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

(0) Obligation:

Clauses:

goal(Name, Code) :- ','(eq(Name, .(First, Others)), ','(reduce(Others, s(0), First, Reduced), eq(Code, Reduced))).
reduce(X1, s(s(s(s(0)))), X2, []) :- !.
reduce([], X3, X4, []) :- !.
reduce(.(Current, Others), Count, Current, Code) :- ','(vowel_h_w_y(Current), ','(!, reduce(Others, Count, Current, Code))).
reduce(.(Letter, Others), Count, Letter, Code) :- ','(!, reduce(Others, Count, Letter, Code)).
reduce(.(Current, Others), Count, X5, .(Current, Code)) :- reduce(Others, s(Count), Current, Code).
vowel_h_w_y(97).
vowel_h_w_y(101).
vowel_h_w_y(105).
vowel_h_w_y(111).
vowel_h_w_y(117).
vowel_h_w_y(104).
vowel_h_w_y(119).
vowel_h_w_y(121).
eq(X, X).

Query: goal(g,a)

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph ICLP10.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: goal(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: goal(T1, T2), SCOPE: 1, CLAUSE: 0\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: ','(eq(T5, .(X10, X11)), ','(reduce(X11, s(0), X10, X12), eq(T7, X12)))\n\nT5 is ground\nX10 is free\nX11 is free\nX12 is free", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: ','(eq(T5, .(X10, X11)), ','(reduce(X11, s(0), X10, X12), eq(T7, X12))), SCOPE: 2, CLAUSE: 14\n\nT5 is ground\nX10 is free\nX11 is free\nX12 is free", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: ','(reduce(T16, s(0), T15, X12), eq(T7, X12))\n\nT15 is ground\nT16 is ground\nX12 is free", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: reduce(T16, s(0), T15, X12)\n\nT15 is ground\nT16 is ground\nX12 is free", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: eq(T7, T18)\n\nT18 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: reduce(T16, s(0), T15, X12), SCOPE: 3, CLAUSE: 1 | reduce(T16, s(0), T15, X12), SCOPE: 3, CLAUSE: 2 | reduce(T16, s(0), T15, X12), SCOPE: 3, CLAUSE: 3 | reduce(T16, s(0), T15, X12), SCOPE: 3, CLAUSE: 4 | reduce(T16, s(0), T15, X12), SCOPE: 3, CLAUSE: 5\n\nT15 is ground\nT16 is ground\nX12 is free", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: reduce(T16, s(0), T15, X12), SCOPE: 3, CLAUSE: 2 | reduce(T16, s(0), T15, X12), SCOPE: 3, CLAUSE: 3 | reduce(T16, s(0), T15, X12), SCOPE: 3, CLAUSE: 4 | reduce(T16, s(0), T15, X12), SCOPE: 3, CLAUSE: 5\n\nT15 is ground\nT16 is ground\nX12 is free", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: !_3 | reduce([], s(0), T23, X12), SCOPE: 3, CLAUSE: 3 | reduce([], s(0), T23, X12), SCOPE: 3, CLAUSE: 4 | reduce([], s(0), T23, X12), SCOPE: 3, CLAUSE: 5\n\nT23 is ground\nX12 is free", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: reduce(T16, s(0), T15, X12), SCOPE: 3, CLAUSE: 3 | reduce(T16, s(0), T15, X12), SCOPE: 3, CLAUSE: 4 | reduce(T16, s(0), T15, X12), SCOPE: 3, CLAUSE: 5\n\nT15 is ground\nT16 is ground\nX12 is free\nX26 is free\nX27 is free\nreduce(T16, s(0), T15, X12) !=? reduce([], X26, X27, [])", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: reduce(T16, s(0), T15, X12), SCOPE: 3, CLAUSE: 3\n\nT15 is ground\nT16 is ground\nX12 is free\nX26 is free\nX27 is free\nreduce(T16, s(0), T15, X12) !=? reduce([], X26, X27, [])", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: reduce(T16, s(0), T15, X12), SCOPE: 3, CLAUSE: 4 | reduce(T16, s(0), T15, X12), SCOPE: 3, CLAUSE: 5\n\nT15 is ground\nT16 is ground\nX12 is free\nX26 is free\nX27 is free\nreduce(T16, s(0), T15, X12) !=? reduce([], X26, X27, [])", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: ','(vowel_h_w_y(T32), ','(!_3, reduce(T33, s(0), T32, X52)))\n\nT32 is ground\nT33 is ground\nX52 is free", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: ','(vowel_h_w_y(T32), ','(!_3, reduce(T33, s(0), T32, X52))), SCOPE: 4, CLAUSE: 6 | ','(vowel_h_w_y(T32), ','(!_3, reduce(T33, s(0), T32, X52))), SCOPE: 4, CLAUSE: 7 | ','(vowel_h_w_y(T32), ','(!_3, reduce(T33, s(0), T32, X52))), SCOPE: 4, CLAUSE: 8 | ','(vowel_h_w_y(T32), ','(!_3, reduce(T33, s(0), T32, X52))), SCOPE: 4, CLAUSE: 9 | ','(vowel_h_w_y(T32), ','(!_3, reduce(T33, s(0), T32, X52))), SCOPE: 4, CLAUSE: 10 | ','(vowel_h_w_y(T32), ','(!_3, reduce(T33, s(0), T32, X52))), SCOPE: 4, CLAUSE: 11 | ','(vowel_h_w_y(T32), ','(!_3, reduce(T33, s(0), T32, X52))), SCOPE: 4, CLAUSE: 12 | ','(vowel_h_w_y(T32), ','(!_3, reduce(T33, s(0), T32, X52))), SCOPE: 4, CLAUSE: 13\n\nT32 is ground\nT33 is ground\nX52 is free", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: ','(!_3, reduce(T33, s(0), 97, X52)) | ','(vowel_h_w_y(97), ','(!_3, reduce(T33, s(0), 97, X52))), SCOPE: 4, CLAUSE: 7 | ','(vowel_h_w_y(97), ','(!_3, reduce(T33, s(0), 97, X52))), SCOPE: 4, CLAUSE: 8 | ','(vowel_h_w_y(97), ','(!_3, reduce(T33, s(0), 97, X52))), SCOPE: 4, CLAUSE: 9 | ','(vowel_h_w_y(97), ','(!_3, reduce(T33, s(0), 97, X52))), SCOPE: 4, CLAUSE: 10 | ','(vowel_h_w_y(97), ','(!_3, reduce(T33, s(0), 97, X52))), SCOPE: 4, CLAUSE: 11 | ','(vowel_h_w_y(97), ','(!_3, reduce(T33, s(0), 97, X52))), SCOPE: 4, CLAUSE: 12 | ','(vowel_h_w_y(97), ','(!_3, reduce(T33, s(0), 97, X52))), SCOPE: 4, CLAUSE: 13\n\nT33 is ground\nX52 is free", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: ','(vowel_h_w_y(T32), ','(!_3, reduce(T33, s(0), T32, X52))), SCOPE: 4, CLAUSE: 7 | ','(vowel_h_w_y(T32), ','(!_3, reduce(T33, s(0), T32, X52))), SCOPE: 4, CLAUSE: 8 | ','(vowel_h_w_y(T32), ','(!_3, reduce(T33, s(0), T32, X52))), SCOPE: 4, CLAUSE: 9 | ','(vowel_h_w_y(T32), ','(!_3, reduce(T33, s(0), T32, X52))), SCOPE: 4, CLAUSE: 10 | ','(vowel_h_w_y(T32), ','(!_3, reduce(T33, s(0), T32, X52))), SCOPE: 4, CLAUSE: 11 | ','(vowel_h_w_y(T32), ','(!_3, reduce(T33, s(0), T32, X52))), SCOPE: 4, CLAUSE: 12 | ','(vowel_h_w_y(T32), ','(!_3, reduce(T33, s(0), T32, X52))), SCOPE: 4, CLAUSE: 13\n\nT32 is ground\nT33 is ground\nX52 is free\nvowel_h_w_y(T32) !=? vowel_h_w_y(97)", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: reduce(T33, s(0), 97, X52)\n\nT33 is ground\nX52 is free", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: ','(!_3, reduce(T33, s(0), 101, X52)) | ','(vowel_h_w_y(101), ','(!_3, reduce(T33, s(0), 101, X52))), SCOPE: 4, CLAUSE: 8 | ','(vowel_h_w_y(101), ','(!_3, reduce(T33, s(0), 101, X52))), SCOPE: 4, CLAUSE: 9 | ','(vowel_h_w_y(101), ','(!_3, reduce(T33, s(0), 101, X52))), SCOPE: 4, CLAUSE: 10 | ','(vowel_h_w_y(101), ','(!_3, reduce(T33, s(0), 101, X52))), SCOPE: 4, CLAUSE: 11 | ','(vowel_h_w_y(101), ','(!_3, reduce(T33, s(0), 101, X52))), SCOPE: 4, CLAUSE: 12 | ','(vowel_h_w_y(101), ','(!_3, reduce(T33, s(0), 101, X52))), SCOPE: 4, CLAUSE: 13\n\nT33 is ground\nX52 is free", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: ','(vowel_h_w_y(T32), ','(!_3, reduce(T33, s(0), T32, X52))), SCOPE: 4, CLAUSE: 8 | ','(vowel_h_w_y(T32), ','(!_3, reduce(T33, s(0), T32, X52))), SCOPE: 4, CLAUSE: 9 | ','(vowel_h_w_y(T32), ','(!_3, reduce(T33, s(0), T32, X52))), SCOPE: 4, CLAUSE: 10 | ','(vowel_h_w_y(T32), ','(!_3, reduce(T33, s(0), T32, X52))), SCOPE: 4, CLAUSE: 11 | ','(vowel_h_w_y(T32), ','(!_3, reduce(T33, s(0), T32, X52))), SCOPE: 4, CLAUSE: 12 | ','(vowel_h_w_y(T32), ','(!_3, reduce(T33, s(0), T32, X52))), SCOPE: 4, CLAUSE: 13\n\nT32 is ground\nT33 is ground\nX52 is free\nvowel_h_w_y(T32) !=? vowel_h_w_y(97)\nvowel_h_w_y(T32) !=? vowel_h_w_y(101)", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: reduce(T33, s(0), 101, X52)\n\nT33 is ground\nX52 is free", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: ','(!_3, reduce(T33, s(0), 105, X52)) | ','(vowel_h_w_y(105), ','(!_3, reduce(T33, s(0), 105, X52))), SCOPE: 4, CLAUSE: 9 | ','(vowel_h_w_y(105), ','(!_3, reduce(T33, s(0), 105, X52))), SCOPE: 4, CLAUSE: 10 | ','(vowel_h_w_y(105), ','(!_3, reduce(T33, s(0), 105, X52))), SCOPE: 4, CLAUSE: 11 | ','(vowel_h_w_y(105), ','(!_3, reduce(T33, s(0), 105, X52))), SCOPE: 4, CLAUSE: 12 | ','(vowel_h_w_y(105), ','(!_3, reduce(T33, s(0), 105, X52))), SCOPE: 4, CLAUSE: 13\n\nT33 is ground\nX52 is free", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: ','(vowel_h_w_y(T32), ','(!_3, reduce(T33, s(0), T32, X52))), SCOPE: 4, CLAUSE: 9 | ','(vowel_h_w_y(T32), ','(!_3, reduce(T33, s(0), T32, X52))), SCOPE: 4, CLAUSE: 10 | ','(vowel_h_w_y(T32), ','(!_3, reduce(T33, s(0), T32, X52))), SCOPE: 4, CLAUSE: 11 | ','(vowel_h_w_y(T32), ','(!_3, reduce(T33, s(0), T32, X52))), SCOPE: 4, CLAUSE: 12 | ','(vowel_h_w_y(T32), ','(!