114.32/31.00 YES 114.75/31.05 proof of /export/starexec/sandbox/benchmark/theBenchmark.pl 114.75/31.05 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 114.75/31.05 114.75/31.05 114.75/31.05 Left Termination of the query pattern 114.75/31.05 114.75/31.05 maxsort(g,a) 114.75/31.05 114.75/31.05 w.r.t. the given Prolog program could successfully be proven: 114.75/31.05 114.75/31.05 (0) Prolog 114.75/31.05 (1) PrologToDTProblemTransformerProof [SOUND, 131 ms] 114.75/31.05 (2) TRIPLES 114.75/31.05 (3) TriplesToPiDPProof [SOUND, 47 ms] 114.75/31.05 (4) PiDP 114.75/31.05 (5) DependencyGraphProof [EQUIVALENT, 0 ms] 114.75/31.05 (6) AND 114.75/31.05 (7) PiDP 114.75/31.05 (8) UsableRulesProof [EQUIVALENT, 0 ms] 114.75/31.05 (9) PiDP 114.75/31.05 (10) PiDPToQDPProof [SOUND, 0 ms] 114.75/31.05 (11) QDP 114.75/31.05 (12) QDPSizeChangeProof [EQUIVALENT, 0 ms] 114.75/31.05 (13) YES 114.75/31.05 (14) PiDP 114.75/31.05 (15) UsableRulesProof [EQUIVALENT, 0 ms] 114.75/31.05 (16) PiDP 114.75/31.05 (17) PiDPToQDPProof [SOUND, 0 ms] 114.75/31.05 (18) QDP 114.75/31.05 (19) QDPSizeChangeProof [EQUIVALENT, 0 ms] 114.75/31.05 (20) YES 114.75/31.05 (21) PiDP 114.75/31.05 (22) UsableRulesProof [EQUIVALENT, 0 ms] 114.75/31.05 (23) PiDP 114.75/31.05 (24) PiDPToQDPProof [SOUND, 0 ms] 114.75/31.05 (25) QDP 114.75/31.05 (26) QDPSizeChangeProof [EQUIVALENT, 0 ms] 114.75/31.05 (27) YES 114.75/31.05 (28) PiDP 114.75/31.05 (29) UsableRulesProof [EQUIVALENT, 0 ms] 114.75/31.05 (30) PiDP 114.75/31.05 (31) PiDPToQDPProof [SOUND, 0 ms] 114.75/31.05 (32) QDP 114.75/31.05 (33) QDPSizeChangeProof [EQUIVALENT, 0 ms] 114.75/31.05 (34) YES 114.75/31.05 (35) PiDP 114.75/31.05 (36) UsableRulesProof [EQUIVALENT, 0 ms] 114.75/31.05 (37) PiDP 114.75/31.05 (38) PiDPToQDPProof [SOUND, 0 ms] 114.75/31.05 (39) QDP 114.75/31.05 (40) TransformationProof [SOUND, 0 ms] 114.75/31.05 (41) QDP 114.75/31.05 (42) UsableRulesProof [EQUIVALENT, 0 ms] 114.75/31.05 (43) QDP 114.75/31.05 (44) QReductionProof [EQUIVALENT, 0 ms] 114.75/31.05 (45) QDP 114.75/31.05 (46) TransformationProof [EQUIVALENT, 0 ms] 114.75/31.05 (47) QDP 114.75/31.05 (48) DependencyGraphProof [EQUIVALENT, 0 ms] 114.75/31.05 (49) QDP 114.75/31.05 (50) QDPQMonotonicMRRProof [EQUIVALENT, 96 ms] 114.75/31.05 (51) QDP 114.75/31.05 (52) QReductionProof [EQUIVALENT, 0 ms] 114.75/31.05 (53) QDP 114.75/31.05 (54) TransformationProof [SOUND, 0 ms] 114.75/31.05 (55) QDP 114.75/31.05 (56) UsableRulesProof [EQUIVALENT, 0 ms] 114.75/31.05 (57) QDP 114.75/31.05 (58) QReductionProof [EQUIVALENT, 0 ms] 114.75/31.05 (59) QDP 114.75/31.05 (60) TransformationProof [EQUIVALENT, 0 ms] 114.75/31.05 (61) QDP 114.75/31.05 (62) QDPQMonotonicMRRProof [EQUIVALENT, 111 ms] 114.75/31.05 (63) QDP 114.75/31.05 (64) UsableRulesProof [EQUIVALENT, 1 ms] 114.75/31.05 (65) QDP 114.75/31.05 (66) QReductionProof [EQUIVALENT, 0 ms] 114.75/31.05 (67) QDP 114.75/31.05 (68) TransformationProof [SOUND, 0 ms] 114.75/31.05 (69) QDP 114.75/31.05 (70) UsableRulesProof [EQUIVALENT, 0 ms] 114.75/31.05 (71) QDP 114.75/31.05 (72) QReductionProof [EQUIVALENT, 0 ms] 114.75/31.05 (73) QDP 114.75/31.05 (74) TransformationProof [EQUIVALENT, 0 ms] 114.75/31.05 (75) QDP 114.75/31.05 (76) QDPQMonotonicMRRProof [EQUIVALENT, 43 ms] 114.75/31.05 (77) QDP 114.75/31.05 (78) DependencyGraphProof [EQUIVALENT, 0 ms] 114.75/31.05 (79) QDP 114.75/31.05 (80) UsableRulesProof [EQUIVALENT, 0 ms] 114.75/31.05 (81) QDP 114.75/31.05 (82) QDPQMonotonicMRRProof [EQUIVALENT, 0 ms] 114.75/31.05 (83) QDP 114.75/31.05 (84) QReductionProof [EQUIVALENT, 0 ms] 114.75/31.05 (85) QDP 114.75/31.05 (86) TransformationProof [SOUND, 0 ms] 114.75/31.05 (87) QDP 114.75/31.05 (88) UsableRulesProof [EQUIVALENT, 0 ms] 114.75/31.05 (89) QDP 114.75/31.05 (90) QReductionProof [EQUIVALENT, 0 ms] 114.75/31.05 (91) QDP 114.75/31.05 (92) TransformationProof [EQUIVALENT, 0 ms] 114.75/31.05 (93) QDP 114.75/31.05 (94) DependencyGraphProof [EQUIVALENT, 0 ms] 114.75/31.05 (95) QDP 114.75/31.05 (96) QDPQMonotonicMRRProof [EQUIVALENT, 30 ms] 114.75/31.05 (97) QDP 114.75/31.05 (98) QDPQMonotonicMRRProof [EQUIVALENT, 11 ms] 114.75/31.05 (99) QDP 114.75/31.05 (100) QDPQMonotonicMRRProof [EQUIVALENT, 15 ms] 114.75/31.05 (101) QDP 114.75/31.05 (102) PisEmptyProof [EQUIVALENT, 0 ms] 114.75/31.05 (103) YES 114.75/31.05 114.75/31.05 114.75/31.05 ---------------------------------------- 114.75/31.05 114.75/31.05 (0) 114.75/31.05 Obligation: 114.75/31.05 Clauses: 114.75/31.05 114.75/31.05 eq(t, t). 114.75/31.05 eq(f, f). 114.75/31.05 neq(t, f). 114.75/31.05 neq(f, t). 114.75/31.05 del(X1, [], []). 114.75/31.05 del(X, .(Y, YS), YS) :- eq(X, Y). 114.75/31.05 del(X, .(Y, YS), .(Y, ZS)) :- ','(neq(X, Y), del(X, YS, ZS)). 114.75/31.05 ge(t, t). 114.75/31.05 ge(t, f). 114.75/31.05 ge(f, f). 114.75/31.05 gt(t, f). 114.75/31.05 max([], f). 114.75/31.05 max(.(X, []), X). 114.75/31.05 max(.(X, .(Y, XS)), Z) :- ','(ge(X, Y), max(.(X, XS), Z)). 114.75/31.05 max(.(X, .(Y, XS)), Z) :- ','(gt(Y, X), max(.(Y, XS), Z)). 114.75/31.05 maxsort([], []). 114.75/31.05 maxsort(.(X, XS), .(Y, YS)) :- ','(max(.(X, XS), Y), ','(del(Y, .(X, XS), ZS), maxsort(ZS, YS))). 114.75/31.05 114.75/31.05 114.75/31.05 Query: maxsort(g,a) 114.75/31.05 ---------------------------------------- 114.75/31.05 114.75/31.05 (1) PrologToDTProblemTransformerProof (SOUND) 114.75/31.05 Built DT problem from termination graph DT10. 114.75/31.05 114.75/31.05 { 114.75/31.05 "root": 3, 114.75/31.05 "program": { 114.75/31.05 "directives": [], 114.75/31.05 "clauses": [ 114.75/31.05 [ 114.75/31.05 "(eq (t) (t))", 114.75/31.05 null 114.75/31.05 ], 114.75/31.05 [ 114.75/31.05 "(eq (f) (f))", 114.75/31.05 null 114.75/31.05 ], 114.75/31.05 [ 114.75/31.05 "(neq (t) (f))", 114.75/31.05 null 114.75/31.05 ], 114.75/31.05 [ 114.75/31.05 "(neq (f) (t))", 114.75/31.05 null 114.75/31.05 ], 114.75/31.05 [ 114.75/31.05 "(del X1 ([]) ([]))", 114.75/31.05 null 114.75/31.05 ], 114.75/31.05 [ 114.75/31.05 "(del X (. Y YS) YS)", 114.75/31.05 "(eq X Y)" 114.75/31.05 ], 114.75/31.05 [ 114.75/31.05 "(del X (. Y YS) (. Y ZS))", 114.75/31.05 "(',' (neq X Y) (del X YS ZS))" 114.75/31.05 ], 114.75/31.05 [ 114.75/31.05 "(ge (t) (t))", 114.75/31.05 null 114.75/31.05 ], 114.75/31.05 [ 114.75/31.05 "(ge (t) (f))", 114.75/31.05 null 114.75/31.05 ], 114.75/31.05 [ 114.75/31.05 "(ge (f) (f))", 114.75/31.05 null 114.75/31.05 ], 114.75/31.05 [ 114.75/31.05 "(gt (t) (f))", 114.75/31.05 null 114.75/31.05 ], 114.75/31.05 [ 114.75/31.05 "(max ([]) (f))", 114.75/31.05 null 114.75/31.05 ], 114.75/31.05 [ 114.75/31.05 "(max (. X ([])) X)", 114.75/31.05 null 114.75/31.05 ], 114.75/31.05 [ 114.75/31.05 "(max (. X (. Y XS)) Z)", 114.75/31.05 "(',' (ge X Y) (max (. X XS) Z))" 114.75/31.05 ], 114.75/31.05 [ 114.75/31.05 "(max (. X (. Y XS)) Z)", 114.75/31.05 "(',' (gt Y X) (max (. Y XS) Z))" 114.75/31.05 ], 114.75/31.05 [ 114.75/31.05 "(maxsort ([]) ([]))", 114.75/31.05 null 114.75/31.05 ], 114.75/31.05 [ 114.75/31.05 "(maxsort (. X XS) (. Y YS))", 114.75/31.05 "(',' (max (. X XS) Y) (',' (del Y (. X XS) ZS) (maxsort ZS YS)))" 114.75/31.05 ] 114.75/31.05 ] 114.75/31.05 }, 114.75/31.05 "graph": { 114.75/31.05 "nodes": { 114.75/31.05 "907": { 114.75/31.05 "goal": [], 114.75/31.05 "kb": { 114.75/31.05 "nonunifying": [], 114.75/31.05 "intvars": {}, 114.75/31.05 "arithmetic": { 114.75/31.05 "type": "PlainIntegerRelationState", 114.75/31.05 "relations": [] 114.75/31.05 }, 114.75/31.05 "ground": [], 114.75/31.05 "free": [], 114.75/31.05 "exprvars": [] 114.75/31.05 } 114.75/31.05 }, 114.75/31.05 "908": { 114.75/31.05 "goal": [ 114.75/31.05 { 114.75/31.05 "clause": 4, 114.75/31.05 "scope": 28, 114.75/31.05 "term": "(del (t) (. (f) T214) X347)" 114.75/31.05 }, 114.75/31.05 { 114.75/31.05 "clause": 5, 114.75/31.05 "scope": 28, 114.75/31.05 "term": "(del (t) (. (f) T214) X347)" 114.75/31.05 }, 114.75/31.05 { 114.75/31.05 "clause": 6, 114.75/31.05 "scope": 28, 114.75/31.05 "term": "(del (t) (. (f) T214) X347)" 114.75/31.05 } 114.75/31.05 ], 114.75/31.05 "kb": { 114.75/31.05 "nonunifying": [], 114.75/31.05 "intvars": {}, 114.75/31.05 "arithmetic": { 114.75/31.05 "type": "PlainIntegerRelationState", 114.75/31.05 "relations": [] 114.75/31.05 }, 114.75/31.05 "ground": ["T214"], 114.75/31.05 "free": ["X347"], 114.75/31.05 "exprvars": [] 114.75/31.05 } 114.75/31.05 }, 114.75/31.05 "909": { 114.75/31.05 "goal": [ 114.75/31.05 { 114.75/31.05 "clause": 5, 114.75/31.05 "scope": 28, 114.75/31.05 "term": "(del (t) (. (f) T214) X347)" 114.75/31.05 }, 114.75/31.05 { 114.75/31.05 "clause": 6, 114.75/31.05 "scope": 28, 114.75/31.05 "term": "(del (t) (. (f) T214) X347)" 114.75/31.05 } 114.75/31.05 ], 114.75/31.05 "kb": { 114.75/31.05 "nonunifying": [], 114.75/31.05 "intvars": {}, 114.75/31.05 "arithmetic": { 114.75/31.05 "type": "PlainIntegerRelationState", 114.75/31.05 "relations": [] 114.75/31.05 }, 114.75/31.05 "ground": ["T214"], 114.75/31.05 "free": ["X347"], 114.75/31.05 "exprvars": [] 114.75/31.05 } 114.75/31.05 }, 114.75/31.05 "910": { 114.75/31.05 "goal": [{ 114.75/31.05 "clause": 5, 114.75/31.05 "scope": 28, 114.75/31.05 "term": "(del (t) (. (f) T214) X347)" 114.75/31.05 }], 114.75/31.05 "kb": { 114.75/31.05 "nonunifying": [], 114.75/31.05 "intvars": {}, 114.75/31.05 "arithmetic": { 114.75/31.05 "type": "PlainIntegerRelationState", 114.75/31.05 "relations": [] 114.75/31.05 }, 114.75/31.05 "ground": ["T214"], 114.75/31.05 "free": ["X347"], 114.75/31.05 "exprvars": [] 114.75/31.05 } 114.75/31.05 }, 114.75/31.05 "911": { 114.75/31.05 "goal": [{ 114.75/31.05 "clause": 6, 114.75/31.05 "scope": 28, 114.75/31.05 "term": "(del (t) (. (f) T214) X347)" 114.75/31.05 }], 114.75/31.05 "kb": { 114.75/31.05 "nonunifying": [], 114.75/31.05 "intvars": {}, 114.75/31.05 "arithmetic": { 114.75/31.05 "type": "PlainIntegerRelationState", 114.75/31.05 "relations": [] 114.75/31.05 }, 114.75/31.05 "ground": ["T214"], 114.75/31.05 "free": ["X347"], 114.75/31.05 "exprvars": [] 114.75/31.05 } 114.75/31.05 }, 114.75/31.05 "912": { 114.75/31.05 "goal": [{ 114.75/31.05 "clause": -1, 114.75/31.05 "scope": -1, 114.75/31.05 "term": "(eq (t) (f))" 114.75/31.05 }], 114.75/31.05 "kb": { 114.75/31.05 "nonunifying": [], 114.75/31.05 "intvars": {}, 114.75/31.05 "arithmetic": { 114.75/31.05 "type": "PlainIntegerRelationState", 114.75/31.05 "relations": [] 114.75/31.05 }, 114.75/31.05 "ground": [], 114.75/31.05 "free": [], 114.75/31.05 "exprvars": [] 114.75/31.05 } 114.75/31.05 }, 114.75/31.05 "913": { 114.75/31.05 "goal": [ 114.75/31.05 { 114.75/31.05 "clause": 0, 114.75/31.05 "scope": 29, 114.75/31.05 "term": "(eq (t) (f))" 114.75/31.05 }, 114.75/31.05 { 114.75/31.05 "clause": 1, 114.75/31.05 "scope": 29, 114.75/31.05 "term": "(eq (t) (f))" 114.75/31.05 } 114.75/31.05 ], 114.75/31.05 "kb": { 114.75/31.05 "nonunifying": [], 114.75/31.05 "intvars": {}, 114.75/31.05 "arithmetic": { 114.75/31.05 "type": "PlainIntegerRelationState", 114.75/31.05 "relations": [] 114.75/31.05 }, 114.75/31.05 "ground": [], 114.75/31.05 "free": [], 114.75/31.05 "exprvars": [] 114.75/31.05 } 114.75/31.05 }, 114.75/31.05 "914": { 114.75/31.05 "goal": [{ 114.75/31.05 "clause": 1, 114.75/31.05 "scope": 29, 114.75/31.05 "term": "(eq (t) (f))" 114.75/31.05 }], 114.75/31.05 "kb": { 114.75/31.05 "nonunifying": [], 114.75/31.05 "intvars": {}, 114.75/31.05 "arithmetic": { 114.75/31.05 "type": "PlainIntegerRelationState", 114.75/31.05 "relations": [] 114.75/31.05 }, 114.75/31.05 "ground": [], 114.75/31.05 "free": [], 114.75/31.05 "exprvars": [] 114.75/31.05 } 114.75/31.05 }, 114.75/31.05 "915": { 114.75/31.05 "goal": [], 114.75/31.05 "kb": { 114.75/31.05 "nonunifying": [], 114.75/31.05 "intvars": {}, 114.75/31.05 "arithmetic": { 114.75/31.05 "type": "PlainIntegerRelationState", 114.75/31.05 "relations": [] 114.75/31.05 }, 114.75/31.05 "ground": [], 114.75/31.05 "free": [], 114.75/31.05 "exprvars": [] 114.75/31.05 } 114.75/31.05 }, 114.75/31.05 "916": { 114.75/31.05 "goal": [{ 114.75/31.05 "clause": -1, 114.75/31.05 "scope": -1, 114.75/31.05 "term": "(',' (neq (t) (f)) (del (t) T224 X384))" 114.75/31.05 }], 114.75/31.05 "kb": { 114.75/31.05 "nonunifying": [], 114.75/31.05 "intvars": {}, 114.75/31.05 "arithmetic": { 114.75/31.05 "type": "PlainIntegerRelationState", 114.75/31.05 "relations": [] 114.75/31.05 }, 114.75/31.05 "ground": ["T224"], 114.75/31.05 "free": ["X384"], 114.75/31.05 "exprvars": [] 114.75/31.05 } 114.75/31.05 }, 114.75/31.05 "917": { 114.75/31.05 "goal": [ 114.75/31.05 { 114.75/31.05 "clause": 2, 114.75/31.05 "scope": 30, 114.75/31.05 "term": "(',' (neq (t) (f)) (del (t) T224 X384))" 114.75/31.05 }, 114.75/31.05 { 114.75/31.05 "clause": 3, 114.75/31.05 "scope": 30, 114.75/31.05 "term": "(',' (neq (t) (f)) (del (t) T224 X384))" 114.75/31.05 } 114.75/31.05 ], 114.75/31.05 "kb": { 114.75/31.05 "nonunifying": [], 114.75/31.05 "intvars": {}, 114.75/31.05 "arithmetic": { 114.75/31.05 "type": "PlainIntegerRelationState", 114.75/31.05 "relations": [] 114.75/31.05 }, 114.75/31.05 "ground": ["T224"], 114.75/31.05 "free": ["X384"], 114.75/31.05 "exprvars": [] 114.75/31.05 } 114.75/31.05 }, 114.75/31.05 "918": { 114.75/31.05 "goal": [{ 114.75/31.05 "clause": 2, 114.75/31.05 "scope": 30, 114.75/31.05 "term": "(',' (neq (t) (f)) (del (t) T224 X384))" 114.75/31.05 }], 114.75/31.05 "kb": { 114.75/31.05 "nonunifying": [], 114.75/31.05 "intvars": {}, 114.75/31.05 "arithmetic": { 114.75/31.05 "type": "PlainIntegerRelationState", 114.75/31.05 "relations": [] 114.75/31.05 }, 114.75/31.05 "ground": ["T224"], 114.75/31.05 "free": ["X384"], 114.75/31.05 "exprvars": [] 114.75/31.05 } 114.75/31.05 }, 114.75/31.05 "919": { 114.75/31.05 "goal": [{ 114.75/31.05 "clause": 3, 114.75/31.05 "scope": 30, 114.75/31.05 "term": "(',' (neq (t) (f)) (del (t) T224 X384))" 114.75/31.05 }], 114.75/31.05 "kb": { 114.75/31.05 "nonunifying": [], 114.75/31.05 "intvars": {}, 114.75/31.05 "arithmetic": { 114.75/31.05 "type": "PlainIntegerRelationState", 114.75/31.05 "relations": [] 114.75/31.05 }, 114.75/31.05 "ground": ["T224"], 114.75/31.05 "free": ["X384"], 114.75/31.05 "exprvars": [] 114.75/31.05 } 114.75/31.05 }, 114.75/31.05 "243": { 114.75/31.05 "goal": [{ 114.75/31.05 "clause": -1, 114.75/31.05 "scope": -1, 114.75/31.05 "term": "(',' (max (. T7 T8) T11) (',' (del T11 (. T7 T8) X14) (maxsort X14 T12)))" 114.75/31.05 }], 114.75/31.05 "kb": { 114.75/31.05 "nonunifying": [], 114.75/31.05 "intvars": {}, 114.75/31.05 "arithmetic": { 114.75/31.05 "type": "PlainIntegerRelationState", 114.75/31.05 "relations": [] 114.75/31.05 }, 114.75/31.05 "ground": [ 114.75/31.05 "T7", 114.75/31.05 "T8" 114.75/31.05 ], 114.75/31.05 "free": ["X14"], 114.75/31.05 "exprvars": [] 114.75/31.05 } 114.75/31.05 }, 114.75/31.05 "244": { 114.75/31.05 "goal": [], 114.75/31.05 "kb": { 114.75/31.05 "nonunifying": [], 114.75/31.05 "intvars": {}, 114.75/31.05 "arithmetic": { 114.75/31.05 "type": "PlainIntegerRelationState", 114.75/31.05 "relations": [] 114.75/31.05 }, 114.75/31.05 "ground": [], 114.75/31.05 "free": [], 114.75/31.05 "exprvars": [] 114.75/31.05 } 114.75/31.05 }, 114.75/31.05 "3": { 114.75/31.05 "goal": [{ 114.75/31.05 "clause": -1, 114.75/31.05 "scope": -1, 114.75/31.05 "term": "(maxsort T1 T2)" 114.75/31.05 }], 114.75/31.05 "kb": { 114.75/31.05 "nonunifying": [], 114.75/31.05 "intvars": {}, 114.75/31.05 "arithmetic": { 114.75/31.05 "type": "PlainIntegerRelationState", 114.75/31.05 "relations": [] 114.75/31.05 }, 114.75/31.05 "ground": ["T1"], 114.75/31.05 "free": [], 114.75/31.05 "exprvars": [] 114.75/31.05 } 114.75/31.05 }, 114.75/31.05 "245": { 114.75/31.05 "goal": [ 114.75/31.05 { 114.75/31.05 "clause": 11, 114.75/31.05 "scope": 2, 114.75/31.05 "term": "(',' (max (. T7 T8) T11) (',' (del T11 (. T7 T8) X14) (maxsort X14 T12)))" 114.75/31.05 }, 114.75/31.05 { 114.75/31.05 "clause": 12, 114.75/31.05 "scope": 2, 114.75/31.05 "term": "(',' (max (. T7 T8) T11) (',' (del T11 (. T7 T8) X14) (maxsort X14 T12)))" 114.75/31.05 }, 114.75/31.05 { 114.75/31.05 "clause": 13, 114.75/31.05 "scope": 2, 114.75/31.05 "term": "(',' (max (. T7 T8) T11) (',' (del T11 (. T7 T8) X14) (maxsort X14 T12)))" 114.75/31.05 }, 114.75/31.05 { 114.75/31.05 "clause": 14, 114.75/31.05 "scope": 2, 114.75/31.05 "term": "(',' (max (. T7 T8) T11) (',' (del T11 (. T7 T8) X14) (maxsort X14 T12)))" 114.75/31.05 } 114.75/31.05 ], 114.75/31.05 "kb": { 114.75/31.05 "nonunifying": [], 114.75/31.05 "intvars": {}, 114.75/31.05 "arithmetic": { 114.75/31.05 "type": "PlainIntegerRelationState", 114.75/31.05 "relations": [] 114.75/31.05 }, 114.75/31.05 "ground": [ 114.75/31.05 "T7", 114.75/31.05 "T8" 114.75/31.05 ], 114.75/31.05 "free": ["X14"], 114.75/31.05 "exprvars": [] 114.75/31.05 } 114.75/31.05 }, 114.75/31.05 "4": { 114.75/31.05 "goal": [ 114.75/31.05 { 114.75/31.05 "clause": 15, 114.75/31.05 "scope": 1, 114.75/31.05 "term": "(maxsort T1 T2)" 114.75/31.05 }, 114.75/31.05 { 114.75/31.05 "clause": 16, 114.75/31.05 "scope": 1, 114.75/31.05 "term": "(maxsort T1 T2)" 114.75/31.05 } 114.75/31.05 ], 114.75/31.05 "kb": { 114.75/31.05 "nonunifying": [], 114.75/31.05 "intvars": {}, 114.75/31.05 "arithmetic": { 114.75/31.05 "type": "PlainIntegerRelationState", 114.75/31.05 "relations": [] 114.75/31.05 }, 114.75/31.05 "ground": ["T1"], 114.75/31.05 "free": [], 114.75/31.05 "exprvars": [] 114.75/31.05 } 114.75/31.05 }, 114.75/31.05 "246": { 114.75/31.05 "goal": [ 114.75/31.05 { 114.75/31.05 "clause": 12, 114.75/31.05 "scope": 2, 114.75/31.05 "term": "(',' (max (. 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(f) T214) X347))" 114.75/31.08 }], 114.75/31.08 "kb": { 114.75/31.08 "nonunifying": [], 114.75/31.08 "intvars": {}, 114.75/31.08 "arithmetic": { 114.75/31.08 "type": "PlainIntegerRelationState", 114.75/31.08 "relations": [] 114.75/31.08 }, 114.75/31.08 "ground": [ 114.75/31.08 "T213", 114.75/31.08 "T214" 114.75/31.08 ], 114.75/31.08 "free": ["X347"], 114.75/31.08 "exprvars": [] 114.75/31.08 } 114.75/31.08 }, 114.75/31.08 "905": { 114.75/31.08 "goal": [{ 114.75/31.08 "clause": 3, 114.75/31.08 "scope": 27, 114.75/31.08 "term": "(',' (neq T213 (f)) (del T213 (. (f) T214) X347))" 114.75/31.08 }], 114.75/31.08 "kb": { 114.75/31.08 "nonunifying": [], 114.75/31.08 "intvars": {}, 114.75/31.08 "arithmetic": { 114.75/31.08 "type": "PlainIntegerRelationState", 114.75/31.08 "relations": [] 114.75/31.08 }, 114.75/31.08 "ground": [ 114.75/31.08 "T213", 114.75/31.08 "T214" 114.75/31.08 ], 114.75/31.08 "free": ["X347"], 114.75/31.08 "exprvars": [] 114.75/31.08 } 114.75/31.08 }, 114.75/31.08 "906": { 114.75/31.08 "goal": [{ 114.75/31.08 "clause": -1, 114.75/31.08 "scope": -1, 114.75/31.08 "term": "(del (t) (. (f) T214) X347)" 114.75/31.08 }], 114.75/31.08 "kb": { 114.75/31.08 "nonunifying": [], 114.75/31.08 "intvars": {}, 114.75/31.08 "arithmetic": { 114.75/31.08 "type": "PlainIntegerRelationState", 114.75/31.08 "relations": [] 114.75/31.08 }, 114.75/31.08 "ground": ["T214"], 114.75/31.08 "free": ["X347"], 114.75/31.08 "exprvars": [] 114.75/31.08 } 114.75/31.08 } 114.75/31.08 }, 114.75/31.08 "edges": [ 114.75/31.08 { 114.75/31.08 "from": 3, 114.75/31.08 "to": 4, 114.75/31.08 "label": "CASE" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 4, 114.75/31.08 "to": 35, 114.75/31.08 "label": "EVAL with clause\nmaxsort([], []).\nand substitutionT1 -> [],\nT2 -> []" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 4, 114.75/31.08 "to": 36, 114.75/31.08 "label": "EVAL-BACKTRACK" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 35, 114.75/31.08 "to": 37, 114.75/31.08 "label": "SUCCESS" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 36, 114.75/31.08 "to": 243, 114.75/31.08 "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" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 36, 114.75/31.08 "to": 244, 114.75/31.08 "label": "EVAL-BACKTRACK" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 37, 114.75/31.08 "to": 38, 114.75/31.08 "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" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 243, 114.75/31.08 "to": 245, 114.75/31.08 "label": "CASE" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 245, 114.75/31.08 "to": 246, 114.75/31.08 "label": "BACKTRACK\nfor clause: max([], f)because of non-unification" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 246, 114.75/31.08 "to": 247, 114.75/31.08 "label": "PARALLEL" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 246, 114.75/31.08 "to": 499, 114.75/31.08 "label": "PARALLEL" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 247, 114.75/31.08 "to": 500, 114.75/31.08 "label": "EVAL with clause\nmax(.(X19, []), X19).\nand substitutionT7 -> T17,\nX19 -> T17,\nT8 -> [],\nT11 -> T17,\nT12 -> T18" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 247, 114.75/31.08 "to": 501, 114.75/31.08 "label": "EVAL-BACKTRACK" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 499, 114.75/31.08 "to": 704, 114.75/31.08 "label": "PARALLEL" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 499, 114.75/31.08 "to": 705, 114.75/31.08 "label": "PARALLEL" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 500, 114.75/31.08 "to": 502, 114.75/31.08 "label": "SPLIT 1" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 500, 114.75/31.08 "to": 503, 114.75/31.08 "label": "SPLIT 2\nnew knowledge:\nT17 is ground\nT21 is ground\nreplacements:X14 -> T21" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 502, 114.75/31.08 "to": 504, 114.75/31.08 "label": "CASE" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 503, 114.75/31.08 "to": 3, 114.75/31.08 "label": "INSTANCE with matching:\nT1 -> T21\nT2 -> T18" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 504, 114.75/31.08 "to": 505, 114.75/31.08 "label": "BACKTRACK\nfor clause: del(X1, [], [])because of non-unification" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 505, 114.75/31.08 "to": 506, 114.75/31.08 "label": "PARALLEL" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 505, 114.75/31.08 "to": 507, 114.75/31.08 "label": "PARALLEL" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 506, 114.75/31.08 "to": 536, 114.75/31.08 "label": "ONLY EVAL with clause\ndel(X37, .(X38, X39), X39) :- eq(X37, X38).\nand substitutionT17 -> T29,\nX37 -> T29,\nX38 -> T29,\nX39 -> [],\nX14 -> []" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 507, 114.75/31.08 "to": 546, 114.75/31.08 "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)" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 536, 114.75/31.08 "to": 537, 114.75/31.08 "label": "CASE" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 537, 114.75/31.08 "to": 538, 114.75/31.08 "label": "PARALLEL" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 537, 114.75/31.08 "to": 539, 114.75/31.08 "label": "PARALLEL" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 538, 114.75/31.08 "to": 540, 114.75/31.08 "label": "EVAL with clause\neq(t, t).\nand substitutionT29 -> t" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 538, 114.75/31.08 "to": 541, 114.75/31.08 "label": "EVAL-BACKTRACK" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 539, 114.75/31.08 "to": 543, 114.75/31.08 "label": "EVAL with clause\neq(f, f).\nand substitutionT29 -> f" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 539, 114.75/31.08 "to": 544, 114.75/31.08 "label": "EVAL-BACKTRACK" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 540, 114.75/31.08 "to": 542, 114.75/31.08 "label": "SUCCESS" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 543, 114.75/31.08 "to": 545, 114.75/31.08 "label": "SUCCESS" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 546, 114.75/31.08 "to": 547, 114.75/31.08 "label": "CASE" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 547, 114.75/31.08 "to": 548, 114.75/31.08 "label": "BACKTRACK\nfor clause: neq(t, f)because of non-unification" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 548, 114.75/31.08 "to": 549, 114.75/31.08 "label": "BACKTRACK\nfor clause: neq(f, t)because of non-unification" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 704, 114.75/31.08 "to": 706, 114.75/31.08 "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" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 704, 114.75/31.08 "to": 707, 114.75/31.08 "label": "EVAL-BACKTRACK" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 705, 114.75/31.08 "to": 948, 114.75/31.08 "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" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 705, 114.75/31.08 "to": 949, 114.75/31.08 "label": "EVAL-BACKTRACK" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 706, 114.75/31.08 "to": 708, 114.75/31.08 "label": "CASE" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 708, 114.75/31.08 "to": 709, 114.75/31.08 "label": "PARALLEL" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 708, 114.75/31.08 "to": 710, 114.75/31.08 "label": "PARALLEL" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 709, 114.75/31.08 "to": 711, 114.75/31.08 "label": "EVAL with clause\nge(t, t).\nand substitutionT51 -> t,\nT52 -> t" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 709, 114.75/31.08 "to": 712, 114.75/31.08 "label": "EVAL-BACKTRACK" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 710, 114.75/31.08 "to": 834, 114.75/31.08 "label": "PARALLEL" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 710, 114.75/31.08 "to": 835, 114.75/31.08 "label": "PARALLEL" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 711, 114.75/31.08 "to": 713, 114.75/31.08 "label": "SPLIT 1" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 711, 114.75/31.08 "to": 714, 114.75/31.08 "label": "SPLIT 2\nnew knowledge:\nT53 is ground\nT57 is ground\nreplacements:T55 -> T57,\nT56 -> T58" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 713, 114.75/31.08 "to": 715, 114.75/31.08 "label": "CASE" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 714, 114.75/31.08 "to": 740, 114.75/31.08 "label": "SPLIT 1" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 714, 114.75/31.08 "to": 741, 114.75/31.08 "label": "SPLIT 2\nnew knowledge:\nT57 is ground\nT53 is ground\nT87 is ground\nreplacements:X14 -> T87" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 715, 114.75/31.08 "to": 716, 114.75/31.08 "label": "BACKTRACK\nfor clause: max([], f)because of non-unification" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 716, 114.75/31.08 "to": 717, 114.75/31.08 "label": "PARALLEL" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 716, 114.75/31.08 "to": 718, 114.75/31.08 "label": "PARALLEL" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 717, 114.75/31.08 "to": 719, 114.75/31.08 "label": "EVAL with clause\nmax(.(X81, []), X81).\nand substitutionX81 -> t,\nT53 -> [],\nT55 -> t" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 717, 114.75/31.08 "to": 720, 114.75/31.08 "label": "EVAL-BACKTRACK" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 718, 114.75/31.08 "to": 722, 114.75/31.08 "label": "PARALLEL" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 718, 114.75/31.08 "to": 723, 114.75/31.08 "label": "PARALLEL" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 719, 114.75/31.08 "to": 721, 114.75/31.08 "label": "SUCCESS" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 722, 114.75/31.08 "to": 724, 114.75/31.08 "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" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 722, 114.75/31.08 "to": 725, 114.75/31.08 "label": "EVAL-BACKTRACK" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 723, 114.75/31.08 "to": 736, 114.75/31.08 "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" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 723, 114.75/31.08 "to": 737, 114.75/31.08 "label": "EVAL-BACKTRACK" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 724, 114.75/31.08 "to": 726, 114.75/31.08 "label": "CASE" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 726, 114.75/31.08 "to": 727, 114.75/31.08 "label": "PARALLEL" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 726, 114.75/31.08 "to": 728, 114.75/31.08 "label": "PARALLEL" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 727, 114.75/31.08 "to": 729, 114.75/31.08 "label": "EVAL with clause\nge(t, t).\nand substitutionT71 -> t" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 727, 114.75/31.08 "to": 730, 114.75/31.08 "label": "EVAL-BACKTRACK" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 728, 114.75/31.08 "to": 731, 114.75/31.08 "label": "PARALLEL" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 728, 114.75/31.08 "to": 732, 114.75/31.08 "label": "PARALLEL" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 729, 114.75/31.08 "to": 713, 114.75/31.08 "label": "INSTANCE with matching:\nT53 -> T72\nT55 -> T74" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 731, 114.75/31.08 "to": 733, 114.75/31.08 "label": "EVAL with clause\nge(t, f).\nand substitutionT71 -> f" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 731, 114.75/31.08 "to": 734, 114.75/31.08 "label": "EVAL-BACKTRACK" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 732, 114.75/31.08 "to": 735, 114.75/31.08 "label": "BACKTRACK\nfor clause: ge(f, f)because of non-unification" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 733, 114.75/31.08 "to": 713, 114.75/31.08 "label": "INSTANCE with matching:\nT53 -> T72\nT55 -> T74" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 736, 114.75/31.08 "to": 738, 114.75/31.08 "label": "CASE" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 738, 114.75/31.08 "to": 739, 114.75/31.08 "label": "BACKTRACK\nfor clause: gt(t, f)because of non-unification" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 740, 114.75/31.08 "to": 742, 114.75/31.08 "label": "CASE" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 741, 114.75/31.08 "to": 3, 114.75/31.08 "label": "INSTANCE with matching:\nT1 -> T87\nT2 -> T58" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 742, 114.75/31.08 "to": 743, 114.75/31.08 "label": "BACKTRACK\nfor clause: del(X1, [], [])because of non-unification" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 743, 114.75/31.08 "to": 744, 114.75/31.08 "label": "PARALLEL" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 743, 114.75/31.08 "to": 745, 114.75/31.08 "label": "PARALLEL" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 744, 114.75/31.08 "to": 746, 114.75/31.08 "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)" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 745, 114.75/31.08 "to": 754, 114.75/31.08 "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)" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 746, 114.75/31.08 "to": 747, 114.75/31.08 "label": "CASE" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 747, 114.75/31.08 "to": 748, 114.75/31.08 "label": "PARALLEL" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 747, 114.75/31.08 "to": 749, 114.75/31.08 "label": "PARALLEL" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 748, 114.75/31.08 "to": 750, 114.75/31.08 "label": "EVAL with clause\neq(t, t).\nand substitutionT99 -> t" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 748, 114.75/31.08 "to": 751, 114.75/31.08 "label": "EVAL-BACKTRACK" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 749, 114.75/31.08 "to": 753, 114.75/31.08 "label": "BACKTRACK\nfor clause: eq(f, f)because of non-unification" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 750, 114.75/31.08 "to": 752, 114.75/31.08 "label": "SUCCESS" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 754, 114.75/31.08 "to": 755, 114.75/31.08 "label": "CASE" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 755, 114.75/31.08 "to": 756, 114.75/31.08 "label": "BACKTRACK\nfor clause: neq(t, f)because of non-unification" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 756, 114.75/31.08 "to": 757, 114.75/31.08 "label": "EVAL with clause\nneq(f, t).