Left Termination proof of /tmp/tmpGJRmZS/permute2.pl

Left Termination of the query pattern
permute(g,a)
w.r.t. the given Prolog program could successfully be proven:

0 Prolog

↳1 PrologToPiTRSViaGraphTransformerProof (⇒, 94 ms)

↳2 PiTRS

↳3 DependencyPairsProof (⇔, 17 ms)

↳4 PiDP

↳5 DependencyGraphProof (⇔, 0 ms)

↳6 AND

    ↳7 PiDP

        ↳8 UsableRulesProof (⇔, 0 ms)

        ↳9 PiDP

        ↳10 PiDPToQDPProof (⇒, 0 ms)

        ↳11 QDP

        ↳12 QDPSizeChangeProof (⇔, 0 ms)

        ↳13 YES

    ↳14 PiDP

        ↳15 UsableRulesProof (⇔, 3 ms)

        ↳16 PiDP

        ↳17 PiDPToQDPProof (⇒, 0 ms)

        ↳18 QDP

        ↳19 QDPOrderProof (⇔, 53 ms)

        ↳20 QDP

        ↳21 DependencyGraphProof (⇔, 0 ms)

        ↳22 TRUE

(0) Obligation:

Clauses:

permute([], []).
permute(.(X, Y), .(U, V)) :- ','(delete(U, .(X, Y), W), permute(W, V)).
delete(X, .(X, Y), Y).
delete(U, .(X, Y), .(X, Z)) :- delete(U, Y, Z).

Query: permute(g,a)

(1) PrologToPiTRSViaGraphTransformerProof (SOUND transformation)

Transformed Prolog program to (Pi-)TRS.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
3 [label="A: permute(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
4 [label="4: permute(T1, T2), SCOPE: 1, CLAUSE: 0 | permute(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: true | permute([], T2), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: permute(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\npermute(T1, T2) !=? permute([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: permute([], T2), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: ','(delete(T11, .(T7, T8), X13), permute(X13, T12))\n\nT7 is ground\nT8 is ground\nX13 is free", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: ','(delete(T11, .(T7, T8), X13), permute(X13, T12)), SCOPE: 2, CLAUSE: 2 | ','(delete(T11, .(T7, T8), X13), permute(X13, T12)), SCOPE: 2, CLAUSE: 3\n\nT7 is ground\nT8 is ground\nX13 is free", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: ','(delete(T11, .(T7, T8), X13), permute(X13, T12)), SCOPE: 2, CLAUSE: 2\n\nT7 is ground\nT8 is ground\nX13 is free", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: ','(delete(T11, .(T7, T8), X13), permute(X13, T12)), SCOPE: 2, CLAUSE: 3\n\nT7 is ground\nT8 is ground\nX13 is free", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: permute(T22, T23)\n\nT22 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="B: ','(delete(T39, T38, X42), permute(.(T37, X42), T40))\n\nT37 is ground\nT38 is ground\nX42 is free", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="C: delete(T39, T38, X42)\n\nT38 is ground\nX42 is free", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: permute(.(T37, T45), T46)\n\nT37 is ground\nT45 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: delete(T39, T38, X42), SCOPE: 3, CLAUSE: 2 | delete(T39, T38, X42), SCOPE: 3, CLAUSE: 3\n\nT38 is ground\nX42 is free", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: delete(T39, T38, X42), SCOPE: 3, CLAUSE: 2\n\nT38 is ground\nX42 is free", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: delete(T39, T38, X42), SCOPE: 3, CLAUSE: 3\n\nT38 is ground\nX42 is free", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: delete(T70, T69, X75)\n\nT69 is ground\nX75 is free", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 75 [arrowhead = none ];
75 -> 4;

76 [label="EVAL with clause\npermute([], []).\nand substitutionT1 -> [],\nT2 -> []", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 76 [arrowhead = none ];
76 -> 13;

77 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 77 [arrowhead = none ];
77 -> 14;

78 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
13 -> 78 [arrowhead = none ];
78 -> 16;