_3, reduce(T33, s(0), T32, X52))), SCOPE: 4, CLAUSE: 13\n\nT32 is ground\nT33 is ground\nX52 is free\nvowel_h_w_y(T32) !=? vowel_h_w_y(97)\nvowel_h_w_y(T32) !=? vowel_h_w_y(101)\nvowel_h_w_y(T32) !=? vowel_h_w_y(105)", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: reduce(T33, s(0), 105, X52)\n\nT33 is ground\nX52 is free", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: ','(!_3, reduce(T33, s(0), 111, X52)) | ','(vowel_h_w_y(111), ','(!_3, reduce(T33, s(0), 111, X52))), SCOPE: 4, CLAUSE: 10 | ','(vowel_h_w_y(111), ','(!_3, reduce(T33, s(0), 111, X52))), SCOPE: 4, CLAUSE: 11 | ','(vowel_h_w_y(111), ','(!_3, reduce(T33, s(0), 111, X52))), SCOPE: 4, CLAUSE: 12 | ','(vowel_h_w_y(111), ','(!_3, reduce(T33, s(0), 111, X52))), SCOPE: 4, CLAUSE: 13\n\nT33 is ground\nX52 is free", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: ','(vowel_h_w_y(T32), ','(!_3, reduce(T33, s(0), T32, X52))), SCOPE: 4, CLAUSE: 10 | ','(vowel_h_w_y(T32), ','(!_3, reduce(T33, s(0), T32, X52))), SCOPE: 4, CLAUSE: 11 | ','(vowel_h_w_y(T32), ','(!_3, reduce(T33, s(0), T32, X52))), SCOPE: 4, CLAUSE: 12 | ','(vowel_h_w_y(T32), ','(!_3, reduce(T33, s(0), T32, X52))), SCOPE: 4, CLAUSE: 13\n\nT32 is ground\nT33 is ground\nX52 is free\nvowel_h_w_y(T32) !=? vowel_h_w_y(97)\nvowel_h_w_y(T32) !=? vowel_h_w_y(101)\nvowel_h_w_y(T32) !=? vowel_h_w_y(105)\nvowel_h_w_y(T32) !=? vowel_h_w_y(111)", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: reduce(T33, s(0), 111, X52)\n\nT33 is ground\nX52 is free", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: ','(!_3, reduce(T33, s(0), 117, X52)) | ','(vowel_h_w_y(117), ','(!_3, reduce(T33, s(0), 117, X52))), SCOPE: 4, CLAUSE: 11 | ','(vowel_h_w_y(117), ','(!_3, reduce(T33, s(0), 117, X52))), SCOPE: 4, CLAUSE: 12 | ','(vowel_h_w_y(117), ','(!_3, reduce(T33, s(0), 117, X52))), SCOPE: 4, CLAUSE: 13\n\nT33 is ground\nX52 is free", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: ','(vowel_h_w_y(T32), ','(!_3, reduce(T33, s(0), T32, X52))), SCOPE: 4, CLAUSE: 11 | ','(vowel_h_w_y(T32), ','(!_3, reduce(T33, s(0), T32, X52))), SCOPE: 4, CLAUSE: 12 | ','(vowel_h_w_y(T32), ','(!_3, reduce(T33, s(0), T32, X52))), SCOPE: 4, CLAUSE: 13\n\nT32 is ground\nT33 is ground\nX52 is free\nvowel_h_w_y(T32) !=? vowel_h_w_y(97)\nvowel_h_w_y(T32) !=? vowel_h_w_y(101)\nvowel_h_w_y(T32) !=? vowel_h_w_y(105)\nvowel_h_w_y(T32) !=? vowel_h_w_y(111)\nvowel_h_w_y(T32) !=? vowel_h_w_y(117)", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: reduce(T33, s(0), 117, X52)\n\nT33 is ground\nX52 is free", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: ','(!_3, reduce(T33, s(0), 104, X52)) | ','(vowel_h_w_y(104), ','(!_3, reduce(T33, s(0), 104, X52))), SCOPE: 4, CLAUSE: 12 | ','(vowel_h_w_y(104), ','(!_3, reduce(T33, s(0), 104, X52))), SCOPE: 4, CLAUSE: 13\n\nT33 is ground\nX52 is free", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: ','(vowel_h_w_y(T32), ','(!_3, reduce(T33, s(0), T32, X52))), SCOPE: 4, CLAUSE: 12 | ','(vowel_h_w_y(T32), ','(!_3, reduce(T33, s(0), T32, X52))), SCOPE: 4, CLAUSE: 13\n\nT32 is ground\nT33 is ground\nX52 is free\nvowel_h_w_y(T32) !=? vowel_h_w_y(97)\nvowel_h_w_y(T32) !=? vowel_h_w_y(101)\nvowel_h_w_y(T32) !=? vowel_h_w_y(105)\nvowel_h_w_y(T32) !=? vowel_h_w_y(111)\nvowel_h_w_y(T32) !=? vowel_h_w_y(117)\nvowel_h_w_y(T32) !=? vowel_h_w_y(104)", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: reduce(T33, s(0), 104, X52)\n\nT33 is ground\nX52 is free", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: ','(!_3, reduce(T33, s(0), 119, X52)) | ','(vowel_h_w_y(119), ','(!_3, reduce(T33, s(0), 119, X52))), SCOPE: 4, CLAUSE: 13\n\nT33 is ground\nX52 is free", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: ','(vowel_h_w_y(T32), ','(!_3, reduce(T33, s(0), T32, X52))), SCOPE: 4, CLAUSE: 13\n\nT32 is ground\nT33 is ground\nX52 is free\nvowel_h_w_y(T32) !=? vowel_h_w_y(97)\nvowel_h_w_y(T32) !=? vowel_h_w_y(101)\nvowel_h_w_y(T32) !=? vowel_h_w_y(105)\nvowel_h_w_y(T32) !=? vowel_h_w_y(111)\nvowel_h_w_y(T32) !=? vowel_h_w_y(117)\nvowel_h_w_y(T32) !=? vowel_h_w_y(104)\nvowel_h_w_y(T32) !=? vowel_h_w_y(119)", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: reduce(T33, s(0), 119, X52)\n\nT33 is ground\nX52 is free", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: ','(!_3, reduce(T33, s(0), 121, X52))\n\nT33 is ground\nX52 is free", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: reduce(T33, s(0), 121, X52)\n\nT33 is ground\nX52 is free", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: reduce(T16, s(0), T15, X12), SCOPE: 3, CLAUSE: 4\n\nT15 is ground\nT16 is ground\nX12 is free\nX26 is free\nX27 is free\nreduce(T16, s(0), T15, X12) !=? reduce([], X26, X27, [])", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: reduce(T16, s(0), T15, X12), SCOPE: 3, CLAUSE: 5\n\nT15 is ground\nT16 is ground\nX12 is free\nX26 is free\nX27 is free\nreduce(T16, s(0), T15, X12) !=? reduce([], X26, X27, [])", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: ','(!_3, reduce(T51, s(0), T50, X93))\n\nT50 is ground\nT51 is ground\nX93 is free", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: reduce(T51, s(0), T50, X93)\n\nT50 is ground\nT51 is ground\nX93 is free", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: reduce(T60, s(s(0)), T59, X113)\n\nT59 is ground\nT60 is ground\nX113 is free", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: reduce(T60, s(s(0)), T59, X113), SCOPE: 5, CLAUSE: 1 | reduce(T60, s(s(0)), T59, X113), SCOPE: 5, CLAUSE: 2 | reduce(T60, s(s(0)), T59, X113), SCOPE: 5, CLAUSE: 3 | reduce(T60, s(s(0)), T59, X113), SCOPE: 5, CLAUSE: 4 | reduce(T60, s(s(0)), T59, X113), SCOPE: 5, CLAUSE: 5\n\nT59 is ground\nT60 is ground\nX113 is free", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: reduce(T60, s(s(0)), T59, X113), SCOPE: 5, CLAUSE: 2 | reduce(T60, s(s(0)), T59, X113), SCOPE: 5, CLAUSE: 3 | reduce(T60, s(s(0)), T59, X113), SCOPE: 5, CLAUSE: 4 | reduce(T60, s(s(0)), T59, X113), SCOPE: 5, CLAUSE: 5\n\nT59 is ground\nT60 is ground\nX113 is free", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: !_5 | reduce([], s(s(0)), T66, X113), SCOPE: 5, CLAUSE: 3 | reduce([], s(s(0)), T66, X113), SCOPE: 5, CLAUSE: 4 | reduce([], s(s(0)), T66, X113), SCOPE: 5, CLAUSE: 5\n\nT66 is ground\nX113 is free", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: reduce(T60, s(s(0)), T59, X113), SCOPE: 5, CLAUSE: 3 | reduce(T60, s(s(0)), T59, X113), SCOPE: 5, CLAUSE: 4 | reduce(T60, s(s(0)), T59, X113), SCOPE: 5, CLAUSE: 5\n\nT59 is ground\nT60 is ground\nX113 is free\nX122 is free\nX123 is free\nreduce(T60, s(s(0)), T59, X113) !=? reduce([], X122, X123, [])", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: reduce(T60, s(s(0)), T59, X113), SCOPE: 5, CLAUSE: 3\n\nT59 is ground\nT60 is ground\nX113 is free\nX122 is free\nX123 is free\nreduce(T60, s(s(0)), T59, X113) !=? reduce([], X122, X123, [])", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: reduce(T60, s(s(0)), T59, X113), SCOPE: 5, CLAUSE: 4 | reduce(T60, s(s(0)), T59, X113), SCOPE: 5, CLAUSE: 5\n\nT59 is ground\nT60 is ground\nX113 is free\nX122 is free\nX123 is free\nreduce(T60, s(s(0)), T59, X113) !=? reduce([], X122, X123, [])", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: ','(vowel_h_w_y(T75), ','(!_5, reduce(T76, s(s(0)), T75, X148)))\n\nT75 is ground\nT76 is ground\nX148 is free", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: ','(vowel_h_w_y(T75), ','(!_5, reduce(T76, s(s(0)), T75, X148))), SCOPE: 6, CLAUSE: 6 | ','(vowel_h_w_y(T75), ','(!_5, reduce(T76, s(s(0)), T75, X148))), SCOPE: 6, CLAUSE: 7 | ','(vowel_h_w_y(T75), ','(!_5, reduce(T76, s(s(0)), T75, X148))), SCOPE: 6, CLAUSE: 8 | ','(vowel_h_w_y(T75), ','(!_5, reduce(T76, s(s(0)), T75, X148))), SCOPE: 6, CLAUSE: 9 | ','(vowel_h_w_y(T75), ','(!_5, reduce(T76, s(s(0)), T75, X148))), SCOPE: 6, CLAUSE: 10 | ','(vowel_h_w_y(T75), ','(!_5, reduce(T76, s(s(0)), T75, X148))), SCOPE: 6, CLAUSE: 11 | ','(vowel_h_w_y(T75), ','(!_5, reduce(T76, s(s(0)), T75, X148))), SCOPE: 6, CLAUSE: 12 | ','(vowel_h_w_y(T75), ','(!_5, reduce(T76, s(s(0)), T75, X148))), SCOPE: 6, CLAUSE: 13\n\nT75 is ground\nT76 is ground\nX148 is free", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: ','(!_5, reduce(T76, s(s(0)), 97, X148)) | ','(vowel_h_w_y(97), ','(!_5, reduce(T76, s(s(0)), 97, X148))), SCOPE: 6, CLAUSE: 7 | ','(vowel_h_w_y(97), ','(!