\nand substitutionT107 -> f" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 756, 114.75/31.08 "to": 758, 114.75/31.08 "label": "EVAL-BACKTRACK" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 757, 114.75/31.08 "to": 759, 114.75/31.08 "label": "CASE" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 759, 114.75/31.08 "to": 760, 114.75/31.08 "label": "BACKTRACK\nfor clause: del(X1, [], [])because of non-unification" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 760, 114.75/31.08 "to": 761, 114.75/31.08 "label": "PARALLEL" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 760, 114.75/31.08 "to": 762, 114.75/31.08 "label": "PARALLEL" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 761, 114.75/31.08 "to": 763, 114.75/31.08 "label": "ONLY EVAL with clause\ndel(X168, .(X169, X170), X170) :- eq(X168, X169).\nand substitutionX168 -> f,\nX169 -> t,\nT108 -> T115,\nX170 -> T115,\nX150 -> T115" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 762, 114.75/31.08 "to": 767, 114.75/31.08 "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)" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 763, 114.75/31.08 "to": 764, 114.75/31.08 "label": "CASE" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 764, 114.75/31.08 "to": 765, 114.75/31.08 "label": "BACKTRACK\nfor clause: eq(t, t)because of non-unification" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 765, 114.75/31.08 "to": 766, 114.75/31.08 "label": "BACKTRACK\nfor clause: eq(f, f)because of non-unification" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 767, 114.75/31.08 "to": 768, 114.75/31.08 "label": "CASE" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 768, 114.75/31.08 "to": 769, 114.75/31.08 "label": "BACKTRACK\nfor clause: neq(t, f)because of non-unification" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 769, 114.75/31.08 "to": 770, 114.75/31.08 "label": "ONLY EVAL with clause\nneq(f, t).\nand substitution" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 770, 114.75/31.08 "to": 771, 114.75/31.08 "label": "CASE" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 771, 114.75/31.08 "to": 772, 114.75/31.08 "label": "PARALLEL" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 771, 114.75/31.08 "to": 773, 114.75/31.08 "label": "PARALLEL" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 772, 114.75/31.08 "to": 774, 114.75/31.08 "label": "EVAL with clause\ndel(X198, [], []).\nand substitutionX198 -> f,\nT118 -> [],\nX187 -> []" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 772, 114.75/31.08 "to": 775, 114.75/31.08 "label": "EVAL-BACKTRACK" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 773, 114.75/31.08 "to": 777, 114.75/31.08 "label": "PARALLEL" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 773, 114.75/31.08 "to": 778, 114.75/31.08 "label": "PARALLEL" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 774, 114.75/31.08 "to": 776, 114.75/31.08 "label": "SUCCESS" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 777, 114.75/31.08 "to": 779, 114.75/31.08 "label": "EVAL with clause\ndel(X211, .(X212, X213), X213) :- eq(X211, X212).\nand substitutionX211 -> f,\nX212 -> T127,\nX213 -> T128,\nT118 -> .(T127, T128),\nX187 -> T128" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 777, 114.75/31.08 "to": 780, 114.75/31.08 "label": "EVAL-BACKTRACK" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 778, 114.75/31.08 "to": 794, 114.75/31.08 "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)" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 778, 114.75/31.08 "to": 796, 114.75/31.08 "label": "EVAL-BACKTRACK" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 779, 114.75/31.08 "to": 781, 114.75/31.08 "label": "CASE" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 781, 114.75/31.08 "to": 782, 114.75/31.08 "label": "BACKTRACK\nfor clause: eq(t, t)because of non-unification" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 782, 114.75/31.08 "to": 783, 114.75/31.08 "label": "EVAL with clause\neq(f, f).\nand substitutionT127 -> f" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 782, 114.75/31.08 "to": 784, 114.75/31.08 "label": "EVAL-BACKTRACK" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 783, 114.75/31.08 "to": 785, 114.75/31.08 "label": "SUCCESS" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 794, 114.75/31.08 "to": 801, 114.75/31.08 "label": "CASE" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 801, 114.75/31.08 "to": 802, 114.75/31.08 "label": "BACKTRACK\nfor clause: neq(t, f)because of non-unification" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 802, 114.75/31.08 "to": 806, 114.75/31.08 "label": "EVAL with clause\nneq(f, t).\nand substitutionT133 -> t" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 802, 114.75/31.08 "to": 808, 114.75/31.08 "label": "EVAL-BACKTRACK" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 806, 114.75/31.08 "to": 770, 114.75/31.08 "label": "INSTANCE with matching:\nT118 -> T134\nX187 -> X228" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 834, 114.75/31.08 "to": 836, 114.75/31.08 "label": "EVAL with clause\nge(t, f).\nand substitutionT51 -> t,\nT52 -> f" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 834, 114.75/31.08 "to": 837, 114.75/31.08 "label": "EVAL-BACKTRACK" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 835, 114.75/31.08 "to": 865, 114.75/31.08 "label": "EVAL with clause\nge(f, f).\nand substitutionT51 -> f,\nT52 -> f" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 835, 114.75/31.08 "to": 866, 114.75/31.08 "label": "EVAL-BACKTRACK" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 836, 114.75/31.08 "to": 844, 114.75/31.08 "label": "SPLIT 1" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 836, 114.75/31.08 "to": 845, 114.75/31.08 "label": "SPLIT 2\nnew knowledge:\nT53 is ground\nT135 is ground\nreplacements:T55 -> T135,\nT56 -> T136" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 844, 114.75/31.08 "to": 713, 114.75/31.08 "label": "INSTANCE" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 845, 114.75/31.08 "to": 846, 114.75/31.08 "label": "SPLIT 1" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 845, 114.75/31.08 "to": 847, 114.75/31.08 "label": "SPLIT 2\nnew knowledge:\nT135 is ground\nT53 is ground\nT139 is ground\nreplacements:X14 -> T139" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 846, 114.75/31.08 "to": 848, 114.75/31.08 "label": "CASE" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 847, 114.75/31.08 "to": 3, 114.75/31.08 "label": "INSTANCE with matching:\nT1 -> T139\nT2 -> T136" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 848, 114.75/31.08 "to": 849, 114.75/31.08 "label": "BACKTRACK\nfor clause: del(X1, [], [])because of non-unification" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 849, 114.75/31.08 "to": 850, 114.75/31.08 "label": "PARALLEL" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 849, 114.75/31.08 "to": 851, 114.75/31.08 "label": "PARALLEL" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 850, 114.75/31.08 "to": 852, 114.75/31.08 "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)" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 851, 114.75/31.08 "to": 860, 114.75/31.08 "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)" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 852, 114.75/31.08 "to": 853, 114.75/31.08 "label": "CASE" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 853, 114.75/31.08 "to": 854, 114.75/31.08 "label": "PARALLEL" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 853, 114.75/31.08 "to": 855, 114.75/31.08 "label": "PARALLEL" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 854, 114.75/31.08 "to": 856, 114.75/31.08 "label": "EVAL with clause\neq(t, t).\nand substitutionT151 -> t" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 854, 114.75/31.08 "to": 857, 114.75/31.08 "label": "EVAL-BACKTRACK" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 855, 114.75/31.08 "to": 859, 114.75/31.08 "label": "BACKTRACK\nfor clause: eq(f, f)because of non-unification" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 856, 114.75/31.08 "to": 858, 114.75/31.08 "label": "SUCCESS" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 860, 114.75/31.08 "to": 861, 114.75/31.08 "label": "CASE" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 861, 114.75/31.08 "to": 862, 114.75/31.08 "label": "BACKTRACK\nfor clause: neq(t, f)because of non-unification" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 862, 114.75/31.08 "to": 863, 114.75/31.08 "label": "EVAL with clause\nneq(f, t).\nand substitutionT159 -> f" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 862, 114.75/31.08 "to": 864, 114.75/31.08 "label": "EVAL-BACKTRACK" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 863, 114.75/31.08 "to": 770, 114.75/31.08 "label": "INSTANCE with matching:\nT118 -> .(f, T160)\nX187 -> X269" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 865, 114.75/31.08 "to": 867, 114.75/31.08 "label": "SPLIT 1" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 865, 114.75/31.08 "to": 868, 114.75/31.08 "label": "SPLIT 2\nnew knowledge:\nT53 is ground\nT163 is ground\nreplacements:T55 -> T163,\nT56 -> T164" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 867, 114.75/31.08 "to": 869, 114.75/31.08 "label": "CASE" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 868, 114.75/31.08 "to": 890, 114.75/31.08 "label": "SPLIT 1" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 868, 114.75/31.08 "to": 891, 114.75/31.08 "label": "SPLIT 2\nnew knowledge:\nT163 is ground\nT53 is ground\nT193 is ground\nreplacements:X14 -> T193" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 869, 114.75/31.08 "to": 870, 114.75/31.08 "label": "BACKTRACK\nfor clause: max([], f)because of non-unification" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 870, 114.75/31.08 "to": 871, 114.75/31.08 "label": "PARALLEL" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 870, 114.75/31.08 "to": 872, 114.75/31.08 "label": "PARALLEL" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 871, 114.75/31.08 "to": 873, 114.75/31.08 "label": "EVAL with clause\nmax(.(X278, []), X278).\nand substitutionX278 -> f,\nT53 -> [],\nT55 -> f" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 871, 114.75/31.08 "to": 874, 114.75/31.08 "label": "EVAL-BACKTRACK" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 872, 114.75/31.08 "to": 876, 114.75/31.08 "label": "PARALLEL" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 872, 114.75/31.08 "to": 877, 114.75/31.08 "label": "PARALLEL" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 873, 114.75/31.08 "to": 875, 114.75/31.08 "label": "SUCCESS" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 876, 114.75/31.08 "to": 878, 114.75/31.08 "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" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 876, 114.75/31.08 "to": 879, 114.75/31.08 "label": "EVAL-BACKTRACK" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 877, 114.75/31.08 "to": 885, 114.75/31.08 "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" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 877, 114.75/31.08 "to": 886, 114.75/31.08 "label": "EVAL-BACKTRACK" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 878, 114.75/31.08 "to": 880, 114.75/31.08 "label": "CASE" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 880, 114.75/31.08 "to": 881, 114.75/31.08 "label": "BACKTRACK\nfor clause: ge(t, t)because of non-unification" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 881, 114.75/31.08 "to": 882, 114.75/31.08 "label": "BACKTRACK\nfor clause: ge(t, f)because of non-unification" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 882, 114.75/31.08 "to": 883, 114.75/31.08 "label": "EVAL with clause\nge(f, f).\nand substitutionT177 -> f" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 882, 114.75/31.08 "to": 884, 114.75/31.08 "label": "EVAL-BACKTRACK" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 883, 114.75/31.08 "to": 867, 114.75/31.08 "label": "INSTANCE with matching:\nT53 -> T178\nT55 -> T180" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 885, 114.75/31.08 "to": 887, 114.75/31.08 "label": "CASE" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 887, 114.75/31.08 "to": 888, 114.75/31.08 "label": "EVAL with clause\ngt(t, f).\nand substitutionT187 -> t" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 887, 114.75/31.08 "to": 889, 114.75/31.08 "label": "EVAL-BACKTRACK" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 888, 114.75/31.08 "to": 713, 114.75/31.08 "label": "INSTANCE with matching:\nT53 -> T188\nT55 -> T190" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 890, 114.75/31.08 "to": 892, 114.75/31.08 "label": "CASE" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 891, 114.75/31.08 "to": 3, 114.75/31.08 "label": "INSTANCE with matching:\nT1 -> T193\nT2 -> T164" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 892, 114.75/31.08 "to": 893, 114.75/31.08 "label": "BACKTRACK\nfor clause: del(X1, [], [])because of non-unification" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 893, 114.75/31.08 "to": 894, 114.75/31.08 "label": "PARALLEL" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 893, 114.75/31.08 "to": 895, 114.75/31.08 "label": "PARALLEL" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 894, 114.75/31.08 "to": 896, 114.75/31.08 "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)" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 895, 114.75/31.08 "to": 902, 114.75/31.08 "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)" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 896, 114.75/31.08 "to": 897, 114.75/31.08 "label": "CASE" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 897, 114.75/31.08 "to": 898, 114.75/31.08 "label": "BACKTRACK\nfor clause: eq(t, t)because of non-unification" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 898, 114.75/31.08 "to": 899, 114.75/31.08 "label": "EVAL with clause\neq(f, f).\nand substitutionT205 -> f" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 898, 114.75/31.08 "to": 900, 114.75/31.08 "label": "EVAL-BACKTRACK" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 899, 114.75/31.08 "to": 901, 114.75/31.08 "label": "SUCCESS" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 902, 114.75/31.08 "to": 903, 114.75/31.08 "label": "CASE" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 903, 114.75/31.08 "to": 904, 114.75/31.08 "label": "PARALLEL" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 903, 114.75/31.08 "to": 905, 114.75/31.08 "label": "PARALLEL" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 904, 114.75/31.08 "to": 906, 114.75/31.08 "label": "EVAL with clause\nneq(t, f).\nand substitutionT213 -> t" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 904, 114.75/31.08 "to": 907, 114.75/31.08 "label": "EVAL-BACKTRACK" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 905, 114.75/31.08 "to": 947, 114.75/31.08 "label": "BACKTRACK\nfor clause: neq(f, t)because of non-unification" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 906, 114.75/31.08 "to": 908, 114.75/31.08 "label": "CASE" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 908, 114.75/31.08 "to": 909, 114.75/31.08 "label": "BACKTRACK\nfor clause: del(X1, [], [])because of non-unification" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 909, 114.75/31.08 "to": 910, 114.75/31.08 "label": "PARALLEL" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 909, 114.75/31.08 "to": 911, 114.75/31.08 "label": "PARALLEL" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 910, 114.75/31.08 "to": 912, 114.75/31.08 "label": "ONLY EVAL with clause\ndel(X365, .(X366, X367), X367) :- eq(X365, X366).\nand substitutionX365 -> t,\nX366 -> f,\nT214 -> T221,\nX367 -> T221,\nX347 -> T221" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 911, 114.75/31.08 "to": 916, 114.75/31.08 "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)" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 912, 114.75/31.08 "to": 913, 114.75/31.08 "label": "CASE" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 913, 114.75/31.08 "to": 914, 114.75/31.08 "label": "BACKTRACK\nfor clause: eq(t, t)because of non-unification" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 914, 114.75/31.08 "to": 915, 114.75/31.08 "label": "BACKTRACK\nfor clause: eq(f, f)because of non-unification" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 916, 114.75/31.08 "to": 917, 114.75/31.08 "label": "CASE" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 917, 114.75/31.08 "to": 918, 114.75/31.08 "label": "PARALLEL" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 917, 114.75/31.08 "to": 919, 114.75/31.08 "label": "PARALLEL" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 918, 114.75/31.08 "to": 920, 114.75/31.08 "label": "ONLY EVAL with clause\nneq(t, f).\nand substitution" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 919, 114.75/31.08 "to": 946, 114.75/31.08 "label": "BACKTRACK\nfor clause: neq(f, t)because of non-unification" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 920, 114.75/31.08 "to": 921, 114.75/31.08 "label": "CASE" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 921, 114.75/31.08 "to": 922, 114.75/31.08 "label": "PARALLEL" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 921, 114.75/31.08 "to": 923, 114.75/31.08 "label": "PARALLEL" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 922, 114.75/31.08 "to": 924, 114.75/31.08 "label": "EVAL with clause\ndel(X397, [], []).\nand substitutionX397 -> t,\nT224 -> [],\nX384 -> []" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 922, 114.75/31.08 "to": 925, 114.75/31.08 "label": "EVAL-BACKTRACK" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 923, 114.75/31.08 "to": 927, 114.75/31.08 "label": "PARALLEL" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 923, 114.75/31.08 "to": 928, 114.75/31.08 "label": "PARALLEL" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 924, 114.75/31.08 "to": 926, 114.75/31.08 "label": "SUCCESS" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 927, 114.75/31.08 "to": 929, 114.75/31.08 "label": "EVAL with clause\ndel(X410, .(X411, X412), X412) :- eq(X410, X411).\nand substitutionX410 -> t,\nX411 -> T233,\nX412 -> T234,\nT224 -> .(T233, T234),\nX384 -> T234" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 927, 114.75/31.08 "to": 930, 114.75/31.08 "label": "EVAL-BACKTRACK" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 928, 114.75/31.08 "to": 938, 114.75/31.08 "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)" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 928, 114.75/31.08 "to": 939, 114.75/31.08 "label": "EVAL-BACKTRACK" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 929, 114.75/31.08 "to": 931, 114.75/31.08 "label": "CASE" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 931, 114.75/31.08 "to": 932, 114.75/31.08 "label": "PARALLEL" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 931, 114.75/31.08 "to": 933, 114.75/31.08 "label": "PARALLEL" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 932, 114.75/31.08 "to": 934, 114.75/31.08 "label": "EVAL with clause\neq(t, t).\nand substitutionT233 -> t" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 932, 114.75/31.08 "to": 935, 114.75/31.08 "label": "EVAL-BACKTRACK" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 933, 114.75/31.08 "to": 937, 114.75/31.08 "label": "BACKTRACK\nfor clause: eq(f, f)because of non-unification" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 934, 114.75/31.08 "to": 936, 114.75/31.08 "label": "SUCCESS" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 938, 114.75/31.08 "to": 940, 114.75/31.08 "label": "CASE" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 940, 114.75/31.08 "to": 941, 114.75/31.08 "label": "PARALLEL" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 940, 114.75/31.08 "to": 942, 114.75/31.08 "label": "PARALLEL" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 941, 114.75/31.08 "to": 943, 114.75/31.08 "label": "EVAL with clause\nneq(t, f).\nand substitutionT239 -> f" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 941, 114.75/31.08 "to": 944, 114.75/31.08 "label": "EVAL-BACKTRACK" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 942, 114.75/31.08 "to": 945, 114.75/31.08 "label": "BACKTRACK\nfor clause: neq(f, t)because of non-unification" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 943, 114.75/31.08 "to": 920, 114.75/31.08 "label": "INSTANCE with matching:\nT224 -> T240\nX384 -> X427" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 948, 114.75/31.08 "to": 950, 114.75/31.08 "label": "CASE" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 950, 114.75/31.08 "to": 951, 114.75/31.08 "label": "EVAL with clause\ngt(t, f).\nand substitutionT250 -> t,\nT249 -> f" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 950, 114.75/31.08 "to": 952, 114.75/31.08 "label": "EVAL-BACKTRACK" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 951, 114.75/31.08 "to": 953, 114.75/31.08 "label": "SPLIT 1" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 951, 114.75/31.08 "to": 954, 114.75/31.08 "label": "SPLIT 2\nnew knowledge:\nT251 is ground\nT255 is ground\nreplacements:T253 -> T255,\nT254 -> T256" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 953, 114.75/31.08 "to": 713, 114.75/31.08 "label": "INSTANCE with matching:\nT53 -> T251\nT55 -> T253" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 954, 114.75/31.08 "to": 955, 114.75/31.08 "label": "SPLIT 1" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 954, 114.75/31.08 "to": 956, 114.75/31.08 "label": "SPLIT 2\nnew knowledge:\nT255 is ground\nT251 is ground\nT259 is ground\nreplacements:X14 -> T259" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 955, 114.75/31.08 "to": 957, 114.75/31.08 "label": "CASE" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 956, 114.75/31.08 "to": 3, 114.75/31.08 "label": "INSTANCE with matching:\nT1 -> T259\nT2 -> T256" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 957, 114.75/31.08 "to": 958, 114.75/31.08 "label": "BACKTRACK\nfor clause: del(X1, [], [])because of non-unification" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 958, 114.75/31.08 "to": 959, 114.75/31.08 "label": "PARALLEL" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 958, 114.75/31.08 "to": 960, 114.75/31.08 "label": "PARALLEL" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 959, 114.75/31.08 "to": 961, 114.75/31.08 "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)" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 960, 114.75/31.08 "to": 969, 114.75/31.08 "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)" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 961, 114.75/31.08 "to": 962, 114.75/31.08 "label": "CASE" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 962, 114.75/31.08 "to": 963, 114.75/31.08 "label": "BACKTRACK\nfor clause: eq(t, t)because of non-unification" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 963, 114.75/31.08 "to": 964, 114.75/31.08 "label": "EVAL with clause\neq(f, f).\nand substitutionT271 -> f" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 963, 114.75/31.08 "to": 965, 114.75/31.08 "label": "EVAL-BACKTRACK" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 964, 114.75/31.08 "to": 966, 114.75/31.08 "label": "SUCCESS" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 969, 114.75/31.08 "to": 970, 114.75/31.08 "label": "CASE" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 970, 114.75/31.08 "to": 971, 114.75/31.08 "label": "PARALLEL" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 970, 114.75/31.08 "to": 972, 114.75/31.08 "label": "PARALLEL" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 971, 114.75/31.08 "to": 973, 114.75/31.08 "label": "EVAL with clause\nneq(t, f).\nand substitutionT279 -> t" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 971, 114.75/31.08 "to": 974, 114.75/31.08 "label": "EVAL-BACKTRACK" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 972, 114.75/31.08 "to": 975, 114.75/31.08 "label": "BACKTRACK\nfor clause: neq(f, t)because of non-unification" 114.75/31.08 }, 114.75/31.08 { 114.75/31.08 "from": 973, 114.75/31.08 "to": 920, 114.75/31.08 "label": "INSTANCE with matching:\nT224 -> .(t, T280)\nX384 -> X480" 114.75/31.08 } 114.75/31.08 ], 114.75/31.08 "type": "Graph" 114.75/31.08 } 114.75/31.08 } 114.75/31.08 114.75/31.08 ---------------------------------------- 114.75/31.08 114.75/31.08 (2) 114.75/31.08 Obligation: 114.75/31.08 Triples: 114.75/31.08 114.75/31.08 maxC(.(t, X1), X2) :- maxC(X1, X2). 114.75/31.08 maxC(.(f, X1), X2) :- maxC(X1, X2). 114.75/31.08 delI(.(t, X1), .(t, X2)) :- delI(X1, X2). 114.75/31.08 maxF(.(f, X1), X2) :- maxF(X1, X2). 114.75/31.08 maxF(.(t, X1), X2) :- maxC(X1, X2). 114.75/31.08 delJ(.(f, X1), .(f, X2)) :- delJ(X1, X2). 114.75/31.08 maxsortA(.(X1, []), .(X1, X2)) :- ','(delcB(X1, X3), maxsortA(X3, X2)). 114.75/31.08 maxsortA(.(t, .(t, X1)), .(X2, X3)) :- maxC(X1, X2). 114.75/31.08 maxsortA(.(t, .(t, X1)), .(f, X2)) :- ','(maxcC(X1, f), delI(X1, X3)). 114.75/31.08 maxsortA(.(t, .(t, X1)), .(X2, X3)) :- ','(maxcC(X1, X2), ','(delcD(X2, X1, X4), maxsortA(X4, X3))). 114.75/31.08 maxsortA(.(t, .(f, X1)), .(X2, X3)) :- maxC(X1, X2). 114.75/31.08 maxsortA(.(t, .(f, X1)), .(f, X2)) :- ','(maxcC(X1, f), delI(.(f, X1), X3)). 114.75/31.08 maxsortA(.(t, .(f, X1)), .(X2, X3)) :- ','(maxcC(X1, X2), ','(delcE(X2, X1, X4), maxsortA(X4, X3))). 114.75/31.08 maxsortA(.(f, .(f, X1)), .(X2, X3)) :- maxF(X1, X2). 114.75/31.08 maxsortA(.(f, .(f, X1)), .(t, X2)) :- ','(maxcF(X1, t), delJ(X1, X3)). 114.75/31.08 maxsortA(.(f, .(f, X1)), .(X2, X3)) :- ','(maxcF(X1, X2), ','(delcG(X2, X1, X4), maxsortA(X4, X3))). 114.75/31.08 maxsortA(.(f, .(t, X1)), .(X2, X3)) :- maxC(X1, X2). 114.75/31.08 maxsortA(.(f, .(t, X1)), .(t, X2)) :- ','(maxcC(X1, t), delJ(.(t, X1), X3)). 114.75/31.08 maxsortA(.(f, .(t, X1)), .(X2, X3)) :- ','(maxcC(X1, X2), ','(delcH(X2, X1, X4), maxsortA(X4, X3))). 114.75/31.08 114.75/31.08 Clauses: 114.75/31.08 114.75/31.08 maxsortcA([], []). 114.75/31.08 maxsortcA(.(X1, []), .(X1, X2)) :- ','(delcB(X1, X3), maxsortcA(X3, X2)). 114.75/31.08 maxsortcA(.(t, .(t, X1)), .(X2, X3)) :- ','(maxcC(X1, X2), ','(delcD(X2, X1, X4), maxsortcA(X4, X3))). 114.75/31.08 maxsortcA(.(t, .(f, X1)), .(X2, X3)) :- ','(maxcC(X1, X2), ','(delcE(X2, X1, X4), maxsortcA(X4, X3))). 114.75/31.08 maxsortcA(.(f, .(f, X1)), .(X2, X3)) :- ','(maxcF(X1, X2), ','(delcG(X2, X1, X4), maxsortcA(X4, X3))). 114.75/31.08 maxsortcA(.(f, .(t, X1)), .(X2, X3)) :- ','(maxcC(X1, X2), ','(delcH(X2, X1, X4), maxsortcA(X4, X3))). 114.75/31.08 maxcC([], t). 114.75/31.08 maxcC(.(t, X1), X2) :- maxcC(X1, X2). 114.75/31.08 maxcC(.(f, X1), X2) :- maxcC(X1, X2). 114.75/31.08 delcI([], []). 114.75/31.08 delcI(.(f, X1), X1). 114.75/31.08 delcI(.(t, X1), .(t, X2)) :- delcI(X1, X2). 114.75/31.08 maxcF([], f). 114.75/31.08 maxcF(.(f, X1), X2) :- maxcF(X1, X2). 114.75/31.08 maxcF(.(t, X1), X2) :- maxcC(X1, X2). 114.75/31.08 delcJ([], []). 114.75/31.08 delcJ(.(t, X1), X1). 114.75/31.08 delcJ(.(f, X1), .(f, X2)) :- delcJ(X1, X2). 114.75/31.08 delcB(t, []). 114.75/31.08 delcB(f, []). 114.75/31.08 delcD(t, X1, .(t, X1)). 114.75/31.08 delcD(f, X1, .(t, .(t, X2))) :- delcI(X1, X2). 114.75/31.08 delcE(t, X1, .(f, X1)). 114.75/31.08 delcE(f, X1, .(t, X2)) :- delcI(.(f, X1), X2). 114.75/31.08 delcG(f, X1, .(f, X1)). 114.75/31.08 delcG(t, X1, .(f, .(f, X2))) :- delcJ(X1, X2). 114.75/31.08 delcH(f, X1, .(t, X1)). 114.75/31.08 delcH(t, X1, .(f, X2)) :- delcJ(.(t, X1), X2). 114.75/31.08 114.75/31.08 Afs: 114.75/31.08 114.75/31.08 maxsortA(x1, x2) = maxsortA(x1) 114.75/31.08 114.75/31.08 114.75/31.08 ---------------------------------------- 114.75/31.08 114.75/31.08 (3) TriplesToPiDPProof (SOUND) 114.75/31.08 We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes: 114.75/31.08 114.75/31.08 maxsortA_in_2: (b,f) 114.75/31.08 114.75/31.08 maxC_in_2: (b,f) 114.75/31.08 114.75/31.08 maxcC_in_2: (b,b) (b,f) 114.75/31.08 114.75/31.08 delI_in_2: (b,f) 114.75/31.08 114.75/31.08 delcD_in_3: (b,b,f) 114.75/31.08 114.75/31.08 delcI_in_2: (b,f) 114.75/31.08 114.75/31.08 delcE_in_3: (b,b,f) 114.75/31.08 114.75/31.08 maxF_in_2: (b,f) 114.75/31.08 114.75/31.08 maxcF_in_2: (b,b) (b,f) 114.75/31.08 114.75/31.08 delJ_in_2: (b,f) 114.75/31.08 114.75/31.08 delcG_in_3: (b,b,f) 114.75/31.08 114.75/31.08 delcJ_in_2: (b,f) 114.75/31.08 114.75/31.08 delcH_in_3: (b,b,f) 114.75/31.08 114.75/31.08 Transforming TRIPLES into the following Term Rewriting System: 114.75/31.08 114.75/31.08 Pi DP problem: 114.75/31.08 The TRS P consists of the following rules: 114.75/31.08 114.75/31.08 MAXSORTA_IN_GA(.(X1, []), .(X1, X2)) -> U7_GA(X1, X2, delcB_in_ga(X1, X3)) 114.75/31.08 U7_GA(X1, X2, delcB_out_ga(X1, X3)) -> U8_GA(X1, X2, maxsortA_in_ga(X3, X2)) 114.75/31.08 U7_GA(X1, X2, delcB_out_ga(X1, X3)) -> MAXSORTA_IN_GA(X3, X2) 114.75/31.08 MAXSORTA_IN_GA(.(t, .(t, X1)), .(X2, X3)) -> U9_GA(X1, X2, X3, maxC_in_ga(X1, X2)) 114.75/31.08 MAXSORTA_IN_GA(.(t, .(t, X1)), .(X2, X3)) -> MAXC_IN_GA(X1, X2) 114.75/31.08 MAXC_IN_GA(.(t, X1), X2) -> U1_GA(X1, X2, maxC_in_ga(X1, X2)) 114.75/31.08 MAXC_IN_GA(.(t, X1), X2) -> MAXC_IN_GA(X1, X2) 114.75/31.08 MAXC_IN_GA(.(f, X1), X2) -> U2_GA(X1, X2, maxC_in_ga(X1, X2)) 114.75/31.08 MAXC_IN_GA(.(f, X1), X2) -> MAXC_IN_GA(X1, X2) 114.75/31.08 MAXSORTA_IN_GA(.(t, .(t, X1)), .(f, X2)) -> U10_GA(X1, X2, maxcC_in_gg(X1, f)) 114.75/31.08 U10_GA(X1, X2, maxcC_out_gg(X1, f)) -> U11_GA(X1, X2, delI_in_ga(X1, X3)) 114.75/31.08 U10_GA(X1, X2, maxcC_out_gg(X1, f)) -> DELI_IN_GA(X1, X3) 114.75/31.08 DELI_IN_GA(.(t, X1), .(t, X2)) -> U3_GA(X1, X2, delI_in_ga(X1, X2)) 114.75/31.08 DELI_IN_GA(.(t, X1), .(t, X2)) -> DELI_IN_GA(X1, X2) 114.75/31.08 MAXSORTA_IN_GA(.(t, .(t, X1)), .(X2, X3)) -> U12_GA(X1, X2, X3, maxcC_in_ga(X1, X2)) 114.75/31.08 U12_GA(X1, X2, X3, maxcC_out_ga(X1, X2)) -> U13_GA(X1, X2, X3, delcD_in_gga(X2, X1, X4)) 114.75/31.08 U13_GA(X1, X2, X3, delcD_out_gga(X2, X1, X4)) -> U14_GA(X1, X2, X3, maxsortA_in_ga(X4, X3)) 114.75/31.08 U13_GA(X1, X2, X3, delcD_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4, X3) 114.75/31.08 MAXSORTA_IN_GA(.(t, .(f, X1)), .(X2, X3)) -> U15_GA(X1, X2, X3, maxC_in_ga(X1, X2)) 114.75/31.08 MAXSORTA_IN_GA(.(t, .(f, X1)), .(X2, X3)) -> MAXC_IN_GA(X1, X2) 114.75/31.08 MAXSORTA_IN_GA(.(t, .(f, X1)), .(f, X2)) -> U16_GA(X1, X2, maxcC_in_gg(X1, f)) 114.75/31.08 U16_GA(X1, X2, maxcC_out_gg(X1, f)) -> U17_GA(X1, X2, delI_in_ga(.(f, X1), X3)) 114.75/31.08 U16_GA(X1, X2, maxcC_out_gg(X1, f)) -> DELI_IN_GA(.(f, X1), X3) 114.75/31.08 MAXSORTA_IN_GA(.(t, .(f, X1)), .(X2, X3)) -> U18_GA(X1, X2, X3, maxcC_in_ga(X1, X2)) 114.75/31.08 U18_GA(X1, X2, X3, maxcC_out_ga(X1, X2)) -> U19_GA(X1, X2, X3, delcE_in_gga(X2, X1, X4)) 114.75/31.08 U19_GA(X1, X2, X3, delcE_out_gga(X2, X1, X4)) -> U20_GA(X1, X2, X3, maxsortA_in_ga(X4, X3)) 114.75/31.08 U19_GA(X1, X2, X3, delcE_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4, X3) 114.75/31.08 MAXSORTA_IN_GA(.(f, .(f, X1)), .(X2, X3)) -> U21_GA(X1, X2, X3, maxF_in_ga(X1, X2)) 114.75/31.08 MAXSORTA_IN_GA(.(f, .(f, X1)), .(X2, X3)) -> MAXF_IN_GA(X1, X2) 114.75/31.08 MAXF_IN_GA(.(f, X1), X2) -> U4_GA(X1, X2, maxF_in_ga(X1, X2)) 114.75/31.08 MAXF_IN_GA(.(f, X1), X2) -> MAXF_IN_GA(X1, X2) 114.75/31.08 MAXF_IN_GA(.(t, X1), X2) -> U5_GA(X1, X2, maxC_in_ga(X1, X2)) 114.75/31.08 MAXF_IN_GA(.(t, X1), X2) -> MAXC_IN_GA(X1, X2) 114.75/31.08 MAXSORTA_IN_GA(.(f, .(f, X1)), .(t, X2)) -> U22_GA(X1, X2, maxcF_in_gg(X1, t)) 114.75/31.08 U22_GA(X1, X2, maxcF_out_gg(X1, t)) -> U23_GA(X1, X2, delJ_in_ga(X1, X3)) 114.75/31.08 U22_GA(X1, X2, maxcF_out_gg(X1, t)) -> DELJ_IN_GA(X1, X3) 114.75/31.08 DELJ_IN_GA(.(f, X1), .(f, X2)) -> U6_GA(X1, X2, delJ_in_ga(X1, X2)) 114.75/31.08 DELJ_IN_GA(.(f, X1), .(f, X2)) -> DELJ_IN_GA(X1, X2) 114.75/31.08 MAXSORTA_IN_GA(.(f, .(f, X1)), .(X2, X3)) -> U24_GA(X1, X2, X3, maxcF_in_ga(X1, X2)) 114.75/31.08 U24_GA(X1, X2, X3, maxcF_out_ga(X1, X2)) -> U25_GA(X1, X2, X3, delcG_in_gga(X2, X1, X4)) 114.75/31.08 U25_GA(X1, X2, X3, delcG_out_gga(X2, X1, X4)) -> U26_GA(X1, X2, X3, maxsortA_in_ga(X4, X3)) 114.75/31.08 U25_GA(X1, X2, X3, delcG_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4, X3) 114.75/31.08 MAXSORTA_IN_GA(.