79 [label="EVAL with clause\npermute(.(X9, X10), .(X11, X12)) :- ','(delete(X11, .(X9, X10), X13), permute(X13, X12)).\nand substitutionX9 -> T7,\nX10 -> T8,\nT1 -> .(T7, T8),\nX11 -> T11,\nX12 -> T12,\nT2 -> .(T11, T12),\nT9 -> T11,\nT10 -> T12", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 79 [arrowhead = none ];
79 -> 22;

80 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
14 -> 80 [arrowhead = none ];
80 -> 24;

81 [label="BACKTRACK\nfor clause: permute(.(X, Y), .(U, V)) :- ','(delete(U, .(X, Y), W), permute(W, V))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
16 -> 81 [arrowhead = none ];
81 -> 19;

82 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
22 -> 82 [arrowhead = none ];
82 -> 26;

83 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
26 -> 83 [arrowhead = none ];
83 -> 28;

84 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
26 -> 84 [arrowhead = none ];
84 -> 30;

85 [label="EVAL with clause\ndelete(X22, .(X22, X23), X23).\nand substitutionT11 -> T21,\nX22 -> T21,\nT7 -> T21,\nT8 -> T22,\nX23 -> T22,\nX13 -> T22,\nT12 -> T23", fontsize=14, style = filled, fillcolor = lightblue];
28 -> 85 [arrowhead = none ];
85 -> 32;

86 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
28 -> 86 [arrowhead = none ];
86 -> 34;

87 [label="ONLY EVAL with clause\ndelete(X38, .(X39, X40), .(X39, X41)) :- delete(X38, X40, X41).\nand substitutionT11 -> T39,\nX38 -> T39,\nT7 -> T37,\nX39 -> T37,\nT8 -> T38,\nX40 -> T38,\nX41 -> X42,\nX13 -> .(T37, X42),\nT36 -> T39,\nT12 -> T40", fontsize=14, style = filled, fillcolor = lightblue];
30 -> 87 [arrowhead = none ];
87 -> 40;

88 [label="INSTANCE with matching:\nT1 -> T22\nT2 -> T23", fontsize=14, style = filled, fillcolor = lightgrey];
32 -> 88 [arrowhead = none , style=dashed];
88 -> 3[style=dashed];

89 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
40 -> 89 [arrowhead = none ];
89 -> 45;

90 [label="SPLIT 2\nnew knowledge:\nT39 is ground\nT38 is ground\nT45 is ground\nreplacements:X42 -> T45,\nT40 -> T46", fontsize=14, style = filled, fillcolor = violet];
40 -> 90 [arrowhead = none ];
90 -> 47;

91 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
45 -> 91 [arrowhead = none ];
91 -> 56;

92 [label="INSTANCE with matching:\nT1 -> .(T37, T45)\nT2 -> T46", fontsize=14, style = filled, fillcolor = lightgrey];
47 -> 92 [arrowhead = none , style=dashed];
92 -> 3[style=dashed];

93 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
56 -> 93 [arrowhead = none ];
93 -> 59;

94 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
56 -> 94 [arrowhead = none ];
94 -> 60;

95 [label="EVAL with clause\ndelete(X59, .(X59, X60), X60).\nand substitutionT39 -> T59,\nX59 -> T59,\nX60 -> T60,\nT38 -> .(T59, T60),\nX42 -> T60", fontsize=14, style = filled, fillcolor = lightblue];
59 -> 95 [arrowhead = none ];
95 -> 63;

96 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
59 -> 96 [arrowhead = none ];
96 -> 64;

97 [label="EVAL with clause\ndelete(X71, .(X72, X73), .(X72, X74)) :- delete(X71, X73, X74).\nand substitutionT39 -> T70,\nX71 -> T70,\nX72 -> T68,\nX73 -> T69,\nT38 -> .(T68, T69),\nX74 -> X75,\nX42 -> .(T68, X75),\nT67 -> T70", fontsize=14, style = filled, fillcolor = lightblue];
60 -> 97 [arrowhead = none ];
97 -> 73;