_5, reduce(T76, s(s(0)), 97, X148))), SCOPE: 6, CLAUSE: 8 | ','(vowel_h_w_y(97), ','(!_5, reduce(T76, s(s(0)), 97, X148))), SCOPE: 6, CLAUSE: 9 | ','(vowel_h_w_y(97), ','(!_5, reduce(T76, s(s(0)), 97, X148))), SCOPE: 6, CLAUSE: 10 | ','(vowel_h_w_y(97), ','(!_5, reduce(T76, s(s(0)), 97, X148))), SCOPE: 6, CLAUSE: 11 | ','(vowel_h_w_y(97), ','(!_5, reduce(T76, s(s(0)), 97, X148))), SCOPE: 6, CLAUSE: 12 | ','(vowel_h_w_y(97), ','(!_5, reduce(T76, s(s(0)), 97, X148))), SCOPE: 6, CLAUSE: 13\n\nT76 is ground\nX148 is free", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: ','(vowel_h_w_y(T75), ','(!_5, reduce(T76, s(s(0)), T75, X148))), SCOPE: 6, CLAUSE: 7 | ','(vowel_h_w_y(T75), ','(!_5, reduce(T76, s(s(0)), T75, X148))), SCOPE: 6, CLAUSE: 8 | ','(vowel_h_w_y(T75), ','(!_5, reduce(T76, s(s(0)), T75, X148))), SCOPE: 6, CLAUSE: 9 | ','(vowel_h_w_y(T75), ','(!_5, reduce(T76, s(s(0)), T75, X148))), SCOPE: 6, CLAUSE: 10 | ','(vowel_h_w_y(T75), ','(!_5, reduce(T76, s(s(0)), T75, X148))), SCOPE: 6, CLAUSE: 11 | ','(vowel_h_w_y(T75), ','(!_5, reduce(T76, s(s(0)), T75, X148))), SCOPE: 6, CLAUSE: 12 | ','(vowel_h_w_y(T75), ','(!_5, reduce(T76, s(s(0)), T75, X148))), SCOPE: 6, CLAUSE: 13\n\nT75 is ground\nT76 is ground\nX148 is free\nvowel_h_w_y(T75) !=? vowel_h_w_y(97)", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: reduce(T76, s(s(0)), 97, X148)\n\nT76 is ground\nX148 is free", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: ','(!_5, reduce(T76, s(s(0)), 101, X148)) | ','(vowel_h_w_y(101), ','(!_5, reduce(T76, s(s(0)), 101, X148))), SCOPE: 6, CLAUSE: 8 | ','(vowel_h_w_y(101), ','(!_5, reduce(T76, s(s(0)), 101, X148))), SCOPE: 6, CLAUSE: 9 | ','(vowel_h_w_y(101), ','(!_5, reduce(T76, s(s(0)), 101, X148))), SCOPE: 6, CLAUSE: 10 | ','(vowel_h_w_y(101), ','(!_5, reduce(T76, s(s(0)), 101, X148))), SCOPE: 6, CLAUSE: 11 | ','(vowel_h_w_y(101), ','(!_5, reduce(T76, s(s(0)), 101, X148))), SCOPE: 6, CLAUSE: 12 | ','(vowel_h_w_y(101), ','(!_5, reduce(T76, s(s(0)), 101, X148))), SCOPE: 6, CLAUSE: 13\n\nT76 is ground\nX148 is free", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: ','(vowel_h_w_y(T75), ','(!_5, reduce(T76, s(s(0)), T75, X148))), SCOPE: 6, CLAUSE: 8 | ','(vowel_h_w_y(T75), ','(!_5, reduce(T76, s(s(0)), T75, X148))), SCOPE: 6, CLAUSE: 9 | ','(vowel_h_w_y(T75), ','(!_5, reduce(T76, s(s(0)), T75, X148))), SCOPE: 6, CLAUSE: 10 | ','(vowel_h_w_y(T75), ','(!_5, reduce(T76, s(s(0)), T75, X148))), SCOPE: 6, CLAUSE: 11 | ','(vowel_h_w_y(T75), ','(!_5, reduce(T76, s(s(0)), T75, X148))), SCOPE: 6, CLAUSE: 12 | ','(vowel_h_w_y(T75), ','(!_5, reduce(T76, s(s(0)), T75, X148))), SCOPE: 6, CLAUSE: 13\n\nT75 is ground\nT76 is ground\nX148 is free\nvowel_h_w_y(T75) !=? vowel_h_w_y(97)\nvowel_h_w_y(T75) !=? vowel_h_w_y(101)", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: reduce(T76, s(s(0)), 101, X148)\n\nT76 is ground\nX148 is free", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: ','(!_5, reduce(T76, s(s(0)), 105, X148)) | ','(vowel_h_w_y(105), ','(!_5, reduce(T76, s(s(0)), 105, X148))), SCOPE: 6, CLAUSE: 9 | ','(vowel_h_w_y(105), ','(!_5, reduce(T76, s(s(0)), 105, X148))), SCOPE: 6, CLAUSE: 10 | ','(vowel_h_w_y(105), ','(!_5, reduce(T76, s(s(0)), 105, X148))), SCOPE: 6, CLAUSE: 11 | ','(vowel_h_w_y(105), ','(!_5, reduce(T76, s(s(0)), 105, X148))), SCOPE: 6, CLAUSE: 12 | ','(vowel_h_w_y(105), ','(!_5, reduce(T76, s(s(0)), 105, X148))), SCOPE: 6, CLAUSE: 13\n\nT76 is ground\nX148 is free", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: ','(vowel_h_w_y(T75), ','(!_5, reduce(T76, s(s(0)), T75, X148))), SCOPE: 6, CLAUSE: 9 | ','(vowel_h_w_y(T75), ','(!_5, reduce(T76, s(s(0)), T75, X148))), SCOPE: 6, CLAUSE: 10 | ','(vowel_h_w_y(T75), ','(!_5, reduce(T76, s(s(0)), T75, X148))), SCOPE: 6, CLAUSE: 11 | ','(vowel_h_w_y(T75), ','(!_5, reduce(T76, s(s(0)), T75, X148))), SCOPE: 6, CLAUSE: 12 | ','(vowel_h_w_y(T75), ','(!_5, reduce(T76, s(s(0)), T75, X148))), SCOPE: 6, CLAUSE: 13\n\nT75 is ground\nT76 is ground\nX148 is free\nvowel_h_w_y(T75) !=? vowel_h_w_y(97)\nvowel_h_w_y(T75) !=? vowel_h_w_y(101)\nvowel_h_w_y(T75) !=? vowel_h_w_y(105)", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: reduce(T76, s(s(0)), 105, X148)\n\nT76 is ground\nX148 is free", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: ','(!_5, reduce(T76, s(s(0)), 111, X148)) | ','(vowel_h_w_y(111), ','(!_5, reduce(T76, s(s(0)), 111, X148))), SCOPE: 6, CLAUSE: 10 | ','(vowel_h_w_y(111), ','(!_5, reduce(T76, s(s(0)), 111, X148))), SCOPE: 6, CLAUSE: 11 | ','(vowel_h_w_y(111), ','(!_5, reduce(T76, s(s(0)), 111, X148))), SCOPE: 6, CLAUSE: 12 | ','(vowel_h_w_y(111), ','(!_5, reduce(T76, s(s(0)), 111, X148))), SCOPE: 6, CLAUSE: 13\n\nT76 is ground\nX148 is free", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: ','(vowel_h_w_y(T75), ','(!_5, reduce(T76, s(s(0)), T75, X148))), SCOPE: 6, CLAUSE: 10 | ','(vowel_h_w_y(T75), ','(!_5, reduce(T76, s(s(0)), T75, X148))), SCOPE: 6, CLAUSE: 11 | ','(vowel_h_w_y(T75), ','(!_5, reduce(T76, s(s(0)), T75, X148))), SCOPE: 6, CLAUSE: 12 | ','(vowel_h_w_y(T75), ','(!_5, reduce(T76, s(s(0)), T75, X148))), SCOPE: 6, CLAUSE: 13\n\nT75 is ground\nT76 is ground\nX148 is free\nvowel_h_w_y(T75) !=? vowel_h_w_y(97)\nvowel_h_w_y(T75) !=? vowel_h_w_y(101)\nvowel_h_w_y(T75) !=? vowel_h_w_y(105)\nvowel_h_w_y(T75) !=? vowel_h_w_y(111)", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: reduce(T76, s(s(0)), 111, X148)\n\nT76 is ground\nX148 is free", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: ','(!_5, reduce(T76, s(s(0)), 117, X148)) | ','(vowel_h_w_y(117), ','(!_5, reduce(T76, s(s(0)), 117, X148))), SCOPE: 6, CLAUSE: 11 | ','(vowel_h_w_y(117), ','(!_5, reduce(T76, s(s(0)), 117, X148))), SCOPE: 6, CLAUSE: 12 | ','(vowel_h_w_y(117), ','(!_5, reduce(T76, s(s(0)), 117, X148))), SCOPE: 6, CLAUSE: 13\n\nT76 is ground\nX148 is free", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: ','(vowel_h_w_y(T75), ','(!_5, reduce(T76, s(s(0)), T75, X148))), SCOPE: 6, CLAUSE: 11 | ','(vowel_h_w_y(T75), ','(!_5, reduce(T76, s(s(0)), T75, X148))), SCOPE: 6, CLAUSE: 12 | ','(vowel_h_w_y(T75), ','(!_5, reduce(T76, s(s(0)), T75, X148))), SCOPE: 6, CLAUSE: 13\n\nT75 is ground\nT76 is ground\nX148 is free\nvowel_h_w_y(T75) !=? vowel_h_w_y(97)\nvowel_h_w_y(T75) !=? vowel_h_w_y(101)\nvowel_h_w_y(T75) !=? vowel_h_w_y(105)\nvowel_h_w_y(T75) !=? vowel_h_w_y(111)\nvowel_h_w_y(T75) !=? vowel_h_w_y(117)", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: reduce(T76, s(s(0)), 117, X148)\n\nT76 is ground\nX148 is free", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="77: ','(!_5, reduce(T76, s(s(0)), 104, X148)) | ','(vowel_h_w_y(104), ','(!_5, reduce(T76, s(s(0)), 104, X148))), SCOPE: 6, CLAUSE: 12 | ','(vowel_h_w_y(104), ','(!_5, reduce(T76, s(s(0)), 104, X148))), SCOPE: 6, CLAUSE: 13\n\nT76 is ground\nX148 is free", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="78: ','(vowel_h_w_y(T75), ','(!_5, reduce(T76, s(s(0)), T75, X148))), SCOPE: 6, CLAUSE: 12 | ','(vowel_h_w_y(T75), ','(!_5, reduce(T76, s(s(0)), T75, X148))), SCOPE: 6, CLAUSE: 13\n\nT75 is ground\nT76 is ground\nX148 is free\nvowel_h_w_y(T75) !=? vowel_h_w_y(97)\nvowel_h_w_y(T75) !=? vowel_h_w_y(101)\nvowel_h_w_y(T75) !=? vowel_h_w_y(105)\nvowel_h_w_y(T75) !=? vowel_h_w_y(111)\nvowel_h_w_y(T75) !=? vowel_h_w_y(117)\nvowel_h_w_y(T75) !=? vowel_h_w_y(104)", fontsize=16, style=filled, fillcolor=lightyellow];
79 [label="79: reduce(T76, s(s(0)), 104, X148)\n\nT76 is ground\nX148 is free", fontsize=16, style=filled, fillcolor=lightyellow];
80 [label="80: ','(!_5, reduce(T76, s(s(0)), 119, X148)) | ','(vowel_h_w_y(119), ','(!_5, reduce(T76, s(s(0)), 119, X148))), SCOPE: 6, CLAUSE: 13\n\nT76 is ground\nX148 is free", fontsize=16, style=filled, fillcolor=lightyellow];
81 [label="81: ','(vowel_h_w_y(T75), ','(!_5, reduce(T76, s(s(0)), T75, X148))), SCOPE: 6, CLAUSE: 13\n\nT75 is ground\nT76 is ground\nX148 is free\nvowel_h_w_y(T75) !=? vowel_h_w_y(97)\nvowel_h_w_y(T75) !=? vowel_h_w_y(101)\nvowel_h_w_y(T75) !=? vowel_h_w_y(105)\nvowel_h_w_y(T75) !=? vowel_h_w_y(111)\nvowel_h_w_y(T75) !=? vowel_h_w_y(117)\nvowel_h_w_y(T75) !=? vowel_h_w_y(104)\nvowel_h_w_y(T75) !=? vowel_h_w_y(119)", fontsize=16, style=filled, fillcolor=lightyellow];
82 [label="82: reduce(T76, s(s(0)), 119, X148)\n\nT76 is ground\nX148 is free", fontsize=16, style=filled, fillcolor=lightyellow];