(f, .(t, X1)), .(X2, X3)) -> U27_GA(X1, X2, X3, maxC_in_ga(X1, X2)) 114.75/31.08 MAXSORTA_IN_GA(.(f, .(t, X1)), .(X2, X3)) -> MAXC_IN_GA(X1, X2) 114.75/31.08 MAXSORTA_IN_GA(.(f, .(t, X1)), .(t, X2)) -> U28_GA(X1, X2, maxcC_in_gg(X1, t)) 114.75/31.08 U28_GA(X1, X2, maxcC_out_gg(X1, t)) -> U29_GA(X1, X2, delJ_in_ga(.(t, X1), X3)) 114.75/31.08 U28_GA(X1, X2, maxcC_out_gg(X1, t)) -> DELJ_IN_GA(.(t, X1), X3) 114.75/31.08 MAXSORTA_IN_GA(.(f, .(t, X1)), .(X2, X3)) -> U30_GA(X1, X2, X3, maxcC_in_ga(X1, X2)) 114.75/31.08 U30_GA(X1, X2, X3, maxcC_out_ga(X1, X2)) -> U31_GA(X1, X2, X3, delcH_in_gga(X2, X1, X4)) 114.75/31.08 U31_GA(X1, X2, X3, delcH_out_gga(X2, X1, X4)) -> U32_GA(X1, X2, X3, maxsortA_in_ga(X4, X3)) 114.75/31.08 U31_GA(X1, X2, X3, delcH_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4, X3) 114.75/31.08 114.75/31.08 The TRS R consists of the following rules: 114.75/31.08 114.75/31.08 delcB_in_ga(t, []) -> delcB_out_ga(t, []) 114.75/31.08 delcB_in_ga(f, []) -> delcB_out_ga(f, []) 114.75/31.08 maxcC_in_gg([], t) -> maxcC_out_gg([], t) 114.75/31.08 maxcC_in_gg(.(t, X1), X2) -> U48_gg(X1, X2, maxcC_in_gg(X1, X2)) 114.75/31.08 maxcC_in_gg(.(f, X1), X2) -> U49_gg(X1, X2, maxcC_in_gg(X1, X2)) 114.75/31.08 U49_gg(X1, X2, maxcC_out_gg(X1, X2)) -> maxcC_out_gg(.(f, X1), X2) 114.75/31.08 U48_gg(X1, X2, maxcC_out_gg(X1, X2)) -> maxcC_out_gg(.(t, X1), X2) 114.75/31.08 maxcC_in_ga([], t) -> maxcC_out_ga([], t) 114.75/31.08 maxcC_in_ga(.(t, X1), X2) -> U48_ga(X1, X2, maxcC_in_ga(X1, X2)) 114.75/31.08 maxcC_in_ga(.(f, X1), X2) -> U49_ga(X1, X2, maxcC_in_ga(X1, X2)) 114.75/31.08 U49_ga(X1, X2, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(f, X1), X2) 114.75/31.08 U48_ga(X1, X2, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(t, X1), X2) 114.75/31.08 delcD_in_gga(t, X1, .(t, X1)) -> delcD_out_gga(t, X1, .(t, X1)) 114.75/31.08 delcD_in_gga(f, X1, .(t, .(t, X2))) -> U54_gga(X1, X2, delcI_in_ga(X1, X2)) 114.75/31.08 delcI_in_ga([], []) -> delcI_out_ga([], []) 114.75/31.08 delcI_in_ga(.(f, X1), X1) -> delcI_out_ga(.(f, X1), X1) 114.75/31.08 delcI_in_ga(.(t, X1), .(t, X2)) -> U50_ga(X1, X2, delcI_in_ga(X1, X2)) 114.75/31.08 U50_ga(X1, X2, delcI_out_ga(X1, X2)) -> delcI_out_ga(.(t, X1), .(t, X2)) 114.75/31.08 U54_gga(X1, X2, delcI_out_ga(X1, X2)) -> delcD_out_gga(f, X1, .(t, .(t, X2))) 114.75/31.08 delcE_in_gga(t, X1, .(f, X1)) -> delcE_out_gga(t, X1, .(f, X1)) 114.75/31.08 delcE_in_gga(f, X1, .(t, X2)) -> U55_gga(X1, X2, delcI_in_ga(.(f, X1), X2)) 114.75/31.08 U55_gga(X1, X2, delcI_out_ga(.(f, X1), X2)) -> delcE_out_gga(f, X1, .(t, X2)) 114.75/31.08 maxcF_in_gg([], f) -> maxcF_out_gg([], f) 114.75/31.08 maxcF_in_gg(.(f, X1), X2) -> U51_gg(X1, X2, maxcF_in_gg(X1, X2)) 114.75/31.08 maxcF_in_gg(.(t, X1), X2) -> U52_gg(X1, X2, maxcC_in_gg(X1, X2)) 114.75/31.08 U52_gg(X1, X2, maxcC_out_gg(X1, X2)) -> maxcF_out_gg(.(t, X1), X2) 114.75/31.08 U51_gg(X1, X2, maxcF_out_gg(X1, X2)) -> maxcF_out_gg(.(f, X1), X2) 114.75/31.08 maxcF_in_ga([], f) -> maxcF_out_ga([], f) 114.75/31.08 maxcF_in_ga(.(f, X1), X2) -> U51_ga(X1, X2, maxcF_in_ga(X1, X2)) 114.75/31.08 maxcF_in_ga(.(t, X1), X2) -> U52_ga(X1, X2, maxcC_in_ga(X1, X2)) 114.75/31.08 U52_ga(X1, X2, maxcC_out_ga(X1, X2)) -> maxcF_out_ga(.(t, X1), X2) 114.75/31.08 U51_ga(X1, X2, maxcF_out_ga(X1, X2)) -> maxcF_out_ga(.(f, X1), X2) 114.75/31.08 delcG_in_gga(f, X1, .(f, X1)) -> delcG_out_gga(f, X1, .(f, X1)) 114.75/31.08 delcG_in_gga(t, X1, .(f, .(f, X2))) -> U56_gga(X1, X2, delcJ_in_ga(X1, X2)) 114.75/31.08 delcJ_in_ga([], []) -> delcJ_out_ga([], []) 114.75/31.08 delcJ_in_ga(.(t, X1), X1) -> delcJ_out_ga(.(t, X1), X1) 114.75/31.08 delcJ_in_ga(.(f, X1), .(f, X2)) -> U53_ga(X1, X2, delcJ_in_ga(X1, X2)) 114.75/31.08 U53_ga(X1, X2, delcJ_out_ga(X1, X2)) -> delcJ_out_ga(.(f, X1), .(f, X2)) 114.75/31.08 U56_gga(X1, X2, delcJ_out_ga(X1, X2)) -> delcG_out_gga(t, X1, .(f, .(f, X2))) 114.75/31.08 delcH_in_gga(f, X1, .(t, X1)) -> delcH_out_gga(f, X1, .(t, X1)) 114.75/31.08 delcH_in_gga(t, X1, .(f, X2)) -> U57_gga(X1, X2, delcJ_in_ga(.(t, X1), X2)) 114.75/31.08 U57_gga(X1, X2, delcJ_out_ga(.(t, X1), X2)) -> delcH_out_gga(t, X1, .(f, X2)) 114.75/31.08 114.75/31.08 The argument filtering Pi contains the following mapping: 114.75/31.08 maxsortA_in_ga(x1, x2) = maxsortA_in_ga(x1) 114.75/31.08 114.75/31.08 .(x1, x2) = .(x1, x2) 114.75/31.08 114.75/31.08 [] = [] 114.75/31.08 114.75/31.08 delcB_in_ga(x1, x2) = delcB_in_ga(x1) 114.75/31.08 114.75/31.08 t = t 114.75/31.08 114.75/31.08 delcB_out_ga(x1, x2) = delcB_out_ga(x1, x2) 114.75/31.08 114.75/31.08 f = f 114.75/31.08 114.75/31.08 maxC_in_ga(x1, x2) = maxC_in_ga(x1) 114.75/31.08 114.75/31.08 maxcC_in_gg(x1, x2) = maxcC_in_gg(x1, x2) 114.75/31.08 114.75/31.08 maxcC_out_gg(x1, x2) = maxcC_out_gg(x1, x2) 114.75/31.08 114.75/31.08 U48_gg(x1, x2, x3) = U48_gg(x1, x2, x3) 114.75/31.08 114.75/31.08 U49_gg(x1, x2, x3) = U49_gg(x1, x2, x3) 114.75/31.08 114.75/31.08 delI_in_ga(x1, x2) = delI_in_ga(x1) 114.75/31.08 114.75/31.08 maxcC_in_ga(x1, x2) = maxcC_in_ga(x1) 114.75/31.08 114.75/31.08 maxcC_out_ga(x1, x2) = maxcC_out_ga(x1, x2) 114.75/31.08 114.75/31.08 U48_ga(x1, x2, x3) = U48_ga(x1, x3) 114.75/31.08 114.75/31.08 U49_ga(x1, x2, x3) = U49_ga(x1, x3) 114.75/31.08 114.75/31.08 delcD_in_gga(x1, x2, x3) = delcD_in_gga(x1, x2) 114.75/31.09 114.75/31.09 delcD_out_gga(x1, x2, x3) = delcD_out_gga(x1, x2, x3) 114.75/31.09 114.75/31.09 U54_gga(x1, x2, x3) = U54_gga(x1, x3) 114.75/31.09 114.75/31.09 delcI_in_ga(x1, x2) = delcI_in_ga(x1) 114.75/31.09 114.75/31.09 delcI_out_ga(x1, x2) = delcI_out_ga(x1, x2) 114.75/31.09 114.75/31.09 U50_ga(x1, x2, x3) = U50_ga(x1, x3) 114.75/31.09 114.75/31.09 delcE_in_gga(x1, x2, x3) = delcE_in_gga(x1, x2) 114.75/31.09 114.75/31.09 delcE_out_gga(x1, x2, x3) = delcE_out_gga(x1, x2, x3) 114.75/31.09 114.75/31.09 U55_gga(x1, x2, x3) = U55_gga(x1, x3) 114.75/31.09 114.75/31.09 maxF_in_ga(x1, x2) = maxF_in_ga(x1) 114.75/31.09 114.75/31.09 maxcF_in_gg(x1, x2) = maxcF_in_gg(x1, x2) 114.75/31.09 114.75/31.09 maxcF_out_gg(x1, x2) = maxcF_out_gg(x1, x2) 114.75/31.09 114.75/31.09 U51_gg(x1, x2, x3) = U51_gg(x1, x2, x3) 114.75/31.09 114.75/31.09 U52_gg(x1, x2, x3) = U52_gg(x1, x2, x3) 114.75/31.09 114.75/31.09 delJ_in_ga(x1, x2) = delJ_in_ga(x1) 114.75/31.09 114.75/31.09 maxcF_in_ga(x1, x2) = maxcF_in_ga(x1) 114.75/31.09 114.75/31.09 maxcF_out_ga(x1, x2) = maxcF_out_ga(x1, x2) 114.75/31.09 114.75/31.09 U51_ga(x1, x2, x3) = U51_ga(x1, x3) 114.75/31.09 114.75/31.09 U52_ga(x1, x2, x3) = U52_ga(x1, x3) 114.75/31.09 114.75/31.09 delcG_in_gga(x1, x2, x3) = delcG_in_gga(x1, x2) 114.75/31.09 114.75/31.09 delcG_out_gga(x1, x2, x3) = delcG_out_gga(x1, x2, x3) 114.75/31.09 114.75/31.09 U56_gga(x1, x2, x3) = U56_gga(x1, x3) 114.75/31.09 114.75/31.09 delcJ_in_ga(x1, x2) = delcJ_in_ga(x1) 114.75/31.09 114.75/31.09 delcJ_out_ga(x1, x2) = delcJ_out_ga(x1, x2) 114.75/31.09 114.75/31.09 U53_ga(x1, x2, x3) = U53_ga(x1, x3) 114.75/31.09 114.75/31.09 delcH_in_gga(x1, x2, x3) = delcH_in_gga(x1, x2) 114.75/31.09 114.75/31.09 delcH_out_gga(x1, x2, x3) = delcH_out_gga(x1, x2, x3) 114.75/31.09 114.75/31.09 U57_gga(x1, x2, x3) = U57_gga(x1, x3) 114.75/31.09 114.75/31.09 MAXSORTA_IN_GA(x1, x2) = MAXSORTA_IN_GA(x1) 114.75/31.09 114.75/31.09 U7_GA(x1, x2, x3) = U7_GA(x1, x3) 114.75/31.09 114.75/31.09 U8_GA(x1, x2, x3) = U8_GA(x1, x3) 114.75/31.09 114.75/31.09 U9_GA(x1, x2, x3, x4) = U9_GA(x1, x4) 114.75/31.09 114.75/31.09 MAXC_IN_GA(x1, x2) = MAXC_IN_GA(x1) 114.75/31.09 114.75/31.09 U1_GA(x1, x2, x3) = U1_GA(x1, x3) 114.75/31.09 114.75/31.09 U2_GA(x1, x2, x3) = U2_GA(x1, x3) 114.75/31.09 114.75/31.09 U10_GA(x1, x2, x3) = U10_GA(x1, x3) 114.75/31.09 114.75/31.09 U11_GA(x1, x2, x3) = U11_GA(x1, x3) 114.75/31.09 114.75/31.09 DELI_IN_GA(x1, x2) = DELI_IN_GA(x1) 114.75/31.09 114.75/31.09 U3_GA(x1, x2, x3) = U3_GA(x1, x3) 114.75/31.09 114.75/31.09 U12_GA(x1, x2, x3, x4) = U12_GA(x1, x4) 114.75/31.09 114.75/31.09 U13_GA(x1, x2, x3, x4) = U13_GA(x1, x4) 114.75/31.09 114.75/31.09 U14_GA(x1, x2, x3, x4) = U14_GA(x1, x4) 114.75/31.09 114.75/31.09 U15_GA(x1, x2, x3, x4) = U15_GA(x1, x4) 114.75/31.09 114.75/31.09 U16_GA(x1, x2, x3) = U16_GA(x1, x3) 114.75/31.09 114.75/31.09 U17_GA(x1, x2, x3) = U17_GA(x1, x3) 114.75/31.09 114.75/31.09 U18_GA(x1, x2, x3, x4) = U18_GA(x1, x4) 114.75/31.09 114.75/31.09 U19_GA(x1, x2, x3, x4) = U19_GA(x1, x4) 114.75/31.09 114.75/31.09 U20_GA(x1, x2, x3, x4) = U20_GA(x1, x4) 114.75/31.09 114.75/31.09 U21_GA(x1, x2, x3, x4) = U21_GA(x1, x4) 114.75/31.09 114.75/31.09 MAXF_IN_GA(x1, x2) = MAXF_IN_GA(x1) 114.75/31.09 114.75/31.09 U4_GA(x1, x2, x3) = U4_GA(x1, x3) 114.75/31.09 114.75/31.09 U5_GA(x1, x2, x3) = U5_GA(x1, x3) 114.75/31.09 114.75/31.09 U22_GA(x1, x2, x3) = U22_GA(x1, x3) 114.75/31.09 114.75/31.09 U23_GA(x1, x2, x3) = U23_GA(x1, x3) 114.75/31.09 114.75/31.09 DELJ_IN_GA(x1, x2) = DELJ_IN_GA(x1) 114.75/31.09 114.75/31.09 U6_GA(x1, x2, x3) = U6_GA(x1, x3) 114.75/31.09 114.75/31.09 U24_GA(x1, x2, x3, x4) = U24_GA(x1, x4) 114.75/31.09 114.75/31.09 U25_GA(x1, x2, x3, x4) = U25_GA(x1, x4) 114.75/31.09 114.75/31.09 U26_GA(x1, x2, x3, x4) = U26_GA(x1, x4) 114.75/31.09 114.75/31.09 U27_GA(x1, x2, x3, x4) = U27_GA(x1, x4) 114.75/31.09 114.75/31.09 U28_GA(x1, x2, x3) = U28_GA(x1, x3) 114.75/31.09 114.75/31.09 U29_GA(x1, x2, x3) = U29_GA(x1, x3) 114.75/31.09 114.75/31.09 U30_GA(x1, x2, x3, x4) = U30_GA(x1, x4) 114.75/31.09 114.75/31.09 U31_GA(x1, x2, x3, x4) = U31_GA(x1, x4) 114.75/31.09 114.75/31.09 U32_GA(x1, x2, x3, x4) = U32_GA(x1, x4) 114.75/31.09 114.75/31.09 114.75/31.09 We have to consider all (P,R,Pi)-chains 114.75/31.09 114.75/31.09 114.75/31.09 Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES 114.75/31.09 114.75/31.09 114.75/31.09 114.75/31.09 ---------------------------------------- 114.75/31.09 114.75/31.09 (4) 114.75/31.09 Obligation: 114.75/31.09 Pi DP problem: 114.75/31.09 The TRS P consists of the following rules: 114.75/31.09 114.75/31.09 MAXSORTA_IN_GA(.(X1, []), .(X1, X2)) -> U7_GA(X1, X2, delcB_in_ga(X1, X3)) 114.75/31.09 U7_GA(X1, X2, delcB_out_ga(X1, X3)) -> U8_GA(X1, X2, maxsortA_in_ga(X3, X2)) 114.75/31.09 U7_GA(X1, X2, delcB_out_ga(X1, X3)) -> MAXSORTA_IN_GA(X3, X2) 114.75/31.09 MAXSORTA_IN_GA(.(t, .(t, X1)), .(X2, X3)) -> U9_GA(X1, X2, X3, maxC_in_ga(X1, X2)) 114.75/31.09 MAXSORTA_IN_GA(.(t, .(t, X1)), .(X2, X3)) -> MAXC_IN_GA(X1, X2) 114.75/31.09 MAXC_IN_GA(.(t, X1), X2) -> U1_GA(X1, X2, maxC_in_ga(X1, X2)) 114.75/31.09 MAXC_IN_GA(.(t, X1), X2) -> MAXC_IN_GA(X1, X2) 114.75/31.09 MAXC_IN_GA(.(f, X1), X2) -> U2_GA(X1, X2, maxC_in_ga(X1, X2)) 114.75/31.09 MAXC_IN_GA(.(f, X1), X2) -> MAXC_IN_GA(X1, X2) 114.75/31.09 MAXSORTA_IN_GA(.(t, .(t, X1)), .(f, X2)) -> U10_GA(X1, X2, maxcC_in_gg(X1, f)) 114.75/31.09 U10_GA(X1, X2, maxcC_out_gg(X1, f)) -> U11_GA(X1, X2, delI_in_ga(X1, X3)) 114.75/31.09 U10_GA(X1, X2, maxcC_out_gg(X1, f)) -> DELI_IN_GA(X1, X3) 114.75/31.09 DELI_IN_GA(.(t, X1), .(t, X2)) -> U3_GA(X1, X2, delI_in_ga(X1, X2)) 114.75/31.09 DELI_IN_GA(.(t, X1), .(t, X2)) -> DELI_IN_GA(X1, X2) 114.75/31.09 MAXSORTA_IN_GA(.(t, .(t, X1)), .(X2, X3)) -> U12_GA(X1, X2, X3, maxcC_in_ga(X1, X2)) 114.75/31.09 U12_GA(X1, X2, X3, maxcC_out_ga(X1, X2)) -> U13_GA(X1, X2, X3, delcD_in_gga(X2, X1, X4)) 114.75/31.09 U13_GA(X1, X2, X3, delcD_out_gga(X2, X1, X4)) -> U14_GA(X1, X2, X3, maxsortA_in_ga(X4, X3)) 114.75/31.09 U13_GA(X1, X2, X3, delcD_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4, X3) 114.75/31.09 MAXSORTA_IN_GA(.(t, .(f, X1)), .(X2, X3)) -> U15_GA(X1, X2, X3, maxC_in_ga(X1, X2)) 114.75/31.09 MAXSORTA_IN_GA(.(t, .(f, X1)), .(X2, X3)) -> MAXC_IN_GA(X1, X2) 114.75/31.09 MAXSORTA_IN_GA(.(t, .(f, X1)), .(f, X2)) -> U16_GA(X1, X2, maxcC_in_gg(X1, f)) 114.75/31.09 U16_GA(X1, X2, maxcC_out_gg(X1, f)) -> U17_GA(X1, X2, delI_in_ga(.(f, X1), X3)) 114.75/31.09 U16_GA(X1, X2, maxcC_out_gg(X1, f)) -> DELI_IN_GA(.(f, X1), X3) 114.75/31.09 MAXSORTA_IN_GA(.(t, .(f, X1)), .(X2, X3)) -> U18_GA(X1, X2, X3, maxcC_in_ga(X1, X2)) 114.75/31.09 U18_GA(X1, X2, X3, maxcC_out_ga(X1, X2)) -> U19_GA(X1, X2, X3, delcE_in_gga(X2, X1, X4)) 114.75/31.09 U19_GA(X1, X2, X3, delcE_out_gga(X2, X1, X4)) -> U20_GA(X1, X2, X3, maxsortA_in_ga(X4, X3)) 114.75/31.09 U19_GA(X1, X2, X3, delcE_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4, X3) 114.75/31.09 MAXSORTA_IN_GA(.(f, .(f, X1)), .(X2, X3)) -> U21_GA(X1, X2, X3, maxF_in_ga(X1, X2)) 114.75/31.09 MAXSORTA_IN_GA(.(f, .(f, X1)), .(X2, X3)) -> MAXF_IN_GA(X1, X2) 114.75/31.09 MAXF_IN_GA(.(f, X1), X2) -> U4_GA(X1, X2, maxF_in_ga(X1, X2)) 114.75/31.09 MAXF_IN_GA(.(f, X1), X2) -> MAXF_IN_GA(X1, X2) 114.75/31.09 MAXF_IN_GA(.(t, X1), X2) -> U5_GA(X1, X2, maxC_in_ga(X1, X2)) 114.75/31.09 MAXF_IN_GA(.(t, X1), X2) -> MAXC_IN_GA(X1, X2) 114.75/31.09 MAXSORTA_IN_GA(.(f, .(f, X1)), .(t, X2)) -> U22_GA(X1, X2, maxcF_in_gg(X1, t)) 114.75/31.09 U22_GA(X1, X2, maxcF_out_gg(X1, t)) -> U23_GA(X1, X2, delJ_in_ga(X1, X3)) 114.75/31.09 U22_GA(X1, X2, maxcF_out_gg(X1, t)) -> DELJ_IN_GA(X1, X3) 114.75/31.09 DELJ_IN_GA(.(f, X1), .(f, X2)) -> U6_GA(X1, X2, delJ_in_ga(X1, X2)) 114.75/31.09 DELJ_IN_GA(.(f, X1), .(f, X2)) -> DELJ_IN_GA(X1, X2) 114.75/31.09 MAXSORTA_IN_GA(.(f, .(f, X1)), .(X2, X3)) -> U24_GA(X1, X2, X3, maxcF_in_ga(X1, X2)) 114.75/31.09 U24_GA(X1, X2, X3, maxcF_out_ga(X1, X2)) -> U25_GA(X1, X2, X3, delcG_in_gga(X2, X1, X4)) 114.75/31.09 U25_GA(X1, X2, X3, delcG_out_gga(X2, X1, X4)) -> U26_GA(X1, X2, X3, maxsortA_in_ga(X4, X3)) 114.75/31.09 U25_GA(X1, X2, X3, delcG_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4, X3) 114.75/31.09 MAXSORTA_IN_GA(.(f, .(t, X1)), .(X2, X3)) -> U27_GA(X1, X2, X3, maxC_in_ga(X1, X2)) 114.75/31.09 MAXSORTA_IN_GA(.(f, .(t, X1)), .(X2, X3)) -> MAXC_IN_GA(X1, X2) 114.75/31.09 MAXSORTA_IN_GA(.(f, .(t, X1)), .(t, X2)) -> U28_GA(X1, X2, maxcC_in_gg(X1, t)) 114.75/31.09 U28_GA(X1, X2, maxcC_out_gg(X1, t)) -> U29_GA(X1, X2, delJ_in_ga(.(t, X1), X3)) 114.75/31.09 U28_GA(X1, X2, maxcC_out_gg(X1, t)) -> DELJ_IN_GA(.(t, X1), X3) 114.75/31.09 MAXSORTA_IN_GA(.(f, .(t, X1)), .(X2, X3)) -> U30_GA(X1, X2, X3, maxcC_in_ga(X1, X2)) 114.75/31.09 U30_GA(X1, X2, X3, maxcC_out_ga(X1, X2)) -> U31_GA(X1, X2, X3, delcH_in_gga(X2, X1, X4)) 114.75/31.09 U31_GA(X1, X2, X3, delcH_out_gga(X2, X1, X4)) -> U32_GA(X1, X2, X3, maxsortA_in_ga(X4, X3)) 114.75/31.09 U31_GA(X1, X2, X3, delcH_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4, X3) 114.75/31.09 114.75/31.09 The TRS R consists of the following rules: 114.75/31.09 114.75/31.09 delcB_in_ga(t, []) -> delcB_out_ga(t, []) 114.75/31.09 delcB_in_ga(f, []) -> delcB_out_ga(f, []) 114.75/31.09 maxcC_in_gg([], t) -> maxcC_out_gg([], t) 114.75/31.09 maxcC_in_gg(.(t, X1), X2) -> U48_gg(X1, X2, maxcC_in_gg(X1, X2)) 114.75/31.09 maxcC_in_gg(.(f, X1), X2) -> U49_gg(X1, X2, maxcC_in_gg(X1, X2)) 114.75/31.09 U49_gg(X1, X2, maxcC_out_gg(X1, X2)) -> maxcC_out_gg(.(f, X1), X2) 114.75/31.09 U48_gg(X1, X2, maxcC_out_gg(X1, X2)) -> maxcC_out_gg(.(t, X1), X2) 114.75/31.09 maxcC_in_ga([], t) -> maxcC_out_ga([], t) 114.75/31.09 maxcC_in_ga(.(t, X1), X2) -> U48_ga(X1, X2, maxcC_in_ga(X1, X2)) 114.75/31.09 maxcC_in_ga(.(f, X1), X2) -> U49_ga(X1, X2, maxcC_in_ga(X1, X2)) 114.75/31.09 U49_ga(X1, X2, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(f, X1), X2) 114.75/31.09 U48_ga(X1, X2, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(t, X1), X2) 114.75/31.09 delcD_in_gga(t, X1, .(t, X1)) -> delcD_out_gga(t, X1, .(t, X1)) 114.75/31.09 delcD_in_gga(f, X1, .(t, .(t, X2))) -> U54_gga(X1, X2, delcI_in_ga(X1, X2)) 114.75/31.09 delcI_in_ga([], []) -> delcI_out_ga([], []) 114.75/31.09 delcI_in_ga(.(f, X1), X1) -> delcI_out_ga(.(f, X1), X1) 114.75/31.09 delcI_in_ga(.(t, X1), .(t, X2)) -> U50_ga(X1, X2, delcI_in_ga(X1, X2)) 114.75/31.09 U50_ga(X1, X2, delcI_out_ga(X1, X2)) -> delcI_out_ga(.(t, X1), .(t, X2)) 114.75/31.09 U54_gga(X1, X2, delcI_out_ga(X1, X2)) -> delcD_out_gga(f, X1, .(t, .(t, X2))) 114.75/31.09 delcE_in_gga(t, X1, .(f, X1)) -> delcE_out_gga(t, X1, .(f, X1)) 114.75/31.09 delcE_in_gga(f, X1, .(t, X2)) -> U55_gga(X1, X2, delcI_in_ga(.(f, X1), X2)) 114.75/31.09 U55_gga(X1, X2, delcI_out_ga(.(f, X1), X2)) -> delcE_out_gga(f, X1, .(t, X2)) 114.75/31.09 maxcF_in_gg([], f) -> maxcF_out_gg([], f) 114.75/31.09 maxcF_in_gg(.(f, X1), X2) -> U51_gg(X1, X2, maxcF_in_gg(X1, X2)) 114.75/31.09 maxcF_in_gg(.(t, X1), X2) -> U52_gg(X1, X2, maxcC_in_gg(X1, X2)) 114.75/31.09 U52_gg(X1, X2, maxcC_out_gg(X1, X2)) -> maxcF_out_gg(.(t, X1), X2) 114.75/31.09 U51_gg(X1, X2, maxcF_out_gg(X1, X2)) -> maxcF_out_gg(.(f, X1), X2) 114.75/31.09 maxcF_in_ga([], f) -> maxcF_out_ga([], f) 114.75/31.09 maxcF_in_ga(.(f, X1), X2) -> U51_ga(X1, X2, maxcF_in_ga(X1, X2)) 114.75/31.09 maxcF_in_ga(.(t, X1), X2) -> U52_ga(X1, X2, maxcC_in_ga(X1, X2)) 114.75/31.09 U52_ga(X1, X2, maxcC_out_ga(X1, X2)) -> maxcF_out_ga(.(t, X1), X2) 114.75/31.09 U51_ga(X1, X2, maxcF_out_ga(X1, X2)) -> maxcF_out_ga(.(f, X1), X2) 114.75/31.09 delcG_in_gga(f, X1, .(f, X1)) -> delcG_out_gga(f, X1, .(f, X1)) 114.75/31.09 delcG_in_gga(t, X1, .(f, .(f, X2))) -> U56_gga(X1, X2, delcJ_in_ga(X1, X2)) 114.75/31.09 delcJ_in_ga([], []) -> delcJ_out_ga([], []) 114.75/31.09 delcJ_in_ga(.(t, X1), X1) -> delcJ_out_ga(.(t, X1), X1) 114.75/31.09 delcJ_in_ga(.(f, X1), .(f, X2)) -> U53_ga(X1, X2, delcJ_in_ga(X1, X2)) 114.75/31.09 U53_ga(X1, X2, delcJ_out_ga(X1, X2)) -> delcJ_out_ga(.(f, X1), .(f, X2)) 114.75/31.09 U56_gga(X1, X2, delcJ_out_ga(X1, X2)) -> delcG_out_gga(t, X1, .(f, .(f, X2))) 114.75/31.09 delcH_in_gga(f, X1, .(t, X1)) -> delcH_out_gga(f, X1, .(t, X1)) 114.75/31.09 delcH_in_gga(t, X1, .(f, X2)) -> U57_gga(X1, X2, delcJ_in_ga(.(t, X1), X2)) 114.75/31.09 U57_gga(X1, X2, delcJ_out_ga(.(t, X1), X2)) -> delcH_out_gga(t, X1, .(f, X2)) 114.75/31.09 114.75/31.09 The argument filtering Pi contains the following mapping: 114.75/31.09 maxsortA_in_ga(x1, x2) = maxsortA_in_ga(x1) 114.75/31.09 114.75/31.09 .(x1, x2) = .(x1, x2) 114.75/31.09 114.75/31.09 [] = [] 114.75/31.09 114.75/31.09 delcB_in_ga(x1, x2) = delcB_in_ga(x1) 114.75/31.09 114.75/31.09 t = t 114.75/31.09 114.75/31.09 delcB_out_ga(x1, x2) = delcB_out_ga(x1, x2) 114.75/31.09 114.75/31.09 f = f 114.75/31.09 114.75/31.09 maxC_in_ga(x1, x2) = maxC_in_ga(x1) 114.75/31.09 114.75/31.09 maxcC_in_gg(x1, x2) = maxcC_in_gg(x1, x2) 114.75/31.09 114.75/31.09 maxcC_out_gg(x1, x2) = maxcC_out_gg(x1, x2) 114.75/31.09 114.75/31.09 U48_gg(x1, x2, x3) = U48_gg(x1, x2, x3) 114.75/31.09 114.75/31.09 U49_gg(x1, x2, x3) = U49_gg(x1, x2, x3) 114.75/31.09 114.75/31.09 delI_in_ga(x1, x2) = delI_in_ga(x1) 114.75/31.09 114.75/31.09 maxcC_in_ga(x1, x2) = maxcC_in_ga(x1) 114.75/31.09 114.75/31.09 maxcC_out_ga(x1, x2) = maxcC_out_ga(x1, x2) 114.75/31.09 114.75/31.09 U48_ga(x1, x2, x3) = U48_ga(x1, x3) 114.75/31.09 114.75/31.09 U49_ga(x1, x2, x3) = U49_ga(x1, x3) 114.75/31.09 114.75/31.09 delcD_in_gga(x1, x2, x3) = delcD_in_gga(x1, x2) 114.75/31.09 114.75/31.09 delcD_out_gga(x1, x2, x3) = delcD_out_gga(x1, x2, x3) 114.75/31.09 114.75/31.09 U54_gga(x1, x2, x3) = U54_gga(x1, x3) 114.75/31.09 114.75/31.09 delcI_in_ga(x1, x2) = delcI_in_ga(x1) 114.75/31.09 114.75/31.09 delcI_out_ga(x1, x2) = delcI_out_ga(x1, x2) 114.75/31.09 114.75/31.09 U50_ga(x1, x2, x3) = U50_ga(x1, x3) 114.75/31.09 114.75/31.09 delcE_in_gga(x1, x2, x3) = delcE_in_gga(x1, x2) 114.75/31.09 114.75/31.09 delcE_out_gga(x1, x2, x3) = delcE_out_gga(x1, x2, x3) 114.75/31.09 114.75/31.09 U55_gga(x1, x2, x3) = U55_gga(x1, x3) 114.75/31.09 114.75/31.09 maxF_in_ga(x1, x2) = maxF_in_ga(x1) 114.75/31.09 114.75/31.09 maxcF_in_gg(x1, x2) = maxcF_in_gg(x1, x2) 114.75/31.09 114.75/31.09 maxcF_out_gg(x1, x2) = maxcF_out_gg(x1, x2) 114.75/31.09 114.75/31.09 U51_gg(x1, x2, x3) = U51_gg(x1, x2, x3) 114.75/31.09 114.75/31.09 U52_gg(x1, x2, x3) = U52_gg(x1, x2, x3) 114.75/31.09 114.75/31.09 delJ_in_ga(x1, x2) = delJ_in_ga(x1) 114.75/31.09 114.75/31.09 maxcF_in_ga(x1, x2) = maxcF_in_ga(x1) 114.75/31.09 114.75/31.09 maxcF_out_ga(x1, x2) = maxcF_out_ga(x1, x2) 114.75/31.09 114.75/31.09 U51_ga(x1, x2, x3) = U51_ga(x1, x3) 114.75/31.09 114.75/31.09 U52_ga(x1, x2, x3) = U52_ga(x1, x3) 114.75/31.09 114.75/31.09 delcG_in_gga(x1, x2, x3) = delcG_in_gga(x1, x2) 114.75/31.09 114.75/31.09 delcG_out_gga(x1, x2, x3) = delcG_out_gga(x1, x2, x3) 114.75/31.09 114.75/31.09 U56_gga(x1, x2, x3) = U56_gga(x1, x3) 114.75/31.09 114.75/31.09 delcJ_in_ga(x1, x2) = delcJ_in_ga(x1) 114.75/31.09 114.75/31.09 delcJ_out_ga(x1, x2) = delcJ_out_ga(x1, x2) 114.75/31.09 114.75/31.09 U53_ga(x1, x2, x3) = U53_ga(x1, x3) 114.75/31.09 114.75/31.09 delcH_in_gga(x1, x2, x3) = delcH_in_gga(x1, x2) 114.75/31.09 114.75/31.09 delcH_out_gga(x1, x2, x3) = delcH_out_gga(x1, x2, x3) 114.75/31.09 114.75/31.09 U57_gga(x1, x2, x3) = U57_gga(x1, x3) 114.75/31.09 114.75/31.09 MAXSORTA_IN_GA(x1, x2) = MAXSORTA_IN_GA(x1) 114.75/31.09 114.75/31.09 U7_GA(x1, x2, x3) = U7_GA(x1, x3) 114.75/31.09 114.75/31.09 U8_GA(x1, x2, x3) = U8_GA(x1, x3) 114.75/31.09 114.75/31.09 U9_GA(x1, x2, x3, x4) = U9_GA(x1, x4) 114.75/31.09 114.75/31.09 MAXC_IN_GA(x1, x2) = MAXC_IN_GA(x1) 114.75/31.09 114.75/31.09 U1_GA(x1, x2, x3) = U1_GA(x1, x3) 114.75/31.09 114.75/31.09 U2_GA(x1, x2, x3) = U2_GA(x1, x3) 114.75/31.09 114.75/31.09 U10_GA(x1, x2, x3) = U10_GA(x1, x3) 114.75/31.09 114.75/31.09 U11_GA(x1, x2, x3) = U11_GA(x1, x3) 114.75/31.09 114.75/31.09 DELI_IN_GA(x1, x2) = DELI_IN_GA(x1) 114.75/31.09 114.75/31.09 U3_GA(x1, x2, x3) = U3_GA(x1, x3) 114.75/31.09 114.75/31.09 U12_GA(x1, x2, x3, x4) = U12_GA(x1, x4) 114.75/31.09 114.75/31.09 U13_GA(x1, x2, x3, x4) = U13_GA(x1, x4) 114.75/31.09 114.75/31.09 U14_GA(x1, x2, x3, x4) = U14_GA(x1, x4) 114.75/31.09 114.75/31.09 U15_GA(x1, x2, x3, x4) = U15_GA(x1, x4) 114.75/31.09 114.75/31.09 U16_GA(x1, x2, x3) = U16_GA(x1, x3) 114.75/31.09 114.75/31.09 U17_GA(x1, x2, x3) = U17_GA(x1, x3) 114.75/31.09 114.75/31.09 U18_GA(x1, x2, x3, x4) = U18_GA(x1, x4) 114.75/31.09 114.75/31.09 U19_GA(x1, x2, x3, x4) = U19_GA(x1, x4) 114.75/31.09 114.75/31.09 U20_GA(x1, x2, x3, x4) = U20_GA(x1, x4) 114.75/31.09 114.75/31.09 U21_GA(x1, x2, x3, x4) = U21_GA(x1, x4) 114.75/31.09 114.75/31.09 MAXF_IN_GA(x1, x2) = MAXF_IN_GA(x1) 114.75/31.09 114.75/31.09 U4_GA(x1, x2, x3) = U4_GA(x1, x3) 114.75/31.09 114.75/31.09 U5_GA(x1, x2, x3) = U5_GA(x1, x3) 114.75/31.09 114.75/31.09 U22_GA(x1, x2, x3) = U22_GA(x1, x3) 114.75/31.09 114.75/31.09 U23_GA(x1, x2, x3) = U23_GA(x1, x3) 114.75/31.09 114.75/31.09 DELJ_IN_GA(x1, x2) = DELJ_IN_GA(x1) 114.75/31.09 114.75/31.09 U6_GA(x1, x2, x3) = U6_GA(x1, x3) 114.75/31.09 114.75/31.09 U24_GA(x1, x2, x3, x4) = U24_GA(x1, x4) 114.75/31.09 114.75/31.09 U25_GA(x1, x2, x3, x4) = U25_GA(x1, x4) 114.75/31.09 114.75/31.09 U26_GA(x1, x2, x3, x4) = U26_GA(x1, x4) 114.75/31.09 114.75/31.09 U27_GA(x1, x2, x3, x4) = U27_GA(x1, x4) 114.75/31.09 114.75/31.09 U28_GA(x1, x2, x3) = U28_GA(x1, x3) 114.75/31.09 114.75/31.09 U29_GA(x1, x2, x3) = U29_GA(x1, x3) 114.75/31.09 114.75/31.09 U30_GA(x1, x2, x3, x4) = U30_GA(x1, x4) 114.75/31.09 114.75/31.09 U31_GA(x1, x2, x3, x4) = U31_GA(x1, x4) 114.75/31.09 114.75/31.09 U32_GA(x1, x2, x3, x4) = U32_GA(x1, x4) 114.75/31.09 114.75/31.09 114.75/31.09 We have to consider all (P,R,Pi)-chains 114.75/31.09 ---------------------------------------- 114.75/31.09 114.75/31.09 (5) DependencyGraphProof (EQUIVALENT) 114.75/31.09 The approximation of the Dependency Graph [LOPSTR] contains 5 SCCs with 32 less nodes. 114.75/31.09 ---------------------------------------- 114.75/31.09 114.75/31.09 (6) 114.75/31.09 Complex Obligation (AND) 114.75/31.09 114.75/31.09 ---------------------------------------- 114.75/31.09 114.75/31.09 (7) 114.75/31.09 Obligation: 114.75/31.09 Pi DP problem: 114.75/31.09 The TRS P consists of the following rules: 114.75/31.09 114.75/31.09 DELJ_IN_GA(.(f, X1), .(f, X2)) -> DELJ_IN_GA(X1, X2) 114.75/31.09 114.75/31.09 The TRS R consists of the following rules: 114.75/31.09 114.75/31.09 delcB_in_ga(t, []) -> delcB_out_ga(t, []) 114.75/31.09 delcB_in_ga(f, []) -> delcB_out_ga(f, []) 114.75/31.09 maxcC_in_gg([], t) -> maxcC_out_gg([], t) 114.75/31.09 maxcC_in_gg(.(t, X1), X2) -> U48_gg(X1, X2, maxcC_in_gg(X1, X2)) 114.75/31.09 maxcC_in_gg(.(f, X1), X2) -> U49_gg(X1, X2, maxcC_in_gg(X1, X2)) 114.75/31.09 U49_gg(X1, X2, maxcC_out_gg(X1, X2)) -> maxcC_out_gg(.(f, X1), X2) 114.75/31.09 U48_gg(X1, X2, maxcC_out_gg(X1, X2)) -> maxcC_out_gg(.(t, X1), X2) 114.75/31.09 maxcC_in_ga([], t) -> maxcC_out_ga([], t) 114.75/31.09 maxcC_in_ga(.(t, X1), X2) -> U48_ga(X1, X2, maxcC_in_ga(X1, X2)) 114.75/31.09 maxcC_in_ga(.(f, X1), X2) -> U49_ga(X1, X2, maxcC_in_ga(X1, X2)) 114.75/31.09 U49_ga(X1, X2, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(f, X1), X2) 114.75/31.09 U48_ga(X1, X2, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(t, X1), X2) 114.75/31.09 delcD_in_gga(t, X1, .(t, X1)) -> delcD_out_gga(t, X1, .(t, X1)) 114.75/31.09 delcD_in_gga(f, X1, .(t, .(t, X2))) -> U54_gga(X1, X2, delcI_in_ga(X1, X2)) 114.75/31.09 delcI_in_ga([], []) -> delcI_out_ga([], []) 114.75/31.09 delcI_in_ga(.(f, X1), X1) -> delcI_out_ga(.(f, X1), X1) 114.75/31.09 delcI_in_ga(.(t, X1), .(t, X2)) -> U50_ga(X1, X2, delcI_in_ga(X1, X2)) 114.75/31.09 U50_ga(X1, X2, delcI_out_ga(X1, X2)) -> delcI_out_ga(.(t, X1), .(t, X2)) 114.75/31.09 U54_gga(X1, X2, delcI_out_ga(X1, X2)) -> delcD_out_gga(f, X1, .(t, .(t, X2))) 114.75/31.09 delcE_in_gga(t, X1, .(f, X1)) -> delcE_out_gga(t, X1, .(f, X1)) 114.75/31.09 delcE_in_gga(f, X1, .(t, X2)) -> U55_gga(X1, X2, delcI_in_ga(.(f, X1), X2)) 114.75/31.09 U55_gga(X1, X2, delcI_out_ga(.(f, X1), X2)) -> delcE_out_gga(f, X1, .(t, X2)) 114.75/31.09 maxcF_in_gg([], f) -> maxcF_out_gg([], f) 114.75/31.09 maxcF_in_gg(.(f, X1), X2) -> U51_gg(X1, X2, maxcF_in_gg(X1, X2)) 114.75/31.09 maxcF_in_gg(.(t, X1), X2) -> U52_gg(X1, X2, maxcC_in_gg(X1, X2)) 114.75/31.09 U52_gg(X1, X2, maxcC_out_gg(X1, X2)) -> maxcF_out_gg(.(t, X1), X2) 114.75/31.09 U51_gg(X1, X2, maxcF_out_gg(X1, X2)) -> maxcF_out_gg(.(f, X1), X2) 114.75/31.09 maxcF_in_ga([], f) -> maxcF_out_ga([], f) 114.75/31.09 maxcF_in_ga(.(f, X1), X2) -> U51_ga(X1, X2, maxcF_in_ga(X1, X2)) 114.75/31.09 maxcF_in_ga(.(t, X1), X2) -> U52_ga(X1, X2, maxcC_in_ga(X1, X2)) 114.75/31.09 U52_ga(X1, X2, maxcC_out_ga(X1, X2)) -> maxcF_out_ga(.(t, X1), X2) 114.75/31.09 U51_ga(X1, X2, maxcF_out_ga(X1, X2)) -> maxcF_out_ga(.(f, X1), X2) 114.75/31.09 delcG_in_gga(f, X1, .(f, X1)) -> delcG_out_gga(f, X1, .(f, X1)) 114.75/31.09 delcG_in_gga(t, X1, .(f, .(f, X2))) -> U56_gga(X1, X2, delcJ_in_ga(X1, X2)) 114.75/31.09 delcJ_in_ga([], []) -> delcJ_out_ga([], []) 114.75/31.09 delcJ_in_ga(.(t, X1), X1) -> delcJ_out_ga(.(t, X1), X1) 114.75/31.09 delcJ_in_ga(.(f, X1), .(f, X2)) -> U53_ga(X1, X2, delcJ_in_ga(X1, X2)) 114.75/31.09 U53_ga(X1, X2, delcJ_out_ga(X1, X2)) -> delcJ_out_ga(.(f, X1), .(f, X2)) 114.75/31.09 U56_gga(X1, X2, delcJ_out_ga(X1, X2)) -> delcG_out_gga(t, X1, .(f, .(f, X2))) 114.75/31.09 delcH_in_gga(f, X1, .(t, X1)) -> delcH_out_gga(f, X1, .(t, X1)) 114.75/31.09 delcH_in_gga(t, X1, .(f, X2)) -> U57_gga(X1, X2, delcJ_in_ga(.(t, X1), X2)) 114.75/31.09 U57_gga(X1, X2, delcJ_out_ga(.(t, X1), X2)) -> delcH_out_gga(t, X1, .(f, X2)) 114.75/31.09 114.75/31.09 The argument filtering Pi contains the following mapping: 114.75/31.09 .(x1, x2) = .(x1, x2) 114.75/31.09 114.75/31.09 [] = [] 114.75/31.09 114.75/31.09 delcB_in_ga(x1, x2) = delcB_in_ga(x1) 114.75/31.09 114.75/31.09 t = t 114.75/31.09 114.75/31.09 delcB_out_ga(x1, x2) = delcB_out_ga(x1, x2) 114.75/31.09 114.75/31.09 f = f 114.75/31.09 114.75/31.09 maxcC_in_gg(x1, x2) = maxcC_in_gg(x1, x2) 114.75/31.09 114.75/31.09 maxcC_out_gg(x1, x2) = maxcC_out_gg(x1, x2) 114.75/31.09 114.75/31.09 U48_gg(x1, x2, x3) = U48_gg(x1, x2, x3) 114.75/31.09 114.75/31.09 U49_gg(x1, x2, x3) = U49_gg(x1, x2, x3) 114.75/31.09 114.75/31.09 maxcC_in_ga(x1, x2) = maxcC_in_ga(x1) 114.75/31.09 114.75/31.09 maxcC_out_ga(x1, x2) = maxcC_out_ga(x1, x2) 114.75/31.09 114.75/31.09 U48_ga(x1, x2, x3) = U48_ga(x1, x3) 114.75/31.09 114.75/31.09 U49_ga(x1, x2, x3) = U49_ga(x1, x3) 114.75/31.09 114.75/31.09 delcD_in_gga(x1, x2, x3) = delcD_in_gga(x1, x2) 114.75/31.09 114.75/31.09 delcD_out_gga(x1, x2, x3) = delcD_out_gga(x1, x2, x3) 114.75/31.09 114.75/31.09 U54_gga(x1, x2, x3) = U54_gga(x1, x3) 114.75/31.09 114.75/31.09 delcI_in_ga(x1, x2) = delcI_in_ga(x1) 114.75/31.09 114.75/31.09 delcI_out_ga(x1, x2) = delcI_out_ga(x1, x2) 114.75/31.09 114.75/31.09 U50_ga(x1, x2, x3) = U50_ga(x1, x3) 114.75/31.09 114.75/31.09 delcE_in_gga(x1, x2, x3) = delcE_in_gga(x1, x2) 114.75/31.09 114.75/31.09 delcE_out_gga(x1, x2, x3) = delcE_out_gga(x1, x2, x3) 114.75/31.09 114.75/31.09 U55_gga(x1, x2, x3) = U55_gga(x1, x3) 114.75/31.09 114.75/31.09 maxcF_in_gg(x1, x2) = maxcF_in_gg(x1, x2) 114.75/31.09 114.75/31.09 maxcF_out_gg(x1, x2) = maxcF_out_gg(x1, x2) 114.75/31.09 114.75/31.09 U51_gg(x1, x2, x3) = U51_gg(x1, x2, x3) 114.75/31.09 114.75/31.09 U52_gg(x1, x2, x3) = U52_gg(x1, x2, x3) 114.75/31.09 114.75/31.09 maxcF_in_ga(x1, x2) = maxcF_in_ga(x1) 114.75/31.09 114.75/31.09 maxcF_out_ga(x1, x2) = maxcF_out_ga(x1, x2) 114.75/31.09 114.75/31.09 U51_ga(x1, x2, x3) = U51_ga(x1, x3) 114.75/31.09 114.75/31.09 U52_ga(x1, x2, x3) = U52_ga(x1, x3) 114.75/31.09 114.75/31.09 delcG_in_gga(x1, x2, x3) = delcG_in_gga(x1, x2) 114.75/31.09 114.75/31.09 delcG_out_gga(x1, x2, x3) = delcG_out_gga(x1, x2, x3) 114.75/31.09 114.75/31.09 U56_gga(x1, x2, x3) = U56_gga(x1, x3) 114.75/31.09 114.75/31.09 delcJ_in_ga(x1, x2) = delcJ_in_ga(x1) 114.75/31.09 114.75/31.09 delcJ_out_ga(x1, x2) = delcJ_out_ga(x1, x2) 114.75/31.09 114.75/31.09 U53_ga(x1, x2, x3) = U53_ga(x1, x3) 114.75/31.09 114.75/31.09 delcH_in_gga(x1, x2, x3) = delcH_in_gga(x1, x2) 114.75/31.09 114.75/31.09 delcH_out_gga(x1, x2, x3) = delcH_out_gga(x1, x2, x3) 114.75/31.09 114.75/31.09 U57_gga(x1, x2, x3) = U57_gga(x1, x3) 114.75/31.09 114.75/31.09 DELJ_IN_GA(x1, x2) = DELJ_IN_GA(x1) 114.75/31.09 114.75/31.09 114.75/31.09 We have to consider all (P,R,Pi)-chains 114.75/31.09 ---------------------------------------- 114.75/31.09 114.75/31.09 (8) UsableRulesProof (EQUIVALENT) 114.75/31.09 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 114.75/31.09 ---------------------------------------- 114.75/31.09 114.75/31.09 (9) 114.75/31.09 Obligation: 114.75/31.09 Pi DP problem: 114.75/31.09 The TRS P consists of the following rules: 114.75/31.09 114.75/31.09 DELJ_IN_GA(.(f, X1), .(f, X2)) -> DELJ_IN_GA(X1, X2) 114.75/31.09 114.75/31.09 R is empty. 114.75/31.09 The argument filtering Pi contains the following mapping: 114.75/31.09 .(x1, x2) = .(x1, x2) 114.75/31.09 114.75/31.09 f = f 114.75/31.09 114.75/31.09 DELJ_IN_GA(x1, x2) = DELJ_IN_GA(x1) 114.75/31.09 114.75/31.09 114.75/31.09 We have to consider all (P,R,Pi)-chains 114.75/31.09 ---------------------------------------- 114.75/31.09 114.75/31.09 (10) PiDPToQDPProof (SOUND) 114.75/31.09 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 114.75/31.09 ---------------------------------------- 114.75/31.09 114.75/31.09 (11) 114.75/31.09 Obligation: 114.75/31.09 Q DP problem: 114.75/31.09 The TRS P consists of the following rules: 114.75/31.09 114.75/31.09 DELJ_IN_GA(.(f, X1)) -> DELJ_IN_GA(X1) 114.75/31.09 114.75/31.09 R is empty. 114.75/31.09 Q is empty. 114.75/31.09 We have to consider all (P,Q,R)-chains. 114.75/31.09 ---------------------------------------- 114.75/31.09 114.75/31.09 (12) QDPSizeChangeProof (EQUIVALENT) 114.75/31.09 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. 114.75/31.09 114.75/31.09 From the DPs we obtained the following set of size-change graphs: 114.75/31.09 *DELJ_IN_GA(.(f, X1)) -> DELJ_IN_GA(X1) 114.75/31.09 The graph contains the following edges 1 > 1 114.75/31.09 114.75/31.09 114.75/31.09 ---------------------------------------- 114.75/31.09 114.75/31.09 (13) 114.75/31.09 YES 114.75/31.09 114.75/31.09 ---------------------------------------- 114.75/31.09 114.75/31.09 (14) 114.75/31.09 Obligation: 114.75/31.09 Pi DP problem: 114.75/31.09 The TRS P consists of the following rules: 114.75/31.09 114.75/31.09 DELI_IN_GA(.(t, X1), .(t, X2)) -> DELI_IN_GA(X1, X2) 114.75/31.09 114.75/31.09 The TRS R consists of the following rules: 114.75/31.09 114.75/31.09 delcB_in_ga(t, []) -> delcB_out_ga(t, []) 114.75/31.09 delcB_in_ga(f, []) -> delcB_out_ga(f, []) 114.75/31.09 maxcC_in_gg([], t) -> maxcC_out_gg([], t) 114.75/31.09 maxcC_in_gg(.