98 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
60 -> 98 [arrowhead = none ];
98 -> 74;

99 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
63 -> 99 [arrowhead = none ];
99 -> 65;

100 [label="INSTANCE with matching:\nT39 -> T70\nT38 -> T69\nX42 -> X75", fontsize=14, style = filled, fillcolor = lightgrey];
73 -> 100 [arrowhead = none , style=dashed];
100 -> 45[style=dashed];

}

(2) Obligation:

Pi-finite rewrite system:
The TRS R consists of the following rules:

permuteA_in_ga([], []) → permuteA_out_ga([], [])
permuteA_in_ga(.(T21, T22), .(T21, T23)) → U1_ga(T21, T22, T23, permuteA_in_ga(T22, T23))
permuteA_in_ga(.(T37, T38), .(T39, T40)) → U2_ga(T37, T38, T39, T40, pB_in_agaga(T39, T38, X42, T37, T40))
pB_in_agaga(T39, T38, T45, T37, T46) → U4_agaga(T39, T38, T45, T37, T46, deleteC_in_aga(T39, T38, T45))
deleteC_in_aga(T59, .(T59, T60), T60) → deleteC_out_aga(T59, .(T59, T60), T60)
deleteC_in_aga(T70, .(T68, T69), .(T68, X75)) → U3_aga(T70, T68, T69, X75, deleteC_in_aga(T70, T69, X75))
U3_aga(T70, T68, T69, X75, deleteC_out_aga(T70, T69, X75)) → deleteC_out_aga(T70, .(T68, T69), .(T68, X75))
U4_agaga(T39, T38, T45, T37, T46, deleteC_out_aga(T39, T38, T45)) → U5_agaga(T39, T38, T45, T37, T46, permuteA_in_ga(.(T37, T45), T46))
U5_agaga(T39, T38, T45, T37, T46, permuteA_out_ga(.(T37, T45), T46)) → pB_out_agaga(T39, T38, T45, T37, T46)
U2_ga(T37, T38, T39, T40, pB_out_agaga(T39, T38, X42, T37, T40)) → permuteA_out_ga(.(T37, T38), .(T39, T40))
U1_ga(T21, T22, T23, permuteA_out_ga(T22, T23)) → permuteA_out_ga(.(T21, T22), .(T21, T23))

(3) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

PERMUTEA_IN_GA(.(T21, T22), .(T21, T23)) → U1_GA(T21, T22, T23, permuteA_in_ga(T22, T23))
PERMUTEA_IN_GA(.(T21, T22), .(T21, T23)) → PERMUTEA_IN_GA(T22, T23)
PERMUTEA_IN_GA(.(T37, T38), .(T39, T40)) → U2_GA(T37, T38, T39, T40, pB_in_agaga(T39, T38, X42, T37, T40))
PERMUTEA_IN_GA(.(T37, T38), .(T39, T40)) → PB_IN_AGAGA(T39, T38, X42, T37, T40)
PB_IN_AGAGA(T39, T38, T45, T37, T46) → U4_AGAGA(T39, T38, T45, T37, T46, deleteC_in_aga(T39, T38, T45))
PB_IN_AGAGA(T39, T38, T45, T37, T46) → DELETEC_IN_AGA(T39, T38, T45)
DELETEC_IN_AGA(T70, .(T68, T69), .(T68, X75)) → U3_AGA(T70, T68, T69, X75, deleteC_in_aga(T70, T69, X75))
DELETEC_IN_AGA(T70, .(T68, T69), .(T68, X75)) → DELETEC_IN_AGA(T70, T69, X75)
U4_AGAGA(T39, T38, T45, T37, T46, deleteC_out_aga(T39, T38, T45)) → U5_AGAGA(T39, T38, T45, T37, T46, permuteA_in_ga(.(T37, T45), T46))
U4_AGAGA(T39, T38, T45, T37, T46, deleteC_out_aga(T39, T38, T45)) → PERMUTEA_IN_GA(.(T37, T45), T46)

