nreplacements:X25 -> T51" 5.91/2.39 }, 5.91/2.39 { 5.91/2.39 "from": 402, 5.91/2.39 "to": 404, 5.91/2.39 "label": "PARALLEL" 5.91/2.39 }, 5.91/2.39 { 5.91/2.39 "from": 402, 5.91/2.39 "to": 405, 5.91/2.39 "label": "PARALLEL" 5.91/2.39 }, 5.91/2.39 { 5.91/2.39 "from": 404, 5.91/2.39 "to": 408, 5.91/2.39 "label": "EVAL with clause\napp([], X42, X42).\nand substitutionX23 -> [],\nT14 -> T34,\nX24 -> T35,\nX42 -> .(T34, T35),\nX43 -> T35,\nT13 -> .(T34, T35)" 5.91/2.39 }, 5.91/2.39 { 5.91/2.39 "from": 404, 5.91/2.39 "to": 409, 5.91/2.39 "label": "EVAL-BACKTRACK" 5.91/2.39 }, 5.91/2.39 { 5.91/2.39 "from": 405, 5.91/2.39 "to": 414, 5.91/2.39 "label": "EVAL with clause\napp(.(X58, X59), X60, .(X58, X61)) :- app(X59, X60, X61).\nand substitutionX58 -> T43,\nX59 -> X63,\nX23 -> .(T43, X63),\nT14 -> T42,\nX24 -> X64,\nX60 -> .(T42, X64),\nX62 -> T43,\nX61 -> T44,\nT13 -> .(T43, T44)" 5.91/2.39 }, 5.91/2.39 { 5.91/2.39 "from": 405, 5.91/2.39 "to": 415, 5.91/2.39 "label": "EVAL-BACKTRACK" 5.91/2.39 }, 5.91/2.39 { 5.91/2.39 "from": 408, 5.91/2.39 "to": 410, 5.91/2.39 "label": "SUCCESS" 5.91/2.39 }, 5.91/2.39 { 5.91/2.39 "from": 414, 5.91/2.39 "to": 398, 5.91/2.39 "label": "INSTANCE with matching:\nX23 -> X63\nT14 -> T42\nX24 -> X64\nT13 -> T44" 5.91/2.39 }, 5.91/2.39 { 5.91/2.39 "from": 440, 5.91/2.39 "to": 442, 5.91/2.39 "label": "CASE" 5.91/2.39 }, 5.91/2.39 { 5.91/2.39 "from": 441, 5.91/2.39 "to": 552, 5.91/2.39 "label": "SPLIT 1" 5.91/2.39 }, 5.91/2.39 { 5.91/2.39 "from": 441, 5.91/2.39 "to": 553, 5.91/2.39 "label": "SPLIT 2\nnew knowledge:\nT51 is ground\nT15 is ground" 5.91/2.39 }, 5.91/2.39 { 5.91/2.39 "from": 442, 5.91/2.39 "to": 445, 5.91/2.39 "label": "PARALLEL" 5.91/2.39 }, 5.91/2.39 { 5.91/2.39 "from": 442, 5.91/2.39 "to": 446, 5.91/2.39 "label": "PARALLEL" 5.91/2.39 }, 5.91/2.39 { 5.91/2.39 "from": 445, 5.91/2.39 "to": 448, 5.91/2.39 "label": "EVAL with clause\napp([], X77, X77).\nand substitutionT20 -> [],\nT21 -> T58,\nX77 -> T58,\nX25 -> T58" 5.91/2.39 }, 5.91/2.39 { 5.91/2.39 "from": 445, 5.91/2.39 "to": 449, 5.91/2.39 "label": "EVAL-BACKTRACK" 5.91/2.39 }, 5.91/2.39 { 5.91/2.39 "from": 446, 5.91/2.39 "to": 542, 5.91/2.39 "label": "EVAL with clause\napp(.(X88, X89), X90, .(X88, X91)) :- app(X89, X90, X91).\nand substitutionX88 -> T65,\nX89 -> T66,\nT20 -> .(T65, T66),\nT21 -> T67,\nX90 -> T67,\nX91 -> X92,\nX25 -> .(T65, X92)" 5.91/2.39 }, 5.91/2.39 { 5.91/2.39 "from": 446, 5.91/2.39 "to": 543, 5.91/2.39 "label": "EVAL-BACKTRACK" 5.91/2.39 }, 5.91/2.39 { 5.91/2.39 "from": 448, 5.91/2.39 "to": 450, 5.91/2.39 "label": "SUCCESS" 5.91/2.39 }, 5.91/2.39 { 5.91/2.39 "from": 542, 5.91/2.39 "to": 440, 5.91/2.39 "label": "INSTANCE with matching:\nT20 -> T66\nT21 -> T67\nX25 -> X92" 5.91/2.39 }, 5.91/2.39 { 5.91/2.39 "from": 552, 5.91/2.39 "to": 554, 5.91/2.39 "label": "CASE" 5.91/2.39 }, 5.91/2.39 { 5.91/2.39 "from": 553, 5.91/2.39 "to": 566, 5.91/2.39 "label": "CASE" 5.91/2.39 }, 5.91/2.39 { 5.91/2.39 "from": 554, 5.91/2.39 "to": 555, 5.91/2.39 "label": "PARALLEL" 5.91/2.39 }, 5.91/2.39 { 5.91/2.39 "from": 554, 5.91/2.39 "to": 556, 5.91/2.39 "label": "PARALLEL" 5.91/2.39 }, 5.91/2.39 { 5.91/2.39 "from": 555, 5.91/2.39 "to": 557, 5.91/2.39 "label": "EVAL with clause\nperm([], []).\nand substitutionT51 -> [],\nT15 -> []" 5.91/2.39 }, 5.91/2.39 { 5.91/2.39 "from": 555, 5.91/2.39 "to": 558, 5.91/2.39 "label": "EVAL-BACKTRACK" 5.91/2.39 }, 5.91/2.39 { 5.91/2.39 "from": 556, 5.91/2.39 "to": 560, 5.91/2.39 "label": "EVAL with clause\nperm(X104, .(X105, X106)) :- ','(app(X107, .(X105, X108), X104), ','(app(X107, X108, X109), perm(X109, X106))).\nand substitutionT51 -> T76,\nX104 -> T76,\nX105 -> T77,\nX106 -> T78,\nT15 -> .(T77, T78)" 5.91/2.39 }, 5.91/2.39 { 5.91/2.39 "from": 556, 5.91/2.39 "to": 561, 5.91/2.39 "label": "EVAL-BACKTRACK" 5.91/2.39 }, 5.91/2.39 { 5.91/2.39 "from": 557, 5.91/2.39 "to": 559, 5.91/2.39 "label": "SUCCESS" 5.91/2.39 }, 5.91/2.39 { 5.91/2.39 "from": 560, 5.91/2.39 "to": 562, 5.91/2.39 "label": "SPLIT 1" 5.91/2.39 }, 5.91/2.39 { 5.91/2.39 "from": 560, 5.91/2.39 "to": 563, 5.91/2.39 "label": "SPLIT 2\nnew knowledge:\nT83 is ground\nT77 is ground\nT84 is ground\nT76 is ground\nreplacements:X107 -> T83,\nX108 -> T84" 5.91/2.39 }, 5.91/2.39 { 5.91/2.39 "from": 562, 5.91/2.39 "to": 398, 5.91/2.39 "label": "INSTANCE with matching:\nX23 -> X107\nT14 -> T77\nX24 -> X108\nT13 -> T76" 5.91/2.39 }, 5.91/2.39 { 5.91/2.39 "from": 563, 5.91/2.39 "to": 564, 5.91/2.39 "label": "SPLIT 1" 5.91/2.39 }, 5.91/2.39 { 5.91/2.39 "from": 563, 5.91/2.39 "to": 565, 5.91/2.39 "label": "SPLIT 2\nnew knowledge:\nT83 is ground\nT84 is ground\nT91 is ground\nreplacements:X109 -> T91" 5.91/2.39 }, 5.91/2.39 { 5.91/2.39 "from": 564, 5.91/2.39 "to": 440, 5.91/2.39 "label": "INSTANCE with matching:\nT20 -> T83\nT21 -> T84\nX25 -> X109" 5.91/2.39 }, 5.91/2.39 { 5.91/2.39 "from": 565, 5.91/2.39 "to": 552, 5.91/2.39 "label": "INSTANCE with matching:\nT51 -> T91\nT15 -> T78" 5.91/2.39 }, 5.91/2.39 { 5.91/2.39 "from": 566, 5.91/2.39 "to": 567, 5.91/2.39 "label": "BACKTRACK\nfor clause: ordered([])because of non-unification" 5.91/2.39 }, 5.91/2.39 { 5.91/2.39 "from": 567, 5.91/2.39 "to": 568, 5.91/2.39 "label": "PARALLEL" 5.91/2.39 }, 5.91/2.39 { 5.91/2.39 "from": 567, 5.91/2.39 "to": 569, 5.91/2.39 "label": "PARALLEL" 5.91/2.39 }, 5.91/2.39 { 5.91/2.39 "from": 568, 5.91/2.39 "to": 570, 5.91/2.39 "label": "EVAL with clause\nordered(.(X126, [])).\nand substitutionT14 -> T98,\nX126 -> T98,\nT15 -> []" 5.91/2.39 }, 5.91/2.39 { 5.91/2.39 "from": 568, 5.91/2.39 "to": 571, 5.91/2.39 "label": "EVAL-BACKTRACK" 5.91/2.39 }, 5.91/2.39 { 5.91/2.39 "from": 569, 5.91/2.39 "to": 573, 5.91/2.39 "label": "EVAL with clause\nordered(.(X133, .(X134, X135))) :- ','(less(X133, s(X134)), ordered(.(X134, X135))).\nand substitutionT14 -> T105,\nX133 -> T105,\nX134 -> T106,\nX135 -> T107,\nT15 -> .(T106, T107)" 5.91/2.39 }, 5.91/2.39 { 5.91/2.39 "from": 569, 5.91/2.39 "to": 574, 5.91/2.39 "label": "EVAL-BACKTRACK" 5.91/2.39 }, 5.91/2.39 { 5.91/2.39 "from": 570, 5.91/2.39 "to": 572, 5.91/2.39 "label": "SUCCESS" 5.91/2.39 }, 5.91/2.39 { 5.91/2.39 "from": 573, 5.91/2.39 "to": 575, 5.91/2.39 "label": "SPLIT 1" 5.91/2.39 }, 5.91/2.39 { 5.91/2.39 "from": 573, 5.91/2.39 "to": 576, 5.91/2.39 "label": "SPLIT 2\nnew knowledge:\nT105 is ground\nT106 is ground" 5.91/2.39 }, 5.91/2.39 { 5.91/2.39 "from": 575, 5.91/2.39 "to": 577, 5.91/2.39 "label": "CASE" 5.91/2.39 }, 5.91/2.39 { 5.91/2.39 "from": 576, 5.91/2.39 "to": 553, 5.91/2.39 "label": "INSTANCE with matching:\nT14 -> T106\nT15 -> T107" 5.91/2.39 }, 5.91/2.39 { 5.91/2.39 "from": 577, 5.91/2.39 "to": 578, 5.91/2.39 "label": "PARALLEL" 5.91/2.39 }, 5.91/2.39 { 5.91/2.39 "from": 577, 5.91/2.39 "to": 579, 5.91/2.39 "label": "PARALLEL" 5.91/2.39 }, 5.91/2.39 { 5.91/2.39 "from": 578, 5.91/2.39 "to": 580, 5.91/2.39 "label": "EVAL with clause\nless(0, s(X144)).\nand substitutionT105 -> 0,\nT106 -> T116,\nX144 -> T116" 5.91/2.39 }, 5.91/2.39 { 5.91/2.39 "from": 578, 5.91/2.39 "to": 581, 5.91/2.39 "label": "EVAL-BACKTRACK" 5.91/2.39 }, 5.91/2.39 { 5.91/2.39 "from": 579, 5.91/2.39 "to": 583, 5.91/2.39 "label": "EVAL with clause\nless(s(X149), s(X150)) :- less(X149, X150).