9.30/3.18 YES 9.64/3.28 proof of /export/starexec/sandbox2/benchmark/theBenchmark.pl 9.64/3.28 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 9.64/3.28 9.64/3.28 9.64/3.28 Left Termination of the query pattern 9.64/3.28 9.64/3.28 shapes(g,a) 9.64/3.28 9.64/3.28 w.r.t. the given Prolog program could successfully be proven: 9.64/3.28 9.64/3.28 (0) Prolog 9.64/3.28 (1) PrologToDTProblemTransformerProof [SOUND, 53 ms] 9.64/3.28 (2) TRIPLES 9.64/3.28 (3) TriplesToPiDPProof [SOUND, 0 ms] 9.64/3.28 (4) PiDP 9.64/3.28 (5) DependencyGraphProof [EQUIVALENT, 0 ms] 9.64/3.28 (6) AND 9.64/3.28 (7) PiDP 9.64/3.28 (8) UsableRulesProof [EQUIVALENT, 0 ms] 9.64/3.28 (9) PiDP 9.64/3.28 (10) PiDPToQDPProof [SOUND, 0 ms] 9.64/3.28 (11) QDP 9.64/3.28 (12) QDPSizeChangeProof [EQUIVALENT, 0 ms] 9.64/3.28 (13) YES 9.64/3.28 (14) PiDP 9.64/3.28 (15) UsableRulesProof [EQUIVALENT, 0 ms] 9.64/3.28 (16) PiDP 9.64/3.28 (17) PiDPToQDPProof [SOUND, 0 ms] 9.64/3.28 (18) QDP 9.64/3.28 (19) QDPQMonotonicMRRProof [EQUIVALENT, 111 ms] 9.64/3.28 (20) QDP 9.64/3.28 (21) DependencyGraphProof [EQUIVALENT, 0 ms] 9.64/3.28 (22) TRUE 9.64/3.28 (23) PiDP 9.64/3.28 (24) UsableRulesProof [EQUIVALENT, 0 ms] 9.64/3.28 (25) PiDP 9.64/3.28 (26) PiDPToQDPProof [SOUND, 0 ms] 9.64/3.28 (27) QDP 9.64/3.28 (28) QDPSizeChangeProof [EQUIVALENT, 0 ms] 9.64/3.28 (29) YES 9.64/3.28 (30) PiDP 9.64/3.28 (31) UsableRulesProof [EQUIVALENT, 0 ms] 9.64/3.28 (32) PiDP 9.64/3.28 (33) PiDPToQDPProof [SOUND, 0 ms] 9.64/3.28 (34) QDP 9.64/3.28 (35) QDPSizeChangeProof [EQUIVALENT, 0 ms] 9.64/3.28 (36) YES 9.64/3.28 9.64/3.28 9.64/3.28 ---------------------------------------- 9.64/3.28 9.64/3.28 (0) 9.64/3.28 Obligation: 9.64/3.28 Clauses: 9.64/3.28 9.64/3.28 shapes(Matrix, N) :- ','(varmat(Matrix, MatrixWithVars), unif_matrx(MatrixWithVars)). 9.64/3.28 varmat([], []). 9.64/3.28 varmat(.(L, Ls), .(VL, VLs)) :- ','(varmat(L, VL), varmat(Ls, VLs)). 9.64/3.28 varmat(.(black, Xs), .(black, VXs)) :- varmat(Xs, VXs). 9.64/3.28 varmat(.(white, Xs), .(w(X1), VXs)) :- varmat(Xs, VXs). 9.64/3.28 unif_matrx(.(L1, .(L2, Ls))) :- ','(unif_lines(L1, L2), unif_matrx(.(L2, Ls))). 9.64/3.28 unif_matrx(.(X2, [])). 9.64/3.28 unif_lines(.(W, .(X, L1s)), .(Y, .(Z, L2s))) :- ','(unif_pairs(.(W, .(X, .(Y, .(Z, .(W, .(Y, .(X, .(Z, .(W, .(Z, .(X, .(Y, []))))))))))))), unif_lines(.(X, L1s), .(Z, L2s))). 9.64/3.28 unif_lines(.(X3, []), .(X4, [])). 9.64/3.28 unif_pairs([]). 9.64/3.28 unif_pairs(.(A, .(B, Pairs))) :- ','(unif(A, B), unif_pairs(Pairs)). 9.64/3.28 unif(w(A), w(A)). 9.64/3.28 unif(black, black). 9.64/3.28 unif(black, w(X5)). 9.64/3.28 unif(w(X6), black). 9.64/3.28 9.64/3.28 9.64/3.28 Query: shapes(g,a) 9.64/3.28 ---------------------------------------- 9.64/3.28 9.64/3.28 (1) PrologToDTProblemTransformerProof (SOUND) 9.64/3.28 Built DT problem from termination graph DT10. 9.64/3.28 9.64/3.28 { 9.64/3.28 "root": 1, 9.64/3.28 "program": { 9.64/3.28 "directives": [], 9.64/3.28 "clauses": [ 9.64/3.28 [ 9.64/3.28 "(shapes Matrix N)", 9.64/3.28 "(',' (varmat Matrix MatrixWithVars) (unif_matrx MatrixWithVars))" 9.64/3.28 ], 9.64/3.28 [ 9.64/3.28 "(varmat ([]) ([]))", 9.64/3.28 null 9.64/3.28 ], 9.64/3.28 [ 9.64/3.28 "(varmat (. L Ls) (. VL VLs))", 9.64/3.28 "(',' (varmat L VL) (varmat Ls VLs))" 9.64/3.28 ], 9.64/3.28 [ 9.64/3.28 "(varmat (. (black) Xs) (. (black) VXs))", 9.64/3.28 "(varmat Xs VXs)" 9.64/3.28 ], 9.64/3.28 [ 9.64/3.28 "(varmat (. (white) Xs) (. (w X1) VXs))", 9.64/3.28 "(varmat Xs VXs)" 9.64/3.28 ], 9.64/3.28 [ 9.64/3.28 "(unif_matrx (. L1 (. L2 Ls)))", 9.64/3.28 "(',' (unif_lines L1 L2) (unif_matrx (. L2 Ls)))" 9.64/3.28 ], 9.64/3.28 [ 9.64/3.28 "(unif_matrx (. X2 ([])))", 9.64/3.28 null 9.64/3.28 ], 9.64/3.28 [ 9.64/3.28 "(unif_lines (. W (. X L1s)) (. Y (. Z L2s)))", 9.64/3.28 "(',' (unif_pairs (. W (. X (. Y (. Z (. W (. Y (. X (. Z (. W (. Z (. X (. Y ([])))))))))))))) (unif_lines (. X L1s) (. Z L2s)))" 9.64/3.28 ], 9.64/3.28 [ 9.64/3.28 "(unif_lines (. X3 ([])) (. X4 ([])))", 9.64/3.28 null 9.64/3.28 ], 9.64/3.28 [ 9.64/3.28 "(unif_pairs ([]))", 9.64/3.28 null 9.64/3.28 ], 9.64/3.28 [ 9.64/3.28 "(unif_pairs (. A (. B Pairs)))", 9.64/3.28 "(',' (unif A B) (unif_pairs Pairs))" 9.64/3.28 ], 9.64/3.28 [ 9.64/3.28 "(unif (w A) (w A))", 9.64/3.28 null 9.64/3.28 ], 9.64/3.28 [ 9.64/3.28 "(unif (black) (black))", 9.64/3.28 null 9.64/3.28 ], 9.64/3.28 [ 9.64/3.28 "(unif (black) (w X5))", 9.64/3.28 null 9.64/3.28 ], 9.64/3.28 [ 9.64/3.28 "(unif (w X6) (black))", 9.64/3.28 null 9.64/3.28 ] 9.64/3.28 ] 9.64/3.28 }, 9.64/3.28 "graph": { 9.64/3.28 "nodes": { 9.64/3.28 "type": "Nodes", 9.64/3.28 "590": { 9.64/3.28 "goal": [], 9.64/3.28 "kb": { 9.64/3.28 "nonunifying": [], 9.64/3.28 "intvars": {}, 9.64/3.28 "arithmetic": { 9.64/3.28 "type": "PlainIntegerRelationState", 9.64/3.28 "relations": [] 9.64/3.28 }, 9.64/3.28 "ground": [], 9.64/3.28 "free": [], 9.64/3.28 "exprvars": [] 9.64/3.28 } 9.64/3.28 }, 9.64/3.28 "592": { 9.64/3.28 "goal": [ 9.64/3.28 { 9.64/3.28 "clause": 11, 9.64/3.28 "scope": 8, 9.64/3.28 "term": "(',' (unif T123 T124) (unif_pairs T125))" 9.64/3.28 }, 9.64/3.28 { 9.64/3.28 "clause": 12, 9.64/3.28 "scope": 8, 9.64/3.28 "term": "(',' (unif T123 T124) (unif_pairs T125))" 9.64/3.28 }, 9.64/3.28 { 9.64/3.28 "clause": 13, 9.64/3.28 "scope": 8, 9.64/3.28 "term": "(',' (unif T123 T124) (unif_pairs T125))" 9.64/3.28 }, 9.64/3.28 { 9.64/3.28 "clause": 14, 9.64/3.28 "scope": 8, 9.64/3.28 "term": "(',' (unif T123 T124) (unif_pairs T125))" 9.64/3.28 } 9.64/3.28 ], 9.64/3.28 "kb": { 9.64/3.28 "nonunifying": [], 9.64/3.28 "intvars": {}, 9.64/3.28 "arithmetic": { 9.64/3.28 "type": "PlainIntegerRelationState", 9.64/3.28 "relations": [] 9.64/3.28 }, 9.64/3.28 "ground": [], 9.64/3.28 "free": [], 9.64/3.28 "exprvars": [] 9.64/3.28 } 9.64/3.28 }, 9.64/3.28 "352": { 9.64/3.28 "goal": [{ 9.64/3.28 "clause": 2, 9.64/3.28 "scope": 4, 9.64/3.28 "term": "(varmat T15 X47)" 9.64/3.28 }], 9.64/3.28 "kb": { 9.64/3.29 "nonunifying": [], 9.64/3.29 "intvars": {}, 9.64/3.29 "arithmetic": { 9.64/3.29 "type": "PlainIntegerRelationState", 9.64/3.29 "relations": [] 9.64/3.29 }, 9.64/3.29 "ground": ["T15"], 9.64/3.29 "free": ["X47"], 9.64/3.29 "exprvars": [] 9.64/3.29 } 9.64/3.29 }, 9.64/3.29 "353": { 9.64/3.29 "goal": [ 9.64/3.29 { 9.64/3.29 "clause": 3, 9.64/3.29 "scope": 4, 9.64/3.29 "term": "(varmat T15 X47)" 9.64/3.29 }, 9.64/3.29 { 9.64/3.29 "clause": 4, 9.64/3.29 "scope": 4, 9.64/3.29 "term": "(varmat T15 X47)" 9.64/3.29 } 9.64/3.29 ], 9.64/3.29 "kb": { 9.64/3.29 "nonunifying": [], 9.64/3.29 "intvars": {}, 9.64/3.29 "arithmetic": { 9.64/3.29 "type": "PlainIntegerRelationState", 9.64/3.29 "relations": [] 9.64/3.29 }, 9.64/3.29 "ground": ["T15"], 9.64/3.29 "free": ["X47"], 9.64/3.29 "exprvars": [] 9.64/3.29 } 9.64/3.29 }, 9.64/3.29 "356": { 9.64/3.29 "goal": [{ 9.64/3.29 "clause": -1, 9.64/3.29 "scope": -1, 9.64/3.29 "term": "(',' (varmat T26 X77) (varmat T27 X78))" 9.64/3.29 }], 9.64/3.29 "kb": { 9.64/3.29 "nonunifying": [], 9.64/3.29 "intvars": {}, 9.64/3.29 "arithmetic": { 9.64/3.29 "type": "PlainIntegerRelationState", 9.64/3.29 "relations": [] 9.64/3.29 }, 9.64/3.29 "ground": [ 9.64/3.29 "T26", 9.64/3.29 "T27" 9.64/3.29 ], 9.64/3.29 "free": [ 9.64/3.29 "X77", 9.64/3.29 "X78" 9.64/3.29 ], 9.64/3.29 "exprvars": [] 9.64/3.29 } 9.64/3.29 }, 9.64/3.29 "357": { 9.64/3.29 "goal": [], 9.64/3.29 "kb": { 9.64/3.29 "nonunifying": [], 9.64/3.29 "intvars": {}, 9.64/3.29 "arithmetic": { 9.64/3.29 "type": "PlainIntegerRelationState", 9.64/3.29 "relations": [] 9.64/3.29 }, 9.64/3.29 "ground": [], 9.64/3.29 "free": [], 9.64/3.29 "exprvars": [] 9.64/3.29 } 9.64/3.29 }, 9.64/3.29 "516": { 9.64/3.29 "goal": [{ 9.64/3.29 "clause": 5, 9.64/3.29 "scope": 5, 9.64/3.29 "term": "(unif_matrx (. T17 T37))" 9.64/3.29 }], 9.64/3.29 "kb": { 9.64/3.29 "nonunifying": [], 9.64/3.29 "intvars": {}, 9.64/3.29 "arithmetic": { 9.64/3.29 "type": "PlainIntegerRelationState", 9.64/3.29 "relations": [] 9.64/3.29 }, 9.64/3.29 "ground": [], 9.64/3.29 "free": [], 9.64/3.29 "exprvars": [] 9.64/3.29 } 9.64/3.29 }, 9.64/3.29 "517": { 9.64/3.29 "goal": [{ 9.64/3.29 "clause": 6, 9.64/3.29 "scope": 5, 9.64/3.29 "term": "(unif_matrx (. 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T103 T109))" 9.64/3.30 } 9.64/3.30 ], 9.64/3.30 "kb": { 9.64/3.30 "nonunifying": [], 9.64/3.30 "intvars": {}, 9.64/3.30 "arithmetic": { 9.64/3.30 "type": "PlainIntegerRelationState", 9.64/3.30 "relations": [] 9.64/3.30 }, 9.64/3.30 "ground": [], 9.64/3.30 "free": [], 9.64/3.30 "exprvars": [] 9.64/3.30 } 9.64/3.30 }, 9.64/3.30 "585": { 9.64/3.30 "goal": [{ 9.64/3.30 "clause": 10, 9.64/3.30 "scope": 7, 9.64/3.30 "term": "(unif_pairs (. 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T17 T37))" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "clause": 6, 9.64/3.30 "scope": 5, 9.64/3.30 "term": "(unif_matrx (. T17 T37))" 9.64/3.30 } 9.64/3.30 ], 9.64/3.30 "kb": { 9.64/3.30 "nonunifying": [], 9.64/3.30 "intvars": {}, 9.64/3.30 "arithmetic": { 9.64/3.30 "type": "PlainIntegerRelationState", 9.64/3.30 "relations": [] 9.64/3.30 }, 9.64/3.30 "ground": [], 9.64/3.30 "free": [], 9.64/3.30 "exprvars": [] 9.64/3.30 } 9.64/3.30 }, 9.64/3.30 "589": { 9.64/3.30 "goal": [{ 9.64/3.30 "clause": -1, 9.64/3.30 "scope": -1, 9.64/3.30 "term": "(',' (unif T123 T124) (unif_pairs T125))" 9.64/3.30 }], 9.64/3.30 "kb": { 9.64/3.30 "nonunifying": [], 9.64/3.30 "intvars": {}, 9.64/3.30 "arithmetic": { 9.64/3.30 "type": "PlainIntegerRelationState", 9.64/3.30 "relations": [] 9.64/3.30 }, 9.64/3.30 "ground": [], 9.64/3.30 "free": [], 9.64/3.30 "exprvars": [] 9.64/3.30 } 9.64/3.30 }, 9.64/3.30 "743": { 9.64/3.30 "goal": [{ 9.64/3.30 "clause": 12, 9.64/3.30 "scope": 8, 9.64/3.30 "term": "(',' (unif T123 T124) (unif_pairs T125))" 9.64/3.30 }], 9.64/3.30 "kb": { 9.64/3.30 "nonunifying": [], 9.64/3.30 "intvars": {}, 9.64/3.30 "arithmetic": { 9.64/3.30 "type": "PlainIntegerRelationState", 9.64/3.30 "relations": [] 9.64/3.30 }, 9.64/3.30 "ground": [], 9.64/3.30 "free": [], 9.64/3.30 "exprvars": [] 9.64/3.30 } 9.64/3.30 }, 9.64/3.30 "744": { 9.64/3.30 "goal": [ 9.64/3.30 { 9.64/3.30 "clause": 13, 9.64/3.30 "scope": 8, 9.64/3.30 "term": "(',' (unif T123 T124) (unif_pairs T125))" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "clause": 14, 9.64/3.30 "scope": 8, 9.64/3.30 "term": "(',' (unif T123 T124) (unif_pairs T125))" 9.64/3.30 } 9.64/3.30 ], 9.64/3.30 "kb": { 9.64/3.30 "nonunifying": [], 9.64/3.30 "intvars": {}, 9.64/3.30 "arithmetic": { 9.64/3.30 "type": "PlainIntegerRelationState", 9.64/3.30 "relations": [] 9.64/3.30 }, 9.64/3.30 "ground": [], 9.64/3.30 "free": [], 9.64/3.30 "exprvars": [] 9.64/3.30 } 9.64/3.30 } 9.64/3.30 }, 9.64/3.30 "edges": [ 9.64/3.30 { 9.64/3.30 "from": 1, 9.64/3.30 "to": 270, 9.64/3.30 "label": "CASE" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 270, 9.64/3.30 "to": 271, 9.64/3.30 "label": "ONLY EVAL with clause\nshapes(X9, X10) :- ','(varmat(X9, X11), unif_matrx(X11)).\nand substitutionT1 -> T5,\nX9 -> T5,\nT2 -> T6,\nX10 -> T6" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 271, 9.64/3.30 "to": 272, 9.64/3.30 "label": "CASE" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 272, 9.64/3.30 "to": 273, 9.64/3.30 "label": "PARALLEL" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 272, 9.64/3.30 "to": 274, 9.64/3.30 "label": "PARALLEL" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 273, 9.64/3.30 "to": 275, 9.64/3.30 "label": "EVAL with clause\nvarmat([], []).\nand substitutionT5 -> [],\nX11 -> []" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 273, 9.64/3.30 "to": 276, 9.64/3.30 "label": "EVAL-BACKTRACK" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 274, 9.64/3.30 "to": 286, 9.64/3.30 "label": "PARALLEL" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 274, 9.64/3.30 "to": 287, 9.64/3.30 "label": "PARALLEL" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 275, 9.64/3.30 "to": 282, 9.64/3.30 "label": "CASE" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 282, 9.64/3.30 "to": 283, 9.64/3.30 "label": "BACKTRACK\nfor clause: unif_matrx(.(L1, .(L2, Ls))) :- ','(unif_lines(L1, L2), unif_matrx(.(L2, Ls)))because of non-unification" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 283, 9.64/3.30 "to": 285, 9.64/3.30 "label": "BACKTRACK\nfor clause: unif_matrx(.(X2, []))because of non-unification" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 286, 9.64/3.30 "to": 290, 9.64/3.30 "label": "EVAL with clause\nvarmat(.(X43, X44), .(X45, X46)) :- ','(varmat(X43, X45), varmat(X44, X46)).\nand substitutionX43 -> T15,\nX44 -> T16,\nT5 -> .(T15, T16),\nX45 -> X47,\nX46 -> X48,\nX11 -> .(X47, X48)" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 286, 9.64/3.30 "to": 291, 9.64/3.30 "label": "EVAL-BACKTRACK" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 287, 9.64/3.30 "to": 786, 9.64/3.30 "label": "PARALLEL" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 287, 9.64/3.30 "to": 787, 9.64/3.30 "label": "PARALLEL" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 290, 9.64/3.30 "to": 295, 9.64/3.30 "label": "SPLIT 1" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 290, 9.64/3.30 "to": 296, 9.64/3.30 "label": "SPLIT 2\nnew knowledge:\nT15 is ground\nreplacements:X47 -> T17" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 295, 9.64/3.30 "to": 300, 9.64/3.30 "label": "CASE" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 296, 9.64/3.30 "to": 496, 9.64/3.30 "label": "SPLIT 1" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 296, 9.64/3.30 "to": 497, 9.64/3.30 "label": "SPLIT 2\nnew knowledge:\nT16 is ground\nreplacements:X48 -> T37" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 300, 9.64/3.30 "to": 301, 9.64/3.30 "label": "PARALLEL" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 300, 9.64/3.30 "to": 302, 9.64/3.30 "label": "PARALLEL" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 301, 9.64/3.30 "to": 303, 9.64/3.30 "label": "EVAL with clause\nvarmat([], []).