(t, X1), X2) -> U48_gg(X1, X2, maxcC_in_gg(X1, X2)) 114.75/31.09 maxcC_in_gg(.(f, X1), X2) -> U49_gg(X1, X2, maxcC_in_gg(X1, X2)) 114.75/31.09 U49_gg(X1, X2, maxcC_out_gg(X1, X2)) -> maxcC_out_gg(.(f, X1), X2) 114.75/31.09 U48_gg(X1, X2, maxcC_out_gg(X1, X2)) -> maxcC_out_gg(.(t, X1), X2) 114.75/31.09 maxcC_in_ga([], t) -> maxcC_out_ga([], t) 114.75/31.09 maxcC_in_ga(.(t, X1), X2) -> U48_ga(X1, X2, maxcC_in_ga(X1, X2)) 114.75/31.09 maxcC_in_ga(.(f, X1), X2) -> U49_ga(X1, X2, maxcC_in_ga(X1, X2)) 114.75/31.09 U49_ga(X1, X2, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(f, X1), X2) 114.75/31.09 U48_ga(X1, X2, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(t, X1), X2) 114.75/31.09 delcD_in_gga(t, X1, .(t, X1)) -> delcD_out_gga(t, X1, .(t, X1)) 114.75/31.09 delcD_in_gga(f, X1, .(t, .(t, X2))) -> U54_gga(X1, X2, delcI_in_ga(X1, X2)) 114.75/31.09 delcI_in_ga([], []) -> delcI_out_ga([], []) 114.75/31.09 delcI_in_ga(.(f, X1), X1) -> delcI_out_ga(.(f, X1), X1) 114.75/31.09 delcI_in_ga(.(t, X1), .(t, X2)) -> U50_ga(X1, X2, delcI_in_ga(X1, X2)) 114.75/31.09 U50_ga(X1, X2, delcI_out_ga(X1, X2)) -> delcI_out_ga(.(t, X1), .(t, X2)) 114.75/31.09 U54_gga(X1, X2, delcI_out_ga(X1, X2)) -> delcD_out_gga(f, X1, .(t, .(t, X2))) 114.75/31.09 delcE_in_gga(t, X1, .(f, X1)) -> delcE_out_gga(t, X1, .(f, X1)) 114.75/31.09 delcE_in_gga(f, X1, .(t, X2)) -> U55_gga(X1, X2, delcI_in_ga(.(f, X1), X2)) 114.75/31.09 U55_gga(X1, X2, delcI_out_ga(.(f, X1), X2)) -> delcE_out_gga(f, X1, .(t, X2)) 114.75/31.09 maxcF_in_gg([], f) -> maxcF_out_gg([], f) 114.75/31.09 maxcF_in_gg(.(f, X1), X2) -> U51_gg(X1, X2, maxcF_in_gg(X1, X2)) 114.75/31.09 maxcF_in_gg(.(t, X1), X2) -> U52_gg(X1, X2, maxcC_in_gg(X1, X2)) 114.75/31.09 U52_gg(X1, X2, maxcC_out_gg(X1, X2)) -> maxcF_out_gg(.(t, X1), X2) 114.75/31.09 U51_gg(X1, X2, maxcF_out_gg(X1, X2)) -> maxcF_out_gg(.(f, X1), X2) 114.75/31.09 maxcF_in_ga([], f) -> maxcF_out_ga([], f) 114.75/31.09 maxcF_in_ga(.(f, X1), X2) -> U51_ga(X1, X2, maxcF_in_ga(X1, X2)) 114.75/31.09 maxcF_in_ga(.(t, X1), X2) -> U52_ga(X1, X2, maxcC_in_ga(X1, X2)) 114.75/31.09 U52_ga(X1, X2, maxcC_out_ga(X1, X2)) -> maxcF_out_ga(.(t, X1), X2) 114.75/31.09 U51_ga(X1, X2, maxcF_out_ga(X1, X2)) -> maxcF_out_ga(.(f, X1), X2) 114.75/31.09 delcG_in_gga(f, X1, .(f, X1)) -> delcG_out_gga(f, X1, .(f, X1)) 114.75/31.09 delcG_in_gga(t, X1, .(f, .(f, X2))) -> U56_gga(X1, X2, delcJ_in_ga(X1, X2)) 114.75/31.09 delcJ_in_ga([], []) -> delcJ_out_ga([], []) 114.75/31.09 delcJ_in_ga(.(t, X1), X1) -> delcJ_out_ga(.(t, X1), X1) 114.75/31.09 delcJ_in_ga(.(f, X1), .(f, X2)) -> U53_ga(X1, X2, delcJ_in_ga(X1, X2)) 114.75/31.09 U53_ga(X1, X2, delcJ_out_ga(X1, X2)) -> delcJ_out_ga(.(f, X1), .(f, X2)) 114.75/31.09 U56_gga(X1, X2, delcJ_out_ga(X1, X2)) -> delcG_out_gga(t, X1, .(f, .(f, X2))) 114.75/31.09 delcH_in_gga(f, X1, .(t, X1)) -> delcH_out_gga(f, X1, .(t, X1)) 114.75/31.09 delcH_in_gga(t, X1, .(f, X2)) -> U57_gga(X1, X2, delcJ_in_ga(.(t, X1), X2)) 114.75/31.09 U57_gga(X1, X2, delcJ_out_ga(.(t, X1), X2)) -> delcH_out_gga(t, X1, .(f, X2)) 114.75/31.09 114.75/31.09 The argument filtering Pi contains the following mapping: 114.75/31.09 .(x1, x2) = .(x1, x2) 114.75/31.09 114.75/31.09 [] = [] 114.75/31.09 114.75/31.09 delcB_in_ga(x1, x2) = delcB_in_ga(x1) 114.75/31.09 114.75/31.09 t = t 114.75/31.09 114.75/31.09 delcB_out_ga(x1, x2) = delcB_out_ga(x1, x2) 114.75/31.09 114.75/31.09 f = f 114.75/31.09 114.75/31.09 maxcC_in_gg(x1, x2) = maxcC_in_gg(x1, x2) 114.75/31.09 114.75/31.09 maxcC_out_gg(x1, x2) = maxcC_out_gg(x1, x2) 114.75/31.09 114.75/31.09 U48_gg(x1, x2, x3) = U48_gg(x1, x2, x3) 114.75/31.09 114.75/31.09 U49_gg(x1, x2, x3) = U49_gg(x1, x2, x3) 114.75/31.09 114.75/31.09 maxcC_in_ga(x1, x2) = maxcC_in_ga(x1) 114.75/31.09 114.75/31.09 maxcC_out_ga(x1, x2) = maxcC_out_ga(x1, x2) 114.75/31.09 114.75/31.09 U48_ga(x1, x2, x3) = U48_ga(x1, x3) 114.75/31.09 114.75/31.09 U49_ga(x1, x2, x3) = U49_ga(x1, x3) 114.75/31.09 114.75/31.09 delcD_in_gga(x1, x2, x3) = delcD_in_gga(x1, x2) 114.75/31.09 114.75/31.09 delcD_out_gga(x1, x2, x3) = delcD_out_gga(x1, x2, x3) 114.75/31.09 114.75/31.09 U54_gga(x1, x2, x3) = U54_gga(x1, x3) 114.75/31.09 114.75/31.09 delcI_in_ga(x1, x2) = delcI_in_ga(x1) 114.75/31.09 114.75/31.09 delcI_out_ga(x1, x2) = delcI_out_ga(x1, x2) 114.75/31.09 114.75/31.09 U50_ga(x1, x2, x3) = U50_ga(x1, x3) 114.75/31.09 114.75/31.09 delcE_in_gga(x1, x2, x3) = delcE_in_gga(x1, x2) 114.75/31.09 114.75/31.09 delcE_out_gga(x1, x2, x3) = delcE_out_gga(x1, x2, x3) 114.75/31.09 114.75/31.09 U55_gga(x1, x2, x3) = U55_gga(x1, x3) 114.75/31.09 114.75/31.09 maxcF_in_gg(x1, x2) = maxcF_in_gg(x1, x2) 114.75/31.09 114.75/31.09 maxcF_out_gg(x1, x2) = maxcF_out_gg(x1, x2) 114.75/31.09 114.75/31.09 U51_gg(x1, x2, x3) = U51_gg(x1, x2, x3) 114.75/31.09 114.75/31.09 U52_gg(x1, x2, x3) = U52_gg(x1, x2, x3) 114.75/31.09 114.75/31.09 maxcF_in_ga(x1, x2) = maxcF_in_ga(x1) 114.75/31.09 114.75/31.09 maxcF_out_ga(x1, x2) = maxcF_out_ga(x1, x2) 114.75/31.09 114.75/31.09 U51_ga(x1, x2, x3) = U51_ga(x1, x3) 114.75/31.09 114.75/31.09 U52_ga(x1, x2, x3) = U52_ga(x1, x3) 114.75/31.09 114.75/31.09 delcG_in_gga(x1, x2, x3) = delcG_in_gga(x1, x2) 114.75/31.09 114.75/31.09 delcG_out_gga(x1, x2, x3) = delcG_out_gga(x1, x2, x3) 114.75/31.09 114.75/31.09 U56_gga(x1, x2, x3) = U56_gga(x1, x3) 114.75/31.09 114.75/31.09 delcJ_in_ga(x1, x2) = delcJ_in_ga(x1) 114.75/31.09 114.75/31.09 delcJ_out_ga(x1, x2) = delcJ_out_ga(x1, x2) 114.75/31.09 114.75/31.09 U53_ga(x1, x2, x3) = U53_ga(x1, x3) 114.75/31.09 114.75/31.09 delcH_in_gga(x1, x2, x3) = delcH_in_gga(x1, x2) 114.75/31.09 114.75/31.09 delcH_out_gga(x1, x2, x3) = delcH_out_gga(x1, x2, x3) 114.75/31.09 114.75/31.09 U57_gga(x1, x2, x3) = U57_gga(x1, x3) 114.75/31.09 114.75/31.09 DELI_IN_GA(x1, x2) = DELI_IN_GA(x1) 114.75/31.09 114.75/31.09 114.75/31.09 We have to consider all (P,R,Pi)-chains 114.75/31.09 ---------------------------------------- 114.75/31.09 114.75/31.09 (15) UsableRulesProof (EQUIVALENT) 114.75/31.09 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 114.75/31.09 ---------------------------------------- 114.75/31.09 114.75/31.09 (16) 114.75/31.09 Obligation: 114.75/31.09 Pi DP problem: 114.75/31.09 The TRS P consists of the following rules: 114.75/31.09 114.75/31.09 DELI_IN_GA(.(t, X1), .(t, X2)) -> DELI_IN_GA(X1, X2) 114.75/31.09 114.75/31.09 R is empty. 114.75/31.09 The argument filtering Pi contains the following mapping: 114.75/31.09 .(x1, x2) = .(x1, x2) 114.75/31.09 114.75/31.09 t = t 114.75/31.09 114.75/31.09 DELI_IN_GA(x1, x2) = DELI_IN_GA(x1) 114.75/31.09 114.75/31.09 114.75/31.09 We have to consider all (P,R,Pi)-chains 114.75/31.09 ---------------------------------------- 114.75/31.09 114.75/31.09 (17) PiDPToQDPProof (SOUND) 114.75/31.09 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 114.75/31.09 ---------------------------------------- 114.75/31.09 114.75/31.09 (18) 114.75/31.09 Obligation: 114.75/31.09 Q DP problem: 114.75/31.09 The TRS P consists of the following rules: 114.75/31.09 114.75/31.09 DELI_IN_GA(.(t, X1)) -> DELI_IN_GA(X1) 114.75/31.09 114.75/31.09 R is empty. 114.75/31.09 Q is empty. 114.75/31.09 We have to consider all (P,Q,R)-chains. 114.75/31.09 ---------------------------------------- 114.75/31.09 114.75/31.09 (19) QDPSizeChangeProof (EQUIVALENT) 114.75/31.09 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. 114.75/31.09 114.75/31.09 From the DPs we obtained the following set of size-change graphs: 114.75/31.09 *DELI_IN_GA(.(t, X1)) -> DELI_IN_GA(X1) 114.75/31.09 The graph contains the following edges 1 > 1 114.75/31.09 114.75/31.09 114.75/31.09 ---------------------------------------- 114.75/31.09 114.75/31.09 (20) 114.75/31.09 YES 114.75/31.09 114.75/31.09 ---------------------------------------- 114.75/31.09 114.75/31.09 (21) 114.75/31.09 Obligation: 114.75/31.09 Pi DP problem: 114.75/31.09 The TRS P consists of the following rules: 114.75/31.09 114.75/31.09 MAXC_IN_GA(.(f, X1), X2) -> MAXC_IN_GA(X1, X2) 114.75/31.09 MAXC_IN_GA(.(t, X1), X2) -> MAXC_IN_GA(X1, X2) 114.75/31.09 114.75/31.09 The TRS R consists of the following rules: 114.75/31.09 114.75/31.09 delcB_in_ga(t, []) -> delcB_out_ga(t, []) 114.75/31.09 delcB_in_ga(f, []) -> delcB_out_ga(f, []) 114.75/31.09 maxcC_in_gg([], t) -> maxcC_out_gg([], t) 114.75/31.09 maxcC_in_gg(.(t, X1), X2) -> U48_gg(X1, X2, maxcC_in_gg(X1, X2)) 114.75/31.09 maxcC_in_gg(.(f, X1), X2) -> U49_gg(X1, X2, maxcC_in_gg(X1, X2)) 114.75/31.09 U49_gg(X1, X2, maxcC_out_gg(X1, X2)) -> maxcC_out_gg(.(f, X1), X2) 114.75/31.09 U48_gg(X1, X2, maxcC_out_gg(X1, X2)) -> maxcC_out_gg(.(t, X1), X2) 114.75/31.09 maxcC_in_ga([], t) -> maxcC_out_ga([], t) 114.75/31.09 maxcC_in_ga(.(t, X1), X2) -> U48_ga(X1, X2, maxcC_in_ga(X1, X2)) 114.75/31.09 maxcC_in_ga(.(f, X1), X2) -> U49_ga(X1, X2, maxcC_in_ga(X1, X2)) 114.75/31.09 U49_ga(X1, X2, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(f, X1), X2) 114.75/31.09 U48_ga(X1, X2, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(t, X1), X2) 114.75/31.09 delcD_in_gga(t, X1, .(t, X1)) -> delcD_out_gga(t, X1, .(t, X1)) 114.75/31.09 delcD_in_gga(f, X1, .(t, .(t, X2))) -> U54_gga(X1, X2, delcI_in_ga(X1, X2)) 114.75/31.09 delcI_in_ga([], []) -> delcI_out_ga([], []) 114.75/31.09 delcI_in_ga(.(f, X1), X1) -> delcI_out_ga(.(f, X1), X1) 114.75/31.09 delcI_in_ga(.(t, X1), .(t, X2)) -> U50_ga(X1, X2, delcI_in_ga(X1, X2)) 114.75/31.09 U50_ga(X1, X2, delcI_out_ga(X1, X2)) -> delcI_out_ga(.(t, X1), .(t, X2)) 114.75/31.09 U54_gga(X1, X2, delcI_out_ga(X1, X2)) -> delcD_out_gga(f, X1, .(t, .(t, X2))) 114.75/31.09 delcE_in_gga(t, X1, .(f, X1)) -> delcE_out_gga(t, X1, .(f, X1)) 114.75/31.09 delcE_in_gga(f, X1, .(t, X2)) -> U55_gga(X1, X2, delcI_in_ga(.(f, X1), X2)) 114.75/31.09 U55_gga(X1, X2, delcI_out_ga(.(f, X1), X2)) -> delcE_out_gga(f, X1, .(t, X2)) 114.75/31.09 maxcF_in_gg([], f) -> maxcF_out_gg([], f) 114.75/31.09 maxcF_in_gg(.(f, X1), X2) -> U51_gg(X1, X2, maxcF_in_gg(X1, X2)) 114.75/31.09 maxcF_in_gg(.(t, X1), X2) -> U52_gg(X1, X2, maxcC_in_gg(X1, X2)) 114.75/31.09 U52_gg(X1, X2, maxcC_out_gg(X1, X2)) -> maxcF_out_gg(.(t, X1), X2) 114.75/31.09 U51_gg(X1, X2, maxcF_out_gg(X1, X2)) -> maxcF_out_gg(.(f, X1), X2) 114.75/31.09 maxcF_in_ga([], f) -> maxcF_out_ga([], f) 114.75/31.09 maxcF_in_ga(.(f, X1), X2) -> U51_ga(X1, X2, maxcF_in_ga(X1, X2)) 114.75/31.09 maxcF_in_ga(.(t, X1), X2) -> U52_ga(X1, X2, maxcC_in_ga(X1, X2)) 114.75/31.09 U52_ga(X1, X2, maxcC_out_ga(X1, X2)) -> maxcF_out_ga(.(t, X1), X2) 114.75/31.09 U51_ga(X1, X2, maxcF_out_ga(X1, X2)) -> maxcF_out_ga(.(f, X1), X2) 114.75/31.09 delcG_in_gga(f, X1, .(f, X1)) -> delcG_out_gga(f, X1, .(f, X1)) 114.75/31.09 delcG_in_gga(t, X1, .(f, .(f, X2))) -> U56_gga(X1, X2, delcJ_in_ga(X1, X2)) 114.75/31.09 delcJ_in_ga([], []) -> delcJ_out_ga([], []) 114.75/31.09 delcJ_in_ga(.(t, X1), X1) -> delcJ_out_ga(.(t, X1), X1) 114.75/31.09 delcJ_in_ga(.(f, X1), .(f, X2)) -> U53_ga(X1, X2, delcJ_in_ga(X1, X2)) 114.75/31.09 U53_ga(X1, X2, delcJ_out_ga(X1, X2)) -> delcJ_out_ga(.(f, X1), .(f, X2)) 114.75/31.09 U56_gga(X1, X2, delcJ_out_ga(X1, X2)) -> delcG_out_gga(t, X1, .(f, .(f, X2))) 114.75/31.09 delcH_in_gga(f, X1, .(t, X1)) -> delcH_out_gga(f, X1, .(t, X1)) 114.75/31.09 delcH_in_gga(t, X1, .(f, X2)) -> U57_gga(X1, X2, delcJ_in_ga(.(t, X1), X2)) 114.75/31.09 U57_gga(X1, X2, delcJ_out_ga(.(t, X1), X2)) -> delcH_out_gga(t, X1, .(f, X2)) 114.75/31.09 114.75/31.09 The argument filtering Pi contains the following mapping: 114.75/31.09 .(x1, x2) = .(x1, x2) 114.75/31.09 114.75/31.09 [] = [] 114.75/31.09 114.75/31.09 delcB_in_ga(x1, x2) = delcB_in_ga(x1) 114.75/31.09 114.75/31.09 t = t 114.75/31.09 114.75/31.09 delcB_out_ga(x1, x2) = delcB_out_ga(x1, x2) 114.75/31.09 114.75/31.09 f = f 114.75/31.09 114.75/31.09 maxcC_in_gg(x1, x2) = maxcC_in_gg(x1, x2) 114.75/31.09 114.75/31.09 maxcC_out_gg(x1, x2) = maxcC_out_gg(x1, x2) 114.75/31.09 114.75/31.09 U48_gg(x1, x2, x3) = U48_gg(x1, x2, x3) 114.75/31.09 114.75/31.09 U49_gg(x1, x2, x3) = U49_gg(x1, x2, x3) 114.75/31.09 114.75/31.09 maxcC_in_ga(x1, x2) = maxcC_in_ga(x1) 114.75/31.09 114.75/31.09 maxcC_out_ga(x1, x2) = maxcC_out_ga(x1, x2) 114.75/31.09 114.75/31.09 U48_ga(x1, x2, x3) = U48_ga(x1, x3) 114.75/31.09 114.75/31.09 U49_ga(x1, x2, x3) = U49_ga(x1, x3) 114.75/31.09 114.75/31.09 delcD_in_gga(x1, x2, x3) = delcD_in_gga(x1, x2) 114.75/31.09 114.75/31.09 delcD_out_gga(x1, x2, x3) = delcD_out_gga(x1, x2, x3) 114.75/31.09 114.75/31.09 U54_gga(x1, x2, x3) = U54_gga(x1, x3) 114.75/31.09 114.75/31.09 delcI_in_ga(x1, x2) = delcI_in_ga(x1) 114.75/31.09 114.75/31.09 delcI_out_ga(x1, x2) = delcI_out_ga(x1, x2) 114.75/31.09 114.75/31.09 U50_ga(x1, x2, x3) = U50_ga(x1, x3) 114.75/31.09 114.75/31.09 delcE_in_gga(x1, x2, x3) = delcE_in_gga(x1, x2) 114.75/31.09 114.75/31.09 delcE_out_gga(x1, x2, x3) = delcE_out_gga(x1, x2, x3) 114.75/31.09 114.75/31.09 U55_gga(x1, x2, x3) = U55_gga(x1, x3) 114.75/31.09 114.75/31.09 maxcF_in_gg(x1, x2) = maxcF_in_gg(x1, x2) 114.75/31.09 114.75/31.09 maxcF_out_gg(x1, x2) = maxcF_out_gg(x1, x2) 114.75/31.09 114.75/31.09 U51_gg(x1, x2, x3) = U51_gg(x1, x2, x3) 114.75/31.09 114.75/31.09 U52_gg(x1, x2, x3) = U52_gg(x1, x2, x3) 114.75/31.09 114.75/31.09 maxcF_in_ga(x1, x2) = maxcF_in_ga(x1) 114.75/31.09 114.75/31.09 maxcF_out_ga(x1, x2) = maxcF_out_ga(x1, x2) 114.75/31.09 114.75/31.09 U51_ga(x1, x2, x3) = U51_ga(x1, x3) 114.75/31.09 114.75/31.09 U52_ga(x1, x2, x3) = U52_ga(x1, x3) 114.75/31.09 114.75/31.09 delcG_in_gga(x1, x2, x3) = delcG_in_gga(x1, x2) 114.75/31.09 114.75/31.09 delcG_out_gga(x1, x2, x3) = delcG_out_gga(x1, x2, x3) 114.75/31.09 114.75/31.09 U56_gga(x1, x2, x3) = U56_gga(x1, x3) 114.75/31.09 114.75/31.09 delcJ_in_ga(x1, x2) = delcJ_in_ga(x1) 114.75/31.09 114.75/31.09 delcJ_out_ga(x1, x2) = delcJ_out_ga(x1, x2) 114.75/31.09 114.75/31.09 U53_ga(x1, x2, x3) = U53_ga(x1, x3) 114.75/31.09 114.75/31.09 delcH_in_gga(x1, x2, x3) = delcH_in_gga(x1, x2) 114.75/31.09 114.75/31.09 delcH_out_gga(x1, x2, x3) = delcH_out_gga(x1, x2, x3) 114.75/31.09 114.75/31.09 U57_gga(x1, x2, x3) = U57_gga(x1, x3) 114.75/31.09 114.75/31.09 MAXC_IN_GA(x1, x2) = MAXC_IN_GA(x1) 114.75/31.09 114.75/31.09 114.75/31.09 We have to consider all (P,R,Pi)-chains 114.75/31.09 ---------------------------------------- 114.75/31.09 114.75/31.09 (22) UsableRulesProof (EQUIVALENT) 114.75/31.09 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 114.75/31.09 ---------------------------------------- 114.75/31.09 114.75/31.09 (23) 114.75/31.09 Obligation: 114.75/31.09 Pi DP problem: 114.75/31.09 The TRS P consists of the following rules: 114.75/31.09 114.75/31.09 MAXC_IN_GA(.(f, X1), X2) -> MAXC_IN_GA(X1, X2) 114.75/31.09 MAXC_IN_GA(.(t, X1), X2) -> MAXC_IN_GA(X1, X2) 114.75/31.09 114.75/31.09 R is empty. 114.75/31.09 The argument filtering Pi contains the following mapping: 114.75/31.09 .(x1, x2) = .(x1, x2) 114.75/31.09 114.75/31.09 t = t 114.75/31.09 114.75/31.09 f = f 114.75/31.09 114.75/31.09 MAXC_IN_GA(x1, x2) = MAXC_IN_GA(x1) 114.75/31.09 114.75/31.09 114.75/31.09 We have to consider all (P,R,Pi)-chains 114.75/31.09 ---------------------------------------- 114.75/31.09 114.75/31.09 (24) PiDPToQDPProof (SOUND) 114.75/31.09 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 114.75/31.09 ---------------------------------------- 114.75/31.09 114.75/31.09 (25) 114.75/31.09 Obligation: 114.75/31.09 Q DP problem: 114.75/31.09 The TRS P consists of the following rules: 114.75/31.09 114.75/31.09 MAXC_IN_GA(.(f, X1)) -> MAXC_IN_GA(X1) 114.75/31.09 MAXC_IN_GA(.(t, X1)) -> MAXC_IN_GA(X1) 114.75/31.09 114.75/31.09 R is empty. 114.75/31.09 Q is empty. 114.75/31.09 We have to consider all (P,Q,R)-chains. 114.75/31.09 ---------------------------------------- 114.75/31.09 114.75/31.09 (26) QDPSizeChangeProof (EQUIVALENT) 114.75/31.09 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. 114.75/31.09 114.75/31.09 From the DPs we obtained the following set of size-change graphs: 114.75/31.09 *MAXC_IN_GA(.(f, X1)) -> MAXC_IN_GA(X1) 114.75/31.09 The graph contains the following edges 1 > 1 114.75/31.09 114.75/31.09 114.75/31.09 *MAXC_IN_GA(.(t, X1)) -> MAXC_IN_GA(X1) 114.75/31.09 The graph contains the following edges 1 > 1 114.75/31.09 114.75/31.09 114.75/31.09 ---------------------------------------- 114.75/31.09 114.75/31.09 (27) 114.75/31.09 YES 114.75/31.09 114.75/31.09 ---------------------------------------- 114.75/31.09 114.75/31.09 (28) 114.75/31.09 Obligation: 114.75/31.09 Pi DP problem: 114.75/31.09 The TRS P consists of the following rules: 114.75/31.09 114.75/31.09 MAXF_IN_GA(.(f, X1), X2) -> MAXF_IN_GA(X1, X2) 114.75/31.09 114.75/31.09 The TRS R consists of the following rules: 114.75/31.09 114.75/31.09 delcB_in_ga(t, []) -> delcB_out_ga(t, []) 114.75/31.09 delcB_in_ga(f, []) -> delcB_out_ga(f, []) 114.75/31.09 maxcC_in_gg([], t) -> maxcC_out_gg([], t) 114.75/31.09 maxcC_in_gg(.(t, X1), X2) -> U48_gg(X1, X2, maxcC_in_gg(X1, X2)) 114.75/31.09 maxcC_in_gg(.(f, X1), X2) -> U49_gg(X1, X2, maxcC_in_gg(X1, X2)) 114.75/31.09 U49_gg(X1, X2, maxcC_out_gg(X1, X2)) -> maxcC_out_gg(.(f, X1), X2) 114.75/31.09 U48_gg(X1, X2, maxcC_out_gg(X1, X2)) -> maxcC_out_gg(.(t, X1), X2) 114.75/31.09 maxcC_in_ga([], t) -> maxcC_out_ga([], t) 114.75/31.09 maxcC_in_ga(.(t, X1), X2) -> U48_ga(X1, X2, maxcC_in_ga(X1, X2)) 114.75/31.09 maxcC_in_ga(.(f, X1), X2) -> U49_ga(X1, X2, maxcC_in_ga(X1, X2)) 114.75/31.09 U49_ga(X1, X2, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(f, X1), X2) 114.75/31.09 U48_ga(X1, X2, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(t, X1), X2) 114.75/31.09 delcD_in_gga(t, X1, .(t, X1)) -> delcD_out_gga(t, X1, .(t, X1)) 114.75/31.09 delcD_in_gga(f, X1, .(t, .(t, X2))) -> U54_gga(X1, X2, delcI_in_ga(X1, X2)) 114.75/31.09 delcI_in_ga([], []) -> delcI_out_ga([], []) 114.75/31.09 delcI_in_ga(.(f, X1), X1) -> delcI_out_ga(.(f, X1), X1) 114.75/31.09 delcI_in_ga(.(t, X1), .(t, X2)) -> U50_ga(X1, X2, delcI_in_ga(X1, X2)) 114.75/31.09 U50_ga(X1, X2, delcI_out_ga(X1, X2)) -> delcI_out_ga(.(t, X1), .(t, X2)) 114.75/31.09 U54_gga(X1, X2, delcI_out_ga(X1, X2)) -> delcD_out_gga(f, X1, .(t, .(t, X2))) 114.75/31.09 delcE_in_gga(t, X1, .(f, X1)) -> delcE_out_gga(t, X1, .(f, X1)) 114.75/31.09 delcE_in_gga(f, X1, .(t, X2)) -> U55_gga(X1, X2, delcI_in_ga(.(f, X1), X2)) 114.75/31.09 U55_gga(X1, X2, delcI_out_ga(.(f, X1), X2)) -> delcE_out_gga(f, X1, .(t, X2)) 114.75/31.09 maxcF_in_gg([], f) -> maxcF_out_gg([], f) 114.75/31.09 maxcF_in_gg(.(f, X1), X2) -> U51_gg(X1, X2, maxcF_in_gg(X1, X2)) 114.75/31.09 maxcF_in_gg(.(t, X1), X2) -> U52_gg(X1, X2, maxcC_in_gg(X1, X2)) 114.75/31.09 U52_gg(X1, X2, maxcC_out_gg(X1, X2)) -> maxcF_out_gg(.(t, X1), X2) 114.75/31.09 U51_gg(X1, X2, maxcF_out_gg(X1, X2)) -> maxcF_out_gg(.(f, X1), X2) 114.75/31.09 maxcF_in_ga([], f) -> maxcF_out_ga([], f) 114.75/31.09 maxcF_in_ga(.(f, X1), X2) -> U51_ga(X1, X2, maxcF_in_ga(X1, X2)) 114.75/31.09 maxcF_in_ga(.(t, X1), X2) -> U52_ga(X1, X2, maxcC_in_ga(X1, X2)) 114.75/31.09 U52_ga(X1, X2, maxcC_out_ga(X1, X2)) -> maxcF_out_ga(.(t, X1), X2) 114.75/31.09 U51_ga(X1, X2, maxcF_out_ga(X1, X2)) -> maxcF_out_ga(.(f, X1), X2) 114.75/31.09 delcG_in_gga(f, X1, .(f, X1)) -> delcG_out_gga(f, X1, .(f, X1)) 114.75/31.09 delcG_in_gga(t, X1, .(f, .(f, X2))) -> U56_gga(X1, X2, delcJ_in_ga(X1, X2)) 114.75/31.09 delcJ_in_ga([], []) -> delcJ_out_ga([], []) 114.75/31.09 delcJ_in_ga(.(t, X1), X1) -> delcJ_out_ga(.(t, X1), X1) 114.75/31.09 delcJ_in_ga(.(f, X1), .(f, X2)) -> U53_ga(X1, X2, delcJ_in_ga(X1, X2)) 114.75/31.09 U53_ga(X1, X2, delcJ_out_ga(X1, X2)) -> delcJ_out_ga(.(f, X1), .(f, X2)) 114.75/31.09 U56_gga(X1, X2, delcJ_out_ga(X1, X2)) -> delcG_out_gga(t, X1, .(f, .(f, X2))) 114.75/31.09 delcH_in_gga(f, X1, .(t, X1)) -> delcH_out_gga(f, X1, .(t, X1)) 114.75/31.09 delcH_in_gga(t, X1, .(f, X2)) -> U57_gga(X1, X2, delcJ_in_ga(.(t, X1), X2)) 114.75/31.09 U57_gga(X1, X2, delcJ_out_ga(.(t, X1), X2)) -> delcH_out_gga(t, X1, .(f, X2)) 114.75/31.09 114.75/31.09 The argument filtering Pi contains the following mapping: 114.75/31.09 .(x1, x2) = .(x1, x2) 114.75/31.09 114.75/31.09 [] = [] 114.75/31.09 114.75/31.09 delcB_in_ga(x1, x2) = delcB_in_ga(x1) 114.75/31.09 114.75/31.09 t = t 114.75/31.09 114.75/31.09 delcB_out_ga(x1, x2) = delcB_out_ga(x1, x2) 114.75/31.09 114.75/31.09 f = f 114.75/31.09 114.75/31.09 maxcC_in_gg(x1, x2) = maxcC_in_gg(x1, x2) 114.75/31.09 114.75/31.09 maxcC_out_gg(x1, x2) = maxcC_out_gg(x1, x2) 114.75/31.09 114.75/31.09 U48_gg(x1, x2, x3) = U48_gg(x1, x2, x3) 114.75/31.09 114.75/31.09 U49_gg(x1, x2, x3) = U49_gg(x1, x2, x3) 114.75/31.09 114.75/31.09 maxcC_in_ga(x1, x2) = maxcC_in_ga(x1) 114.75/31.09 114.75/31.09 maxcC_out_ga(x1, x2) = maxcC_out_ga(x1, x2) 114.75/31.09 114.75/31.09 U48_ga(x1, x2, x3) = U48_ga(x1, x3) 114.75/31.09 114.75/31.09 U49_ga(x1, x2, x3) = U49_ga(x1, x3) 114.75/31.09 114.75/31.09 delcD_in_gga(x1, x2, x3) = delcD_in_gga(x1, x2) 114.75/31.09 114.75/31.09 delcD_out_gga(x1, x2, x3) = delcD_out_gga(x1, x2, x3) 114.75/31.09 114.75/31.09 U54_gga(x1, x2, x3) = U54_gga(x1, x3) 114.75/31.09 114.75/31.09 delcI_in_ga(x1, x2) = delcI_in_ga(x1) 114.75/31.09 114.75/31.09 delcI_out_ga(x1, x2) = delcI_out_ga(x1, x2) 114.75/31.09 114.75/31.09 U50_ga(x1, x2, x3) = U50_ga(x1, x3) 114.75/31.09 114.75/31.09 delcE_in_gga(x1, x2, x3) = delcE_in_gga(x1, x2) 114.75/31.09 114.75/31.09 delcE_out_gga(x1, x2, x3) = delcE_out_gga(x1, x2, x3) 114.75/31.09 114.75/31.09 U55_gga(x1, x2, x3) = U55_gga(x1, x3) 114.75/31.09 114.75/31.09 maxcF_in_gg(x1, x2) = maxcF_in_gg(x1, x2) 114.75/31.09 114.75/31.09 maxcF_out_gg(x1, x2) = maxcF_out_gg(x1, x2) 114.75/31.09 114.75/31.09 U51_gg(x1, x2, x3) = U51_gg(x1, x2, x3) 114.75/31.09 114.75/31.09 U52_gg(x1, x2, x3) = U52_gg(x1, x2, x3) 114.75/31.09 114.75/31.09 maxcF_in_ga(x1, x2) = maxcF_in_ga(x1) 114.75/31.09 114.75/31.09 maxcF_out_ga(x1, x2) = maxcF_out_ga(x1, x2) 114.75/31.09 114.75/31.09 U51_ga(x1, x2, x3) = U51_ga(x1, x3) 114.75/31.09 114.75/31.09 U52_ga(x1, x2, x3) = U52_ga(x1, x3) 114.75/31.09 114.75/31.09 delcG_in_gga(x1, x2, x3) = delcG_in_gga(x1, x2) 114.75/31.09 114.75/31.09 delcG_out_gga(x1, x2, x3) = delcG_out_gga(x1, x2, x3) 114.75/31.09 114.75/31.09 U56_gga(x1, x2, x3) = U56_gga(x1, x3) 114.75/31.09 114.75/31.09 delcJ_in_ga(x1, x2) = delcJ_in_ga(x1) 114.75/31.09 114.75/31.09 delcJ_out_ga(x1, x2) = delcJ_out_ga(x1, x2) 114.75/31.09 114.75/31.09 U53_ga(x1, x2, x3) = U53_ga(x1, x3) 114.75/31.09 114.75/31.09 delcH_in_gga(x1, x2, x3) = delcH_in_gga(x1, x2) 114.75/31.09 114.75/31.09 delcH_out_gga(x1, x2, x3) = delcH_out_gga(x1, x2, x3) 114.75/31.09 114.75/31.09 U57_gga(x1, x2, x3) = U57_gga(x1, x3) 114.75/31.09 114.75/31.09 MAXF_IN_GA(x1, x2) = MAXF_IN_GA(x1) 114.75/31.09 114.75/31.09 114.75/31.09 We have to consider all (P,R,Pi)-chains 114.75/31.09 ---------------------------------------- 114.75/31.09 114.75/31.09 (29) UsableRulesProof (EQUIVALENT) 114.75/31.09 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 114.75/31.09 ---------------------------------------- 114.75/31.09 114.75/31.09 (30) 114.75/31.09 Obligation: 114.75/31.09 Pi DP problem: 114.75/31.09 The TRS P consists of the following rules: 114.75/31.09 114.75/31.09 MAXF_IN_GA(.(f, X1), X2) -> MAXF_IN_GA(X1, X2) 114.75/31.09 114.75/31.09 R is empty. 114.75/31.09 The argument filtering Pi contains the following mapping: 114.75/31.09 .(x1, x2) = .(x1, x2) 114.75/31.09 114.75/31.09 f = f 114.75/31.09 114.75/31.09 MAXF_IN_GA(x1, x2) = MAXF_IN_GA(x1) 114.75/31.09 114.75/31.09 114.75/31.09 We have to consider all (P,R,Pi)-chains 114.75/31.09 ---------------------------------------- 114.75/31.09 114.75/31.09 (31) PiDPToQDPProof (SOUND) 114.75/31.09 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 114.75/31.09 ---------------------------------------- 114.75/31.09 114.75/31.09 (32) 114.75/31.09 Obligation: 114.75/31.09 Q DP problem: 114.75/31.09 The TRS P consists of the following rules: 114.75/31.09 114.75/31.09 MAXF_IN_GA(.(f, X1)) -> MAXF_IN_GA(X1) 114.75/31.09 114.75/31.09 R is empty. 114.75/31.09 Q is empty. 114.75/31.09 We have to consider all (P,Q,R)-chains. 114.75/31.09 ---------------------------------------- 114.75/31.09 114.75/31.09 (33) QDPSizeChangeProof (EQUIVALENT) 114.75/31.09 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. 114.75/31.09 114.75/31.09 From the DPs we obtained the following set of size-change graphs: 114.75/31.09 *MAXF_IN_GA(.(f, X1)) -> MAXF_IN_GA(X1) 114.75/31.09 The graph contains the following edges 1 > 1 114.75/31.09 114.75/31.09 114.75/31.09 ---------------------------------------- 114.75/31.09 114.75/31.09 (34) 114.75/31.09 YES 114.75/31.09 114.75/31.09 ---------------------------------------- 114.75/31.09 114.75/31.09 (35) 114.75/31.09 Obligation: 114.75/31.09 Pi DP problem: 114.75/31.09 The TRS P consists of the following rules: 114.75/31.09 114.75/31.09 U7_GA(X1, X2, delcB_out_ga(X1, X3)) -> MAXSORTA_IN_GA(X3, X2) 114.75/31.09 MAXSORTA_IN_GA(.(X1, []), .(X1, X2)) -> U7_GA(X1, X2, delcB_in_ga(X1, X3)) 114.75/31.09 MAXSORTA_IN_GA(.(t, .(t, X1)), .(X2, X3)) -> U12_GA(X1, X2, X3, maxcC_in_ga(X1, X2)) 114.75/31.09 U12_GA(X1, X2, X3, maxcC_out_ga(X1, X2)) -> U13_GA(X1, X2, X3, delcD_in_gga(X2, X1, X4)) 114.75/31.09 U13_GA(X1, X2, X3, delcD_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4, X3) 114.75/31.09 MAXSORTA_IN_GA(.(t, .(f, X1)), .(X2, X3)) -> U18_GA(X1, X2, X3, maxcC_in_ga(X1, X2)) 114.75/31.09 U18_GA(X1, X2, X3, maxcC_out_ga(X1, X2)) -> U19_GA(X1, X2, X3, delcE_in_gga(X2, X1, X4)) 114.75/31.09 U19_GA(X1, X2, X3, delcE_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4, X3) 114.75/31.09 MAXSORTA_IN_GA(.(f, .(f, X1)), .(X2, X3)) -> U24_GA(X1, X2, X3, maxcF_in_ga(X1, X2)) 114.75/31.09 U24_GA(X1, X2, X3, maxcF_out_ga(X1, X2)) -> U25_GA(X1, X2, X3, delcG_in_gga(X2, X1, X4)) 114.75/31.09 U25_GA(X1, X2, X3, delcG_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4, X3) 114.75/31.09 MAXSORTA_IN_GA(.(f, .(t, X1)), .(X2, X3)) -> U30_GA(X1, X2, X3, maxcC_in_ga(X1, X2)) 114.75/31.09 U30_GA(X1, X2, X3, maxcC_out_ga(X1, X2)) -> U31_GA(X1, X2, X3, delcH_in_gga(X2, X1, X4)) 114.75/31.09 U31_GA(X1, X2, X3, delcH_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4, X3) 114.75/31.09 114.75/31.09 The TRS R consists of the following rules: 114.75/31.09 114.75/31.09 delcB_in_ga(t, []) -> delcB_out_ga(t, []) 114.75/31.09 delcB_in_ga(f, []) -> delcB_out_ga(f, []) 114.75/31.09 maxcC_in_gg([], t) -> maxcC_out_gg([], t) 114.75/31.09 maxcC_in_gg(.(t, X1), X2) -> U48_gg(X1, X2, maxcC_in_gg(X1, X2)) 114.75/31.09 maxcC_in_gg(.(f, X1), X2) -> U49_gg(X1, X2, maxcC_in_gg(X1, X2)) 114.75/31.09 U49_gg(X1, X2, maxcC_out_gg(X1, X2)) -> maxcC_out_gg(.(f, X1), X2) 114.75/31.09 U48_gg(X1, X2, maxcC_out_gg(X1, X2)) -> maxcC_out_gg(.(t, X1), X2) 114.75/31.09 maxcC_in_ga([], t) -> maxcC_out_ga([], t) 114.75/31.09 maxcC_in_ga(.(t, X1), X2) -> U48_ga(X1, X2, maxcC_in_ga(X1, X2)) 114.75/31.09 maxcC_in_ga(.(f, X1), X2) -> U49_ga(X1, X2, maxcC_in_ga(X1, X2)) 114.75/31.09 U49_ga(X1, X2, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(f, X1), X2) 114.75/31.09 U48_ga(X1, X2, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(t, X1), X2) 114.75/31.09 delcD_in_gga(t, X1, .(t, X1)) -> delcD_out_gga(t, X1, .(t, X1)) 114.75/31.09 delcD_in_gga(f, X1, .(t, .(t, X2))) -> U54_gga(X1, X2, delcI_in_ga(X1, X2)) 114.75/31.09 delcI_in_ga([], []) -> delcI_out_ga([], []) 114.75/31.09 delcI_in_ga(.(f, X1), X1) -> delcI_out_ga(.(f, X1), X1) 114.75/31.09 delcI_in_ga(.(t, X1), .(t, X2)) -> U50_ga(X1, X2, delcI_in_ga(X1, X2)) 114.75/31.09 U50_ga(X1, X2, delcI_out_ga(X1, X2)) -> delcI_out_ga(.(t, X1), .(t, X2)) 114.75/31.09 U54_gga(X1, X2, delcI_out_ga(X1, X2)) -> delcD_out_gga(f, X1, .(t, .(t, X2))) 114.75/31.09 delcE_in_gga(t, X1, .(f, X1)) -> delcE_out_gga(t, X1, .(f, X1)) 114.75/31.09 delcE_in_gga(f, X1, .(t, X2)) -> U55_gga(X1, X2, delcI_in_ga(.(f, X1), X2)) 114.75/31.09 U55_gga(X1, X2, delcI_out_ga(.(f, X1), X2)) -> delcE_out_gga(f, X1, .(t, X2)) 114.75/31.09 maxcF_in_gg([], f) -> maxcF_out_gg([], f) 114.75/31.09 maxcF_in_gg(.(f, X1), X2) -> U51_gg(X1, X2, maxcF_in_gg(X1, X2)) 114.75/31.09 maxcF_in_gg(.(t, X1), X2) -> U52_gg(X1, X2, maxcC_in_gg(X1, X2)) 114.75/31.09 U52_gg(X1, X2, maxcC_out_gg(X1, X2)) -> maxcF_out_gg(.(t, X1), X2) 114.75/31.09 U51_gg(X1, X2, maxcF_out_gg(X1, X2)) -> maxcF_out_gg(.(f, X1), X2) 114.75/31.09 maxcF_in_ga([], f) -> maxcF_out_ga([], f) 114.75/31.09 maxcF_in_ga(.(f, X1), X2) -> U51_ga(X1, X2, maxcF_in_ga(X1, X2)) 114.75/31.09 maxcF_in_ga(.(t, X1), X2) -> U52_ga(X1, X2, maxcC_in_ga(X1, X2)) 114.75/31.09 U52_ga(X1, X2, maxcC_out_ga(X1, X2)) -> maxcF_out_ga(.(t, X1), X2) 114.75/31.09 U51_ga(X1, X2, maxcF_out_ga(X1, X2)) -> maxcF_out_ga(.(f, X1), X2) 114.75/31.09 delcG_in_gga(f, X1, .(f, X1)) -> delcG_out_gga(f, X1, .(f, X1)) 114.75/31.09 delcG_in_gga(t, X1, .(f, .(f, X2))) -> U56_gga(X1, X2, delcJ_in_ga(X1, X2)) 114.75/31.09 delcJ_in_ga([], []) -> delcJ_out_ga([], []) 114.75/31.09 delcJ_in_ga(.(t, X1), X1) -> delcJ_out_ga(.(t, X1), X1) 114.75/31.09 delcJ_in_ga(.(f, X1), .(f, X2)) -> U53_ga(X1, X2, delcJ_in_ga(X1, X2)) 114.75/31.09 U53_ga(X1, X2, delcJ_out_ga(X1, X2)) -> delcJ_out_ga(.(f, X1), .(f, X2)) 114.75/31.09 U56_gga(X1, X2, delcJ_out_ga(X1, X2)) -> delcG_out_gga(t, X1, .(f, .(f, X2))) 114.75/31.09 delcH_in_gga(f, X1, .(t, X1)) -> delcH_out_gga(f, X1, .(t, X1)) 114.75/31.09 delcH_in_gga(t, X1, .(f, X2)) -> U57_gga(X1, X2, delcJ_in_ga(.(t, X1), X2)) 114.75/31.09 U57_gga(X1, X2, delcJ_out_ga(.(t, X1), X2)) -> delcH_out_gga(t, X1, .(f, X2)) 114.75/31.09 114.75/31.09 The argument filtering Pi contains the following mapping: 114.75/31.09 .(x1, x2) = .(x1, x2) 114.75/31.09 114.75/31.09 [] = [] 114.75/31.09 114.75/31.09 delcB_in_ga(x1, x2) = delcB_in_ga(x1) 114.75/31.09 114.75/31.09 t = t 114.75/31.09 114.75/31.09 delcB_out_ga(x1, x2) = delcB_out_ga(x1, x2) 114.75/31.09 114.75/31.09 f = f 114.75/31.09 114.75/31.09 maxcC_in_gg(x1, x2) = maxcC_in_gg(x1, x2) 114.75/31.09 114.75/31.09 maxcC_out_gg(x1, x2) = maxcC_out_gg(x1, x2) 114.75/31.09 114.75/31.09 U48_gg(x1, x2, x3) = U48_gg(x1, x2, x3) 114.75/31.09 114.75/31.09 U49_gg(x1, x2, x3) = U49_gg(x1, x2, x3) 114.75/31.09 114.75/31.09 maxcC_in_ga(x1, x2) = maxcC_in_ga(x1) 114.75/31.09 114.75/31.09 maxcC_out_ga(x1, x2) = maxcC_out_ga(x1, x2) 114.75/31.09 114.75/31.09 U48_ga(x1, x2, x3) = U48_ga(x1, x3) 114.75/31.09 114.75/31.09 U49_ga(x1, x2, x3) = U49_ga(x1, x3) 114.75/31.09 114.75/31.09 delcD_in_gga(x1, x2, x3) = delcD_in_gga(x1, x2) 114.75/31.09 114.75/31.09 delcD_out_gga(x1, x2, x3) = delcD_out_gga(x1, x2, x3) 114.75/31.09 114.75/31.09 U54_gga(x1, x2, x3) = U54_gga(x1, x3) 114.75/31.09 114.75/31.09 delcI_in_ga(x1, x2) = delcI_in_ga(x1) 114.75/31.09 114.75/31.09 delcI_out_ga(x1, x2) = delcI_out_ga(x1, x2) 114.75/31.09 114.75/31.09 U50_ga(x1, x2, x3) = U50_ga(x1, x3) 114.75/31.09 114.75/31.09 delcE_in_gga(x1, x2, x3) = delcE_in_gga(x1, x2) 114.75/31.09 114.75/31.09 delcE_out_gga(x1, x2, x3) = delcE_out_gga(x1, x2, x3) 114.75/31.09 114.75/31.09 U55_gga(x1, x2, x3) = U55_gga(x1, x3) 114.75/31.09 114.75/31.09 maxcF_in_gg(x1, x2) = maxcF_in_gg(x1, x2) 114.75/31.09 114.75/31.09 maxcF_out_gg(x1, x2) = maxcF_out_gg(x1, x2) 114.75/31.09 114.75/31.09 U51_gg(x1, x2, x3) = U51_gg(x1, x2, x3) 114.75/31.09 114.75/31.09 U52_gg(x1, x2, x3) = U52_gg(x1, x2, x3) 114.75/31.09 114.75/31.09 maxcF_in_ga(x1, x2) = maxcF_in_ga(x1) 114.75/31.09 114.75/31.09 maxcF_out_ga(x1, x2) = maxcF_out_ga(x1, x2) 114.75/31.09 114.75/31.09 U51_ga(x1, x2, x3) = U51_ga(x1, x3) 114.75/31.09 114.75/31.09 U52_ga(x1, x2, x3) = U52_ga(x1, x3) 114.75/31.09 114.75/31.09 delcG_in_gga(x1, x2, x3) = delcG_in_gga(x1, x2) 114.75/31.09 114.75/31.09 delcG_out_gga(x1, x2, x3) = delcG_out_gga(x1, x2, x3) 114.75/31.09 114.75/31.09 U56_gga(x1, x2, x3) = U56_gga(x1, x3) 114.75/31.09 114.75/31.09 delcJ_in_ga(x1, x2) = delcJ_in_ga(x1) 114.75/31.09 114.75/31.09 delcJ_out_ga(x1, x2) = delcJ_out_ga(x1, x2) 114.75/31.09 114.75/31.09 U53_ga(x1, x2, x3) = U53_ga(x1, x3) 114.75/31.09 114.75/31.09 delcH_in_gga(x1, x2, x3) = delcH_in_gga(x1, x2) 114.75/31.09 114.75/31.09 delcH_out_gga(x1, x2, x3) = delcH_out_gga(x1, x2, x3) 114.75/31.09 114.75/31.09 U57_gga(x1, x2, x3) = U57_gga(x1, x3) 114.75/31.09 114.75/31.09 MAXSORTA_IN_GA(x1, x2) = MAXSORTA_IN_GA(x1) 114.75/31.09 114.75/31.09 U7_GA(x1, x2, x3) = U7_GA(x1, x3) 114.75/31.09 114.75/31.09 U12_GA(x1, x2, x3, x4) = U12_GA(x1, x4) 114.75/31.09 114.75/31.09 U13_GA(x1, x2, x3, x4) = U13_GA(x1, x4) 114.75/31.09 114.75/31.09 U18_GA(x1, x2, x3, x4) = U18_GA(x1, x4) 114.75/31.09 114.75/31.09 U19_GA(x1, x2, x3, x4) = U19_GA(x1, x4) 114.75/31.09 114.75/31.09 U24_GA(x1, x2, x3, x4) = U24_GA(x1, x4) 114.75/31.09 114.75/31.09 U25_GA(x1, x2, x3, x4) = U25_GA(x1, x4) 114.75/31.09 114.75/31.09 U30_GA(x1, x2, x3, x4) = U30_GA(x1, x4) 114.75/31.09 114.75/31.09 U31_GA(x1, x2, x3, x4) = U31_GA(x1, x4) 114.75/31.09 114.75/31.09 114.75/31.09 We have to consider all (P,R,Pi)-chains 114.75/31.09 ---------------------------------------- 114.75/31.09 114.75/31.09 (36) UsableRulesProof (EQUIVALENT) 114.75/31.09 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 114.75/31.09 ---------------------------------------- 114.75/31.09 114.75/31.09 (37) 114.75/31.09 Obligation: 114.75/31.09 Pi DP problem: 114.75/31.09 The TRS P consists of the following rules: 114.75/31.09 114.75/31.09 U7_GA(X1, X2, delcB_out_ga(X1, X3)) -> MAXSORTA_IN_GA(X3, X2) 114.75/31.09 MAXSORTA_IN_GA(.