\nand substitutionX149 -> T121,\nT105 -> s(T121),\nT106 -> T122,\nX150 -> T122" 5.91/2.39 }, 5.91/2.39 { 5.91/2.39 "from": 579, 5.91/2.39 "to": 584, 5.91/2.39 "label": "EVAL-BACKTRACK" 5.91/2.39 }, 5.91/2.39 { 5.91/2.39 "from": 580, 5.91/2.39 "to": 582, 5.91/2.39 "label": "SUCCESS" 5.91/2.39 }, 5.91/2.39 { 5.91/2.39 "from": 583, 5.91/2.39 "to": 585, 5.91/2.39 "label": "CASE" 5.91/2.39 }, 5.91/2.39 { 5.91/2.39 "from": 585, 5.91/2.39 "to": 586, 5.91/2.39 "label": "PARALLEL" 5.91/2.39 }, 5.91/2.39 { 5.91/2.39 "from": 585, 5.91/2.39 "to": 587, 5.91/2.39 "label": "PARALLEL" 5.91/2.39 }, 5.91/2.39 { 5.91/2.39 "from": 586, 5.91/2.39 "to": 588, 5.91/2.39 "label": "EVAL with clause\nless(0, s(X157)).\nand substitutionT121 -> 0,\nX157 -> T129,\nT122 -> s(T129)" 5.91/2.39 }, 5.91/2.39 { 5.91/2.39 "from": 586, 5.91/2.39 "to": 589, 5.91/2.39 "label": "EVAL-BACKTRACK" 5.91/2.39 }, 5.91/2.39 { 5.91/2.39 "from": 587, 5.91/2.39 "to": 591, 5.91/2.39 "label": "EVAL with clause\nless(s(X162), s(X163)) :- less(X162, X163).\nand substitutionX162 -> T134,\nT121 -> s(T134),\nX163 -> T135,\nT122 -> s(T135)" 5.91/2.39 }, 5.91/2.39 { 5.91/2.39 "from": 587, 5.91/2.39 "to": 592, 5.91/2.39 "label": "EVAL-BACKTRACK" 5.91/2.39 }, 5.91/2.39 { 5.91/2.39 "from": 588, 5.91/2.39 "to": 590, 5.91/2.39 "label": "SUCCESS" 5.91/2.39 }, 5.91/2.39 { 5.91/2.39 "from": 591, 5.91/2.39 "to": 583, 5.91/2.39 "label": "INSTANCE with matching:\nT121 -> T134\nT122 -> T135" 5.91/2.39 } 5.91/2.39 ], 5.91/2.39 "type": "Graph" 5.91/2.39 } 5.91/2.39 } 5.91/2.39 5.91/2.39 ---------------------------------------- 5.91/2.39 5.91/2.39 (2) 5.91/2.39 Obligation: 5.91/2.39 Triples: 5.91/2.39 5.91/2.39 appA(.(X1, X2), X3, X4, .(X1, X5)) :- appA(X2, X3, X4, X5). 5.91/2.39 appB(.(X1, X2), X3, .(X1, X4)) :- appB(X2, X3, X4). 5.91/2.39 permC(X1, .(X2, X3)) :- appA(X4, X2, X5, X1). 5.91/2.39 permC(X1, .(X2, X3)) :- ','(appcA(X4, X2, X5, X1), appB(X4, X5, X6)). 5.91/2.39 permC(X1, .(X2, X3)) :- ','(appcA(X4, X2, X5, X1), ','(appcB(X4, X5, X6), permC(X6, X3))). 5.91/2.39 orderedD(s(X1), .(X2, X3)) :- lessF(X1, X2). 5.91/2.39 orderedD(X1, .(X2, X3)) :- ','(lesscE(X1, X2), orderedD(X2, X3)). 5.91/2.39 lessF(s(X1), s(X2)) :- lessF(X1, X2). 5.91/2.39 ssG(X1, .(X2, X3)) :- appA(X4, X2, X5, X1). 5.91/2.39 ssG(X1, .(X2, X3)) :- ','(appcA(X4, X2, X5, X1), appB(X4, X5, X6)). 5.91/2.39 ssG(X1, .(X2, X3)) :- ','(appcA(X4, X2, X5, X1), ','(appcB(X4, X5, X6), permC(X6, X3))). 5.91/2.39 ssG(X1, .(X2, X3)) :- ','(appcA(X4, X2, X5, X1), ','(appcB(X4, X5, X6), ','(permcC(X6, X3), orderedD(X2, X3)))). 5.91/2.39 5.91/2.39 Clauses: 5.91/2.39 5.91/2.39 appcA([], X1, X2, .(X1, X2)). 5.91/2.39 appcA(.(X1, X2), X3, X4, .(X1, X5)) :- appcA(X2, X3, X4, X5). 5.91/2.39 appcB([], X1, X1). 5.91/2.39 appcB(.(X1, X2), X3, .(X1, X4)) :- appcB(X2, X3, X4). 5.91/2.39 permcC([], []). 5.91/2.39 permcC(X1, .(X2, X3)) :- ','(appcA(X4, X2, X5, X1), ','(appcB(X4, X5, X6), permcC(X6, X3))). 5.91/2.39 orderedcD(X1, []). 5.91/2.39 orderedcD(X1, .(X2, X3)) :- ','(lesscE(X1, X2), orderedcD(X2, X3)). 5.91/2.39 lesscF(0, s(X1)). 5.91/2.39 lesscF(s(X1), s(X2)) :- lesscF(X1, X2). 5.91/2.39 lesscE(0, X1). 5.91/2.39 lesscE(s(X1), X2) :- lesscF(X1, X2). 5.91/2.39 5.91/2.39 Afs: 5.91/2.39 5.91/2.39 ssG(x1, x2) = ssG(x1, x2) 5.91/2.39 5.91/2.39 5.91/2.39 ---------------------------------------- 5.91/2.39 5.91/2.39 (3) TriplesToPiDPProof (SOUND) 5.91/2.39 We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes: 5.91/2.39 5.91/2.39 ssG_in_2: (b,b) 5.91/2.39 5.91/2.39 appA_in_4: (f,b,f,b) 5.91/2.39 5.91/2.39 appcA_in_4: (f,b,f,b) 5.91/2.39 5.91/2.39 appB_in_3: (b,b,f) 5.91/2.39 5.91/2.39 appcB_in_3: (b,b,f) 5.91/2.39 5.91/2.39 permC_in_2: (b,b) 5.91/2.39 5.91/2.39 permcC_in_2: (b,b) 5.91/2.39 5.91/2.39 orderedD_in_2: (b,b) 5.91/2.39 5.91/2.39 lessF_in_2: (b,b) 5.91/2.39 5.91/2.39 lesscE_in_2: (b,b) 5.91/2.39 5.91/2.39 lesscF_in_2: (b,b) 5.91/2.39 5.91/2.39 Transforming TRIPLES into the following Term Rewriting System: 5.91/2.39 5.91/2.39 Pi DP problem: 5.91/2.39 The TRS P consists of the following rules: 5.91/2.39 5.91/2.39 SSG_IN_GG(X1, .(X2, X3)) -> U12_GG(X1, X2, X3, appA_in_agag(X4, X2, X5, X1)) 5.91/2.39 SSG_IN_GG(X1, .(X2, X3)) -> APPA_IN_AGAG(X4, X2, X5, X1) 5.91/2.39 APPA_IN_AGAG(.(X1, X2), X3, X4, .(X1, X5)) -> U1_AGAG(X1, X2, X3, X4, X5, appA_in_agag(X2, X3, X4, X5)) 5.91/2.39 APPA_IN_AGAG(.(X1, X2), X3, X4, .(X1, X5)) -> APPA_IN_AGAG(X2, X3, X4, X5) 5.91/2.39 SSG_IN_GG(X1, .(X2, X3)) -> U13_GG(X1, X2, X3, appcA_in_agag(X4, X2, X5, X1)) 5.91/2.39 U13_GG(X1, X2, X3, appcA_out_agag(X4, X2, X5, X1)) -> U14_GG(X1, X2, X3, appB_in_gga(X4, X5, X6)) 5.91/2.39 U13_GG(X1, X2, X3, appcA_out_agag(X4, X2, X5, X1)) -> APPB_IN_GGA(X4, X5, X6) 5.91/2.39 APPB_IN_GGA(.(X1, X2), X3, .(X1, X4)) -> U2_GGA(X1, X2, X3, X4, appB_in_gga(X2, X3, X4)) 5.91/2.39 APPB_IN_GGA(.(X1, X2), X3, .(X1, X4)) -> APPB_IN_GGA(X2, X3, X4) 5.91/2.39 U13_GG(X1, X2, X3, appcA_out_agag(X4, X2, X5, X1)) -> U15_GG(X1, X2, X3, appcB_in_gga(X4, X5, X6)) 5.91/2.39 U15_GG(X1, X2, X3, appcB_out_gga(X4, X5, X6)) -> U16_GG(X1, X2, X3, permC_in_gg(X6, X3)) 5.91/2.39 U15_GG(X1, X2, X3, appcB_out_gga(X4, X5, X6)) -> PERMC_IN_GG(X6, X3) 5.91/2.39 PERMC_IN_GG(X1, .(X2, X3)) -> U3_GG(X1, X2, X3, appA_in_agag(X4, X2, X5, X1)) 5.91/2.39 PERMC_IN_GG(X1, .(X2, X3)) -> APPA_IN_AGAG(X4, X2, X5, X1) 5.91/2.39 PERMC_IN_GG(X1, .(X2, X3)) -> U4_GG(X1, X2, X3, appcA_in_agag(X4, X2, X5, X1)) 5.91/2.39 U4_GG(X1, X2, X3, appcA_out_agag(X4, X2, X5, X1)) -> U5_GG(X1, X2, X3, appB_in_gga(X4, X5, X6)) 5.91/2.39 U4_GG(X1, X2, X3, appcA_out_agag(X4, X2, X5, X1)) -> APPB_IN_GGA(X4, X5, X6) 5.91/2.39 U4_GG(X1, X2, X3, appcA_out_agag(X4, X2, X5, X1)) -> U6_GG(X1, X2, X3, appcB_in_gga(X4, X5, X6)) 5.91/2.39 U6_GG(X1, X2, X3, appcB_out_gga(X4, X5, X6)) -> U7_GG(X1, X2, X3, permC_in_gg(X6, X3)) 5.91/2.39 U6_GG(X1, X2, X3, appcB_out_gga(X4, X5, X6)) -> PERMC_IN_GG(X6, X3) 5.91/2.39 U15_GG(X1, X2, X3, appcB_out_gga(X4, X5, X6)) -> U17_GG(X1, X2, X3, permcC_in_gg(X6, X3)) 5.91/2.39 U17_GG(X1, X2, X3, permcC_out_gg(X6, X3)) -> U18_GG(X1, X2, X3, orderedD_in_gg(X2, X3)) 5.91/2.39 U17_GG(X1, X2, X3, permcC_out_gg(X6, X3)) -> ORDEREDD_IN_GG(X2, X3) 5.91/2.39 ORDEREDD_IN_GG(s(X1), .(X2, X3)) -> U8_GG(X1, X2, X3, lessF_in_gg(X1, X2)) 5.91/2.39 ORDEREDD_IN_GG(s(X1), .(X2, X3)) -> LESSF_IN_GG(X1, X2) 5.91/2.39 LESSF_IN_GG(s(X1), s(X2)) -> U11_GG(X1, X2, lessF_in_gg(X1, X2)) 5.91/2.39 LESSF_IN_GG(s(X1), s(X2)) -> LESSF_IN_GG(X1, X2) 5.91/2.39 ORDEREDD_IN_GG(X1, .(X2, X3)) -> U9_GG(X1, X2, X3, lesscE_in_gg(X1, X2)) 5.91/2.39 U9_GG(X1, X2, X3, lesscE_out_gg(X1, X2)) -> U10_GG(X1, X2, X3, orderedD_in_gg(X2, X3)) 5.91/2.39 U9_GG(X1, X2, X3, lesscE_out_gg(X1, X2)) -> ORDEREDD_IN_GG(X2, X3) 5.91/2.39 5.91/2.39 The TRS R consists of the following rules: 5.91/2.39 5.91/2.39 appcA_in_agag([], X1, X2, .(X1, X2)) -> appcA_out_agag([], X1, X2, .(X1, X2)) 5.91/2.39 appcA_in_agag(.(X1, X2), X3, X4, .