\nand substitutionT15 -> [],\nX47 -> []" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 301, 9.64/3.30 "to": 304, 9.64/3.30 "label": "EVAL-BACKTRACK" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 302, 9.64/3.30 "to": 352, 9.64/3.30 "label": "PARALLEL" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 302, 9.64/3.30 "to": 353, 9.64/3.30 "label": "PARALLEL" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 303, 9.64/3.30 "to": 307, 9.64/3.30 "label": "SUCCESS" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 352, 9.64/3.30 "to": 356, 9.64/3.30 "label": "EVAL with clause\nvarmat(.(X73, X74), .(X75, X76)) :- ','(varmat(X73, X75), varmat(X74, X76)).\nand substitutionX73 -> T26,\nX74 -> T27,\nT15 -> .(T26, T27),\nX75 -> X77,\nX76 -> X78,\nX47 -> .(X77, X78)" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 352, 9.64/3.30 "to": 357, 9.64/3.30 "label": "EVAL-BACKTRACK" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 353, 9.64/3.30 "to": 365, 9.64/3.30 "label": "PARALLEL" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 353, 9.64/3.30 "to": 366, 9.64/3.30 "label": "PARALLEL" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 356, 9.64/3.30 "to": 361, 9.64/3.30 "label": "SPLIT 1" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 356, 9.64/3.30 "to": 362, 9.64/3.30 "label": "SPLIT 2\nnew knowledge:\nT26 is ground\nreplacements:X77 -> T28" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 361, 9.64/3.30 "to": 295, 9.64/3.30 "label": "INSTANCE with matching:\nT15 -> T26\nX47 -> X77" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 362, 9.64/3.30 "to": 295, 9.64/3.30 "label": "INSTANCE with matching:\nT15 -> T27\nX47 -> X78" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 365, 9.64/3.30 "to": 367, 9.64/3.30 "label": "EVAL with clause\nvarmat(.(black, X91), .(black, X92)) :- varmat(X91, X92).\nand substitutionX91 -> T33,\nT15 -> .(black, T33),\nX92 -> X93,\nX47 -> .(black, X93)" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 365, 9.64/3.30 "to": 370, 9.64/3.30 "label": "EVAL-BACKTRACK" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 366, 9.64/3.30 "to": 415, 9.64/3.30 "label": "EVAL with clause\nvarmat(.(white, X104), .(w(X105), X106)) :- varmat(X104, X106).\nand substitutionX104 -> T36,\nT15 -> .(white, T36),\nX105 -> X107,\nX106 -> X108,\nX47 -> .(w(X107), X108)" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 366, 9.64/3.30 "to": 425, 9.64/3.30 "label": "EVAL-BACKTRACK" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 367, 9.64/3.30 "to": 295, 9.64/3.30 "label": "INSTANCE with matching:\nT15 -> T33\nX47 -> X93" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 415, 9.64/3.30 "to": 295, 9.64/3.30 "label": "INSTANCE with matching:\nT15 -> T36\nX47 -> X108" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 496, 9.64/3.30 "to": 295, 9.64/3.30 "label": "INSTANCE with matching:\nT15 -> T16\nX47 -> X48" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 497, 9.64/3.30 "to": 501, 9.64/3.30 "label": "CASE" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 501, 9.64/3.30 "to": 516, 9.64/3.30 "label": "PARALLEL" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 501, 9.64/3.30 "to": 517, 9.64/3.30 "label": "PARALLEL" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 516, 9.64/3.30 "to": 534, 9.64/3.30 "label": "EVAL with clause\nunif_matrx(.(X124, .(X125, X126))) :- ','(unif_lines(X124, X125), unif_matrx(.(X125, X126))).\nand substitutionT17 -> T56,\nX124 -> T56,\nX125 -> T57,\nX126 -> T58,\nT37 -> .(T57, T58),\nT53 -> T56,\nT54 -> T57,\nT55 -> T58" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 516, 9.64/3.30 "to": 536, 9.64/3.30 "label": "EVAL-BACKTRACK" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 517, 9.64/3.30 "to": 783, 9.64/3.30 "label": "EVAL with clause\nunif_matrx(.(X230, [])).\nand substitutionT17 -> T189,\nX230 -> T189,\nT37 -> []" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 517, 9.64/3.30 "to": 784, 9.64/3.30 "label": "EVAL-BACKTRACK" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 534, 9.64/3.30 "to": 560, 9.64/3.30 "label": "SPLIT 1" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 534, 9.64/3.30 "to": 561, 9.64/3.30 "label": "SPLIT 2\nreplacements:T57 -> T65,\nT58 -> T66" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 560, 9.64/3.30 "to": 564, 9.64/3.30 "label": "CASE" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 561, 9.64/3.30 "to": 497, 9.64/3.30 "label": "INSTANCE with matching:\nT17 -> T65\nT37 -> T66" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 564, 9.64/3.30 "to": 568, 9.64/3.30 "label": "PARALLEL" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 564, 9.64/3.30 "to": 569, 9.64/3.30 "label": "PARALLEL" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 568, 9.64/3.30 "to": 575, 9.64/3.30 "label": "EVAL with clause\nunif_lines(.(X163, .(X164, X165)), .(X166, .(X167, X168))) :- ','(unif_pairs(.(X163, .(X164, .(X166, .(X167, .(X163, .(X166, .(X164, .(X167, .(X163, .(X167, .(X164, .(X166, []))))))))))))), unif_lines(.(X164, X165), .(X167, X168))).\nand substitutionX163 -> T103,\nX164 -> T104,\nX165 -> T107,\nT56 -> .(T103, .(T104, T107)),\nX166 -> T105,\nX167 -> T106,\nX168 -> T108,\nT57 -> .(T105, .(T106, T108)),\nT97 -> T103,\nT98 -> T104,\nT100 -> T105,\nT101 -> T106,\nT99 -> T107,\nT102 -> T108" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 568, 9.64/3.30 "to": 576, 9.64/3.30 "label": "EVAL-BACKTRACK" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 569, 9.64/3.30 "to": 780, 9.64/3.30 "label": "EVAL with clause\nunif_lines(.(X223, []), .(X224, [])).\nand substitutionX223 -> T182,\nT56 -> .(T182, []),\nX224 -> T183,\nT57 -> .(T183, [])" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 569, 9.64/3.30 "to": 781, 9.64/3.30 "label": "EVAL-BACKTRACK" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 575, 9.64/3.30 "to": 579, 9.64/3.30 "label": "GENERALIZATION\nT109 <-- .(T104, .(T105, .(T106, .(T103, .(T105, .(T104, .(T106, .(T103, .(T106, .(T104, .(T105, [])))))))))))" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 579, 9.64/3.30 "to": 580, 9.64/3.30 "label": "SPLIT 1" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 579, 9.64/3.30 "to": 582, 9.64/3.30 "label": "SPLIT 2\nreplacements:T104 -> T110,\nT107 -> T111,\nT106 -> T112,\nT108 -> T113" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 580, 9.64/3.30 "to": 584, 9.64/3.30 "label": "CASE" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 582, 9.64/3.30 "to": 560, 9.64/3.30 "label": "INSTANCE with matching:\nT56 -> .(T110, T111)\nT57 -> .(T112, T113)" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 584, 9.64/3.30 "to": 585, 9.64/3.30 "label": "BACKTRACK\nfor clause: unif_pairs([])because of non-unification" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 585, 9.64/3.30 "to": 589, 9.64/3.30 "label": "EVAL with clause\nunif_pairs(.(X175, .(X176, X177))) :- ','(unif(X175, X176), unif_pairs(X177)).\nand substitutionT103 -> T123,\nX175 -> T123,\nX176 -> T124,\nX177 -> T125,\nT109 -> .(T124, T125),\nT120 -> T123,\nT121 -> T124,\nT122 -> T125" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 585, 9.64/3.30 "to": 590, 9.64/3.30 "label": "EVAL-BACKTRACK" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 589, 9.64/3.30 "to": 592, 9.64/3.30 "label": "CASE" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 592, 9.64/3.30 "to": 702, 9.64/3.30 "label": "PARALLEL" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 592, 9.64/3.30 "to": 704, 9.64/3.30 "label": "PARALLEL" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 702, 9.64/3.30 "to": 705, 9.64/3.30 "label": "EVAL with clause\nunif(w(X182), w(X182)).\nand substitutionX182 -> T130,\nT123 -> w(T130),\nT124 -> w(T130),\nT125 -> T131" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 702, 9.64/3.30 "to": 708, 9.64/3.30 "label": "EVAL-BACKTRACK" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 704, 9.64/3.30 "to": 743, 9.64/3.30 "label": "PARALLEL" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 704, 9.64/3.30 "to": 744, 9.64/3.30 "label": "PARALLEL" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 705, 9.64/3.30 "to": 711, 9.64/3.30 "label": "CASE" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 711, 9.64/3.30 "to": 712, 9.64/3.30 "label": "PARALLEL" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 711, 9.64/3.30 "to": 713, 9.64/3.30 "label": "PARALLEL" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 712, 9.64/3.30 "to": 714, 9.64/3.30 "label": "EVAL with clause\nunif_pairs([]).\nand substitutionT131 -> []" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 712, 9.64/3.30 "to": 715, 9.64/3.30 "label": "EVAL-BACKTRACK" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 713, 9.64/3.30 "to": 721, 9.64/3.30 "label": "EVAL with clause\nunif_pairs(.(X189, .(X190, X191))) :- ','(unif(X189, X190), unif_pairs(X191)).\nand substitutionX189 -> T141,\nX190 -> T142,\nX191 -> T143,\nT131 -> .(T141, .(T142, T143)),\nT138 -> T141,\nT139 -> T142,\nT140 -> T143" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 713, 9.64/3.30 "to": 722, 9.64/3.30 "label": "EVAL-BACKTRACK" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 714, 9.64/3.30 "to": 716, 9.64/3.30 "label": "SUCCESS" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 721, 9.64/3.30 "to": 723, 9.64/3.30 "label": "CASE" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 723, 9.64/3.30 "to": 724, 9.64/3.30 "label": "PARALLEL" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 723, 9.64/3.30 "to": 725, 9.64/3.30 "label": "PARALLEL" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 724, 9.64/3.30 "to": 728, 9.64/3.30 "label": "EVAL with clause\nunif(w(X196), w(X196)).\nand substitutionX196 -> T148,\nT141 -> w(T148),\nT142 -> w(T148),\nT143 -> T149" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 724, 9.64/3.30 "to": 729, 9.64/3.30 "label": "EVAL-BACKTRACK" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 725, 9.64/3.30 "to": 730, 9.64/3.30 "label": "PARALLEL" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 725, 9.64/3.30 "to": 731, 9.64/3.30 "label": "PARALLEL" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 728, 9.64/3.30 "to": 705, 9.64/3.30 "label": "INSTANCE with matching:\nT131 -> T149" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 730, 9.64/3.30 "to": 732, 9.64/3.30 "label": "EVAL with clause\nunif(black, black).\nand substitutionT141 -> black,\nT142 -> black,\nT143 -> T150" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 730, 9.64/3.30 "to": 733, 9.64/3.30 "label": "EVAL-BACKTRACK" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 731, 9.64/3.30 "to": 734, 9.64/3.30 "label": "PARALLEL" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 731, 9.64/3.30 "to": 735, 9.64/3.30 "label": "PARALLEL" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 732, 9.64/3.30 "to": 705, 9.64/3.30 "label": "INSTANCE with matching:\nT131 -> T150" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 734, 9.64/3.30 "to": 739, 9.64/3.30 "label": "EVAL with clause\nunif(black, w(X201)).\nand substitutionT141 -> black,\nX201 -> T155,\nT142 -> w(T155),\nT143 -> T156" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 734, 9.64/3.30 "to": 740, 9.64/3.30 "label": "EVAL-BACKTRACK" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 735, 9.64/3.30 "to": 741, 9.64/3.30 "label": "EVAL with clause\nunif(w(X204), black).\nand substitutionX204 -> T159,\nT141 -> w(T159),\nT142 -> black,\nT143 -> T160" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 735, 9.64/3.30 "to": 742, 9.64/3.30 "label": "EVAL-BACKTRACK" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 739, 9.64/3.30 "to": 705, 9.64/3.30 "label": "INSTANCE with matching:\nT131 -> T156" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 741, 9.64/3.30 "to": 705, 9.64/3.30 "label": "INSTANCE with matching:\nT131 -> T160" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 743, 9.64/3.30 "to": 769, 9.64/3.30 "label": "EVAL with clause\nunif(black, black).\nand substitutionT123 -> black,\nT124 -> black,\nT125 -> T161" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 743, 9.64/3.30 "to": 770, 9.64/3.30 "label": "EVAL-BACKTRACK" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 744, 9.64/3.30 "to": 771, 9.64/3.30 "label": "PARALLEL" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 744, 9.64/3.30 "to": 772, 9.64/3.30 "label": "PARALLEL" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 769, 9.64/3.30 "to": 705, 9.64/3.30 "label": "INSTANCE with matching:\nT131 -> T161" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 771, 9.64/3.30 "to": 773, 9.64/3.30 "label": "EVAL with clause\nunif(black, w(X209)).\nand substitutionT123 -> black,\nX209 -> T166,\nT124 -> w(T166),\nT125 -> T167" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 771, 9.64/3.30 "to": 774, 9.64/3.30 "label": "EVAL-BACKTRACK" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 772, 9.64/3.30 "to": 776, 9.64/3.30 "label": "EVAL with clause\nunif(w(X212), black).\nand substitutionX212 -> T170,\nT123 -> w(T170),\nT124 -> black,\nT125 -> T171" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 772, 9.64/3.30 "to": 777, 9.64/3.30 "label": "EVAL-BACKTRACK" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 773, 9.64/3.30 "to": 705, 9.64/3.30 "label": "INSTANCE with matching:\nT131 -> T167" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 776, 9.64/3.30 "to": 705, 9.64/3.30 "label": "INSTANCE with matching:\nT131 -> T171" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 780, 9.64/3.30 "to": 782, 9.64/3.30 "label": "SUCCESS" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 783, 9.64/3.30 "to": 785, 9.64/3.30 "label": "SUCCESS" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 786, 9.64/3.30 "to": 788, 9.64/3.30 "label": "EVAL with clause\nvarmat(.