The TRS R consists of the following rules:

permuteA_in_ga([], []) → permuteA_out_ga([], [])
permuteA_in_ga(.(T21, T22), .(T21, T23)) → U1_ga(T21, T22, T23, permuteA_in_ga(T22, T23))
permuteA_in_ga(.(T37, T38), .(T39, T40)) → U2_ga(T37, T38, T39, T40, pB_in_agaga(T39, T38, X42, T37, T40))
pB_in_agaga(T39, T38, T45, T37, T46) → U4_agaga(T39, T38, T45, T37, T46, deleteC_in_aga(T39, T38, T45))
deleteC_in_aga(T59, .(T59, T60), T60) → deleteC_out_aga(T59, .(T59, T60), T60)
deleteC_in_aga(T70, .(T68, T69), .(T68, X75)) → U3_aga(T70, T68, T69, X75, deleteC_in_aga(T70, T69, X75))
U3_aga(T70, T68, T69, X75, deleteC_out_aga(T70, T69, X75)) → deleteC_out_aga(T70, .(T68, T69), .(T68, X75))
U4_agaga(T39, T38, T45, T37, T46, deleteC_out_aga(T39, T38, T45)) → U5_agaga(T39, T38, T45, T37, T46, permuteA_in_ga(.(T37, T45), T46))
U5_agaga(T39, T38, T45, T37, T46, permuteA_out_ga(.(T37, T45), T46)) → pB_out_agaga(T39, T38, T45, T37, T46)
U2_ga(T37, T38, T39, T40, pB_out_agaga(T39, T38, X42, T37, T40)) → permuteA_out_ga(.(T37, T38), .(T39, T40))
U1_ga(T21, T22, T23, permuteA_out_ga(T22, T23)) → permuteA_out_ga(.(T21, T22), .(T21, T23))

The argument filtering Pi contains the following mapping:
permuteA_in_ga(x1, x2) = permuteA_in_ga(x1)
[] = []
permuteA_out_ga(x1, x2) = permuteA_out_ga(x1, x2)
.(x1, x2) = .(x1, x2)
U1_ga(x1, x2, x3, x4) = U1_ga(x1, x2, x4)
U2_ga(x1, x2, x3, x4, x5) = U2_ga(x1, x2, x5)
pB_in_agaga(x1, x2, x3, x4, x5) = pB_in_agaga(x2, x4)
U4_agaga(x1, x2, x3, x4, x5, x6) = U4_agaga(x2, x4, x6)
deleteC_in_aga(x1, x2, x3) = deleteC_in_aga(x2)
deleteC_out_aga(x1, x2, x3) = deleteC_out_aga(x1, x2, x3)
U3_aga(x1, x2, x3, x4, x5) = U3_aga(x2, x3, x5)
U5_agaga(x1, x2, x3, x4, x5, x6) = U5_agaga(x1, x2, x3, x4, x6)
pB_out_agaga(x1, x2, x3, x4, x5) = pB_out_agaga(x1, x2, x3, x4, x5)
PERMUTEA_IN_GA(x1, x2) = PERMUTEA_IN_GA(x1)
U1_GA(x1, x2, x3, x4) = U1_GA(x1, x2, x4)
U2_GA(x1, x2, x3, x4, x5) = U2_GA(x1, x2, x5)
PB_IN_AGAGA(x1, x2, x3, x4, x5) = PB_IN_AGAGA(x2, x4)
U4_AGAGA(x1, x2, x3, x4, x5, x6) = U4_AGAGA(x2, x4, x6)
DELETEC_IN_AGA(x1, x2, x3) = DELETEC_IN_AGA(x2)
U3_AGA(x1, x2, x3, x4, x5) = U3_AGA(x2, x3, x5)
U5_AGAGA(x1, x2, x3, x4, x5, x6) = U5_AGAGA(x1, x2, x3, x4, x6)