83 [label="83: ','(!_5, reduce(T76, s(s(0)), 121, X148))\n\nT76 is ground\nX148 is free", fontsize=16, style=filled, fillcolor=lightyellow];
84 [label="84: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
85 [label="85: reduce(T76, s(s(0)), 121, X148)\n\nT76 is ground\nX148 is free", fontsize=16, style=filled, fillcolor=lightyellow];
86 [label="86: reduce(T60, s(s(0)), T59, X113), SCOPE: 5, CLAUSE: 4\n\nT59 is ground\nT60 is ground\nX113 is free\nX122 is free\nX123 is free\nreduce(T60, s(s(0)), T59, X113) !=? reduce([], X122, X123, [])", fontsize=16, style=filled, fillcolor=lightyellow];
87 [label="87: reduce(T60, s(s(0)), T59, X113), SCOPE: 5, CLAUSE: 5\n\nT59 is ground\nT60 is ground\nX113 is free\nX122 is free\nX123 is free\nreduce(T60, s(s(0)), T59, X113) !=? reduce([], X122, X123, [])", fontsize=16, style=filled, fillcolor=lightyellow];
88 [label="88: ','(!_5, reduce(T94, s(s(0)), T93, X189))\n\nT93 is ground\nT94 is ground\nX189 is free", fontsize=16, style=filled, fillcolor=lightyellow];
89 [label="89: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
90 [label="90: reduce(T94, s(s(0)), T93, X189)\n\nT93 is ground\nT94 is ground\nX189 is free", fontsize=16, style=filled, fillcolor=lightyellow];
91 [label="91: reduce(T103, s(s(s(0))), T102, X209)\n\nT102 is ground\nT103 is ground\nX209 is free", fontsize=16, style=filled, fillcolor=lightyellow];
92 [label="92: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
93 [label="93: reduce(T103, s(s(s(0))), T102, X209), SCOPE: 7, CLAUSE: 1 | reduce(T103, s(s(s(0))), T102, X209), SCOPE: 7, CLAUSE: 2 | reduce(T103, s(s(s(0))), T102, X209), SCOPE: 7, CLAUSE: 3 | reduce(T103, s(s(s(0))), T102, X209), SCOPE: 7, CLAUSE: 4 | reduce(T103, s(s(s(0))), T102, X209), SCOPE: 7, CLAUSE: 5\n\nT102 is ground\nT103 is ground\nX209 is free", fontsize=16, style=filled, fillcolor=lightyellow];
94 [label="94: reduce(T103, s(s(s(0))), T102, X209), SCOPE: 7, CLAUSE: 2 | reduce(T103, s(s(s(0))), T102, X209), SCOPE: 7, CLAUSE: 3 | reduce(T103, s(s(s(0))), T102, X209), SCOPE: 7, CLAUSE: 4 | reduce(T103, s(s(s(0))), T102, X209), SCOPE: 7, CLAUSE: 5\n\nT102 is ground\nT103 is ground\nX209 is free", fontsize=16, style=filled, fillcolor=lightyellow];
95 [label="95: !_7 | reduce([], s(s(s(0))), T109, X209), SCOPE: 7, CLAUSE: 3 | reduce([], s(s(s(0))), T109, X209), SCOPE: 7, CLAUSE: 4 | reduce([], s(s(s(0))), T109, X209), SCOPE: 7, CLAUSE: 5\n\nT109 is ground\nX209 is free", fontsize=16, style=filled, fillcolor=lightyellow];
96 [label="96: reduce(T103, s(s(s(0))), T102, X209), SCOPE: 7, CLAUSE: 3 | reduce(T103, s(s(s(0))), T102, X209), SCOPE: 7, CLAUSE: 4 | reduce(T103, s(s(s(0))), T102, X209), SCOPE: 7, CLAUSE: 5\n\nT102 is ground\nT103 is ground\nX209 is free\nX218 is free\nX219 is free\nreduce(T103, s(s(s(0))), T102, X209) !=? reduce([], X218, X219, [])", fontsize=16, style=filled, fillcolor=lightyellow];
97 [label="97: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
98 [label="98: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
99 [label="99: reduce(T103, s(s(s(0))), T102, X209), SCOPE: 7, CLAUSE: 3\n\nT102 is ground\nT103 is ground\nX209 is free\nX218 is free\nX219 is free\nreduce(T103, s(s(s(0))), T102, X209) !=? reduce([], X218, X219, [])", fontsize=16, style=filled, fillcolor=lightyellow];
100 [label="100: reduce(T103, s(s(s(0))), T102, X209), SCOPE: 7, CLAUSE: 4 | reduce(T103, s(s(s(0))), T102, X209), SCOPE: 7, CLAUSE: 5\n\nT102 is ground\nT103 is ground\nX209 is free\nX218 is free\nX219 is free\nreduce(T103, s(s(s(0))), T102, X209) !=? reduce([], X218, X219, [])", fontsize=16, style=filled, fillcolor=lightyellow];
101 [label="101: ','(vowel_h_w_y(T118), ','(!_7, reduce(T119, s(s(s(0))), T118, X244)))\n\nT118 is ground\nT119 is ground\nX244 is free", fontsize=16, style=filled, fillcolor=lightyellow];
102 [label="102: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
103 [label="103: ','(vowel_h_w_y(T118), ','(!_7, reduce(T119, s(s(s(0))), T118, X244))), SCOPE: 8, CLAUSE: 6 | ','(vowel_h_w_y(T118), ','(!_7, reduce(T119, s(s(s(0))), T118, X244))), SCOPE: 8, CLAUSE: 7 | ','(vowel_h_w_y(T118), ','(!_7, reduce(T119, s(s(s(0))), T118, X244))), SCOPE: 8, CLAUSE: 8 | ','(vowel_h_w_y(T118), ','(!_7, reduce(T119, s(s(s(0))), T118, X244))), SCOPE: 8, CLAUSE: 9 | ','(vowel_h_w_y(T118), ','(!_7, reduce(T119, s(s(s(0))), T118, X244))), SCOPE: 8, CLAUSE: 10 | ','(vowel_h_w_y(T118), ','(!_7, reduce(T119, s(s(s(0))), T118, X244))), SCOPE: 8, CLAUSE: 11 | ','(vowel_h_w_y(T118), ','(!_7, reduce(T119, s(s(s(0))), T118, X244))), SCOPE: 8, CLAUSE: 12 | ','(vowel_h_w_y(T118), ','(!_7, reduce(T119, s(s(s(0))), T118, X244))), SCOPE: 8, CLAUSE: 13\n\nT118 is ground\nT119 is ground\nX244 is free", fontsize=16, style=filled, fillcolor=lightyellow];
104 [label="104: ','(!_7, reduce(T119, s(s(s(0))), 97, X244)) | ','(vowel_h_w_y(97), ','(!_7, reduce(T119, s(s(s(0))), 97, X244))), SCOPE: 8, CLAUSE: 7 | ','(vowel_h_w_y(97), ','(!_7, reduce(T119, s(s(s(0))), 97, X244))), SCOPE: 8, CLAUSE: 8 | ','(vowel_h_w_y(97), ','(!_7, reduce(T119, s(s(s(0))), 97, X244))), SCOPE: 8, CLAUSE: 9 | ','(vowel_h_w_y(97), ','(!_7, reduce(T119, s(s(s(0))), 97, X244))), SCOPE: 8, CLAUSE: 10 | ','(vowel_h_w_y(97), ','(!_7, reduce(T119, s(s(s(0))), 97, X244))), SCOPE: 8, CLAUSE: 11 | ','(vowel_h_w_y(97), ','(!_7, reduce(T119, s(s(s(0))), 97, X244))), SCOPE: 8, CLAUSE: 12 | ','(vowel_h_w_y(97), ','(!_7, reduce(T119, s(s(s(0))), 97, X244))), SCOPE: 8, CLAUSE: 13\n\nT119 is ground\nX244 is free", fontsize=16, style=filled, fillcolor=lightyellow];
105 [label="105: ','(vowel_h_w_y(T118), ','(!_7, reduce(T119, s(s(s(0))), T118, X244))), SCOPE: 8, CLAUSE: 7 | ','(vowel_h_w_y(T118), ','(!_7, reduce(T119, s(s(s(0))), T118, X244))), SCOPE: 8, CLAUSE: 8 | ','(vowel_h_w_y(T118), ','(!_7, reduce(T119, s(s(s(0))), T118, X244))), SCOPE: 8, CLAUSE: 9 | ','(vowel_h_w_y(T118), ','(!_7, reduce(T119, s(s(s(0))), T118, X244))), SCOPE: 8, CLAUSE: 10 | ','(vowel_h_w_y(T118), ','(!_7, reduce(T119, s(s(s(0))), T118, X244))), SCOPE: 8, CLAUSE: 11 | ','(vowel_h_w_y(T118), ','(!_7, reduce(T119, s(s(s(0))), T118, X244))), SCOPE: 8, CLAUSE: 12 | ','(vowel_h_w_y(T118), ','(!_7, reduce(T119, s(s(s(0))), T118, X244))), SCOPE: 8, CLAUSE: 13\n\nT118 is ground\nT119 is ground\nX244 is free\nvowel_h_w_y(T118) !=? vowel_h_w_y(97)", fontsize=16, style=filled, fillcolor=lightyellow];
106 [label="106: reduce(T119, s(s(s(0))), 97, X244)\n\nT119 is ground\nX244 is free", fontsize=16, style=filled, fillcolor=lightyellow];
107 [label="107: ','(!_7, reduce(T119, s(s(s(0))), 101, X244)) | ','(vowel_h_w_y(101), ','(!_7, reduce(T119, s(s(s(0))), 101, X244))), SCOPE: 8, CLAUSE: 8 | ','(vowel_h_w_y(101), ','(!_7, reduce(T119, s(s(s(0))), 101, X244))), SCOPE: 8, CLAUSE: 9 | ','(vowel_h_w_y(101), ','(!_7, reduce(T119, s(s(s(0))), 101, X244))), SCOPE: 8, CLAUSE: 10 | ','(vowel_h_w_y(101), ','(!_7, reduce(T119, s(s(s(0))), 101, X244))), SCOPE: 8, CLAUSE: 11 | ','(vowel_h_w_y(101), ','(!_7, reduce(T119, s(s(s(0))), 101, X244))), SCOPE: 8, CLAUSE: 12 | ','(vowel_h_w_y(101), ','(!_7, reduce(T119, s(s(s(0))), 101, X244))), SCOPE: 8, CLAUSE: 13\n\nT119 is ground\nX244 is free", fontsize=16, style=filled, fillcolor=lightyellow];
108 [label="108: ','(vowel_h_w_y(T118), ','(!_7, reduce(T119, s(s(s(0))), T118, X244))), SCOPE: 8, CLAUSE: 8 | ','(vowel_h_w_y(T118), ','(!_7, reduce(T119, s(s(s(0))), T118, X244))), SCOPE: 8, CLAUSE: 9 | ','(vowel_h_w_y(T118), ','(!_7, reduce(T119, s(s(s(0))), T118, X244))), SCOPE: 8, CLAUSE: 10 | ','(vowel_h_w_y(T118), ','(!_7, reduce(T119, s(s(s(0))), T118, X244))), SCOPE: 8, CLAUSE: 11 | ','(vowel_h_w_y(T118), ','(!_7, reduce(T119, s(s(s(0))), T118, X244))), SCOPE: 8, CLAUSE: 12 | ','(vowel_h_w_y(T118), ','(!_7, reduce(T119, s(s(s(0))), T118, X244))), SCOPE: 8, CLAUSE: 13\n\nT118 is ground\nT119 is ground\nX244 is free\nvowel_h_w_y(T118) !=? vowel_h_w_y(97)\nvowel_h_w_y(T118) !=? vowel_h_w_y(101)", fontsize=16, style=filled, fillcolor=lightyellow];
109 [label="109: reduce(T119, s(s(s(0))), 101, X244)\n\nT119 is ground\nX244 is free", fontsize=16, style=filled, fillcolor=lightyellow];
110 [label="110: ','(!_7, reduce(T119, s(s(s(0))), 105, X244)) | ','(vowel_h_w_y(105), ','(!_7, reduce(T119, s(s(s(0))), 105, X244))), SCOPE: 8, CLAUSE: 9 | ','(vowel_h_w_y(105), ','(!_7, reduce(T119, s(s(s(0))), 105, X244))), SCOPE: 8, CLAUSE: 10 | ','(vowel_h_w_y(105), ','(!_7, reduce(T119, s(s(s(0))), 105, X244))), SCOPE: 8, CLAUSE: 11 | ','(vowel_h_w_y(105), ','(!_7, reduce(T119, s(s(s(0))), 105, X244))), SCOPE: 8, CLAUSE: 12 | ','(vowel_h_w_y(105), ','(!_7, reduce(T119, s(s(s(0))), 105, X244))), SCOPE: 8, CLAUSE: 13\n\nT119 is ground\nX244 is free", fontsize=16, style=filled, fillcolor=lightyellow];