(black, X243), .(black, X244)) :- varmat(X243, X244).\nand substitutionX243 -> T194,\nT5 -> .(black, T194),\nX244 -> X245,\nX11 -> .(black, X245)" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 786, 9.64/3.30 "to": 789, 9.64/3.30 "label": "EVAL-BACKTRACK" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 787, 9.64/3.30 "to": 790, 9.64/3.30 "label": "EVAL with clause\nvarmat(.(white, X256), .(w(X257), X258)) :- varmat(X256, X258).\nand substitutionX256 -> T197,\nT5 -> .(white, T197),\nX257 -> X259,\nX258 -> X260,\nX11 -> .(w(X259), X260)" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 787, 9.64/3.30 "to": 791, 9.64/3.30 "label": "EVAL-BACKTRACK" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 788, 9.64/3.30 "to": 296, 9.64/3.30 "label": "INSTANCE with matching:\nT16 -> T194\nX48 -> X245\nT17 -> black" 9.64/3.30 }, 9.64/3.30 { 9.64/3.30 "from": 790, 9.64/3.30 "to": 296, 9.64/3.30 "label": "INSTANCE with matching:\nT16 -> T197\nX48 -> X260\nT17 -> w(X259)" 9.64/3.30 } 9.64/3.30 ], 9.64/3.30 "type": "Graph" 9.64/3.30 } 9.64/3.30 } 9.64/3.30 9.64/3.30 ---------------------------------------- 9.64/3.30 9.64/3.30 (2) 9.64/3.30 Obligation: 9.64/3.30 Triples: 9.64/3.30 9.64/3.30 varmatA(.(X1, X2), .(X3, X4)) :- varmatA(X1, X3). 9.64/3.30 varmatA(.(X1, X2), .(X3, X4)) :- ','(varmatcA(X1, X3), varmatA(X2, X4)). 9.64/3.30 varmatA(.(black, X1), .(black, X2)) :- varmatA(X1, X2). 9.64/3.30 varmatA(.(white, X1), .(w(X2), X3)) :- varmatA(X1, X3). 9.64/3.30 unif_matrxB(X1, .(X2, X3)) :- unif_linesC(X1, X2). 9.64/3.30 unif_matrxB(X1, .(X2, X3)) :- ','(unif_linescC(X1, X2), unif_matrxB(X2, X3)). 9.64/3.30 unif_linesC(.(X1, .(X2, X3)), .(X4, .(X5, X6))) :- pD(X1, .(X2, .(X4, .(X5, .(X1, .(X4, .(X2, .(X5, .(X1, .(X5, .(X2, .(X4, []))))))))))), X2, X3, X5, X6). 9.64/3.30 unif_pairsE(.(w(X1), .(w(X1), X2))) :- unif_pairsE(X2). 9.64/3.30 unif_pairsE(.(black, .(black, X1))) :- unif_pairsE(X1). 9.64/3.30 unif_pairsE(.(black, .(w(X1), X2))) :- unif_pairsE(X2). 9.64/3.30 unif_pairsE(.(w(X1), .(black, X2))) :- unif_pairsE(X2). 9.64/3.30 pF(X1, X2, X3) :- varmatA(X1, X2). 9.64/3.30 pF(X1, X2, X3) :- ','(varmatcA(X1, X2), unif_matrxB(X3, X2)). 9.64/3.30 pD(w(X1), .(w(X1), X2), X3, X4, X5, X6) :- unif_pairsE(X2). 9.64/3.30 pD(black, .(black, X1), X2, X3, X4, X5) :- unif_pairsE(X1). 9.64/3.30 pD(black, .(w(X1), X2), X3, X4, X5, X6) :- unif_pairsE(X2). 9.64/3.30 pD(w(X1), .(black, X2), X3, X4, X5, X6) :- unif_pairsE(X2). 9.64/3.30 pD(X1, X2, X3, X4, X5, X6) :- ','(unif_pairscG(X1, X2), unif_linesC(.(X3, X4), .(X5, X6))). 9.64/3.30 shapesH(.(X1, X2), X3) :- varmatA(X1, X4). 9.64/3.30 shapesH(.(X1, X2), X3) :- ','(varmatcA(X1, X4), pF(X2, X5, X4)). 9.64/3.30 shapesH(.(black, X1), X2) :- pF(X1, X3, black). 9.64/3.30 shapesH(.(white, X1), X2) :- pF(X1, X3, w(X4)). 9.64/3.30 9.64/3.30 Clauses: 9.64/3.30 9.64/3.30 varmatcA([], []). 9.64/3.30 varmatcA(.(X1, X2), .(X3, X4)) :- ','(varmatcA(X1, X3), varmatcA(X2, X4)). 9.64/3.30 varmatcA(.(black, X1), .(black, X2)) :- varmatcA(X1, X2). 9.64/3.30 varmatcA(.(white, X1), .(w(X2), X3)) :- varmatcA(X1, X3). 9.64/3.30 unif_matrxcB(X1, .(X2, X3)) :- ','(unif_linescC(X1, X2), unif_matrxcB(X2, X3)). 9.64/3.30 unif_matrxcB(X1, []). 9.64/3.30 unif_linescC(.(X1, .(X2, X3)), .(X4, .(X5, X6))) :- qcD(X1, .(X2, .(X4, .(X5, .(X1, .(X4, .(X2, .(X5, .(X1, .(X5, .(X2, .(X4, []))))))))))), X2, X3, X5, X6). 9.64/3.30 unif_linescC(.(X1, []), .(X2, [])). 9.64/3.30 unif_pairscE([]). 9.64/3.30 unif_pairscE(.(w(X1), .(w(X1), X2))) :- unif_pairscE(X2). 9.64/3.30 unif_pairscE(.(black, .(black, X1))) :- unif_pairscE(X1). 9.64/3.30 unif_pairscE(.(black, .(w(X1), X2))) :- unif_pairscE(X2). 9.64/3.30 unif_pairscE(.(w(X1), .(black, X2))) :- unif_pairscE(X2). 9.64/3.30 qcF(X1, X2, X3) :- ','(varmatcA(X1, X2), unif_matrxcB(X3, X2)). 9.64/3.30 qcD(X1, X2, X3, X4, X5, X6) :- ','(unif_pairscG(X1, X2), unif_linescC(.(X3, X4), .(X5, X6))). 9.64/3.30 unif_pairscG(w(X1), .(w(X1), X2)) :- unif_pairscE(X2). 9.64/3.30 unif_pairscG(black, .(black, X1)) :- unif_pairscE(X1). 9.64/3.30 unif_pairscG(black, .(w(X1), X2)) :- unif_pairscE(X2). 9.64/3.30 unif_pairscG(w(X1), .(black, X2)) :- unif_pairscE(X2). 9.64/3.30 9.64/3.30 Afs: 9.64/3.30 9.64/3.30 shapesH(x1, x2) = shapesH(x1) 9.64/3.30 9.64/3.30 9.64/3.30 ---------------------------------------- 9.64/3.30 9.64/3.30 (3) TriplesToPiDPProof (SOUND) 9.64/3.30 We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes: 9.64/3.30 9.64/3.30 shapesH_in_2: (b,f) 9.64/3.30 9.64/3.30 varmatA_in_2: (b,f) 9.64/3.30 9.64/3.30 varmatcA_in_2: (b,f) 9.64/3.30 9.64/3.30 pF_in_3: (b,f,b) 9.64/3.30 9.64/3.30 unif_matrxB_in_2: (b,b) 9.64/3.30 9.64/3.30 unif_linesC_in_2: (b,b) 9.64/3.30 9.64/3.30 pD_in_6: (b,b,b,b,b,b) 9.64/3.30 9.64/3.30 unif_pairsE_in_1: (b) 9.64/3.30 9.64/3.30 unif_pairscG_in_2: (b,b) 9.64/3.30 9.64/3.30 unif_pairscE_in_1: (b) 9.64/3.30 9.64/3.30 unif_linescC_in_2: (b,b) 9.64/3.30 9.64/3.30 qcD_in_6: (b,b,b,b,b,b) 9.64/3.30 9.64/3.30 Transforming TRIPLES into the following Term Rewriting System: 9.64/3.30 9.64/3.30 Pi DP problem: 9.64/3.30 The TRS P consists of the following rules: 9.64/3.30 9.64/3.30 SHAPESH_IN_GA(.(X1, X2), X3) -> U23_GA(X1, X2, X3, varmatA_in_ga(X1, X4)) 9.64/3.30 SHAPESH_IN_GA(.(X1, X2), X3) -> VARMATA_IN_GA(X1, X4) 9.64/3.30 VARMATA_IN_GA(.(X1, X2), .(X3, X4)) -> U1_GA(X1, X2, X3, X4, varmatA_in_ga(X1, X3)) 9.64/3.30 VARMATA_IN_GA(.(X1, X2), .(X3, X4)) -> VARMATA_IN_GA(X1, X3) 9.64/3.30 VARMATA_IN_GA(.(X1, X2), .(X3, X4)) -> U2_GA(X1, X2, X3, X4, varmatcA_in_ga(X1, X3)) 9.64/3.30 U2_GA(X1, X2, X3, X4, varmatcA_out_ga(X1, X3)) -> U3_GA(X1, X2, X3, X4, varmatA_in_ga(X2, X4)) 9.64/3.30 U2_GA(X1, X2, X3, X4, varmatcA_out_ga(X1, X3)) -> VARMATA_IN_GA(X2, X4) 9.64/3.30 VARMATA_IN_GA(.(black, X1), .(black, X2)) -> U4_GA(X1, X2, varmatA_in_ga(X1, X2)) 9.64/3.30 VARMATA_IN_GA(.(black, X1), .(black, X2)) -> VARMATA_IN_GA(X1, X2) 9.64/3.30 VARMATA_IN_GA(.(white, X1), .(w(X2), X3)) -> U5_GA(X1, X2, X3, varmatA_in_ga(X1, X3)) 9.64/3.30 VARMATA_IN_GA(.(white, X1), .(w(X2), X3)) -> VARMATA_IN_GA(X1, X3) 9.64/3.30 SHAPESH_IN_GA(.(X1, X2), X3) -> U24_GA(X1, X2, X3, varmatcA_in_ga(X1, X4)) 9.64/3.30 U24_GA(X1, X2, X3, varmatcA_out_ga(X1, X4)) -> U25_GA(X1, X2, X3, pF_in_gag(X2, X5, X4)) 9.64/3.30 U24_GA(X1, X2, X3, varmatcA_out_ga(X1, X4)) -> PF_IN_GAG(X2, X5, X4) 9.64/3.30 PF_IN_GAG(X1, X2, X3) -> U14_GAG(X1, X2, X3, varmatA_in_ga(X1, X2)) 9.64/3.30 PF_IN_GAG(X1, X2, X3) -> VARMATA_IN_GA(X1, X2) 9.64/3.30 PF_IN_GAG(X1, X2, X3) -> U15_GAG(X1, X2, X3, varmatcA_in_ga(X1, X2)) 9.64/3.30 U15_GAG(X1, X2, X3, varmatcA_out_ga(X1, X2)) -> U16_GAG(X1, X2, X3, unif_matrxB_in_gg(X3, X2)) 9.64/3.30 U15_GAG(X1, X2, X3, varmatcA_out_ga(X1, X2)) -> UNIF_MATRXB_IN_GG(X3, X2) 9.64/3.30 UNIF_MATRXB_IN_GG(X1, .(X2, X3)) -> U6_GG(X1, X2, X3, unif_linesC_in_gg(X1, X2)) 9.64/3.30 UNIF_MATRXB_IN_GG(X1, .(X2, X3)) -> UNIF_LINESC_IN_GG(X1, X2) 9.64/3.30 UNIF_LINESC_IN_GG(.(X1, .(X2, X3)), .(X4, .(X5, X6))) -> U9_GG(X1, X2, X3, X4, X5, X6, pD_in_gggggg(X1, .(X2, .(X4, .(X5, .(X1, .(X4, .(X2, .(X5, .(X1, .(X5, .(X2, .(X4, []))))))))))), X2, X3, X5, X6)) 9.64/3.30 UNIF_LINESC_IN_GG(.(X1, .(X2, X3)), .(X4, .(X5, X6))) -> PD_IN_GGGGGG(X1, .(X2, .(X4, .(X5, .(X1, .(X4, .(X2, .(X5, .(X1, .(X5, .(X2, .(X4, []))))))))))), X2, X3, X5, X6) 9.64/3.30 PD_IN_GGGGGG(w(X1), .(w(X1), X2), X3, X4, X5, X6) -> U17_GGGGGG(X1, X2, X3, X4, X5, X6, unif_pairsE_in_g(X2)) 9.64/3.30 PD_IN_GGGGGG(w(X1), .(w(X1), X2), X3, X4, X5, X6) -> UNIF_PAIRSE_IN_G(X2) 9.64/3.30 UNIF_PAIRSE_IN_G(.(w(X1), .(w(X1), X2))) -> U10_G(X1, X2, unif_pairsE_in_g(X2)) 9.64/3.30 UNIF_PAIRSE_IN_G(.(w(X1), .(w(X1), X2))) -> UNIF_PAIRSE_IN_G(X2) 9.64/3.30 UNIF_PAIRSE_IN_G(.(black, .(black, X1))) -> U11_G(X1, unif_pairsE_in_g(X1)) 9.64/3.30 UNIF_PAIRSE_IN_G(.(black, .(black, X1))) -> UNIF_PAIRSE_IN_G(X1) 9.64/3.30 UNIF_PAIRSE_IN_G(.(black, .(w(X1), X2))) -> U12_G(X1, X2, unif_pairsE_in_g(X2)) 9.64/3.30 UNIF_PAIRSE_IN_G(.(black, .(w(X1), X2))) -> UNIF_PAIRSE_IN_G(X2) 9.64/3.30 UNIF_PAIRSE_IN_G(.(w(X1), .(black, X2))) -> U13_G(X1, X2, unif_pairsE_in_g(X2)) 9.64/3.30 UNIF_PAIRSE_IN_G(.(w(X1), .(black, X2))) -> UNIF_PAIRSE_IN_G(X2) 9.64/3.30 PD_IN_GGGGGG(black, .(black, X1), X2, X3, X4, X5) -> U18_GGGGGG(X1, X2, X3, X4, X5, unif_pairsE_in_g(X1)) 9.64/3.30 PD_IN_GGGGGG(black, .(black, X1), X2, X3, X4, X5) -> UNIF_PAIRSE_IN_G(X1) 9.64/3.30 PD_IN_GGGGGG(black, .(w(X1), X2), X3, X4, X5, X6) -> U19_GGGGGG(X1, X2, X3, X4, X5, X6, unif_pairsE_in_g(X2)) 9.64/3.30 PD_IN_GGGGGG(black, .(w(X1), X2), X3, X4, X5, X6) -> UNIF_PAIRSE_IN_G(X2) 9.64/3.30 PD_IN_GGGGGG(w(X1), .(black, X2), X3, X4, X5, X6) -> U20_GGGGGG(X1, X2, X3, X4, X5, X6, unif_pairsE_in_g(X2)) 9.64/3.30 PD_IN_GGGGGG(w(X1), .(black, X2), X3, X4, X5, X6) -> UNIF_PAIRSE_IN_G(X2) 9.64/3.30 PD_IN_GGGGGG(X1, X2, X3, X4, X5, X6) -> U21_GGGGGG(X1, X2, X3, X4, X5, X6, unif_pairscG_in_gg(X1, X2)) 9.64/3.30 U21_GGGGGG(X1, X2, X3, X4, X5, X6, unif_pairscG_out_gg(X1, X2)) -> U22_GGGGGG(X1, X2, X3, X4, X5, X6, unif_linesC_in_gg(.(X3, X4), .(X5, X6))) 9.64/3.30 U21_GGGGGG(X1, X2, X3, X4, X5, X6, unif_pairscG_out_gg(X1, X2)) -> UNIF_LINESC_IN_GG(.(X3, X4), .(X5, X6)) 9.64/3.30 UNIF_MATRXB_IN_GG(X1, .(X2, X3)) -> U7_GG(X1, X2, X3, unif_linescC_in_gg(X1, X2)) 9.64/3.30 U7_GG(X1, X2, X3, unif_linescC_out_gg(X1, X2)) -> U8_GG(X1, X2, X3, unif_matrxB_in_gg(X2, X3)) 9.64/3.30 U7_GG(X1, X2, X3, unif_linescC_out_gg(X1, X2)) -> UNIF_MATRXB_IN_GG(X2, X3) 9.64/3.30 SHAPESH_IN_GA(.(black, X1), X2) -> U26_GA(X1, X2, pF_in_gag(X1, X3, black)) 9.64/3.30 SHAPESH_IN_GA(.(black, X1), X2) -> PF_IN_GAG(X1, X3, black) 9.64/3.30 SHAPESH_IN_GA(.(white, X1), X2) -> U27_GA(X1, X2, pF_in_gag(X1, X3, w(X4))) 9.64/3.30 SHAPESH_IN_GA(.(white, X1), X2) -> PF_IN_GAG(X1, X3, w(X4)) 9.64/3.30 9.64/3.30 The TRS R consists of the following rules: 9.64/3.30 9.64/3.30 varmatcA_in_ga([], []) -> varmatcA_out_ga([], []) 9.64/3.30 varmatcA_in_ga(.(X1, X2), .(X3, X4)) -> U29_ga(X1, X2, X3, X4, varmatcA_in_ga(X1, X3)) 9.64/3.30 varmatcA_in_ga(.(black, X1), .(black, X2)) -> U31_ga(X1, X2, varmatcA_in_ga(X1, X2)) 9.64/3.30 varmatcA_in_ga(.(white, X1), .(w(X2), X3)) -> U32_ga(X1, X2, X3, varmatcA_in_ga(X1, X3)) 9.64/3.30 U32_ga(X1, X2, X3, varmatcA_out_ga(X1, X3)) -> varmatcA_out_ga(.(white, X1), .(w(X2), X3)) 9.64/3.30 U31_ga(X1, X2, varmatcA_out_ga(X1, X2)) -> varmatcA_out_ga(.(black, X1), .(black, X2)) 9.64/3.30 U29_ga(X1, X2, X3, X4, varmatcA_out_ga(X1, X3)) -> U30_ga(X1, X2, X3, X4, varmatcA_in_ga(X2, X4)) 9.64/3.30 U30_ga(X1, X2, X3, X4, varmatcA_out_ga(X2, X4)) -> varmatcA_out_ga(.(X1, X2), .(X3, X4)) 9.64/3.30 unif_pairscG_in_gg(w(X1), .(w(X1), X2)) -> U44_gg(X1, X2, unif_pairscE_in_g(X2)) 9.64/3.30 unif_pairscE_in_g([]) -> unif_pairscE_out_g([]) 9.64/3.30 unif_pairscE_in_g(.(w(X1), .(w(X1), X2))) -> U36_g(X1, X2, unif_pairscE_in_g(X2)) 9.64/3.30 unif_pairscE_in_g(.(black, .(black, X1))) -> U37_g(X1, unif_pairscE_in_g(X1)) 9.64/3.30 unif_pairscE_in_g(.(black, .(w(X1), X2))) -> U38_g(X1, X2, unif_pairscE_in_g(X2)) 9.64/3.30 unif_pairscE_in_g(.(w(X1), .(black, X2))) -> U39_g(X1, X2, unif_pairscE_in_g(X2)) 9.64/3.30 U39_g(X1, X2, unif_pairscE_out_g(X2)) -> unif_pairscE_out_g(.(w(X1), .(black, X2))) 9.64/3.30 U38_g(X1, X2, unif_pairscE_out_g(X2)) -> unif_pairscE_out_g(.(black, .(w(X1), X2))) 9.64/3.30 U37_g(X1, unif_pairscE_out_g(X1)) -> unif_pairscE_out_g(.(black, .(black, X1))) 9.64/3.30 U36_g(X1, X2, unif_pairscE_out_g(X2)) -> unif_pairscE_out_g(.(w(X1), .(w(X1), X2))) 9.64/3.30 U44_gg(X1, X2, unif_pairscE_out_g(X2)) -> unif_pairscG_out_gg(w(X1), .(w(X1), X2)) 9.64/3.30 unif_pairscG_in_gg(black, .(black, X1)) -> U45_gg(X1, unif_pairscE_in_g(X1)) 9.64/3.30 U45_gg(X1, unif_pairscE_out_g(X1)) -> unif_pairscG_out_gg(black, .(black, X1)) 9.64/3.30 unif_pairscG_in_gg(black, .(w(X1), X2)) -> U46_gg(X1, X2, unif_pairscE_in_g(X2)) 9.64/3.30 U46_gg(X1, X2, unif_pairscE_out_g(X2)) -> unif_pairscG_out_gg(black, .(w(X1), X2)) 9.64/3.30 unif_pairscG_in_gg(w(X1), .(black, X2)) -> U47_gg(X1, X2, unif_pairscE_in_g(X2)) 9.64/3.30 U47_gg(X1, X2, unif_pairscE_out_g(X2)) -> unif_pairscG_out_gg(w(X1), .(black, X2)) 9.64/3.30 unif_linescC_in_gg(.(X1, .(X2, X3)), .(X4, .(X5, X6))) -> U35_gg(X1, X2, X3, X4, X5, X6, qcD_in_gggggg(X1, .(X2, .(X4, .(X5, .(X1, .(X4, .(X2, .(X5, .(X1, .(X5, .(X2, .(X4, []))))))))))), X2, X3, X5, X6)) 9.64/3.30 qcD_in_gggggg(X1, X2, X3, X4, X5, X6) -> U42_gggggg(X1, X2, X3, X4, X5, X6, unif_pairscG_in_gg(X1, X2)) 9.64/3.30 U42_gggggg(X1, X2, X3, X4, X5, X6, unif_pairscG_out_gg(X1, X2)) -> U43_gggggg(X1, X2, X3, X4, X5, X6, unif_linescC_in_gg(.(X3, X4), .(X5, X6))) 9.64/3.30 unif_linescC_in_gg(.(X1, []), .(X2, [])) -> unif_linescC_out_gg(.(X1, []), .(X2, [])) 9.64/3.30 U43_gggggg(X1, X2, X3, X4, X5, X6, unif_linescC_out_gg(.(X3, X4), .(X5, X6))) -> qcD_out_gggggg(X1, X2, X3, X4, X5, X6) 9.64/3.30 U35_gg(X1, X2, X3, X4, X5, X6, qcD_out_gggggg(X1, .(X2, .(X4, .(X5, .(X1, .(X4, .(X2, .(X5, .(X1, .(X5, .(X2, .(X4, []))))))))))), X2, X3, X5, X6)) -> unif_linescC_out_gg(.(X1, .(X2, X3)), .(X4, .(X5, X6))) 9.64/3.30 9.64/3.30 The argument filtering Pi contains the following mapping: 9.64/3.30 .(x1, x2) = .