(X1, []), .(X1, X2)) -> U7_GA(X1, X2, delcB_in_ga(X1, X3)) 114.75/31.09 MAXSORTA_IN_GA(.(t, .(t, X1)), .(X2, X3)) -> U12_GA(X1, X2, X3, maxcC_in_ga(X1, X2)) 114.75/31.09 U12_GA(X1, X2, X3, maxcC_out_ga(X1, X2)) -> U13_GA(X1, X2, X3, delcD_in_gga(X2, X1, X4)) 114.75/31.09 U13_GA(X1, X2, X3, delcD_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4, X3) 114.75/31.09 MAXSORTA_IN_GA(.(t, .(f, X1)), .(X2, X3)) -> U18_GA(X1, X2, X3, maxcC_in_ga(X1, X2)) 114.75/31.09 U18_GA(X1, X2, X3, maxcC_out_ga(X1, X2)) -> U19_GA(X1, X2, X3, delcE_in_gga(X2, X1, X4)) 114.75/31.09 U19_GA(X1, X2, X3, delcE_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4, X3) 114.75/31.09 MAXSORTA_IN_GA(.(f, .(f, X1)), .(X2, X3)) -> U24_GA(X1, X2, X3, maxcF_in_ga(X1, X2)) 114.75/31.09 U24_GA(X1, X2, X3, maxcF_out_ga(X1, X2)) -> U25_GA(X1, X2, X3, delcG_in_gga(X2, X1, X4)) 114.75/31.09 U25_GA(X1, X2, X3, delcG_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4, X3) 114.75/31.09 MAXSORTA_IN_GA(.(f, .(t, X1)), .(X2, X3)) -> U30_GA(X1, X2, X3, maxcC_in_ga(X1, X2)) 114.75/31.09 U30_GA(X1, X2, X3, maxcC_out_ga(X1, X2)) -> U31_GA(X1, X2, X3, delcH_in_gga(X2, X1, X4)) 114.75/31.09 U31_GA(X1, X2, X3, delcH_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4, X3) 114.75/31.09 114.75/31.09 The TRS R consists of the following rules: 114.75/31.09 114.75/31.09 delcB_in_ga(t, []) -> delcB_out_ga(t, []) 114.75/31.09 delcB_in_ga(f, []) -> delcB_out_ga(f, []) 114.75/31.09 maxcC_in_ga([], t) -> maxcC_out_ga([], t) 114.75/31.09 maxcC_in_ga(.(t, X1), X2) -> U48_ga(X1, X2, maxcC_in_ga(X1, X2)) 114.75/31.09 maxcC_in_ga(.(f, X1), X2) -> U49_ga(X1, X2, maxcC_in_ga(X1, X2)) 114.75/31.09 delcD_in_gga(t, X1, .(t, X1)) -> delcD_out_gga(t, X1, .(t, X1)) 114.75/31.09 delcD_in_gga(f, X1, .(t, .(t, X2))) -> U54_gga(X1, X2, delcI_in_ga(X1, X2)) 114.75/31.09 delcE_in_gga(t, X1, .(f, X1)) -> delcE_out_gga(t, X1, .(f, X1)) 114.75/31.09 delcE_in_gga(f, X1, .(t, X2)) -> U55_gga(X1, X2, delcI_in_ga(.(f, X1), X2)) 114.75/31.09 maxcF_in_ga([], f) -> maxcF_out_ga([], f) 114.75/31.09 maxcF_in_ga(.(f, X1), X2) -> U51_ga(X1, X2, maxcF_in_ga(X1, X2)) 114.75/31.09 maxcF_in_ga(.(t, X1), X2) -> U52_ga(X1, X2, maxcC_in_ga(X1, X2)) 114.75/31.09 delcG_in_gga(f, X1, .(f, X1)) -> delcG_out_gga(f, X1, .(f, X1)) 114.75/31.09 delcG_in_gga(t, X1, .(f, .(f, X2))) -> U56_gga(X1, X2, delcJ_in_ga(X1, X2)) 114.75/31.09 delcH_in_gga(f, X1, .(t, X1)) -> delcH_out_gga(f, X1, .(t, X1)) 114.75/31.09 delcH_in_gga(t, X1, .(f, X2)) -> U57_gga(X1, X2, delcJ_in_ga(.(t, X1), X2)) 114.75/31.09 U48_ga(X1, X2, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(t, X1), X2) 114.75/31.09 U49_ga(X1, X2, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(f, X1), X2) 114.75/31.09 U54_gga(X1, X2, delcI_out_ga(X1, X2)) -> delcD_out_gga(f, X1, .(t, .(t, X2))) 114.75/31.09 U55_gga(X1, X2, delcI_out_ga(.(f, X1), X2)) -> delcE_out_gga(f, X1, .(t, X2)) 114.75/31.09 U51_ga(X1, X2, maxcF_out_ga(X1, X2)) -> maxcF_out_ga(.(f, X1), X2) 114.75/31.09 U52_ga(X1, X2, maxcC_out_ga(X1, X2)) -> maxcF_out_ga(.(t, X1), X2) 114.75/31.09 U56_gga(X1, X2, delcJ_out_ga(X1, X2)) -> delcG_out_gga(t, X1, .(f, .(f, X2))) 114.75/31.09 U57_gga(X1, X2, delcJ_out_ga(.(t, X1), X2)) -> delcH_out_gga(t, X1, .(f, X2)) 114.75/31.09 delcI_in_ga([], []) -> delcI_out_ga([], []) 114.75/31.09 delcI_in_ga(.(f, X1), X1) -> delcI_out_ga(.(f, X1), X1) 114.75/31.09 delcI_in_ga(.(t, X1), .(t, X2)) -> U50_ga(X1, X2, delcI_in_ga(X1, X2)) 114.75/31.09 delcJ_in_ga([], []) -> delcJ_out_ga([], []) 114.75/31.09 delcJ_in_ga(.(t, X1), X1) -> delcJ_out_ga(.(t, X1), X1) 114.75/31.09 delcJ_in_ga(.(f, X1), .(f, X2)) -> U53_ga(X1, X2, delcJ_in_ga(X1, X2)) 114.75/31.09 U50_ga(X1, X2, delcI_out_ga(X1, X2)) -> delcI_out_ga(.(t, X1), .(t, X2)) 114.75/31.09 U53_ga(X1, X2, delcJ_out_ga(X1, X2)) -> delcJ_out_ga(.(f, X1), .(f, X2)) 114.75/31.09 114.75/31.09 The argument filtering Pi contains the following mapping: 114.75/31.09 .(x1, x2) = .(x1, x2) 114.75/31.09 114.75/31.09 [] = [] 114.75/31.09 114.75/31.09 delcB_in_ga(x1, x2) = delcB_in_ga(x1) 114.75/31.09 114.75/31.09 t = t 114.75/31.09 114.75/31.09 delcB_out_ga(x1, x2) = delcB_out_ga(x1, x2) 114.75/31.09 114.75/31.09 f = f 114.75/31.09 114.75/31.09 maxcC_in_ga(x1, x2) = maxcC_in_ga(x1) 114.75/31.09 114.75/31.09 maxcC_out_ga(x1, x2) = maxcC_out_ga(x1, x2) 114.75/31.09 114.75/31.09 U48_ga(x1, x2, x3) = U48_ga(x1, x3) 114.75/31.09 114.75/31.09 U49_ga(x1, x2, x3) = U49_ga(x1, x3) 114.75/31.09 114.75/31.09 delcD_in_gga(x1, x2, x3) = delcD_in_gga(x1, x2) 114.75/31.09 114.75/31.09 delcD_out_gga(x1, x2, x3) = delcD_out_gga(x1, x2, x3) 114.75/31.09 114.75/31.09 U54_gga(x1, x2, x3) = U54_gga(x1, x3) 114.75/31.09 114.75/31.09 delcI_in_ga(x1, x2) = delcI_in_ga(x1) 114.75/31.09 114.75/31.09 delcI_out_ga(x1, x2) = delcI_out_ga(x1, x2) 114.75/31.09 114.75/31.09 U50_ga(x1, x2, x3) = U50_ga(x1, x3) 114.75/31.09 114.75/31.09 delcE_in_gga(x1, x2, x3) = delcE_in_gga(x1, x2) 114.75/31.09 114.75/31.09 delcE_out_gga(x1, x2, x3) = delcE_out_gga(x1, x2, x3) 114.75/31.09 114.75/31.09 U55_gga(x1, x2, x3) = U55_gga(x1, x3) 114.75/31.09 114.75/31.09 maxcF_in_ga(x1, x2) = maxcF_in_ga(x1) 114.75/31.09 114.75/31.09 maxcF_out_ga(x1, x2) = maxcF_out_ga(x1, x2) 114.75/31.09 114.75/31.09 U51_ga(x1, x2, x3) = U51_ga(x1, x3) 114.75/31.09 114.75/31.09 U52_ga(x1, x2, x3) = U52_ga(x1, x3) 114.75/31.09 114.75/31.09 delcG_in_gga(x1, x2, x3) = delcG_in_gga(x1, x2) 114.75/31.09 114.75/31.09 delcG_out_gga(x1, x2, x3) = delcG_out_gga(x1, x2, x3) 114.75/31.09 114.75/31.09 U56_gga(x1, x2, x3) = U56_gga(x1, x3) 114.75/31.09 114.75/31.09 delcJ_in_ga(x1, x2) = delcJ_in_ga(x1) 114.75/31.09 114.75/31.09 delcJ_out_ga(x1, x2) = delcJ_out_ga(x1, x2) 114.75/31.09 114.75/31.09 U53_ga(x1, x2, x3) = U53_ga(x1, x3) 114.75/31.09 114.75/31.09 delcH_in_gga(x1, x2, x3) = delcH_in_gga(x1, x2) 114.75/31.09 114.75/31.09 delcH_out_gga(x1, x2, x3) = delcH_out_gga(x1, x2, x3) 114.75/31.09 114.75/31.09 U57_gga(x1, x2, x3) = U57_gga(x1, x3) 114.75/31.09 114.75/31.09 MAXSORTA_IN_GA(x1, x2) = MAXSORTA_IN_GA(x1) 114.75/31.09 114.75/31.09 U7_GA(x1, x2, x3) = U7_GA(x1, x3) 114.75/31.09 114.75/31.09 U12_GA(x1, x2, x3, x4) = U12_GA(x1, x4) 114.75/31.09 114.75/31.09 U13_GA(x1, x2, x3, x4) = U13_GA(x1, x4) 114.75/31.09 114.75/31.09 U18_GA(x1, x2, x3, x4) = U18_GA(x1, x4) 114.75/31.09 114.75/31.09 U19_GA(x1, x2, x3, x4) = U19_GA(x1, x4) 114.75/31.09 114.75/31.09 U24_GA(x1, x2, x3, x4) = U24_GA(x1, x4) 114.75/31.09 114.75/31.09 U25_GA(x1, x2, x3, x4) = U25_GA(x1, x4) 114.75/31.09 114.75/31.09 U30_GA(x1, x2, x3, x4) = U30_GA(x1, x4) 114.75/31.09 114.75/31.09 U31_GA(x1, x2, x3, x4) = U31_GA(x1, x4) 114.75/31.09 114.75/31.09 114.75/31.09 We have to consider all (P,R,Pi)-chains 114.75/31.09 ---------------------------------------- 114.75/31.09 114.75/31.09 (38) PiDPToQDPProof (SOUND) 114.75/31.09 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 114.75/31.09 ---------------------------------------- 114.75/31.09 114.75/31.09 (39) 114.75/31.09 Obligation: 114.75/31.09 Q DP problem: 114.75/31.09 The TRS P consists of the following rules: 114.75/31.09 114.75/31.09 U7_GA(X1, delcB_out_ga(X1, X3)) -> MAXSORTA_IN_GA(X3) 114.75/31.09 MAXSORTA_IN_GA(.(X1, [])) -> U7_GA(X1, delcB_in_ga(X1)) 114.75/31.09 MAXSORTA_IN_GA(.(t, .(t, X1))) -> U12_GA(X1, maxcC_in_ga(X1)) 114.75/31.09 U12_GA(X1, maxcC_out_ga(X1, X2)) -> U13_GA(X1, delcD_in_gga(X2, X1)) 114.75/31.09 U13_GA(X1, delcD_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) 114.75/31.09 MAXSORTA_IN_GA(.(t, .(f, X1))) -> U18_GA(X1, maxcC_in_ga(X1)) 114.75/31.09 U18_GA(X1, maxcC_out_ga(X1, X2)) -> U19_GA(X1, delcE_in_gga(X2, X1)) 114.75/31.09 U19_GA(X1, delcE_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) 114.75/31.09 MAXSORTA_IN_GA(.(f, .(f, X1))) -> U24_GA(X1, maxcF_in_ga(X1)) 114.75/31.09 U24_GA(X1, maxcF_out_ga(X1, X2)) -> U25_GA(X1, delcG_in_gga(X2, X1)) 114.75/31.09 U25_GA(X1, delcG_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) 114.75/31.09 MAXSORTA_IN_GA(.(f, .(t, X1))) -> U30_GA(X1, maxcC_in_ga(X1)) 114.75/31.09 U30_GA(X1, maxcC_out_ga(X1, X2)) -> U31_GA(X1, delcH_in_gga(X2, X1)) 114.75/31.09 U31_GA(X1, delcH_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) 114.75/31.09 114.75/31.09 The TRS R consists of the following rules: 114.75/31.09 114.75/31.09 delcB_in_ga(t) -> delcB_out_ga(t, []) 114.75/31.09 delcB_in_ga(f) -> delcB_out_ga(f, []) 114.75/31.09 maxcC_in_ga([]) -> maxcC_out_ga([], t) 114.75/31.09 maxcC_in_ga(.(t, X1)) -> U48_ga(X1, maxcC_in_ga(X1)) 114.75/31.09 maxcC_in_ga(.(f, X1)) -> U49_ga(X1, maxcC_in_ga(X1)) 114.75/31.09 delcD_in_gga(t, X1) -> delcD_out_gga(t, X1, .(t, X1)) 114.75/31.09 delcD_in_gga(f, X1) -> U54_gga(X1, delcI_in_ga(X1)) 114.75/31.09 delcE_in_gga(t, X1) -> delcE_out_gga(t, X1, .(f, X1)) 114.75/31.09 delcE_in_gga(f, X1) -> U55_gga(X1, delcI_in_ga(.(f, X1))) 114.75/31.09 maxcF_in_ga([]) -> maxcF_out_ga([], f) 114.75/31.09 maxcF_in_ga(.(f, X1)) -> U51_ga(X1, maxcF_in_ga(X1)) 114.75/31.09 maxcF_in_ga(.(t, X1)) -> U52_ga(X1, maxcC_in_ga(X1)) 114.75/31.09 delcG_in_gga(f, X1) -> delcG_out_gga(f, X1, .(f, X1)) 114.75/31.09 delcG_in_gga(t, X1) -> U56_gga(X1, delcJ_in_ga(X1)) 114.75/31.09 delcH_in_gga(f, X1) -> delcH_out_gga(f, X1, .(t, X1)) 114.75/31.09 delcH_in_gga(t, X1) -> U57_gga(X1, delcJ_in_ga(.(t, X1))) 114.75/31.09 U48_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(t, X1), X2) 114.75/31.09 U49_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(f, X1), X2) 114.75/31.09 U54_gga(X1, delcI_out_ga(X1, X2)) -> delcD_out_gga(f, X1, .(t, .(t, X2))) 114.75/31.09 U55_gga(X1, delcI_out_ga(.(f, X1), X2)) -> delcE_out_gga(f, X1, .(t, X2)) 114.75/31.09 U51_ga(X1, maxcF_out_ga(X1, X2)) -> maxcF_out_ga(.(f, X1), X2) 114.75/31.09 U52_ga(X1, maxcC_out_ga(X1, X2)) -> maxcF_out_ga(.(t, X1), X2) 114.75/31.09 U56_gga(X1, delcJ_out_ga(X1, X2)) -> delcG_out_gga(t, X1, .(f, .(f, X2))) 114.75/31.09 U57_gga(X1, delcJ_out_ga(.(t, X1), X2)) -> delcH_out_gga(t, X1, .(f, X2)) 114.75/31.09 delcI_in_ga([]) -> delcI_out_ga([], []) 114.75/31.09 delcI_in_ga(.(f, X1)) -> delcI_out_ga(.(f, X1), X1) 114.75/31.09 delcI_in_ga(.(t, X1)) -> U50_ga(X1, delcI_in_ga(X1)) 114.75/31.09 delcJ_in_ga([]) -> delcJ_out_ga([], []) 114.75/31.09 delcJ_in_ga(.(t, X1)) -> delcJ_out_ga(.(t, X1), X1) 114.75/31.09 delcJ_in_ga(.(f, X1)) -> U53_ga(X1, delcJ_in_ga(X1)) 114.75/31.09 U50_ga(X1, delcI_out_ga(X1, X2)) -> delcI_out_ga(.(t, X1), .(t, X2)) 114.75/31.09 U53_ga(X1, delcJ_out_ga(X1, X2)) -> delcJ_out_ga(.(f, X1), .(f, X2)) 114.75/31.09 114.75/31.09 The set Q consists of the following terms: 114.75/31.09 114.75/31.09 delcB_in_ga(x0) 114.75/31.09 maxcC_in_ga(x0) 114.75/31.09 delcD_in_gga(x0, x1) 114.75/31.09 delcE_in_gga(x0, x1) 114.75/31.09 maxcF_in_ga(x0) 114.75/31.09 delcG_in_gga(x0, x1) 114.75/31.09 delcH_in_gga(x0, x1) 114.75/31.09 U48_ga(x0, x1) 114.75/31.09 U49_ga(x0, x1) 114.75/31.09 U54_gga(x0, x1) 114.75/31.09 U55_gga(x0, x1) 114.75/31.09 U51_ga(x0, x1) 114.75/31.09 U52_ga(x0, x1) 114.75/31.09 U56_gga(x0, x1) 114.75/31.09 U57_gga(x0, x1) 114.75/31.09 delcI_in_ga(x0) 114.75/31.09 delcJ_in_ga(x0) 114.75/31.09 U50_ga(x0, x1) 114.75/31.09 U53_ga(x0, x1) 114.75/31.09 114.75/31.09 We have to consider all (P,Q,R)-chains. 114.75/31.09 ---------------------------------------- 114.75/31.09 114.75/31.09 (40) TransformationProof (SOUND) 114.75/31.09 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]: 114.75/31.09 114.75/31.09 (MAXSORTA_IN_GA(.(t, [])) -> U7_GA(t, delcB_out_ga(t, [])),MAXSORTA_IN_GA(.(t, [])) -> U7_GA(t, delcB_out_ga(t, []))) 114.75/31.09 (MAXSORTA_IN_GA(.(f, [])) -> U7_GA(f, delcB_out_ga(f, [])),MAXSORTA_IN_GA(.(f, [])) -> U7_GA(f, delcB_out_ga(f, []))) 114.75/31.09 114.75/31.09 114.75/31.09 ---------------------------------------- 114.75/31.09 114.75/31.09 (41) 114.75/31.09 Obligation: 114.75/31.09 Q DP problem: 114.75/31.09 The TRS P consists of the following rules: 114.75/31.09 114.75/31.09 U7_GA(X1, delcB_out_ga(X1, X3)) -> MAXSORTA_IN_GA(X3) 114.75/31.09 MAXSORTA_IN_GA(.(t, .(t, X1))) -> U12_GA(X1, maxcC_in_ga(X1)) 114.75/31.09 U12_GA(X1, maxcC_out_ga(X1, X2)) -> U13_GA(X1, delcD_in_gga(X2, X1)) 114.75/31.09 U13_GA(X1, delcD_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) 114.75/31.09 MAXSORTA_IN_GA(.(t, .(f, X1))) -> U18_GA(X1, maxcC_in_ga(X1)) 114.75/31.09 U18_GA(X1, maxcC_out_ga(X1, X2)) -> U19_GA(X1, delcE_in_gga(X2, X1)) 114.75/31.09 U19_GA(X1, delcE_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) 114.75/31.09 MAXSORTA_IN_GA(.(f, .(f, X1))) -> U24_GA(X1, maxcF_in_ga(X1)) 114.75/31.09 U24_GA(X1, maxcF_out_ga(X1, X2)) -> U25_GA(X1, delcG_in_gga(X2, X1)) 114.75/31.09 U25_GA(X1, delcG_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) 114.75/31.09 MAXSORTA_IN_GA(.(f, .(t, X1))) -> U30_GA(X1, maxcC_in_ga(X1)) 114.75/31.09 U30_GA(X1, maxcC_out_ga(X1, X2)) -> U31_GA(X1, delcH_in_gga(X2, X1)) 114.75/31.09 U31_GA(X1, delcH_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) 114.75/31.09 MAXSORTA_IN_GA(.(t, [])) -> U7_GA(t, delcB_out_ga(t, [])) 114.75/31.09 MAXSORTA_IN_GA(.(f, [])) -> U7_GA(f, delcB_out_ga(f, [])) 114.75/31.09 114.75/31.09 The TRS R consists of the following rules: 114.75/31.09 114.75/31.09 delcB_in_ga(t) -> delcB_out_ga(t, []) 114.75/31.09 delcB_in_ga(f) -> delcB_out_ga(f, []) 114.75/31.09 maxcC_in_ga([]) -> maxcC_out_ga([], t) 114.75/31.09 maxcC_in_ga(.(t, X1)) -> U48_ga(X1, maxcC_in_ga(X1)) 114.75/31.09 maxcC_in_ga(.(f, X1)) -> U49_ga(X1, maxcC_in_ga(X1)) 114.75/31.09 delcD_in_gga(t, X1) -> delcD_out_gga(t, X1, .(t, X1)) 114.75/31.09 delcD_in_gga(f, X1) -> U54_gga(X1, delcI_in_ga(X1)) 114.75/31.09 delcE_in_gga(t, X1) -> delcE_out_gga(t, X1, .(f, X1)) 114.75/31.09 delcE_in_gga(f, X1) -> U55_gga(X1, delcI_in_ga(.(f, X1))) 114.75/31.09 maxcF_in_ga([]) -> maxcF_out_ga([], f) 114.75/31.09 maxcF_in_ga(.(f, X1)) -> U51_ga(X1, maxcF_in_ga(X1)) 114.75/31.09 maxcF_in_ga(.(t, X1)) -> U52_ga(X1, maxcC_in_ga(X1)) 114.75/31.09 delcG_in_gga(f, X1) -> delcG_out_gga(f, X1, .(f, X1)) 114.75/31.09 delcG_in_gga(t, X1) -> U56_gga(X1, delcJ_in_ga(X1)) 114.75/31.09 delcH_in_gga(f, X1) -> delcH_out_gga(f, X1, .(t, X1)) 114.75/31.09 delcH_in_gga(t, X1) -> U57_gga(X1, delcJ_in_ga(.(t, X1))) 114.75/31.09 U48_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(t, X1), X2) 114.75/31.09 U49_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(f, X1), X2) 114.75/31.09 U54_gga(X1, delcI_out_ga(X1, X2)) -> delcD_out_gga(f, X1, .(t, .(t, X2))) 114.75/31.09 U55_gga(X1, delcI_out_ga(.(f, X1), X2)) -> delcE_out_gga(f, X1, .(t, X2)) 114.75/31.09 U51_ga(X1, maxcF_out_ga(X1, X2)) -> maxcF_out_ga(.(f, X1), X2) 114.75/31.09 U52_ga(X1, maxcC_out_ga(X1, X2)) -> maxcF_out_ga(.(t, X1), X2) 114.75/31.09 U56_gga(X1, delcJ_out_ga(X1, X2)) -> delcG_out_gga(t, X1, .(f, .(f, X2))) 114.75/31.09 U57_gga(X1, delcJ_out_ga(.(t, X1), X2)) -> delcH_out_gga(t, X1, .(f, X2)) 114.75/31.09 delcI_in_ga([]) -> delcI_out_ga([], []) 114.75/31.09 delcI_in_ga(.(f, X1)) -> delcI_out_ga(.(f, X1), X1) 114.75/31.09 delcI_in_ga(.(t, X1)) -> U50_ga(X1, delcI_in_ga(X1)) 114.75/31.09 delcJ_in_ga([]) -> delcJ_out_ga([], []) 114.75/31.09 delcJ_in_ga(.(t, X1)) -> delcJ_out_ga(.(t, X1), X1) 114.75/31.09 delcJ_in_ga(.(f, X1)) -> U53_ga(X1, delcJ_in_ga(X1)) 114.75/31.09 U50_ga(X1, delcI_out_ga(X1, X2)) -> delcI_out_ga(.(t, X1), .(t, X2)) 114.75/31.09 U53_ga(X1, delcJ_out_ga(X1, X2)) -> delcJ_out_ga(.(f, X1), .(f, X2)) 114.75/31.09 114.75/31.09 The set Q consists of the following terms: 114.75/31.09 114.75/31.09 delcB_in_ga(x0) 114.75/31.09 maxcC_in_ga(x0) 114.75/31.09 delcD_in_gga(x0, x1) 114.75/31.09 delcE_in_gga(x0, x1) 114.75/31.09 maxcF_in_ga(x0) 114.75/31.09 delcG_in_gga(x0, x1) 114.75/31.09 delcH_in_gga(x0, x1) 114.75/31.09 U48_ga(x0, x1) 114.75/31.09 U49_ga(x0, x1) 114.75/31.09 U54_gga(x0, x1) 114.75/31.09 U55_gga(x0, x1) 114.75/31.09 U51_ga(x0, x1) 114.75/31.09 U52_ga(x0, x1) 114.75/31.09 U56_gga(x0, x1) 114.75/31.09 U57_gga(x0, x1) 114.75/31.09 delcI_in_ga(x0) 114.75/31.09 delcJ_in_ga(x0) 114.75/31.09 U50_ga(x0, x1) 114.75/31.09 U53_ga(x0, x1) 114.75/31.09 114.75/31.09 We have to consider all (P,Q,R)-chains. 114.75/31.09 ---------------------------------------- 114.75/31.09 114.75/31.09 (42) UsableRulesProof (EQUIVALENT) 114.75/31.09 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. 114.75/31.09 ---------------------------------------- 114.75/31.09 114.75/31.09 (43) 114.75/31.09 Obligation: 114.75/31.09 Q DP problem: 114.75/31.09 The TRS P consists of the following rules: 114.75/31.09 114.75/31.09 U7_GA(X1, delcB_out_ga(X1, X3)) -> MAXSORTA_IN_GA(X3) 114.75/31.09 MAXSORTA_IN_GA(.(t, .(t, X1))) -> U12_GA(X1, maxcC_in_ga(X1)) 114.75/31.09 U12_GA(X1, maxcC_out_ga(X1, X2)) -> U13_GA(X1, delcD_in_gga(X2, X1)) 114.75/31.09 U13_GA(X1, delcD_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) 114.75/31.09 MAXSORTA_IN_GA(.(t, .(f, X1))) -> U18_GA(X1, maxcC_in_ga(X1)) 114.75/31.09 U18_GA(X1, maxcC_out_ga(X1, X2)) -> U19_GA(X1, delcE_in_gga(X2, X1)) 114.75/31.09 U19_GA(X1, delcE_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) 114.75/31.09 MAXSORTA_IN_GA(.(f, .(f, X1))) -> U24_GA(X1, maxcF_in_ga(X1)) 114.75/31.09 U24_GA(X1, maxcF_out_ga(X1, X2)) -> U25_GA(X1, delcG_in_gga(X2, X1)) 114.75/31.09 U25_GA(X1, delcG_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) 114.75/31.09 MAXSORTA_IN_GA(.(f, .(t, X1))) -> U30_GA(X1, maxcC_in_ga(X1)) 114.75/31.09 U30_GA(X1, maxcC_out_ga(X1, X2)) -> U31_GA(X1, delcH_in_gga(X2, X1)) 114.75/31.09 U31_GA(X1, delcH_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) 114.75/31.09 MAXSORTA_IN_GA(.(t, [])) -> U7_GA(t, delcB_out_ga(t, [])) 114.75/31.09 MAXSORTA_IN_GA(.(f, [])) -> U7_GA(f, delcB_out_ga(f, [])) 114.75/31.09 114.75/31.09 The TRS R consists of the following rules: 114.75/31.09 114.75/31.09 delcH_in_gga(f, X1) -> delcH_out_gga(f, X1, .(t, X1)) 114.75/31.09 delcH_in_gga(t, X1) -> U57_gga(X1, delcJ_in_ga(.(t, X1))) 114.75/31.09 delcJ_in_ga(.(t, X1)) -> delcJ_out_ga(.(t, X1), X1) 114.75/31.09 U57_gga(X1, delcJ_out_ga(.(t, X1), X2)) -> delcH_out_gga(t, X1, .(f, X2)) 114.75/31.09 maxcC_in_ga([]) -> maxcC_out_ga([], t) 114.75/31.09 maxcC_in_ga(.(t, X1)) -> U48_ga(X1, maxcC_in_ga(X1)) 114.75/31.09 maxcC_in_ga(.(f, X1)) -> U49_ga(X1, maxcC_in_ga(X1)) 114.75/31.09 U49_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(f, X1), X2) 114.75/31.09 U48_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(t, X1), X2) 114.75/31.09 delcG_in_gga(f, X1) -> delcG_out_gga(f, X1, .(f, X1)) 114.75/31.09 delcG_in_gga(t, X1) -> U56_gga(X1, delcJ_in_ga(X1)) 114.75/31.09 delcJ_in_ga([]) -> delcJ_out_ga([], []) 114.75/31.09 delcJ_in_ga(.(f, X1)) -> U53_ga(X1, delcJ_in_ga(X1)) 114.75/31.09 U56_gga(X1, delcJ_out_ga(X1, X2)) -> delcG_out_gga(t, X1, .(f, .(f, X2))) 114.75/31.09 U53_ga(X1, delcJ_out_ga(X1, X2)) -> delcJ_out_ga(.(f, X1), .(f, X2)) 114.75/31.09 maxcF_in_ga([]) -> maxcF_out_ga([], f) 114.75/31.09 maxcF_in_ga(.(f, X1)) -> U51_ga(X1, maxcF_in_ga(X1)) 114.75/31.09 maxcF_in_ga(.(t, X1)) -> U52_ga(X1, maxcC_in_ga(X1)) 114.75/31.09 U52_ga(X1, maxcC_out_ga(X1, X2)) -> maxcF_out_ga(.(t, X1), X2) 114.75/31.09 U51_ga(X1, maxcF_out_ga(X1, X2)) -> maxcF_out_ga(.(f, X1), X2) 114.75/31.09 delcE_in_gga(t, X1) -> delcE_out_gga(t, X1, .(f, X1)) 114.75/31.09 delcE_in_gga(f, X1) -> U55_gga(X1, delcI_in_ga(.(f, X1))) 114.75/31.09 delcI_in_ga(.(f, X1)) -> delcI_out_ga(.(f, X1), X1) 114.75/31.09 U55_gga(X1, delcI_out_ga(.(f, X1), X2)) -> delcE_out_gga(f, X1, .(t, X2)) 114.75/31.09 delcD_in_gga(t, X1) -> delcD_out_gga(t, X1, .(t, X1)) 114.75/31.09 delcD_in_gga(f, X1) -> U54_gga(X1, delcI_in_ga(X1)) 114.75/31.09 delcI_in_ga([]) -> delcI_out_ga([], []) 114.75/31.09 delcI_in_ga(.(t, X1)) -> U50_ga(X1, delcI_in_ga(X1)) 114.75/31.09 U54_gga(X1, delcI_out_ga(X1, X2)) -> delcD_out_gga(f, X1, .(t, .(t, X2))) 114.75/31.09 U50_ga(X1, delcI_out_ga(X1, X2)) -> delcI_out_ga(.(t, X1), .(t, X2)) 114.75/31.09 114.75/31.09 The set Q consists of the following terms: 114.75/31.09 114.75/31.09 delcB_in_ga(x0) 114.75/31.09 maxcC_in_ga(x0) 114.75/31.09 delcD_in_gga(x0, x1) 114.75/31.09 delcE_in_gga(x0, x1) 114.75/31.09 maxcF_in_ga(x0) 114.75/31.09 delcG_in_gga(x0, x1) 114.75/31.09 delcH_in_gga(x0, x1) 114.75/31.09 U48_ga(x0, x1) 114.75/31.09 U49_ga(x0, x1) 114.75/31.09 U54_gga(x0, x1) 114.75/31.09 U55_gga(x0, x1) 114.75/31.09 U51_ga(x0, x1) 114.75/31.09 U52_ga(x0, x1) 114.75/31.09 U56_gga(x0, x1) 114.75/31.09 U57_gga(x0, x1) 114.75/31.09 delcI_in_ga(x0) 114.75/31.09 delcJ_in_ga(x0) 114.75/31.09 U50_ga(x0, x1) 114.75/31.09 U53_ga(x0, x1) 114.75/31.09 114.75/31.09 We have to consider all (P,Q,R)-chains. 114.75/31.09 ---------------------------------------- 114.75/31.09 114.75/31.09 (44) QReductionProof (EQUIVALENT) 114.75/31.09 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 114.75/31.09 114.75/31.09 delcB_in_ga(x0) 114.75/31.09 114.75/31.09 114.75/31.09 ---------------------------------------- 114.75/31.09 114.75/31.09 (45) 114.75/31.09 Obligation: 114.75/31.09 Q DP problem: 114.75/31.09 The TRS P consists of the following rules: 114.75/31.09 114.75/31.09 U7_GA(X1, delcB_out_ga(X1, X3)) -> MAXSORTA_IN_GA(X3) 114.75/31.09 MAXSORTA_IN_GA(.(t, .(t, X1))) -> U12_GA(X1, maxcC_in_ga(X1)) 114.75/31.09 U12_GA(X1, maxcC_out_ga(X1, X2)) -> U13_GA(X1, delcD_in_gga(X2, X1)) 114.75/31.09 U13_GA(X1, delcD_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) 114.75/31.09 MAXSORTA_IN_GA(.(t, .(f, X1))) -> U18_GA(X1, maxcC_in_ga(X1)) 114.75/31.09 U18_GA(X1, maxcC_out_ga(X1, X2)) -> U19_GA(X1, delcE_in_gga(X2, X1)) 114.75/31.09 U19_GA(X1, delcE_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) 114.75/31.09 MAXSORTA_IN_GA(.(f, .(f, X1))) -> U24_GA(X1, maxcF_in_ga(X1)) 114.75/31.09 U24_GA(X1, maxcF_out_ga(X1, X2)) -> U25_GA(X1, delcG_in_gga(X2, X1)) 114.75/31.09 U25_GA(X1, delcG_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) 114.75/31.09 MAXSORTA_IN_GA(.(f, .(t, X1))) -> U30_GA(X1, maxcC_in_ga(X1)) 114.75/31.09 U30_GA(X1, maxcC_out_ga(X1, X2)) -> U31_GA(X1, delcH_in_gga(X2, X1)) 114.75/31.09 U31_GA(X1, delcH_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) 114.75/31.09 MAXSORTA_IN_GA(.(t, [])) -> U7_GA(t, delcB_out_ga(t, [])) 114.75/31.09 MAXSORTA_IN_GA(.(f, [])) -> U7_GA(f, delcB_out_ga(f, [])) 114.75/31.09 114.75/31.09 The TRS R consists of the following rules: 114.75/31.09 114.75/31.09 delcH_in_gga(f, X1) -> delcH_out_gga(f, X1, .(t, X1)) 114.75/31.09 delcH_in_gga(t, X1) -> U57_gga(X1, delcJ_in_ga(.(t, X1))) 114.75/31.09 delcJ_in_ga(.(t, X1)) -> delcJ_out_ga(.(t, X1), X1) 114.75/31.09 U57_gga(X1, delcJ_out_ga(.(t, X1), X2)) -> delcH_out_gga(t, X1, .(f, X2)) 114.75/31.09 maxcC_in_ga([]) -> maxcC_out_ga([], t) 114.75/31.09 maxcC_in_ga(.(t, X1)) -> U48_ga(X1, maxcC_in_ga(X1)) 114.75/31.09 maxcC_in_ga(.(f, X1)) -> U49_ga(X1, maxcC_in_ga(X1)) 114.75/31.09 U49_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(f, X1), X2) 114.75/31.09 U48_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(t, X1), X2) 114.75/31.09 delcG_in_gga(f, X1) -> delcG_out_gga(f, X1, .(f, X1)) 114.75/31.09 delcG_in_gga(t, X1) -> U56_gga(X1, delcJ_in_ga(X1)) 114.75/31.09 delcJ_in_ga([]) -> delcJ_out_ga([], []) 114.75/31.09 delcJ_in_ga(.(f, X1)) -> U53_ga(X1, delcJ_in_ga(X1)) 114.75/31.09 U56_gga(X1, delcJ_out_ga(X1, X2)) -> delcG_out_gga(t, X1, .(f, .(f, X2))) 114.75/31.09 U53_ga(X1, delcJ_out_ga(X1, X2)) -> delcJ_out_ga(.(f, X1), .(f, X2)) 114.75/31.09 maxcF_in_ga([]) -> maxcF_out_ga([], f) 114.75/31.09 maxcF_in_ga(.(f, X1)) -> U51_ga(X1, maxcF_in_ga(X1)) 114.75/31.09 maxcF_in_ga(.(t, X1)) -> U52_ga(X1, maxcC_in_ga(X1)) 114.75/31.09 U52_ga(X1, maxcC_out_ga(X1, X2)) -> maxcF_out_ga(.(t, X1), X2) 114.75/31.09 U51_ga(X1, maxcF_out_ga(X1, X2)) -> maxcF_out_ga(.(f, X1), X2) 114.75/31.09 delcE_in_gga(t, X1) -> delcE_out_gga(t, X1, .(f, X1)) 114.75/31.09 delcE_in_gga(f, X1) -> U55_gga(X1, delcI_in_ga(.(f, X1))) 114.75/31.09 delcI_in_ga(.(f, X1)) -> delcI_out_ga(.(f, X1), X1) 114.75/31.09 U55_gga(X1, delcI_out_ga(.(f, X1), X2)) -> delcE_out_gga(f, X1, .(t, X2)) 114.75/31.09 delcD_in_gga(t, X1) -> delcD_out_gga(t, X1, .(t, X1)) 114.75/31.09 delcD_in_gga(f, X1) -> U54_gga(X1, delcI_in_ga(X1)) 114.75/31.09 delcI_in_ga([]) -> delcI_out_ga([], []) 114.75/31.09 delcI_in_ga(.(t, X1)) -> U50_ga(X1, delcI_in_ga(X1)) 114.75/31.09 U54_gga(X1, delcI_out_ga(X1, X2)) -> delcD_out_gga(f, X1, .(t, .(t, X2))) 114.75/31.09 U50_ga(X1, delcI_out_ga(X1, X2)) -> delcI_out_ga(.(t, X1), .(t, X2)) 114.75/31.09 114.75/31.09 The set Q consists of the following terms: 114.75/31.09 114.75/31.09 maxcC_in_ga(x0) 114.75/31.09 delcD_in_gga(x0, x1) 114.75/31.09 delcE_in_gga(x0, x1) 114.75/31.09 maxcF_in_ga(x0) 114.75/31.09 delcG_in_gga(x0, x1) 114.75/31.09 delcH_in_gga(x0, x1) 114.75/31.09 U48_ga(x0, x1) 114.75/31.09 U49_ga(x0, x1) 114.75/31.09 U54_gga(x0, x1) 114.75/31.09 U55_gga(x0, x1) 114.75/31.09 U51_ga(x0, x1) 114.75/31.09 U52_ga(x0, x1) 114.75/31.09 U56_gga(x0, x1) 114.75/31.09 U57_gga(x0, x1) 114.75/31.09 delcI_in_ga(x0) 114.75/31.09 delcJ_in_ga(x0) 114.75/31.09 U50_ga(x0, x1) 114.75/31.09 U53_ga(x0, x1) 114.75/31.09 114.75/31.09 We have to consider all (P,Q,R)-chains. 114.75/31.09 ---------------------------------------- 114.75/31.09 114.75/31.09 (46) TransformationProof (EQUIVALENT) 114.75/31.09 By instantiating [LPAR04] the rule U7_GA(X1, delcB_out_ga(X1, X3)) -> MAXSORTA_IN_GA(X3) we obtained the following new rules [LPAR04]: 114.75/31.09 114.75/31.09 (U7_GA(t, delcB_out_ga(t, [])) -> MAXSORTA_IN_GA([]),U7_GA(t, delcB_out_ga(t, [])) -> MAXSORTA_IN_GA([])) 114.75/31.09 (U7_GA(f, delcB_out_ga(f, [])) -> MAXSORTA_IN_GA([]),U7_GA(f, delcB_out_ga(f, [])) -> MAXSORTA_IN_GA([])) 114.75/31.09 114.75/31.09 114.75/31.09 ---------------------------------------- 114.75/31.09 114.75/31.09 (47) 114.75/31.09 Obligation: 114.75/31.09 Q DP problem: 114.75/31.09 The TRS P consists of the following rules: 114.75/31.09 114.75/31.09 MAXSORTA_IN_GA(.(t, .(t, X1))) -> U12_GA(X1, maxcC_in_ga(X1)) 114.75/31.09 U12_GA(X1, maxcC_out_ga(X1, X2)) -> U13_GA(X1, delcD_in_gga(X2, X1)) 114.75/31.09 U13_GA(X1, delcD_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) 114.75/31.09 MAXSORTA_IN_GA(.(t, .(f, X1))) -> U18_GA(X1, maxcC_in_ga(X1)) 114.75/31.09 U18_GA(X1, maxcC_out_ga(X1, X2)) -> U19_GA(X1, delcE_in_gga(X2, X1)) 114.75/31.09 U19_GA(X1, delcE_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) 114.75/31.09 MAXSORTA_IN_GA(.(f, .(f, X1))) -> U24_GA(X1, maxcF_in_ga(X1)) 114.75/31.09 U24_GA(X1, maxcF_out_ga(X1, X2)) -> U25_GA(X1, delcG_in_gga(X2, X1)) 114.75/31.09 U25_GA(X1, delcG_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) 114.75/31.09 MAXSORTA_IN_GA(.(f, .(t, X1))) -> U30_GA(X1, maxcC_in_ga(X1)) 114.75/31.09 U30_GA(X1, maxcC_out_ga(X1, X2)) -> U31_GA(X1, delcH_in_gga(X2, X1)) 114.75/31.09 U31_GA(X1, delcH_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) 114.75/31.09 MAXSORTA_IN_GA(.(t, [])) -> U7_GA(t, delcB_out_ga(t, [])) 114.75/31.09 MAXSORTA_IN_GA(.(f, [])) -> U7_GA(f, delcB_out_ga(f, [])) 114.75/31.09 U7_GA(t, delcB_out_ga(t, [])) -> MAXSORTA_IN_GA([]) 114.75/31.09 U7_GA(f, delcB_out_ga(f, [])) -> MAXSORTA_IN_GA([]) 114.75/31.09 114.75/31.09 The TRS R consists of the following rules: 114.75/31.09 114.75/31.09 delcH_in_gga(f, X1) -> delcH_out_gga(f, X1, .(t, X1)) 114.75/31.09 delcH_in_gga(t, X1) -> U57_gga(X1, delcJ_in_ga(.(t, X1))) 114.75/31.09 delcJ_in_ga(.(t, X1)) -> delcJ_out_ga(.(t, X1), X1) 114.75/31.09 U57_gga(X1, delcJ_out_ga(.(t, X1), X2)) -> delcH_out_gga(t, X1, .(f, X2)) 114.75/31.09 maxcC_in_ga([]) -> maxcC_out_ga([], t) 114.75/31.09 maxcC_in_ga(.(t, X1)) -> U48_ga(X1, maxcC_in_ga(X1)) 114.75/31.09 maxcC_in_ga(.(f, X1)) -> U49_ga(X1, maxcC_in_ga(X1)) 114.75/31.09 U49_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(f, X1), X2) 114.75/31.09 U48_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(t, X1), X2) 114.75/31.09 delcG_in_gga(f, X1) -> delcG_out_gga(f, X1, .(f, X1)) 114.75/31.09 delcG_in_gga(t, X1) -> U56_gga(X1, delcJ_in_ga(X1)) 114.75/31.09 delcJ_in_ga([]) -> delcJ_out_ga([], []) 114.75/31.09 delcJ_in_ga(.(f, X1)) -> U53_ga(X1, delcJ_in_ga(X1)) 114.75/31.09 U56_gga(X1, delcJ_out_ga(X1, X2)) -> delcG_out_gga(t, X1, .(f, .(f, X2))) 114.75/31.09 U53_ga(X1, delcJ_out_ga(X1, X2)) -> delcJ_out_ga(.(f, X1), .(f, X2)) 114.75/31.09 maxcF_in_ga([]) -> maxcF_out_ga([], f) 114.75/31.09 maxcF_in_ga(.(f, X1)) -> U51_ga(X1, maxcF_in_ga(X1)) 114.75/31.09 maxcF_in_ga(.(t, X1)) -> U52_ga(X1, maxcC_in_ga(X1)) 114.75/31.09 U52_ga(X1, maxcC_out_ga(X1, X2)) -> maxcF_out_ga(.(t, X1), X2) 114.75/31.09 U51_ga(X1, maxcF_out_ga(X1, X2)) -> maxcF_out_ga(.(f, X1), X2) 114.75/31.09 delcE_in_gga(t, X1) -> delcE_out_gga(t, X1, .(f, X1)) 114.75/31.09 delcE_in_gga(f, X1) -> U55_gga(X1, delcI_in_ga(.(f, X1))) 114.75/31.09 delcI_in_ga(.(f, X1)) -> delcI_out_ga(.(f, X1), X1) 114.75/31.09 U55_gga(X1, delcI_out_ga(.(f, X1), X2)) -> delcE_out_gga(f, X1, .(t, X2)) 114.75/31.09 delcD_in_gga(t, X1) -> delcD_out_gga(t, X1, .(t, X1)) 114.75/31.09 delcD_in_gga(f, X1) -> U54_gga(X1, delcI_in_ga(X1)) 114.75/31.09 delcI_in_ga([]) -> delcI_out_ga([], []) 114.75/31.09 delcI_in_ga(.(t, X1)) -> U50_ga(X1, delcI_in_ga(X1)) 114.75/31.09 U54_gga(X1, delcI_out_ga(X1, X2)) -> delcD_out_gga(f, X1, .(t, .(t, X2))) 114.75/31.09 U50_ga(X1, delcI_out_ga(X1, X2)) -> delcI_out_ga(.(t, X1), .(t, X2)) 114.75/31.09 114.75/31.09 The set Q consists of the following terms: 114.75/31.09 114.75/31.09 maxcC_in_ga(x0) 114.75/31.09 delcD_in_gga(x0, x1) 114.75/31.09 delcE_in_gga(x0, x1) 114.75/31.09 maxcF_in_ga(x0) 114.75/31.09 delcG_in_gga(x0, x1) 114.75/31.09 delcH_in_gga(x0, x1) 114.75/31.09 U48_ga(x0, x1) 114.75/31.09 U49_ga(x0, x1) 114.75/31.09 U54_gga(x0, x1) 114.75/31.09 U55_gga(x0, x1) 114.75/31.09 U51_ga(x0, x1) 114.75/31.09 U52_ga(x0, x1) 114.75/31.09 U56_gga(x0, x1) 114.75/31.09 U57_gga(x0, x1) 114.75/31.09 delcI_in_ga(x0) 114.75/31.09 delcJ_in_ga(x0) 114.75/31.09 U50_ga(x0, x1) 114.75/31.09 U53_ga(x0, x1) 114.75/31.09 114.75/31.09 We have to consider all (P,Q,R)-chains. 114.75/31.09 ---------------------------------------- 114.75/31.09 114.75/31.09 (48) DependencyGraphProof (EQUIVALENT) 114.75/31.09 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 4 less nodes. 114.75/31.09 ---------------------------------------- 114.75/31.09 114.75/31.09 (49) 114.75/31.09 Obligation: 114.75/31.09 Q DP problem: 114.75/31.09 The TRS P consists of the following rules: 114.75/31.09 114.75/31.09 U12_GA(X1, maxcC_out_ga(X1, X2)) -> U13_GA(X1, delcD_in_gga(X2, X1)) 114.75/31.09 U13_GA(X1, delcD_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) 114.75/31.09 MAXSORTA_IN_GA(.(t, .(t, X1))) -> U12_GA(X1, maxcC_in_ga(X1)) 114.75/31.09 MAXSORTA_IN_GA(.(t, .(f, X1))) -> U18_GA(X1, maxcC_in_ga(X1)) 114.75/31.09 U18_GA(X1, maxcC_out_ga(X1, X2)) -> U19_GA(X1, delcE_in_gga(X2, X1)) 114.75/31.09 U19_GA(X1, delcE_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) 114.75/31.09 MAXSORTA_IN_GA(.