111 [label="111: ','(vowel_h_w_y(T118), ','(!_7, reduce(T119, s(s(s(0))), T118, X244))), SCOPE: 8, CLAUSE: 9 | ','(vowel_h_w_y(T118), ','(!_7, reduce(T119, s(s(s(0))), T118, X244))), SCOPE: 8, CLAUSE: 10 | ','(vowel_h_w_y(T118), ','(!_7, reduce(T119, s(s(s(0))), T118, X244))), SCOPE: 8, CLAUSE: 11 | ','(vowel_h_w_y(T118), ','(!_7, reduce(T119, s(s(s(0))), T118, X244))), SCOPE: 8, CLAUSE: 12 | ','(vowel_h_w_y(T118), ','(!_7, reduce(T119, s(s(s(0))), T118, X244))), SCOPE: 8, CLAUSE: 13\n\nT118 is ground\nT119 is ground\nX244 is free\nvowel_h_w_y(T118) !=? vowel_h_w_y(97)\nvowel_h_w_y(T118) !=? vowel_h_w_y(101)\nvowel_h_w_y(T118) !=? vowel_h_w_y(105)", fontsize=16, style=filled, fillcolor=lightyellow];
112 [label="112: reduce(T119, s(s(s(0))), 105, X244)\n\nT119 is ground\nX244 is free", fontsize=16, style=filled, fillcolor=lightyellow];
113 [label="113: ','(!_7, reduce(T119, s(s(s(0))), 111, X244)) | ','(vowel_h_w_y(111), ','(!_7, reduce(T119, s(s(s(0))), 111, X244))), SCOPE: 8, CLAUSE: 10 | ','(vowel_h_w_y(111), ','(!_7, reduce(T119, s(s(s(0))), 111, X244))), SCOPE: 8, CLAUSE: 11 | ','(vowel_h_w_y(111), ','(!_7, reduce(T119, s(s(s(0))), 111, X244))), SCOPE: 8, CLAUSE: 12 | ','(vowel_h_w_y(111), ','(!_7, reduce(T119, s(s(s(0))), 111, X244))), SCOPE: 8, CLAUSE: 13\n\nT119 is ground\nX244 is free", fontsize=16, style=filled, fillcolor=lightyellow];
114 [label="114: ','(vowel_h_w_y(T118), ','(!_7, reduce(T119, s(s(s(0))), T118, X244))), SCOPE: 8, CLAUSE: 10 | ','(vowel_h_w_y(T118), ','(!_7, reduce(T119, s(s(s(0))), T118, X244))), SCOPE: 8, CLAUSE: 11 | ','(vowel_h_w_y(T118), ','(!_7, reduce(T119, s(s(s(0))), T118, X244))), SCOPE: 8, CLAUSE: 12 | ','(vowel_h_w_y(T118), ','(!_7, reduce(T119, s(s(s(0))), T118, X244))), SCOPE: 8, CLAUSE: 13\n\nT118 is ground\nT119 is ground\nX244 is free\nvowel_h_w_y(T118) !=? vowel_h_w_y(97)\nvowel_h_w_y(T118) !=? vowel_h_w_y(101)\nvowel_h_w_y(T118) !=? vowel_h_w_y(105)\nvowel_h_w_y(T118) !=? vowel_h_w_y(111)", fontsize=16, style=filled, fillcolor=lightyellow];
115 [label="115: reduce(T119, s(s(s(0))), 111, X244)\n\nT119 is ground\nX244 is free", fontsize=16, style=filled, fillcolor=lightyellow];
116 [label="116: ','(!_7, reduce(T119, s(s(s(0))), 117, X244)) | ','(vowel_h_w_y(117), ','(!_7, reduce(T119, s(s(s(0))), 117, X244))), SCOPE: 8, CLAUSE: 11 | ','(vowel_h_w_y(117), ','(!_7, reduce(T119, s(s(s(0))), 117, X244))), SCOPE: 8, CLAUSE: 12 | ','(vowel_h_w_y(117), ','(!_7, reduce(T119, s(s(s(0))), 117, X244))), SCOPE: 8, CLAUSE: 13\n\nT119 is ground\nX244 is free", fontsize=16, style=filled, fillcolor=lightyellow];
117 [label="117: ','(vowel_h_w_y(T118), ','(!_7, reduce(T119, s(s(s(0))), T118, X244))), SCOPE: 8, CLAUSE: 11 | ','(vowel_h_w_y(T118), ','(!_7, reduce(T119, s(s(s(0))), T118, X244))), SCOPE: 8, CLAUSE: 12 | ','(vowel_h_w_y(T118), ','(!_7, reduce(T119, s(s(s(0))), T118, X244))), SCOPE: 8, CLAUSE: 13\n\nT118 is ground\nT119 is ground\nX244 is free\nvowel_h_w_y(T118) !=? vowel_h_w_y(97)\nvowel_h_w_y(T118) !=? vowel_h_w_y(101)\nvowel_h_w_y(T118) !=? vowel_h_w_y(105)\nvowel_h_w_y(T118) !=? vowel_h_w_y(111)\nvowel_h_w_y(T118) !=? vowel_h_w_y(117)", fontsize=16, style=filled, fillcolor=lightyellow];
118 [label="118: reduce(T119, s(s(s(0))), 117, X244)\n\nT119 is ground\nX244 is free", fontsize=16, style=filled, fillcolor=lightyellow];
119 [label="119: ','(!_7, reduce(T119, s(s(s(0))), 104, X244)) | ','(vowel_h_w_y(104), ','(!_7, reduce(T119, s(s(s(0))), 104, X244))), SCOPE: 8, CLAUSE: 12 | ','(vowel_h_w_y(104), ','(!_7, reduce(T119, s(s(s(0))), 104, X244))), SCOPE: 8, CLAUSE: 13\n\nT119 is ground\nX244 is free", fontsize=16, style=filled, fillcolor=lightyellow];
120 [label="120: ','(vowel_h_w_y(T118), ','(!_7, reduce(T119, s(s(s(0))), T118, X244))), SCOPE: 8, CLAUSE: 12 | ','(vowel_h_w_y(T118), ','(!_7, reduce(T119, s(s(s(0))), T118, X244))), SCOPE: 8, CLAUSE: 13\n\nT118 is ground\nT119 is ground\nX244 is free\nvowel_h_w_y(T118) !=? vowel_h_w_y(97)\nvowel_h_w_y(T118) !=? vowel_h_w_y(101)\nvowel_h_w_y(T118) !=? vowel_h_w_y(105)\nvowel_h_w_y(T118) !=? vowel_h_w_y(111)\nvowel_h_w_y(T118) !=? vowel_h_w_y(117)\nvowel_h_w_y(T118) !=? vowel_h_w_y(104)", fontsize=16, style=filled, fillcolor=lightyellow];
121 [label="121: reduce(T119, s(s(s(0))), 104, X244)\n\nT119 is ground\nX244 is free", fontsize=16, style=filled, fillcolor=lightyellow];
122 [label="122: ','(!_7, reduce(T119, s(s(s(0))), 119, X244)) | ','(vowel_h_w_y(119), ','(!_7, reduce(T119, s(s(s(0))), 119, X244))), SCOPE: 8, CLAUSE: 13\n\nT119 is ground\nX244 is free", fontsize=16, style=filled, fillcolor=lightyellow];
123 [label="123: ','(vowel_h_w_y(T118), ','(!_7, reduce(T119, s(s(s(0))), T118, X244))), SCOPE: 8, CLAUSE: 13\n\nT118 is ground\nT119 is ground\nX244 is free\nvowel_h_w_y(T118) !=? vowel_h_w_y(97)\nvowel_h_w_y(T118) !=? vowel_h_w_y(101)\nvowel_h_w_y(T118) !=? vowel_h_w_y(105)\nvowel_h_w_y(T118) !=? vowel_h_w_y(111)\nvowel_h_w_y(T118) !=? vowel_h_w_y(117)\nvowel_h_w_y(T118) !=? vowel_h_w_y(104)\nvowel_h_w_y(T118) !=? vowel_h_w_y(119)", fontsize=16, style=filled, fillcolor=lightyellow];
124 [label="124: reduce(T119, s(s(s(0))), 119, X244)\n\nT119 is ground\nX244 is free", fontsize=16, style=filled, fillcolor=lightyellow];
125 [label="125: ','(!_7, reduce(T119, s(s(s(0))), 121, X244))\n\nT119 is ground\nX244 is free", fontsize=16, style=filled, fillcolor=lightyellow];
126 [label="126: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
127 [label="127: reduce(T119, s(s(s(0))), 121, X244)\n\nT119 is ground\nX244 is free", fontsize=16, style=filled, fillcolor=lightyellow];
128 [label="128: reduce(T103, s(s(s(0))), T102, X209), SCOPE: 7, CLAUSE: 4\n\nT102 is ground\nT103 is ground\nX209 is free\nX218 is free\nX219 is free\nreduce(T103, s(s(s(0))), T102, X209) !=? reduce([], X218, X219, [])", fontsize=16, style=filled, fillcolor=lightyellow];
129 [label="129: reduce(T103, s(s(s(0))), T102, X209), SCOPE: 7, CLAUSE: 5\n\nT102 is ground\nT103 is ground\nX209 is free\nX218 is free\nX219 is free\nreduce(T103, s(s(s(0))), T102, X209) !=? reduce([], X218, X219, [])", fontsize=16, style=filled, fillcolor=lightyellow];
130 [label="130: ','(!_7, reduce(T137, s(s(s(0))), T136, X285))\n\nT136 is ground\nT137 is ground\nX285 is free", fontsize=16, style=filled, fillcolor=lightyellow];
131 [label="131: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
132 [label="132: reduce(T137, s(s(s(0))), T136, X285)\n\nT136 is ground\nT137 is ground\nX285 is free", fontsize=16, style=filled, fillcolor=lightyellow];
133 [label="133: reduce(T146, s(s(s(s(0)))), T145, X305)\n\nT145 is ground\nT146 is ground\nX305 is free", fontsize=16, style=filled, fillcolor=lightyellow];
134 [label="134: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
135 [label="135: reduce(T146, s(s(s(s(0)))), T145, X305), SCOPE: 9, CLAUSE: 1 | reduce(T146, s(s(s(s(0)))), T145, X305), SCOPE: 9, CLAUSE: 2 | reduce(T146, s(s(s(s(0)))), T145, X305), SCOPE: 9, CLAUSE: 3 | reduce(T146, s(s(s(s(0)))), T145, X305), SCOPE: 9, CLAUSE: 4 | reduce(T146, s(s(s(s(0)))), T145, X305), SCOPE: 9, CLAUSE: 5\n\nT145 is ground\nT146 is ground\nX305 is free", fontsize=16, style=filled, fillcolor=lightyellow];
136 [label="136: !_9 | reduce(T154, s(s(s(s(0)))), T155, X305), SCOPE: 9, CLAUSE: 2 | reduce(T154, s(s(s(s(0)))), T155, X305), SCOPE: 9, CLAUSE: 3 | reduce(T154, s(s(s(s(0)))), T155, X305), SCOPE: 9, CLAUSE: 4 | reduce(T154, s(s(s(s(0)))), T155, X305), SCOPE: 9, CLAUSE: 5\n\nT154 is ground\nT155 is ground\nX305 is free", fontsize=16, style=filled, fillcolor=lightyellow];
137 [label="137: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
138 [label="138: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
139 [label="139: eq(T7, T18), SCOPE: 10, CLAUSE: 14\n\nT18 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
140 [label="140: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
141 [label="141: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
142 [label="142: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
143 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 143 [arrowhead = none ];
143 -> 2;