(x1, x2) 9.64/3.30 9.64/3.30 varmatA_in_ga(x1, x2) = varmatA_in_ga(x1) 9.64/3.30 9.64/3.30 varmatcA_in_ga(x1, x2) = varmatcA_in_ga(x1) 9.64/3.30 9.64/3.30 [] = [] 9.64/3.30 9.64/3.30 varmatcA_out_ga(x1, x2) = varmatcA_out_ga(x1, x2) 9.64/3.30 9.64/3.30 U29_ga(x1, x2, x3, x4, x5) = U29_ga(x1, x2, x5) 9.64/3.30 9.64/3.30 black = black 9.64/3.30 9.64/3.30 U31_ga(x1, x2, x3) = U31_ga(x1, x3) 9.64/3.30 9.64/3.30 white = white 9.64/3.30 9.64/3.30 U32_ga(x1, x2, x3, x4) = U32_ga(x1, x4) 9.64/3.30 9.64/3.30 w(x1) = w 9.64/3.30 9.64/3.30 U30_ga(x1, x2, x3, x4, x5) = U30_ga(x1, x2, x3, x5) 9.64/3.30 9.64/3.30 pF_in_gag(x1, x2, x3) = pF_in_gag(x1, x3) 9.64/3.30 9.64/3.30 unif_matrxB_in_gg(x1, x2) = unif_matrxB_in_gg(x1, x2) 9.64/3.30 9.64/3.30 unif_linesC_in_gg(x1, x2) = unif_linesC_in_gg(x1, x2) 9.64/3.30 9.64/3.30 pD_in_gggggg(x1, x2, x3, x4, x5, x6) = pD_in_gggggg(x1, x2, x3, x4, x5, x6) 9.64/3.30 9.64/3.30 unif_pairsE_in_g(x1) = unif_pairsE_in_g(x1) 9.64/3.30 9.64/3.30 unif_pairscG_in_gg(x1, x2) = unif_pairscG_in_gg(x1, x2) 9.64/3.30 9.64/3.30 U44_gg(x1, x2, x3) = U44_gg(x2, x3) 9.64/3.30 9.64/3.30 unif_pairscE_in_g(x1) = unif_pairscE_in_g(x1) 9.64/3.30 9.64/3.30 unif_pairscE_out_g(x1) = unif_pairscE_out_g(x1) 9.64/3.30 9.64/3.30 U36_g(x1, x2, x3) = U36_g(x2, x3) 9.64/3.30 9.64/3.30 U37_g(x1, x2) = U37_g(x1, x2) 9.64/3.30 9.64/3.30 U38_g(x1, x2, x3) = U38_g(x2, x3) 9.64/3.30 9.64/3.30 U39_g(x1, x2, x3) = U39_g(x2, x3) 9.64/3.30 9.64/3.30 unif_pairscG_out_gg(x1, x2) = unif_pairscG_out_gg(x1, x2) 9.64/3.30 9.64/3.30 U45_gg(x1, x2) = U45_gg(x1, x2) 9.64/3.30 9.64/3.30 U46_gg(x1, x2, x3) = U46_gg(x2, x3) 9.64/3.30 9.64/3.30 U47_gg(x1, x2, x3) = U47_gg(x2, x3) 9.64/3.30 9.64/3.30 unif_linescC_in_gg(x1, x2) = unif_linescC_in_gg(x1, x2) 9.64/3.30 9.64/3.30 U35_gg(x1, x2, x3, x4, x5, x6, x7) = U35_gg(x1, x2, x3, x4, x5, x6, x7) 9.64/3.30 9.64/3.30 qcD_in_gggggg(x1, x2, x3, x4, x5, x6) = qcD_in_gggggg(x1, x2, x3, x4, x5, x6) 9.64/3.30 9.64/3.30 U42_gggggg(x1, x2, x3, x4, x5, x6, x7) = U42_gggggg(x1, x2, x3, x4, x5, x6, x7) 9.64/3.30 9.64/3.30 U43_gggggg(x1, x2, x3, x4, x5, x6, x7) = U43_gggggg(x1, x2, x3, x4, x5, x6, x7) 9.64/3.30 9.64/3.30 unif_linescC_out_gg(x1, x2) = unif_linescC_out_gg(x1, x2) 9.64/3.30 9.64/3.30 qcD_out_gggggg(x1, x2, x3, x4, x5, x6) = qcD_out_gggggg(x1, x2, x3, x4, x5, x6) 9.64/3.30 9.64/3.30 SHAPESH_IN_GA(x1, x2) = SHAPESH_IN_GA(x1) 9.64/3.30 9.64/3.30 U23_GA(x1, x2, x3, x4) = U23_GA(x1, x2, x4) 9.64/3.30 9.64/3.30 VARMATA_IN_GA(x1, x2) = VARMATA_IN_GA(x1) 9.64/3.30 9.64/3.30 U1_GA(x1, x2, x3, x4, x5) = U1_GA(x1, x2, x5) 9.64/3.30 9.64/3.30 U2_GA(x1, x2, x3, x4, x5) = U2_GA(x1, x2, x5) 9.64/3.30 9.64/3.30 U3_GA(x1, x2, x3, x4, x5) = U3_GA(x1, x2, x5) 9.64/3.30 9.64/3.30 U4_GA(x1, x2, x3) = U4_GA(x1, x3) 9.64/3.30 9.64/3.30 U5_GA(x1, x2, x3, x4) = U5_GA(x1, x4) 9.64/3.30 9.64/3.30 U24_GA(x1, x2, x3, x4) = U24_GA(x1, x2, x4) 9.64/3.30 9.64/3.30 U25_GA(x1, x2, x3, x4) = U25_GA(x1, x2, x4) 9.64/3.30 9.64/3.30 PF_IN_GAG(x1, x2, x3) = PF_IN_GAG(x1, x3) 9.64/3.30 9.64/3.30 U14_GAG(x1, x2, x3, x4) = U14_GAG(x1, x3, x4) 9.64/3.30 9.64/3.30 U15_GAG(x1, x2, x3, x4) = U15_GAG(x1, x3, x4) 9.64/3.30 9.64/3.30 U16_GAG(x1, x2, x3, x4) = U16_GAG(x1, x3, x4) 9.64/3.30 9.64/3.30 UNIF_MATRXB_IN_GG(x1, x2) = UNIF_MATRXB_IN_GG(x1, x2) 9.64/3.30 9.64/3.30 U6_GG(x1, x2, x3, x4) = U6_GG(x1, x2, x3, x4) 9.64/3.30 9.64/3.30 UNIF_LINESC_IN_GG(x1, x2) = UNIF_LINESC_IN_GG(x1, x2) 9.64/3.30 9.64/3.30 U9_GG(x1, x2, x3, x4, x5, x6, x7) = U9_GG(x1, x2, x3, x4, x5, x6, x7) 9.64/3.30 9.64/3.30 PD_IN_GGGGGG(x1, x2, x3, x4, x5, x6) = PD_IN_GGGGGG(x1, x2, x3, x4, x5, x6) 9.64/3.30 9.64/3.30 U17_GGGGGG(x1, x2, x3, x4, x5, x6, x7) = U17_GGGGGG(x2, x3, x4, x5, x6, x7) 9.64/3.30 9.64/3.30 UNIF_PAIRSE_IN_G(x1) = UNIF_PAIRSE_IN_G(x1) 9.64/3.30 9.64/3.30 U10_G(x1, x2, x3) = U10_G(x2, x3) 9.64/3.30 9.64/3.30 U11_G(x1, x2) = U11_G(x1, x2) 9.64/3.30 9.64/3.30 U12_G(x1, x2, x3) = U12_G(x2, x3) 9.64/3.30 9.64/3.30 U13_G(x1, x2, x3) = U13_G(x2, x3) 9.64/3.30 9.64/3.30 U18_GGGGGG(x1, x2, x3, x4, x5, x6) = U18_GGGGGG(x1, x2, x3, x4, x5, x6) 9.64/3.30 9.64/3.30 U19_GGGGGG(x1, x2, x3, x4, x5, x6, x7) = U19_GGGGGG(x2, x3, x4, x5, x6, x7) 9.64/3.30 9.64/3.30 U20_GGGGGG(x1, x2, x3, x4, x5, x6, x7) = U20_GGGGGG(x2, x3, x4, x5, x6, x7) 9.64/3.30 9.64/3.30 U21_GGGGGG(x1, x2, x3, x4, x5, x6, x7) = U21_GGGGGG(x1, x2, x3, x4, x5, x6, x7) 9.64/3.30 9.64/3.30 U22_GGGGGG(x1, x2, x3, x4, x5, x6, x7) = U22_GGGGGG(x1, x2, x3, x4, x5, x6, x7) 9.64/3.30 9.64/3.30 U7_GG(x1, x2, x3, x4) = U7_GG(x1, x2, x3, x4) 9.64/3.30 9.64/3.30 U8_GG(x1, x2, x3, x4) = U8_GG(x1, x2, x3, x4) 9.64/3.30 9.64/3.30 U26_GA(x1, x2, x3) = U26_GA(x1, x3) 9.64/3.30 9.64/3.30 U27_GA(x1, x2, x3) = U27_GA(x1, x3) 9.64/3.30 9.64/3.30 9.64/3.30 We have to consider all (P,R,Pi)-chains 9.64/3.30 9.64/3.30 9.64/3.30 Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES 9.64/3.30 9.64/3.30 9.64/3.30 9.64/3.30 ---------------------------------------- 9.64/3.30 9.64/3.30 (4) 9.64/3.30 Obligation: 9.64/3.30 Pi DP problem: 9.64/3.30 The TRS P consists of the following rules: 9.64/3.30 9.64/3.30 SHAPESH_IN_GA(.(X1, X2), X3) -> U23_GA(X1, X2, X3, varmatA_in_ga(X1, X4)) 9.64/3.30 SHAPESH_IN_GA(.(X1, X2), X3) -> VARMATA_IN_GA(X1, X4) 9.64/3.30 VARMATA_IN_GA(.(X1, X2), .(X3, X4)) -> U1_GA(X1, X2, X3, X4, varmatA_in_ga(X1, X3)) 9.64/3.30 VARMATA_IN_GA(.(X1, X2), .(X3, X4)) -> VARMATA_IN_GA(X1, X3) 9.64/3.30 VARMATA_IN_GA(.(X1, X2), .(X3, X4)) -> U2_GA(X1, X2, X3, X4, varmatcA_in_ga(X1, X3)) 9.64/3.30 U2_GA(X1, X2, X3, X4, varmatcA_out_ga(X1, X3)) -> U3_GA(X1, X2, X3, X4, varmatA_in_ga(X2, X4)) 9.64/3.30 U2_GA(X1, X2, X3, X4, varmatcA_out_ga(X1, X3)) -> VARMATA_IN_GA(X2, X4) 9.64/3.30 VARMATA_IN_GA(.(black, X1), .(black, X2)) -> U4_GA(X1, X2, varmatA_in_ga(X1, X2)) 9.64/3.30 VARMATA_IN_GA(.(black, X1), .(black, X2)) -> VARMATA_IN_GA(X1, X2) 9.64/3.30 VARMATA_IN_GA(.(white, X1), .(w(X2), X3)) -> U5_GA(X1, X2, X3, varmatA_in_ga(X1, X3)) 9.64/3.30 VARMATA_IN_GA(.(white, X1), .(w(X2), X3)) -> VARMATA_IN_GA(X1, X3) 9.64/3.30 SHAPESH_IN_GA(.(X1, X2), X3) -> U24_GA(X1, X2, X3, varmatcA_in_ga(X1, X4)) 9.64/3.30 U24_GA(X1, X2, X3, varmatcA_out_ga(X1, X4)) -> U25_GA(X1, X2, X3, pF_in_gag(X2, X5, X4)) 9.64/3.30 U24_GA(X1, X2, X3, varmatcA_out_ga(X1, X4)) -> PF_IN_GAG(X2, X5, X4) 9.64/3.30 PF_IN_GAG(X1, X2, X3) -> U14_GAG(X1, X2, X3, varmatA_in_ga(X1, X2)) 9.64/3.30 PF_IN_GAG(X1, X2, X3) -> VARMATA_IN_GA(X1, X2) 9.64/3.30 PF_IN_GAG(X1, X2, X3) -> U15_GAG(X1, X2, X3, varmatcA_in_ga(X1, X2)) 9.64/3.30 U15_GAG(X1, X2, X3, varmatcA_out_ga(X1, X2)) -> U16_GAG(X1, X2, X3, unif_matrxB_in_gg(X3, X2)) 9.64/3.30 U15_GAG(X1, X2, X3, varmatcA_out_ga(X1, X2)) -> UNIF_MATRXB_IN_GG(X3, X2) 9.64/3.30 UNIF_MATRXB_IN_GG(X1, .(X2, X3)) -> U6_GG(X1, X2, X3, unif_linesC_in_gg(X1, X2)) 9.64/3.30 UNIF_MATRXB_IN_GG(X1, .(X2, X3)) -> UNIF_LINESC_IN_GG(X1, X2) 9.64/3.30 UNIF_LINESC_IN_GG(.(X1, .(X2, X3)), .(X4, .(X5, X6))) -> U9_GG(X1, X2, X3, X4, X5, X6, pD_in_gggggg(X1, .(X2, .(X4, .(X5, .(X1, .(X4, .(X2, .(X5, .(X1, .(X5, .(X2, .(X4, []))))))))))), X2, X3, X5, X6)) 9.64/3.30 UNIF_LINESC_IN_GG(.(X1, .(X2, X3)), .(X4, .(X5, X6))) -> PD_IN_GGGGGG(X1, .(X2, .(X4, .(X5, .(X1, .(X4, .(X2, .(X5, .(X1, .(X5, .(X2, .(X4, []))))))))))), X2, X3, X5, X6) 9.64/3.30 PD_IN_GGGGGG(w(X1), .(w(X1), X2), X3, X4, X5, X6) -> U17_GGGGGG(X1, X2, X3, X4, X5, X6, unif_pairsE_in_g(X2)) 9.64/3.30 PD_IN_GGGGGG(w(X1), .(w(X1), X2), X3, X4, X5, X6) -> UNIF_PAIRSE_IN_G(X2) 9.64/3.30 UNIF_PAIRSE_IN_G(.(w(X1), .(w(X1), X2))) -> U10_G(X1, X2, unif_pairsE_in_g(X2)) 9.64/3.30 UNIF_PAIRSE_IN_G(.(w(X1), .(w(X1), X2))) -> UNIF_PAIRSE_IN_G(X2) 9.64/3.30 UNIF_PAIRSE_IN_G(.(black, .(black, X1))) -> U11_G(X1, unif_pairsE_in_g(X1)) 9.64/3.30 UNIF_PAIRSE_IN_G(.(black, .(black, X1))) -> UNIF_PAIRSE_IN_G(X1) 9.64/3.30 UNIF_PAIRSE_IN_G(.(black, .(w(X1), X2))) -> U12_G(X1, X2, unif_pairsE_in_g(X2)) 9.64/3.30 UNIF_PAIRSE_IN_G(.(black, .(w(X1), X2))) -> UNIF_PAIRSE_IN_G(X2) 9.64/3.30 UNIF_PAIRSE_IN_G(.(w(X1), .(black, X2))) -> U13_G(X1, X2, unif_pairsE_in_g(X2)) 9.64/3.30 UNIF_PAIRSE_IN_G(.(w(X1), .(black, X2))) -> UNIF_PAIRSE_IN_G(X2) 9.64/3.30 PD_IN_GGGGGG(black, .(black, X1), X2, X3, X4, X5) -> U18_GGGGGG(X1, X2, X3, X4, X5, unif_pairsE_in_g(X1)) 9.64/3.30 PD_IN_GGGGGG(black, .(black, X1), X2, X3, X4, X5) -> UNIF_PAIRSE_IN_G(X1) 9.64/3.30 PD_IN_GGGGGG(black, .(w(X1), X2), X3, X4, X5, X6) -> U19_GGGGGG(X1, X2, X3, X4, X5, X6, unif_pairsE_in_g(X2)) 9.64/3.30 PD_IN_GGGGGG(black, .(w(X1), X2), X3, X4, X5, X6) -> UNIF_PAIRSE_IN_G(X2) 9.64/3.30 PD_IN_GGGGGG(w(X1), .(black, X2), X3, X4, X5, X6) -> U20_GGGGGG(X1, X2, X3, X4, X5, X6, unif_pairsE_in_g(X2)) 9.64/3.30 PD_IN_GGGGGG(w(X1), .(black, X2), X3, X4, X5, X6) -> UNIF_PAIRSE_IN_G(X2) 9.64/3.30 PD_IN_GGGGGG(X1, X2, X3, X4, X5, X6) -> U21_GGGGGG(X1, X2, X3, X4, X5, X6, unif_pairscG_in_gg(X1, X2)) 9.64/3.30 U21_GGGGGG(X1, X2, X3, X4, X5, X6, unif_pairscG_out_gg(X1, X2)) -> U22_GGGGGG(X1, X2, X3, X4, X5, X6, unif_linesC_in_gg(.(X3, X4), .(X5, X6))) 9.64/3.30 U21_GGGGGG(X1, X2, X3, X4, X5, X6, unif_pairscG_out_gg(X1, X2)) -> UNIF_LINESC_IN_GG(.(X3, X4), .(X5, X6)) 9.64/3.30 UNIF_MATRXB_IN_GG(X1, .(X2, X3)) -> U7_GG(X1, X2, X3, unif_linescC_in_gg(X1, X2)) 9.64/3.30 U7_GG(X1, X2, X3, unif_linescC_out_gg(X1, X2)) -> U8_GG(X1, X2, X3, unif_matrxB_in_gg(X2, X3)) 9.64/3.30 U7_GG(X1, X2, X3, unif_linescC_out_gg(X1, X2)) -> UNIF_MATRXB_IN_GG(X2, X3) 9.64/3.30 SHAPESH_IN_GA(.(black, X1), X2) -> U26_GA(X1, X2, pF_in_gag(X1, X3, black)) 9.64/3.30 SHAPESH_IN_GA(.(black, X1), X2) -> PF_IN_GAG(X1, X3, black) 9.64/3.30 SHAPESH_IN_GA(.(white, X1), X2) -> U27_GA(X1, X2, pF_in_gag(X1, X3, w(X4))) 9.64/3.30 SHAPESH_IN_GA(.(white, X1), X2) -> PF_IN_GAG(X1, X3, w(X4)) 9.64/3.30 9.64/3.30 The TRS R consists of the following rules: 9.64/3.30 9.64/3.30 varmatcA_in_ga([], []) -> varmatcA_out_ga([], []) 9.64/3.30 varmatcA_in_ga(.(X1, X2), .(X3, X4)) -> U29_ga(X1, X2, X3, X4, varmatcA_in_ga(X1, X3)) 9.64/3.30 varmatcA_in_ga(.(black, X1), .(black, X2)) -> U31_ga(X1, X2, varmatcA_in_ga(X1, X2)) 9.64/3.30 varmatcA_in_ga(.(white, X1), .(w(X2), X3)) -> U32_ga(X1, X2, X3, varmatcA_in_ga(X1, X3)) 9.64/3.30 U32_ga(X1, X2, X3, varmatcA_out_ga(X1, X3)) -> varmatcA_out_ga(.(white, X1), .(w(X2), X3)) 9.64/3.30 U31_ga(X1, X2, varmatcA_out_ga(X1, X2)) -> varmatcA_out_ga(.(black, X1), .(black, X2)) 9.64/3.30 U29_ga(X1, X2, X3, X4, varmatcA_out_ga(X1, X3)) -> U30_ga(X1, X2, X3, X4, varmatcA_in_ga(X2, X4)) 9.64/3.30 U30_ga(X1, X2, X3, X4, varmatcA_out_ga(X2, X4)) -> varmatcA_out_ga(.(X1, X2), .(X3, X4)) 9.64/3.30 unif_pairscG_in_gg(w(X1), .(w(X1), X2)) -> U44_gg(X1, X2, unif_pairscE_in_g(X2)) 9.64/3.30 unif_pairscE_in_g([]) -> unif_pairscE_out_g([]) 9.64/3.30 unif_pairscE_in_g(.(w(X1), .(w(X1), X2))) -> U36_g(X1, X2, unif_pairscE_in_g(X2)) 9.64/3.30 unif_pairscE_in_g(.(black, .(black, X1))) -> U37_g(X1, unif_pairscE_in_g(X1)) 9.64/3.30 unif_pairscE_in_g(.(black, .(w(X1), X2))) -> U38_g(X1, X2, unif_pairscE_in_g(X2)) 9.64/3.30 unif_pairscE_in_g(.(w(X1), .(black, X2))) -> U39_g(X1, X2, unif_pairscE_in_g(X2)) 9.64/3.30 U39_g(X1, X2, unif_pairscE_out_g(X2)) -> unif_pairscE_out_g(.(w(X1), .(black, X2))) 9.64/3.30 U38_g(X1, X2, unif_pairscE_out_g(X2)) -> unif_pairscE_out_g(.(black, .(w(X1), X2))) 9.64/3.30 U37_g(X1, unif_pairscE_out_g(X1)) -> unif_pairscE_out_g(.(black, .(black, X1))) 9.64/3.30 U36_g(X1, X2, unif_pairscE_out_g(X2)) -> unif_pairscE_out_g(.(w(X1), .(w(X1), X2))) 9.64/3.30 U44_gg(X1, X2, unif_pairscE_out_g(X2)) -> unif_pairscG_out_gg(w(X1), .(w(X1), X2)) 9.64/3.30 unif_pairscG_in_gg(black, .(black, X1)) -> U45_gg(X1, unif_pairscE_in_g(X1)) 9.64/3.30 U45_gg(X1, unif_pairscE_out_g(X1)) -> unif_pairscG_out_gg(black, .(black, X1)) 9.64/3.30 unif_pairscG_in_gg(black, .(w(X1), X2)) -> U46_gg(X1, X2, unif_pairscE_in_g(X2)) 9.64/3.30 U46_gg(X1, X2, unif_pairscE_out_g(X2)) -> unif_pairscG_out_gg(black, .(w(X1), X2)) 9.64/3.30 unif_pairscG_in_gg(w(X1), .(black, X2)) -> U47_gg(X1, X2, unif_pairscE_in_g(X2)) 9.64/3.30 U47_gg(X1, X2, unif_pairscE_out_g(X2)) -> unif_pairscG_out_gg(w(X1), .(black, X2)) 9.64/3.30 unif_linescC_in_gg(.(X1, .(X2, X3)), .(X4, .(X5, X6))) -> U35_gg(X1, X2, X3, X4, X5, X6, qcD_in_gggggg(X1, .(X2, .(X4, .(X5, .(X1, .(X4, .(X2, .(X5, .(X1, .(X5, .(X2, .(X4, []))))))))))), X2, X3, X5, X6)) 9.64/3.30 qcD_in_gggggg(X1, X2, X3, X4, X5, X6) -> U42_gggggg(X1, X2, X3, X4, X5, X6, unif_pairscG_in_gg(X1, X2)) 9.64/3.30 U42_gggggg(X1, X2, X3, X4, X5, X6, unif_pairscG_out_gg(X1, X2)) -> U43_gggggg(X1, X2, X3, X4, X5, X6, unif_linescC_in_gg(.(X3, X4), .(X5, X6))) 9.64/3.30 unif_linescC_in_gg(.(X1, []), .(X2, [])) -> unif_linescC_out_gg(.(X1, []), .(X2, [])) 9.64/3.30 U43_gggggg(X1, X2, X3, X4, X5, X6, unif_linescC_out_gg(.(X3, X4), .(X5, X6))) -> qcD_out_gggggg(X1, X2, X3, X4, X5, X6) 9.64/3.30 U35_gg(X1, X2, X3, X4, X5, X6, qcD_out_gggggg(X1, .(X2, .(X4, .(X5, .(X1, .(X4, .(X2, .(X5, .(X1, .(X5, .(X2, .(X4, []))))))))))), X2, X3, X5, X6)) -> unif_linescC_out_gg(.(X1, .(X2, X3)), .(X4, .(X5, X6))) 9.64/3.30 9.64/3.30 The argument filtering Pi contains the following mapping: 9.64/3.30 .(x1, x2) = .