(f, .(f, X1))) -> U24_GA(X1, maxcF_in_ga(X1)) 114.75/31.09 U24_GA(X1, maxcF_out_ga(X1, X2)) -> U25_GA(X1, delcG_in_gga(X2, X1)) 114.75/31.09 U25_GA(X1, delcG_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) 114.75/31.09 MAXSORTA_IN_GA(.(f, .(t, X1))) -> U30_GA(X1, maxcC_in_ga(X1)) 114.75/31.09 U30_GA(X1, maxcC_out_ga(X1, X2)) -> U31_GA(X1, delcH_in_gga(X2, X1)) 114.75/31.09 U31_GA(X1, delcH_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) 114.75/31.09 114.75/31.09 The TRS R consists of the following rules: 114.75/31.09 114.75/31.09 delcH_in_gga(f, X1) -> delcH_out_gga(f, X1, .(t, X1)) 114.75/31.09 delcH_in_gga(t, X1) -> U57_gga(X1, delcJ_in_ga(.(t, X1))) 114.75/31.09 delcJ_in_ga(.(t, X1)) -> delcJ_out_ga(.(t, X1), X1) 114.75/31.09 U57_gga(X1, delcJ_out_ga(.(t, X1), X2)) -> delcH_out_gga(t, X1, .(f, X2)) 114.75/31.09 maxcC_in_ga([]) -> maxcC_out_ga([], t) 114.75/31.09 maxcC_in_ga(.(t, X1)) -> U48_ga(X1, maxcC_in_ga(X1)) 114.75/31.09 maxcC_in_ga(.(f, X1)) -> U49_ga(X1, maxcC_in_ga(X1)) 114.75/31.09 U49_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(f, X1), X2) 114.75/31.09 U48_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(t, X1), X2) 114.75/31.09 delcG_in_gga(f, X1) -> delcG_out_gga(f, X1, .(f, X1)) 114.75/31.09 delcG_in_gga(t, X1) -> U56_gga(X1, delcJ_in_ga(X1)) 114.75/31.09 delcJ_in_ga([]) -> delcJ_out_ga([], []) 114.75/31.09 delcJ_in_ga(.(f, X1)) -> U53_ga(X1, delcJ_in_ga(X1)) 114.75/31.09 U56_gga(X1, delcJ_out_ga(X1, X2)) -> delcG_out_gga(t, X1, .(f, .(f, X2))) 114.75/31.09 U53_ga(X1, delcJ_out_ga(X1, X2)) -> delcJ_out_ga(.(f, X1), .(f, X2)) 114.75/31.09 maxcF_in_ga([]) -> maxcF_out_ga([], f) 114.75/31.09 maxcF_in_ga(.(f, X1)) -> U51_ga(X1, maxcF_in_ga(X1)) 114.75/31.09 maxcF_in_ga(.(t, X1)) -> U52_ga(X1, maxcC_in_ga(X1)) 114.75/31.09 U52_ga(X1, maxcC_out_ga(X1, X2)) -> maxcF_out_ga(.(t, X1), X2) 114.75/31.09 U51_ga(X1, maxcF_out_ga(X1, X2)) -> maxcF_out_ga(.(f, X1), X2) 114.75/31.09 delcE_in_gga(t, X1) -> delcE_out_gga(t, X1, .(f, X1)) 114.75/31.09 delcE_in_gga(f, X1) -> U55_gga(X1, delcI_in_ga(.(f, X1))) 114.75/31.09 delcI_in_ga(.(f, X1)) -> delcI_out_ga(.(f, X1), X1) 114.75/31.09 U55_gga(X1, delcI_out_ga(.(f, X1), X2)) -> delcE_out_gga(f, X1, .(t, X2)) 114.75/31.09 delcD_in_gga(t, X1) -> delcD_out_gga(t, X1, .(t, X1)) 114.75/31.09 delcD_in_gga(f, X1) -> U54_gga(X1, delcI_in_ga(X1)) 114.75/31.10 delcI_in_ga([]) -> delcI_out_ga([], []) 114.75/31.10 delcI_in_ga(.(t, X1)) -> U50_ga(X1, delcI_in_ga(X1)) 114.75/31.10 U54_gga(X1, delcI_out_ga(X1, X2)) -> delcD_out_gga(f, X1, .(t, .(t, X2))) 114.75/31.10 U50_ga(X1, delcI_out_ga(X1, X2)) -> delcI_out_ga(.(t, X1), .(t, X2)) 114.75/31.10 114.75/31.10 The set Q consists of the following terms: 114.75/31.10 114.75/31.10 maxcC_in_ga(x0) 114.75/31.10 delcD_in_gga(x0, x1) 114.75/31.10 delcE_in_gga(x0, x1) 114.75/31.10 maxcF_in_ga(x0) 114.75/31.10 delcG_in_gga(x0, x1) 114.75/31.10 delcH_in_gga(x0, x1) 114.75/31.10 U48_ga(x0, x1) 114.75/31.10 U49_ga(x0, x1) 114.75/31.10 U54_gga(x0, x1) 114.75/31.10 U55_gga(x0, x1) 114.75/31.10 U51_ga(x0, x1) 114.75/31.10 U52_ga(x0, x1) 114.75/31.10 U56_gga(x0, x1) 114.75/31.10 U57_gga(x0, x1) 114.75/31.10 delcI_in_ga(x0) 114.75/31.10 delcJ_in_ga(x0) 114.75/31.10 U50_ga(x0, x1) 114.75/31.10 U53_ga(x0, x1) 114.75/31.10 114.75/31.10 We have to consider all (P,Q,R)-chains. 114.75/31.10 ---------------------------------------- 114.75/31.10 114.75/31.10 (50) QDPQMonotonicMRRProof (EQUIVALENT) 114.75/31.10 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. 114.75/31.10 114.75/31.10 114.75/31.10 Strictly oriented rules of the TRS R: 114.75/31.10 114.75/31.10 delcE_in_gga(f, X1) -> U55_gga(X1, delcI_in_ga(.(f, X1))) 114.75/31.10 U55_gga(X1, delcI_out_ga(.(f, X1), X2)) -> delcE_out_gga(f, X1, .(t, X2)) 114.75/31.10 114.75/31.10 Used ordering: Polynomial interpretation [POLO]: 114.75/31.10 114.75/31.10 POL(.(x_1, x_2)) = 0 114.75/31.10 POL(MAXSORTA_IN_GA(x_1)) = 0 114.75/31.10 POL(U12_GA(x_1, x_2)) = 0 114.75/31.10 POL(U13_GA(x_1, x_2)) = 0 114.75/31.10 POL(U18_GA(x_1, x_2)) = x_2 114.75/31.10 POL(U19_GA(x_1, x_2)) = 2*x_2 114.75/31.10 POL(U24_GA(x_1, x_2)) = 0 114.75/31.10 POL(U25_GA(x_1, x_2)) = 0 114.75/31.10 POL(U30_GA(x_1, x_2)) = 0 114.75/31.10 POL(U31_GA(x_1, x_2)) = 0 114.75/31.10 POL(U48_ga(x_1, x_2)) = 2*x_2 114.75/31.10 POL(U49_ga(x_1, x_2)) = x_2 114.75/31.10 POL(U50_ga(x_1, x_2)) = 0 114.75/31.10 POL(U51_ga(x_1, x_2)) = 0 114.75/31.10 POL(U52_ga(x_1, x_2)) = 0 114.75/31.10 POL(U53_ga(x_1, x_2)) = 0 114.75/31.10 POL(U54_gga(x_1, x_2)) = 0 114.75/31.10 POL(U55_gga(x_1, x_2)) = 1 114.75/31.10 POL(U56_gga(x_1, x_2)) = 0 114.75/31.10 POL(U57_gga(x_1, x_2)) = 0 114.75/31.10 POL([]) = 0 114.75/31.10 POL(delcD_in_gga(x_1, x_2)) = 0 114.75/31.10 POL(delcD_out_gga(x_1, x_2, x_3)) = 0 114.75/31.10 POL(delcE_in_gga(x_1, x_2)) = x_1 114.75/31.10 POL(delcE_out_gga(x_1, x_2, x_3)) = 0 114.75/31.10 POL(delcG_in_gga(x_1, x_2)) = 2*x_1 114.75/31.10 POL(delcG_out_gga(x_1, x_2, x_3)) = x_1 114.75/31.10 POL(delcH_in_gga(x_1, x_2)) = 0 114.75/31.10 POL(delcH_out_gga(x_1, x_2, x_3)) = 0 114.75/31.10 POL(delcI_in_ga(x_1)) = 0 114.75/31.10 POL(delcI_out_ga(x_1, x_2)) = 0 114.75/31.10 POL(delcJ_in_ga(x_1)) = 0 114.75/31.10 POL(delcJ_out_ga(x_1, x_2)) = 0 114.75/31.10 POL(f) = 2 114.75/31.10 POL(maxcC_in_ga(x_1)) = 0 114.75/31.10 POL(maxcC_out_ga(x_1, x_2)) = 2*x_2 114.75/31.10 POL(maxcF_in_ga(x_1)) = 1 114.75/31.10 POL(maxcF_out_ga(x_1, x_2)) = 0 114.75/31.10 POL(t) = 0 114.75/31.10 114.75/31.10 114.75/31.10 ---------------------------------------- 114.75/31.10 114.75/31.10 (51) 114.75/31.10 Obligation: 114.75/31.10 Q DP problem: 114.75/31.10 The TRS P consists of the following rules: 114.75/31.10 114.75/31.10 U12_GA(X1, maxcC_out_ga(X1, X2)) -> U13_GA(X1, delcD_in_gga(X2, X1)) 114.75/31.10 U13_GA(X1, delcD_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) 114.75/31.10 MAXSORTA_IN_GA(.(t, .(t, X1))) -> U12_GA(X1, maxcC_in_ga(X1)) 114.75/31.10 MAXSORTA_IN_GA(.(t, .(f, X1))) -> U18_GA(X1, maxcC_in_ga(X1)) 114.75/31.10 U18_GA(X1, maxcC_out_ga(X1, X2)) -> U19_GA(X1, delcE_in_gga(X2, X1)) 114.75/31.10 U19_GA(X1, delcE_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) 114.75/31.10 MAXSORTA_IN_GA(.(f, .(f, X1))) -> U24_GA(X1, maxcF_in_ga(X1)) 114.75/31.10 U24_GA(X1, maxcF_out_ga(X1, X2)) -> U25_GA(X1, delcG_in_gga(X2, X1)) 114.75/31.10 U25_GA(X1, delcG_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) 114.75/31.10 MAXSORTA_IN_GA(.(f, .(t, X1))) -> U30_GA(X1, maxcC_in_ga(X1)) 114.75/31.10 U30_GA(X1, maxcC_out_ga(X1, X2)) -> U31_GA(X1, delcH_in_gga(X2, X1)) 114.75/31.10 U31_GA(X1, delcH_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) 114.75/31.10 114.75/31.10 The TRS R consists of the following rules: 114.75/31.10 114.75/31.10 delcH_in_gga(f, X1) -> delcH_out_gga(f, X1, .(t, X1)) 114.75/31.10 delcH_in_gga(t, X1) -> U57_gga(X1, delcJ_in_ga(.(t, X1))) 114.75/31.10 delcJ_in_ga(.(t, X1)) -> delcJ_out_ga(.(t, X1), X1) 114.75/31.10 U57_gga(X1, delcJ_out_ga(.(t, X1), X2)) -> delcH_out_gga(t, X1, .(f, X2)) 114.75/31.10 maxcC_in_ga([]) -> maxcC_out_ga([], t) 114.75/31.10 maxcC_in_ga(.(t, X1)) -> U48_ga(X1, maxcC_in_ga(X1)) 114.75/31.10 maxcC_in_ga(.(f, X1)) -> U49_ga(X1, maxcC_in_ga(X1)) 114.75/31.10 U49_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(f, X1), X2) 114.75/31.10 U48_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(t, X1), X2) 114.75/31.10 delcG_in_gga(f, X1) -> delcG_out_gga(f, X1, .(f, X1)) 114.75/31.10 delcG_in_gga(t, X1) -> U56_gga(X1, delcJ_in_ga(X1)) 114.75/31.10 delcJ_in_ga([]) -> delcJ_out_ga([], []) 114.75/31.10 delcJ_in_ga(.(f, X1)) -> U53_ga(X1, delcJ_in_ga(X1)) 114.75/31.10 U56_gga(X1, delcJ_out_ga(X1, X2)) -> delcG_out_gga(t, X1, .(f, .(f, X2))) 114.75/31.10 U53_ga(X1, delcJ_out_ga(X1, X2)) -> delcJ_out_ga(.(f, X1), .(f, X2)) 114.75/31.10 maxcF_in_ga([]) -> maxcF_out_ga([], f) 114.75/31.10 maxcF_in_ga(.(f, X1)) -> U51_ga(X1, maxcF_in_ga(X1)) 114.75/31.10 maxcF_in_ga(.(t, X1)) -> U52_ga(X1, maxcC_in_ga(X1)) 114.75/31.10 U52_ga(X1, maxcC_out_ga(X1, X2)) -> maxcF_out_ga(.(t, X1), X2) 114.75/31.10 U51_ga(X1, maxcF_out_ga(X1, X2)) -> maxcF_out_ga(.(f, X1), X2) 114.75/31.10 delcE_in_gga(t, X1) -> delcE_out_gga(t, X1, .(f, X1)) 114.75/31.10 delcI_in_ga(.(f, X1)) -> delcI_out_ga(.(f, X1), X1) 114.75/31.10 delcD_in_gga(t, X1) -> delcD_out_gga(t, X1, .(t, X1)) 114.75/31.10 delcD_in_gga(f, X1) -> U54_gga(X1, delcI_in_ga(X1)) 114.75/31.10 delcI_in_ga([]) -> delcI_out_ga([], []) 114.75/31.10 delcI_in_ga(.(t, X1)) -> U50_ga(X1, delcI_in_ga(X1)) 114.75/31.10 U54_gga(X1, delcI_out_ga(X1, X2)) -> delcD_out_gga(f, X1, .(t, .(t, X2))) 114.75/31.10 U50_ga(X1, delcI_out_ga(X1, X2)) -> delcI_out_ga(.(t, X1), .(t, X2)) 114.75/31.10 114.75/31.10 The set Q consists of the following terms: 114.75/31.10 114.75/31.10 maxcC_in_ga(x0) 114.75/31.10 delcD_in_gga(x0, x1) 114.75/31.10 delcE_in_gga(x0, x1) 114.75/31.10 maxcF_in_ga(x0) 114.75/31.10 delcG_in_gga(x0, x1) 114.75/31.10 delcH_in_gga(x0, x1) 114.75/31.10 U48_ga(x0, x1) 114.75/31.10 U49_ga(x0, x1) 114.75/31.10 U54_gga(x0, x1) 114.75/31.10 U55_gga(x0, x1) 114.75/31.10 U51_ga(x0, x1) 114.75/31.10 U52_ga(x0, x1) 114.75/31.10 U56_gga(x0, x1) 114.75/31.10 U57_gga(x0, x1) 114.75/31.10 delcI_in_ga(x0) 114.75/31.10 delcJ_in_ga(x0) 114.75/31.10 U50_ga(x0, x1) 114.75/31.10 U53_ga(x0, x1) 114.75/31.10 114.75/31.10 We have to consider all (P,Q,R)-chains. 114.75/31.10 ---------------------------------------- 114.75/31.10 114.75/31.10 (52) QReductionProof (EQUIVALENT) 114.75/31.10 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 114.75/31.10 114.75/31.10 U55_gga(x0, x1) 114.75/31.10 114.75/31.10 114.75/31.10 ---------------------------------------- 114.75/31.10 114.75/31.10 (53) 114.75/31.10 Obligation: 114.75/31.10 Q DP problem: 114.75/31.10 The TRS P consists of the following rules: 114.75/31.10 114.75/31.10 U12_GA(X1, maxcC_out_ga(X1, X2)) -> U13_GA(X1, delcD_in_gga(X2, X1)) 114.75/31.10 U13_GA(X1, delcD_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) 114.75/31.10 MAXSORTA_IN_GA(.(t, .(t, X1))) -> U12_GA(X1, maxcC_in_ga(X1)) 114.75/31.10 MAXSORTA_IN_GA(.(t, .(f, X1))) -> U18_GA(X1, maxcC_in_ga(X1)) 114.75/31.10 U18_GA(X1, maxcC_out_ga(X1, X2)) -> U19_GA(X1, delcE_in_gga(X2, X1)) 114.75/31.10 U19_GA(X1, delcE_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) 114.75/31.10 MAXSORTA_IN_GA(.(f, .(f, X1))) -> U24_GA(X1, maxcF_in_ga(X1)) 114.75/31.10 U24_GA(X1, maxcF_out_ga(X1, X2)) -> U25_GA(X1, delcG_in_gga(X2, X1)) 114.75/31.10 U25_GA(X1, delcG_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) 114.75/31.10 MAXSORTA_IN_GA(.(f, .(t, X1))) -> U30_GA(X1, maxcC_in_ga(X1)) 114.75/31.10 U30_GA(X1, maxcC_out_ga(X1, X2)) -> U31_GA(X1, delcH_in_gga(X2, X1)) 114.75/31.10 U31_GA(X1, delcH_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) 114.75/31.10 114.75/31.10 The TRS R consists of the following rules: 114.75/31.10 114.75/31.10 delcH_in_gga(f, X1) -> delcH_out_gga(f, X1, .(t, X1)) 114.75/31.10 delcH_in_gga(t, X1) -> U57_gga(X1, delcJ_in_ga(.(t, X1))) 114.75/31.10 delcJ_in_ga(.(t, X1)) -> delcJ_out_ga(.(t, X1), X1) 114.75/31.10 U57_gga(X1, delcJ_out_ga(.(t, X1), X2)) -> delcH_out_gga(t, X1, .(f, X2)) 114.75/31.10 maxcC_in_ga([]) -> maxcC_out_ga([], t) 114.75/31.10 maxcC_in_ga(.(t, X1)) -> U48_ga(X1, maxcC_in_ga(X1)) 114.75/31.10 maxcC_in_ga(.(f, X1)) -> U49_ga(X1, maxcC_in_ga(X1)) 114.75/31.10 U49_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(f, X1), X2) 114.75/31.10 U48_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(t, X1), X2) 114.75/31.10 delcG_in_gga(f, X1) -> delcG_out_gga(f, X1, .(f, X1)) 114.75/31.10 delcG_in_gga(t, X1) -> U56_gga(X1, delcJ_in_ga(X1)) 114.75/31.10 delcJ_in_ga([]) -> delcJ_out_ga([], []) 114.75/31.10 delcJ_in_ga(.(f, X1)) -> U53_ga(X1, delcJ_in_ga(X1)) 114.75/31.10 U56_gga(X1, delcJ_out_ga(X1, X2)) -> delcG_out_gga(t, X1, .(f, .(f, X2))) 114.75/31.10 U53_ga(X1, delcJ_out_ga(X1, X2)) -> delcJ_out_ga(.(f, X1), .(f, X2)) 114.75/31.10 maxcF_in_ga([]) -> maxcF_out_ga([], f) 114.75/31.10 maxcF_in_ga(.(f, X1)) -> U51_ga(X1, maxcF_in_ga(X1)) 114.75/31.10 maxcF_in_ga(.(t, X1)) -> U52_ga(X1, maxcC_in_ga(X1)) 114.75/31.10 U52_ga(X1, maxcC_out_ga(X1, X2)) -> maxcF_out_ga(.(t, X1), X2) 114.75/31.10 U51_ga(X1, maxcF_out_ga(X1, X2)) -> maxcF_out_ga(.(f, X1), X2) 114.75/31.10 delcE_in_gga(t, X1) -> delcE_out_gga(t, X1, .(f, X1)) 114.75/31.10 delcI_in_ga(.(f, X1)) -> delcI_out_ga(.(f, X1), X1) 114.75/31.10 delcD_in_gga(t, X1) -> delcD_out_gga(t, X1, .(t, X1)) 114.75/31.10 delcD_in_gga(f, X1) -> U54_gga(X1, delcI_in_ga(X1)) 114.75/31.10 delcI_in_ga([]) -> delcI_out_ga([], []) 114.75/31.10 delcI_in_ga(.(t, X1)) -> U50_ga(X1, delcI_in_ga(X1)) 114.75/31.10 U54_gga(X1, delcI_out_ga(X1, X2)) -> delcD_out_gga(f, X1, .(t, .(t, X2))) 114.75/31.10 U50_ga(X1, delcI_out_ga(X1, X2)) -> delcI_out_ga(.(t, X1), .(t, X2)) 114.75/31.10 114.75/31.10 The set Q consists of the following terms: 114.75/31.10 114.75/31.10 maxcC_in_ga(x0) 114.75/31.10 delcD_in_gga(x0, x1) 114.75/31.10 delcE_in_gga(x0, x1) 114.75/31.10 maxcF_in_ga(x0) 114.75/31.10 delcG_in_gga(x0, x1) 114.75/31.10 delcH_in_gga(x0, x1) 114.75/31.10 U48_ga(x0, x1) 114.75/31.10 U49_ga(x0, x1) 114.75/31.10 U54_gga(x0, x1) 114.75/31.10 U51_ga(x0, x1) 114.75/31.10 U52_ga(x0, x1) 114.75/31.10 U56_gga(x0, x1) 114.75/31.10 U57_gga(x0, x1) 114.75/31.10 delcI_in_ga(x0) 114.75/31.10 delcJ_in_ga(x0) 114.75/31.10 U50_ga(x0, x1) 114.75/31.10 U53_ga(x0, x1) 114.75/31.10 114.75/31.10 We have to consider all (P,Q,R)-chains. 114.75/31.10 ---------------------------------------- 114.75/31.10 114.75/31.10 (54) TransformationProof (SOUND) 114.75/31.10 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]: 114.75/31.10 114.75/31.10 (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)))) 114.75/31.10 114.75/31.10 114.75/31.10 ---------------------------------------- 114.75/31.10 114.75/31.10 (55) 114.75/31.10 Obligation: 114.75/31.10 Q DP problem: 114.75/31.10 The TRS P consists of the following rules: 114.75/31.10 114.75/31.10 U12_GA(X1, maxcC_out_ga(X1, X2)) -> U13_GA(X1, delcD_in_gga(X2, X1)) 114.75/31.10 U13_GA(X1, delcD_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) 114.75/31.10 MAXSORTA_IN_GA(.(t, .(t, X1))) -> U12_GA(X1, maxcC_in_ga(X1)) 114.75/31.10 MAXSORTA_IN_GA(.(t, .(f, X1))) -> U18_GA(X1, maxcC_in_ga(X1)) 114.75/31.10 U19_GA(X1, delcE_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) 114.75/31.10 MAXSORTA_IN_GA(.(f, .(f, X1))) -> U24_GA(X1, maxcF_in_ga(X1)) 114.75/31.10 U24_GA(X1, maxcF_out_ga(X1, X2)) -> U25_GA(X1, delcG_in_gga(X2, X1)) 114.75/31.10 U25_GA(X1, delcG_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) 114.75/31.10 MAXSORTA_IN_GA(.(f, .(t, X1))) -> U30_GA(X1, maxcC_in_ga(X1)) 114.75/31.10 U30_GA(X1, maxcC_out_ga(X1, X2)) -> U31_GA(X1, delcH_in_gga(X2, X1)) 114.75/31.10 U31_GA(X1, delcH_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) 114.75/31.10 U18_GA(x0, maxcC_out_ga(x0, t)) -> U19_GA(x0, delcE_out_gga(t, x0, .(f, x0))) 114.75/31.10 114.75/31.10 The TRS R consists of the following rules: 114.75/31.10 114.75/31.10 delcH_in_gga(f, X1) -> delcH_out_gga(f, X1, .(t, X1)) 114.75/31.10 delcH_in_gga(t, X1) -> U57_gga(X1, delcJ_in_ga(.(t, X1))) 114.75/31.10 delcJ_in_ga(.(t, X1)) -> delcJ_out_ga(.(t, X1), X1) 114.75/31.10 U57_gga(X1, delcJ_out_ga(.(t, X1), X2)) -> delcH_out_gga(t, X1, .(f, X2)) 114.75/31.10 maxcC_in_ga([]) -> maxcC_out_ga([], t) 114.75/31.10 maxcC_in_ga(.(t, X1)) -> U48_ga(X1, maxcC_in_ga(X1)) 114.75/31.10 maxcC_in_ga(.(f, X1)) -> U49_ga(X1, maxcC_in_ga(X1)) 114.75/31.10 U49_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(f, X1), X2) 114.75/31.10 U48_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(t, X1), X2) 114.75/31.10 delcG_in_gga(f, X1) -> delcG_out_gga(f, X1, .(f, X1)) 114.75/31.10 delcG_in_gga(t, X1) -> U56_gga(X1, delcJ_in_ga(X1)) 114.75/31.10 delcJ_in_ga([]) -> delcJ_out_ga([], []) 114.75/31.10 delcJ_in_ga(.(f, X1)) -> U53_ga(X1, delcJ_in_ga(X1)) 114.75/31.10 U56_gga(X1, delcJ_out_ga(X1, X2)) -> delcG_out_gga(t, X1, .(f, .(f, X2))) 114.75/31.10 U53_ga(X1, delcJ_out_ga(X1, X2)) -> delcJ_out_ga(.(f, X1), .(f, X2)) 114.75/31.10 maxcF_in_ga([]) -> maxcF_out_ga([], f) 114.75/31.10 maxcF_in_ga(.(f, X1)) -> U51_ga(X1, maxcF_in_ga(X1)) 114.75/31.10 maxcF_in_ga(.(t, X1)) -> U52_ga(X1, maxcC_in_ga(X1)) 114.75/31.10 U52_ga(X1, maxcC_out_ga(X1, X2)) -> maxcF_out_ga(.(t, X1), X2) 114.75/31.10 U51_ga(X1, maxcF_out_ga(X1, X2)) -> maxcF_out_ga(.(f, X1), X2) 114.75/31.10 delcE_in_gga(t, X1) -> delcE_out_gga(t, X1, .(f, X1)) 114.75/31.10 delcI_in_ga(.(f, X1)) -> delcI_out_ga(.(f, X1), X1) 114.75/31.10 delcD_in_gga(t, X1) -> delcD_out_gga(t, X1, .(t, X1)) 114.75/31.10 delcD_in_gga(f, X1) -> U54_gga(X1, delcI_in_ga(X1)) 114.75/31.10 delcI_in_ga([]) -> delcI_out_ga([], []) 114.75/31.10 delcI_in_ga(.(t, X1)) -> U50_ga(X1, delcI_in_ga(X1)) 114.75/31.10 U54_gga(X1, delcI_out_ga(X1, X2)) -> delcD_out_gga(f, X1, .(t, .(t, X2))) 114.75/31.10 U50_ga(X1, delcI_out_ga(X1, X2)) -> delcI_out_ga(.(t, X1), .(t, X2)) 114.75/31.10 114.75/31.10 The set Q consists of the following terms: 114.75/31.10 114.75/31.10 maxcC_in_ga(x0) 114.75/31.10 delcD_in_gga(x0, x1) 114.75/31.10 delcE_in_gga(x0, x1) 114.75/31.10 maxcF_in_ga(x0) 114.75/31.10 delcG_in_gga(x0, x1) 114.75/31.10 delcH_in_gga(x0, x1) 114.75/31.10 U48_ga(x0, x1) 114.75/31.10 U49_ga(x0, x1) 114.75/31.10 U54_gga(x0, x1) 114.75/31.10 U51_ga(x0, x1) 114.75/31.10 U52_ga(x0, x1) 114.75/31.10 U56_gga(x0, x1) 114.75/31.10 U57_gga(x0, x1) 114.75/31.10 delcI_in_ga(x0) 114.75/31.10 delcJ_in_ga(x0) 114.75/31.10 U50_ga(x0, x1) 114.75/31.10 U53_ga(x0, x1) 114.75/31.10 114.75/31.10 We have to consider all (P,Q,R)-chains. 114.75/31.10 ---------------------------------------- 114.75/31.10 114.75/31.10 (56) UsableRulesProof (EQUIVALENT) 114.75/31.10 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. 114.75/31.10 ---------------------------------------- 114.75/31.10 114.75/31.10 (57) 114.75/31.10 Obligation: 114.75/31.10 Q DP problem: 114.75/31.10 The TRS P consists of the following rules: 114.75/31.10 114.75/31.10 U12_GA(X1, maxcC_out_ga(X1, X2)) -> U13_GA(X1, delcD_in_gga(X2, X1)) 114.75/31.10 U13_GA(X1, delcD_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) 114.75/31.10 MAXSORTA_IN_GA(.(t, .(t, X1))) -> U12_GA(X1, maxcC_in_ga(X1)) 114.75/31.10 MAXSORTA_IN_GA(.(t, .(f, X1))) -> U18_GA(X1, maxcC_in_ga(X1)) 114.75/31.10 U19_GA(X1, delcE_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) 114.75/31.10 MAXSORTA_IN_GA(.(f, .(f, X1))) -> U24_GA(X1, maxcF_in_ga(X1)) 114.75/31.10 U24_GA(X1, maxcF_out_ga(X1, X2)) -> U25_GA(X1, delcG_in_gga(X2, X1)) 114.75/31.10 U25_GA(X1, delcG_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) 114.75/31.10 MAXSORTA_IN_GA(.(f, .(t, X1))) -> U30_GA(X1, maxcC_in_ga(X1)) 114.75/31.10 U30_GA(X1, maxcC_out_ga(X1, X2)) -> U31_GA(X1, delcH_in_gga(X2, X1)) 114.75/31.10 U31_GA(X1, delcH_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) 114.75/31.10 U18_GA(x0, maxcC_out_ga(x0, t)) -> U19_GA(x0, delcE_out_gga(t, x0, .(f, x0))) 114.75/31.10 114.75/31.10 The TRS R consists of the following rules: 114.75/31.10 114.75/31.10 delcH_in_gga(f, X1) -> delcH_out_gga(f, X1, .(t, X1)) 114.75/31.10 delcH_in_gga(t, X1) -> U57_gga(X1, delcJ_in_ga(.(t, X1))) 114.75/31.10 delcJ_in_ga(.(t, X1)) -> delcJ_out_ga(.(t, X1), X1) 114.75/31.10 U57_gga(X1, delcJ_out_ga(.(t, X1), X2)) -> delcH_out_gga(t, X1, .(f, X2)) 114.75/31.10 maxcC_in_ga([]) -> maxcC_out_ga([], t) 114.75/31.10 maxcC_in_ga(.(t, X1)) -> U48_ga(X1, maxcC_in_ga(X1)) 114.75/31.10 maxcC_in_ga(.(f, X1)) -> U49_ga(X1, maxcC_in_ga(X1)) 114.75/31.10 U49_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(f, X1), X2) 114.75/31.10 U48_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(t, X1), X2) 114.75/31.10 delcG_in_gga(f, X1) -> delcG_out_gga(f, X1, .(f, X1)) 114.75/31.10 delcG_in_gga(t, X1) -> U56_gga(X1, delcJ_in_ga(X1)) 114.75/31.10 delcJ_in_ga([]) -> delcJ_out_ga([], []) 114.75/31.10 delcJ_in_ga(.(f, X1)) -> U53_ga(X1, delcJ_in_ga(X1)) 114.75/31.10 U56_gga(X1, delcJ_out_ga(X1, X2)) -> delcG_out_gga(t, X1, .(f, .(f, X2))) 114.75/31.10 U53_ga(X1, delcJ_out_ga(X1, X2)) -> delcJ_out_ga(.(f, X1), .(f, X2)) 114.75/31.10 maxcF_in_ga([]) -> maxcF_out_ga([], f) 114.75/31.10 maxcF_in_ga(.(f, X1)) -> U51_ga(X1, maxcF_in_ga(X1)) 114.75/31.10 maxcF_in_ga(.(t, X1)) -> U52_ga(X1, maxcC_in_ga(X1)) 114.75/31.10 U52_ga(X1, maxcC_out_ga(X1, X2)) -> maxcF_out_ga(.(t, X1), X2) 114.75/31.10 U51_ga(X1, maxcF_out_ga(X1, X2)) -> maxcF_out_ga(.(f, X1), X2) 114.75/31.10 delcD_in_gga(t, X1) -> delcD_out_gga(t, X1, .(t, X1)) 114.75/31.10 delcD_in_gga(f, X1) -> U54_gga(X1, delcI_in_ga(X1)) 114.75/31.10 delcI_in_ga(.(f, X1)) -> delcI_out_ga(.(f, X1), X1) 114.75/31.10 delcI_in_ga([]) -> delcI_out_ga([], []) 114.75/31.10 delcI_in_ga(.(t, X1)) -> U50_ga(X1, delcI_in_ga(X1)) 114.75/31.10 U54_gga(X1, delcI_out_ga(X1, X2)) -> delcD_out_gga(f, X1, .(t, .(t, X2))) 114.75/31.10 U50_ga(X1, delcI_out_ga(X1, X2)) -> delcI_out_ga(.(t, X1), .(t, X2)) 114.75/31.10 114.75/31.10 The set Q consists of the following terms: 114.75/31.10 114.75/31.10 maxcC_in_ga(x0) 114.75/31.10 delcD_in_gga(x0, x1) 114.75/31.10 delcE_in_gga(x0, x1) 114.75/31.10 maxcF_in_ga(x0) 114.75/31.10 delcG_in_gga(x0, x1) 114.75/31.10 delcH_in_gga(x0, x1) 114.75/31.10 U48_ga(x0, x1) 114.75/31.10 U49_ga(x0, x1) 114.75/31.10 U54_gga(x0, x1) 114.75/31.10 U51_ga(x0, x1) 114.75/31.10 U52_ga(x0, x1) 114.75/31.10 U56_gga(x0, x1) 114.75/31.10 U57_gga(x0, x1) 114.75/31.10 delcI_in_ga(x0) 114.75/31.10 delcJ_in_ga(x0) 114.75/31.10 U50_ga(x0, x1) 114.75/31.10 U53_ga(x0, x1) 114.75/31.10 114.75/31.10 We have to consider all (P,Q,R)-chains. 114.75/31.10 ---------------------------------------- 114.75/31.10 114.75/31.10 (58) QReductionProof (EQUIVALENT) 114.75/31.10 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 114.75/31.10 114.75/31.10 delcE_in_gga(x0, x1) 114.75/31.10 114.75/31.10 114.75/31.10 ---------------------------------------- 114.75/31.10 114.75/31.10 (59) 114.75/31.10 Obligation: 114.75/31.10 Q DP problem: 114.75/31.10 The TRS P consists of the following rules: 114.75/31.10 114.75/31.10 U12_GA(X1, maxcC_out_ga(X1, X2)) -> U13_GA(X1, delcD_in_gga(X2, X1)) 114.75/31.10 U13_GA(X1, delcD_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) 114.75/31.10 MAXSORTA_IN_GA(.(t, .(t, X1))) -> U12_GA(X1, maxcC_in_ga(X1)) 114.75/31.10 MAXSORTA_IN_GA(.(t, .(f, X1))) -> U18_GA(X1, maxcC_in_ga(X1)) 114.75/31.10 U19_GA(X1, delcE_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) 114.75/31.10 MAXSORTA_IN_GA(.(f, .(f, X1))) -> U24_GA(X1, maxcF_in_ga(X1)) 114.75/31.10 U24_GA(X1, maxcF_out_ga(X1, X2)) -> U25_GA(X1, delcG_in_gga(X2, X1)) 114.75/31.10 U25_GA(X1, delcG_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) 114.75/31.10 MAXSORTA_IN_GA(.(f, .(t, X1))) -> U30_GA(X1, maxcC_in_ga(X1)) 114.75/31.10 U30_GA(X1, maxcC_out_ga(X1, X2)) -> U31_GA(X1, delcH_in_gga(X2, X1)) 114.75/31.10 U31_GA(X1, delcH_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) 114.75/31.10 U18_GA(x0, maxcC_out_ga(x0, t)) -> U19_GA(x0, delcE_out_gga(t, x0, .(f, x0))) 114.75/31.10 114.75/31.10 The TRS R consists of the following rules: 114.75/31.10 114.75/31.10 delcH_in_gga(f, X1) -> delcH_out_gga(f, X1, .(t, X1)) 114.75/31.10 delcH_in_gga(t, X1) -> U57_gga(X1, delcJ_in_ga(.(t, X1))) 114.75/31.10 delcJ_in_ga(.(t, X1)) -> delcJ_out_ga(.(t, X1), X1) 114.75/31.10 U57_gga(X1, delcJ_out_ga(.(t, X1), X2)) -> delcH_out_gga(t, X1, .(f, X2)) 114.75/31.10 maxcC_in_ga([]) -> maxcC_out_ga([], t) 114.75/31.10 maxcC_in_ga(.(t, X1)) -> U48_ga(X1, maxcC_in_ga(X1)) 114.75/31.10 maxcC_in_ga(.(f, X1)) -> U49_ga(X1, maxcC_in_ga(X1)) 114.75/31.10 U49_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(f, X1), X2) 114.75/31.10 U48_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(t, X1), X2) 114.75/31.10 delcG_in_gga(f, X1) -> delcG_out_gga(f, X1, .(f, X1)) 114.75/31.10 delcG_in_gga(t, X1) -> U56_gga(X1, delcJ_in_ga(X1)) 114.75/31.10 delcJ_in_ga([]) -> delcJ_out_ga([], []) 114.75/31.10 delcJ_in_ga(.(f, X1)) -> U53_ga(X1, delcJ_in_ga(X1)) 114.75/31.10 U56_gga(X1, delcJ_out_ga(X1, X2)) -> delcG_out_gga(t, X1, .(f, .(f, X2))) 114.75/31.10 U53_ga(X1, delcJ_out_ga(X1, X2)) -> delcJ_out_ga(.(f, X1), .(f, X2)) 114.75/31.10 maxcF_in_ga([]) -> maxcF_out_ga([], f) 114.75/31.10 maxcF_in_ga(.(f, X1)) -> U51_ga(X1, maxcF_in_ga(X1)) 114.75/31.10 maxcF_in_ga(.(t, X1)) -> U52_ga(X1, maxcC_in_ga(X1)) 114.75/31.10 U52_ga(X1, maxcC_out_ga(X1, X2)) -> maxcF_out_ga(.(t, X1), X2) 114.75/31.10 U51_ga(X1, maxcF_out_ga(X1, X2)) -> maxcF_out_ga(.(f, X1), X2) 114.75/31.10 delcD_in_gga(t, X1) -> delcD_out_gga(t, X1, .(t, X1)) 114.75/31.10 delcD_in_gga(f, X1) -> U54_gga(X1, delcI_in_ga(X1)) 114.75/31.10 delcI_in_ga(.(f, X1)) -> delcI_out_ga(.(f, X1), X1) 114.75/31.10 delcI_in_ga([]) -> delcI_out_ga([], []) 114.75/31.10 delcI_in_ga(.(t, X1)) -> U50_ga(X1, delcI_in_ga(X1)) 114.75/31.10 U54_gga(X1, delcI_out_ga(X1, X2)) -> delcD_out_gga(f, X1, .(t, .(t, X2))) 114.75/31.10 U50_ga(X1, delcI_out_ga(X1, X2)) -> delcI_out_ga(.(t, X1), .(t, X2)) 114.75/31.10 114.75/31.10 The set Q consists of the following terms: 114.75/31.10 114.75/31.10 maxcC_in_ga(x0) 114.75/31.10 delcD_in_gga(x0, x1) 114.75/31.10 maxcF_in_ga(x0) 114.75/31.10 delcG_in_gga(x0, x1) 114.75/31.10 delcH_in_gga(x0, x1) 114.75/31.10 U48_ga(x0, x1) 114.75/31.10 U49_ga(x0, x1) 114.75/31.10 U54_gga(x0, x1) 114.75/31.10 U51_ga(x0, x1) 114.75/31.10 U52_ga(x0, x1) 114.75/31.10 U56_gga(x0, x1) 114.75/31.10 U57_gga(x0, x1) 114.75/31.10 delcI_in_ga(x0) 114.75/31.10 delcJ_in_ga(x0) 114.75/31.10 U50_ga(x0, x1) 114.75/31.10 U53_ga(x0, x1) 114.75/31.10 114.75/31.10 We have to consider all (P,Q,R)-chains. 114.75/31.10 ---------------------------------------- 114.75/31.10 114.75/31.10 (60) TransformationProof (EQUIVALENT) 114.75/31.10 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]: 114.75/31.10 114.75/31.10 (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))) 114.75/31.10 114.75/31.10 114.75/31.10 ---------------------------------------- 114.75/31.10 114.75/31.10 (61) 114.75/31.10 Obligation: 114.75/31.10 Q DP problem: 114.75/31.10 The TRS P consists of the following rules: 114.75/31.10 114.75/31.10 U12_GA(X1, maxcC_out_ga(X1, X2)) -> U13_GA(X1, delcD_in_gga(X2, X1)) 114.75/31.10 U13_GA(X1, delcD_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) 114.75/31.10 MAXSORTA_IN_GA(.(t, .(t, X1))) -> U12_GA(X1, maxcC_in_ga(X1)) 114.75/31.10 MAXSORTA_IN_GA(.(t, .(f, X1))) -> U18_GA(X1, maxcC_in_ga(X1)) 114.75/31.10 MAXSORTA_IN_GA(.(f, .(f, X1))) -> U24_GA(X1, maxcF_in_ga(X1)) 114.75/31.10 U24_GA(X1, maxcF_out_ga(X1, X2)) -> U25_GA(X1, delcG_in_gga(X2, X1)) 114.75/31.10 U25_GA(X1, delcG_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) 114.75/31.10 MAXSORTA_IN_GA(.(f, .(t, X1))) -> U30_GA(X1, maxcC_in_ga(X1)) 114.75/31.10 U30_GA(X1, maxcC_out_ga(X1, X2)) -> U31_GA(X1, delcH_in_gga(X2, X1)) 114.75/31.10 U31_GA(X1, delcH_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) 114.75/31.10 U18_GA(x0, maxcC_out_ga(x0, t)) -> U19_GA(x0, delcE_out_gga(t, x0, .(f, x0))) 114.75/31.10 U19_GA(z0, delcE_out_gga(t, z0, .(f, z0))) -> MAXSORTA_IN_GA(.(f, z0)) 114.75/31.10 114.75/31.10 The TRS R consists of the following rules: 114.75/31.10 114.75/31.10 delcH_in_gga(f, X1) -> delcH_out_gga(f, X1, .(t, X1)) 114.75/31.10 delcH_in_gga(t, X1) -> U57_gga(X1, delcJ_in_ga(.(t, X1))) 114.75/31.10 delcJ_in_ga(.(t, X1)) -> delcJ_out_ga(.(t, X1), X1) 114.75/31.10 U57_gga(X1, delcJ_out_ga(.(t, X1), X2)) -> delcH_out_gga(t, X1, .(f, X2)) 114.75/31.10 maxcC_in_ga([]) -> maxcC_out_ga([], t) 114.75/31.10 maxcC_in_ga(.(t, X1)) -> U48_ga(X1, maxcC_in_ga(X1)) 114.75/31.10 maxcC_in_ga(.(f, X1)) -> U49_ga(X1, maxcC_in_ga(X1)) 114.75/31.10 U49_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(f, X1), X2) 114.75/31.10 U48_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(t, X1), X2) 114.75/31.10 delcG_in_gga(f, X1) -> delcG_out_gga(f, X1, .(f, X1)) 114.75/31.10 delcG_in_gga(t, X1) -> U56_gga(X1, delcJ_in_ga(X1)) 114.75/31.10 delcJ_in_ga([]) -> delcJ_out_ga([], []) 114.75/31.10 delcJ_in_ga(.(f, X1)) -> U53_ga(X1, delcJ_in_ga(X1)) 114.75/31.10 U56_gga(X1, delcJ_out_ga(X1, X2)) -> delcG_out_gga(t, X1, .(f, .(f, X2))) 114.75/31.10 U53_ga(X1, delcJ_out_ga(X1, X2)) -> delcJ_out_ga(.(f, X1), .(f, X2)) 114.75/31.10 maxcF_in_ga([]) -> maxcF_out_ga([], f) 114.75/31.10 maxcF_in_ga(.(f, X1)) -> U51_ga(X1, maxcF_in_ga(X1)) 114.75/31.10 maxcF_in_ga(.(t, X1)) -> U52_ga(X1, maxcC_in_ga(X1)) 114.75/31.10 U52_ga(X1, maxcC_out_ga(X1, X2)) -> maxcF_out_ga(.(t, X1), X2) 114.75/31.10 U51_ga(X1, maxcF_out_ga(X1, X2)) -> maxcF_out_ga(.(f, X1), X2) 114.75/31.10 delcD_in_gga(t, X1) -> delcD_out_gga(t, X1, .(t, X1)) 114.75/31.10 delcD_in_gga(f, X1) -> U54_gga(X1, delcI_in_ga(X1)) 114.75/31.10 delcI_in_ga(.(f, X1)) -> delcI_out_ga(.(f, X1), X1) 114.75/31.10 delcI_in_ga([]) -> delcI_out_ga([], []) 114.75/31.10 delcI_in_ga(.(t, X1)) -> U50_ga(X1, delcI_in_ga(X1)) 114.75/31.10 U54_gga(X1, delcI_out_ga(X1, X2)) -> delcD_out_gga(f, X1, .(t, .(t, X2))) 114.75/31.10 U50_ga(X1, delcI_out_ga(X1, X2)) -> delcI_out_ga(.(t, X1), .(t, X2)) 114.75/31.10 114.75/31.10 The set Q consists of the following terms: 114.75/31.10 114.75/31.10 maxcC_in_ga(x0) 114.75/31.10 delcD_in_gga(x0, x1) 114.75/31.10 maxcF_in_ga(x0) 114.75/31.10 delcG_in_gga(x0, x1) 114.75/31.10 delcH_in_gga(x0, x1) 114.75/31.10 U48_ga(x0, x1) 114.75/31.10 U49_ga(x0, x1) 114.75/31.10 U54_gga(x0, x1) 114.75/31.10 U51_ga(x0, x1) 114.75/31.10 U52_ga(x0, x1) 114.75/31.10 U56_gga(x0, x1) 114.75/31.10 U57_gga(x0, x1) 114.75/31.10 delcI_in_ga(x0) 114.75/31.10 delcJ_in_ga(x0) 114.75/31.10 U50_ga(x0, x1) 114.75/31.10 U53_ga(x0, x1) 114.75/31.10 114.75/31.10 We have to consider all (P,Q,R)-chains. 114.75/31.10 ---------------------------------------- 114.75/31.10 114.75/31.10 (62) QDPQMonotonicMRRProof (EQUIVALENT) 114.75/31.10 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. 114.75/31.10 114.75/31.10 114.75/31.10 Strictly oriented rules of the TRS R: 114.75/31.10 114.75/31.10 delcD_in_gga(f, X1) -> U54_gga(X1, delcI_in_ga(X1)) 114.75/31.10 114.75/31.10 Used ordering: Polynomial interpretation [POLO]: 114.75/31.10 114.75/31.10 POL(.(x_1, x_2)) = 0 114.75/31.10 POL(MAXSORTA_IN_GA(x_1)) = 0 114.75/31.10 POL(U12_GA(x_1, x_2)) = x_2 114.75/31.10 POL(U13_GA(x_1, x_2)) = x_2 114.75/31.10 POL(U18_GA(x_1, x_2)) = 0 114.75/31.10 POL(U19_GA(x_1, x_2)) = 0 114.75/31.10 POL(U24_GA(x_1, x_2)) = 0 114.75/31.10 POL(U25_GA(x_1, x_2)) = 0 114.75/31.10 POL(U30_GA(x_1, x_2)) = 0 114.75/31.10 POL(U31_GA(x_1, x_2)) = 0 114.75/31.10 POL(U48_ga(x_1, x_2)) = 2*x_2 114.75/31.10 POL(U49_ga(x_1, x_2)) = 2*x_2 114.75/31.10 POL(U50_ga(x_1, x_2)) = 0 114.75/31.10 POL(U51_ga(x_1, x_2)) = 0 114.75/31.10 POL(U52_ga(x_1, x_2)) = 0 114.75/31.10 POL(U53_ga(x_1, x_2)) = 0 114.75/31.10 POL(U54_gga(x_1, x_2)) = 0 114.75/31.10 POL(U56_gga(x_1, x_2)) = 0 114.75/31.10 POL(U57_gga(x_1, x_2)) = 0 114.75/31.10 POL([]) = 0 114.75/31.10 POL(delcD_in_gga(x_1, x_2)) = x_1 114.75/31.10 POL(delcD_out_gga(x_1, x_2, x_3)) = 0 114.75/31.10 POL(delcE_out_gga(x_1, x_2, x_3)) = 2 + 2*x_1 + x_2 114.75/31.10 POL(delcG_in_gga(x_1, x_2)) = 0 114.75/31.10 POL(delcG_out_gga(x_1, x_2, x_3)) = 0 114.75/31.10 POL(delcH_in_gga(x_1, x_2)) = 0 114.75/31.10 POL(delcH_out_gga(x_1, x_2, x_3)) = 0 114.75/31.10 POL(delcI_in_ga(x_1)) = 0 114.75/31.10 POL(delcI_out_ga(x_1, x_2)) = 0 114.75/31.10 POL(delcJ_in_ga(x_1)) = 0 114.75/31.10 POL(delcJ_out_ga(x_1, x_2)) = 0 114.75/31.10 POL(f) = 2 114.75/31.10 POL(maxcC_in_ga(x_1)) = 0 114.75/31.10 POL(maxcC_out_ga(x_1, x_2)) = x_2 114.75/31.10 POL(maxcF_in_ga(x_1)) = 0 114.75/31.10 POL(maxcF_out_ga(x_1, x_2)) = 0 114.75/31.10 POL(t) = 0 114.75/31.10 114.75/31.10 114.75/31.10 ---------------------------------------- 114.75/31.10 114.75/31.10 (63) 114.75/31.10 Obligation: 114.75/31.10 Q DP problem: 114.75/31.10 The TRS P consists of the following rules: 114.75/31.10 114.75/31.10 U12_GA(X1, maxcC_out_ga(X1, X2)) -> U13_GA(X1, delcD_in_gga(X2, X1)) 114.75/31.10 U13_GA(X1, delcD_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) 114.75/31.10 MAXSORTA_IN_GA(.(t, .