Left Termination proof of /tmp/tmpnNZnyE/som.pl

Left Termination of the query pattern
som3(g,a,a)
w.r.t. the given Prolog program could successfully be proven:

0 Prolog

↳1 PrologToPrologProblemTransformerProof (⇒, 89 ms)

↳2 Prolog

↳3 PrologToPiTRSProof (⇒, 0 ms)

↳4 PiTRS

↳5 DependencyPairsProof (⇔, 13 ms)

↳6 PiDP

↳7 DependencyGraphProof (⇔, 0 ms)

↳8 PiDP

↳9 UsableRulesProof (⇔, 0 ms)

↳10 PiDP

↳11 PiDPToQDPProof (⇒, 0 ms)

↳12 QDP

↳13 QDPSizeChangeProof (⇔, 0 ms)

↳14 YES

(0) Obligation:

Clauses:

som3([], Bs, Bs).
som3(As, [], As).
som3(.(A, As), .(B, Bs), .(+(A, B), Cs)) :- som3(As, Bs, Cs).
som4_1(As, Bs, Cs, Ds) :- ','(som3(As, Bs, Es), som3(Es, Cs, Ds)).
som4_2(As, Bs, Cs, Ds) :- ','(som3(Es, Cs, Ds), som3(As, Bs, Es)).

Query: som3(g,a,a)

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph ICLP10.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: som3(T1, T2, T3)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: som3(T1, T2, T3), SCOPE: 1, CLAUSE: 0 | som3(T1, T2, T3), SCOPE: 1, CLAUSE: 1 | som3(T1, T2, T3), SCOPE: 1, CLAUSE: 2\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: true | som3([], T2, T3), SCOPE: 1, CLAUSE: 1 | som3([], T2, T3), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: som3(T1, T2, T3), SCOPE: 1, CLAUSE: 1 | som3(T1, T2, T3), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nX2 is free\nsom3(T1, T2, T3) !=? som3([], X2, X2)", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: som3([], T2, T3), SCOPE: 1, CLAUSE: 1 | som3([], T2, T3), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: true | som3([], T2, T3), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: som3([], T2, T3), SCOPE: 1, CLAUSE: 2\n\nX4 is free\nsom3([], T2, T3) !=? som3(X4, [], X4)", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: som3([], T2, T3), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: true | som3(T7, T2, T3), SCOPE: 1, CLAUSE: 2\n\nT7 is ground\nX2 is free\nsom3(T7, T2, T3) !=? som3([], X2, X2)", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: som3(T1, T2, T3), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nX2 is free\nX16 is free\nsom3(T1, T2, T3) !=? som3([], X2, X2)\nsom3(T1, T2, T3) !=? som3(X16, [], X16)", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: som3(T7, T2, T3), SCOPE: 1, CLAUSE: 2\n\nT7 is ground\nX2 is free\nsom3(T7, T2, T3) !=? som3([], X2, X2)", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: som3(T14, T18, T19)\n\nT14 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: som3(T14, T18, T19), SCOPE: 2, CLAUSE: 0 | som3(T14, T18, T19), SCOPE: 2, CLAUSE: 1 | som3(T14, T18, T19), SCOPE: 2, CLAUSE: 2\n\nT14 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: som3(T14, T18, T19), SCOPE: 2, CLAUSE: 0\n\nT14 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: som3(T14, T18, T19), SCOPE: 2, CLAUSE: 1 | som3(T14, T18, T19), SCOPE: 2, CLAUSE: 2\n\nT14 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: som3(T14, T18, T19), SCOPE: 2, CLAUSE: 1\n\nT14 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: som3(T14, T18, T19), SCOPE: 2, CLAUSE: 2\n\nT14 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: som3(T41, T45, T46)\n\nT41 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: som3(T55, T59, T60)\n\nT54 is ground\nT55 is ground\nX16 is free\nsom3(.(T54, T55), T2, T3) !=? som3(X16, [], X16)", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: som3(T55, T59, T60), SCOPE: 3, CLAUSE: 0 | som3(T55, T59, T60), SCOPE: 3, CLAUSE: 1 | som3(T55, T59, T60), SCOPE: 3, CLAUSE: 2\n\nT54 is ground\nT55 is ground\nX16 is free\nsom3(.(T54, T55), T2, T3) !=? som3(X16, [], X16)", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: som3(T55, T59, T60), SCOPE: 3, CLAUSE: 0\n\nT54 is ground\nT55 is ground\nX16 is free\nsom3(.(T54, T55), T2, T3) !=? som3(X16, [], X16)", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: som3(T55, T59, T60), SCOPE: 3, CLAUSE: 1 | som3(T55, T59, T60), SCOPE: 3, CLAUSE: 2\n\nT54 is ground\nT55 is ground\nX16 is free\nsom3(.(T54, T55), T2, T3) !=? som3(X16, [], X16)", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: som3(T55, T59, T60), SCOPE: 3, CLAUSE: 1\n\nT54 is ground\nT55 is ground\nX16 is free\nsom3(.(T54, T55), T2, T3) !=? som3(X16, [], X16)", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: som3(T55, T59, T60), SCOPE: 3, CLAUSE: 2\n\nT54 is ground\nT55 is ground\nX16 is free\nsom3(.(T54, T55), T2, T3) !=? som3(X16, [], X16)", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: som3(T82, T86, T87)\n\nT54 is ground\nT81 is ground\nT82 is ground\nX16 is free\nsom3(.(T54, .(T81, T82)), T2, T3) !=? som3(X16, [], X16)", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 44 [arrowhead = none ];
44 -> 2;