(x1, x2) 9.64/3.30 9.64/3.30 varmatA_in_ga(x1, x2) = varmatA_in_ga(x1) 9.64/3.30 9.64/3.30 varmatcA_in_ga(x1, x2) = varmatcA_in_ga(x1) 9.64/3.30 9.64/3.30 [] = [] 9.64/3.30 9.64/3.30 varmatcA_out_ga(x1, x2) = varmatcA_out_ga(x1, x2) 9.64/3.30 9.64/3.30 U29_ga(x1, x2, x3, x4, x5) = U29_ga(x1, x2, x5) 9.64/3.30 9.64/3.30 black = black 9.64/3.30 9.64/3.30 U31_ga(x1, x2, x3) = U31_ga(x1, x3) 9.64/3.30 9.64/3.30 white = white 9.64/3.30 9.64/3.30 U32_ga(x1, x2, x3, x4) = U32_ga(x1, x4) 9.64/3.30 9.64/3.30 w(x1) = w 9.64/3.30 9.64/3.30 U30_ga(x1, x2, x3, x4, x5) = U30_ga(x1, x2, x3, x5) 9.64/3.30 9.64/3.30 pF_in_gag(x1, x2, x3) = pF_in_gag(x1, x3) 9.64/3.30 9.64/3.30 unif_matrxB_in_gg(x1, x2) = unif_matrxB_in_gg(x1, x2) 9.64/3.30 9.64/3.30 unif_linesC_in_gg(x1, x2) = unif_linesC_in_gg(x1, x2) 9.64/3.30 9.64/3.30 pD_in_gggggg(x1, x2, x3, x4, x5, x6) = pD_in_gggggg(x1, x2, x3, x4, x5, x6) 9.64/3.30 9.64/3.30 unif_pairsE_in_g(x1) = unif_pairsE_in_g(x1) 9.64/3.30 9.64/3.30 unif_pairscG_in_gg(x1, x2) = unif_pairscG_in_gg(x1, x2) 9.64/3.30 9.64/3.30 U44_gg(x1, x2, x3) = U44_gg(x2, x3) 9.64/3.30 9.64/3.30 unif_pairscE_in_g(x1) = unif_pairscE_in_g(x1) 9.64/3.30 9.64/3.30 unif_pairscE_out_g(x1) = unif_pairscE_out_g(x1) 9.64/3.30 9.64/3.30 U36_g(x1, x2, x3) = U36_g(x2, x3) 9.64/3.30 9.64/3.30 U37_g(x1, x2) = U37_g(x1, x2) 9.64/3.30 9.64/3.30 U38_g(x1, x2, x3) = U38_g(x2, x3) 9.64/3.30 9.64/3.30 U39_g(x1, x2, x3) = U39_g(x2, x3) 9.64/3.30 9.64/3.30 unif_pairscG_out_gg(x1, x2) = unif_pairscG_out_gg(x1, x2) 9.64/3.30 9.64/3.30 U45_gg(x1, x2) = U45_gg(x1, x2) 9.64/3.30 9.64/3.30 U46_gg(x1, x2, x3) = U46_gg(x2, x3) 9.64/3.30 9.64/3.30 U47_gg(x1, x2, x3) = U47_gg(x2, x3) 9.64/3.30 9.64/3.30 unif_linescC_in_gg(x1, x2) = unif_linescC_in_gg(x1, x2) 9.64/3.30 9.64/3.30 U35_gg(x1, x2, x3, x4, x5, x6, x7) = U35_gg(x1, x2, x3, x4, x5, x6, x7) 9.64/3.30 9.64/3.30 qcD_in_gggggg(x1, x2, x3, x4, x5, x6) = qcD_in_gggggg(x1, x2, x3, x4, x5, x6) 9.64/3.30 9.64/3.30 U42_gggggg(x1, x2, x3, x4, x5, x6, x7) = U42_gggggg(x1, x2, x3, x4, x5, x6, x7) 9.64/3.30 9.64/3.30 U43_gggggg(x1, x2, x3, x4, x5, x6, x7) = U43_gggggg(x1, x2, x3, x4, x5, x6, x7) 9.64/3.30 9.64/3.30 unif_linescC_out_gg(x1, x2) = unif_linescC_out_gg(x1, x2) 9.64/3.30 9.64/3.30 qcD_out_gggggg(x1, x2, x3, x4, x5, x6) = qcD_out_gggggg(x1, x2, x3, x4, x5, x6) 9.64/3.30 9.64/3.30 SHAPESH_IN_GA(x1, x2) = SHAPESH_IN_GA(x1) 9.64/3.30 9.64/3.30 U23_GA(x1, x2, x3, x4) = U23_GA(x1, x2, x4) 9.64/3.30 9.64/3.30 VARMATA_IN_GA(x1, x2) = VARMATA_IN_GA(x1) 9.64/3.30 9.64/3.30 U1_GA(x1, x2, x3, x4, x5) = U1_GA(x1, x2, x5) 9.64/3.30 9.64/3.30 U2_GA(x1, x2, x3, x4, x5) = U2_GA(x1, x2, x5) 9.64/3.30 9.64/3.30 U3_GA(x1, x2, x3, x4, x5) = U3_GA(x1, x2, x5) 9.64/3.30 9.64/3.30 U4_GA(x1, x2, x3) = U4_GA(x1, x3) 9.64/3.30 9.64/3.30 U5_GA(x1, x2, x3, x4) = U5_GA(x1, x4) 9.64/3.30 9.64/3.30 U24_GA(x1, x2, x3, x4) = U24_GA(x1, x2, x4) 9.64/3.30 9.64/3.30 U25_GA(x1, x2, x3, x4) = U25_GA(x1, x2, x4) 9.64/3.30 9.64/3.30 PF_IN_GAG(x1, x2, x3) = PF_IN_GAG(x1, x3) 9.64/3.30 9.64/3.30 U14_GAG(x1, x2, x3, x4) = U14_GAG(x1, x3, x4) 9.64/3.30 9.64/3.30 U15_GAG(x1, x2, x3, x4) = U15_GAG(x1, x3, x4) 9.64/3.30 9.64/3.30 U16_GAG(x1, x2, x3, x4) = U16_GAG(x1, x3, x4) 9.64/3.30 9.64/3.30 UNIF_MATRXB_IN_GG(x1, x2) = UNIF_MATRXB_IN_GG(x1, x2) 9.64/3.30 9.64/3.30 U6_GG(x1, x2, x3, x4) = U6_GG(x1, x2, x3, x4) 9.64/3.30 9.64/3.30 UNIF_LINESC_IN_GG(x1, x2) = UNIF_LINESC_IN_GG(x1, x2) 9.64/3.30 9.64/3.30 U9_GG(x1, x2, x3, x4, x5, x6, x7) = U9_GG(x1, x2, x3, x4, x5, x6, x7) 9.64/3.30 9.64/3.30 PD_IN_GGGGGG(x1, x2, x3, x4, x5, x6) = PD_IN_GGGGGG(x1, x2, x3, x4, x5, x6) 9.64/3.30 9.64/3.30 U17_GGGGGG(x1, x2, x3, x4, x5, x6, x7) = U17_GGGGGG(x2, x3, x4, x5, x6, x7) 9.64/3.30 9.64/3.30 UNIF_PAIRSE_IN_G(x1) = UNIF_PAIRSE_IN_G(x1) 9.64/3.30 9.64/3.30 U10_G(x1, x2, x3) = U10_G(x2, x3) 9.64/3.30 9.64/3.30 U11_G(x1, x2) = U11_G(x1, x2) 9.64/3.30 9.64/3.30 U12_G(x1, x2, x3) = U12_G(x2, x3) 9.64/3.30 9.64/3.30 U13_G(x1, x2, x3) = U13_G(x2, x3) 9.64/3.30 9.64/3.30 U18_GGGGGG(x1, x2, x3, x4, x5, x6) = U18_GGGGGG(x1, x2, x3, x4, x5, x6) 9.64/3.30 9.64/3.30 U19_GGGGGG(x1, x2, x3, x4, x5, x6, x7) = U19_GGGGGG(x2, x3, x4, x5, x6, x7) 9.64/3.30 9.64/3.30 U20_GGGGGG(x1, x2, x3, x4, x5, x6, x7) = U20_GGGGGG(x2, x3, x4, x5, x6, x7) 9.64/3.30 9.64/3.30 U21_GGGGGG(x1, x2, x3, x4, x5, x6, x7) = U21_GGGGGG(x1, x2, x3, x4, x5, x6, x7) 9.64/3.30 9.64/3.30 U22_GGGGGG(x1, x2, x3, x4, x5, x6, x7) = U22_GGGGGG(x1, x2, x3, x4, x5, x6, x7) 9.64/3.30 9.64/3.30 U7_GG(x1, x2, x3, x4) = U7_GG(x1, x2, x3, x4) 9.64/3.30 9.64/3.30 U8_GG(x1, x2, x3, x4) = U8_GG(x1, x2, x3, x4) 9.64/3.30 9.64/3.30 U26_GA(x1, x2, x3) = U26_GA(x1, x3) 9.64/3.30 9.64/3.30 U27_GA(x1, x2, x3) = U27_GA(x1, x3) 9.64/3.30 9.64/3.30 9.64/3.30 We have to consider all (P,R,Pi)-chains 9.64/3.30 ---------------------------------------- 9.64/3.30 9.64/3.30 (5) DependencyGraphProof (EQUIVALENT) 9.64/3.30 The approximation of the Dependency Graph [LOPSTR] contains 4 SCCs with 35 less nodes. 9.64/3.30 ---------------------------------------- 9.64/3.30 9.64/3.30 (6) 9.64/3.30 Complex Obligation (AND) 9.64/3.30 9.64/3.30 ---------------------------------------- 9.64/3.30 9.64/3.30 (7) 9.64/3.30 Obligation: 9.64/3.30 Pi DP problem: 9.64/3.30 The TRS P consists of the following rules: 9.64/3.30 9.64/3.30 UNIF_PAIRSE_IN_G(.(black, .(black, X1))) -> UNIF_PAIRSE_IN_G(X1) 9.64/3.30 UNIF_PAIRSE_IN_G(.(w(X1), .(w(X1), X2))) -> UNIF_PAIRSE_IN_G(X2) 9.64/3.30 UNIF_PAIRSE_IN_G(.(black, .(w(X1), X2))) -> UNIF_PAIRSE_IN_G(X2) 9.64/3.30 UNIF_PAIRSE_IN_G(.(w(X1), .(black, X2))) -> UNIF_PAIRSE_IN_G(X2) 9.64/3.30 9.64/3.30 The TRS R consists of the following rules: 9.64/3.30 9.64/3.30 varmatcA_in_ga([], []) -> varmatcA_out_ga([], []) 9.64/3.30 varmatcA_in_ga(.(X1, X2), .(X3, X4)) -> U29_ga(X1, X2, X3, X4, varmatcA_in_ga(X1, X3)) 9.64/3.30 varmatcA_in_ga(.(black, X1), .(black, X2)) -> U31_ga(X1, X2, varmatcA_in_ga(X1, X2)) 9.64/3.30 varmatcA_in_ga(.(white, X1), .(w(X2), X3)) -> U32_ga(X1, X2, X3, varmatcA_in_ga(X1, X3)) 9.64/3.30 U32_ga(X1, X2, X3, varmatcA_out_ga(X1, X3)) -> varmatcA_out_ga(.(white, X1), .(w(X2), X3)) 9.64/3.30 U31_ga(X1, X2, varmatcA_out_ga(X1, X2)) -> varmatcA_out_ga(.(black, X1), .(black, X2)) 9.64/3.30 U29_ga(X1, X2, X3, X4, varmatcA_out_ga(X1, X3)) -> U30_ga(X1, X2, X3, X4, varmatcA_in_ga(X2, X4)) 9.64/3.30 U30_ga(X1, X2, X3, X4, varmatcA_out_ga(X2, X4)) -> varmatcA_out_ga(.(X1, X2), .(X3, X4)) 9.64/3.30 unif_pairscG_in_gg(w(X1), .(w(X1), X2)) -> U44_gg(X1, X2, unif_pairscE_in_g(X2)) 9.64/3.30 unif_pairscE_in_g([]) -> unif_pairscE_out_g([]) 9.64/3.30 unif_pairscE_in_g(.(w(X1), .(w(X1), X2))) -> U36_g(X1, X2, unif_pairscE_in_g(X2)) 9.64/3.30 unif_pairscE_in_g(.(black, .(black, X1))) -> U37_g(X1, unif_pairscE_in_g(X1)) 9.64/3.30 unif_pairscE_in_g(.(black, .(w(X1), X2))) -> U38_g(X1, X2, unif_pairscE_in_g(X2)) 9.64/3.30 unif_pairscE_in_g(.(w(X1), .(black, X2))) -> U39_g(X1, X2, unif_pairscE_in_g(X2)) 9.64/3.30 U39_g(X1, X2, unif_pairscE_out_g(X2)) -> unif_pairscE_out_g(.(w(X1), .(black, X2))) 9.64/3.30 U38_g(X1, X2, unif_pairscE_out_g(X2)) -> unif_pairscE_out_g(.(black, .(w(X1), X2))) 9.64/3.30 U37_g(X1, unif_pairscE_out_g(X1)) -> unif_pairscE_out_g(.(black, .(black, X1))) 9.64/3.30 U36_g(X1, X2, unif_pairscE_out_g(X2)) -> unif_pairscE_out_g(.(w(X1), .(w(X1), X2))) 9.64/3.30 U44_gg(X1, X2, unif_pairscE_out_g(X2)) -> unif_pairscG_out_gg(w(X1), .(w(X1), X2)) 9.64/3.30 unif_pairscG_in_gg(black, .(black, X1)) -> U45_gg(X1, unif_pairscE_in_g(X1)) 9.64/3.30 U45_gg(X1, unif_pairscE_out_g(X1)) -> unif_pairscG_out_gg(black, .(black, X1)) 9.64/3.30 unif_pairscG_in_gg(black, .(w(X1), X2)) -> U46_gg(X1, X2, unif_pairscE_in_g(X2)) 9.64/3.30 U46_gg(X1, X2, unif_pairscE_out_g(X2)) -> unif_pairscG_out_gg(black, .(w(X1), X2)) 9.64/3.30 unif_pairscG_in_gg(w(X1), .(black, X2)) -> U47_gg(X1, X2, unif_pairscE_in_g(X2)) 9.64/3.30 U47_gg(X1, X2, unif_pairscE_out_g(X2)) -> unif_pairscG_out_gg(w(X1), .(black, X2)) 9.64/3.30 unif_linescC_in_gg(.(X1, .(X2, X3)), .(X4, .(X5, X6))) -> U35_gg(X1, X2, X3, X4, X5, X6, qcD_in_gggggg(X1, .(X2, .(X4, .(X5, .(X1, .(X4, .(X2, .(X5, .(X1, .(X5, .(X2, .(X4, []))))))))))), X2, X3, X5, X6)) 9.64/3.30 qcD_in_gggggg(X1, X2, X3, X4, X5, X6) -> U42_gggggg(X1, X2, X3, X4, X5, X6, unif_pairscG_in_gg(X1, X2)) 9.64/3.30 U42_gggggg(X1, X2, X3, X4, X5, X6, unif_pairscG_out_gg(X1, X2)) -> U43_gggggg(X1, X2, X3, X4, X5, X6, unif_linescC_in_gg(.(X3, X4), .(X5, X6))) 9.64/3.30 unif_linescC_in_gg(.(X1, []), .(X2, [])) -> unif_linescC_out_gg(.(X1, []), .(X2, [])) 9.64/3.30 U43_gggggg(X1, X2, X3, X4, X5, X6, unif_linescC_out_gg(.(X3, X4), .(X5, X6))) -> qcD_out_gggggg(X1, X2, X3, X4, X5, X6) 9.64/3.30 U35_gg(X1, X2, X3, X4, X5, X6, qcD_out_gggggg(X1, .(X2, .(X4, .(X5, .(X1, .(X4, .(X2, .(X5, .(X1, .(X5, .(X2, .(X4, []))))))))))), X2, X3, X5, X6)) -> unif_linescC_out_gg(.(X1, .(X2, X3)), .(X4, .(X5, X6))) 9.64/3.30 9.64/3.30 The argument filtering Pi contains the following mapping: 9.64/3.30 .(x1, x2) = .(x1, x2) 9.64/3.30 9.64/3.30 varmatcA_in_ga(x1, x2) = varmatcA_in_ga(x1) 9.64/3.30 9.64/3.30 [] = [] 9.64/3.30 9.64/3.30 varmatcA_out_ga(x1, x2) = varmatcA_out_ga(x1, x2) 9.64/3.30 9.64/3.30 U29_ga(x1, x2, x3, x4, x5) = U29_ga(x1, x2, x5) 9.64/3.30 9.64/3.30 black = black 9.64/3.30 9.64/3.30 U31_ga(x1, x2, x3) = U31_ga(x1, x3) 9.64/3.30 9.64/3.30 white = white 9.64/3.30 9.64/3.30 U32_ga(x1, x2, x3, x4) = U32_ga(x1, x4) 9.64/3.30 9.64/3.30 w(x1) = w 9.64/3.30 9.64/3.30 U30_ga(x1, x2, x3, x4, x5) = U30_ga(x1, x2, x3, x5) 9.64/3.30 9.64/3.30 unif_pairscG_in_gg(x1, x2) = unif_pairscG_in_gg(x1, x2) 9.64/3.30 9.64/3.30 U44_gg(x1, x2, x3) = U44_gg(x2, x3) 9.64/3.30 9.64/3.30 unif_pairscE_in_g(x1) = unif_pairscE_in_g(x1) 9.64/3.30 9.64/3.30 unif_pairscE_out_g(x1) = unif_pairscE_out_g(x1) 9.64/3.30 9.64/3.30 U36_g(x1, x2, x3) = U36_g(x2, x3) 9.64/3.30 9.64/3.30 U37_g(x1, x2) = U37_g(x1, x2) 9.64/3.30 9.64/3.30 U38_g(x1, x2, x3) = U38_g(x2, x3) 9.64/3.30 9.64/3.30 U39_g(x1, x2, x3) = U39_g(x2, x3) 9.64/3.30 9.64/3.30 unif_pairscG_out_gg(x1, x2) = unif_pairscG_out_gg(x1, x2) 9.64/3.30 9.64/3.30 U45_gg(x1, x2) = U45_gg(x1, x2) 9.64/3.30 9.64/3.30 U46_gg(x1, x2, x3) = U46_gg(x2, x3) 9.64/3.30 9.64/3.30 U47_gg(x1, x2, x3) = U47_gg(x2, x3) 9.64/3.30 9.64/3.30 unif_linescC_in_gg(x1, x2) = unif_linescC_in_gg(x1, x2) 9.64/3.30 9.64/3.30 U35_gg(x1, x2, x3, x4, x5, x6, x7) = U35_gg(x1, x2, x3, x4, x5, x6, x7) 9.64/3.30 9.64/3.30 qcD_in_gggggg(x1, x2, x3, x4, x5, x6) = qcD_in_gggggg(x1, x2, x3, x4, x5, x6) 9.64/3.30 9.64/3.30 U42_gggggg(x1, x2, x3, x4, x5, x6, x7) = U42_gggggg(x1, x2, x3, x4, x5, x6, x7) 9.64/3.30 9.64/3.30 U43_gggggg(x1, x2, x3, x4, x5, x6, x7) = U43_gggggg(x1, x2, x3, x4, x5, x6, x7) 9.64/3.30 9.64/3.30 unif_linescC_out_gg(x1, x2) = unif_linescC_out_gg(x1, x2) 9.64/3.30 9.64/3.30 qcD_out_gggggg(x1, x2, x3, x4, x5, x6) = qcD_out_gggggg(x1, x2, x3, x4, x5, x6) 9.64/3.30 9.64/3.30 UNIF_PAIRSE_IN_G(x1) = UNIF_PAIRSE_IN_G(x1) 9.64/3.30 9.64/3.30 9.64/3.30 We have to consider all (P,R,Pi)-chains 9.64/3.30 ---------------------------------------- 9.64/3.30 9.64/3.30 (8) UsableRulesProof (EQUIVALENT) 9.64/3.30 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 9.64/3.30 ---------------------------------------- 9.64/3.30 9.64/3.30 (9) 9.64/3.30 Obligation: 9.64/3.30 Pi DP problem: 9.64/3.30 The TRS P consists of the following rules: 9.64/3.30 9.64/3.30 UNIF_PAIRSE_IN_G(.(black, .(black, X1))) -> UNIF_PAIRSE_IN_G(X1) 9.64/3.30 UNIF_PAIRSE_IN_G(.(w(X1), .(w(X1), X2))) -> UNIF_PAIRSE_IN_G(X2) 9.64/3.30 UNIF_PAIRSE_IN_G(.(black, .(w(X1), X2))) -> UNIF_PAIRSE_IN_G(X2) 9.64/3.30 UNIF_PAIRSE_IN_G(.(w(X1), .(black, X2))) -> UNIF_PAIRSE_IN_G(X2) 9.64/3.30 9.64/3.30 R is empty. 9.64/3.30 The argument filtering Pi contains the following mapping: 9.64/3.30 .(x1, x2) = .(x1, x2) 9.64/3.30 9.64/3.30 black = black 9.64/3.30 9.64/3.30 w(x1) = w 9.64/3.30 9.64/3.30 UNIF_PAIRSE_IN_G(x1) = UNIF_PAIRSE_IN_G(x1) 9.64/3.30 9.64/3.30 9.64/3.30 We have to consider all (P,R,Pi)-chains 9.64/3.30 ---------------------------------------- 9.64/3.30 9.64/3.30 (10) PiDPToQDPProof (SOUND) 9.64/3.30 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 9.64/3.30 ---------------------------------------- 9.64/3.30 9.64/3.30 (11) 9.64/3.30 Obligation: 9.64/3.30 Q DP problem: 9.64/3.30 The TRS P consists of the following rules: 9.64/3.30 9.64/3.30 UNIF_PAIRSE_IN_G(.(black, .(black, X1))) -> UNIF_PAIRSE_IN_G(X1) 9.64/3.30 UNIF_PAIRSE_IN_G(.(w, .(w, X2))) -> UNIF_PAIRSE_IN_G(X2) 9.64/3.30 UNIF_PAIRSE_IN_G(.(black, .(w, X2))) -> UNIF_PAIRSE_IN_G(X2) 9.64/3.30 UNIF_PAIRSE_IN_G(.(w, .(black, X2))) -> UNIF_PAIRSE_IN_G(X2) 9.64/3.30 9.64/3.30 R is empty. 9.64/3.30 Q is empty. 9.64/3.30 We have to consider all (P,Q,R)-chains. 9.64/3.30 ---------------------------------------- 9.64/3.30 9.64/3.30 (12) QDPSizeChangeProof (EQUIVALENT) 9.64/3.30 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. 9.64/3.30 9.64/3.30 From the DPs we obtained the following set of size-change graphs: 9.64/3.30 *UNIF_PAIRSE_IN_G(.(black, .(black, X1))) -> UNIF_PAIRSE_IN_G(X1) 9.64/3.30 The graph contains the following edges 1 > 1 9.64/3.30 9.64/3.30 9.64/3.30 *UNIF_PAIRSE_IN_G(.(w, .(w, X2))) -> UNIF_PAIRSE_IN_G(X2) 9.64/3.30 The graph contains the following edges 1 > 1 9.64/3.30 9.64/3.30 9.64/3.30 *UNIF_PAIRSE_IN_G(.(black, .(w, X2))) -> UNIF_PAIRSE_IN_G(X2) 9.64/3.30 The graph contains the following edges 1 > 1 9.64/3.30 9.64/3.30 9.64/3.30 *UNIF_PAIRSE_IN_G(.(w, .(black, X2))) -> UNIF_PAIRSE_IN_G(X2) 9.64/3.30 The graph contains the following edges 1 > 1 9.64/3.30 9.64/3.30 9.64/3.30 ---------------------------------------- 9.64/3.30 9.64/3.30 (13) 9.64/3.30 YES 9.64/3.30 9.64/3.30 ---------------------------------------- 9.64/3.30 9.64/3.30 (14) 9.64/3.30 Obligation: 9.64/3.30 Pi DP problem: 9.64/3.30 The TRS P consists of the following rules: 9.64/3.31 9.64/3.31 UNIF_LINESC_IN_GG(.(X1, .(X2, X3)), .(X4, .(X5, X6))) -> PD_IN_GGGGGG(X1, .(X2, .(X4, .(X5, .(X1, .(X4, .(X2, .(X5, .(X1, .(X5, .(X2, .(X4, []))))))))))), X2, X3, X5, X6) 9.64/3.31 PD_IN_GGGGGG(X1, X2, X3, X4, X5, X6) -> U21_GGGGGG(X1, X2, X3, X4, X5, X6, unif_pairscG_in_gg(X1, X2)) 9.64/3.31 U21_GGGGGG(X1, X2, X3, X4, X5, X6, unif_pairscG_out_gg(X1, X2)) -> UNIF_LINESC_IN_GG(.(X3, X4), .(X5, X6)) 9.64/3.31 9.64/3.31 The TRS R consists of the following rules: 9.64/3.31 9.64/3.31 varmatcA_in_ga([], []) -> varmatcA_out_ga([], []) 9.64/3.31 varmatcA_in_ga(.(X1, X2), .(X3, X4)) -> U29_ga(X1, X2, X3, X4, varmatcA_in_ga(X1, X3)) 9.64/3.31 varmatcA_in_ga(.(black, X1), .(black, X2)) -> U31_ga(X1, X2, varmatcA_in_ga(X1, X2)) 9.64/3.31 varmatcA_in_ga(.(white, X1), .(w(X2), X3)) -> U32_ga(X1, X2, X3, varmatcA_in_ga(X1, X3)) 9.64/3.31 U32_ga(X1, X2, X3, varmatcA_out_ga(X1, X3)) -> varmatcA_out_ga(.(white, X1), .(w(X2), X3)) 9.64/3.31 U31_ga(X1, X2, varmatcA_out_ga(X1, X2)) -> varmatcA_out_ga(.(black, X1), .