144 [label="ONLY EVAL with clause\ngoal(X8, X9) :- ','(eq(X8, .(X10, X11)), ','(reduce(X11, s(0), X10, X12), eq(X9, X12))).\nand substitutionT1 -> T5,\nX8 -> T5,\nT2 -> T7,\nX9 -> T7,\nT6 -> T7", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 144 [arrowhead = none ];
144 -> 3;

145 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 145 [arrowhead = none ];
145 -> 4;

146 [label="EVAL with clause\neq(X15, X15).\nand substitutionT5 -> .(T15, T16),\nX15 -> .(T15, T16),\nX10 -> T15,\nX11 -> T16,\nT14 -> .(T15, T16)", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 146 [arrowhead = none ];
146 -> 5;

147 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 147 [arrowhead = none ];
147 -> 6;

148 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
5 -> 148 [arrowhead = none ];
148 -> 7;

149 [label="SPLIT 2\nnew knowledge:\nT16 is ground\nT15 is ground\nT18 is ground\nreplacements:X12 -> T18", fontsize=14, style = filled, fillcolor = violet];
5 -> 149 [arrowhead = none ];
149 -> 8;

150 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
7 -> 150 [arrowhead = none ];
150 -> 9;

151 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
8 -> 151 [arrowhead = none ];
151 -> 139;

152 [label="BACKTRACK\nfor clause: reduce(X1, s(s(s(s(0)))), X2, []) :- !because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 152 [arrowhead = none ];
152 -> 10;

153 [label="EVAL with clause\nreduce([], X26, X27, []) :- !_3.\nand substitutionT16 -> [],\nX26 -> s(0),\nT15 -> T23,\nX27 -> T23,\nX12 -> []", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 153 [arrowhead = none ];
153 -> 11;

154 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 154 [arrowhead = none ];
154 -> 12;

155 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 155 [arrowhead = none ];
155 -> 13;

156 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
12 -> 156 [arrowhead = none ];
156 -> 15;

157 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
12 -> 157 [arrowhead = none ];
157 -> 16;

158 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
13 -> 158 [arrowhead = none ];
158 -> 14;

159 [label="EVAL with clause\nreduce(.(X48, X49), X50, X48, X51) :- ','(vowel_h_w_y(X48), ','(!_3, reduce(X49, X50, X48, X51))).\nand substitutionX48 -> T32,\nX49 -> T33,\nT16 -> .(T32, T33),\nX50 -> s(0),\nT15 -> T32,\nX12 -> X52,\nX51 -> X52", fontsize=14, style = filled, fillcolor = lightblue];
15 -> 159 [arrowhead = none ];
159 -> 17;

160 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
15 -> 160 [arrowhead = none ];
160 -> 18;

161 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
16 -> 161 [arrowhead = none ];
161 -> 44;

162 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
16 -> 162 [arrowhead = none ];
162 -> 45;

163 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
17 -> 163 [arrowhead = none ];
163 -> 19;

164 [label="EVAL with clause\nvowel_h_w_y(97).\nand substitutionT32 -> 97", fontsize=14, style = filled, fillcolor = lightblue];
19 -> 164 [arrowhead = none ];
164 -> 20;

165 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
19 -> 165 [arrowhead = none ];
165 -> 21;

166 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
20 -> 166 [arrowhead = none ];
166 -> 22;

167 [label="EVAL with clause\nvowel_h_w_y(101).\nand substitutionT32 -> 101", fontsize=14, style = filled, fillcolor = lightblue];
21 -> 167 [arrowhead = none ];
167 -> 23;

168 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
21 -> 168 [arrowhead = none ];
168 -> 24;

169 [label="INSTANCE with matching:\nT16 -> T33\nT15 -> 97\nX12 -> X52", fontsize=14, style = filled, fillcolor = lightgrey];
22 -> 169 [arrowhead = none , style=dashed];
169 -> 7[style=dashed];

170 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
23 -> 170 [arrowhead = none ];
170 -> 25;