(X1, X5)) -> U20_agag(X1, X2, X3, X4, X5, appcA_in_agag(X2, X3, X4, X5)) 5.91/2.39 U20_agag(X1, X2, X3, X4, X5, appcA_out_agag(X2, X3, X4, X5)) -> appcA_out_agag(.(X1, X2), X3, X4, .(X1, X5)) 5.91/2.39 appcB_in_gga([], X1, X1) -> appcB_out_gga([], X1, X1) 5.91/2.39 appcB_in_gga(.(X1, X2), X3, .(X1, X4)) -> U21_gga(X1, X2, X3, X4, appcB_in_gga(X2, X3, X4)) 5.91/2.39 U21_gga(X1, X2, X3, X4, appcB_out_gga(X2, X3, X4)) -> appcB_out_gga(.(X1, X2), X3, .(X1, X4)) 5.91/2.39 permcC_in_gg([], []) -> permcC_out_gg([], []) 5.91/2.39 permcC_in_gg(X1, .(X2, X3)) -> U22_gg(X1, X2, X3, appcA_in_agag(X4, X2, X5, X1)) 5.91/2.39 U22_gg(X1, X2, X3, appcA_out_agag(X4, X2, X5, X1)) -> U23_gg(X1, X2, X3, appcB_in_gga(X4, X5, X6)) 5.91/2.39 U23_gg(X1, X2, X3, appcB_out_gga(X4, X5, X6)) -> U24_gg(X1, X2, X3, permcC_in_gg(X6, X3)) 5.91/2.39 U24_gg(X1, X2, X3, permcC_out_gg(X6, X3)) -> permcC_out_gg(X1, .(X2, X3)) 5.91/2.39 lesscE_in_gg(0, X1) -> lesscE_out_gg(0, X1) 5.91/2.39 lesscE_in_gg(s(X1), X2) -> U28_gg(X1, X2, lesscF_in_gg(X1, X2)) 5.91/2.39 lesscF_in_gg(0, s(X1)) -> lesscF_out_gg(0, s(X1)) 5.91/2.39 lesscF_in_gg(s(X1), s(X2)) -> U27_gg(X1, X2, lesscF_in_gg(X1, X2)) 5.91/2.39 U27_gg(X1, X2, lesscF_out_gg(X1, X2)) -> lesscF_out_gg(s(X1), s(X2)) 5.91/2.39 U28_gg(X1, X2, lesscF_out_gg(X1, X2)) -> lesscE_out_gg(s(X1), X2) 5.91/2.39 5.91/2.39 The argument filtering Pi contains the following mapping: 5.91/2.39 .(x1, x2) = .(x1, x2) 5.91/2.39 5.91/2.39 appA_in_agag(x1, x2, x3, x4) = appA_in_agag(x2, x4) 5.91/2.39 5.91/2.39 appcA_in_agag(x1, x2, x3, x4) = appcA_in_agag(x2, x4) 5.91/2.39 5.91/2.39 appcA_out_agag(x1, x2, x3, x4) = appcA_out_agag(x1, x2, x3, x4) 5.91/2.39 5.91/2.39 U20_agag(x1, x2, x3, x4, x5, x6) = U20_agag(x1, x3, x5, x6) 5.91/2.39 5.91/2.39 appB_in_gga(x1, x2, x3) = appB_in_gga(x1, x2) 5.91/2.39 5.91/2.39 appcB_in_gga(x1, x2, x3) = appcB_in_gga(x1, x2) 5.91/2.39 5.91/2.39 [] = [] 5.91/2.39 5.91/2.39 appcB_out_gga(x1, x2, x3) = appcB_out_gga(x1, x2, x3) 5.91/2.39 5.91/2.39 U21_gga(x1, x2, x3, x4, x5) = U21_gga(x1, x2, x3, x5) 5.91/2.39 5.91/2.39 permC_in_gg(x1, x2) = permC_in_gg(x1, x2) 5.91/2.39 5.91/2.39 permcC_in_gg(x1, x2) = permcC_in_gg(x1, x2) 5.91/2.39 5.91/2.39 permcC_out_gg(x1, x2) = permcC_out_gg(x1, x2) 5.91/2.39 5.91/2.39 U22_gg(x1, x2, x3, x4) = U22_gg(x1, x2, x3, x4) 5.91/2.39 5.91/2.39 U23_gg(x1, x2, x3, x4) = U23_gg(x1, x2, x3, x4) 5.91/2.39 5.91/2.39 U24_gg(x1, x2, x3, x4) = U24_gg(x1, x2, x3, x4) 5.91/2.39 5.91/2.39 orderedD_in_gg(x1, x2) = orderedD_in_gg(x1, x2) 5.91/2.39 5.91/2.39 s(x1) = s(x1) 5.91/2.39 5.91/2.39 lessF_in_gg(x1, x2) = lessF_in_gg(x1, x2) 5.91/2.39 5.91/2.39 lesscE_in_gg(x1, x2) = lesscE_in_gg(x1, x2) 5.91/2.39 5.91/2.39 0 = 0 5.91/2.39 5.91/2.39 lesscE_out_gg(x1, x2) = lesscE_out_gg(x1, x2) 5.91/2.39 5.91/2.39 U28_gg(x1, x2, x3) = U28_gg(x1, x2, x3) 5.91/2.39 5.91/2.39 lesscF_in_gg(x1, x2) = lesscF_in_gg(x1, x2) 5.91/2.39 5.91/2.39 lesscF_out_gg(x1, x2) = lesscF_out_gg(x1, x2) 5.91/2.39 5.91/2.39 U27_gg(x1, x2, x3) = U27_gg(x1, x2, x3) 5.91/2.39 5.91/2.39 SSG_IN_GG(x1, x2) = SSG_IN_GG(x1, x2) 5.91/2.39 5.91/2.39 U12_GG(x1, x2, x3, x4) = U12_GG(x1, x2, x3, x4) 5.91/2.39 5.91/2.39 APPA_IN_AGAG(x1, x2, x3, x4) = APPA_IN_AGAG(x2, x4) 5.91/2.39 5.91/2.39 U1_AGAG(x1, x2, x3, x4, x5, x6) = U1_AGAG(x1, x3, x5, x6) 5.91/2.39 5.91/2.39 U13_GG(x1, x2, x3, x4) = U13_GG(x1, x2, x3, x4) 5.91/2.39 5.91/2.39 U14_GG(x1, x2, x3, x4) = U14_GG(x1, x2, x3, x4) 5.91/2.39 5.91/2.39 APPB_IN_GGA(x1, x2, x3) = APPB_IN_GGA(x1, x2) 5.91/2.39 5.91/2.39 U2_GGA(x1, x2, x3, x4, x5) = U2_GGA(x1, x2, x3, x5) 5.91/2.39 5.91/2.39 U15_GG(x1, x2, x3, x4) = U15_GG(x1, x2, x3, x4) 5.91/2.39 5.91/2.39 U16_GG(x1, x2, x3, x4) = U16_GG(x1, x2, x3, x4) 5.91/2.39 5.91/2.39 PERMC_IN_GG(x1, x2) = PERMC_IN_GG(x1, x2) 5.91/2.39 5.91/2.39 U3_GG(x1, x2, x3, x4) = U3_GG(x1, x2, x3, x4) 5.91/2.39 5.91/2.39 U4_GG(x1, x2, x3, x4) = U4_GG(x1, x2, x3, x4) 5.91/2.39 5.91/2.39 U5_GG(x1, x2, x3, x4) = U5_GG(x1, x2, x3, x4) 5.91/2.39 5.91/2.39 U6_GG(x1, x2, x3, x4) = U6_GG(x1, x2, x3, x4) 5.91/2.39 5.91/2.39 U7_GG(x1, x2, x3, x4) = U7_GG(x1, x2, x3, x4) 5.91/2.39 5.91/2.39 U17_GG(x1, x2, x3, x4) = U17_GG(x1, x2, x3, x4) 5.91/2.39 5.91/2.39 U18_GG(x1, x2, x3, x4) = U18_GG(x1, x2, x3, x4) 5.91/2.39 5.91/2.39 ORDEREDD_IN_GG(x1, x2) = ORDEREDD_IN_GG(x1, x2) 5.91/2.39 5.91/2.39 U8_GG(x1, x2, x3, x4) = U8_GG(x1, x2, x3, x4) 5.91/2.39 5.91/2.39 LESSF_IN_GG(x1, x2) = LESSF_IN_GG(x1, x2) 5.91/2.39 5.91/2.39 U11_GG(x1, x2, x3) = U11_GG(x1, x2, x3) 5.91/2.39 5.91/2.39 U9_GG(x1, x2, x3, x4) = U9_GG(x1, x2, x3, x4) 5.91/2.39 5.91/2.39 U10_GG(x1, x2, x3, x4) = U10_GG(x1, x2, x3, x4) 5.91/2.39 5.91/2.39 5.91/2.39 We have to consider all (P,R,Pi)-chains 5.91/2.39 5.91/2.39 5.91/2.39 Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES 5.91/2.39 5.91/2.39 5.91/2.39 5.91/2.39 ---------------------------------------- 5.91/2.39 5.91/2.39 (4) 5.91/2.39 Obligation: 5.91/2.39 Pi DP problem: 5.91/2.39 The TRS P consists of the following rules: 5.91/2.39 5.91/2.39 SSG_IN_GG(X1, .(X2, X3)) -> U12_GG(X1, X2, X3, appA_in_agag(X4, X2, X5, X1)) 5.91/2.39 SSG_IN_GG(X1, .(X2, X3)) -> APPA_IN_AGAG(X4, X2, X5, X1) 5.91/2.39 APPA_IN_AGAG(.(X1, X2), X3, X4, .(X1, X5)) -> U1_AGAG(X1, X2, X3, X4, X5, appA_in_agag(X2, X3, X4, X5)) 5.91/2.39 APPA_IN_AGAG(.(X1, X2), X3, X4, .(X1, X5)) -> APPA_IN_AGAG(X2, X3, X4, X5) 5.91/2.39 SSG_IN_GG(X1, .(X2, X3)) -> U13_GG(X1, X2, X3, appcA_in_agag(X4, X2, X5, X1)) 5.91/2.39 U13_GG(X1, X2, X3, appcA_out_agag(X4, X2, X5, X1)) -> U14_GG(X1, X2, X3, appB_in_gga(X4, X5, X6)) 5.91/2.39 U13_GG(X1, X2, X3, appcA_out_agag(X4, X2, X5, X1)) -> APPB_IN_GGA(X4, X5, X6) 5.91/2.39 APPB_IN_GGA(.(X1, X2), X3, .(X1, X4)) -> U2_GGA(X1, X2, X3, X4, appB_in_gga(X2, X3, X4)) 5.91/2.39 APPB_IN_GGA(.(X1, X2), X3, .(X1, X4)) -> APPB_IN_GGA(X2, X3, X4) 5.91/2.39 U13_GG(X1, X2, X3, appcA_out_agag(X4, X2, X5, X1)) -> U15_GG(X1, X2, X3, appcB_in_gga(X4, X5, X6)) 5.91/2.39 U15_GG(X1, X2, X3, appcB_out_gga(X4, X5, X6)) -> U16_GG(X1, X2, X3, permC_in_gg(X6, X3)) 5.91/2.39 U15_GG(X1, X2, X3, appcB_out_gga(X4, X5, X6)) -> PERMC_IN_GG(X6, X3) 5.91/2.39 PERMC_IN_GG(X1, .(X2, X3)) -> U3_GG(X1, X2, X3, appA_in_agag(X4, X2, X5, X1)) 5.91/2.39 PERMC_IN_GG(X1, .(X2, X3)) -> APPA_IN_AGAG(X4, X2, X5, X1) 5.91/2.39 PERMC_IN_GG(X1, .(X2, X3)) -> U4_GG(X1, X2, X3, appcA_in_agag(X4, X2, X5, X1)) 5.91/2.39 U4_GG(X1, X2, X3, appcA_out_agag(X4, X2, X5, X1)) -> U5_GG(X1, X2, X3, appB_in_gga(X4, X5, X6)) 5.91/2.39 U4_GG(X1, X2, X3, appcA_out_agag(X4, X2, X5, X1)) -> APPB_IN_GGA(X4, X5, X6) 5.91/2.39 U4_GG(X1, X2, X3, appcA_out_agag(X4, X2, X5, X1)) -> U6_GG(X1, X2, X3, appcB_in_gga(X4, X5, X6)) 5.91/2.39 U6_GG(X1, X2, X3, appcB_out_gga(X4, X5, X6)) -> U7_GG(X1, X2, X3, permC_in_gg(X6, X3)) 5.91/2.39 U6_GG(X1, X2, X3, appcB_out_gga(X4, X5, X6)) -> PERMC_IN_GG(X6, X3) 5.91/2.39 U15_GG(X1, X2, X3, appcB_out_gga(X4, X5, X6)) -> U17_GG(X1, X2, X3, permcC_in_gg(X6, X3)) 5.91/2.39 U17_GG(X1, X2, X3, permcC_out_gg(X6, X3)) -> U18_GG(X1, X2, X3, orderedD_in_gg(X2, X3)) 5.91/2.39 U17_GG(X1, X2, X3, permcC_out_gg(X6, X3)) -> ORDEREDD_IN_GG(X2, X3) 5.91/2.39 ORDEREDD_IN_GG(s(X1), .