(black, X2)) 9.64/3.31 U29_ga(X1, X2, X3, X4, varmatcA_out_ga(X1, X3)) -> U30_ga(X1, X2, X3, X4, varmatcA_in_ga(X2, X4)) 9.64/3.31 U30_ga(X1, X2, X3, X4, varmatcA_out_ga(X2, X4)) -> varmatcA_out_ga(.(X1, X2), .(X3, X4)) 9.64/3.31 unif_pairscG_in_gg(w(X1), .(w(X1), X2)) -> U44_gg(X1, X2, unif_pairscE_in_g(X2)) 9.64/3.31 unif_pairscE_in_g([]) -> unif_pairscE_out_g([]) 9.64/3.31 unif_pairscE_in_g(.(w(X1), .(w(X1), X2))) -> U36_g(X1, X2, unif_pairscE_in_g(X2)) 9.64/3.31 unif_pairscE_in_g(.(black, .(black, X1))) -> U37_g(X1, unif_pairscE_in_g(X1)) 9.64/3.31 unif_pairscE_in_g(.(black, .(w(X1), X2))) -> U38_g(X1, X2, unif_pairscE_in_g(X2)) 9.64/3.31 unif_pairscE_in_g(.(w(X1), .(black, X2))) -> U39_g(X1, X2, unif_pairscE_in_g(X2)) 9.64/3.31 U39_g(X1, X2, unif_pairscE_out_g(X2)) -> unif_pairscE_out_g(.(w(X1), .(black, X2))) 9.64/3.31 U38_g(X1, X2, unif_pairscE_out_g(X2)) -> unif_pairscE_out_g(.(black, .(w(X1), X2))) 9.64/3.31 U37_g(X1, unif_pairscE_out_g(X1)) -> unif_pairscE_out_g(.(black, .(black, X1))) 9.64/3.31 U36_g(X1, X2, unif_pairscE_out_g(X2)) -> unif_pairscE_out_g(.(w(X1), .(w(X1), X2))) 9.64/3.31 U44_gg(X1, X2, unif_pairscE_out_g(X2)) -> unif_pairscG_out_gg(w(X1), .(w(X1), X2)) 9.64/3.31 unif_pairscG_in_gg(black, .(black, X1)) -> U45_gg(X1, unif_pairscE_in_g(X1)) 9.64/3.31 U45_gg(X1, unif_pairscE_out_g(X1)) -> unif_pairscG_out_gg(black, .(black, X1)) 9.64/3.31 unif_pairscG_in_gg(black, .(w(X1), X2)) -> U46_gg(X1, X2, unif_pairscE_in_g(X2)) 9.64/3.31 U46_gg(X1, X2, unif_pairscE_out_g(X2)) -> unif_pairscG_out_gg(black, .(w(X1), X2)) 9.64/3.31 unif_pairscG_in_gg(w(X1), .(black, X2)) -> U47_gg(X1, X2, unif_pairscE_in_g(X2)) 9.64/3.31 U47_gg(X1, X2, unif_pairscE_out_g(X2)) -> unif_pairscG_out_gg(w(X1), .(black, X2)) 9.64/3.31 unif_linescC_in_gg(.(X1, .(X2, X3)), .(X4, .(X5, X6))) -> U35_gg(X1, X2, X3, X4, X5, X6, qcD_in_gggggg(X1, .(X2, .(X4, .(X5, .(X1, .(X4, .(X2, .(X5, .(X1, .(X5, .(X2, .(X4, []))))))))))), X2, X3, X5, X6)) 9.64/3.31 qcD_in_gggggg(X1, X2, X3, X4, X5, X6) -> U42_gggggg(X1, X2, X3, X4, X5, X6, unif_pairscG_in_gg(X1, X2)) 9.64/3.31 U42_gggggg(X1, X2, X3, X4, X5, X6, unif_pairscG_out_gg(X1, X2)) -> U43_gggggg(X1, X2, X3, X4, X5, X6, unif_linescC_in_gg(.(X3, X4), .(X5, X6))) 9.64/3.31 unif_linescC_in_gg(.(X1, []), .(X2, [])) -> unif_linescC_out_gg(.(X1, []), .(X2, [])) 9.64/3.31 U43_gggggg(X1, X2, X3, X4, X5, X6, unif_linescC_out_gg(.(X3, X4), .(X5, X6))) -> qcD_out_gggggg(X1, X2, X3, X4, X5, X6) 9.64/3.31 U35_gg(X1, X2, X3, X4, X5, X6, qcD_out_gggggg(X1, .(X2, .(X4, .(X5, .(X1, .(X4, .(X2, .(X5, .(X1, .(X5, .(X2, .(X4, []))))))))))), X2, X3, X5, X6)) -> unif_linescC_out_gg(.(X1, .(X2, X3)), .(X4, .(X5, X6))) 9.64/3.31 9.64/3.31 The argument filtering Pi contains the following mapping: 9.64/3.31 .(x1, x2) = .(x1, x2) 9.64/3.31 9.64/3.31 varmatcA_in_ga(x1, x2) = varmatcA_in_ga(x1) 9.64/3.31 9.64/3.31 [] = [] 9.64/3.31 9.64/3.31 varmatcA_out_ga(x1, x2) = varmatcA_out_ga(x1, x2) 9.64/3.31 9.64/3.31 U29_ga(x1, x2, x3, x4, x5) = U29_ga(x1, x2, x5) 9.64/3.31 9.64/3.31 black = black 9.64/3.31 9.64/3.31 U31_ga(x1, x2, x3) = U31_ga(x1, x3) 9.64/3.31 9.64/3.31 white = white 9.64/3.31 9.64/3.31 U32_ga(x1, x2, x3, x4) = U32_ga(x1, x4) 9.64/3.31 9.64/3.31 w(x1) = w 9.64/3.31 9.64/3.31 U30_ga(x1, x2, x3, x4, x5) = U30_ga(x1, x2, x3, x5) 9.64/3.31 9.64/3.31 unif_pairscG_in_gg(x1, x2) = unif_pairscG_in_gg(x1, x2) 9.64/3.31 9.64/3.31 U44_gg(x1, x2, x3) = U44_gg(x2, x3) 9.64/3.31 9.64/3.31 unif_pairscE_in_g(x1) = unif_pairscE_in_g(x1) 9.64/3.31 9.64/3.31 unif_pairscE_out_g(x1) = unif_pairscE_out_g(x1) 9.64/3.31 9.64/3.31 U36_g(x1, x2, x3) = U36_g(x2, x3) 9.64/3.31 9.64/3.31 U37_g(x1, x2) = U37_g(x1, x2) 9.64/3.31 9.64/3.31 U38_g(x1, x2, x3) = U38_g(x2, x3) 9.64/3.31 9.64/3.31 U39_g(x1, x2, x3) = U39_g(x2, x3) 9.64/3.31 9.64/3.31 unif_pairscG_out_gg(x1, x2) = unif_pairscG_out_gg(x1, x2) 9.64/3.31 9.64/3.31 U45_gg(x1, x2) = U45_gg(x1, x2) 9.64/3.31 9.64/3.31 U46_gg(x1, x2, x3) = U46_gg(x2, x3) 9.64/3.31 9.64/3.31 U47_gg(x1, x2, x3) = U47_gg(x2, x3) 9.64/3.31 9.64/3.31 unif_linescC_in_gg(x1, x2) = unif_linescC_in_gg(x1, x2) 9.64/3.31 9.64/3.31 U35_gg(x1, x2, x3, x4, x5, x6, x7) = U35_gg(x1, x2, x3, x4, x5, x6, x7) 9.64/3.31 9.64/3.31 qcD_in_gggggg(x1, x2, x3, x4, x5, x6) = qcD_in_gggggg(x1, x2, x3, x4, x5, x6) 9.64/3.31 9.64/3.31 U42_gggggg(x1, x2, x3, x4, x5, x6, x7) = U42_gggggg(x1, x2, x3, x4, x5, x6, x7) 9.64/3.31 9.64/3.31 U43_gggggg(x1, x2, x3, x4, x5, x6, x7) = U43_gggggg(x1, x2, x3, x4, x5, x6, x7) 9.64/3.31 9.64/3.31 unif_linescC_out_gg(x1, x2) = unif_linescC_out_gg(x1, x2) 9.64/3.31 9.64/3.31 qcD_out_gggggg(x1, x2, x3, x4, x5, x6) = qcD_out_gggggg(x1, x2, x3, x4, x5, x6) 9.64/3.31 9.64/3.31 UNIF_LINESC_IN_GG(x1, x2) = UNIF_LINESC_IN_GG(x1, x2) 9.64/3.31 9.64/3.31 PD_IN_GGGGGG(x1, x2, x3, x4, x5, x6) = PD_IN_GGGGGG(x1, x2, x3, x4, x5, x6) 9.64/3.31 9.64/3.31 U21_GGGGGG(x1, x2, x3, x4, x5, x6, x7) = U21_GGGGGG(x1, x2, x3, x4, x5, x6, x7) 9.64/3.31 9.64/3.31 9.64/3.31 We have to consider all (P,R,Pi)-chains 9.64/3.31 ---------------------------------------- 9.64/3.31 9.64/3.31 (15) UsableRulesProof (EQUIVALENT) 9.64/3.31 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 9.64/3.31 ---------------------------------------- 9.64/3.31 9.64/3.31 (16) 9.64/3.31 Obligation: 9.64/3.31 Pi DP problem: 9.64/3.31 The TRS P consists of the following rules: 9.64/3.31 9.64/3.31 UNIF_LINESC_IN_GG(.(X1, .(X2, X3)), .(X4, .(X5, X6))) -> PD_IN_GGGGGG(X1, .(X2, .(X4, .(X5, .(X1, .(X4, .(X2, .(X5, .(X1, .(X5, .(X2, .(X4, []))))))))))), X2, X3, X5, X6) 9.64/3.31 PD_IN_GGGGGG(X1, X2, X3, X4, X5, X6) -> U21_GGGGGG(X1, X2, X3, X4, X5, X6, unif_pairscG_in_gg(X1, X2)) 9.64/3.31 U21_GGGGGG(X1, X2, X3, X4, X5, X6, unif_pairscG_out_gg(X1, X2)) -> UNIF_LINESC_IN_GG(.(X3, X4), .(X5, X6)) 9.64/3.31 9.64/3.31 The TRS R consists of the following rules: 9.64/3.31 9.64/3.31 unif_pairscG_in_gg(w(X1), .(w(X1), X2)) -> U44_gg(X1, X2, unif_pairscE_in_g(X2)) 9.64/3.31 unif_pairscG_in_gg(black, .(black, X1)) -> U45_gg(X1, unif_pairscE_in_g(X1)) 9.64/3.31 unif_pairscG_in_gg(black, .(w(X1), X2)) -> U46_gg(X1, X2, unif_pairscE_in_g(X2)) 9.64/3.31 unif_pairscG_in_gg(w(X1), .(black, X2)) -> U47_gg(X1, X2, unif_pairscE_in_g(X2)) 9.64/3.31 U44_gg(X1, X2, unif_pairscE_out_g(X2)) -> unif_pairscG_out_gg(w(X1), .(w(X1), X2)) 9.64/3.31 U45_gg(X1, unif_pairscE_out_g(X1)) -> unif_pairscG_out_gg(black, .(black, X1)) 9.64/3.31 U46_gg(X1, X2, unif_pairscE_out_g(X2)) -> unif_pairscG_out_gg(black, .(w(X1), X2)) 9.64/3.31 U47_gg(X1, X2, unif_pairscE_out_g(X2)) -> unif_pairscG_out_gg(w(X1), .(black, X2)) 9.64/3.31 unif_pairscE_in_g([]) -> unif_pairscE_out_g([]) 9.64/3.31 unif_pairscE_in_g(.(w(X1), .(w(X1), X2))) -> U36_g(X1, X2, unif_pairscE_in_g(X2)) 9.64/3.31 unif_pairscE_in_g(.(black, .(black, X1))) -> U37_g(X1, unif_pairscE_in_g(X1)) 9.64/3.31 unif_pairscE_in_g(.(black, .(w(X1), X2))) -> U38_g(X1, X2, unif_pairscE_in_g(X2)) 9.64/3.31 unif_pairscE_in_g(.(w(X1), .(black, X2))) -> U39_g(X1, X2, unif_pairscE_in_g(X2)) 9.64/3.31 U36_g(X1, X2, unif_pairscE_out_g(X2)) -> unif_pairscE_out_g(.(w(X1), .(w(X1), X2))) 9.64/3.31 U37_g(X1, unif_pairscE_out_g(X1)) -> unif_pairscE_out_g(.(black, .(black, X1))) 9.64/3.31 U38_g(X1, X2, unif_pairscE_out_g(X2)) -> unif_pairscE_out_g(.(black, .(w(X1), X2))) 9.64/3.31 U39_g(X1, X2, unif_pairscE_out_g(X2)) -> unif_pairscE_out_g(.(w(X1), .(black, X2))) 9.64/3.31 9.64/3.31 The argument filtering Pi contains the following mapping: 9.64/3.31 .(x1, x2) = .(x1, x2) 9.64/3.31 9.64/3.31 [] = [] 9.64/3.31 9.64/3.31 black = black 9.64/3.31 9.64/3.31 w(x1) = w 9.64/3.31 9.64/3.31 unif_pairscG_in_gg(x1, x2) = unif_pairscG_in_gg(x1, x2) 9.64/3.31 9.64/3.31 U44_gg(x1, x2, x3) = U44_gg(x2, x3) 9.64/3.31 9.64/3.31 unif_pairscE_in_g(x1) = unif_pairscE_in_g(x1) 9.64/3.31 9.64/3.31 unif_pairscE_out_g(x1) = unif_pairscE_out_g(x1) 9.64/3.31 9.64/3.31 U36_g(x1, x2, x3) = U36_g(x2, x3) 9.64/3.31 9.64/3.31 U37_g(x1, x2) = U37_g(x1, x2) 9.64/3.31 9.64/3.31 U38_g(x1, x2, x3) = U38_g(x2, x3) 9.64/3.31 9.64/3.31 U39_g(x1, x2, x3) = U39_g(x2, x3) 9.64/3.31 9.64/3.31 unif_pairscG_out_gg(x1, x2) = unif_pairscG_out_gg(x1, x2) 9.64/3.31 9.64/3.31 U45_gg(x1, x2) = U45_gg(x1, x2) 9.64/3.31 9.64/3.31 U46_gg(x1, x2, x3) = U46_gg(x2, x3) 9.64/3.31 9.64/3.31 U47_gg(x1, x2, x3) = U47_gg(x2, x3) 9.64/3.31 9.64/3.31 UNIF_LINESC_IN_GG(x1, x2) = UNIF_LINESC_IN_GG(x1, x2) 9.64/3.31 9.64/3.31 PD_IN_GGGGGG(x1, x2, x3, x4, x5, x6) = PD_IN_GGGGGG(x1, x2, x3, x4, x5, x6) 9.64/3.31 9.64/3.31 U21_GGGGGG(x1, x2, x3, x4, x5, x6, x7) = U21_GGGGGG(x1, x2, x3, x4, x5, x6, x7) 9.64/3.31 9.64/3.31 9.64/3.31 We have to consider all (P,R,Pi)-chains 9.64/3.31 ---------------------------------------- 9.64/3.31 9.64/3.31 (17) PiDPToQDPProof (SOUND) 9.64/3.31 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 9.64/3.31 ---------------------------------------- 9.64/3.31 9.64/3.31 (18) 9.64/3.31 Obligation: 9.64/3.31 Q DP problem: 9.64/3.31 The TRS P consists of the following rules: 9.64/3.31 9.64/3.31 UNIF_LINESC_IN_GG(.(X1, .(X2, X3)), .(X4, .(X5, X6))) -> PD_IN_GGGGGG(X1, .(X2, .(X4, .(X5, .(X1, .(X4, .(X2, .(X5, .(X1, .(X5, .(X2, .(X4, []))))))))))), X2, X3, X5, X6) 9.64/3.31 PD_IN_GGGGGG(X1, X2, X3, X4, X5, X6) -> U21_GGGGGG(X1, X2, X3, X4, X5, X6, unif_pairscG_in_gg(X1, X2)) 9.64/3.31 U21_GGGGGG(X1, X2, X3, X4, X5, X6, unif_pairscG_out_gg(X1, X2)) -> UNIF_LINESC_IN_GG(.(X3, X4), .(X5, X6)) 9.64/3.31 9.64/3.31 The TRS R consists of the following rules: 9.64/3.31 9.64/3.31 unif_pairscG_in_gg(w, .(w, X2)) -> U44_gg(X2, unif_pairscE_in_g(X2)) 9.64/3.31 unif_pairscG_in_gg(black, .(black, X1)) -> U45_gg(X1, unif_pairscE_in_g(X1)) 9.64/3.31 unif_pairscG_in_gg(black, .(w, X2)) -> U46_gg(X2, unif_pairscE_in_g(X2)) 9.64/3.31 unif_pairscG_in_gg(w, .(black, X2)) -> U47_gg(X2, unif_pairscE_in_g(X2)) 9.64/3.31 U44_gg(X2, unif_pairscE_out_g(X2)) -> unif_pairscG_out_gg(w, .(w, X2)) 9.64/3.31 U45_gg(X1, unif_pairscE_out_g(X1)) -> unif_pairscG_out_gg(black, .(black, X1)) 9.64/3.31 U46_gg(X2, unif_pairscE_out_g(X2)) -> unif_pairscG_out_gg(black, .(w, X2)) 9.64/3.31 U47_gg(X2, unif_pairscE_out_g(X2)) -> unif_pairscG_out_gg(w, .(black, X2)) 9.64/3.31 unif_pairscE_in_g([]) -> unif_pairscE_out_g([]) 9.64/3.31 unif_pairscE_in_g(.(w, .(w, X2))) -> U36_g(X2, unif_pairscE_in_g(X2)) 9.64/3.31 unif_pairscE_in_g(.(black, .(black, X1))) -> U37_g(X1, unif_pairscE_in_g(X1)) 9.64/3.31 unif_pairscE_in_g(.(black, .(w, X2))) -> U38_g(X2, unif_pairscE_in_g(X2)) 9.64/3.31 unif_pairscE_in_g(.(w, .(black, X2))) -> U39_g(X2, unif_pairscE_in_g(X2)) 9.64/3.31 U36_g(X2, unif_pairscE_out_g(X2)) -> unif_pairscE_out_g(.(w, .(w, X2))) 9.64/3.31 U37_g(X1, unif_pairscE_out_g(X1)) -> unif_pairscE_out_g(.(black, .(black, X1))) 9.64/3.31 U38_g(X2, unif_pairscE_out_g(X2)) -> unif_pairscE_out_g(.(black, .(w, X2))) 9.64/3.31 U39_g(X2, unif_pairscE_out_g(X2)) -> unif_pairscE_out_g(.(w, .(black, X2))) 9.64/3.31 9.64/3.31 The set Q consists of the following terms: 9.64/3.31 9.64/3.31 unif_pairscG_in_gg(x0, x1) 9.64/3.31 U44_gg(x0, x1) 9.64/3.31 U45_gg(x0, x1) 9.64/3.31 U46_gg(x0, x1) 9.64/3.31 U47_gg(x0, x1) 9.64/3.31 unif_pairscE_in_g(x0) 9.64/3.31 U36_g(x0, x1) 9.64/3.31 U37_g(x0, x1) 9.64/3.31 U38_g(x0, x1) 9.64/3.31 U39_g(x0, x1) 9.64/3.31 9.64/3.31 We have to consider all (P,Q,R)-chains. 9.64/3.31 ---------------------------------------- 9.64/3.31 9.64/3.31 (19) QDPQMonotonicMRRProof (EQUIVALENT) 9.64/3.31 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. 9.64/3.31 9.64/3.31 Strictly oriented dependency pairs: 9.64/3.31 9.64/3.31 UNIF_LINESC_IN_GG(.(X1, .(X2, X3)), .(X4, .(X5, X6))) -> PD_IN_GGGGGG(X1, .(X2, .(X4, .(X5, .(X1, .(X4, .(X2, .(X5, .(X1, .(X5, .(X2, .(X4, []))))))))))), X2, X3, X5, X6) 9.64/3.31 9.64/3.31 9.64/3.31 Used ordering: Polynomial interpretation [POLO]: 9.64/3.31 9.64/3.31 POL(.(x_1, x_2)) = 1 + 2*x_1 + 2*x_2 9.64/3.31 POL(PD_IN_GGGGGG(x_1, x_2, x_3, x_4, x_5, x_6)) = 1 + 2*x_5 + 2*x_6 9.64/3.31 POL(U21_GGGGGG(x_1, x_2, x_3, x_4, x_5, x_6, x_7)) = 1 + 2*x_5 + 2*x_6 9.64/3.31 POL(U36_g(x_1, x_2)) = 0 9.64/3.31 POL(U37_g(x_1, x_2)) = 0 9.64/3.31 POL(U38_g(x_1, x_2)) = 0 9.64/3.31 POL(U39_g(x_1, x_2)) = 0 9.64/3.31 POL(U44_gg(x_1, x_2)) = 0 9.64/3.31 POL(U45_gg(x_1, x_2)) = 1 9.64/3.31 POL(U46_gg(x_1, x_2)) = 0 9.64/3.31 POL(U47_gg(x_1, x_2)) = 2 9.64/3.31 POL(UNIF_LINESC_IN_GG(x_1, x_2)) = x_2 9.64/3.31 POL([]) = 0 9.64/3.31 POL(black) = 2 9.64/3.31 POL(unif_pairscE_in_g(x_1)) = 0 9.64/3.31 POL(unif_pairscE_out_g(x_1)) = 0 9.64/3.31 POL(unif_pairscG_in_gg(x_1, x_2)) = x_2 9.64/3.31 POL(unif_pairscG_out_gg(x_1, x_2)) = 0 9.64/3.31 POL(w) = 0 9.64/3.31 9.64/3.31 9.64/3.31 ---------------------------------------- 9.64/3.31 9.64/3.31 (20) 9.64/3.31 Obligation: 9.64/3.31 Q DP problem: 9.64/3.31 The TRS P consists of the following rules: 9.64/3.31 9.64/3.31 PD_IN_GGGGGG(X1, X2, X3, X4, X5, X6) -> U21_GGGGGG(X1, X2, X3, X4, X5, X6, unif_pairscG_in_gg(X1, X2)) 9.64/3.31 U21_GGGGGG(X1, X2, X3, X4, X5, X6, unif_pairscG_out_gg(X1, X2)) -> UNIF_LINESC_IN_GG(.(X3, X4), .(X5, X6)) 9.64/3.31 9.64/3.31 The TRS R consists of the following rules: 9.64/3.31 9.64/3.31 unif_pairscG_in_gg(w, .(w, X2)) -> U44_gg(X2, unif_pairscE_in_g(X2)) 9.64/3.31 unif_pairscG_in_gg(black, .(black, X1)) -> U45_gg(X1, unif_pairscE_in_g(X1)) 9.64/3.31 unif_pairscG_in_gg(black, .(w, X2)) -> U46_gg(X2, unif_pairscE_in_g(X2)) 9.64/3.31 unif_pairscG_in_gg(w, .(black, X2)) -> U47_gg(X2, unif_pairscE_in_g(X2)) 9.64/3.31 U44_gg(X2, unif_pairscE_out_g(X2)) -> unif_pairscG_out_gg(w, .(w, X2)) 9.64/3.31 U45_gg(X1, unif_pairscE_out_g(X1)) -> unif_pairscG_out_gg(black, .(black, X1)) 9.64/3.31 U46_gg(X2, unif_pairscE_out_g(X2)) -> unif_pairscG_out_gg(black, .(w, X2)) 9.64/3.31 U47_gg(X2, unif_pairscE_out_g(X2)) -> unif_pairscG_out_gg(w, .