We have to consider all (P,R,Pi)-chains

(4) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

PERMUTEA_IN_GA(.(T21, T22), .(T21, T23)) → U1_GA(T21, T22, T23, permuteA_in_ga(T22, T23))
PERMUTEA_IN_GA(.(T21, T22), .(T21, T23)) → PERMUTEA_IN_GA(T22, T23)
PERMUTEA_IN_GA(.(T37, T38), .(T39, T40)) → U2_GA(T37, T38, T39, T40, pB_in_agaga(T39, T38, X42, T37, T40))
PERMUTEA_IN_GA(.(T37, T38), .(T39, T40)) → PB_IN_AGAGA(T39, T38, X42, T37, T40)
PB_IN_AGAGA(T39, T38, T45, T37, T46) → U4_AGAGA(T39, T38, T45, T37, T46, deleteC_in_aga(T39, T38, T45))
PB_IN_AGAGA(T39, T38, T45, T37, T46) → DELETEC_IN_AGA(T39, T38, T45)
DELETEC_IN_AGA(T70, .(T68, T69), .(T68, X75)) → U3_AGA(T70, T68, T69, X75, deleteC_in_aga(T70, T69, X75))
DELETEC_IN_AGA(T70, .(T68, T69), .(T68, X75)) → DELETEC_IN_AGA(T70, T69, X75)
U4_AGAGA(T39, T38, T45, T37, T46, deleteC_out_aga(T39, T38, T45)) → U5_AGAGA(T39, T38, T45, T37, T46, permuteA_in_ga(.(T37, T45), T46))
U4_AGAGA(T39, T38, T45, T37, T46, deleteC_out_aga(T39, T38, T45)) → PERMUTEA_IN_GA(.(T37, T45), T46)

The TRS R consists of the following rules:

permuteA_in_ga([], []) → permuteA_out_ga([], [])
permuteA_in_ga(.(T21, T22), .(T21, T23)) → U1_ga(T21, T22, T23, permuteA_in_ga(T22, T23))
permuteA_in_ga(.(T37, T38), .(T39, T40)) → U2_ga(T37, T38, T39, T40, pB_in_agaga(T39, T38, X42, T37, T40))
pB_in_agaga(T39, T38, T45, T37, T46) → U4_agaga(T39, T38, T45, T37, T46, deleteC_in_aga(T39, T38, T45))
deleteC_in_aga(T59, .(T59, T60), T60) → deleteC_out_aga(T59, .(T59, T60), T60)
deleteC_in_aga(T70, .(T68, T69), .(T68, X75)) → U3_aga(T70, T68, T69, X75, deleteC_in_aga(T70, T69, X75))
U3_aga(T70, T68, T69, X75, deleteC_out_aga(T70, T69, X75)) → deleteC_out_aga(T70, .(T68, T69), .(T68, X75))
U4_agaga(T39, T38, T45, T37, T46, deleteC_out_aga(T39, T38, T45)) → U5_agaga(T39, T38, T45, T37, T46, permuteA_in_ga(.(T37, T45), T46))
U5_agaga(T39, T38, T45, T37, T46, permuteA_out_ga(.(T37, T45), T46)) → pB_out_agaga(T39, T38, T45, T37, T46)
U2_ga(T37, T38, T39, T40, pB_out_agaga(T39, T38, X42, T37, T40)) → permuteA_out_ga(.(T37, T38), .(T39, T40))
U1_ga(T21, T22, T23, permuteA_out_ga(T22, T23)) → permuteA_out_ga(.(T21, T22), .(T21, T23))

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 5 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

DELETEC_IN_AGA(T70, .(T68, T69), .(T68, X75)) → DELETEC_IN_AGA(T70, T69, X75)