45 [label="EVAL with clause\nsom3([], X2, X2).\nand substitutionT1 -> [],\nT2 -> T5,\nX2 -> T5,\nT3 -> T5", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 45 [arrowhead = none ];
45 -> 3;

46 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
2 -> 46 [arrowhead = none ];
46 -> 4;

47 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 47 [arrowhead = none ];
47 -> 5;

48 [label="EVAL with clause\nsom3(X16, [], X16).\nand substitutionT1 -> T7,\nX16 -> T7,\nT2 -> [],\nT3 -> T7", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 48 [arrowhead = none ];
48 -> 11;

49 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 49 [arrowhead = none ];
49 -> 12;

50 [label="EVAL with clause\nsom3(X4, [], X4).\nand substitutionX4 -> [],\nT2 -> [],\nT3 -> []", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 50 [arrowhead = none ];
50 -> 6;

51 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 51 [arrowhead = none ];
51 -> 7;

52 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 52 [arrowhead = none ];
52 -> 8;

53 [label="BACKTRACK\nfor clause: som3(.(A, As), .(B, Bs), .(+(A, B), Cs)) :- som3(As, Bs, Cs)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 53 [arrowhead = none ];
53 -> 10;

54 [label="BACKTRACK\nfor clause: som3(.(A, As), .(B, Bs), .(+(A, B), Cs)) :- som3(As, Bs, Cs)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
8 -> 54 [arrowhead = none ];
54 -> 9;

55 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 55 [arrowhead = none ];
55 -> 13;

56 [label="EVAL with clause\nsom3(.(X59, X60), .(X61, X62), .(+(X59, X61), X63)) :- som3(X60, X62, X63).\nand substitutionX59 -> T54,\nX60 -> T55,\nT1 -> .(T54, T55),\nX61 -> T56,\nX62 -> T59,\nT2 -> .(T56, T59),\nX63 -> T60,\nT3 -> .(+(T54, T56), T60),\nT57 -> T59,\nT58 -> T60", fontsize=14, style = filled, fillcolor = lightblue];
12 -> 56 [arrowhead = none ];
56 -> 29;

57 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
12 -> 57 [arrowhead = none ];
57 -> 30;

58 [label="EVAL with clause\nsom3(.(X22, X23), .(X24, X25), .(+(X22, X24), X26)) :- som3(X23, X25, X26).\nand substitutionX22 -> T13,\nX23 -> T14,\nT7 -> .(T13, T14),\nX24 -> T15,\nX25 -> T18,\nT2 -> .(T15, T18),\nX26 -> T19,\nT3 -> .(+(T13, T15), T19),\nT16 -> T18,\nT17 -> T19", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 58 [arrowhead = none ];
58 -> 14;

59 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
13 -> 59 [arrowhead = none ];
59 -> 15;

60 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
14 -> 60 [arrowhead = none ];
60 -> 16;

61 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
16 -> 61 [arrowhead = none ];
61 -> 17;

62 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
16 -> 62 [arrowhead = none ];
62 -> 18;

63 [label="EVAL with clause\nsom3([], X31, X31).\nand substitutionT14 -> [],\nT18 -> T24,\nX31 -> T24,\nT19 -> T24", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 63 [arrowhead = none ];
63 -> 19;

64 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
17 -> 64 [arrowhead = none ];
64 -> 20;