(X2, X3)) -> U8_GG(X1, X2, X3, lessF_in_gg(X1, X2)) 5.91/2.39 ORDEREDD_IN_GG(s(X1), .(X2, X3)) -> LESSF_IN_GG(X1, X2) 5.91/2.39 LESSF_IN_GG(s(X1), s(X2)) -> U11_GG(X1, X2, lessF_in_gg(X1, X2)) 5.91/2.39 LESSF_IN_GG(s(X1), s(X2)) -> LESSF_IN_GG(X1, X2) 5.91/2.39 ORDEREDD_IN_GG(X1, .(X2, X3)) -> U9_GG(X1, X2, X3, lesscE_in_gg(X1, X2)) 5.91/2.39 U9_GG(X1, X2, X3, lesscE_out_gg(X1, X2)) -> U10_GG(X1, X2, X3, orderedD_in_gg(X2, X3)) 5.91/2.39 U9_GG(X1, X2, X3, lesscE_out_gg(X1, X2)) -> ORDEREDD_IN_GG(X2, X3) 5.91/2.39 5.91/2.39 The TRS R consists of the following rules: 5.91/2.39 5.91/2.39 appcA_in_agag([], X1, X2, .(X1, X2)) -> appcA_out_agag([], X1, X2, .(X1, X2)) 5.91/2.39 appcA_in_agag(.(X1, X2), X3, X4, .(X1, X5)) -> U20_agag(X1, X2, X3, X4, X5, appcA_in_agag(X2, X3, X4, X5)) 5.91/2.39 U20_agag(X1, X2, X3, X4, X5, appcA_out_agag(X2, X3, X4, X5)) -> appcA_out_agag(.(X1, X2), X3, X4, .(X1, X5)) 5.91/2.39 appcB_in_gga([], X1, X1) -> appcB_out_gga([], X1, X1) 5.91/2.39 appcB_in_gga(.(X1, X2), X3, .(X1, X4)) -> U21_gga(X1, X2, X3, X4, appcB_in_gga(X2, X3, X4)) 5.91/2.39 U21_gga(X1, X2, X3, X4, appcB_out_gga(X2, X3, X4)) -> appcB_out_gga(.(X1, X2), X3, .(X1, X4)) 5.91/2.39 permcC_in_gg([], []) -> permcC_out_gg([], []) 5.91/2.39 permcC_in_gg(X1, .(X2, X3)) -> U22_gg(X1, X2, X3, appcA_in_agag(X4, X2, X5, X1)) 5.91/2.39 U22_gg(X1, X2, X3, appcA_out_agag(X4, X2, X5, X1)) -> U23_gg(X1, X2, X3, appcB_in_gga(X4, X5, X6)) 5.91/2.39 U23_gg(X1, X2, X3, appcB_out_gga(X4, X5, X6)) -> U24_gg(X1, X2, X3, permcC_in_gg(X6, X3)) 5.91/2.39 U24_gg(X1, X2, X3, permcC_out_gg(X6, X3)) -> permcC_out_gg(X1, .(X2, X3)) 5.91/2.39 lesscE_in_gg(0, X1) -> lesscE_out_gg(0, X1) 5.91/2.39 lesscE_in_gg(s(X1), X2) -> U28_gg(X1, X2, lesscF_in_gg(X1, X2)) 5.91/2.39 lesscF_in_gg(0, s(X1)) -> lesscF_out_gg(0, s(X1)) 5.91/2.39 lesscF_in_gg(s(X1), s(X2)) -> U27_gg(X1, X2, lesscF_in_gg(X1, X2)) 5.91/2.39 U27_gg(X1, X2, lesscF_out_gg(X1, X2)) -> lesscF_out_gg(s(X1), s(X2)) 5.91/2.39 U28_gg(X1, X2, lesscF_out_gg(X1, X2)) -> lesscE_out_gg(s(X1), X2) 5.91/2.39 5.91/2.39 The argument filtering Pi contains the following mapping: 5.91/2.39 .(x1, x2) = .(x1, x2) 5.91/2.39 5.91/2.39 appA_in_agag(x1, x2, x3, x4) = appA_in_agag(x2, x4) 5.91/2.39 5.91/2.39 appcA_in_agag(x1, x2, x3, x4) = appcA_in_agag(x2, x4) 5.91/2.39 5.91/2.39 appcA_out_agag(x1, x2, x3, x4) = appcA_out_agag(x1, x2, x3, x4) 5.91/2.39 5.91/2.39 U20_agag(x1, x2, x3, x4, x5, x6) = U20_agag(x1, x3, x5, x6) 5.91/2.39 5.91/2.39 appB_in_gga(x1, x2, x3) = appB_in_gga(x1, x2) 5.91/2.39 5.91/2.39 appcB_in_gga(x1, x2, x3) = appcB_in_gga(x1, x2) 5.91/2.39 5.91/2.39 [] = [] 5.91/2.39 5.91/2.39 appcB_out_gga(x1, x2, x3) = appcB_out_gga(x1, x2, x3) 5.91/2.39 5.91/2.39 U21_gga(x1, x2, x3, x4, x5) = U21_gga(x1, x2, x3, x5) 5.91/2.39 5.91/2.39 permC_in_gg(x1, x2) = permC_in_gg(x1, x2) 5.91/2.39 5.91/2.39 permcC_in_gg(x1, x2) = permcC_in_gg(x1, x2) 5.91/2.39 5.91/2.39 permcC_out_gg(x1, x2) = permcC_out_gg(x1, x2) 5.91/2.39 5.91/2.39 U22_gg(x1, x2, x3, x4) = U22_gg(x1, x2, x3, x4) 5.91/2.39 5.91/2.39 U23_gg(x1, x2, x3, x4) = U23_gg(x1, x2, x3, x4) 5.91/2.39 5.91/2.39 U24_gg(x1, x2, x3, x4) = U24_gg(x1, x2, x3, x4) 5.91/2.39 5.91/2.39 orderedD_in_gg(x1, x2) = orderedD_in_gg(x1, x2) 5.91/2.39 5.91/2.39 s(x1) = s(x1) 5.91/2.39 5.91/2.39 lessF_in_gg(x1, x2) = lessF_in_gg(x1, x2) 5.91/2.39 5.91/2.39 lesscE_in_gg(x1, x2) = lesscE_in_gg(x1, x2) 5.91/2.39 5.91/2.39 0 = 0 5.91/2.39 5.91/2.39 lesscE_out_gg(x1, x2) = lesscE_out_gg(x1, x2) 5.91/2.39 5.91/2.39 U28_gg(x1, x2, x3) = U28_gg(x1, x2, x3) 5.91/2.39 5.91/2.39 lesscF_in_gg(x1, x2) = lesscF_in_gg(x1, x2) 5.91/2.39 5.91/2.39 lesscF_out_gg(x1, x2) = lesscF_out_gg(x1, x2) 5.91/2.39 5.91/2.39 U27_gg(x1, x2, x3) = U27_gg(x1, x2, x3) 5.91/2.39 5.91/2.39 SSG_IN_GG(x1, x2) = SSG_IN_GG(x1, x2) 5.91/2.39 5.91/2.39 U12_GG(x1, x2, x3, x4) = U12_GG(x1, x2, x3, x4) 5.91/2.39 5.91/2.39 APPA_IN_AGAG(x1, x2, x3, x4) = APPA_IN_AGAG(x2, x4) 5.91/2.39 5.91/2.39 U1_AGAG(x1, x2, x3, x4, x5, x6) = U1_AGAG(x1, x3, x5, x6) 5.91/2.39 5.91/2.39 U13_GG(x1, x2, x3, x4) = U13_GG(x1, x2, x3, x4) 5.91/2.39 5.91/2.39 U14_GG(x1, x2, x3, x4) = U14_GG(x1, x2, x3, x4) 5.91/2.39 5.91/2.39 APPB_IN_GGA(x1, x2, x3) = APPB_IN_GGA(x1, x2) 5.91/2.39 5.91/2.39 U2_GGA(x1, x2, x3, x4, x5) = U2_GGA(x1, x2, x3, x5) 5.91/2.39 5.91/2.39 U15_GG(x1, x2, x3, x4) = U15_GG(x1, x2, x3, x4) 5.91/2.40 5.91/2.40 U16_GG(x1, x2, x3, x4) = U16_GG(x1, x2, x3, x4) 5.91/2.40 5.91/2.40 PERMC_IN_GG(x1, x2) = PERMC_IN_GG(x1, x2) 5.91/2.40 5.91/2.40 U3_GG(x1, x2, x3, x4) = U3_GG(x1, x2, x3, x4) 5.91/2.40 5.91/2.40 U4_GG(x1, x2, x3, x4) = U4_GG(x1, x2, x3, x4) 5.91/2.40 5.91/2.40 U5_GG(x1, x2, x3, x4) = U5_GG(x1, x2, x3, x4) 5.91/2.40 5.91/2.40 U6_GG(x1, x2, x3, x4) = U6_GG(x1, x2, x3, x4) 5.91/2.40 5.91/2.40 U7_GG(x1, x2, x3, x4) = U7_GG(x1, x2, x3, x4) 5.91/2.40 5.91/2.40 U17_GG(x1, x2, x3, x4) = U17_GG(x1, x2, x3, x4) 5.91/2.40 5.91/2.40 U18_GG(x1, x2, x3, x4) = U18_GG(x1, x2, x3, x4) 5.91/2.40 5.91/2.40 ORDEREDD_IN_GG(x1, x2) = ORDEREDD_IN_GG(x1, x2) 5.91/2.40 5.91/2.40 U8_GG(x1, x2, x3, x4) = U8_GG(x1, x2, x3, x4) 5.91/2.40 5.91/2.40 LESSF_IN_GG(x1, x2) = LESSF_IN_GG(x1, x2) 5.91/2.40 5.91/2.40 U11_GG(x1, x2, x3) = U11_GG(x1, x2, x3) 5.91/2.40 5.91/2.40 U9_GG(x1, x2, x3, x4) = U9_GG(x1, x2, x3, x4) 5.91/2.40 5.91/2.40 U10_GG(x1, x2, x3, x4) = U10_GG(x1, x2, x3, x4) 5.91/2.40 5.91/2.40 5.91/2.40 We have to consider all (P,R,Pi)-chains 5.91/2.40 ---------------------------------------- 5.91/2.40 5.91/2.40 (5) DependencyGraphProof (EQUIVALENT) 5.91/2.40 The approximation of the Dependency Graph [LOPSTR] contains 5 SCCs with 22 less nodes. 5.91/2.40 ---------------------------------------- 5.91/2.40 5.91/2.40 (6) 5.91/2.40 Complex Obligation (AND) 5.91/2.40 5.91/2.40 ---------------------------------------- 5.91/2.40 5.91/2.40 (7) 5.91/2.40 Obligation: 5.91/2.40 Pi DP problem: 5.91/2.40 The TRS P consists of the following rules: 5.91/2.40 5.91/2.40 LESSF_IN_GG(s(X1), s(X2)) -> LESSF_IN_GG(X1, X2) 5.91/2.40 5.91/2.40 The TRS R consists of the following rules: 5.91/2.40 5.91/2.40 appcA_in_agag([], X1, X2, .(X1, X2)) -> appcA_out_agag([], X1, X2, .(X1, X2)) 5.91/2.40 appcA_in_agag(.(X1, X2), X3, X4, .(X1, X5)) -> U20_agag(X1, X2, X3, X4, X5, appcA_in_agag(X2, X3, X4, X5)) 5.91/2.40 U20_agag(X1, X2, X3, X4, X5, appcA_out_agag(X2, X3, X4, X5)) -> appcA_out_agag(.(X1, X2), X3, X4, .(X1, X5)) 5.91/2.40 appcB_in_gga([], X1, X1) -> appcB_out_gga([], X1, X1) 5.91/2.40 appcB_in_gga(.(X1, X2), X3, .(X1, X4)) -> U21_gga(X1, X2, X3, X4, appcB_in_gga(X2, X3, X4)) 5.91/2.40 U21_gga(X1, X2, X3, X4, appcB_out_gga(X2, X3, X4)) -> appcB_out_gga(.(X1, X2), X3, .(X1, X4)) 5.91/2.40 permcC_in_gg([], []) -> permcC_out_gg([], []) 5.91/2.40 permcC_in_gg(X1, .(X2, X3)) -> U22_gg(X1, X2, X3, appcA_in_agag(X4, X2, X5, X1)) 5.91/2.40 U22_gg(X1, X2, X3, appcA_out_agag(X4, X2, X5, X1)) -> U23_gg(X1, X2, X3, appcB_in_gga(X4, X5, X6)) 5.91/2.40 U23_gg(X1, X2, X3, appcB_out_gga(X4, X5, X6)) -> U24_gg(X1, X2, X3, permcC_in_gg(X6, X3)) 5.91/2.40 U24_gg(X1, X2, X3, permcC_out_gg(X6, X3)) -> permcC_out_gg(X1, .