The TRS R consists of the following rules:

permuteA_in_ga([], []) → permuteA_out_ga([], [])
permuteA_in_ga(.(T21, T22), .(T21, T23)) → U1_ga(T21, T22, T23, permuteA_in_ga(T22, T23))
permuteA_in_ga(.(T37, T38), .(T39, T40)) → U2_ga(T37, T38, T39, T40, pB_in_agaga(T39, T38, X42, T37, T40))
pB_in_agaga(T39, T38, T45, T37, T46) → U4_agaga(T39, T38, T45, T37, T46, deleteC_in_aga(T39, T38, T45))
deleteC_in_aga(T59, .(T59, T60), T60) → deleteC_out_aga(T59, .(T59, T60), T60)
deleteC_in_aga(T70, .(T68, T69), .(T68, X75)) → U3_aga(T70, T68, T69, X75, deleteC_in_aga(T70, T69, X75))
U3_aga(T70, T68, T69, X75, deleteC_out_aga(T70, T69, X75)) → deleteC_out_aga(T70, .(T68, T69), .(T68, X75))
U4_agaga(T39, T38, T45, T37, T46, deleteC_out_aga(T39, T38, T45)) → U5_agaga(T39, T38, T45, T37, T46, permuteA_in_ga(.(T37, T45), T46))
U5_agaga(T39, T38, T45, T37, T46, permuteA_out_ga(.(T37, T45), T46)) → pB_out_agaga(T39, T38, T45, T37, T46)
U2_ga(T37, T38, T39, T40, pB_out_agaga(T39, T38, X42, T37, T40)) → permuteA_out_ga(.(T37, T38), .(T39, T40))
U1_ga(T21, T22, T23, permuteA_out_ga(T22, T23)) → permuteA_out_ga(.(T21, T22), .(T21, T23))

(8) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(9) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

DELETEC_IN_AGA(T70, .(T68, T69), .(T68, X75)) → DELETEC_IN_AGA(T70, T69, X75)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
DELETEC_IN_AGA(x1, x2, x3) = DELETEC_IN_AGA(x2)

We have to consider all (P,R,Pi)-chains

(10) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(11) Obligation:

Q DP problem:
The TRS P consists of the following rules:

DELETEC_IN_AGA(.(T68, T69)) → DELETEC_IN_AGA(T69)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(12) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

DELETEC_IN_AGA(.(T68, T69)) → DELETEC_IN_AGA(T69)
The graph contains the following edges 1 > 1

(13) YES

(14) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

PERMUTEA_IN_GA(.(T37, T38), .(T39, T40)) → PB_IN_AGAGA(T39, T38, X42, T37, T40)
PB_IN_AGAGA(T39, T38, T45, T37, T46) → U4_AGAGA(T39, T38, T45, T37, T46, deleteC_in_aga(T39, T38, T45))
U4_AGAGA(T39, T38, T45, T37, T46, deleteC_out_aga(T39, T38, T45)) → PERMUTEA_IN_GA(.(T37, T45), T46)
PERMUTEA_IN_GA(.(T21, T22), .(T21, T23)) → PERMUTEA_IN_GA(T22, T23)

The TRS R consists of the following rules:

permuteA_in_ga([], []) → permuteA_out_ga([], [])
permuteA_in_ga(.(T21, T22), .(T21, T23)) → U1_ga(T21, T22, T23, permuteA_in_ga(T22, T23))
permuteA_in_ga(.(T37, T38), .(T39, T40)) → U2_ga(T37, T38, T39, T40, pB_in_agaga(T39, T38, X42, T37, T40))
pB_in_agaga(T39, T38, T45, T37, T46) → U4_agaga(T39, T38, T45, T37, T46, deleteC_in_aga(T39, T38, T45))
deleteC_in_aga(T59, .(T59, T60), T60) → deleteC_out_aga(T59, .(T59, T60), T60)
deleteC_in_aga(T70, .(T68, T69), .(T68, X75)) → U3_aga(T70, T68, T69, X75, deleteC_in_aga(T70, T69, X75))
U3_aga(T70, T68, T69, X75, deleteC_out_aga(T70, T69, X75)) → deleteC_out_aga(T70, .(T68, T69), .(T68, X75))
U4_agaga(T39, T38, T45, T37, T46, deleteC_out_aga(T39, T38, T45)) → U5_agaga(T39, T38, T45, T37, T46, permuteA_in_ga(.(T37, T45), T46))
U5_agaga(T39, T38, T45, T37, T46, permuteA_out_ga(.(T37, T45), T46)) → pB_out_agaga(T39, T38, T45, T37, T46)
U2_ga(T37, T38, T39, T40, pB_out_agaga(T39, T38, X42, T37, T40)) → permuteA_out_ga(.(T37, T38), .(T39, T40))
U1_ga(T21, T22, T23, permuteA_out_ga(T22, T23)) → permuteA_out_ga(.(T21, T22), .(T21, T23))