(t, X1))) -> U12_GA(X1, maxcC_in_ga(X1)) 114.75/31.10 MAXSORTA_IN_GA(.(t, .(f, X1))) -> U18_GA(X1, maxcC_in_ga(X1)) 114.75/31.10 MAXSORTA_IN_GA(.(f, .(f, X1))) -> U24_GA(X1, maxcF_in_ga(X1)) 114.75/31.10 U24_GA(X1, maxcF_out_ga(X1, X2)) -> U25_GA(X1, delcG_in_gga(X2, X1)) 114.75/31.10 U25_GA(X1, delcG_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) 114.75/31.10 MAXSORTA_IN_GA(.(f, .(t, X1))) -> U30_GA(X1, maxcC_in_ga(X1)) 114.75/31.10 U30_GA(X1, maxcC_out_ga(X1, X2)) -> U31_GA(X1, delcH_in_gga(X2, X1)) 114.75/31.10 U31_GA(X1, delcH_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) 114.75/31.10 U18_GA(x0, maxcC_out_ga(x0, t)) -> U19_GA(x0, delcE_out_gga(t, x0, .(f, x0))) 114.75/31.10 U19_GA(z0, delcE_out_gga(t, z0, .(f, z0))) -> MAXSORTA_IN_GA(.(f, z0)) 114.75/31.10 114.75/31.10 The TRS R consists of the following rules: 114.75/31.10 114.75/31.10 delcH_in_gga(f, X1) -> delcH_out_gga(f, X1, .(t, X1)) 114.75/31.10 delcH_in_gga(t, X1) -> U57_gga(X1, delcJ_in_ga(.(t, X1))) 114.75/31.10 delcJ_in_ga(.(t, X1)) -> delcJ_out_ga(.(t, X1), X1) 114.75/31.10 U57_gga(X1, delcJ_out_ga(.(t, X1), X2)) -> delcH_out_gga(t, X1, .(f, X2)) 114.75/31.10 maxcC_in_ga([]) -> maxcC_out_ga([], t) 114.75/31.10 maxcC_in_ga(.(t, X1)) -> U48_ga(X1, maxcC_in_ga(X1)) 114.75/31.10 maxcC_in_ga(.(f, X1)) -> U49_ga(X1, maxcC_in_ga(X1)) 114.75/31.10 U49_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(f, X1), X2) 114.75/31.10 U48_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(t, X1), X2) 114.75/31.10 delcG_in_gga(f, X1) -> delcG_out_gga(f, X1, .(f, X1)) 114.75/31.10 delcG_in_gga(t, X1) -> U56_gga(X1, delcJ_in_ga(X1)) 114.75/31.10 delcJ_in_ga([]) -> delcJ_out_ga([], []) 114.75/31.10 delcJ_in_ga(.(f, X1)) -> U53_ga(X1, delcJ_in_ga(X1)) 114.75/31.10 U56_gga(X1, delcJ_out_ga(X1, X2)) -> delcG_out_gga(t, X1, .(f, .(f, X2))) 114.75/31.10 U53_ga(X1, delcJ_out_ga(X1, X2)) -> delcJ_out_ga(.(f, X1), .(f, X2)) 114.75/31.10 maxcF_in_ga([]) -> maxcF_out_ga([], f) 114.75/31.10 maxcF_in_ga(.(f, X1)) -> U51_ga(X1, maxcF_in_ga(X1)) 114.75/31.10 maxcF_in_ga(.(t, X1)) -> U52_ga(X1, maxcC_in_ga(X1)) 114.75/31.10 U52_ga(X1, maxcC_out_ga(X1, X2)) -> maxcF_out_ga(.(t, X1), X2) 114.75/31.10 U51_ga(X1, maxcF_out_ga(X1, X2)) -> maxcF_out_ga(.(f, X1), X2) 114.75/31.10 delcD_in_gga(t, X1) -> delcD_out_gga(t, X1, .(t, X1)) 114.75/31.10 delcI_in_ga(.(f, X1)) -> delcI_out_ga(.(f, X1), X1) 114.75/31.10 delcI_in_ga([]) -> delcI_out_ga([], []) 114.75/31.10 delcI_in_ga(.(t, X1)) -> U50_ga(X1, delcI_in_ga(X1)) 114.75/31.10 U54_gga(X1, delcI_out_ga(X1, X2)) -> delcD_out_gga(f, X1, .(t, .(t, X2))) 114.75/31.10 U50_ga(X1, delcI_out_ga(X1, X2)) -> delcI_out_ga(.(t, X1), .(t, X2)) 114.75/31.10 114.75/31.10 The set Q consists of the following terms: 114.75/31.10 114.75/31.10 maxcC_in_ga(x0) 114.75/31.10 delcD_in_gga(x0, x1) 114.75/31.10 maxcF_in_ga(x0) 114.75/31.10 delcG_in_gga(x0, x1) 114.75/31.10 delcH_in_gga(x0, x1) 114.75/31.10 U48_ga(x0, x1) 114.75/31.10 U49_ga(x0, x1) 114.75/31.10 U54_gga(x0, x1) 114.75/31.10 U51_ga(x0, x1) 114.75/31.10 U52_ga(x0, x1) 114.75/31.10 U56_gga(x0, x1) 114.75/31.10 U57_gga(x0, x1) 114.75/31.10 delcI_in_ga(x0) 114.75/31.10 delcJ_in_ga(x0) 114.75/31.10 U50_ga(x0, x1) 114.75/31.10 U53_ga(x0, x1) 114.75/31.10 114.75/31.10 We have to consider all (P,Q,R)-chains. 114.75/31.10 ---------------------------------------- 114.75/31.10 114.75/31.10 (64) UsableRulesProof (EQUIVALENT) 114.75/31.10 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. 114.75/31.10 ---------------------------------------- 114.75/31.10 114.75/31.10 (65) 114.75/31.10 Obligation: 114.75/31.10 Q DP problem: 114.75/31.10 The TRS P consists of the following rules: 114.75/31.10 114.75/31.10 U12_GA(X1, maxcC_out_ga(X1, X2)) -> U13_GA(X1, delcD_in_gga(X2, X1)) 114.75/31.10 U13_GA(X1, delcD_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) 114.75/31.10 MAXSORTA_IN_GA(.(t, .(t, X1))) -> U12_GA(X1, maxcC_in_ga(X1)) 114.75/31.10 MAXSORTA_IN_GA(.(t, .(f, X1))) -> U18_GA(X1, maxcC_in_ga(X1)) 114.75/31.10 MAXSORTA_IN_GA(.(f, .(f, X1))) -> U24_GA(X1, maxcF_in_ga(X1)) 114.75/31.10 U24_GA(X1, maxcF_out_ga(X1, X2)) -> U25_GA(X1, delcG_in_gga(X2, X1)) 114.75/31.10 U25_GA(X1, delcG_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) 114.75/31.10 MAXSORTA_IN_GA(.(f, .(t, X1))) -> U30_GA(X1, maxcC_in_ga(X1)) 114.75/31.10 U30_GA(X1, maxcC_out_ga(X1, X2)) -> U31_GA(X1, delcH_in_gga(X2, X1)) 114.75/31.10 U31_GA(X1, delcH_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) 114.75/31.10 U18_GA(x0, maxcC_out_ga(x0, t)) -> U19_GA(x0, delcE_out_gga(t, x0, .(f, x0))) 114.75/31.10 U19_GA(z0, delcE_out_gga(t, z0, .(f, z0))) -> MAXSORTA_IN_GA(.(f, z0)) 114.75/31.10 114.75/31.10 The TRS R consists of the following rules: 114.75/31.10 114.75/31.10 delcH_in_gga(f, X1) -> delcH_out_gga(f, X1, .(t, X1)) 114.75/31.10 delcH_in_gga(t, X1) -> U57_gga(X1, delcJ_in_ga(.(t, X1))) 114.75/31.10 delcJ_in_ga(.(t, X1)) -> delcJ_out_ga(.(t, X1), X1) 114.75/31.10 U57_gga(X1, delcJ_out_ga(.(t, X1), X2)) -> delcH_out_gga(t, X1, .(f, X2)) 114.75/31.10 maxcC_in_ga([]) -> maxcC_out_ga([], t) 114.75/31.10 maxcC_in_ga(.(t, X1)) -> U48_ga(X1, maxcC_in_ga(X1)) 114.75/31.10 maxcC_in_ga(.(f, X1)) -> U49_ga(X1, maxcC_in_ga(X1)) 114.75/31.10 U49_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(f, X1), X2) 114.75/31.10 U48_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(t, X1), X2) 114.75/31.10 delcG_in_gga(f, X1) -> delcG_out_gga(f, X1, .(f, X1)) 114.75/31.10 delcG_in_gga(t, X1) -> U56_gga(X1, delcJ_in_ga(X1)) 114.75/31.10 delcJ_in_ga([]) -> delcJ_out_ga([], []) 114.75/31.10 delcJ_in_ga(.(f, X1)) -> U53_ga(X1, delcJ_in_ga(X1)) 114.75/31.10 U56_gga(X1, delcJ_out_ga(X1, X2)) -> delcG_out_gga(t, X1, .(f, .(f, X2))) 114.75/31.10 U53_ga(X1, delcJ_out_ga(X1, X2)) -> delcJ_out_ga(.(f, X1), .(f, X2)) 114.75/31.10 maxcF_in_ga([]) -> maxcF_out_ga([], f) 114.75/31.10 maxcF_in_ga(.(f, X1)) -> U51_ga(X1, maxcF_in_ga(X1)) 114.75/31.10 maxcF_in_ga(.(t, X1)) -> U52_ga(X1, maxcC_in_ga(X1)) 114.75/31.10 U52_ga(X1, maxcC_out_ga(X1, X2)) -> maxcF_out_ga(.(t, X1), X2) 114.75/31.10 U51_ga(X1, maxcF_out_ga(X1, X2)) -> maxcF_out_ga(.(f, X1), X2) 114.75/31.10 delcD_in_gga(t, X1) -> delcD_out_gga(t, X1, .(t, X1)) 114.75/31.10 114.75/31.10 The set Q consists of the following terms: 114.75/31.10 114.75/31.10 maxcC_in_ga(x0) 114.75/31.10 delcD_in_gga(x0, x1) 114.75/31.10 maxcF_in_ga(x0) 114.75/31.10 delcG_in_gga(x0, x1) 114.75/31.10 delcH_in_gga(x0, x1) 114.75/31.10 U48_ga(x0, x1) 114.75/31.10 U49_ga(x0, x1) 114.75/31.10 U54_gga(x0, x1) 114.75/31.10 U51_ga(x0, x1) 114.75/31.10 U52_ga(x0, x1) 114.75/31.10 U56_gga(x0, x1) 114.75/31.10 U57_gga(x0, x1) 114.75/31.10 delcI_in_ga(x0) 114.75/31.10 delcJ_in_ga(x0) 114.75/31.10 U50_ga(x0, x1) 114.75/31.10 U53_ga(x0, x1) 114.75/31.10 114.75/31.10 We have to consider all (P,Q,R)-chains. 114.75/31.10 ---------------------------------------- 114.75/31.10 114.75/31.10 (66) QReductionProof (EQUIVALENT) 114.75/31.10 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 114.75/31.10 114.75/31.10 U54_gga(x0, x1) 114.75/31.10 delcI_in_ga(x0) 114.75/31.10 U50_ga(x0, x1) 114.75/31.10 114.75/31.10 114.75/31.10 ---------------------------------------- 114.75/31.10 114.75/31.10 (67) 114.75/31.10 Obligation: 114.75/31.10 Q DP problem: 114.75/31.10 The TRS P consists of the following rules: 114.75/31.10 114.75/31.10 U12_GA(X1, maxcC_out_ga(X1, X2)) -> U13_GA(X1, delcD_in_gga(X2, X1)) 114.75/31.10 U13_GA(X1, delcD_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) 114.75/31.10 MAXSORTA_IN_GA(.(t, .(t, X1))) -> U12_GA(X1, maxcC_in_ga(X1)) 114.75/31.10 MAXSORTA_IN_GA(.(t, .(f, X1))) -> U18_GA(X1, maxcC_in_ga(X1)) 114.75/31.10 MAXSORTA_IN_GA(.(f, .(f, X1))) -> U24_GA(X1, maxcF_in_ga(X1)) 114.75/31.10 U24_GA(X1, maxcF_out_ga(X1, X2)) -> U25_GA(X1, delcG_in_gga(X2, X1)) 114.75/31.10 U25_GA(X1, delcG_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) 114.75/31.10 MAXSORTA_IN_GA(.(f, .(t, X1))) -> U30_GA(X1, maxcC_in_ga(X1)) 114.75/31.10 U30_GA(X1, maxcC_out_ga(X1, X2)) -> U31_GA(X1, delcH_in_gga(X2, X1)) 114.75/31.10 U31_GA(X1, delcH_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) 114.75/31.10 U18_GA(x0, maxcC_out_ga(x0, t)) -> U19_GA(x0, delcE_out_gga(t, x0, .(f, x0))) 114.75/31.10 U19_GA(z0, delcE_out_gga(t, z0, .(f, z0))) -> MAXSORTA_IN_GA(.(f, z0)) 114.75/31.10 114.75/31.10 The TRS R consists of the following rules: 114.75/31.10 114.75/31.10 delcH_in_gga(f, X1) -> delcH_out_gga(f, X1, .(t, X1)) 114.75/31.10 delcH_in_gga(t, X1) -> U57_gga(X1, delcJ_in_ga(.(t, X1))) 114.75/31.10 delcJ_in_ga(.(t, X1)) -> delcJ_out_ga(.(t, X1), X1) 114.75/31.10 U57_gga(X1, delcJ_out_ga(.(t, X1), X2)) -> delcH_out_gga(t, X1, .(f, X2)) 114.75/31.10 maxcC_in_ga([]) -> maxcC_out_ga([], t) 114.75/31.10 maxcC_in_ga(.(t, X1)) -> U48_ga(X1, maxcC_in_ga(X1)) 114.75/31.10 maxcC_in_ga(.(f, X1)) -> U49_ga(X1, maxcC_in_ga(X1)) 114.75/31.10 U49_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(f, X1), X2) 114.75/31.10 U48_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(t, X1), X2) 114.75/31.10 delcG_in_gga(f, X1) -> delcG_out_gga(f, X1, .(f, X1)) 114.75/31.10 delcG_in_gga(t, X1) -> U56_gga(X1, delcJ_in_ga(X1)) 114.75/31.10 delcJ_in_ga([]) -> delcJ_out_ga([], []) 114.75/31.10 delcJ_in_ga(.(f, X1)) -> U53_ga(X1, delcJ_in_ga(X1)) 114.75/31.10 U56_gga(X1, delcJ_out_ga(X1, X2)) -> delcG_out_gga(t, X1, .(f, .(f, X2))) 114.75/31.10 U53_ga(X1, delcJ_out_ga(X1, X2)) -> delcJ_out_ga(.(f, X1), .(f, X2)) 114.75/31.10 maxcF_in_ga([]) -> maxcF_out_ga([], f) 114.75/31.10 maxcF_in_ga(.(f, X1)) -> U51_ga(X1, maxcF_in_ga(X1)) 114.75/31.10 maxcF_in_ga(.(t, X1)) -> U52_ga(X1, maxcC_in_ga(X1)) 114.75/31.10 U52_ga(X1, maxcC_out_ga(X1, X2)) -> maxcF_out_ga(.(t, X1), X2) 114.75/31.10 U51_ga(X1, maxcF_out_ga(X1, X2)) -> maxcF_out_ga(.(f, X1), X2) 114.75/31.10 delcD_in_gga(t, X1) -> delcD_out_gga(t, X1, .(t, X1)) 114.75/31.10 114.75/31.10 The set Q consists of the following terms: 114.75/31.10 114.75/31.10 maxcC_in_ga(x0) 114.75/31.10 delcD_in_gga(x0, x1) 114.75/31.10 maxcF_in_ga(x0) 114.75/31.10 delcG_in_gga(x0, x1) 114.75/31.10 delcH_in_gga(x0, x1) 114.75/31.10 U48_ga(x0, x1) 114.75/31.10 U49_ga(x0, x1) 114.75/31.10 U51_ga(x0, x1) 114.75/31.10 U52_ga(x0, x1) 114.75/31.10 U56_gga(x0, x1) 114.75/31.10 U57_gga(x0, x1) 114.75/31.10 delcJ_in_ga(x0) 114.75/31.10 U53_ga(x0, x1) 114.75/31.10 114.75/31.10 We have to consider all (P,Q,R)-chains. 114.75/31.10 ---------------------------------------- 114.75/31.10 114.75/31.10 (68) TransformationProof (SOUND) 114.75/31.10 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]: 114.75/31.10 114.75/31.10 (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)))) 114.75/31.10 114.75/31.10 114.75/31.10 ---------------------------------------- 114.75/31.10 114.75/31.10 (69) 114.75/31.10 Obligation: 114.75/31.10 Q DP problem: 114.75/31.10 The TRS P consists of the following rules: 114.75/31.10 114.75/31.10 U13_GA(X1, delcD_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) 114.75/31.10 MAXSORTA_IN_GA(.(t, .(t, X1))) -> U12_GA(X1, maxcC_in_ga(X1)) 114.75/31.10 MAXSORTA_IN_GA(.(t, .(f, X1))) -> U18_GA(X1, maxcC_in_ga(X1)) 114.75/31.10 MAXSORTA_IN_GA(.(f, .(f, X1))) -> U24_GA(X1, maxcF_in_ga(X1)) 114.75/31.10 U24_GA(X1, maxcF_out_ga(X1, X2)) -> U25_GA(X1, delcG_in_gga(X2, X1)) 114.75/31.10 U25_GA(X1, delcG_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) 114.75/31.10 MAXSORTA_IN_GA(.(f, .(t, X1))) -> U30_GA(X1, maxcC_in_ga(X1)) 114.75/31.10 U30_GA(X1, maxcC_out_ga(X1, X2)) -> U31_GA(X1, delcH_in_gga(X2, X1)) 114.75/31.10 U31_GA(X1, delcH_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) 114.75/31.10 U18_GA(x0, maxcC_out_ga(x0, t)) -> U19_GA(x0, delcE_out_gga(t, x0, .(f, x0))) 114.75/31.10 U19_GA(z0, delcE_out_gga(t, z0, .(f, z0))) -> MAXSORTA_IN_GA(.(f, z0)) 114.75/31.10 U12_GA(x0, maxcC_out_ga(x0, t)) -> U13_GA(x0, delcD_out_gga(t, x0, .(t, x0))) 114.75/31.10 114.75/31.10 The TRS R consists of the following rules: 114.75/31.10 114.75/31.10 delcH_in_gga(f, X1) -> delcH_out_gga(f, X1, .(t, X1)) 114.75/31.10 delcH_in_gga(t, X1) -> U57_gga(X1, delcJ_in_ga(.(t, X1))) 114.75/31.10 delcJ_in_ga(.(t, X1)) -> delcJ_out_ga(.(t, X1), X1) 114.75/31.10 U57_gga(X1, delcJ_out_ga(.(t, X1), X2)) -> delcH_out_gga(t, X1, .(f, X2)) 114.75/31.10 maxcC_in_ga([]) -> maxcC_out_ga([], t) 114.75/31.10 maxcC_in_ga(.(t, X1)) -> U48_ga(X1, maxcC_in_ga(X1)) 114.75/31.10 maxcC_in_ga(.(f, X1)) -> U49_ga(X1, maxcC_in_ga(X1)) 114.75/31.10 U49_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(f, X1), X2) 114.75/31.10 U48_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(t, X1), X2) 114.75/31.10 delcG_in_gga(f, X1) -> delcG_out_gga(f, X1, .(f, X1)) 114.75/31.10 delcG_in_gga(t, X1) -> U56_gga(X1, delcJ_in_ga(X1)) 114.75/31.10 delcJ_in_ga([]) -> delcJ_out_ga([], []) 114.75/31.10 delcJ_in_ga(.(f, X1)) -> U53_ga(X1, delcJ_in_ga(X1)) 114.75/31.10 U56_gga(X1, delcJ_out_ga(X1, X2)) -> delcG_out_gga(t, X1, .(f, .(f, X2))) 114.75/31.10 U53_ga(X1, delcJ_out_ga(X1, X2)) -> delcJ_out_ga(.(f, X1), .(f, X2)) 114.75/31.10 maxcF_in_ga([]) -> maxcF_out_ga([], f) 114.75/31.10 maxcF_in_ga(.(f, X1)) -> U51_ga(X1, maxcF_in_ga(X1)) 114.75/31.10 maxcF_in_ga(.(t, X1)) -> U52_ga(X1, maxcC_in_ga(X1)) 114.75/31.10 U52_ga(X1, maxcC_out_ga(X1, X2)) -> maxcF_out_ga(.(t, X1), X2) 114.75/31.10 U51_ga(X1, maxcF_out_ga(X1, X2)) -> maxcF_out_ga(.(f, X1), X2) 114.75/31.10 delcD_in_gga(t, X1) -> delcD_out_gga(t, X1, .(t, X1)) 114.75/31.10 114.75/31.10 The set Q consists of the following terms: 114.75/31.10 114.75/31.10 maxcC_in_ga(x0) 114.75/31.10 delcD_in_gga(x0, x1) 114.75/31.10 maxcF_in_ga(x0) 114.75/31.10 delcG_in_gga(x0, x1) 114.75/31.10 delcH_in_gga(x0, x1) 114.75/31.10 U48_ga(x0, x1) 114.75/31.10 U49_ga(x0, x1) 114.75/31.10 U51_ga(x0, x1) 114.75/31.10 U52_ga(x0, x1) 114.75/31.10 U56_gga(x0, x1) 114.75/31.10 U57_gga(x0, x1) 114.75/31.10 delcJ_in_ga(x0) 114.75/31.10 U53_ga(x0, x1) 114.75/31.10 114.75/31.10 We have to consider all (P,Q,R)-chains. 114.75/31.10 ---------------------------------------- 114.75/31.10 114.75/31.10 (70) UsableRulesProof (EQUIVALENT) 114.75/31.10 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. 114.75/31.10 ---------------------------------------- 114.75/31.10 114.75/31.10 (71) 114.75/31.10 Obligation: 114.75/31.10 Q DP problem: 114.75/31.10 The TRS P consists of the following rules: 114.75/31.10 114.75/31.10 U13_GA(X1, delcD_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) 114.75/31.10 MAXSORTA_IN_GA(.(t, .(t, X1))) -> U12_GA(X1, maxcC_in_ga(X1)) 114.75/31.10 MAXSORTA_IN_GA(.(t, .(f, X1))) -> U18_GA(X1, maxcC_in_ga(X1)) 114.75/31.10 MAXSORTA_IN_GA(.(f, .(f, X1))) -> U24_GA(X1, maxcF_in_ga(X1)) 114.75/31.10 U24_GA(X1, maxcF_out_ga(X1, X2)) -> U25_GA(X1, delcG_in_gga(X2, X1)) 114.75/31.10 U25_GA(X1, delcG_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) 114.75/31.10 MAXSORTA_IN_GA(.(f, .(t, X1))) -> U30_GA(X1, maxcC_in_ga(X1)) 114.75/31.10 U30_GA(X1, maxcC_out_ga(X1, X2)) -> U31_GA(X1, delcH_in_gga(X2, X1)) 114.75/31.10 U31_GA(X1, delcH_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) 114.75/31.10 U18_GA(x0, maxcC_out_ga(x0, t)) -> U19_GA(x0, delcE_out_gga(t, x0, .(f, x0))) 114.75/31.10 U19_GA(z0, delcE_out_gga(t, z0, .(f, z0))) -> MAXSORTA_IN_GA(.(f, z0)) 114.75/31.10 U12_GA(x0, maxcC_out_ga(x0, t)) -> U13_GA(x0, delcD_out_gga(t, x0, .(t, x0))) 114.75/31.10 114.75/31.10 The TRS R consists of the following rules: 114.75/31.10 114.75/31.10 delcH_in_gga(f, X1) -> delcH_out_gga(f, X1, .(t, X1)) 114.75/31.10 delcH_in_gga(t, X1) -> U57_gga(X1, delcJ_in_ga(.(t, X1))) 114.75/31.10 delcJ_in_ga(.(t, X1)) -> delcJ_out_ga(.(t, X1), X1) 114.75/31.10 U57_gga(X1, delcJ_out_ga(.(t, X1), X2)) -> delcH_out_gga(t, X1, .(f, X2)) 114.75/31.10 maxcC_in_ga([]) -> maxcC_out_ga([], t) 114.75/31.10 maxcC_in_ga(.(t, X1)) -> U48_ga(X1, maxcC_in_ga(X1)) 114.75/31.10 maxcC_in_ga(.(f, X1)) -> U49_ga(X1, maxcC_in_ga(X1)) 114.75/31.10 U49_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(f, X1), X2) 114.75/31.10 U48_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(t, X1), X2) 114.75/31.10 delcG_in_gga(f, X1) -> delcG_out_gga(f, X1, .(f, X1)) 114.75/31.10 delcG_in_gga(t, X1) -> U56_gga(X1, delcJ_in_ga(X1)) 114.75/31.10 delcJ_in_ga([]) -> delcJ_out_ga([], []) 114.75/31.10 delcJ_in_ga(.(f, X1)) -> U53_ga(X1, delcJ_in_ga(X1)) 114.75/31.10 U56_gga(X1, delcJ_out_ga(X1, X2)) -> delcG_out_gga(t, X1, .(f, .(f, X2))) 114.75/31.10 U53_ga(X1, delcJ_out_ga(X1, X2)) -> delcJ_out_ga(.(f, X1), .(f, X2)) 114.75/31.10 maxcF_in_ga([]) -> maxcF_out_ga([], f) 114.75/31.10 maxcF_in_ga(.(f, X1)) -> U51_ga(X1, maxcF_in_ga(X1)) 114.75/31.10 maxcF_in_ga(.(t, X1)) -> U52_ga(X1, maxcC_in_ga(X1)) 114.75/31.10 U52_ga(X1, maxcC_out_ga(X1, X2)) -> maxcF_out_ga(.(t, X1), X2) 114.75/31.10 U51_ga(X1, maxcF_out_ga(X1, X2)) -> maxcF_out_ga(.(f, X1), X2) 114.75/31.10 114.75/31.10 The set Q consists of the following terms: 114.75/31.10 114.75/31.10 maxcC_in_ga(x0) 114.75/31.10 delcD_in_gga(x0, x1) 114.75/31.10 maxcF_in_ga(x0) 114.75/31.10 delcG_in_gga(x0, x1) 114.75/31.10 delcH_in_gga(x0, x1) 114.75/31.10 U48_ga(x0, x1) 114.75/31.10 U49_ga(x0, x1) 114.75/31.10 U51_ga(x0, x1) 114.75/31.10 U52_ga(x0, x1) 114.75/31.10 U56_gga(x0, x1) 114.75/31.10 U57_gga(x0, x1) 114.75/31.10 delcJ_in_ga(x0) 114.75/31.10 U53_ga(x0, x1) 114.75/31.10 114.75/31.10 We have to consider all (P,Q,R)-chains. 114.75/31.10 ---------------------------------------- 114.75/31.10 114.75/31.10 (72) QReductionProof (EQUIVALENT) 114.75/31.10 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 114.75/31.10 114.75/31.10 delcD_in_gga(x0, x1) 114.75/31.10 114.75/31.10 114.75/31.10 ---------------------------------------- 114.75/31.10 114.75/31.10 (73) 114.75/31.10 Obligation: 114.75/31.10 Q DP problem: 114.75/31.10 The TRS P consists of the following rules: 114.75/31.10 114.75/31.10 U13_GA(X1, delcD_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) 114.75/31.10 MAXSORTA_IN_GA(.(t, .(t, X1))) -> U12_GA(X1, maxcC_in_ga(X1)) 114.75/31.10 MAXSORTA_IN_GA(.(t, .(f, X1))) -> U18_GA(X1, maxcC_in_ga(X1)) 114.75/31.10 MAXSORTA_IN_GA(.(f, .(f, X1))) -> U24_GA(X1, maxcF_in_ga(X1)) 114.75/31.10 U24_GA(X1, maxcF_out_ga(X1, X2)) -> U25_GA(X1, delcG_in_gga(X2, X1)) 114.75/31.10 U25_GA(X1, delcG_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) 114.75/31.10 MAXSORTA_IN_GA(.(f, .(t, X1))) -> U30_GA(X1, maxcC_in_ga(X1)) 114.75/31.10 U30_GA(X1, maxcC_out_ga(X1, X2)) -> U31_GA(X1, delcH_in_gga(X2, X1)) 114.75/31.10 U31_GA(X1, delcH_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) 114.75/31.10 U18_GA(x0, maxcC_out_ga(x0, t)) -> U19_GA(x0, delcE_out_gga(t, x0, .(f, x0))) 114.75/31.10 U19_GA(z0, delcE_out_gga(t, z0, .(f, z0))) -> MAXSORTA_IN_GA(.(f, z0)) 114.75/31.10 U12_GA(x0, maxcC_out_ga(x0, t)) -> U13_GA(x0, delcD_out_gga(t, x0, .(t, x0))) 114.75/31.10 114.75/31.10 The TRS R consists of the following rules: 114.75/31.10 114.75/31.10 delcH_in_gga(f, X1) -> delcH_out_gga(f, X1, .(t, X1)) 114.75/31.10 delcH_in_gga(t, X1) -> U57_gga(X1, delcJ_in_ga(.(t, X1))) 114.75/31.10 delcJ_in_ga(.(t, X1)) -> delcJ_out_ga(.(t, X1), X1) 114.75/31.10 U57_gga(X1, delcJ_out_ga(.(t, X1), X2)) -> delcH_out_gga(t, X1, .(f, X2)) 114.75/31.10 maxcC_in_ga([]) -> maxcC_out_ga([], t) 114.75/31.10 maxcC_in_ga(.(t, X1)) -> U48_ga(X1, maxcC_in_ga(X1)) 114.75/31.10 maxcC_in_ga(.(f, X1)) -> U49_ga(X1, maxcC_in_ga(X1)) 114.75/31.10 U49_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(f, X1), X2) 114.75/31.10 U48_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(t, X1), X2) 114.75/31.10 delcG_in_gga(f, X1) -> delcG_out_gga(f, X1, .(f, X1)) 114.75/31.10 delcG_in_gga(t, X1) -> U56_gga(X1, delcJ_in_ga(X1)) 114.75/31.10 delcJ_in_ga([]) -> delcJ_out_ga([], []) 114.75/31.10 delcJ_in_ga(.(f, X1)) -> U53_ga(X1, delcJ_in_ga(X1)) 114.75/31.10 U56_gga(X1, delcJ_out_ga(X1, X2)) -> delcG_out_gga(t, X1, .(f, .(f, X2))) 114.75/31.10 U53_ga(X1, delcJ_out_ga(X1, X2)) -> delcJ_out_ga(.(f, X1), .(f, X2)) 114.75/31.10 maxcF_in_ga([]) -> maxcF_out_ga([], f) 114.75/31.10 maxcF_in_ga(.(f, X1)) -> U51_ga(X1, maxcF_in_ga(X1)) 114.75/31.10 maxcF_in_ga(.(t, X1)) -> U52_ga(X1, maxcC_in_ga(X1)) 114.75/31.10 U52_ga(X1, maxcC_out_ga(X1, X2)) -> maxcF_out_ga(.(t, X1), X2) 114.75/31.10 U51_ga(X1, maxcF_out_ga(X1, X2)) -> maxcF_out_ga(.(f, X1), X2) 114.75/31.10 114.75/31.10 The set Q consists of the following terms: 114.75/31.10 114.75/31.10 maxcC_in_ga(x0) 114.75/31.10 maxcF_in_ga(x0) 114.75/31.10 delcG_in_gga(x0, x1) 114.75/31.10 delcH_in_gga(x0, x1) 114.75/31.10 U48_ga(x0, x1) 114.75/31.10 U49_ga(x0, x1) 114.75/31.10 U51_ga(x0, x1) 114.75/31.10 U52_ga(x0, x1) 114.75/31.10 U56_gga(x0, x1) 114.75/31.10 U57_gga(x0, x1) 114.75/31.10 delcJ_in_ga(x0) 114.75/31.10 U53_ga(x0, x1) 114.75/31.10 114.75/31.10 We have to consider all (P,Q,R)-chains. 114.75/31.10 ---------------------------------------- 114.75/31.10 114.75/31.10 (74) TransformationProof (EQUIVALENT) 114.75/31.10 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]: 114.75/31.10 114.75/31.10 (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))) 114.75/31.10 114.75/31.10 114.75/31.10 ---------------------------------------- 114.75/31.10 114.75/31.10 (75) 114.75/31.10 Obligation: 114.75/31.10 Q DP problem: 114.75/31.10 The TRS P consists of the following rules: 114.75/31.10 114.75/31.10 MAXSORTA_IN_GA(.(t, .(t, X1))) -> U12_GA(X1, maxcC_in_ga(X1)) 114.75/31.10 MAXSORTA_IN_GA(.(t, .(f, X1))) -> U18_GA(X1, maxcC_in_ga(X1)) 114.75/31.10 MAXSORTA_IN_GA(.(f, .(f, X1))) -> U24_GA(X1, maxcF_in_ga(X1)) 114.75/31.10 U24_GA(X1, maxcF_out_ga(X1, X2)) -> U25_GA(X1, delcG_in_gga(X2, X1)) 114.75/31.10 U25_GA(X1, delcG_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) 114.75/31.10 MAXSORTA_IN_GA(.(f, .(t, X1))) -> U30_GA(X1, maxcC_in_ga(X1)) 114.75/31.10 U30_GA(X1, maxcC_out_ga(X1, X2)) -> U31_GA(X1, delcH_in_gga(X2, X1)) 114.75/31.10 U31_GA(X1, delcH_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) 114.75/31.10 U18_GA(x0, maxcC_out_ga(x0, t)) -> U19_GA(x0, delcE_out_gga(t, x0, .(f, x0))) 114.75/31.10 U19_GA(z0, delcE_out_gga(t, z0, .(f, z0))) -> MAXSORTA_IN_GA(.(f, z0)) 114.75/31.10 U12_GA(x0, maxcC_out_ga(x0, t)) -> U13_GA(x0, delcD_out_gga(t, x0, .(t, x0))) 114.75/31.10 U13_GA(z0, delcD_out_gga(t, z0, .(t, z0))) -> MAXSORTA_IN_GA(.(t, z0)) 114.75/31.10 114.75/31.10 The TRS R consists of the following rules: 114.75/31.10 114.75/31.10 delcH_in_gga(f, X1) -> delcH_out_gga(f, X1, .(t, X1)) 114.75/31.10 delcH_in_gga(t, X1) -> U57_gga(X1, delcJ_in_ga(.(t, X1))) 114.75/31.10 delcJ_in_ga(.(t, X1)) -> delcJ_out_ga(.(t, X1), X1) 114.75/31.10 U57_gga(X1, delcJ_out_ga(.(t, X1), X2)) -> delcH_out_gga(t, X1, .(f, X2)) 114.75/31.10 maxcC_in_ga([]) -> maxcC_out_ga([], t) 114.75/31.10 maxcC_in_ga(.(t, X1)) -> U48_ga(X1, maxcC_in_ga(X1)) 114.75/31.10 maxcC_in_ga(.(f, X1)) -> U49_ga(X1, maxcC_in_ga(X1)) 114.75/31.10 U49_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(f, X1), X2) 114.75/31.10 U48_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(t, X1), X2) 114.75/31.10 delcG_in_gga(f, X1) -> delcG_out_gga(f, X1, .(f, X1)) 114.75/31.10 delcG_in_gga(t, X1) -> U56_gga(X1, delcJ_in_ga(X1)) 114.75/31.10 delcJ_in_ga([]) -> delcJ_out_ga([], []) 114.75/31.10 delcJ_in_ga(.(f, X1)) -> U53_ga(X1, delcJ_in_ga(X1)) 114.75/31.10 U56_gga(X1, delcJ_out_ga(X1, X2)) -> delcG_out_gga(t, X1, .(f, .(f, X2))) 114.75/31.10 U53_ga(X1, delcJ_out_ga(X1, X2)) -> delcJ_out_ga(.(f, X1), .(f, X2)) 114.75/31.10 maxcF_in_ga([]) -> maxcF_out_ga([], f) 114.75/31.10 maxcF_in_ga(.(f, X1)) -> U51_ga(X1, maxcF_in_ga(X1)) 114.75/31.10 maxcF_in_ga(.(t, X1)) -> U52_ga(X1, maxcC_in_ga(X1)) 114.75/31.10 U52_ga(X1, maxcC_out_ga(X1, X2)) -> maxcF_out_ga(.(t, X1), X2) 114.75/31.10 U51_ga(X1, maxcF_out_ga(X1, X2)) -> maxcF_out_ga(.(f, X1), X2) 114.75/31.10 114.75/31.10 The set Q consists of the following terms: 114.75/31.10 114.75/31.10 maxcC_in_ga(x0) 114.75/31.10 maxcF_in_ga(x0) 114.75/31.10 delcG_in_gga(x0, x1) 114.75/31.10 delcH_in_gga(x0, x1) 114.75/31.10 U48_ga(x0, x1) 114.75/31.10 U49_ga(x0, x1) 114.75/31.10 U51_ga(x0, x1) 114.75/31.10 U52_ga(x0, x1) 114.75/31.10 U56_gga(x0, x1) 114.75/31.10 U57_gga(x0, x1) 114.75/31.10 delcJ_in_ga(x0) 114.75/31.10 U53_ga(x0, x1) 114.75/31.10 114.75/31.10 We have to consider all (P,Q,R)-chains. 114.75/31.10 ---------------------------------------- 114.75/31.10 114.75/31.10 (76) QDPQMonotonicMRRProof (EQUIVALENT) 114.75/31.10 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. 114.75/31.10 114.75/31.10 Strictly oriented dependency pairs: 114.75/31.10 114.75/31.10 MAXSORTA_IN_GA(.(t, .(t, X1))) -> U12_GA(X1, maxcC_in_ga(X1)) 114.75/31.10 MAXSORTA_IN_GA(.(t, .(f, X1))) -> U18_GA(X1, maxcC_in_ga(X1)) 114.75/31.10 U18_GA(x0, maxcC_out_ga(x0, t)) -> U19_GA(x0, delcE_out_gga(t, x0, .(f, x0))) 114.75/31.10 U12_GA(x0, maxcC_out_ga(x0, t)) -> U13_GA(x0, delcD_out_gga(t, x0, .(t, x0))) 114.75/31.10 114.75/31.10 Strictly oriented rules of the TRS R: 114.75/31.10 114.75/31.10 delcJ_in_ga(.(t, X1)) -> delcJ_out_ga(.(t, X1), X1) 114.75/31.10 114.75/31.10 Used ordering: Polynomial interpretation [POLO]: 114.75/31.10 114.75/31.10 POL(.(x_1, x_2)) = x_1 + x_2 114.75/31.10 POL(MAXSORTA_IN_GA(x_1)) = 2 + x_1 114.75/31.10 POL(U12_GA(x_1, x_2)) = 1 + x_1 + 2*x_2 114.75/31.10 POL(U13_GA(x_1, x_2)) = x_2 114.75/31.10 POL(U18_GA(x_1, x_2)) = 1 + x_1 + x_2 114.75/31.10 POL(U19_GA(x_1, x_2)) = 2 + x_1 114.75/31.10 POL(U24_GA(x_1, x_2)) = 2 + x_1 114.75/31.10 POL(U25_GA(x_1, x_2)) = 1 + x_2 114.75/31.10 POL(U30_GA(x_1, x_2)) = x_1 + 2*x_2 114.75/31.10 POL(U31_GA(x_1, x_2)) = 2 + x_2 114.75/31.10 POL(U48_ga(x_1, x_2)) = 2 114.75/31.10 POL(U49_ga(x_1, x_2)) = 2 114.75/31.10 POL(U51_ga(x_1, x_2)) = 0 114.75/31.10 POL(U52_ga(x_1, x_2)) = 0 114.75/31.10 POL(U53_ga(x_1, x_2)) = x_2 114.75/31.10 POL(U56_gga(x_1, x_2)) = 1 + x_2 114.75/31.10 POL(U57_gga(x_1, x_2)) = x_2 114.75/31.10 POL([]) = 0 114.75/31.10 POL(delcD_out_gga(x_1, x_2, x_3)) = 2*x_1 + x_2 114.75/31.10 POL(delcE_out_gga(x_1, x_2, x_3)) = 2 + x_2 114.75/31.10 POL(delcG_in_gga(x_1, x_2)) = 1 + x_2 114.75/31.10 POL(delcG_out_gga(x_1, x_2, x_3)) = 1 + x_3 114.75/31.10 POL(delcH_in_gga(x_1, x_2)) = 2 + x_2 114.75/31.10 POL(delcH_out_gga(x_1, x_2, x_3)) = x_3 114.75/31.10 POL(delcJ_in_ga(x_1)) = x_1 114.75/31.10 POL(delcJ_out_ga(x_1, x_2)) = x_2 114.75/31.10 POL(f) = 0 114.75/31.10 POL(maxcC_in_ga(x_1)) = 2 114.75/31.10 POL(maxcC_out_ga(x_1, x_2)) = 2 114.75/31.10 POL(maxcF_in_ga(x_1)) = 0 114.75/31.10 POL(maxcF_out_ga(x_1, x_2)) = 0 114.75/31.10 POL(t) = 2 114.75/31.10 114.75/31.10 114.75/31.10 ---------------------------------------- 114.75/31.10 114.75/31.10 (77) 114.75/31.10 Obligation: 114.75/31.10 Q DP problem: 114.75/31.10 The TRS P consists of the following rules: 114.75/31.10 114.75/31.10 MAXSORTA_IN_GA(.(f, .(f, X1))) -> U24_GA(X1, maxcF_in_ga(X1)) 114.75/31.10 U24_GA(X1, maxcF_out_ga(X1, X2)) -> U25_GA(X1, delcG_in_gga(X2, X1)) 114.75/31.10 U25_GA(X1, delcG_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) 114.75/31.10 MAXSORTA_IN_GA(.(f, .(t, X1))) -> U30_GA(X1, maxcC_in_ga(X1)) 114.75/31.10 U30_GA(X1, maxcC_out_ga(X1, X2)) -> U31_GA(X1, delcH_in_gga(X2, X1)) 114.75/31.10 U31_GA(X1, delcH_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) 114.75/31.10 U19_GA(z0, delcE_out_gga(t, z0, .(f, z0))) -> MAXSORTA_IN_GA(.(f, z0)) 114.88/31.10 U13_GA(z0, delcD_out_gga(t, z0, .(t, z0))) -> MAXSORTA_IN_GA(.(t, z0)) 114.88/31.10 114.88/31.10 The TRS R consists of the following rules: 114.88/31.10 114.88/31.10 delcH_in_gga(f, X1) -> delcH_out_gga(f, X1, .(t, X1)) 114.88/31.10 delcH_in_gga(t, X1) -> U57_gga(X1, delcJ_in_ga(.(t, X1))) 114.88/31.10 U57_gga(X1, delcJ_out_ga(.(t, X1), X2)) -> delcH_out_gga(t, X1, .(f, X2)) 114.88/31.10 maxcC_in_ga([]) -> maxcC_out_ga([], t) 114.88/31.10 maxcC_in_ga(.(t, X1)) -> U48_ga(X1, maxcC_in_ga(X1)) 114.88/31.10 maxcC_in_ga(.(f, X1)) -> U49_ga(X1, maxcC_in_ga(X1)) 114.88/31.10 U49_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(f, X1), X2) 114.88/31.10 U48_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(t, X1), X2) 114.88/31.10 delcG_in_gga(f, X1) -> delcG_out_gga(f, X1, .(f, X1)) 114.88/31.10 delcG_in_gga(t, X1) -> U56_gga(X1, delcJ_in_ga(X1)) 114.88/31.10 delcJ_in_ga([]) -> delcJ_out_ga([], []) 114.88/31.10 delcJ_in_ga(.(f, X1)) -> U53_ga(X1, delcJ_in_ga(X1)) 114.88/31.10 U56_gga(X1, delcJ_out_ga(X1, X2)) -> delcG_out_gga(t, X1, .(f, .(f, X2))) 114.88/31.10 U53_ga(X1, delcJ_out_ga(X1, X2)) -> delcJ_out_ga(.(f, X1), .(f, X2)) 114.88/31.10 maxcF_in_ga([]) -> maxcF_out_ga([], f) 114.88/31.10 maxcF_in_ga(.(f, X1)) -> U51_ga(X1, maxcF_in_ga(X1)) 114.88/31.10 maxcF_in_ga(.(t, X1)) -> U52_ga(X1, maxcC_in_ga(X1)) 114.88/31.10 U52_ga(X1, maxcC_out_ga(X1, X2)) -> maxcF_out_ga(.(t, X1), X2) 114.88/31.10 U51_ga(X1, maxcF_out_ga(X1, X2)) -> maxcF_out_ga(.(f, X1), X2) 114.88/31.10 114.88/31.10 The set Q consists of the following terms: 114.88/31.10 114.88/31.10 maxcC_in_ga(x0) 114.88/31.10 maxcF_in_ga(x0) 114.88/31.10 delcG_in_gga(x0, x1) 114.88/31.10 delcH_in_gga(x0, x1) 114.88/31.10 U48_ga(x0, x1) 114.88/31.10 U49_ga(x0, x1) 114.88/31.10 U51_ga(x0, x1) 114.88/31.10 U52_ga(x0, x1) 114.88/31.10 U56_gga(x0, x1) 114.88/31.10 U57_gga(x0, x1) 114.88/31.10 delcJ_in_ga(x0) 114.88/31.10 U53_ga(x0, x1) 114.88/31.10 114.88/31.10 We have to consider all (P,Q,R)-chains. 114.88/31.10 ---------------------------------------- 114.88/31.10 114.88/31.10 (78) DependencyGraphProof (EQUIVALENT) 114.88/31.10 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes. 114.88/31.10 ---------------------------------------- 114.88/31.10 114.88/31.10 (79) 114.88/31.10 Obligation: 114.88/31.10 Q DP problem: 114.88/31.10 The TRS P consists of the following rules: 114.88/31.10 114.88/31.10 U24_GA(X1, maxcF_out_ga(X1, X2)) -> U25_GA(X1, delcG_in_gga(X2, X1)) 114.88/31.10 U25_GA(X1, delcG_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) 114.88/31.10 MAXSORTA_IN_GA(.(f, .(f, X1))) -> U24_GA(X1, maxcF_in_ga(X1)) 114.88/31.10 MAXSORTA_IN_GA(.(f, .(t, X1))) -> U30_GA(X1, maxcC_in_ga(X1)) 114.88/31.10 U30_GA(X1, maxcC_out_ga(X1, X2)) -> U31_GA(X1, delcH_in_gga(X2, X1)) 114.88/31.10 U31_GA(X1, delcH_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) 114.88/31.10 114.88/31.10 The TRS R consists of the following rules: 114.88/31.10 114.88/31.10 delcH_in_gga(f, X1) -> delcH_out_gga(f, X1, .(t, X1)) 114.88/31.10 delcH_in_gga(t, X1) -> U57_gga(X1, delcJ_in_ga(.(t, X1))) 114.88/31.10 U57_gga(X1, delcJ_out_ga(.(t, X1), X2)) -> delcH_out_gga(t, X1, .(f, X2)) 114.88/31.10 maxcC_in_ga([]) -> maxcC_out_ga([], t) 114.88/31.10 maxcC_in_ga(.(t, X1)) -> U48_ga(X1, maxcC_in_ga(X1)) 114.88/31.10 maxcC_in_ga(.(f, X1)) -> U49_ga(X1, maxcC_in_ga(X1)) 114.88/31.10 U49_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(f, X1), X2) 114.88/31.10 U48_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(t, X1), X2) 114.88/31.10 delcG_in_gga(f, X1) -> delcG_out_gga(f, X1, .(f, X1)) 114.88/31.10 delcG_in_gga(t, X1) -> U56_gga(X1, delcJ_in_ga(X1)) 114.88/31.10 delcJ_in_ga([]) -> delcJ_out_ga([], []) 114.88/31.10 delcJ_in_ga(.(f, X1)) -> U53_ga(X1, delcJ_in_ga(X1)) 114.88/31.10 U56_gga(X1, delcJ_out_ga(X1, X2)) -> delcG_out_gga(t, X1, .(f, .(f, X2))) 114.88/31.10 U53_ga(X1, delcJ_out_ga(X1, X2)) -> delcJ_out_ga(.(f, X1), .(f, X2)) 114.88/31.10 maxcF_in_ga([]) -> maxcF_out_ga([], f) 114.88/31.10 maxcF_in_ga(.(f, X1)) -> U51_ga(X1, maxcF_in_ga(X1)) 114.88/31.10 maxcF_in_ga(.(t, X1)) -> U52_ga(X1, maxcC_in_ga(X1)) 114.88/31.10 U52_ga(X1, maxcC_out_ga(X1, X2)) -> maxcF_out_ga(.(t, X1), X2) 114.88/31.10 U51_ga(X1, maxcF_out_ga(X1, X2)) -> maxcF_out_ga(.(f, X1), X2) 114.88/31.10 114.88/31.10 The set Q consists of the following terms: 114.88/31.10 114.88/31.10 maxcC_in_ga(x0) 114.88/31.10 maxcF_in_ga(x0) 114.88/31.10 delcG_in_gga(x0, x1) 114.88/31.10 delcH_in_gga(x0, x1) 114.88/31.10 U48_ga(x0, x1) 114.88/31.10 U49_ga(x0, x1) 114.88/31.10 U51_ga(x0, x1) 114.88/31.10 U52_ga(x0, x1) 114.88/31.10 U56_gga(x0, x1) 114.88/31.10 U57_gga(x0, x1) 114.88/31.10 delcJ_in_ga(x0) 114.88/31.10 U53_ga(x0, x1) 114.88/31.10 114.88/31.10 We have to consider all (P,Q,R)-chains. 