65 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
18 -> 65 [arrowhead = none ];
65 -> 22;

66 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
18 -> 66 [arrowhead = none ];
66 -> 23;

67 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
19 -> 67 [arrowhead = none ];
67 -> 21;

68 [label="EVAL with clause\nsom3(X36, [], X36).\nand substitutionT14 -> T29,\nX36 -> T29,\nT18 -> [],\nT19 -> T29", fontsize=14, style = filled, fillcolor = lightblue];
22 -> 68 [arrowhead = none ];
68 -> 24;

69 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
22 -> 69 [arrowhead = none ];
69 -> 25;

70 [label="EVAL with clause\nsom3(.(X47, X48), .(X49, X50), .(+(X47, X49), X51)) :- som3(X48, X50, X51).\nand substitutionX47 -> T40,\nX48 -> T41,\nT14 -> .(T40, T41),\nX49 -> T42,\nX50 -> T45,\nT18 -> .(T42, T45),\nX51 -> T46,\nT19 -> .(+(T40, T42), T46),\nT43 -> T45,\nT44 -> T46", fontsize=14, style = filled, fillcolor = lightblue];
23 -> 70 [arrowhead = none ];
70 -> 27;

71 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
23 -> 71 [arrowhead = none ];
71 -> 28;

72 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
24 -> 72 [arrowhead = none ];
72 -> 26;

73 [label="INSTANCE with matching:\nT1 -> T41\nT2 -> T45\nT3 -> T46", fontsize=14, style = filled, fillcolor = lightgrey];
27 -> 73 [arrowhead = none , style=dashed];
73 -> 1[style=dashed];

74 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
29 -> 74 [arrowhead = none ];
74 -> 31;

75 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
31 -> 75 [arrowhead = none ];
75 -> 32;

76 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
31 -> 76 [arrowhead = none ];
76 -> 33;

77 [label="EVAL with clause\nsom3([], X68, X68).\nand substitutionT55 -> [],\nT59 -> T65,\nX68 -> T65,\nT60 -> T65", fontsize=14, style = filled, fillcolor = lightblue];
32 -> 77 [arrowhead = none ];
77 -> 34;

78 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
32 -> 78 [arrowhead = none ];
78 -> 35;

79 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
33 -> 79 [arrowhead = none ];
79 -> 37;

80 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
33 -> 80 [arrowhead = none ];
80 -> 38;

81 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
34 -> 81 [arrowhead = none ];
81 -> 36;

82 [label="EVAL with clause\nsom3(X73, [], X73).\nand substitutionT55 -> T70,\nX73 -> T70,\nT59 -> [],\nT60 -> T70", fontsize=14, style = filled, fillcolor = lightblue];
37 -> 82 [arrowhead = none ];
82 -> 39;

83 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
37 -> 83 [arrowhead = none ];
83 -> 40;

84 [label="EVAL with clause\nsom3(.(X84, X85), .(X86, X87), .(+(X84, X86), X88)) :- som3(X85, X87, X88).\nand substitutionX84 -> T81,\nX85 -> T82,\nT55 -> .(T81, T82),\nX86 -> T83,\nX87 -> T86,\nT59 -> .(T83, T86),\nX88 -> T87,\nT60 -> .(+(T81, T83), T87),\nT84 -> T86,\nT85 -> T87", fontsize=14, style = filled, fillcolor = lightblue];
38 -> 84 [arrowhead = none ];
84 -> 42;

85 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
38 -> 85 [arrowhead = none ];
85 -> 43;

86 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
39 -> 86 [arrowhead = none ];
86 -> 41;

87 [label="INSTANCE with matching:\nT1 -> T82\nT2 -> T86\nT3 -> T87", fontsize=14, style = filled, fillcolor = lightgrey];
42 -> 87 [arrowhead = none , style=dashed];
87 -> 1[style=dashed];

}

(2) Obligation:

Clauses:

som3A([], T5, T5).
som3A([], [], []).
som3A(T7, [], T7).
som3A(.(T13, []), .(T15, T24), .(+(T13, T15), T24)).
som3A(.(T13, T29), .(T15, []), .(+(T13, T15), T29)).
som3A(.(T13, .(T40, T41)), .(T15, .(T42, T45)), .(+(T13, T15), .(+(T40, T42), T46))) :- som3A(T41, T45, T46).
som3A(.(T54, []), .(T56, T65), .(+(T54, T56), T65)).
som3A(.(T54, T70), .(T56, []), .(+(T54, T56), T70)).
som3A(.(T54, .(T81, T82)), .(T56, .(T83, T86)), .(+(T54, T56), .(+(T81, T83), T87))) :- som3A(T82, T86, T87).