(X2, X3)) 5.91/2.40 lesscE_in_gg(0, X1) -> lesscE_out_gg(0, X1) 5.91/2.40 lesscE_in_gg(s(X1), X2) -> U28_gg(X1, X2, lesscF_in_gg(X1, X2)) 5.91/2.40 lesscF_in_gg(0, s(X1)) -> lesscF_out_gg(0, s(X1)) 5.91/2.40 lesscF_in_gg(s(X1), s(X2)) -> U27_gg(X1, X2, lesscF_in_gg(X1, X2)) 5.91/2.40 U27_gg(X1, X2, lesscF_out_gg(X1, X2)) -> lesscF_out_gg(s(X1), s(X2)) 5.91/2.40 U28_gg(X1, X2, lesscF_out_gg(X1, X2)) -> lesscE_out_gg(s(X1), X2) 5.91/2.40 5.91/2.40 The argument filtering Pi contains the following mapping: 5.91/2.40 .(x1, x2) = .(x1, x2) 5.91/2.40 5.91/2.40 appcA_in_agag(x1, x2, x3, x4) = appcA_in_agag(x2, x4) 5.91/2.40 5.91/2.40 appcA_out_agag(x1, x2, x3, x4) = appcA_out_agag(x1, x2, x3, x4) 5.91/2.40 5.91/2.40 U20_agag(x1, x2, x3, x4, x5, x6) = U20_agag(x1, x3, x5, x6) 5.91/2.40 5.91/2.40 appcB_in_gga(x1, x2, x3) = appcB_in_gga(x1, x2) 5.91/2.40 5.91/2.40 [] = [] 5.91/2.40 5.91/2.40 appcB_out_gga(x1, x2, x3) = appcB_out_gga(x1, x2, x3) 5.91/2.40 5.91/2.40 U21_gga(x1, x2, x3, x4, x5) = U21_gga(x1, x2, x3, x5) 5.91/2.40 5.91/2.40 permcC_in_gg(x1, x2) = permcC_in_gg(x1, x2) 5.91/2.40 5.91/2.40 permcC_out_gg(x1, x2) = permcC_out_gg(x1, x2) 5.91/2.40 5.91/2.40 U22_gg(x1, x2, x3, x4) = U22_gg(x1, x2, x3, x4) 5.91/2.40 5.91/2.40 U23_gg(x1, x2, x3, x4) = U23_gg(x1, x2, x3, x4) 5.91/2.40 5.91/2.40 U24_gg(x1, x2, x3, x4) = U24_gg(x1, x2, x3, x4) 5.91/2.40 5.91/2.40 s(x1) = s(x1) 5.91/2.40 5.91/2.40 lesscE_in_gg(x1, x2) = lesscE_in_gg(x1, x2) 5.91/2.40 5.91/2.40 0 = 0 5.91/2.40 5.91/2.40 lesscE_out_gg(x1, x2) = lesscE_out_gg(x1, x2) 5.91/2.40 5.91/2.40 U28_gg(x1, x2, x3) = U28_gg(x1, x2, x3) 5.91/2.40 5.91/2.40 lesscF_in_gg(x1, x2) = lesscF_in_gg(x1, x2) 5.91/2.40 5.91/2.40 lesscF_out_gg(x1, x2) = lesscF_out_gg(x1, x2) 5.91/2.40 5.91/2.40 U27_gg(x1, x2, x3) = U27_gg(x1, x2, x3) 5.91/2.40 5.91/2.40 LESSF_IN_GG(x1, x2) = LESSF_IN_GG(x1, x2) 5.91/2.40 5.91/2.40 5.91/2.40 We have to consider all (P,R,Pi)-chains 5.91/2.40 ---------------------------------------- 5.91/2.40 5.91/2.40 (8) UsableRulesProof (EQUIVALENT) 5.91/2.40 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 5.91/2.40 ---------------------------------------- 5.91/2.40 5.91/2.40 (9) 5.91/2.40 Obligation: 5.91/2.40 Pi DP problem: 5.91/2.40 The TRS P consists of the following rules: 5.91/2.40 5.91/2.40 LESSF_IN_GG(s(X1), s(X2)) -> LESSF_IN_GG(X1, X2) 5.91/2.40 5.91/2.40 R is empty. 5.91/2.40 Pi is empty. 5.91/2.40 We have to consider all (P,R,Pi)-chains 5.91/2.40 ---------------------------------------- 5.91/2.40 5.91/2.40 (10) PiDPToQDPProof (EQUIVALENT) 5.91/2.40 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 5.91/2.40 ---------------------------------------- 5.91/2.40 5.91/2.40 (11) 5.91/2.40 Obligation: 5.91/2.40 Q DP problem: 5.91/2.40 The TRS P consists of the following rules: 5.91/2.40 5.91/2.40 LESSF_IN_GG(s(X1), s(X2)) -> LESSF_IN_GG(X1, X2) 5.91/2.40 5.91/2.40 R is empty. 5.91/2.40 Q is empty. 5.91/2.40 We have to consider all (P,Q,R)-chains. 5.91/2.40 ---------------------------------------- 5.91/2.40 5.91/2.40 (12) QDPSizeChangeProof (EQUIVALENT) 5.91/2.40 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. 5.91/2.40 5.91/2.40 From the DPs we obtained the following set of size-change graphs: 5.91/2.40 *LESSF_IN_GG(s(X1), s(X2)) -> LESSF_IN_GG(X1, X2) 5.91/2.40 The graph contains the following edges 1 > 1, 2 > 2 5.91/2.40 5.91/2.40 5.91/2.40 ---------------------------------------- 5.91/2.40 5.91/2.40 (13) 5.91/2.40 YES 5.91/2.40 5.91/2.40 ---------------------------------------- 5.91/2.40 5.91/2.40 (14) 5.91/2.40 Obligation: 5.91/2.40 Pi DP problem: 5.91/2.40 The TRS P consists of the following rules: 5.91/2.40 5.91/2.40 ORDEREDD_IN_GG(X1, .(X2, X3)) -> U9_GG(X1, X2, X3, lesscE_in_gg(X1, X2)) 5.91/2.40 U9_GG(X1, X2, X3, lesscE_out_gg(X1, X2)) -> ORDEREDD_IN_GG(X2, X3) 5.91/2.40 5.91/2.40 The TRS R consists of the following rules: 5.91/2.40 5.91/2.40 appcA_in_agag([], X1, X2, .(X1, X2)) -> appcA_out_agag([], X1, X2, .(X1, X2)) 5.91/2.40 appcA_in_agag(.(X1, X2), X3, X4, .(X1, X5)) -> U20_agag(X1, X2, X3, X4, X5, appcA_in_agag(X2, X3, X4, X5)) 5.91/2.40 U20_agag(X1, X2, X3, X4, X5, appcA_out_agag(X2, X3, X4, X5)) -> appcA_out_agag(.(X1, X2), X3, X4, .(X1, X5)) 5.91/2.40 appcB_in_gga([], X1, X1) -> appcB_out_gga([], X1, X1) 5.91/2.40 appcB_in_gga(.(X1, X2), X3, .(X1, X4)) -> U21_gga(X1, X2, X3, X4, appcB_in_gga(X2, X3, X4)) 5.91/2.40 U21_gga(X1, X2, X3, X4, appcB_out_gga(X2, X3, X4)) -> appcB_out_gga(.(X1, X2), X3, .(X1, X4)) 5.91/2.40 permcC_in_gg([], []) -> permcC_out_gg([], []) 5.91/2.40 permcC_in_gg(X1, .(X2, X3)) -> U22_gg(X1, X2, X3, appcA_in_agag(X4, X2, X5, X1)) 5.91/2.40 U22_gg(X1, X2, X3, appcA_out_agag(X4, X2, X5, X1)) -> U23_gg(X1, X2, X3, appcB_in_gga(X4, X5, X6)) 5.91/2.40 U23_gg(X1, X2, X3, appcB_out_gga(X4, X5, X6)) -> U24_gg(X1, X2, X3, permcC_in_gg(X6, X3)) 5.91/2.40 U24_gg(X1, X2, X3, permcC_out_gg(X6, X3)) -> permcC_out_gg(X1, .(X2, X3)) 5.91/2.40 lesscE_in_gg(0, X1) -> lesscE_out_gg(0, X1) 5.91/2.40 lesscE_in_gg(s(X1), X2) -> U28_gg(X1, X2, lesscF_in_gg(X1, X2)) 5.91/2.40 lesscF_in_gg(0, s(X1)) -> lesscF_out_gg(0, s(X1)) 5.91/2.40 lesscF_in_gg(s(X1), s(X2)) -> U27_gg(X1, X2, lesscF_in_gg(X1, X2)) 5.91/2.40 U27_gg(X1, X2, lesscF_out_gg(X1, X2)) -> lesscF_out_gg(s(X1), s(X2)) 5.91/2.40 U28_gg(X1, X2, lesscF_out_gg(X1, X2)) -> lesscE_out_gg(s(X1), X2) 5.91/2.40 5.91/2.40 The argument filtering Pi contains the following mapping: 5.91/2.40 .(x1, x2) = .(x1, x2) 5.91/2.40 5.91/2.40 appcA_in_agag(x1, x2, x3, x4) = appcA_in_agag(x2, x4) 5.91/2.40 5.91/2.40 appcA_out_agag(x1, x2, x3, x4) = appcA_out_agag(x1, x2, x3, x4) 5.91/2.40 5.91/2.40 U20_agag(x1, x2, x3, x4, x5, x6) = U20_agag(x1, x3, x5, x6) 5.91/2.40 5.91/2.40 appcB_in_gga(x1, x2, x3) = appcB_in_gga(x1, x2) 5.91/2.40 5.91/2.40 [] = [] 5.91/2.40 5.91/2.40 appcB_out_gga(x1, x2, x3) = appcB_out_gga(x1, x2, x3) 5.91/2.40 5.91/2.40 U21_gga(x1, x2, x3, x4, x5) = U21_gga(x1, x2, x3, x5) 5.91/2.40 5.91/2.40 permcC_in_gg(x1, x2) = permcC_in_gg(x1, x2) 5.91/2.40 5.91/2.40 permcC_out_gg(x1, x2) = permcC_out_gg(x1, x2) 5.91/2.40 5.91/2.40 U22_gg(x1, x2, x3, x4) = U22_gg(x1, x2, x3, x4) 5.91/2.40 5.91/2.40 U23_gg(x1, x2, x3, x4) = U23_gg(x1, x2, x3, x4) 5.91/2.40 5.91/2.40 U24_gg(x1, x2, x3, x4) = U24_gg(x1, x2, x3, x4) 5.91/2.40 5.91/2.40 s(x1) = s(x1) 5.91/2.40 5.91/2.40 lesscE_in_gg(x1, x2) = lesscE_in_gg(x1, x2) 5.91/2.40 5.91/2.40 0 = 0 5.91/2.40 5.91/2.40 lesscE_out_gg(x1, x2) = lesscE_out_gg(x1, x2) 5.91/2.40 5.91/2.40 U28_gg(x1, x2, x3) = U28_gg(x1, x2, x3) 5.91/2.40 5.91/2.40 lesscF_in_gg(x1, x2) = lesscF_in_gg(x1, x2) 5.91/2.40 5.91/2.40 lesscF_out_gg(x1, x2) = lesscF_out_gg(x1, x2) 5.91/2.40 5.91/2.40 U27_gg(x1, x2, x3) = U27_gg(x1, x2, x3) 5.91/2.40 5.91/2.40 ORDEREDD_IN_GG(x1, x2) = ORDEREDD_IN_GG(x1, x2) 5.91/2.40 5.91/2.40 U9_GG(x1, x2, x3, x4) = U9_GG(x1, x2, x3, x4) 5.91/2.40 5.91/2.40 5.91/2.40 We have to consider all (P,R,Pi)-chains 5.91/2.40 ---------------------------------------- 5.91/2.40 5.91/2.40 (15) UsableRulesProof (EQUIVALENT) 5.91/2.40 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 5.91/2.40 ---------------------------------------- 5.91/2.40 5.91/2.40 (16) 5.91/2.40 Obligation: 5.91/2.40 Pi DP problem: 5.91/2.40 The TRS P consists of the following rules: 5.91/2.40 5.91/2.40 ORDEREDD_IN_GG(X1, .