(black, X2)) 9.64/3.31 unif_pairscE_in_g([]) -> unif_pairscE_out_g([]) 9.64/3.31 unif_pairscE_in_g(.(w, .(w, X2))) -> U36_g(X2, unif_pairscE_in_g(X2)) 9.64/3.31 unif_pairscE_in_g(.(black, .(black, X1))) -> U37_g(X1, unif_pairscE_in_g(X1)) 9.64/3.31 unif_pairscE_in_g(.(black, .(w, X2))) -> U38_g(X2, unif_pairscE_in_g(X2)) 9.64/3.31 unif_pairscE_in_g(.(w, .(black, X2))) -> U39_g(X2, unif_pairscE_in_g(X2)) 9.64/3.31 U36_g(X2, unif_pairscE_out_g(X2)) -> unif_pairscE_out_g(.(w, .(w, X2))) 9.64/3.31 U37_g(X1, unif_pairscE_out_g(X1)) -> unif_pairscE_out_g(.(black, .(black, X1))) 9.64/3.31 U38_g(X2, unif_pairscE_out_g(X2)) -> unif_pairscE_out_g(.(black, .(w, X2))) 9.64/3.31 U39_g(X2, unif_pairscE_out_g(X2)) -> unif_pairscE_out_g(.(w, .(black, X2))) 9.64/3.31 9.64/3.31 The set Q consists of the following terms: 9.64/3.31 9.64/3.31 unif_pairscG_in_gg(x0, x1) 9.64/3.31 U44_gg(x0, x1) 9.64/3.31 U45_gg(x0, x1) 9.64/3.31 U46_gg(x0, x1) 9.64/3.31 U47_gg(x0, x1) 9.64/3.31 unif_pairscE_in_g(x0) 9.64/3.31 U36_g(x0, x1) 9.64/3.31 U37_g(x0, x1) 9.64/3.31 U38_g(x0, x1) 9.64/3.31 U39_g(x0, x1) 9.64/3.31 9.64/3.31 We have to consider all (P,Q,R)-chains. 9.64/3.31 ---------------------------------------- 9.64/3.31 9.64/3.31 (21) DependencyGraphProof (EQUIVALENT) 9.64/3.31 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes. 9.64/3.31 ---------------------------------------- 9.64/3.31 9.64/3.31 (22) 9.64/3.31 TRUE 9.64/3.31 9.64/3.31 ---------------------------------------- 9.64/3.31 9.64/3.31 (23) 9.64/3.31 Obligation: 9.64/3.31 Pi DP problem: 9.64/3.31 The TRS P consists of the following rules: 9.64/3.31 9.64/3.31 UNIF_MATRXB_IN_GG(X1, .(X2, X3)) -> U7_GG(X1, X2, X3, unif_linescC_in_gg(X1, X2)) 9.64/3.31 U7_GG(X1, X2, X3, unif_linescC_out_gg(X1, X2)) -> UNIF_MATRXB_IN_GG(X2, X3) 9.64/3.31 9.64/3.31 The TRS R consists of the following rules: 9.64/3.31 9.64/3.31 varmatcA_in_ga([], []) -> varmatcA_out_ga([], []) 9.64/3.31 varmatcA_in_ga(.(X1, X2), .(X3, X4)) -> U29_ga(X1, X2, X3, X4, varmatcA_in_ga(X1, X3)) 9.64/3.31 varmatcA_in_ga(.(black, X1), .(black, X2)) -> U31_ga(X1, X2, varmatcA_in_ga(X1, X2)) 9.64/3.31 varmatcA_in_ga(.(white, X1), .(w(X2), X3)) -> U32_ga(X1, X2, X3, varmatcA_in_ga(X1, X3)) 9.64/3.31 U32_ga(X1, X2, X3, varmatcA_out_ga(X1, X3)) -> varmatcA_out_ga(.(white, X1), .(w(X2), X3)) 9.64/3.31 U31_ga(X1, X2, varmatcA_out_ga(X1, X2)) -> varmatcA_out_ga(.(black, X1), .(black, X2)) 9.64/3.31 U29_ga(X1, X2, X3, X4, varmatcA_out_ga(X1, X3)) -> U30_ga(X1, X2, X3, X4, varmatcA_in_ga(X2, X4)) 9.64/3.31 U30_ga(X1, X2, X3, X4, varmatcA_out_ga(X2, X4)) -> varmatcA_out_ga(.(X1, X2), .(X3, X4)) 9.64/3.31 unif_pairscG_in_gg(w(X1), .(w(X1), X2)) -> U44_gg(X1, X2, unif_pairscE_in_g(X2)) 9.64/3.31 unif_pairscE_in_g([]) -> unif_pairscE_out_g([]) 9.64/3.31 unif_pairscE_in_g(.(w(X1), .(w(X1), X2))) -> U36_g(X1, X2, unif_pairscE_in_g(X2)) 9.64/3.31 unif_pairscE_in_g(.(black, .(black, X1))) -> U37_g(X1, unif_pairscE_in_g(X1)) 9.64/3.31 unif_pairscE_in_g(.(black, .(w(X1), X2))) -> U38_g(X1, X2, unif_pairscE_in_g(X2)) 9.64/3.31 unif_pairscE_in_g(.(w(X1), .(black, X2))) -> U39_g(X1, X2, unif_pairscE_in_g(X2)) 9.64/3.31 U39_g(X1, X2, unif_pairscE_out_g(X2)) -> unif_pairscE_out_g(.(w(X1), .(black, X2))) 9.64/3.31 U38_g(X1, X2, unif_pairscE_out_g(X2)) -> unif_pairscE_out_g(.(black, .(w(X1), X2))) 9.64/3.31 U37_g(X1, unif_pairscE_out_g(X1)) -> unif_pairscE_out_g(.(black, .(black, X1))) 9.64/3.31 U36_g(X1, X2, unif_pairscE_out_g(X2)) -> unif_pairscE_out_g(.(w(X1), .(w(X1), X2))) 9.64/3.31 U44_gg(X1, X2, unif_pairscE_out_g(X2)) -> unif_pairscG_out_gg(w(X1), .(w(X1), X2)) 9.64/3.31 unif_pairscG_in_gg(black, .(black, X1)) -> U45_gg(X1, unif_pairscE_in_g(X1)) 9.64/3.31 U45_gg(X1, unif_pairscE_out_g(X1)) -> unif_pairscG_out_gg(black, .(black, X1)) 9.64/3.31 unif_pairscG_in_gg(black, .(w(X1), X2)) -> U46_gg(X1, X2, unif_pairscE_in_g(X2)) 9.64/3.31 U46_gg(X1, X2, unif_pairscE_out_g(X2)) -> unif_pairscG_out_gg(black, .(w(X1), X2)) 9.64/3.31 unif_pairscG_in_gg(w(X1), .(black, X2)) -> U47_gg(X1, X2, unif_pairscE_in_g(X2)) 9.64/3.31 U47_gg(X1, X2, unif_pairscE_out_g(X2)) -> unif_pairscG_out_gg(w(X1), .(black, X2)) 9.64/3.31 unif_linescC_in_gg(.(X1, .(X2, X3)), .(X4, .(X5, X6))) -> U35_gg(X1, X2, X3, X4, X5, X6, qcD_in_gggggg(X1, .(X2, .(X4, .(X5, .(X1, .(X4, .(X2, .(X5, .(X1, .(X5, .(X2, .(X4, []))))))))))), X2, X3, X5, X6)) 9.64/3.31 qcD_in_gggggg(X1, X2, X3, X4, X5, X6) -> U42_gggggg(X1, X2, X3, X4, X5, X6, unif_pairscG_in_gg(X1, X2)) 9.64/3.31 U42_gggggg(X1, X2, X3, X4, X5, X6, unif_pairscG_out_gg(X1, X2)) -> U43_gggggg(X1, X2, X3, X4, X5, X6, unif_linescC_in_gg(.(X3, X4), .(X5, X6))) 9.64/3.31 unif_linescC_in_gg(.(X1, []), .(X2, [])) -> unif_linescC_out_gg(.(X1, []), .(X2, [])) 9.64/3.31 U43_gggggg(X1, X2, X3, X4, X5, X6, unif_linescC_out_gg(.(X3, X4), .(X5, X6))) -> qcD_out_gggggg(X1, X2, X3, X4, X5, X6) 9.64/3.31 U35_gg(X1, X2, X3, X4, X5, X6, qcD_out_gggggg(X1, .(X2, .(X4, .(X5, .(X1, .(X4, .(X2, .(X5, .(X1, .(X5, .(X2, .(X4, []))))))))))), X2, X3, X5, X6)) -> unif_linescC_out_gg(.(X1, .(X2, X3)), .(X4, .(X5, X6))) 9.64/3.31 9.64/3.31 The argument filtering Pi contains the following mapping: 9.64/3.31 .(x1, x2) = .(x1, x2) 9.64/3.31 9.64/3.31 varmatcA_in_ga(x1, x2) = varmatcA_in_ga(x1) 9.64/3.31 9.64/3.31 [] = [] 9.64/3.31 9.64/3.31 varmatcA_out_ga(x1, x2) = varmatcA_out_ga(x1, x2) 9.64/3.31 9.64/3.31 U29_ga(x1, x2, x3, x4, x5) = U29_ga(x1, x2, x5) 9.64/3.31 9.64/3.31 black = black 9.64/3.31 9.64/3.31 U31_ga(x1, x2, x3) = U31_ga(x1, x3) 9.64/3.31 9.64/3.31 white = white 9.64/3.31 9.64/3.31 U32_ga(x1, x2, x3, x4) = U32_ga(x1, x4) 9.64/3.31 9.64/3.31 w(x1) = w 9.64/3.31 9.64/3.31 U30_ga(x1, x2, x3, x4, x5) = U30_ga(x1, x2, x3, x5) 9.64/3.31 9.64/3.31 unif_pairscG_in_gg(x1, x2) = unif_pairscG_in_gg(x1, x2) 9.64/3.31 9.64/3.31 U44_gg(x1, x2, x3) = U44_gg(x2, x3) 9.64/3.31 9.64/3.31 unif_pairscE_in_g(x1) = unif_pairscE_in_g(x1) 9.64/3.31 9.64/3.31 unif_pairscE_out_g(x1) = unif_pairscE_out_g(x1) 9.64/3.31 9.64/3.31 U36_g(x1, x2, x3) = U36_g(x2, x3) 9.64/3.31 9.64/3.31 U37_g(x1, x2) = U37_g(x1, x2) 9.64/3.31 9.64/3.31 U38_g(x1, x2, x3) = U38_g(x2, x3) 9.64/3.31 9.64/3.31 U39_g(x1, x2, x3) = U39_g(x2, x3) 9.64/3.31 9.64/3.31 unif_pairscG_out_gg(x1, x2) = unif_pairscG_out_gg(x1, x2) 9.64/3.31 9.64/3.31 U45_gg(x1, x2) = U45_gg(x1, x2) 9.64/3.31 9.64/3.31 U46_gg(x1, x2, x3) = U46_gg(x2, x3) 9.64/3.31 9.64/3.31 U47_gg(x1, x2, x3) = U47_gg(x2, x3) 9.64/3.31 9.64/3.31 unif_linescC_in_gg(x1, x2) = unif_linescC_in_gg(x1, x2) 9.64/3.31 9.64/3.31 U35_gg(x1, x2, x3, x4, x5, x6, x7) = U35_gg(x1, x2, x3, x4, x5, x6, x7) 9.64/3.31 9.64/3.31 qcD_in_gggggg(x1, x2, x3, x4, x5, x6) = qcD_in_gggggg(x1, x2, x3, x4, x5, x6) 9.64/3.31 9.64/3.31 U42_gggggg(x1, x2, x3, x4, x5, x6, x7) = U42_gggggg(x1, x2, x3, x4, x5, x6, x7) 9.64/3.31 9.64/3.31 U43_gggggg(x1, x2, x3, x4, x5, x6, x7) = U43_gggggg(x1, x2, x3, x4, x5, x6, x7) 9.64/3.31 9.64/3.31 unif_linescC_out_gg(x1, x2) = unif_linescC_out_gg(x1, x2) 9.64/3.31 9.64/3.31 qcD_out_gggggg(x1, x2, x3, x4, x5, x6) = qcD_out_gggggg(x1, x2, x3, x4, x5, x6) 9.64/3.31 9.64/3.31 UNIF_MATRXB_IN_GG(x1, x2) = UNIF_MATRXB_IN_GG(x1, x2) 9.64/3.31 9.64/3.31 U7_GG(x1, x2, x3, x4) = U7_GG(x1, x2, x3, x4) 9.64/3.31 9.64/3.31 9.64/3.31 We have to consider all (P,R,Pi)-chains 9.64/3.31 ---------------------------------------- 9.64/3.31 9.64/3.31 (24) UsableRulesProof (EQUIVALENT) 9.64/3.31 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 9.64/3.31 ---------------------------------------- 9.64/3.31 9.64/3.31 (25) 9.64/3.31 Obligation: 9.64/3.31 Pi DP problem: 9.64/3.31 The TRS P consists of the following rules: 9.64/3.31 9.64/3.31 UNIF_MATRXB_IN_GG(X1, .(X2, X3)) -> U7_GG(X1, X2, X3, unif_linescC_in_gg(X1, X2)) 9.64/3.31 U7_GG(X1, X2, X3, unif_linescC_out_gg(X1, X2)) -> UNIF_MATRXB_IN_GG(X2, X3) 9.64/3.31 9.64/3.31 The TRS R consists of the following rules: 9.64/3.31 9.64/3.31 unif_linescC_in_gg(.(X1, .(X2, X3)), .(X4, .(X5, X6))) -> U35_gg(X1, X2, X3, X4, X5, X6, qcD_in_gggggg(X1, .(X2, .(X4, .(X5, .(X1, .(X4, .(X2, .(X5, .(X1, .(X5, .(X2, .(X4, []))))))))))), X2, X3, X5, X6)) 9.64/3.31 unif_linescC_in_gg(.(X1, []), .(X2, [])) -> unif_linescC_out_gg(.(X1, []), .(X2, [])) 9.64/3.31 U35_gg(X1, X2, X3, X4, X5, X6, qcD_out_gggggg(X1, .(X2, .(X4, .(X5, .(X1, .(X4, .(X2, .(X5, .(X1, .(X5, .(X2, .(X4, []))))))))))), X2, X3, X5, X6)) -> unif_linescC_out_gg(.(X1, .(X2, X3)), .(X4, .(X5, X6))) 9.64/3.31 qcD_in_gggggg(X1, X2, X3, X4, X5, X6) -> U42_gggggg(X1, X2, X3, X4, X5, X6, unif_pairscG_in_gg(X1, X2)) 9.64/3.31 U42_gggggg(X1, X2, X3, X4, X5, X6, unif_pairscG_out_gg(X1, X2)) -> U43_gggggg(X1, X2, X3, X4, X5, X6, unif_linescC_in_gg(.(X3, X4), .(X5, X6))) 9.64/3.31 unif_pairscG_in_gg(w(X1), .(w(X1), X2)) -> U44_gg(X1, X2, unif_pairscE_in_g(X2)) 9.64/3.31 unif_pairscG_in_gg(black, .(black, X1)) -> U45_gg(X1, unif_pairscE_in_g(X1)) 9.64/3.31 unif_pairscG_in_gg(black, .(w(X1), X2)) -> U46_gg(X1, X2, unif_pairscE_in_g(X2)) 9.64/3.31 unif_pairscG_in_gg(w(X1), .(black, X2)) -> U47_gg(X1, X2, unif_pairscE_in_g(X2)) 9.64/3.31 U43_gggggg(X1, X2, X3, X4, X5, X6, unif_linescC_out_gg(.(X3, X4), .(X5, X6))) -> qcD_out_gggggg(X1, X2, X3, X4, X5, X6) 9.64/3.31 U44_gg(X1, X2, unif_pairscE_out_g(X2)) -> unif_pairscG_out_gg(w(X1), .(w(X1), X2)) 9.64/3.31 U45_gg(X1, unif_pairscE_out_g(X1)) -> unif_pairscG_out_gg(black, .(black, X1)) 9.64/3.31 U46_gg(X1, X2, unif_pairscE_out_g(X2)) -> unif_pairscG_out_gg(black, .(w(X1), X2)) 9.64/3.31 U47_gg(X1, X2, unif_pairscE_out_g(X2)) -> unif_pairscG_out_gg(w(X1), .(black, X2)) 9.64/3.31 unif_pairscE_in_g([]) -> unif_pairscE_out_g([]) 9.64/3.31 unif_pairscE_in_g(.(w(X1), .(w(X1), X2))) -> U36_g(X1, X2, unif_pairscE_in_g(X2)) 9.64/3.31 unif_pairscE_in_g(.(black, .(black, X1))) -> U37_g(X1, unif_pairscE_in_g(X1)) 9.64/3.31 unif_pairscE_in_g(.(black, .(w(X1), X2))) -> U38_g(X1, X2, unif_pairscE_in_g(X2)) 9.64/3.31 unif_pairscE_in_g(.(w(X1), .(black, X2))) -> U39_g(X1, X2, unif_pairscE_in_g(X2)) 9.64/3.31 U36_g(X1, X2, unif_pairscE_out_g(X2)) -> unif_pairscE_out_g(.(w(X1), .(w(X1), X2))) 9.64/3.31 U37_g(X1, unif_pairscE_out_g(X1)) -> unif_pairscE_out_g(.(black, .(black, X1))) 9.64/3.31 U38_g(X1, X2, unif_pairscE_out_g(X2)) -> unif_pairscE_out_g(.(black, .(w(X1), X2))) 9.64/3.31 U39_g(X1, X2, unif_pairscE_out_g(X2)) -> unif_pairscE_out_g(.(w(X1), .(black, X2))) 9.64/3.31 9.64/3.31 The argument filtering Pi contains the following mapping: 9.64/3.31 .(x1, x2) = .(x1, x2) 9.64/3.31 9.64/3.31 [] = [] 9.64/3.31 9.64/3.31 black = black 9.64/3.31 9.64/3.31 w(x1) = w 9.64/3.31 9.64/3.31 unif_pairscG_in_gg(x1, x2) = unif_pairscG_in_gg(x1, x2) 9.64/3.31 9.64/3.31 U44_gg(x1, x2, x3) = U44_gg(x2, x3) 9.64/3.31 9.64/3.31 unif_pairscE_in_g(x1) = unif_pairscE_in_g(x1) 9.64/3.31 9.64/3.31 unif_pairscE_out_g(x1) = unif_pairscE_out_g(x1) 9.64/3.31 9.64/3.31 U36_g(x1, x2, x3) = U36_g(x2, x3) 9.64/3.31 9.64/3.31 U37_g(x1, x2) = U37_g(x1, x2) 9.64/3.31 9.64/3.31 U38_g(x1, x2, x3) = U38_g(x2, x3) 9.64/3.31 9.64/3.31 U39_g(x1, x2, x3) = U39_g(x2, x3) 9.64/3.31 9.64/3.31 unif_pairscG_out_gg(x1, x2) = unif_pairscG_out_gg(x1, x2) 9.64/3.31 9.64/3.31 U45_gg(x1, x2) = U45_gg(x1, x2) 9.64/3.31 9.64/3.31 U46_gg(x1, x2, x3) = U46_gg(x2, x3) 9.64/3.31 9.64/3.31 U47_gg(x1, x2, x3) = U47_gg(x2, x3) 9.64/3.31 9.64/3.31 unif_linescC_in_gg(x1, x2) = unif_linescC_in_gg(x1, x2) 9.64/3.31 9.64/3.31 U35_gg(x1, x2, x3, x4, x5, x6, x7) = U35_gg(x1, x2, x3, x4, x5, x6, x7) 9.64/3.31 9.64/3.31 qcD_in_gggggg(x1, x2, x3, x4, x5, x6) = qcD_in_gggggg(x1, x2, x3, x4, x5, x6) 9.64/3.31 9.64/3.31 U42_gggggg(x1, x2, x3, x4, x5, x6, x7) = U42_gggggg(x1, x2, x3, x4, x5, x6, x7) 9.64/3.31 9.64/3.31 U43_gggggg(x1, x2, x3, x4, x5, x6, x7) = U43_gggggg(x1, x2, x3, x4, x5, x6, x7) 9.64/3.31 9.64/3.31 unif_linescC_out_gg(x1, x2) = unif_linescC_out_gg(x1, x2) 9.64/3.31 9.64/3.31 qcD_out_gggggg(x1, x2, x3, x4, x5, x6) = qcD_out_gggggg(x1, x2, x3, x4, x5, x6) 9.64/3.31 9.64/3.31 UNIF_MATRXB_IN_GG(x1, x2) = UNIF_MATRXB_IN_GG(x1, x2) 9.64/3.31 9.64/3.31 U7_GG(x1, x2, x3, x4) = U7_GG(x1, x2, x3, x4) 9.64/3.31 9.64/3.31 9.64/3.31 We have to consider all (P,R,Pi)-chains 9.64/3.31 ---------------------------------------- 9.64/3.31 9.64/3.31 (26) PiDPToQDPProof (SOUND) 9.64/3.31 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 9.64/3.31 ---------------------------------------- 9.64/3.31 9.64/3.31 (27) 9.64/3.31 Obligation: 9.64/3.31 Q DP problem: 9.64/3.31 The TRS P consists of the following rules: 9.64/3.31 9.64/3.31 UNIF_MATRXB_IN_GG(X1, .(X2, X3)) -> U7_GG(X1, X2, X3, unif_linescC_in_gg(X1, X2)) 9.64/3.31 U7_GG(X1, X2, X3, unif_linescC_out_gg(X1, X2)) -> UNIF_MATRXB_IN_GG(X2, X3) 9.64/3.31 9.64/3.31 The TRS R consists of the following rules: 9.64/3.31 9.64/3.31 unif_linescC_in_gg(.(X1, .(X2, X3)), .(X4, .(X5, X6))) -> U35_gg(X1, X2, X3, X4, X5, X6, qcD_in_gggggg(X1, .(X2, .(X4, .(X5, .(X1, .(X4, .(X2, .(X5, .(X1, .(X5, .(X2, .(X4, []))))))))))), X2, X3, X5, X6)) 9.64/3.31 unif_linescC_in_gg(.(X1, []), .(X2, [])) -> unif_linescC_out_gg(.(X1, []), .(X2, [])) 9.64/3.31 U35_gg(X1, X2, X3, X4, X5, X6, qcD_out_gggggg(X1, .(X2, .(X4, .(X5, .(X1, .(X4, .(X2, .(X5, .(X1, .(X5, .(X2, .(X4, []))))))))))), X2, X3, X5, X6)) -> unif_linescC_out_gg(.(X1, .(X2, X3)), .(X4, .(X5, X6))) 9.64/3.31 qcD_in_gggggg(X1, X2, X3, X4, X5, X6) -> U42_gggggg(X1, X2, X3, X4, X5, X6, unif_pairscG_in_gg(X1, X2)) 9.64/3.31 U42_gggggg(X1, X2, X3, X4, X5, X6, unif_pairscG_out_gg(X1, X2)) -> U43_gggggg(X1, X2, X3, X4, X5, X6, unif_linescC_in_gg(.(X3, X4), .(X5, X6))) 9.64/3.31 unif_pairscG_in_gg(w, .(w, X2)) -> U44_gg(X2, unif_pairscE_in_g(X2)) 9.64/3.31 unif_pairscG_in_gg(black, .(black, X1)) -> U45_gg(X1, unif_pairscE_in_g(X1)) 9.64/3.31 unif_pairscG_in_gg(black, .(w, X2)) -> U46_gg(X2, unif_pairscE_in_g(X2)) 9.64/3.31 unif_pairscG_in_gg(w, .