Query: som3A(g,a,a)

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
som3A_in: (b,f,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

som3A_in_gaa([], T5, T5) → som3A_out_gaa([], T5, T5)
som3A_in_gaa([], [], []) → som3A_out_gaa([], [], [])
som3A_in_gaa(T7, [], T7) → som3A_out_gaa(T7, [], T7)
som3A_in_gaa(.(T13, []), .(T15, T24), .(+(T13, T15), T24)) → som3A_out_gaa(.(T13, []), .(T15, T24), .(+(T13, T15), T24))
som3A_in_gaa(.(T13, T29), .(T15, []), .(+(T13, T15), T29)) → som3A_out_gaa(.(T13, T29), .(T15, []), .(+(T13, T15), T29))
som3A_in_gaa(.(T13, .(T40, T41)), .(T15, .(T42, T45)), .(+(T13, T15), .(+(T40, T42), T46))) → U1_gaa(T13, T40, T41, T15, T42, T45, T46, som3A_in_gaa(T41, T45, T46))
som3A_in_gaa(.(T54, .(T81, T82)), .(T56, .(T83, T86)), .(+(T54, T56), .(+(T81, T83), T87))) → U2_gaa(T54, T81, T82, T56, T83, T86, T87, som3A_in_gaa(T82, T86, T87))
U2_gaa(T54, T81, T82, T56, T83, T86, T87, som3A_out_gaa(T82, T86, T87)) → som3A_out_gaa(.(T54, .(T81, T82)), .(T56, .(T83, T86)), .(+(T54, T56), .(+(T81, T83), T87)))
U1_gaa(T13, T40, T41, T15, T42, T45, T46, som3A_out_gaa(T41, T45, T46)) → som3A_out_gaa(.(T13, .(T40, T41)), .(T15, .(T42, T45)), .(+(T13, T15), .(+(T40, T42), T46)))

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

som3A_in_gaa([], T5, T5) → som3A_out_gaa([], T5, T5)
som3A_in_gaa([], [], []) → som3A_out_gaa([], [], [])
som3A_in_gaa(T7, [], T7) → som3A_out_gaa(T7, [], T7)
som3A_in_gaa(.(T13, []), .(T15, T24), .(+(T13, T15), T24)) → som3A_out_gaa(.(T13, []), .(T15, T24), .(+(T13, T15), T24))
som3A_in_gaa(.(T13, T29), .(T15, []), .(+(T13, T15), T29)) → som3A_out_gaa(.(T13, T29), .(T15, []), .(+(T13, T15), T29))
som3A_in_gaa(.(T13, .(T40, T41)), .(T15, .(T42, T45)), .(+(T13, T15), .(+(T40, T42), T46))) → U1_gaa(T13, T40, T41, T15, T42, T45, T46, som3A_in_gaa(T41, T45, T46))
som3A_in_gaa(.(T54, .(T81, T82)), .(T56, .(T83, T86)), .(+(T54, T56), .(+(T81, T83), T87))) → U2_gaa(T54, T81, T82, T56, T83, T86, T87, som3A_in_gaa(T82, T86, T87))
U2_gaa(T54, T81, T82, T56, T83, T86, T87, som3A_out_gaa(T82, T86, T87)) → som3A_out_gaa(.(T54, .(T81, T82)), .(T56, .(T83, T86)), .(+(T54, T56), .(+(T81, T83), T87)))
U1_gaa(T13, T40, T41, T15, T42, T45, T46, som3A_out_gaa(T41, T45, T46)) → som3A_out_gaa(.(T13, .(T40, T41)), .(T15, .(T42, T45)), .(+(T13, T15), .(+(T40, T42), T46)))

(5) 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:

SOM3A_IN_GAA(.(T13, .(T40, T41)), .(T15, .(T42, T45)), .(+(T13, T15), .(+(T40, T42), T46))) → U1_GAA(T13, T40, T41, T15, T42, T45, T46, som3A_in_gaa(T41, T45, T46))
SOM3A_IN_GAA(.(T13, .(T40, T41)), .(T15, .(T42, T45)), .(+(T13, T15), .(+(T40, T42), T46))) → SOM3A_IN_GAA(T41, T45, T46)
SOM3A_IN_GAA(.(T54, .(T81, T82)), .(T56, .(T83, T86)), .(+(T54, T56), .(+(T81, T83), T87))) → U2_GAA(T54, T81, T82, T56, T83, T86, T87, som3A_in_gaa(T82, T86, T87))