(X2, X3)) -> U9_GG(X1, X2, X3, lesscE_in_gg(X1, X2)) 5.91/2.40 U9_GG(X1, X2, X3, lesscE_out_gg(X1, X2)) -> ORDEREDD_IN_GG(X2, X3) 5.91/2.40 5.91/2.40 The TRS R consists of the following rules: 5.91/2.40 5.91/2.40 lesscE_in_gg(0, X1) -> lesscE_out_gg(0, X1) 5.91/2.40 lesscE_in_gg(s(X1), X2) -> U28_gg(X1, X2, lesscF_in_gg(X1, X2)) 5.91/2.40 U28_gg(X1, X2, lesscF_out_gg(X1, X2)) -> lesscE_out_gg(s(X1), X2) 5.91/2.40 lesscF_in_gg(0, s(X1)) -> lesscF_out_gg(0, s(X1)) 5.91/2.40 lesscF_in_gg(s(X1), s(X2)) -> U27_gg(X1, X2, lesscF_in_gg(X1, X2)) 5.91/2.40 U27_gg(X1, X2, lesscF_out_gg(X1, X2)) -> lesscF_out_gg(s(X1), s(X2)) 5.91/2.40 5.91/2.40 Pi is empty. 5.91/2.40 We have to consider all (P,R,Pi)-chains 5.91/2.40 ---------------------------------------- 5.91/2.40 5.91/2.40 (17) PiDPToQDPProof (EQUIVALENT) 5.91/2.40 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 5.91/2.40 ---------------------------------------- 5.91/2.40 5.91/2.40 (18) 5.91/2.40 Obligation: 5.91/2.40 Q DP problem: 5.91/2.40 The TRS P consists of the following rules: 5.91/2.40 5.91/2.40 ORDEREDD_IN_GG(X1, .(X2, X3)) -> U9_GG(X1, X2, X3, lesscE_in_gg(X1, X2)) 5.91/2.40 U9_GG(X1, X2, X3, lesscE_out_gg(X1, X2)) -> ORDEREDD_IN_GG(X2, X3) 5.91/2.40 5.91/2.40 The TRS R consists of the following rules: 5.91/2.40 5.91/2.40 lesscE_in_gg(0, X1) -> lesscE_out_gg(0, X1) 5.91/2.40 lesscE_in_gg(s(X1), X2) -> U28_gg(X1, X2, lesscF_in_gg(X1, X2)) 5.91/2.40 U28_gg(X1, X2, lesscF_out_gg(X1, X2)) -> lesscE_out_gg(s(X1), X2) 5.91/2.40 lesscF_in_gg(0, s(X1)) -> lesscF_out_gg(0, s(X1)) 5.91/2.40 lesscF_in_gg(s(X1), s(X2)) -> U27_gg(X1, X2, lesscF_in_gg(X1, X2)) 5.91/2.40 U27_gg(X1, X2, lesscF_out_gg(X1, X2)) -> lesscF_out_gg(s(X1), s(X2)) 5.91/2.40 5.91/2.40 The set Q consists of the following terms: 5.91/2.40 5.91/2.40 lesscE_in_gg(x0, x1) 5.91/2.40 U28_gg(x0, x1, x2) 5.91/2.40 lesscF_in_gg(x0, x1) 5.91/2.40 U27_gg(x0, x1, x2) 5.91/2.40 5.91/2.40 We have to consider all (P,Q,R)-chains. 5.91/2.40 ---------------------------------------- 5.91/2.40 5.91/2.40 (19) QDPSizeChangeProof (EQUIVALENT) 5.91/2.40 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. 5.91/2.40 5.91/2.40 From the DPs we obtained the following set of size-change graphs: 5.91/2.40 *U9_GG(X1, X2, X3, lesscE_out_gg(X1, X2)) -> ORDEREDD_IN_GG(X2, X3) 5.91/2.40 The graph contains the following edges 2 >= 1, 4 > 1, 3 >= 2 5.91/2.40 5.91/2.40 5.91/2.40 *ORDEREDD_IN_GG(X1, .(X2, X3)) -> U9_GG(X1, X2, X3, lesscE_in_gg(X1, X2)) 5.91/2.40 The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3 5.91/2.40 5.91/2.40 5.91/2.40 ---------------------------------------- 5.91/2.40 5.91/2.40 (20) 5.91/2.40 YES 5.91/2.40 5.91/2.40 ---------------------------------------- 5.91/2.40 5.91/2.40 (21) 5.91/2.40 Obligation: 5.91/2.40 Pi DP problem: 5.91/2.40 The TRS P consists of the following rules: 5.91/2.40 5.91/2.40 APPB_IN_GGA(.(X1, X2), X3, .(X1, X4)) -> APPB_IN_GGA(X2, X3, X4) 5.91/2.40 5.91/2.40 The TRS R consists of the following rules: 5.91/2.40 5.91/2.40 appcA_in_agag([], X1, X2, .(X1, X2)) -> appcA_out_agag([], X1, X2, .(X1, X2)) 5.91/2.40 appcA_in_agag(.(X1, X2), X3, X4, .(X1, X5)) -> U20_agag(X1, X2, X3, X4, X5, appcA_in_agag(X2, X3, X4, X5)) 5.91/2.40 U20_agag(X1, X2, X3, X4, X5, appcA_out_agag(X2, X3, X4, X5)) -> appcA_out_agag(.(X1, X2), X3, X4, .(X1, X5)) 5.91/2.40 appcB_in_gga([], X1, X1) -> appcB_out_gga([], X1, X1) 5.91/2.40 appcB_in_gga(.(X1, X2), X3, .(X1, X4)) -> U21_gga(X1, X2, X3, X4, appcB_in_gga(X2, X3, X4)) 5.91/2.40 U21_gga(X1, X2, X3, X4, appcB_out_gga(X2, X3, X4)) -> appcB_out_gga(.(X1, X2), X3, .(X1, X4)) 5.91/2.40 permcC_in_gg([], []) -> permcC_out_gg([], []) 5.91/2.40 permcC_in_gg(X1, .(X2, X3)) -> U22_gg(X1, X2, X3, appcA_in_agag(X4, X2, X5, X1)) 5.91/2.40 U22_gg(X1, X2, X3, appcA_out_agag(X4, X2, X5, X1)) -> U23_gg(X1, X2, X3, appcB_in_gga(X4, X5, X6)) 5.91/2.40 U23_gg(X1, X2, X3, appcB_out_gga(X4, X5, X6)) -> U24_gg(X1, X2, X3, permcC_in_gg(X6, X3)) 5.91/2.40 U24_gg(X1, X2, X3, permcC_out_gg(X6, X3)) -> permcC_out_gg(X1, .(X2, X3)) 5.91/2.40 lesscE_in_gg(0, X1) -> lesscE_out_gg(0, X1) 5.91/2.40 lesscE_in_gg(s(X1), X2) -> U28_gg(X1, X2, lesscF_in_gg(X1, X2)) 5.91/2.40 lesscF_in_gg(0, s(X1)) -> lesscF_out_gg(0, s(X1)) 5.91/2.40 lesscF_in_gg(s(X1), s(X2)) -> U27_gg(X1, X2, lesscF_in_gg(X1, X2)) 5.91/2.40 U27_gg(X1, X2, lesscF_out_gg(X1, X2)) -> lesscF_out_gg(s(X1), s(X2)) 5.91/2.40 U28_gg(X1, X2, lesscF_out_gg(X1, X2)) -> lesscE_out_gg(s(X1), X2) 5.91/2.40 5.91/2.40 The argument filtering Pi contains the following mapping: 5.91/2.40 .(x1, x2) = .(x1, x2) 5.91/2.40 5.91/2.40 appcA_in_agag(x1, x2, x3, x4) = appcA_in_agag(x2, x4) 5.91/2.40 5.91/2.40 appcA_out_agag(x1, x2, x3, x4) = appcA_out_agag(x1, x2, x3, x4) 5.91/2.40 5.91/2.40 U20_agag(x1, x2, x3, x4, x5, x6) = U20_agag(x1, x3, x5, x6) 5.91/2.40 5.91/2.40 appcB_in_gga(x1, x2, x3) = appcB_in_gga(x1, x2) 5.91/2.40 5.91/2.40 [] = [] 5.91/2.40 5.91/2.40 appcB_out_gga(x1, x2, x3) = appcB_out_gga(x1, x2, x3) 5.91/2.40 5.91/2.40 U21_gga(x1, x2, x3, x4, x5) = U21_gga(x1, x2, x3, x5) 5.91/2.40 5.91/2.40 permcC_in_gg(x1, x2) = permcC_in_gg(x1, x2) 5.91/2.40 5.91/2.40 permcC_out_gg(x1, x2) = permcC_out_gg(x1, x2) 5.91/2.40 5.91/2.40 U22_gg(x1, x2, x3, x4) = U22_gg(x1, x2, x3, x4) 5.91/2.40 5.91/2.40 U23_gg(x1, x2, x3, x4) = U23_gg(x1, x2, x3, x4) 5.91/2.40 5.91/2.40 U24_gg(x1, x2, x3, x4) = U24_gg(x1, x2, x3, x4) 5.91/2.40 5.91/2.40 s(x1) = s(x1) 5.91/2.40 5.91/2.40 lesscE_in_gg(x1, x2) = lesscE_in_gg(x1, x2) 5.91/2.40 5.91/2.40 0 = 0 5.91/2.40 5.91/2.40 lesscE_out_gg(x1, x2) = lesscE_out_gg(x1, x2) 5.91/2.40 5.91/2.40 U28_gg(x1, x2, x3) = U28_gg(x1, x2, x3) 5.91/2.40 5.91/2.40 lesscF_in_gg(x1, x2) = lesscF_in_gg(x1, x2) 5.91/2.40 5.91/2.40 lesscF_out_gg(x1, x2) = lesscF_out_gg(x1, x2) 5.91/2.40 5.91/2.40 U27_gg(x1, x2, x3) = U27_gg(x1, x2, x3) 5.91/2.40 5.91/2.40 APPB_IN_GGA(x1, x2, x3) = APPB_IN_GGA(x1, x2) 5.91/2.40 5.91/2.40 5.91/2.40 We have to consider all (P,R,Pi)-chains 5.91/2.40 ---------------------------------------- 5.91/2.40 5.91/2.40 (22) UsableRulesProof (EQUIVALENT) 5.91/2.40 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 5.91/2.40 ---------------------------------------- 5.91/2.40 5.91/2.40 (23) 5.91/2.40 Obligation: 5.91/2.40 Pi DP problem: 5.91/2.40 The TRS P consists of the following rules: 5.91/2.40 5.91/2.40 APPB_IN_GGA(.(X1, X2), X3, .(X1, X4)) -> APPB_IN_GGA(X2, X3, X4) 5.91/2.40 5.91/2.40 R is empty. 5.91/2.40 The argument filtering Pi contains the following mapping: 5.91/2.40 .(x1, x2) = .