171 [label="EVAL with clause\nvowel_h_w_y(105).\nand substitutionT32 -> 105", fontsize=14, style = filled, fillcolor = lightblue];
24 -> 171 [arrowhead = none ];
171 -> 26;

172 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
24 -> 172 [arrowhead = none ];
172 -> 27;

173 [label="INSTANCE with matching:\nT16 -> T33\nT15 -> 101\nX12 -> X52", fontsize=14, style = filled, fillcolor = lightgrey];
25 -> 173 [arrowhead = none , style=dashed];
173 -> 7[style=dashed];

174 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
26 -> 174 [arrowhead = none ];
174 -> 28;

175 [label="EVAL with clause\nvowel_h_w_y(111).\nand substitutionT32 -> 111", fontsize=14, style = filled, fillcolor = lightblue];
27 -> 175 [arrowhead = none ];
175 -> 29;

176 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
27 -> 176 [arrowhead = none ];
176 -> 30;

177 [label="INSTANCE with matching:\nT16 -> T33\nT15 -> 105\nX12 -> X52", fontsize=14, style = filled, fillcolor = lightgrey];
28 -> 177 [arrowhead = none , style=dashed];
177 -> 7[style=dashed];

178 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
29 -> 178 [arrowhead = none ];
178 -> 31;

179 [label="EVAL with clause\nvowel_h_w_y(117).\nand substitutionT32 -> 117", fontsize=14, style = filled, fillcolor = lightblue];
30 -> 179 [arrowhead = none ];
179 -> 32;

180 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
30 -> 180 [arrowhead = none ];
180 -> 33;

181 [label="INSTANCE with matching:\nT16 -> T33\nT15 -> 111\nX12 -> X52", fontsize=14, style = filled, fillcolor = lightgrey];
31 -> 181 [arrowhead = none , style=dashed];
181 -> 7[style=dashed];

182 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
32 -> 182 [arrowhead = none ];
182 -> 34;

183 [label="EVAL with clause\nvowel_h_w_y(104).\nand substitutionT32 -> 104", fontsize=14, style = filled, fillcolor = lightblue];
33 -> 183 [arrowhead = none ];
183 -> 35;

184 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
33 -> 184 [arrowhead = none ];
184 -> 36;

185 [label="INSTANCE with matching:\nT16 -> T33\nT15 -> 117\nX12 -> X52", fontsize=14, style = filled, fillcolor = lightgrey];
34 -> 185 [arrowhead = none , style=dashed];
185 -> 7[style=dashed];

186 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
35 -> 186 [arrowhead = none ];
186 -> 37;

187 [label="EVAL with clause\nvowel_h_w_y(119).\nand substitutionT32 -> 119", fontsize=14, style = filled, fillcolor = lightblue];
36 -> 187 [arrowhead = none ];
187 -> 38;

188 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
36 -> 188 [arrowhead = none ];
188 -> 39;

189 [label="INSTANCE with matching:\nT16 -> T33\nT15 -> 104\nX12 -> X52", fontsize=14, style = filled, fillcolor = lightgrey];
37 -> 189 [arrowhead = none , style=dashed];
189 -> 7[style=dashed];

190 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
38 -> 190 [arrowhead = none ];
190 -> 40;

191 [label="EVAL with clause\nvowel_h_w_y(121).\nand substitutionT32 -> 121", fontsize=14, style = filled, fillcolor = lightblue];
39 -> 191 [arrowhead = none ];
191 -> 41;

192 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
39 -> 192 [arrowhead = none ];
192 -> 42;

193 [label="INSTANCE with matching:\nT16 -> T33\nT15 -> 119\nX12 -> X52", fontsize=14, style = filled, fillcolor = lightgrey];
40 -> 193 [arrowhead = none , style=dashed];
193 -> 7[style=dashed];

194 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
41 -> 194 [arrowhead = none ];
194 -> 43;

195 [label="INSTANCE with matching:\nT16 -> T33\nT15 -> 121\nX12 -> X52", fontsize=14, style = filled, fillcolor = lightgrey];
43 -> 195 [arrowhead = none , style=dashed];
195 -> 7[style=dashed];

196 [label="EVAL with clause\nreduce(.(X89, X90), X91, X89, X92) :- ','(!_3, reduce(X90, X91, X89, X92)).\nand substitutionX89 -> T50,\nX90 -> T51,\nT16 -> .(T50, T51),\nX91 -> s(0),\nT15 -> T50,\nX12 -> X93,\nX92 -> X93", fontsize=14, style = filled, fillcolor = lightblue];
44 -> 196 [arrowhead = none ];
196 -> 46;

197 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
44 -> 197 [arrowhead = none ];
197 -> 47;

198 [label="EVAL with clause\nreduce(.(X108, X109), X110, X111, .(X108, X112)) :- reduce(X109, s(X110), X108, X112).\nand substitutionX108 -> T59,\nX109 -> T60,\nT16 -> .(T59, T60),\nX110 -> s(0),\nT15 -> T61,\nX111 -> T61,\nX112 -> X113,\nX12 -> .(T59, X113)", fontsize=14, style = filled, fillcolor = lightblue];
45 -> 198 [arrowhead = none ];
198 -> 49;

199 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
45 -> 199 [arrowhead = none ];
199 -> 50;

200 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
46 -> 200 [arrowhead = none ];
200 -> 48;

201 [label="INSTANCE with matching:\nT16 -> T51\nT15 -> T50\nX12 -> X93", fontsize=14, style = filled, fillcolor = lightgrey];
48 -> 201 [arrowhead = none , style=dashed];
201 -> 7[style=dashed];

202 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
49 -> 202 [arrowhead = none ];
202 -> 51;

203 [label="BACKTRACK\nfor clause: reduce(X1, s(s(s(s(0)))), X2, []) :- !because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
51 -> 203 [arrowhead = none ];
203 -> 52;

204 [label="EVAL with clause\nreduce([], X122, X123, []) :- !_5.\nand substitutionT60 -> [],\nX122 -> s(s(0)),\nT59 -> T66,\nX123 -> T66,\nX113 -> []", fontsize=14, style = filled, fillcolor = lightblue];
52 -> 204 [arrowhead = none ];
204 -> 53;

205 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
52 -> 205 [arrowhead = none ];
205 -> 54;

206 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
53 -> 206 [arrowhead = none ];
206 -> 55;

207 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
54 -> 207 [arrowhead = none ];
207 -> 57;

208 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
54 -> 208 [arrowhead = none ];
208 -> 58;

209 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
55 -> 209 [arrowhead = none ];
209 -> 56;

210 [label="EVAL with clause\nreduce(.(X144, X145), X146, X144, X147) :- ','(vowel_h_w_y(X144), ','(!_5, reduce(X145, X146, X144, X147))).\nand substitutionX144 -> T75,\nX145 -> T76,\nT60 -> .(T75, T76),\nX146 -> s(s(0)),\nT59 -> T75,\nX113 -> X148,\nX147 -> X148", fontsize=14, style = filled, fillcolor = lightblue];
57 -> 210 [arrowhead = none ];
210 -> 59;

211 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
57 -> 211 [arrowhead = none ];
211 -> 60;

212 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
58 -> 212 [arrowhead = none ];
212 -> 86;

213 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
58 -> 213 [arrowhead = none ];
213 -> 87;

214 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
59 -> 214 [arrowhead = none ];
214 -> 61;

215 [label="EVAL with clause\nvowel_h_w_y(97).\nand substitutionT75 -> 97", fontsize=14, style = filled, fillcolor = lightblue];
61 -> 215 [arrowhead = none ];
215 -> 62;

216 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
61 -> 216 [arrowhead = none ];
216 -> 63;

217 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
62 -> 217 [arrowhead = none ];
217 -> 64;

218 [label="EVAL with clause\nvowel_h_w_y(101).\nand substitutionT75 -> 101", fontsize=14, style = filled, fillcolor = lightblue];
63 -> 218 [arrowhead = none ];
218 -> 65;

219 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
63 -> 219 [arrowhead = none ];
219 -> 66;

220 [label="INSTANCE with matching:\nT60 -> T76\nT59 -> 97\nX113 -> X148", fontsize=14, style = filled, fillcolor = lightgrey];
64 -> 220 [arrowhead = none , style=dashed];
220 -> 49[style=dashed];

221 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
65 -> 221 [arrowhead = none ];
221 -> 67;

222 [label="EVAL with clause\nvowel_h_w_y(105).\nand substitutionT75 -> 105", fontsize=14, style = filled, fillcolor = lightblue];
66 -> 222 [arrowhead = none ];
222 -> 68;

223 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
66 -> 223 [arrowhead = none ];
223 -> 69;

224 [label="INSTANCE with matching:\nT60 -> T76\nT59 -> 101\nX113 -> X148", fontsize=14, style = filled, fillcolor = lightgrey];
67 -> 224 [arrowhead = none , style=dashed];
224 -> 49[style=dashed];

225 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
68 -> 225 [arrowhead = none ];
225 -> 70;

226 [label="EVAL with clause\nvowel_h_w_y(111).\nand substitutionT75 -> 111", fontsize=14, style = filled, fillcolor = lightblue];
69 -> 226 [arrowhead = none ];
226 -> 71;

227 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
69 -> 227 [arrowhead = none ];
227 -> 72;

228 [label="INSTANCE with matching:\nT60 -> T76\nT59 -> 105\nX113 -> X148", fontsize=14, style = filled, fillcolor = lightgrey];
70 -> 228 [arrowhead = none , style=dashed];
228 -> 49[style=dashed];

229 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
71 -> 229 [arrowhead = none ];
229 -> 73;

230 [label="EVAL with clause\nvowel_h_w_y(117).\nand substitutionT75 -> 117", fontsize=14, style = filled, fillcolor = lightblue];
72 -> 230 [arrowhead = none ];
230 -> 74;

231 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
72 -> 231 [arrowhead = none ];
231 -> 75;

232 [label="INSTANCE with matching:\nT60 -> T76\nT59 -> 111\nX113 -> X148", fontsize=14, style = filled, fillcolor = lightgrey];
73 -> 232 [arrowhead = none , style=dashed];
232 -> 49[style=dashed];

233 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
74 -> 233 [arrowhead = none ];
233 -> 76;

234 [label="EVAL with clause\nvowel_h_w_y(104).\nand substitutionT75 -> 104", fontsize=14, style = filled, fillcolor = lightblue];
75 -> 234 [arrowhead = none ];
234 -> 77;

235 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
75 -> 235 [arrowhead = none ];
235 -> 78;

236 [label="INSTANCE with matching:\nT60 -> T76\nT59 -> 117\nX113 -> X148", fontsize=14, style = filled, fillcolor = lightgrey];
76 -> 236 [arrowhead = none , style=dashed];
236 -> 49[style=dashed];

237 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
77 -> 237 [arrowhead = none ];
237 -> 79;