114.88/31.10 ---------------------------------------- 114.88/31.10 114.88/31.10 (80) UsableRulesProof (EQUIVALENT) 114.88/31.10 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. 114.88/31.10 ---------------------------------------- 114.88/31.10 114.88/31.10 (81) 114.88/31.10 Obligation: 114.88/31.10 Q DP problem: 114.88/31.10 The TRS P consists of the following rules: 114.88/31.10 114.88/31.10 U24_GA(X1, maxcF_out_ga(X1, X2)) -> U25_GA(X1, delcG_in_gga(X2, X1)) 114.88/31.10 U25_GA(X1, delcG_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) 114.88/31.10 MAXSORTA_IN_GA(.(f, .(f, X1))) -> U24_GA(X1, maxcF_in_ga(X1)) 114.88/31.10 MAXSORTA_IN_GA(.(f, .(t, X1))) -> U30_GA(X1, maxcC_in_ga(X1)) 114.88/31.10 U30_GA(X1, maxcC_out_ga(X1, X2)) -> U31_GA(X1, delcH_in_gga(X2, X1)) 114.88/31.10 U31_GA(X1, delcH_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) 114.88/31.10 114.88/31.10 The TRS R consists of the following rules: 114.88/31.10 114.88/31.10 delcH_in_gga(f, X1) -> delcH_out_gga(f, X1, .(t, X1)) 114.88/31.10 delcH_in_gga(t, X1) -> U57_gga(X1, delcJ_in_ga(.(t, X1))) 114.88/31.10 maxcC_in_ga([]) -> maxcC_out_ga([], t) 114.88/31.10 maxcC_in_ga(.(t, X1)) -> U48_ga(X1, maxcC_in_ga(X1)) 114.88/31.10 maxcC_in_ga(.(f, X1)) -> U49_ga(X1, maxcC_in_ga(X1)) 114.88/31.10 U49_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(f, X1), X2) 114.88/31.10 U48_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(t, X1), X2) 114.88/31.10 maxcF_in_ga([]) -> maxcF_out_ga([], f) 114.88/31.10 maxcF_in_ga(.(f, X1)) -> U51_ga(X1, maxcF_in_ga(X1)) 114.88/31.10 maxcF_in_ga(.(t, X1)) -> U52_ga(X1, maxcC_in_ga(X1)) 114.88/31.10 U52_ga(X1, maxcC_out_ga(X1, X2)) -> maxcF_out_ga(.(t, X1), X2) 114.88/31.10 U51_ga(X1, maxcF_out_ga(X1, X2)) -> maxcF_out_ga(.(f, X1), X2) 114.88/31.10 delcG_in_gga(f, X1) -> delcG_out_gga(f, X1, .(f, X1)) 114.88/31.10 delcG_in_gga(t, X1) -> U56_gga(X1, delcJ_in_ga(X1)) 114.88/31.10 delcJ_in_ga([]) -> delcJ_out_ga([], []) 114.88/31.10 delcJ_in_ga(.(f, X1)) -> U53_ga(X1, delcJ_in_ga(X1)) 114.88/31.10 U56_gga(X1, delcJ_out_ga(X1, X2)) -> delcG_out_gga(t, X1, .(f, .(f, X2))) 114.88/31.10 U53_ga(X1, delcJ_out_ga(X1, X2)) -> delcJ_out_ga(.(f, X1), .(f, X2)) 114.88/31.10 114.88/31.10 The set Q consists of the following terms: 114.88/31.10 114.88/31.10 maxcC_in_ga(x0) 114.88/31.10 maxcF_in_ga(x0) 114.88/31.10 delcG_in_gga(x0, x1) 114.88/31.10 delcH_in_gga(x0, x1) 114.88/31.10 U48_ga(x0, x1) 114.88/31.10 U49_ga(x0, x1) 114.88/31.10 U51_ga(x0, x1) 114.88/31.10 U52_ga(x0, x1) 114.88/31.10 U56_gga(x0, x1) 114.88/31.10 U57_gga(x0, x1) 114.88/31.10 delcJ_in_ga(x0) 114.88/31.10 U53_ga(x0, x1) 114.88/31.10 114.88/31.10 We have to consider all (P,Q,R)-chains. 114.88/31.10 ---------------------------------------- 114.88/31.10 114.88/31.10 (82) QDPQMonotonicMRRProof (EQUIVALENT) 114.88/31.10 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. 114.88/31.10 114.88/31.10 114.88/31.10 Strictly oriented rules of the TRS R: 114.88/31.10 114.88/31.10 delcH_in_gga(t, X1) -> U57_gga(X1, delcJ_in_ga(.(t, X1))) 114.88/31.10 114.88/31.10 Used ordering: Polynomial interpretation [POLO]: 114.88/31.10 114.88/31.10 POL(.(x_1, x_2)) = 2 + 2*x_2 114.88/31.10 POL(MAXSORTA_IN_GA(x_1)) = 2 114.88/31.10 POL(U24_GA(x_1, x_2)) = 2 114.88/31.10 POL(U25_GA(x_1, x_2)) = 2 114.88/31.10 POL(U30_GA(x_1, x_2)) = 2 114.88/31.10 POL(U31_GA(x_1, x_2)) = x_2 114.88/31.10 POL(U48_ga(x_1, x_2)) = 0 114.88/31.10 POL(U49_ga(x_1, x_2)) = 2 114.88/31.10 POL(U51_ga(x_1, x_2)) = 2 + 2*x_1 114.88/31.10 POL(U52_ga(x_1, x_2)) = 2 + 2*x_1 114.88/31.10 POL(U53_ga(x_1, x_2)) = 0 114.88/31.10 POL(U56_gga(x_1, x_2)) = 2 114.88/31.10 POL(U57_gga(x_1, x_2)) = 0 114.88/31.10 POL([]) = 0 114.88/31.10 POL(delcG_in_gga(x_1, x_2)) = 2*x_1 114.88/31.10 POL(delcG_out_gga(x_1, x_2, x_3)) = 0 114.88/31.10 POL(delcH_in_gga(x_1, x_2)) = 2 114.88/31.10 POL(delcH_out_gga(x_1, x_2, x_3)) = 2 + 2*x_1 114.88/31.10 POL(delcJ_in_ga(x_1)) = 0 114.88/31.10 POL(delcJ_out_ga(x_1, x_2)) = 0 114.88/31.10 POL(f) = 0 114.88/31.10 POL(maxcC_in_ga(x_1)) = 2 + x_1 114.88/31.10 POL(maxcC_out_ga(x_1, x_2)) = 0 114.88/31.10 POL(maxcF_in_ga(x_1)) = 2*x_1 114.88/31.10 POL(maxcF_out_ga(x_1, x_2)) = 0 114.88/31.10 POL(t) = 2 114.88/31.10 114.88/31.10 114.88/31.10 ---------------------------------------- 114.88/31.10 114.88/31.10 (83) 114.88/31.10 Obligation: 114.88/31.10 Q DP problem: 114.88/31.10 The TRS P consists of the following rules: 114.88/31.10 114.88/31.10 U24_GA(X1, maxcF_out_ga(X1, X2)) -> U25_GA(X1, delcG_in_gga(X2, X1)) 114.88/31.10 U25_GA(X1, delcG_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) 114.88/31.10 MAXSORTA_IN_GA(.(f, .(f, X1))) -> U24_GA(X1, maxcF_in_ga(X1)) 114.88/31.10 MAXSORTA_IN_GA(.(f, .(t, X1))) -> U30_GA(X1, maxcC_in_ga(X1)) 114.88/31.10 U30_GA(X1, maxcC_out_ga(X1, X2)) -> U31_GA(X1, delcH_in_gga(X2, X1)) 114.88/31.10 U31_GA(X1, delcH_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) 114.88/31.10 114.88/31.10 The TRS R consists of the following rules: 114.88/31.10 114.88/31.10 delcH_in_gga(f, X1) -> delcH_out_gga(f, X1, .(t, X1)) 114.88/31.10 maxcC_in_ga([]) -> maxcC_out_ga([], t) 114.88/31.10 maxcC_in_ga(.(t, X1)) -> U48_ga(X1, maxcC_in_ga(X1)) 114.88/31.10 maxcC_in_ga(.(f, X1)) -> U49_ga(X1, maxcC_in_ga(X1)) 114.88/31.10 U49_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(f, X1), X2) 114.88/31.10 U48_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(t, X1), X2) 114.88/31.10 maxcF_in_ga([]) -> maxcF_out_ga([], f) 114.88/31.10 maxcF_in_ga(.(f, X1)) -> U51_ga(X1, maxcF_in_ga(X1)) 114.88/31.10 maxcF_in_ga(.(t, X1)) -> U52_ga(X1, maxcC_in_ga(X1)) 114.88/31.10 U52_ga(X1, maxcC_out_ga(X1, X2)) -> maxcF_out_ga(.(t, X1), X2) 114.88/31.10 U51_ga(X1, maxcF_out_ga(X1, X2)) -> maxcF_out_ga(.(f, X1), X2) 114.88/31.10 delcG_in_gga(f, X1) -> delcG_out_gga(f, X1, .(f, X1)) 114.88/31.10 delcG_in_gga(t, X1) -> U56_gga(X1, delcJ_in_ga(X1)) 114.88/31.10 delcJ_in_ga([]) -> delcJ_out_ga([], []) 114.88/31.10 delcJ_in_ga(.(f, X1)) -> U53_ga(X1, delcJ_in_ga(X1)) 114.88/31.10 U56_gga(X1, delcJ_out_ga(X1, X2)) -> delcG_out_gga(t, X1, .(f, .(f, X2))) 114.88/31.10 U53_ga(X1, delcJ_out_ga(X1, X2)) -> delcJ_out_ga(.(f, X1), .(f, X2)) 114.88/31.10 114.88/31.10 The set Q consists of the following terms: 114.88/31.10 114.88/31.10 maxcC_in_ga(x0) 114.88/31.10 maxcF_in_ga(x0) 114.88/31.10 delcG_in_gga(x0, x1) 114.88/31.10 delcH_in_gga(x0, x1) 114.88/31.10 U48_ga(x0, x1) 114.88/31.10 U49_ga(x0, x1) 114.88/31.10 U51_ga(x0, x1) 114.88/31.10 U52_ga(x0, x1) 114.88/31.10 U56_gga(x0, x1) 114.88/31.10 U57_gga(x0, x1) 114.88/31.10 delcJ_in_ga(x0) 114.88/31.10 U53_ga(x0, x1) 114.88/31.10 114.88/31.10 We have to consider all (P,Q,R)-chains. 114.88/31.10 ---------------------------------------- 114.88/31.10 114.88/31.10 (84) QReductionProof (EQUIVALENT) 114.88/31.10 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 114.88/31.10 114.88/31.10 U57_gga(x0, x1) 114.88/31.10 114.88/31.10 114.88/31.10 ---------------------------------------- 114.88/31.10 114.88/31.10 (85) 114.88/31.10 Obligation: 114.88/31.10 Q DP problem: 114.88/31.10 The TRS P consists of the following rules: 114.88/31.10 114.88/31.10 U24_GA(X1, maxcF_out_ga(X1, X2)) -> U25_GA(X1, delcG_in_gga(X2, X1)) 114.88/31.10 U25_GA(X1, delcG_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) 114.88/31.10 MAXSORTA_IN_GA(.(f, .(f, X1))) -> U24_GA(X1, maxcF_in_ga(X1)) 114.88/31.10 MAXSORTA_IN_GA(.(f, .(t, X1))) -> U30_GA(X1, maxcC_in_ga(X1)) 114.88/31.10 U30_GA(X1, maxcC_out_ga(X1, X2)) -> U31_GA(X1, delcH_in_gga(X2, X1)) 114.88/31.10 U31_GA(X1, delcH_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) 114.88/31.10 114.88/31.10 The TRS R consists of the following rules: 114.88/31.10 114.88/31.10 delcH_in_gga(f, X1) -> delcH_out_gga(f, X1, .(t, X1)) 114.88/31.10 maxcC_in_ga([]) -> maxcC_out_ga([], t) 114.88/31.10 maxcC_in_ga(.(t, X1)) -> U48_ga(X1, maxcC_in_ga(X1)) 114.88/31.10 maxcC_in_ga(.(f, X1)) -> U49_ga(X1, maxcC_in_ga(X1)) 114.88/31.10 U49_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(f, X1), X2) 114.88/31.10 U48_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(t, X1), X2) 114.88/31.10 maxcF_in_ga([]) -> maxcF_out_ga([], f) 114.88/31.10 maxcF_in_ga(.(f, X1)) -> U51_ga(X1, maxcF_in_ga(X1)) 114.88/31.10 maxcF_in_ga(.(t, X1)) -> U52_ga(X1, maxcC_in_ga(X1)) 114.88/31.10 U52_ga(X1, maxcC_out_ga(X1, X2)) -> maxcF_out_ga(.(t, X1), X2) 114.88/31.10 U51_ga(X1, maxcF_out_ga(X1, X2)) -> maxcF_out_ga(.(f, X1), X2) 114.88/31.10 delcG_in_gga(f, X1) -> delcG_out_gga(f, X1, .(f, X1)) 114.88/31.10 delcG_in_gga(t, X1) -> U56_gga(X1, delcJ_in_ga(X1)) 114.88/31.10 delcJ_in_ga([]) -> delcJ_out_ga([], []) 114.88/31.10 delcJ_in_ga(.(f, X1)) -> U53_ga(X1, delcJ_in_ga(X1)) 114.88/31.10 U56_gga(X1, delcJ_out_ga(X1, X2)) -> delcG_out_gga(t, X1, .(f, .(f, X2))) 114.88/31.10 U53_ga(X1, delcJ_out_ga(X1, X2)) -> delcJ_out_ga(.(f, X1), .(f, X2)) 114.88/31.10 114.88/31.10 The set Q consists of the following terms: 114.88/31.10 114.88/31.10 maxcC_in_ga(x0) 114.88/31.10 maxcF_in_ga(x0) 114.88/31.10 delcG_in_gga(x0, x1) 114.88/31.10 delcH_in_gga(x0, x1) 114.88/31.10 U48_ga(x0, x1) 114.88/31.10 U49_ga(x0, x1) 114.88/31.10 U51_ga(x0, x1) 114.88/31.10 U52_ga(x0, x1) 114.88/31.10 U56_gga(x0, x1) 114.88/31.10 delcJ_in_ga(x0) 114.88/31.10 U53_ga(x0, x1) 114.88/31.10 114.88/31.10 We have to consider all (P,Q,R)-chains. 114.88/31.10 ---------------------------------------- 114.88/31.10 114.88/31.10 (86) TransformationProof (SOUND) 114.88/31.10 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]: 114.88/31.10 114.88/31.10 (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)))) 114.88/31.10 114.88/31.10 114.88/31.10 ---------------------------------------- 114.88/31.10 114.88/31.10 (87) 114.88/31.10 Obligation: 114.88/31.10 Q DP problem: 114.88/31.10 The TRS P consists of the following rules: 114.88/31.10 114.88/31.10 U24_GA(X1, maxcF_out_ga(X1, X2)) -> U25_GA(X1, delcG_in_gga(X2, X1)) 114.88/31.10 U25_GA(X1, delcG_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) 114.88/31.10 MAXSORTA_IN_GA(.(f, .(f, X1))) -> U24_GA(X1, maxcF_in_ga(X1)) 114.88/31.10 MAXSORTA_IN_GA(.(f, .(t, X1))) -> U30_GA(X1, maxcC_in_ga(X1)) 114.88/31.10 U31_GA(X1, delcH_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) 114.88/31.10 U30_GA(x0, maxcC_out_ga(x0, f)) -> U31_GA(x0, delcH_out_gga(f, x0, .(t, x0))) 114.88/31.10 114.88/31.10 The TRS R consists of the following rules: 114.88/31.10 114.88/31.10 delcH_in_gga(f, X1) -> delcH_out_gga(f, X1, .(t, X1)) 114.88/31.10 maxcC_in_ga([]) -> maxcC_out_ga([], t) 114.88/31.10 maxcC_in_ga(.(t, X1)) -> U48_ga(X1, maxcC_in_ga(X1)) 114.88/31.10 maxcC_in_ga(.(f, X1)) -> U49_ga(X1, maxcC_in_ga(X1)) 114.88/31.10 U49_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(f, X1), X2) 114.88/31.10 U48_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(t, X1), X2) 114.88/31.10 maxcF_in_ga([]) -> maxcF_out_ga([], f) 114.88/31.10 maxcF_in_ga(.(f, X1)) -> U51_ga(X1, maxcF_in_ga(X1)) 114.88/31.10 maxcF_in_ga(.(t, X1)) -> U52_ga(X1, maxcC_in_ga(X1)) 114.88/31.10 U52_ga(X1, maxcC_out_ga(X1, X2)) -> maxcF_out_ga(.(t, X1), X2) 114.88/31.10 U51_ga(X1, maxcF_out_ga(X1, X2)) -> maxcF_out_ga(.(f, X1), X2) 114.88/31.10 delcG_in_gga(f, X1) -> delcG_out_gga(f, X1, .(f, X1)) 114.88/31.10 delcG_in_gga(t, X1) -> U56_gga(X1, delcJ_in_ga(X1)) 114.88/31.10 delcJ_in_ga([]) -> delcJ_out_ga([], []) 114.88/31.10 delcJ_in_ga(.(f, X1)) -> U53_ga(X1, delcJ_in_ga(X1)) 114.88/31.10 U56_gga(X1, delcJ_out_ga(X1, X2)) -> delcG_out_gga(t, X1, .(f, .(f, X2))) 114.88/31.10 U53_ga(X1, delcJ_out_ga(X1, X2)) -> delcJ_out_ga(.(f, X1), .(f, X2)) 114.88/31.10 114.88/31.10 The set Q consists of the following terms: 114.88/31.10 114.88/31.10 maxcC_in_ga(x0) 114.88/31.10 maxcF_in_ga(x0) 114.88/31.10 delcG_in_gga(x0, x1) 114.88/31.10 delcH_in_gga(x0, x1) 114.88/31.10 U48_ga(x0, x1) 114.88/31.10 U49_ga(x0, x1) 114.88/31.10 U51_ga(x0, x1) 114.88/31.10 U52_ga(x0, x1) 114.88/31.10 U56_gga(x0, x1) 114.88/31.10 delcJ_in_ga(x0) 114.88/31.10 U53_ga(x0, x1) 114.88/31.10 114.88/31.10 We have to consider all (P,Q,R)-chains. 114.88/31.10 ---------------------------------------- 114.88/31.10 114.88/31.10 (88) UsableRulesProof (EQUIVALENT) 114.88/31.10 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. 114.88/31.10 ---------------------------------------- 114.88/31.10 114.88/31.10 (89) 114.88/31.10 Obligation: 114.88/31.10 Q DP problem: 114.88/31.10 The TRS P consists of the following rules: 114.88/31.10 114.88/31.10 U24_GA(X1, maxcF_out_ga(X1, X2)) -> U25_GA(X1, delcG_in_gga(X2, X1)) 114.88/31.10 U25_GA(X1, delcG_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) 114.88/31.10 MAXSORTA_IN_GA(.(f, .(f, X1))) -> U24_GA(X1, maxcF_in_ga(X1)) 114.88/31.10 MAXSORTA_IN_GA(.(f, .(t, X1))) -> U30_GA(X1, maxcC_in_ga(X1)) 114.88/31.10 U31_GA(X1, delcH_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) 114.88/31.10 U30_GA(x0, maxcC_out_ga(x0, f)) -> U31_GA(x0, delcH_out_gga(f, x0, .(t, x0))) 114.88/31.10 114.88/31.10 The TRS R consists of the following rules: 114.88/31.10 114.88/31.10 maxcC_in_ga([]) -> maxcC_out_ga([], t) 114.88/31.10 maxcC_in_ga(.(t, X1)) -> U48_ga(X1, maxcC_in_ga(X1)) 114.88/31.10 maxcC_in_ga(.(f, X1)) -> U49_ga(X1, maxcC_in_ga(X1)) 114.88/31.10 U49_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(f, X1), X2) 114.88/31.10 U48_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(t, X1), X2) 114.88/31.10 maxcF_in_ga([]) -> maxcF_out_ga([], f) 114.88/31.10 maxcF_in_ga(.(f, X1)) -> U51_ga(X1, maxcF_in_ga(X1)) 114.88/31.10 maxcF_in_ga(.(t, X1)) -> U52_ga(X1, maxcC_in_ga(X1)) 114.88/31.10 U52_ga(X1, maxcC_out_ga(X1, X2)) -> maxcF_out_ga(.(t, X1), X2) 114.88/31.10 U51_ga(X1, maxcF_out_ga(X1, X2)) -> maxcF_out_ga(.(f, X1), X2) 114.88/31.10 delcG_in_gga(f, X1) -> delcG_out_gga(f, X1, .(f, X1)) 114.88/31.10 delcG_in_gga(t, X1) -> U56_gga(X1, delcJ_in_ga(X1)) 114.88/31.10 delcJ_in_ga([]) -> delcJ_out_ga([], []) 114.88/31.10 delcJ_in_ga(.(f, X1)) -> U53_ga(X1, delcJ_in_ga(X1)) 114.88/31.10 U56_gga(X1, delcJ_out_ga(X1, X2)) -> delcG_out_gga(t, X1, .(f, .(f, X2))) 114.88/31.10 U53_ga(X1, delcJ_out_ga(X1, X2)) -> delcJ_out_ga(.(f, X1), .(f, X2)) 114.88/31.10 114.88/31.10 The set Q consists of the following terms: 114.88/31.10 114.88/31.10 maxcC_in_ga(x0) 114.88/31.10 maxcF_in_ga(x0) 114.88/31.10 delcG_in_gga(x0, x1) 114.88/31.10 delcH_in_gga(x0, x1) 114.88/31.10 U48_ga(x0, x1) 114.88/31.10 U49_ga(x0, x1) 114.88/31.10 U51_ga(x0, x1) 114.88/31.10 U52_ga(x0, x1) 114.88/31.10 U56_gga(x0, x1) 114.88/31.10 delcJ_in_ga(x0) 114.88/31.10 U53_ga(x0, x1) 114.88/31.10 114.88/31.10 We have to consider all (P,Q,R)-chains. 114.88/31.10 ---------------------------------------- 114.88/31.10 114.88/31.10 (90) QReductionProof (EQUIVALENT) 114.88/31.10 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 114.88/31.10 114.88/31.10 delcH_in_gga(x0, x1) 114.88/31.10 114.88/31.10 114.88/31.10 ---------------------------------------- 114.88/31.10 114.88/31.10 (91) 114.88/31.10 Obligation: 114.88/31.10 Q DP problem: 114.88/31.10 The TRS P consists of the following rules: 114.88/31.10 114.88/31.10 U24_GA(X1, maxcF_out_ga(X1, X2)) -> U25_GA(X1, delcG_in_gga(X2, X1)) 114.88/31.10 U25_GA(X1, delcG_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) 114.88/31.10 MAXSORTA_IN_GA(.(f, .(f, X1))) -> U24_GA(X1, maxcF_in_ga(X1)) 114.88/31.10 MAXSORTA_IN_GA(.(f, .(t, X1))) -> U30_GA(X1, maxcC_in_ga(X1)) 114.88/31.10 U31_GA(X1, delcH_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) 114.88/31.10 U30_GA(x0, maxcC_out_ga(x0, f)) -> U31_GA(x0, delcH_out_gga(f, x0, .(t, x0))) 114.88/31.10 114.88/31.10 The TRS R consists of the following rules: 114.88/31.10 114.88/31.10 maxcC_in_ga([]) -> maxcC_out_ga([], t) 114.88/31.10 maxcC_in_ga(.(t, X1)) -> U48_ga(X1, maxcC_in_ga(X1)) 114.88/31.10 maxcC_in_ga(.(f, X1)) -> U49_ga(X1, maxcC_in_ga(X1)) 114.88/31.10 U49_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(f, X1), X2) 114.88/31.10 U48_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(t, X1), X2) 114.88/31.10 maxcF_in_ga([]) -> maxcF_out_ga([], f) 114.88/31.10 maxcF_in_ga(.(f, X1)) -> U51_ga(X1, maxcF_in_ga(X1)) 114.88/31.10 maxcF_in_ga(.(t, X1)) -> U52_ga(X1, maxcC_in_ga(X1)) 114.88/31.10 U52_ga(X1, maxcC_out_ga(X1, X2)) -> maxcF_out_ga(.(t, X1), X2) 114.88/31.10 U51_ga(X1, maxcF_out_ga(X1, X2)) -> maxcF_out_ga(.(f, X1), X2) 114.88/31.10 delcG_in_gga(f, X1) -> delcG_out_gga(f, X1, .(f, X1)) 114.88/31.10 delcG_in_gga(t, X1) -> U56_gga(X1, delcJ_in_ga(X1)) 114.88/31.10 delcJ_in_ga([]) -> delcJ_out_ga([], []) 114.88/31.10 delcJ_in_ga(.(f, X1)) -> U53_ga(X1, delcJ_in_ga(X1)) 114.88/31.10 U56_gga(X1, delcJ_out_ga(X1, X2)) -> delcG_out_gga(t, X1, .(f, .(f, X2))) 114.88/31.10 U53_ga(X1, delcJ_out_ga(X1, X2)) -> delcJ_out_ga(.(f, X1), .(f, X2)) 114.88/31.10 114.88/31.10 The set Q consists of the following terms: 114.88/31.10 114.88/31.10 maxcC_in_ga(x0) 114.88/31.10 maxcF_in_ga(x0) 114.88/31.10 delcG_in_gga(x0, x1) 114.88/31.10 U48_ga(x0, x1) 114.88/31.10 U49_ga(x0, x1) 114.88/31.10 U51_ga(x0, x1) 114.88/31.10 U52_ga(x0, x1) 114.88/31.10 U56_gga(x0, x1) 114.88/31.10 delcJ_in_ga(x0) 114.88/31.10 U53_ga(x0, x1) 114.88/31.10 114.88/31.10 We have to consider all (P,Q,R)-chains. 114.88/31.10 ---------------------------------------- 114.88/31.10 114.88/31.10 (92) TransformationProof (EQUIVALENT) 114.88/31.10 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]: 114.88/31.10 114.88/31.10 (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))) 114.88/31.10 114.88/31.10 114.88/31.10 ---------------------------------------- 114.88/31.10 114.88/31.10 (93) 114.88/31.10 Obligation: 114.88/31.10 Q DP problem: 114.88/31.10 The TRS P consists of the following rules: 114.88/31.10 114.88/31.10 U24_GA(X1, maxcF_out_ga(X1, X2)) -> U25_GA(X1, delcG_in_gga(X2, X1)) 114.88/31.10 U25_GA(X1, delcG_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) 114.88/31.10 MAXSORTA_IN_GA(.(f, .(f, X1))) -> U24_GA(X1, maxcF_in_ga(X1)) 114.88/31.10 MAXSORTA_IN_GA(.(f, .(t, X1))) -> U30_GA(X1, maxcC_in_ga(X1)) 114.88/31.10 U30_GA(x0, maxcC_out_ga(x0, f)) -> U31_GA(x0, delcH_out_gga(f, x0, .(t, x0))) 114.88/31.10 U31_GA(z0, delcH_out_gga(f, z0, .(t, z0))) -> MAXSORTA_IN_GA(.(t, z0)) 114.88/31.10 114.88/31.10 The TRS R consists of the following rules: 114.88/31.10 114.88/31.10 maxcC_in_ga([]) -> maxcC_out_ga([], t) 114.88/31.10 maxcC_in_ga(.(t, X1)) -> U48_ga(X1, maxcC_in_ga(X1)) 114.88/31.10 maxcC_in_ga(.(f, X1)) -> U49_ga(X1, maxcC_in_ga(X1)) 114.88/31.10 U49_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(f, X1), X2) 114.88/31.10 U48_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(t, X1), X2) 114.88/31.10 maxcF_in_ga([]) -> maxcF_out_ga([], f) 114.88/31.10 maxcF_in_ga(.(f, X1)) -> U51_ga(X1, maxcF_in_ga(X1)) 114.88/31.10 maxcF_in_ga(.(t, X1)) -> U52_ga(X1, maxcC_in_ga(X1)) 114.88/31.10 U52_ga(X1, maxcC_out_ga(X1, X2)) -> maxcF_out_ga(.(t, X1), X2) 114.88/31.10 U51_ga(X1, maxcF_out_ga(X1, X2)) -> maxcF_out_ga(.(f, X1), X2) 114.88/31.10 delcG_in_gga(f, X1) -> delcG_out_gga(f, X1, .(f, X1)) 114.88/31.10 delcG_in_gga(t, X1) -> U56_gga(X1, delcJ_in_ga(X1)) 114.88/31.10 delcJ_in_ga([]) -> delcJ_out_ga([], []) 114.88/31.10 delcJ_in_ga(.(f, X1)) -> U53_ga(X1, delcJ_in_ga(X1)) 114.88/31.10 U56_gga(X1, delcJ_out_ga(X1, X2)) -> delcG_out_gga(t, X1, .(f, .(f, X2))) 114.88/31.10 U53_ga(X1, delcJ_out_ga(X1, X2)) -> delcJ_out_ga(.(f, X1), .(f, X2)) 114.88/31.10 114.88/31.10 The set Q consists of the following terms: 114.88/31.10 114.88/31.10 maxcC_in_ga(x0) 114.88/31.10 maxcF_in_ga(x0) 114.88/31.10 delcG_in_gga(x0, x1) 114.88/31.10 U48_ga(x0, x1) 114.88/31.10 U49_ga(x0, x1) 114.88/31.10 U51_ga(x0, x1) 114.88/31.10 U52_ga(x0, x1) 114.88/31.10 U56_gga(x0, x1) 114.88/31.10 delcJ_in_ga(x0) 114.88/31.10 U53_ga(x0, x1) 114.88/31.10 114.88/31.10 We have to consider all (P,Q,R)-chains. 114.88/31.10 ---------------------------------------- 114.88/31.10 114.88/31.10 (94) DependencyGraphProof (EQUIVALENT) 114.88/31.10 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes. 114.88/31.10 ---------------------------------------- 114.88/31.10 114.88/31.10 (95) 114.88/31.10 Obligation: 114.88/31.10 Q DP problem: 114.88/31.10 The TRS P consists of the following rules: 114.88/31.10 114.88/31.10 U25_GA(X1, delcG_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) 114.88/31.10 MAXSORTA_IN_GA(.(f, .(f, X1))) -> U24_GA(X1, maxcF_in_ga(X1)) 114.88/31.10 U24_GA(X1, maxcF_out_ga(X1, X2)) -> U25_GA(X1, delcG_in_gga(X2, X1)) 114.88/31.10 114.88/31.10 The TRS R consists of the following rules: 114.88/31.10 114.88/31.10 maxcC_in_ga([]) -> maxcC_out_ga([], t) 114.88/31.10 maxcC_in_ga(.(t, X1)) -> U48_ga(X1, maxcC_in_ga(X1)) 114.88/31.10 maxcC_in_ga(.(f, X1)) -> U49_ga(X1, maxcC_in_ga(X1)) 114.88/31.10 U49_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(f, X1), X2) 114.88/31.10 U48_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(t, X1), X2) 114.88/31.10 maxcF_in_ga([]) -> maxcF_out_ga([], f) 114.88/31.10 maxcF_in_ga(.(f, X1)) -> U51_ga(X1, maxcF_in_ga(X1)) 114.88/31.10 maxcF_in_ga(.(t, X1)) -> U52_ga(X1, maxcC_in_ga(X1)) 114.88/31.10 U52_ga(X1, maxcC_out_ga(X1, X2)) -> maxcF_out_ga(.(t, X1), X2) 114.88/31.10 U51_ga(X1, maxcF_out_ga(X1, X2)) -> maxcF_out_ga(.(f, X1), X2) 114.88/31.10 delcG_in_gga(f, X1) -> delcG_out_gga(f, X1, .(f, X1)) 114.88/31.10 delcG_in_gga(t, X1) -> U56_gga(X1, delcJ_in_ga(X1)) 114.88/31.10 delcJ_in_ga([]) -> delcJ_out_ga([], []) 114.88/31.10 delcJ_in_ga(.(f, X1)) -> U53_ga(X1, delcJ_in_ga(X1)) 114.88/31.10 U56_gga(X1, delcJ_out_ga(X1, X2)) -> delcG_out_gga(t, X1, .(f, .(f, X2))) 114.88/31.10 U53_ga(X1, delcJ_out_ga(X1, X2)) -> delcJ_out_ga(.(f, X1), .(f, X2)) 114.88/31.10 114.88/31.10 The set Q consists of the following terms: 114.88/31.10 114.88/31.10 maxcC_in_ga(x0) 114.88/31.10 maxcF_in_ga(x0) 114.88/31.10 delcG_in_gga(x0, x1) 114.88/31.10 U48_ga(x0, x1) 114.88/31.10 U49_ga(x0, x1) 114.88/31.10 U51_ga(x0, x1) 114.88/31.10 U52_ga(x0, x1) 114.88/31.10 U56_gga(x0, x1) 114.88/31.10 delcJ_in_ga(x0) 114.88/31.10 U53_ga(x0, x1) 114.88/31.10 114.88/31.10 We have to consider all (P,Q,R)-chains. 114.88/31.10 ---------------------------------------- 114.88/31.10 114.88/31.10 (96) QDPQMonotonicMRRProof (EQUIVALENT) 114.88/31.10 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. 114.88/31.10 114.88/31.10 114.88/31.10 Strictly oriented rules of the TRS R: 114.88/31.10 114.88/31.10 maxcC_in_ga([]) -> maxcC_out_ga([], t) 114.88/31.10 114.88/31.10 Used ordering: Polynomial interpretation [POLO]: 114.88/31.10 114.88/31.10 POL(.(x_1, x_2)) = x_1 + 2*x_2 114.88/31.10 POL(MAXSORTA_IN_GA(x_1)) = 2*x_1 114.88/31.10 POL(U24_GA(x_1, x_2)) = 2*x_1 + 2*x_2 114.88/31.10 POL(U25_GA(x_1, x_2)) = 2*x_2 114.88/31.10 POL(U48_ga(x_1, x_2)) = 2 + x_1 + x_2 114.88/31.10 POL(U49_ga(x_1, x_2)) = x_1 + x_2 114.88/31.10 POL(U51_ga(x_1, x_2)) = 2*x_1 + x_2 114.88/31.10 POL(U52_ga(x_1, x_2)) = 2*x_1 + 2*x_2 114.88/31.10 POL(U53_ga(x_1, x_2)) = 2*x_2 114.88/31.10 POL(U56_gga(x_1, x_2)) = 2*x_2 114.88/31.10 POL([]) = 0 114.88/31.10 POL(delcG_in_gga(x_1, x_2)) = 2*x_2 114.88/31.10 POL(delcG_out_gga(x_1, x_2, x_3)) = x_3 114.88/31.10 POL(delcJ_in_ga(x_1)) = 0 114.88/31.10 POL(delcJ_out_ga(x_1, x_2)) = 2*x_2 114.88/31.10 POL(f) = 0 114.88/31.10 POL(maxcC_in_ga(x_1)) = 2 + x_1 114.88/31.10 POL(maxcC_out_ga(x_1, x_2)) = 1 + x_1 114.88/31.10 POL(maxcF_in_ga(x_1)) = 2*x_1 114.88/31.10 POL(maxcF_out_ga(x_1, x_2)) = x_1 114.88/31.10 POL(t) = 2 114.88/31.10 114.88/31.10 114.88/31.10 ---------------------------------------- 114.88/31.10 114.88/31.10 (97) 114.88/31.10 Obligation: 114.88/31.10 Q DP problem: 114.88/31.10 The TRS P consists of the following rules: 114.88/31.10 114.88/31.10 U25_GA(X1, delcG_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) 114.88/31.10 MAXSORTA_IN_GA(.(f, .(f, X1))) -> U24_GA(X1, maxcF_in_ga(X1)) 114.88/31.10 U24_GA(X1, maxcF_out_ga(X1, X2)) -> U25_GA(X1, delcG_in_gga(X2, X1)) 114.88/31.10 114.88/31.10 The TRS R consists of the following rules: 114.88/31.10 114.88/31.10 maxcC_in_ga(.(t, X1)) -> U48_ga(X1, maxcC_in_ga(X1)) 114.88/31.10 maxcC_in_ga(.(f, X1)) -> U49_ga(X1, maxcC_in_ga(X1)) 114.88/31.10 U49_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(f, X1), X2) 114.88/31.10 U48_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(t, X1), X2) 114.88/31.10 maxcF_in_ga([]) -> maxcF_out_ga([], f) 114.88/31.10 maxcF_in_ga(.(f, X1)) -> U51_ga(X1, maxcF_in_ga(X1)) 114.88/31.10 maxcF_in_ga(.(t, X1)) -> U52_ga(X1, maxcC_in_ga(X1)) 114.88/31.10 U52_ga(X1, maxcC_out_ga(X1, X2)) -> maxcF_out_ga(.(t, X1), X2) 114.88/31.10 U51_ga(X1, maxcF_out_ga(X1, X2)) -> maxcF_out_ga(.(f, X1), X2) 114.88/31.10 delcG_in_gga(f, X1) -> delcG_out_gga(f, X1, .(f, X1)) 114.88/31.11 delcG_in_gga(t, X1) -> U56_gga(X1, delcJ_in_ga(X1)) 114.88/31.11 delcJ_in_ga([]) -> delcJ_out_ga([], []) 114.88/31.11 delcJ_in_ga(.(f, X1)) -> U53_ga(X1, delcJ_in_ga(X1)) 114.88/31.11 U56_gga(X1, delcJ_out_ga(X1, X2)) -> delcG_out_gga(t, X1, .(f, .(f, X2))) 114.88/31.11 U53_ga(X1, delcJ_out_ga(X1, X2)) -> delcJ_out_ga(.(f, X1), .(f, X2)) 114.88/31.11 114.88/31.11 The set Q consists of the following terms: 114.88/31.11 114.88/31.11 maxcC_in_ga(x0) 114.88/31.11 maxcF_in_ga(x0) 114.88/31.11 delcG_in_gga(x0, x1) 114.88/31.11 U48_ga(x0, x1) 114.88/31.11 U49_ga(x0, x1) 114.88/31.11 U51_ga(x0, x1) 114.88/31.11 U52_ga(x0, x1) 114.88/31.11 U56_gga(x0, x1) 114.88/31.11 delcJ_in_ga(x0) 114.88/31.11 U53_ga(x0, x1) 114.88/31.11 114.88/31.11 We have to consider all (P,Q,R)-chains. 114.88/31.11 ---------------------------------------- 114.88/31.11 114.88/31.11 (98) QDPQMonotonicMRRProof (EQUIVALENT) 114.88/31.11 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. 114.88/31.11 114.88/31.11 114.88/31.11 Strictly oriented rules of the TRS R: 114.88/31.11 114.88/31.11 U52_ga(X1, maxcC_out_ga(X1, X2)) -> maxcF_out_ga(.(t, X1), X2) 114.88/31.11 114.88/31.11 Used ordering: Polynomial interpretation [POLO]: 114.88/31.11 114.88/31.11 POL(.(x_1, x_2)) = 2*x_2 114.88/31.11 POL(MAXSORTA_IN_GA(x_1)) = 0 114.88/31.11 POL(U24_GA(x_1, x_2)) = 2*x_2 114.88/31.11 POL(U25_GA(x_1, x_2)) = 0 114.88/31.11 POL(U48_ga(x_1, x_2)) = x_2 114.88/31.11 POL(U49_ga(x_1, x_2)) = x_2 114.88/31.11 POL(U51_ga(x_1, x_2)) = x_2 114.88/31.11 POL(U52_ga(x_1, x_2)) = 2*x_2 114.88/31.11 POL(U53_ga(x_1, x_2)) = 1 + 2*x_1 114.88/31.11 POL(U56_gga(x_1, x_2)) = 0 114.88/31.11 POL([]) = 0 114.88/31.11 POL(delcG_in_gga(x_1, x_2)) = 0 114.88/31.11 POL(delcG_out_gga(x_1, x_2, x_3)) = 0 114.88/31.11 POL(delcJ_in_ga(x_1)) = 2 + 2*x_1 114.88/31.11 POL(delcJ_out_ga(x_1, x_2)) = 0 114.88/31.11 POL(f) = 0 114.88/31.11 POL(maxcC_in_ga(x_1)) = 0 114.88/31.11 POL(maxcC_out_ga(x_1, x_2)) = 1 114.88/31.11 POL(maxcF_in_ga(x_1)) = 0 114.88/31.11 POL(maxcF_out_ga(x_1, x_2)) = 0 114.88/31.11 POL(t) = 0 114.88/31.11 114.88/31.11 114.88/31.11 ---------------------------------------- 114.88/31.11 114.88/31.11 (99) 114.88/31.11 Obligation: 114.88/31.11 Q DP problem: 114.88/31.11 The TRS P consists of the following rules: 114.88/31.11 114.88/31.11 U25_GA(X1, delcG_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) 114.88/31.11 MAXSORTA_IN_GA(.(f, .(f, X1))) -> U24_GA(X1, maxcF_in_ga(X1)) 114.88/31.11 U24_GA(X1, maxcF_out_ga(X1, X2)) -> U25_GA(X1, delcG_in_gga(X2, X1)) 114.88/31.11 114.88/31.11 The TRS R consists of the following rules: 114.88/31.11 114.88/31.11 maxcC_in_ga(.(t, X1)) -> U48_ga(X1, maxcC_in_ga(X1)) 114.88/31.11 maxcC_in_ga(.(f, X1)) -> U49_ga(X1, maxcC_in_ga(X1)) 114.88/31.11 U49_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(f, X1), X2) 114.88/31.11 U48_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(t, X1), X2) 114.88/31.11 maxcF_in_ga([]) -> maxcF_out_ga([], f) 114.88/31.11 maxcF_in_ga(.(f, X1)) -> U51_ga(X1, maxcF_in_ga(X1)) 114.88/31.11 maxcF_in_ga(.(t, X1)) -> U52_ga(X1, maxcC_in_ga(X1)) 114.88/31.11 U51_ga(X1, maxcF_out_ga(X1, X2)) -> maxcF_out_ga(.(f, X1), X2) 114.88/31.11 delcG_in_gga(f, X1) -> delcG_out_gga(f, X1, .(f, X1)) 114.88/31.11 delcG_in_gga(t, X1) -> U56_gga(X1, delcJ_in_ga(X1)) 114.88/31.11 delcJ_in_ga([]) -> delcJ_out_ga([], []) 114.88/31.11 delcJ_in_ga(.(f, X1)) -> U53_ga(X1, delcJ_in_ga(X1)) 114.88/31.11 U56_gga(X1, delcJ_out_ga(X1, X2)) -> delcG_out_gga(t, X1, .(f, .(f, X2))) 114.88/31.11 U53_ga(X1, delcJ_out_ga(X1, X2)) -> delcJ_out_ga(.(f, X1), .(f, X2)) 114.88/31.11 114.88/31.11 The set Q consists of the following terms: 114.88/31.11 114.88/31.11 maxcC_in_ga(x0) 114.88/31.11 maxcF_in_ga(x0) 114.88/31.11 delcG_in_gga(x0, x1) 114.88/31.11 U48_ga(x0, x1) 114.88/31.11 U49_ga(x0, x1) 114.88/31.11 U51_ga(x0, x1) 114.88/31.11 U52_ga(x0, x1) 114.88/31.11 U56_gga(x0, x1) 114.88/31.11 delcJ_in_ga(x0) 114.88/31.11 U53_ga(x0, x1) 114.88/31.11 114.88/31.11 We have to consider all (P,Q,R)-chains. 114.88/31.11 ---------------------------------------- 114.88/31.11 114.88/31.11 (100) QDPQMonotonicMRRProof (EQUIVALENT) 114.88/31.11 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. 114.88/31.11 114.88/31.11 Strictly oriented dependency pairs: 114.88/31.11 114.88/31.11 U25_GA(X1, delcG_out_gga(X2, X1, X4)) -> MAXSORTA_IN_GA(X4) 114.88/31.11 MAXSORTA_IN_GA(.(f, .(f, X1))) -> U24_GA(X1, maxcF_in_ga(X1)) 114.88/31.11 U24_GA(X1, maxcF_out_ga(X1, X2)) -> U25_GA(X1, delcG_in_gga(X2, X1)) 114.88/31.11 114.88/31.11 114.88/31.11 Used ordering: Polynomial interpretation [POLO]: 114.88/31.11 114.88/31.11 POL(.(x_1, x_2)) = 2 + x_2 114.88/31.11 POL(MAXSORTA_IN_GA(x_1)) = 2*x_1 114.88/31.11 POL(U24_GA(x_1, x_2)) = 2 + 2*x_1 + 2*x_2 114.88/31.11 POL(U25_GA(x_1, x_2)) = 1 + 2*x_2 114.88/31.11 POL(U48_ga(x_1, x_2)) = 0 114.88/31.11 POL(U49_ga(x_1, x_2)) = 2 114.88/31.11 POL(U51_ga(x_1, x_2)) = x_2 114.88/31.11 POL(U52_ga(x_1, x_2)) = 2 114.88/31.11 POL(U53_ga(x_1, x_2)) = 2 + x_2 114.88/31.11 POL(U56_gga(x_1, x_2)) = 2 + x_2 114.88/31.11 POL([]) = 0 114.88/31.11 POL(delcG_in_gga(x_1, x_2)) = 2 + x_1 + x_2 114.88/31.11 POL(delcG_out_gga(x_1, x_2, x_3)) = x_3 114.88/31.11 POL(delcJ_in_ga(x_1)) = 2 + x_1 114.88/31.11 POL(delcJ_out_ga(x_1, x_2)) = 2 + x_2 114.88/31.11 POL(f) = 0 114.88/31.11 POL(maxcC_in_ga(x_1)) = 2*x_1 114.88/31.11 POL(maxcC_out_ga(x_1, x_2)) = 0 114.88/31.11 POL(maxcF_in_ga(x_1)) = 2 114.88/31.11 POL(maxcF_out_ga(x_1, x_2)) = 2 + x_2 114.88/31.11 POL(t) = 2 114.88/31.11 114.88/31.11 114.88/31.11 ---------------------------------------- 114.88/31.11 114.88/31.11 (101) 114.88/31.11 Obligation: 114.88/31.11 Q DP problem: 114.88/31.11 P is empty. 114.88/31.11 The TRS R consists of the following rules: 114.88/31.11 114.88/31.11 maxcC_in_ga(.(t, X1)) -> U48_ga(X1, maxcC_in_ga(X1)) 114.88/31.11 maxcC_in_ga(.(f, X1)) -> U49_ga(X1, maxcC_in_ga(X1)) 114.88/31.11 U49_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(f, X1), X2) 114.88/31.11 U48_ga(X1, maxcC_out_ga(X1, X2)) -> maxcC_out_ga(.(t, X1), X2) 114.88/31.11 maxcF_in_ga([]) -> maxcF_out_ga([], f) 114.88/31.11 maxcF_in_ga(.(f, X1)) -> U51_ga(X1, maxcF_in_ga(X1)) 114.88/31.11 maxcF_in_ga(.(t, X1)) -> U52_ga(X1, maxcC_in_ga(X1)) 114.88/31.11 U51_ga(X1, maxcF_out_ga(X1, X2)) -> maxcF_out_ga(.(f, X1), X2) 114.88/31.11 delcG_in_gga(f, X1) -> delcG_out_gga(f, X1, .(f, X1)) 114.88/31.11 delcG_in_gga(t, X1) -> U56_gga(X1, delcJ_in_ga(X1)) 114.88/31.11 delcJ_in_ga([]) -> delcJ_out_ga([], []) 114.88/31.11 delcJ_in_ga(.(f, X1)) -> U53_ga(X1, delcJ_in_ga(X1)) 114.88/31.11 U56_gga(X1, delcJ_out_ga(X1, X2)) -> delcG_out_gga(t, X1, .(f, .(f, X2))) 114.88/31.11 U53_ga(X1, delcJ_out_ga(X1, X2)) -> delcJ_out_ga(.(f, X1), .(f, X2)) 114.88/31.11 114.88/31.11 The set Q consists of the following terms: 114.88/31.11 114.88/31.11 maxcC_in_ga(x0) 114.88/31.11 maxcF_in_ga(x0) 114.88/31.11 delcG_in_gga(x0, x1) 114.88/31.11 U48_ga(x0, x1) 114.88/31.11 U49_ga(x0, x1) 114.88/31.11 U51_ga(x0, x1) 114.88/31.11 U52_ga(x0, x1) 114.88/31.11 U56_gga(x0, x1) 114.88/31.11 delcJ_in_ga(x0) 114.88/31.11 U53_ga(x0, x1) 114.88/31.11 114.88/31.11 We have to consider all (P,Q,R)-chains. 114.88/31.11 ---------------------------------------- 114.88/31.11 114.88/31.11 (102) PisEmptyProof (EQUIVALENT) 114.88/31.11 The TRS P is empty. Hence, there is no (P,Q,R) chain. 114.88/31.11 ---------------------------------------- 114.88/31.11 114.88/31.11 (103) 114.88/31.11 YES 114.93/31.19 EOF