(x1, x2) 5.91/2.40 5.91/2.40 APPB_IN_GGA(x1, x2, x3) = APPB_IN_GGA(x1, x2) 5.91/2.40 5.91/2.40 5.91/2.40 We have to consider all (P,R,Pi)-chains 5.91/2.40 ---------------------------------------- 5.91/2.40 5.91/2.40 (24) PiDPToQDPProof (SOUND) 5.91/2.40 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 5.91/2.40 ---------------------------------------- 5.91/2.40 5.91/2.40 (25) 5.91/2.40 Obligation: 5.91/2.40 Q DP problem: 5.91/2.40 The TRS P consists of the following rules: 5.91/2.40 5.91/2.40 APPB_IN_GGA(.(X1, X2), X3) -> APPB_IN_GGA(X2, X3) 5.91/2.40 5.91/2.40 R is empty. 5.91/2.40 Q is empty. 5.91/2.40 We have to consider all (P,Q,R)-chains. 5.91/2.40 ---------------------------------------- 5.91/2.40 5.91/2.40 (26) QDPSizeChangeProof (EQUIVALENT) 5.91/2.40 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. 5.91/2.40 5.91/2.40 From the DPs we obtained the following set of size-change graphs: 5.91/2.40 *APPB_IN_GGA(.(X1, X2), X3) -> APPB_IN_GGA(X2, X3) 5.91/2.40 The graph contains the following edges 1 > 1, 2 >= 2 5.91/2.40 5.91/2.40 5.91/2.40 ---------------------------------------- 5.91/2.40 5.91/2.40 (27) 5.91/2.40 YES 5.91/2.40 5.91/2.40 ---------------------------------------- 5.91/2.40 5.91/2.40 (28) 5.91/2.40 Obligation: 5.91/2.40 Pi DP problem: 5.91/2.40 The TRS P consists of the following rules: 5.91/2.40 5.91/2.40 APPA_IN_AGAG(.(X1, X2), X3, X4, .(X1, X5)) -> APPA_IN_AGAG(X2, X3, X4, X5) 5.91/2.40 5.91/2.40 The TRS R consists of the following rules: 5.91/2.40 5.91/2.40 appcA_in_agag([], X1, X2, .(X1, X2)) -> appcA_out_agag([], X1, X2, .(X1, X2)) 5.91/2.40 appcA_in_agag(.(X1, X2), X3, X4, .(X1, X5)) -> U20_agag(X1, X2, X3, X4, X5, appcA_in_agag(X2, X3, X4, X5)) 5.91/2.40 U20_agag(X1, X2, X3, X4, X5, appcA_out_agag(X2, X3, X4, X5)) -> appcA_out_agag(.(X1, X2), X3, X4, .(X1, X5)) 5.91/2.40 appcB_in_gga([], X1, X1) -> appcB_out_gga([], X1, X1) 5.91/2.40 appcB_in_gga(.(X1, X2), X3, .(X1, X4)) -> U21_gga(X1, X2, X3, X4, appcB_in_gga(X2, X3, X4)) 5.91/2.40 U21_gga(X1, X2, X3, X4, appcB_out_gga(X2, X3, X4)) -> appcB_out_gga(.(X1, X2), X3, .(X1, X4)) 5.91/2.40 permcC_in_gg([], []) -> permcC_out_gg([], []) 5.91/2.40 permcC_in_gg(X1, .(X2, X3)) -> U22_gg(X1, X2, X3, appcA_in_agag(X4, X2, X5, X1)) 5.91/2.40 U22_gg(X1, X2, X3, appcA_out_agag(X4, X2, X5, X1)) -> U23_gg(X1, X2, X3, appcB_in_gga(X4, X5, X6)) 5.91/2.40 U23_gg(X1, X2, X3, appcB_out_gga(X4, X5, X6)) -> U24_gg(X1, X2, X3, permcC_in_gg(X6, X3)) 5.91/2.40 U24_gg(X1, X2, X3, permcC_out_gg(X6, X3)) -> permcC_out_gg(X1, .(X2, X3)) 5.91/2.40 lesscE_in_gg(0, X1) -> lesscE_out_gg(0, X1) 5.91/2.40 lesscE_in_gg(s(X1), X2) -> U28_gg(X1, X2, lesscF_in_gg(X1, X2)) 5.91/2.40 lesscF_in_gg(0, s(X1)) -> lesscF_out_gg(0, s(X1)) 5.91/2.40 lesscF_in_gg(s(X1), s(X2)) -> U27_gg(X1, X2, lesscF_in_gg(X1, X2)) 5.91/2.40 U27_gg(X1, X2, lesscF_out_gg(X1, X2)) -> lesscF_out_gg(s(X1), s(X2)) 5.91/2.40 U28_gg(X1, X2, lesscF_out_gg(X1, X2)) -> lesscE_out_gg(s(X1), X2) 5.91/2.40 5.91/2.40 The argument filtering Pi contains the following mapping: 5.91/2.40 .(x1, x2) = .(x1, x2) 5.91/2.40 5.91/2.40 appcA_in_agag(x1, x2, x3, x4) = appcA_in_agag(x2, x4) 5.91/2.40 5.91/2.40 appcA_out_agag(x1, x2, x3, x4) = appcA_out_agag(x1, x2, x3, x4) 5.91/2.40 5.91/2.40 U20_agag(x1, x2, x3, x4, x5, x6) = U20_agag(x1, x3, x5, x6) 5.91/2.40 5.91/2.40 appcB_in_gga(x1, x2, x3) = appcB_in_gga(x1, x2) 5.91/2.40 5.91/2.40 [] = [] 5.91/2.40 5.91/2.40 appcB_out_gga(x1, x2, x3) = appcB_out_gga(x1, x2, x3) 5.91/2.40 5.91/2.40 U21_gga(x1, x2, x3, x4, x5) = U21_gga(x1, x2, x3, x5) 5.91/2.40 5.91/2.40 permcC_in_gg(x1, x2) = permcC_in_gg(x1, x2) 5.91/2.40 5.91/2.40 permcC_out_gg(x1, x2) = permcC_out_gg(x1, x2) 5.91/2.40 5.91/2.40 U22_gg(x1, x2, x3, x4) = U22_gg(x1, x2, x3, x4) 5.91/2.40 5.91/2.40 U23_gg(x1, x2, x3, x4) = U23_gg(x1, x2, x3, x4) 5.91/2.40 5.91/2.40 U24_gg(x1, x2, x3, x4) = U24_gg(x1, x2, x3, x4) 5.91/2.40 5.91/2.40 s(x1) = s(x1) 5.91/2.40 5.91/2.40 lesscE_in_gg(x1, x2) = lesscE_in_gg(x1, x2) 5.91/2.40 5.91/2.40 0 = 0 5.91/2.40 5.91/2.40 lesscE_out_gg(x1, x2) = lesscE_out_gg(x1, x2) 5.91/2.40 5.91/2.40 U28_gg(x1, x2, x3) = U28_gg(x1, x2, x3) 5.91/2.40 5.91/2.40 lesscF_in_gg(x1, x2) = lesscF_in_gg(x1, x2) 5.91/2.40 5.91/2.40 lesscF_out_gg(x1, x2) = lesscF_out_gg(x1, x2) 5.91/2.40 5.91/2.40 U27_gg(x1, x2, x3) = U27_gg(x1, x2, x3) 5.91/2.40 5.91/2.40 APPA_IN_AGAG(x1, x2, x3, x4) = APPA_IN_AGAG(x2, x4) 5.91/2.40 5.91/2.40 5.91/2.40 We have to consider all (P,R,Pi)-chains 5.91/2.40 ---------------------------------------- 5.91/2.40 5.91/2.40 (29) UsableRulesProof (EQUIVALENT) 5.91/2.40 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 5.91/2.40 ---------------------------------------- 5.91/2.40 5.91/2.40 (30) 5.91/2.40 Obligation: 5.91/2.40 Pi DP problem: 5.91/2.40 The TRS P consists of the following rules: 5.91/2.40 5.91/2.40 APPA_IN_AGAG(.(X1, X2), X3, X4, .(X1, X5)) -> APPA_IN_AGAG(X2, X3, X4, X5) 5.91/2.40 5.91/2.40 R is empty. 5.91/2.40 The argument filtering Pi contains the following mapping: 5.91/2.40 .(x1, x2) = .(x1, x2) 5.91/2.40 5.91/2.40 APPA_IN_AGAG(x1, x2, x3, x4) = APPA_IN_AGAG(x2, x4) 5.91/2.40 5.91/2.40 5.91/2.40 We have to consider all (P,R,Pi)-chains 5.91/2.40 ---------------------------------------- 5.91/2.40 5.91/2.40 (31) PiDPToQDPProof (SOUND) 5.91/2.40 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 5.91/2.40 ---------------------------------------- 5.91/2.40 5.91/2.40 (32) 5.91/2.40 Obligation: 5.91/2.40 Q DP problem: 5.91/2.40 The TRS P consists of the following rules: 5.91/2.40 5.91/2.40 APPA_IN_AGAG(X3, .(X1, X5)) -> APPA_IN_AGAG(X3, X5) 5.91/2.40 5.91/2.40 R is empty. 5.91/2.40 Q is empty. 5.91/2.40 We have to consider all (P,Q,R)-chains. 5.91/2.40 ---------------------------------------- 5.91/2.40 5.91/2.40 (33) QDPSizeChangeProof (EQUIVALENT) 5.91/2.40 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. 5.91/2.40 5.91/2.40 From the DPs we obtained the following set of size-change graphs: 5.91/2.40 *APPA_IN_AGAG(X3, .(X1, X5)) -> APPA_IN_AGAG(X3, X5) 5.91/2.40 The graph contains the following edges 1 >= 1, 2 > 2 5.91/2.40 5.91/2.40 5.91/2.40 ---------------------------------------- 5.91/2.40 5.91/2.40 (34) 5.91/2.40 YES 5.91/2.40 5.91/2.40 ---------------------------------------- 5.91/2.40 5.91/2.40 (35) 5.91/2.40 Obligation: 5.91/2.40 Pi DP problem: 5.91/2.40 The TRS P consists of the following rules: 5.91/2.40 5.91/2.40 PERMC_IN_GG(X1, .(X2, X3)) -> U4_GG(X1, X2, X3, appcA_in_agag(X4, X2, X5, X1)) 5.91/2.40 U4_GG(X1, X2, X3, appcA_out_agag(X4, X2, X5, X1)) -> U6_GG(X1, X2, X3, appcB_in_gga(X4, X5, X6)) 5.91/2.40 U6_GG(X1, X2, X3, appcB_out_gga(X4, X5, X6)) -> PERMC_IN_GG(X6, X3) 5.91/2.40 5.91/2.40 The TRS R consists of the following rules: 5.91/2.40 5.91/2.40 appcA_in_agag([], X1, X2, .(X1, X2)) -> appcA_out_agag([], X1, X2, .(X1, X2)) 5.91/2.40 appcA_in_agag(.(X1, X2), X3, X4, .(X1, X5)) -> U20_agag(X1, X2, X3, X4, X5, appcA_in_agag(X2, X3, X4, X5)) 5.91/2.40 U20_agag(X1, X2, X3, X4, X5, appcA_out_agag(X2, X3, X4, X5)) -> appcA_out_agag(.(X1, X2), X3, X4, .(X1, X5)) 5.91/2.40 appcB_in_gga([], X1, X1) -> appcB_out_gga([], X1, X1) 5.91/2.40 appcB_in_gga(.(X1, X2), X3, .(X1, X4)) -> U21_gga(X1, X2, X3, X4, appcB_in_gga(X2, X3, X4)) 5.91/2.40 U21_gga(X1, X2, X3, X4, appcB_out_gga(X2, X3, X4)) -> appcB_out_gga(.