The TRS R consists of the following rules:

som3A_in_gaa([], T5, T5) → som3A_out_gaa([], T5, T5)
som3A_in_gaa([], [], []) → som3A_out_gaa([], [], [])
som3A_in_gaa(T7, [], T7) → som3A_out_gaa(T7, [], T7)
som3A_in_gaa(.(T13, []), .(T15, T24), .(+(T13, T15), T24)) → som3A_out_gaa(.(T13, []), .(T15, T24), .(+(T13, T15), T24))
som3A_in_gaa(.(T13, T29), .(T15, []), .(+(T13, T15), T29)) → som3A_out_gaa(.(T13, T29), .(T15, []), .(+(T13, T15), T29))
som3A_in_gaa(.(T13, .(T40, T41)), .(T15, .(T42, T45)), .(+(T13, T15), .(+(T40, T42), T46))) → U1_gaa(T13, T40, T41, T15, T42, T45, T46, som3A_in_gaa(T41, T45, T46))
som3A_in_gaa(.(T54, .(T81, T82)), .(T56, .(T83, T86)), .(+(T54, T56), .(+(T81, T83), T87))) → U2_gaa(T54, T81, T82, T56, T83, T86, T87, som3A_in_gaa(T82, T86, T87))
U2_gaa(T54, T81, T82, T56, T83, T86, T87, som3A_out_gaa(T82, T86, T87)) → som3A_out_gaa(.(T54, .(T81, T82)), .(T56, .(T83, T86)), .(+(T54, T56), .(+(T81, T83), T87)))
U1_gaa(T13, T40, T41, T15, T42, T45, T46, som3A_out_gaa(T41, T45, T46)) → som3A_out_gaa(.(T13, .(T40, T41)), .(T15, .(T42, T45)), .(+(T13, T15), .(+(T40, T42), T46)))

The argument filtering Pi contains the following mapping:
som3A_in_gaa(x1, x2, x3) = som3A_in_gaa(x1)
[] = []
som3A_out_gaa(x1, x2, x3) = som3A_out_gaa
.(x1, x2) = .(x1, x2)
U1_gaa(x1, x2, x3, x4, x5, x6, x7, x8) = U1_gaa(x8)
U2_gaa(x1, x2, x3, x4, x5, x6, x7, x8) = U2_gaa(x8)
SOM3A_IN_GAA(x1, x2, x3) = SOM3A_IN_GAA(x1)
U1_GAA(x1, x2, x3, x4, x5, x6, x7, x8) = U1_GAA(x8)
U2_GAA(x1, x2, x3, x4, x5, x6, x7, x8) = U2_GAA(x8)

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

(6) Obligation:

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

SOM3A_IN_GAA(.(T13, .(T40, T41)), .(T15, .(T42, T45)), .(+(T13, T15), .(+(T40, T42), T46))) → U1_GAA(T13, T40, T41, T15, T42, T45, T46, som3A_in_gaa(T41, T45, T46))
SOM3A_IN_GAA(.(T13, .(T40, T41)), .(T15, .(T42, T45)), .(+(T13, T15), .(+(T40, T42), T46))) → SOM3A_IN_GAA(T41, T45, T46)
SOM3A_IN_GAA(.(T54, .(T81, T82)), .(T56, .(T83, T86)), .(+(T54, T56), .(+(T81, T83), T87))) → U2_GAA(T54, T81, T82, T56, T83, T86, T87, som3A_in_gaa(T82, T86, T87))

The TRS R consists of the following rules:

som3A_in_gaa([], T5, T5) → som3A_out_gaa([], T5, T5)
som3A_in_gaa([], [], []) → som3A_out_gaa([], [], [])
som3A_in_gaa(T7, [], T7) → som3A_out_gaa(T7, [], T7)
som3A_in_gaa(.(T13, []), .(T15, T24), .(+(T13, T15), T24)) → som3A_out_gaa(.(T13, []), .(T15, T24), .(+(T13, T15), T24))
som3A_in_gaa(.(T13, T29), .(T15, []), .(+(T13, T15), T29)) → som3A_out_gaa(.(T13, T29), .(T15, []), .(+(T13, T15), T29))
som3A_in_gaa(.(T13, .(T40, T41)), .(T15, .(T42, T45)), .(+(T13, T15), .(+(T40, T42), T46))) → U1_gaa(T13, T40, T41, T15, T42, T45, T46, som3A_in_gaa(T41, T45, T46))
som3A_in_gaa(.(T54, .(T81, T82)), .(T56, .(T83, T86)), .(+(T54, T56), .(+(T81, T83), T87))) → U2_gaa(T54, T81, T82, T56, T83, T86, T87, som3A_in_gaa(T82, T86, T87))
U2_gaa(T54, T81, T82, T56, T83, T86, T87, som3A_out_gaa(T82, T86, T87)) → som3A_out_gaa(.(T54, .(T81, T82)), .(T56, .(T83, T86)), .(+(T54, T56), .(+(T81, T83), T87)))
U1_gaa(T13, T40, T41, T15, T42, T45, T46, som3A_out_gaa(T41, T45, T46)) → som3A_out_gaa(.(T13, .(T40, T41)), .(T15, .(T42, T45)), .(+(T13, T15), .(+(T40, T42), T46)))

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 2 less nodes.

(8) Obligation:

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

SOM3A_IN_GAA(.(T13, .(T40, T41)), .(T15, .(T42, T45)), .(+(T13, T15), .(+(T40, T42), T46))) → SOM3A_IN_GAA(T41, T45, T46)

The TRS R consists of the following rules:

som3A_in_gaa([], T5, T5) → som3A_out_gaa([], T5, T5)
som3A_in_gaa([], [], []) → som3A_out_gaa([], [], [])
som3A_in_gaa(T7, [], T7) → som3A_out_gaa(T7, [], T7)
som3A_in_gaa(.(T13, []), .(T15, T24), .(+(T13, T15), T24)) → som3A_out_gaa(.(T13, []), .(T15, T24), .(+(T13, T15), T24))
som3A_in_gaa(.(T13, T29), .(T15, []), .(+(T13, T15), T29)) → som3A_out_gaa(.(T13, T29), .(T15, []), .(+(T13, T15), T29))
som3A_in_gaa(.(T13, .(T40, T41)), .(T15, .(T42, T45)), .(+(T13, T15), .(+(T40, T42), T46))) → U1_gaa(T13, T40, T41, T15, T42, T45, T46, som3A_in_gaa(T41, T45, T46))
som3A_in_gaa(.(T54, .(T81, T82)), .(T56, .(T83, T86)), .(+(T54, T56), .(+(T81, T83), T87))) → U2_gaa(T54, T81, T82, T56, T83, T86, T87, som3A_in_gaa(T82, T86, T87))
U2_gaa(T54, T81, T82, T56, T83, T86, T87, som3A_out_gaa(T82, T86, T87)) → som3A_out_gaa(.(T54, .(T81, T82)), .(T56, .(T83, T86)), .(+(T54, T56), .(+(T81, T83), T87)))
U1_gaa(T13, T40, T41, T15, T42, T45, T46, som3A_out_gaa(T41, T45, T46)) → som3A_out_gaa(.(T13, .(T40, T41)), .(T15, .(T42, T45)), .(+(T13, T15), .(+(T40, T42), T46)))

(9) UsableRulesProof (EQUIVALENT transformation)

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

(10) Obligation:

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

SOM3A_IN_GAA(.(T13, .(T40, T41)), .(T15, .(T42, T45)), .(+(T13, T15), .(+(T40, T42), T46))) → SOM3A_IN_GAA(T41, T45, T46)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
SOM3A_IN_GAA(x1, x2, x3) = SOM3A_IN_GAA(x1)

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

(11) PiDPToQDPProof (SOUND transformation)

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

(12) Obligation:

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

SOM3A_IN_GAA(.(T13, .(T40, T41))) → SOM3A_IN_GAA(T41)

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

(13) 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:

SOM3A_IN_GAA(.(T13, .(T40, T41))) → SOM3A_IN_GAA(T41)
The graph contains the following edges 1 > 1

(0) Obligation:

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

(2) Obligation:

(3) PrologToPiTRSProof (SOUND transformation)

(4) Obligation:

(5) DependencyPairsProof (EQUIVALENT transformation)

(6) Obligation:

(7) DependencyGraphProof (EQUIVALENT transformation)

(8) Obligation:

(9) UsableRulesProof (EQUIVALENT transformation)

(10) Obligation:

(11) PiDPToQDPProof (SOUND transformation)

(12) Obligation:

(13) QDPSizeChangeProof (EQUIVALENT transformation)

(14) YES