The argument filtering Pi contains the following mapping:
permuteA_in_ga(x1, x2) = permuteA_in_ga(x1)
[] = []
permuteA_out_ga(x1, x2) = permuteA_out_ga(x1, x2)
.(x1, x2) = .(x1, x2)
U1_ga(x1, x2, x3, x4) = U1_ga(x1, x2, x4)
U2_ga(x1, x2, x3, x4, x5) = U2_ga(x1, x2, x5)
pB_in_agaga(x1, x2, x3, x4, x5) = pB_in_agaga(x2, x4)
U4_agaga(x1, x2, x3, x4, x5, x6) = U4_agaga(x2, x4, x6)
deleteC_in_aga(x1, x2, x3) = deleteC_in_aga(x2)
deleteC_out_aga(x1, x2, x3) = deleteC_out_aga(x1, x2, x3)
U3_aga(x1, x2, x3, x4, x5) = U3_aga(x2, x3, x5)
U5_agaga(x1, x2, x3, x4, x5, x6) = U5_agaga(x1, x2, x3, x4, x6)
pB_out_agaga(x1, x2, x3, x4, x5) = pB_out_agaga(x1, x2, x3, x4, x5)
PERMUTEA_IN_GA(x1, x2) = PERMUTEA_IN_GA(x1)
PB_IN_AGAGA(x1, x2, x3, x4, x5) = PB_IN_AGAGA(x2, x4)
U4_AGAGA(x1, x2, x3, x4, x5, x6) = U4_AGAGA(x2, x4, x6)

We have to consider all (P,R,Pi)-chains

(15) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(16) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

PERMUTEA_IN_GA(.(T37, T38), .(T39, T40)) → PB_IN_AGAGA(T39, T38, X42, T37, T40)
PB_IN_AGAGA(T39, T38, T45, T37, T46) → U4_AGAGA(T39, T38, T45, T37, T46, deleteC_in_aga(T39, T38, T45))
U4_AGAGA(T39, T38, T45, T37, T46, deleteC_out_aga(T39, T38, T45)) → PERMUTEA_IN_GA(.(T37, T45), T46)
PERMUTEA_IN_GA(.(T21, T22), .(T21, T23)) → PERMUTEA_IN_GA(T22, T23)

The TRS R consists of the following rules:

deleteC_in_aga(T59, .(T59, T60), T60) → deleteC_out_aga(T59, .(T59, T60), T60)
deleteC_in_aga(T70, .(T68, T69), .(T68, X75)) → U3_aga(T70, T68, T69, X75, deleteC_in_aga(T70, T69, X75))
U3_aga(T70, T68, T69, X75, deleteC_out_aga(T70, T69, X75)) → deleteC_out_aga(T70, .(T68, T69), .(T68, X75))

The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
deleteC_in_aga(x1, x2, x3) = deleteC_in_aga(x2)
deleteC_out_aga(x1, x2, x3) = deleteC_out_aga(x1, x2, x3)
U3_aga(x1, x2, x3, x4, x5) = U3_aga(x2, x3, x5)
PERMUTEA_IN_GA(x1, x2) = PERMUTEA_IN_GA(x1)
PB_IN_AGAGA(x1, x2, x3, x4, x5) = PB_IN_AGAGA(x2, x4)
U4_AGAGA(x1, x2, x3, x4, x5, x6) = U4_AGAGA(x2, x4, x6)

We have to consider all (P,R,Pi)-chains

(17) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(18) Obligation:

Q DP problem:
The TRS P consists of the following rules:

PERMUTEA_IN_GA(.(T37, T38)) → PB_IN_AGAGA(T38, T37)
PB_IN_AGAGA(T38, T37) → U4_AGAGA(T38, T37, deleteC_in_aga(T38))
U4_AGAGA(T38, T37, deleteC_out_aga(T39, T38, T45)) → PERMUTEA_IN_GA(.(T37, T45))
PERMUTEA_IN_GA(.(T21, T22)) → PERMUTEA_IN_GA(T22)

The TRS R consists of the following rules:

deleteC_in_aga(.(T59, T60)) → deleteC_out_aga(T59, .(T59, T60), T60)
deleteC_in_aga(.(T68, T69)) → U3_aga(T68, T69, deleteC_in_aga(T69))
U3_aga(T68, T69, deleteC_out_aga(T70, T69, X75)) → deleteC_out_aga(T70, .(T68, T69), .(T68, X75))

The set Q consists of the following terms:

deleteC_in_aga(x0)
U3_aga(x0, x1, x2)

We have to consider all (P,Q,R)-chains.

(19) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04,JAR06].

The following pairs can be oriented strictly and are deleted.

PB_IN_AGAGA(T38, T37) → U4_AGAGA(T38, T37, deleteC_in_aga(T38))
PERMUTEA_IN_GA(.(T21, T22)) → PERMUTEA_IN_GA(T22)

The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(.(x₁, x₂)) = 1 + x₂
POL(PB_IN_AGAGA(x₁, x₂)) = 1 + x₁
POL(PERMUTEA_IN_GA(x₁)) = x₁
POL(U3_aga(x₁, x₂, x₃)) = 1 + x₃
POL(U4_AGAGA(x₁, x₂, x₃)) = x₃
POL(deleteC_in_aga(x₁)) = x₁
POL(deleteC_out_aga(x₁, x₂, x₃)) = 1 + x₃

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

deleteC_in_aga(.(T59, T60)) → deleteC_out_aga(T59, .(T59, T60), T60)
deleteC_in_aga(.(T68, T69)) → U3_aga(T68, T69, deleteC_in_aga(T69))
U3_aga(T68, T69, deleteC_out_aga(T70, T69, X75)) → deleteC_out_aga(T70, .(T68, T69), .(T68, X75))

(20) Obligation:

Q DP problem:
The TRS P consists of the following rules:

PERMUTEA_IN_GA(.(T37, T38)) → PB_IN_AGAGA(T38, T37)
U4_AGAGA(T38, T37, deleteC_out_aga(T39, T38, T45)) → PERMUTEA_IN_GA(.(T37, T45))

The TRS R consists of the following rules:

deleteC_in_aga(.(T59, T60)) → deleteC_out_aga(T59, .(T59, T60), T60)
deleteC_in_aga(.(T68, T69)) → U3_aga(T68, T69, deleteC_in_aga(T69))
U3_aga(T68, T69, deleteC_out_aga(T70, T69, X75)) → deleteC_out_aga(T70, .(T68, T69), .(T68, X75))

The set Q consists of the following terms:

deleteC_in_aga(x0)
U3_aga(x0, x1, x2)

We have to consider all (P,Q,R)-chains.

(21) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.

(0) Obligation:

(1) PrologToPiTRSViaGraphTransformerProof (SOUND transformation)

(2) Obligation:

(3) DependencyPairsProof (EQUIVALENT transformation)

(4) Obligation:

(5) DependencyGraphProof (EQUIVALENT transformation)

(6) Complex Obligation (AND)

(7) Obligation:

(8) UsableRulesProof (EQUIVALENT transformation)

(9) Obligation:

(10) PiDPToQDPProof (SOUND transformation)

(11) Obligation:

(12) QDPSizeChangeProof (EQUIVALENT transformation)

(13) YES

(14) Obligation:

(15) UsableRulesProof (EQUIVALENT transformation)

(16) Obligation:

(17) PiDPToQDPProof (SOUND transformation)

(18) Obligation:

(19) QDPOrderProof (EQUIVALENT transformation)

(20) Obligation:

(21) DependencyGraphProof (EQUIVALENT transformation)

(22) TRUE