(X1, X2), X3, .(X1, X4)) 5.91/2.40 permcC_in_gg([], []) -> permcC_out_gg([], []) 5.91/2.40 permcC_in_gg(X1, .(X2, X3)) -> U22_gg(X1, X2, X3, appcA_in_agag(X4, X2, X5, X1)) 5.91/2.40 U22_gg(X1, X2, X3, appcA_out_agag(X4, X2, X5, X1)) -> U23_gg(X1, X2, X3, appcB_in_gga(X4, X5, X6)) 5.91/2.40 U23_gg(X1, X2, X3, appcB_out_gga(X4, X5, X6)) -> U24_gg(X1, X2, X3, permcC_in_gg(X6, X3)) 5.91/2.40 U24_gg(X1, X2, X3, permcC_out_gg(X6, X3)) -> permcC_out_gg(X1, .(X2, X3)) 5.91/2.40 lesscE_in_gg(0, X1) -> lesscE_out_gg(0, X1) 5.91/2.40 lesscE_in_gg(s(X1), X2) -> U28_gg(X1, X2, lesscF_in_gg(X1, X2)) 5.91/2.40 lesscF_in_gg(0, s(X1)) -> lesscF_out_gg(0, s(X1)) 5.91/2.40 lesscF_in_gg(s(X1), s(X2)) -> U27_gg(X1, X2, lesscF_in_gg(X1, X2)) 5.91/2.40 U27_gg(X1, X2, lesscF_out_gg(X1, X2)) -> lesscF_out_gg(s(X1), s(X2)) 5.91/2.40 U28_gg(X1, X2, lesscF_out_gg(X1, X2)) -> lesscE_out_gg(s(X1), X2) 5.91/2.40 5.91/2.40 The argument filtering Pi contains the following mapping: 5.91/2.40 .(x1, x2) = .(x1, x2) 5.91/2.40 5.91/2.40 appcA_in_agag(x1, x2, x3, x4) = appcA_in_agag(x2, x4) 5.91/2.40 5.91/2.40 appcA_out_agag(x1, x2, x3, x4) = appcA_out_agag(x1, x2, x3, x4) 5.91/2.40 5.91/2.40 U20_agag(x1, x2, x3, x4, x5, x6) = U20_agag(x1, x3, x5, x6) 5.91/2.40 5.91/2.40 appcB_in_gga(x1, x2, x3) = appcB_in_gga(x1, x2) 5.91/2.40 5.91/2.40 [] = [] 5.91/2.40 5.91/2.40 appcB_out_gga(x1, x2, x3) = appcB_out_gga(x1, x2, x3) 5.91/2.40 5.91/2.40 U21_gga(x1, x2, x3, x4, x5) = U21_gga(x1, x2, x3, x5) 5.91/2.40 5.91/2.40 permcC_in_gg(x1, x2) = permcC_in_gg(x1, x2) 5.91/2.40 5.91/2.40 permcC_out_gg(x1, x2) = permcC_out_gg(x1, x2) 5.91/2.40 5.91/2.40 U22_gg(x1, x2, x3, x4) = U22_gg(x1, x2, x3, x4) 5.91/2.40 5.91/2.40 U23_gg(x1, x2, x3, x4) = U23_gg(x1, x2, x3, x4) 5.91/2.40 5.91/2.40 U24_gg(x1, x2, x3, x4) = U24_gg(x1, x2, x3, x4) 5.91/2.40 5.91/2.40 s(x1) = s(x1) 5.91/2.40 5.91/2.40 lesscE_in_gg(x1, x2) = lesscE_in_gg(x1, x2) 5.91/2.40 5.91/2.40 0 = 0 5.91/2.40 5.91/2.40 lesscE_out_gg(x1, x2) = lesscE_out_gg(x1, x2) 5.91/2.40 5.91/2.40 U28_gg(x1, x2, x3) = U28_gg(x1, x2, x3) 5.91/2.40 5.91/2.40 lesscF_in_gg(x1, x2) = lesscF_in_gg(x1, x2) 5.91/2.40 5.91/2.40 lesscF_out_gg(x1, x2) = lesscF_out_gg(x1, x2) 5.91/2.40 5.91/2.40 U27_gg(x1, x2, x3) = U27_gg(x1, x2, x3) 5.91/2.40 5.91/2.40 PERMC_IN_GG(x1, x2) = PERMC_IN_GG(x1, x2) 5.91/2.40 5.91/2.40 U4_GG(x1, x2, x3, x4) = U4_GG(x1, x2, x3, x4) 5.91/2.40 5.91/2.40 U6_GG(x1, x2, x3, x4) = U6_GG(x1, x2, x3, x4) 5.91/2.40 5.91/2.40 5.91/2.40 We have to consider all (P,R,Pi)-chains 5.91/2.40 ---------------------------------------- 5.91/2.40 5.91/2.40 (36) UsableRulesProof (EQUIVALENT) 5.91/2.40 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 5.91/2.40 ---------------------------------------- 5.91/2.40 5.91/2.40 (37) 5.91/2.40 Obligation: 5.91/2.40 Pi DP problem: 5.91/2.40 The TRS P consists of the following rules: 5.91/2.40 5.91/2.40 PERMC_IN_GG(X1, .(X2, X3)) -> U4_GG(X1, X2, X3, appcA_in_agag(X4, X2, X5, X1)) 5.91/2.40 U4_GG(X1, X2, X3, appcA_out_agag(X4, X2, X5, X1)) -> U6_GG(X1, X2, X3, appcB_in_gga(X4, X5, X6)) 5.91/2.40 U6_GG(X1, X2, X3, appcB_out_gga(X4, X5, X6)) -> PERMC_IN_GG(X6, X3) 5.91/2.40 5.91/2.40 The TRS R consists of the following rules: 5.91/2.40 5.91/2.40 appcA_in_agag([], X1, X2, .(X1, X2)) -> appcA_out_agag([], X1, X2, .(X1, X2)) 5.91/2.40 appcA_in_agag(.(X1, X2), X3, X4, .(X1, X5)) -> U20_agag(X1, X2, X3, X4, X5, appcA_in_agag(X2, X3, X4, X5)) 5.91/2.40 appcB_in_gga([], X1, X1) -> appcB_out_gga([], X1, X1) 5.91/2.40 appcB_in_gga(.(X1, X2), X3, .(X1, X4)) -> U21_gga(X1, X2, X3, X4, appcB_in_gga(X2, X3, X4)) 5.91/2.40 U20_agag(X1, X2, X3, X4, X5, appcA_out_agag(X2, X3, X4, X5)) -> appcA_out_agag(.(X1, X2), X3, X4, .(X1, X5)) 5.91/2.40 U21_gga(X1, X2, X3, X4, appcB_out_gga(X2, X3, X4)) -> appcB_out_gga(.(X1, X2), X3, .(X1, X4)) 5.91/2.40 5.91/2.40 The argument filtering Pi contains the following mapping: 5.91/2.40 .(x1, x2) = .(x1, x2) 5.91/2.40 5.91/2.40 appcA_in_agag(x1, x2, x3, x4) = appcA_in_agag(x2, x4) 5.91/2.40 5.91/2.40 appcA_out_agag(x1, x2, x3, x4) = appcA_out_agag(x1, x2, x3, x4) 5.91/2.40 5.91/2.40 U20_agag(x1, x2, x3, x4, x5, x6) = U20_agag(x1, x3, x5, x6) 5.91/2.40 5.91/2.40 appcB_in_gga(x1, x2, x3) = appcB_in_gga(x1, x2) 5.91/2.40 5.91/2.40 [] = [] 5.91/2.40 5.91/2.40 appcB_out_gga(x1, x2, x3) = appcB_out_gga(x1, x2, x3) 5.91/2.40 5.91/2.40 U21_gga(x1, x2, x3, x4, x5) = U21_gga(x1, x2, x3, x5) 5.91/2.40 5.91/2.40 PERMC_IN_GG(x1, x2) = PERMC_IN_GG(x1, x2) 5.91/2.40 5.91/2.40 U4_GG(x1, x2, x3, x4) = U4_GG(x1, x2, x3, x4) 5.91/2.40 5.91/2.40 U6_GG(x1, x2, x3, x4) = U6_GG(x1, x2, x3, x4) 5.91/2.40 5.91/2.40 5.91/2.40 We have to consider all (P,R,Pi)-chains 5.91/2.40 ---------------------------------------- 5.91/2.40 5.91/2.40 (38) PiDPToQDPProof (SOUND) 5.91/2.40 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 5.91/2.40 ---------------------------------------- 5.91/2.40 5.91/2.40 (39) 5.91/2.40 Obligation: 5.91/2.40 Q DP problem: 5.91/2.40 The TRS P consists of the following rules: 5.91/2.40 5.91/2.40 PERMC_IN_GG(X1, .(X2, X3)) -> U4_GG(X1, X2, X3, appcA_in_agag(X2, X1)) 5.91/2.40 U4_GG(X1, X2, X3, appcA_out_agag(X4, X2, X5, X1)) -> U6_GG(X1, X2, X3, appcB_in_gga(X4, X5)) 5.91/2.40 U6_GG(X1, X2, X3, appcB_out_gga(X4, X5, X6)) -> PERMC_IN_GG(X6, X3) 5.91/2.40 5.91/2.40 The TRS R consists of the following rules: 5.91/2.40 5.91/2.40 appcA_in_agag(X1, .(X1, X2)) -> appcA_out_agag([], X1, X2, .(X1, X2)) 5.91/2.40 appcA_in_agag(X3, .(X1, X5)) -> U20_agag(X1, X3, X5, appcA_in_agag(X3, X5)) 5.91/2.40 appcB_in_gga([], X1) -> appcB_out_gga([], X1, X1) 5.91/2.40 appcB_in_gga(.(X1, X2), X3) -> U21_gga(X1, X2, X3, appcB_in_gga(X2, X3)) 5.91/2.40 U20_agag(X1, X3, X5, appcA_out_agag(X2, X3, X4, X5)) -> appcA_out_agag(.(X1, X2), X3, X4, .(X1, X5)) 5.91/2.40 U21_gga(X1, X2, X3, appcB_out_gga(X2, X3, X4)) -> appcB_out_gga(.(X1, X2), X3, .(X1, X4)) 5.91/2.40 5.91/2.40 The set Q consists of the following terms: 5.91/2.40 5.91/2.40 appcA_in_agag(x0, x1) 5.91/2.40 appcB_in_gga(x0, x1) 5.91/2.40 U20_agag(x0, x1, x2, x3) 5.91/2.40 U21_gga(x0, x1, x2, x3) 5.91/2.40 5.91/2.40 We have to consider all (P,Q,R)-chains. 5.91/2.40 ---------------------------------------- 5.91/2.40 5.91/2.40 (40) QDPSizeChangeProof (EQUIVALENT) 5.91/2.40 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. 5.91/2.40 5.91/2.40 From the DPs we obtained the following set of size-change graphs: 5.91/2.40 *U4_GG(X1, X2, X3, appcA_out_agag(X4, X2, X5, X1)) -> U6_GG(X1, X2, X3, appcB_in_gga(X4, X5)) 5.91/2.40 The graph contains the following edges 1 >= 1, 4 > 1, 2 >= 2, 4 > 2, 3 >= 3 5.91/2.40 5.91/2.40 5.91/2.40 *U6_GG(X1, X2, X3, appcB_out_gga(X4, X5, X6)) -> PERMC_IN_GG(X6, X3) 5.91/2.40 The graph contains the following edges 4 > 1, 3 >= 2 5.91/2.40 5.91/2.40 5.91/2.40 *PERMC_IN_GG(X1, .(X2, X3)) -> U4_GG(X1, X2, X3, appcA_in_agag(X2, X1)) 5.91/2.40 The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3 5.91/2.40 5.91/2.40 5.91/2.40 ---------------------------------------- 5.91/2.40 5.91/2.40 (41) 5.91/2.40 YES 6.07/2.46 EOF