(black, X2)) -> U47_gg(X2, unif_pairscE_in_g(X2)) 9.64/3.31 U43_gggggg(X1, X2, X3, X4, X5, X6, unif_linescC_out_gg(.(X3, X4), .(X5, X6))) -> qcD_out_gggggg(X1, X2, X3, X4, X5, X6) 9.64/3.31 U44_gg(X2, unif_pairscE_out_g(X2)) -> unif_pairscG_out_gg(w, .(w, X2)) 9.64/3.31 U45_gg(X1, unif_pairscE_out_g(X1)) -> unif_pairscG_out_gg(black, .(black, X1)) 9.64/3.31 U46_gg(X2, unif_pairscE_out_g(X2)) -> unif_pairscG_out_gg(black, .(w, X2)) 9.64/3.31 U47_gg(X2, unif_pairscE_out_g(X2)) -> unif_pairscG_out_gg(w, .(black, X2)) 9.64/3.31 unif_pairscE_in_g([]) -> unif_pairscE_out_g([]) 9.64/3.31 unif_pairscE_in_g(.(w, .(w, X2))) -> U36_g(X2, unif_pairscE_in_g(X2)) 9.64/3.31 unif_pairscE_in_g(.(black, .(black, X1))) -> U37_g(X1, unif_pairscE_in_g(X1)) 9.64/3.31 unif_pairscE_in_g(.(black, .(w, X2))) -> U38_g(X2, unif_pairscE_in_g(X2)) 9.64/3.31 unif_pairscE_in_g(.(w, .(black, X2))) -> U39_g(X2, unif_pairscE_in_g(X2)) 9.64/3.31 U36_g(X2, unif_pairscE_out_g(X2)) -> unif_pairscE_out_g(.(w, .(w, X2))) 9.64/3.31 U37_g(X1, unif_pairscE_out_g(X1)) -> unif_pairscE_out_g(.(black, .(black, X1))) 9.64/3.31 U38_g(X2, unif_pairscE_out_g(X2)) -> unif_pairscE_out_g(.(black, .(w, X2))) 9.64/3.31 U39_g(X2, unif_pairscE_out_g(X2)) -> unif_pairscE_out_g(.(w, .(black, X2))) 9.64/3.31 9.64/3.31 The set Q consists of the following terms: 9.64/3.31 9.64/3.31 unif_linescC_in_gg(x0, x1) 9.64/3.31 U35_gg(x0, x1, x2, x3, x4, x5, x6) 9.64/3.31 qcD_in_gggggg(x0, x1, x2, x3, x4, x5) 9.64/3.31 U42_gggggg(x0, x1, x2, x3, x4, x5, x6) 9.64/3.31 unif_pairscG_in_gg(x0, x1) 9.64/3.31 U43_gggggg(x0, x1, x2, x3, x4, x5, x6) 9.64/3.31 U44_gg(x0, x1) 9.64/3.31 U45_gg(x0, x1) 9.64/3.31 U46_gg(x0, x1) 9.64/3.31 U47_gg(x0, x1) 9.64/3.31 unif_pairscE_in_g(x0) 9.64/3.31 U36_g(x0, x1) 9.64/3.31 U37_g(x0, x1) 9.64/3.31 U38_g(x0, x1) 9.64/3.31 U39_g(x0, x1) 9.64/3.31 9.64/3.31 We have to consider all (P,Q,R)-chains. 9.64/3.31 ---------------------------------------- 9.64/3.31 9.64/3.31 (28) QDPSizeChangeProof (EQUIVALENT) 9.64/3.31 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. 9.64/3.31 9.64/3.31 From the DPs we obtained the following set of size-change graphs: 9.64/3.31 *U7_GG(X1, X2, X3, unif_linescC_out_gg(X1, X2)) -> UNIF_MATRXB_IN_GG(X2, X3) 9.64/3.31 The graph contains the following edges 2 >= 1, 4 > 1, 3 >= 2 9.64/3.31 9.64/3.31 9.64/3.31 *UNIF_MATRXB_IN_GG(X1, .(X2, X3)) -> U7_GG(X1, X2, X3, unif_linescC_in_gg(X1, X2)) 9.64/3.31 The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3 9.64/3.31 9.64/3.31 9.64/3.31 ---------------------------------------- 9.64/3.31 9.64/3.31 (29) 9.64/3.31 YES 9.64/3.31 9.64/3.31 ---------------------------------------- 9.64/3.31 9.64/3.31 (30) 9.64/3.31 Obligation: 9.64/3.31 Pi DP problem: 9.64/3.31 The TRS P consists of the following rules: 9.64/3.31 9.64/3.31 VARMATA_IN_GA(.(X1, X2), .(X3, X4)) -> U2_GA(X1, X2, X3, X4, varmatcA_in_ga(X1, X3)) 9.64/3.31 U2_GA(X1, X2, X3, X4, varmatcA_out_ga(X1, X3)) -> VARMATA_IN_GA(X2, X4) 9.64/3.31 VARMATA_IN_GA(.(X1, X2), .(X3, X4)) -> VARMATA_IN_GA(X1, X3) 9.64/3.31 VARMATA_IN_GA(.(black, X1), .(black, X2)) -> VARMATA_IN_GA(X1, X2) 9.64/3.31 VARMATA_IN_GA(.(white, X1), .(w(X2), X3)) -> VARMATA_IN_GA(X1, X3) 9.64/3.31 9.64/3.31 The TRS R consists of the following rules: 9.64/3.31 9.64/3.31 varmatcA_in_ga([], []) -> varmatcA_out_ga([], []) 9.64/3.31 varmatcA_in_ga(.(X1, X2), .(X3, X4)) -> U29_ga(X1, X2, X3, X4, varmatcA_in_ga(X1, X3)) 9.64/3.31 varmatcA_in_ga(.(black, X1), .(black, X2)) -> U31_ga(X1, X2, varmatcA_in_ga(X1, X2)) 9.64/3.31 varmatcA_in_ga(.(white, X1), .(w(X2), X3)) -> U32_ga(X1, X2, X3, varmatcA_in_ga(X1, X3)) 9.64/3.31 U32_ga(X1, X2, X3, varmatcA_out_ga(X1, X3)) -> varmatcA_out_ga(.(white, X1), .(w(X2), X3)) 9.64/3.31 U31_ga(X1, X2, varmatcA_out_ga(X1, X2)) -> varmatcA_out_ga(.(black, X1), .(black, X2)) 9.64/3.31 U29_ga(X1, X2, X3, X4, varmatcA_out_ga(X1, X3)) -> U30_ga(X1, X2, X3, X4, varmatcA_in_ga(X2, X4)) 9.64/3.31 U30_ga(X1, X2, X3, X4, varmatcA_out_ga(X2, X4)) -> varmatcA_out_ga(.(X1, X2), .(X3, X4)) 9.64/3.31 unif_pairscG_in_gg(w(X1), .(w(X1), X2)) -> U44_gg(X1, X2, unif_pairscE_in_g(X2)) 9.64/3.31 unif_pairscE_in_g([]) -> unif_pairscE_out_g([]) 9.64/3.31 unif_pairscE_in_g(.(w(X1), .(w(X1), X2))) -> U36_g(X1, X2, unif_pairscE_in_g(X2)) 9.64/3.31 unif_pairscE_in_g(.(black, .(black, X1))) -> U37_g(X1, unif_pairscE_in_g(X1)) 9.64/3.31 unif_pairscE_in_g(.(black, .(w(X1), X2))) -> U38_g(X1, X2, unif_pairscE_in_g(X2)) 9.64/3.31 unif_pairscE_in_g(.(w(X1), .(black, X2))) -> U39_g(X1, X2, unif_pairscE_in_g(X2)) 9.64/3.31 U39_g(X1, X2, unif_pairscE_out_g(X2)) -> unif_pairscE_out_g(.(w(X1), .(black, X2))) 9.64/3.31 U38_g(X1, X2, unif_pairscE_out_g(X2)) -> unif_pairscE_out_g(.(black, .(w(X1), X2))) 9.64/3.31 U37_g(X1, unif_pairscE_out_g(X1)) -> unif_pairscE_out_g(.(black, .(black, X1))) 9.64/3.31 U36_g(X1, X2, unif_pairscE_out_g(X2)) -> unif_pairscE_out_g(.(w(X1), .(w(X1), X2))) 9.64/3.31 U44_gg(X1, X2, unif_pairscE_out_g(X2)) -> unif_pairscG_out_gg(w(X1), .(w(X1), X2)) 9.64/3.31 unif_pairscG_in_gg(black, .(black, X1)) -> U45_gg(X1, unif_pairscE_in_g(X1)) 9.64/3.31 U45_gg(X1, unif_pairscE_out_g(X1)) -> unif_pairscG_out_gg(black, .(black, X1)) 9.64/3.31 unif_pairscG_in_gg(black, .(w(X1), X2)) -> U46_gg(X1, X2, unif_pairscE_in_g(X2)) 9.64/3.31 U46_gg(X1, X2, unif_pairscE_out_g(X2)) -> unif_pairscG_out_gg(black, .(w(X1), X2)) 9.64/3.31 unif_pairscG_in_gg(w(X1), .(black, X2)) -> U47_gg(X1, X2, unif_pairscE_in_g(X2)) 9.64/3.31 U47_gg(X1, X2, unif_pairscE_out_g(X2)) -> unif_pairscG_out_gg(w(X1), .(black, X2)) 9.64/3.31 unif_linescC_in_gg(.(X1, .(X2, X3)), .(X4, .(X5, X6))) -> U35_gg(X1, X2, X3, X4, X5, X6, qcD_in_gggggg(X1, .(X2, .(X4, .(X5, .(X1, .(X4, .(X2, .(X5, .(X1, .(X5, .(X2, .(X4, []))))))))))), X2, X3, X5, X6)) 9.64/3.31 qcD_in_gggggg(X1, X2, X3, X4, X5, X6) -> U42_gggggg(X1, X2, X3, X4, X5, X6, unif_pairscG_in_gg(X1, X2)) 9.64/3.31 U42_gggggg(X1, X2, X3, X4, X5, X6, unif_pairscG_out_gg(X1, X2)) -> U43_gggggg(X1, X2, X3, X4, X5, X6, unif_linescC_in_gg(.(X3, X4), .(X5, X6))) 9.64/3.31 unif_linescC_in_gg(.(X1, []), .(X2, [])) -> unif_linescC_out_gg(.(X1, []), .(X2, [])) 9.64/3.31 U43_gggggg(X1, X2, X3, X4, X5, X6, unif_linescC_out_gg(.(X3, X4), .(X5, X6))) -> qcD_out_gggggg(X1, X2, X3, X4, X5, X6) 9.64/3.31 U35_gg(X1, X2, X3, X4, X5, X6, qcD_out_gggggg(X1, .(X2, .(X4, .(X5, .(X1, .(X4, .(X2, .(X5, .(X1, .(X5, .(X2, .(X4, []))))))))))), X2, X3, X5, X6)) -> unif_linescC_out_gg(.(X1, .(X2, X3)), .(X4, .(X5, X6))) 9.64/3.31 9.64/3.31 The argument filtering Pi contains the following mapping: 9.64/3.31 .(x1, x2) = .(x1, x2) 9.64/3.31 9.64/3.31 varmatcA_in_ga(x1, x2) = varmatcA_in_ga(x1) 9.64/3.31 9.64/3.31 [] = [] 9.64/3.31 9.64/3.31 varmatcA_out_ga(x1, x2) = varmatcA_out_ga(x1, x2) 9.64/3.31 9.64/3.31 U29_ga(x1, x2, x3, x4, x5) = U29_ga(x1, x2, x5) 9.64/3.31 9.64/3.31 black = black 9.64/3.31 9.64/3.31 U31_ga(x1, x2, x3) = U31_ga(x1, x3) 9.64/3.31 9.64/3.31 white = white 9.64/3.31 9.64/3.31 U32_ga(x1, x2, x3, x4) = U32_ga(x1, x4) 9.64/3.31 9.64/3.31 w(x1) = w 9.64/3.31 9.64/3.31 U30_ga(x1, x2, x3, x4, x5) = U30_ga(x1, x2, x3, x5) 9.64/3.31 9.64/3.31 unif_pairscG_in_gg(x1, x2) = unif_pairscG_in_gg(x1, x2) 9.64/3.31 9.64/3.31 U44_gg(x1, x2, x3) = U44_gg(x2, x3) 9.64/3.31 9.64/3.31 unif_pairscE_in_g(x1) = unif_pairscE_in_g(x1) 9.64/3.31 9.64/3.31 unif_pairscE_out_g(x1) = unif_pairscE_out_g(x1) 9.64/3.31 9.64/3.31 U36_g(x1, x2, x3) = U36_g(x2, x3) 9.64/3.31 9.64/3.31 U37_g(x1, x2) = U37_g(x1, x2) 9.64/3.31 9.64/3.31 U38_g(x1, x2, x3) = U38_g(x2, x3) 9.64/3.31 9.64/3.31 U39_g(x1, x2, x3) = U39_g(x2, x3) 9.64/3.31 9.64/3.31 unif_pairscG_out_gg(x1, x2) = unif_pairscG_out_gg(x1, x2) 9.64/3.31 9.64/3.31 U45_gg(x1, x2) = U45_gg(x1, x2) 9.64/3.31 9.64/3.31 U46_gg(x1, x2, x3) = U46_gg(x2, x3) 9.64/3.31 9.64/3.31 U47_gg(x1, x2, x3) = U47_gg(x2, x3) 9.64/3.31 9.64/3.31 unif_linescC_in_gg(x1, x2) = unif_linescC_in_gg(x1, x2) 9.64/3.31 9.64/3.31 U35_gg(x1, x2, x3, x4, x5, x6, x7) = U35_gg(x1, x2, x3, x4, x5, x6, x7) 9.64/3.31 9.64/3.31 qcD_in_gggggg(x1, x2, x3, x4, x5, x6) = qcD_in_gggggg(x1, x2, x3, x4, x5, x6) 9.64/3.31 9.64/3.31 U42_gggggg(x1, x2, x3, x4, x5, x6, x7) = U42_gggggg(x1, x2, x3, x4, x5, x6, x7) 9.64/3.31 9.64/3.31 U43_gggggg(x1, x2, x3, x4, x5, x6, x7) = U43_gggggg(x1, x2, x3, x4, x5, x6, x7) 9.64/3.31 9.64/3.31 unif_linescC_out_gg(x1, x2) = unif_linescC_out_gg(x1, x2) 9.64/3.31 9.64/3.31 qcD_out_gggggg(x1, x2, x3, x4, x5, x6) = qcD_out_gggggg(x1, x2, x3, x4, x5, x6) 9.64/3.31 9.64/3.31 VARMATA_IN_GA(x1, x2) = VARMATA_IN_GA(x1) 9.64/3.31 9.64/3.31 U2_GA(x1, x2, x3, x4, x5) = U2_GA(x1, x2, x5) 9.64/3.31 9.64/3.31 9.64/3.31 We have to consider all (P,R,Pi)-chains 9.64/3.31 ---------------------------------------- 9.64/3.31 9.64/3.31 (31) UsableRulesProof (EQUIVALENT) 9.64/3.31 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 9.64/3.31 ---------------------------------------- 9.64/3.31 9.64/3.31 (32) 9.64/3.31 Obligation: 9.64/3.31 Pi DP problem: 9.64/3.31 The TRS P consists of the following rules: 9.64/3.31 9.64/3.31 VARMATA_IN_GA(.(X1, X2), .(X3, X4)) -> U2_GA(X1, X2, X3, X4, varmatcA_in_ga(X1, X3)) 9.64/3.31 U2_GA(X1, X2, X3, X4, varmatcA_out_ga(X1, X3)) -> VARMATA_IN_GA(X2, X4) 9.64/3.31 VARMATA_IN_GA(.(X1, X2), .(X3, X4)) -> VARMATA_IN_GA(X1, X3) 9.64/3.31 VARMATA_IN_GA(.(black, X1), .(black, X2)) -> VARMATA_IN_GA(X1, X2) 9.64/3.31 VARMATA_IN_GA(.(white, X1), .(w(X2), X3)) -> VARMATA_IN_GA(X1, X3) 9.64/3.31 9.64/3.31 The TRS R consists of the following rules: 9.64/3.31 9.64/3.31 varmatcA_in_ga([], []) -> varmatcA_out_ga([], []) 9.64/3.31 varmatcA_in_ga(.(X1, X2), .(X3, X4)) -> U29_ga(X1, X2, X3, X4, varmatcA_in_ga(X1, X3)) 9.64/3.31 varmatcA_in_ga(.(black, X1), .(black, X2)) -> U31_ga(X1, X2, varmatcA_in_ga(X1, X2)) 9.64/3.31 varmatcA_in_ga(.(white, X1), .(w(X2), X3)) -> U32_ga(X1, X2, X3, varmatcA_in_ga(X1, X3)) 9.64/3.31 U29_ga(X1, X2, X3, X4, varmatcA_out_ga(X1, X3)) -> U30_ga(X1, X2, X3, X4, varmatcA_in_ga(X2, X4)) 9.64/3.31 U31_ga(X1, X2, varmatcA_out_ga(X1, X2)) -> varmatcA_out_ga(.(black, X1), .(black, X2)) 9.64/3.31 U32_ga(X1, X2, X3, varmatcA_out_ga(X1, X3)) -> varmatcA_out_ga(.(white, X1), .(w(X2), X3)) 9.64/3.31 U30_ga(X1, X2, X3, X4, varmatcA_out_ga(X2, X4)) -> varmatcA_out_ga(.(X1, X2), .(X3, X4)) 9.64/3.31 9.64/3.31 The argument filtering Pi contains the following mapping: 9.64/3.31 .(x1, x2) = .(x1, x2) 9.64/3.31 9.64/3.31 varmatcA_in_ga(x1, x2) = varmatcA_in_ga(x1) 9.64/3.31 9.64/3.31 [] = [] 9.64/3.31 9.64/3.31 varmatcA_out_ga(x1, x2) = varmatcA_out_ga(x1, x2) 9.64/3.31 9.64/3.31 U29_ga(x1, x2, x3, x4, x5) = U29_ga(x1, x2, x5) 9.64/3.31 9.64/3.31 black = black 9.64/3.31 9.64/3.31 U31_ga(x1, x2, x3) = U31_ga(x1, x3) 9.64/3.31 9.64/3.31 white = white 9.64/3.31 9.64/3.31 U32_ga(x1, x2, x3, x4) = U32_ga(x1, x4) 9.64/3.31 9.64/3.31 w(x1) = w 9.64/3.31 9.64/3.31 U30_ga(x1, x2, x3, x4, x5) = U30_ga(x1, x2, x3, x5) 9.64/3.31 9.64/3.31 VARMATA_IN_GA(x1, x2) = VARMATA_IN_GA(x1) 9.64/3.31 9.64/3.31 U2_GA(x1, x2, x3, x4, x5) = U2_GA(x1, x2, x5) 9.64/3.31 9.64/3.31 9.64/3.31 We have to consider all (P,R,Pi)-chains 9.64/3.31 ---------------------------------------- 9.64/3.31 9.64/3.31 (33) PiDPToQDPProof (SOUND) 9.64/3.31 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 9.64/3.31 ---------------------------------------- 9.64/3.31 9.64/3.31 (34) 9.64/3.31 Obligation: 9.64/3.31 Q DP problem: 9.64/3.31 The TRS P consists of the following rules: 9.64/3.31 9.64/3.31 VARMATA_IN_GA(.(X1, X2)) -> U2_GA(X1, X2, varmatcA_in_ga(X1)) 9.64/3.31 U2_GA(X1, X2, varmatcA_out_ga(X1, X3)) -> VARMATA_IN_GA(X2) 9.64/3.31 VARMATA_IN_GA(.(X1, X2)) -> VARMATA_IN_GA(X1) 9.64/3.31 VARMATA_IN_GA(.(black, X1)) -> VARMATA_IN_GA(X1) 9.64/3.31 VARMATA_IN_GA(.(white, X1)) -> VARMATA_IN_GA(X1) 9.64/3.31 9.64/3.31 The TRS R consists of the following rules: 9.64/3.31 9.64/3.31 varmatcA_in_ga([]) -> varmatcA_out_ga([], []) 9.64/3.31 varmatcA_in_ga(.(X1, X2)) -> U29_ga(X1, X2, varmatcA_in_ga(X1)) 9.64/3.31 varmatcA_in_ga(.(black, X1)) -> U31_ga(X1, varmatcA_in_ga(X1)) 9.64/3.31 varmatcA_in_ga(.(white, X1)) -> U32_ga(X1, varmatcA_in_ga(X1)) 9.64/3.31 U29_ga(X1, X2, varmatcA_out_ga(X1, X3)) -> U30_ga(X1, X2, X3, varmatcA_in_ga(X2)) 9.64/3.31 U31_ga(X1, varmatcA_out_ga(X1, X2)) -> varmatcA_out_ga(.(black, X1), .(black, X2)) 9.64/3.31 U32_ga(X1, varmatcA_out_ga(X1, X3)) -> varmatcA_out_ga(.(white, X1), .(w, X3)) 9.64/3.31 U30_ga(X1, X2, X3, varmatcA_out_ga(X2, X4)) -> varmatcA_out_ga(.(X1, X2), .(X3, X4)) 9.64/3.31 9.64/3.31 The set Q consists of the following terms: 9.64/3.31 9.64/3.31 varmatcA_in_ga(x0) 9.64/3.31 U29_ga(x0, x1, x2) 9.64/3.31 U31_ga(x0, x1) 9.64/3.31 U32_ga(x0, x1) 9.64/3.31 U30_ga(x0, x1, x2, x3) 9.64/3.31 9.64/3.31 We have to consider all (P,Q,R)-chains. 9.64/3.31 ---------------------------------------- 9.64/3.31 9.64/3.31 (35) QDPSizeChangeProof (EQUIVALENT) 9.64/3.31 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. 9.64/3.31 9.64/3.31 From the DPs we obtained the following set of size-change graphs: 9.64/3.31 *U2_GA(X1, X2, varmatcA_out_ga(X1, X3)) -> VARMATA_IN_GA(X2) 9.64/3.31 The graph contains the following edges 2 >= 1 9.64/3.31 9.64/3.31 9.64/3.31 *VARMATA_IN_GA(.(X1, X2)) -> U2_GA(X1, X2, varmatcA_in_ga(X1)) 9.64/3.31 The graph contains the following edges 1 > 1, 1 > 2 9.64/3.31 9.64/3.31 9.64/3.31 *VARMATA_IN_GA(.(X1, X2)) -> VARMATA_IN_GA(X1) 9.64/3.31 The graph contains the following edges 1 > 1 9.64/3.31 9.64/3.31 9.64/3.31 *VARMATA_IN_GA(.(black, X1)) -> VARMATA_IN_GA(X1) 9.64/3.31 The graph contains the following edges 1 > 1 9.64/3.31 9.64/3.31 9.64/3.31 *VARMATA_IN_GA(.(white, X1)) -> VARMATA_IN_GA(X1) 9.64/3.31 The graph contains the following edges 1 > 1 9.64/3.31 9.64/3.31 9.64/3.31 ---------------------------------------- 9.64/3.31 9.64/3.31 (36) 9.64/3.31 YES 9.78/3.34 EOF