238 [label="EVAL with clause\nvowel_h_w_y(119).\nand substitutionT75 -> 119", fontsize=14, style = filled, fillcolor = lightblue];
78 -> 238 [arrowhead = none ];
238 -> 80;

239 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
78 -> 239 [arrowhead = none ];
239 -> 81;

240 [label="INSTANCE with matching:\nT60 -> T76\nT59 -> 104\nX113 -> X148", fontsize=14, style = filled, fillcolor = lightgrey];
79 -> 240 [arrowhead = none , style=dashed];
240 -> 49[style=dashed];

241 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
80 -> 241 [arrowhead = none ];
241 -> 82;

242 [label="EVAL with clause\nvowel_h_w_y(121).\nand substitutionT75 -> 121", fontsize=14, style = filled, fillcolor = lightblue];
81 -> 242 [arrowhead = none ];
242 -> 83;

243 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
81 -> 243 [arrowhead = none ];
243 -> 84;

244 [label="INSTANCE with matching:\nT60 -> T76\nT59 -> 119\nX113 -> X148", fontsize=14, style = filled, fillcolor = lightgrey];
82 -> 244 [arrowhead = none , style=dashed];
244 -> 49[style=dashed];

245 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
83 -> 245 [arrowhead = none ];
245 -> 85;

246 [label="INSTANCE with matching:\nT60 -> T76\nT59 -> 121\nX113 -> X148", fontsize=14, style = filled, fillcolor = lightgrey];
85 -> 246 [arrowhead = none , style=dashed];
246 -> 49[style=dashed];

247 [label="EVAL with clause\nreduce(.(X185, X186), X187, X185, X188) :- ','(!_5, reduce(X186, X187, X185, X188)).\nand substitutionX185 -> T93,\nX186 -> T94,\nT60 -> .(T93, T94),\nX187 -> s(s(0)),\nT59 -> T93,\nX113 -> X189,\nX188 -> X189", fontsize=14, style = filled, fillcolor = lightblue];
86 -> 247 [arrowhead = none ];
247 -> 88;

248 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
86 -> 248 [arrowhead = none ];
248 -> 89;

249 [label="EVAL with clause\nreduce(.(X204, X205), X206, X207, .(X204, X208)) :- reduce(X205, s(X206), X204, X208).\nand substitutionX204 -> T102,\nX205 -> T103,\nT60 -> .(T102, T103),\nX206 -> s(s(0)),\nT59 -> T104,\nX207 -> T104,\nX208 -> X209,\nX113 -> .(T102, X209)", fontsize=14, style = filled, fillcolor = lightblue];
87 -> 249 [arrowhead = none ];
249 -> 91;

250 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
87 -> 250 [arrowhead = none ];
250 -> 92;

251 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
88 -> 251 [arrowhead = none ];
251 -> 90;

252 [label="INSTANCE with matching:\nT60 -> T94\nT59 -> T93\nX113 -> X189", fontsize=14, style = filled, fillcolor = lightgrey];
90 -> 252 [arrowhead = none , style=dashed];
252 -> 49[style=dashed];

253 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
91 -> 253 [arrowhead = none ];
253 -> 93;

254 [label="BACKTRACK\nfor clause: reduce(X1, s(s(s(s(0)))), X2, []) :- !because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
93 -> 254 [arrowhead = none ];
254 -> 94;

255 [label="EVAL with clause\nreduce([], X218, X219, []) :- !_7.\nand substitutionT103 -> [],\nX218 -> s(s(s(0))),\nT102 -> T109,\nX219 -> T109,\nX209 -> []", fontsize=14, style = filled, fillcolor = lightblue];
94 -> 255 [arrowhead = none ];
255 -> 95;

256 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
94 -> 256 [arrowhead = none ];
256 -> 96;

257 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
95 -> 257 [arrowhead = none ];
257 -> 97;

258 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
96 -> 258 [arrowhead = none ];
258 -> 99;

259 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
96 -> 259 [arrowhead = none ];
259 -> 100;

260 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
97 -> 260 [arrowhead = none ];
260 -> 98;

261 [label="EVAL with clause\nreduce(.(X240, X241), X242, X240, X243) :- ','(vowel_h_w_y(X240), ','(!_7, reduce(X241, X242, X240, X243))).\nand substitutionX240 -> T118,\nX241 -> T119,\nT103 -> .(T118, T119),\nX242 -> s(s(s(0))),\nT102 -> T118,\nX209 -> X244,\nX243 -> X244", fontsize=14, style = filled, fillcolor = lightblue];
99 -> 261 [arrowhead = none ];
261 -> 101;

262 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
99 -> 262 [arrowhead = none ];
262 -> 102;

263 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
100 -> 263 [arrowhead = none ];
263 -> 128;

264 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
100 -> 264 [arrowhead = none ];
264 -> 129;

265 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
101 -> 265 [arrowhead = none ];
265 -> 103;

266 [label="EVAL with clause\nvowel_h_w_y(97).\nand substitutionT118 -> 97", fontsize=14, style = filled, fillcolor = lightblue];
103 -> 266 [arrowhead = none ];
266 -> 104;

267 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
103 -> 267 [arrowhead = none ];
267 -> 105;

268 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
104 -> 268 [arrowhead = none ];
268 -> 106;

269 [label="EVAL with clause\nvowel_h_w_y(101).\nand substitutionT118 -> 101", fontsize=14, style = filled, fillcolor = lightblue];
105 -> 269 [arrowhead = none ];
269 -> 107;

270 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
105 -> 270 [arrowhead = none ];
270 -> 108;

271 [label="INSTANCE with matching:\nT103 -> T119\nT102 -> 97\nX209 -> X244", fontsize=14, style = filled, fillcolor = lightgrey];
106 -> 271 [arrowhead = none , style=dashed];
271 -> 91[style=dashed];

272 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
107 -> 272 [arrowhead = none ];
272 -> 109;

273 [label="EVAL with clause\nvowel_h_w_y(105).\nand substitutionT118 -> 105", fontsize=14, style = filled, fillcolor = lightblue];
108 -> 273 [arrowhead = none ];
273 -> 110;

274 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
108 -> 274 [arrowhead = none ];
274 -> 111;

275 [label="INSTANCE with matching:\nT103 -> T119\nT102 -> 101\nX209 -> X244", fontsize=14, style = filled, fillcolor = lightgrey];
109 -> 275 [arrowhead = none , style=dashed];
275 -> 91[style=dashed];

276 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
110 -> 276 [arrowhead = none ];
276 -> 112;

277 [label="EVAL with clause\nvowel_h_w_y(111).\nand substitutionT118 -> 111", fontsize=14, style = filled, fillcolor = lightblue];
111 -> 277 [arrowhead = none ];
277 -> 113;

278 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
111 -> 278 [arrowhead = none ];
278 -> 114;

279 [label="INSTANCE with matching:\nT103 -> T119\nT102 -> 105\nX209 -> X244", fontsize=14, style = filled, fillcolor = lightgrey];
112 -> 279 [arrowhead = none , style=dashed];
279 -> 91[style=dashed];

280 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
113 -> 280 [arrowhead = none ];
280 -> 115;

281 [label="EVAL with clause\nvowel_h_w_y(117).\nand substitutionT118 -> 117", fontsize=14, style = filled, fillcolor = lightblue];
114 -> 281 [arrowhead = none ];
281 -> 116;

282 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
114 -> 282 [arrowhead = none ];
282 -> 117;

283 [label="INSTANCE with matching:\nT103 -> T119\nT102 -> 111\nX209 -> X244", fontsize=14, style = filled, fillcolor = lightgrey];
115 -> 283 [arrowhead = none , style=dashed];
283 -> 91[style=dashed];

284 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
116 -> 284 [arrowhead = none ];
284 -> 118;

285 [label="EVAL with clause\nvowel_h_w_y(104).\nand substitutionT118 -> 104", fontsize=14, style = filled, fillcolor = lightblue];
117 -> 285 [arrowhead = none ];
285 -> 119;

286 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
117 -> 286 [arrowhead = none ];
286 -> 120;

287 [label="INSTANCE with matching:\nT103 -> T119\nT102 -> 117\nX209 -> X244", fontsize=14, style = filled, fillcolor = lightgrey];
118 -> 287 [arrowhead = none , style=dashed];
287 -> 91[style=dashed];

288 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
119 -> 288 [arrowhead = none ];
288 -> 121;

289 [label="EVAL with clause\nvowel_h_w_y(119).\nand substitutionT118 -> 119", fontsize=14, style = filled, fillcolor = lightblue];
120 -> 289 [arrowhead = none ];
289 -> 122;

290 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
120 -> 290 [arrowhead = none ];
290 -> 123;

291 [label="INSTANCE with matching:\nT103 -> T119\nT102 -> 104\nX209 -> X244", fontsize=14, style = filled, fillcolor = lightgrey];
121 -> 291 [arrowhead = none , style=dashed];
291 -> 91[style=dashed];

292 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
122 -> 292 [arrowhead = none ];
292 -> 124;

293 [label="EVAL with clause\nvowel_h_w_y(121).\nand substitutionT118 -> 121", fontsize=14, style = filled, fillcolor = lightblue];
123 -> 293 [arrowhead = none ];
293 -> 125;

294 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
123 -> 294 [arrowhead = none ];
294 -> 126;

295 [label="INSTANCE with matching:\nT103 -> T119\nT102 -> 119\nX209 -> X244", fontsize=14, style = filled, fillcolor = lightgrey];
124 -> 295 [arrowhead = none , style=dashed];
295 -> 91[style=dashed];

296 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
125 -> 296 [arrowhead = none ];
296 -> 127;

297 [label="INSTANCE with matching:\nT103 -> T119\nT102 -> 121\nX209 -> X244", fontsize=14, style = filled, fillcolor = lightgrey];
127 -> 297 [arrowhead = none , style=dashed];
297 -> 91[style=dashed];

298 [label="EVAL with clause\nreduce(.(X281, X282), X283, X281, X284) :- ','(!_7, reduce(X282, X283, X281, X284)).\nand substitutionX281 -> T136,\nX282 -> T137,\nT103 -> .(T136, T137),\nX283 -> s(s(s(0))),\nT102 -> T136,\nX209 -> X285,\nX284 -> X285", fontsize=14, style = filled, fillcolor = lightblue];
128 -> 298 [arrowhead = none ];
298 -> 130;

299 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
128 -> 299 [arrowhead = none ];
299 -> 131;

300 [label="EVAL with clause\nreduce(.(X300, X301), X302, X303, .(X300, X304)) :- reduce(X301, s(X302), X300, X304).\nand substitutionX300 -> T145,\nX301 -> T146,\nT103 -> .(T145, T146),\nX302 -> s(s(s(0))),\nT102 -> T147,\nX303 -> T147,\nX304 -> X305,\nX209 -> .(T145, X305)", fontsize=14, style = filled, fillcolor = lightblue];
129 -> 300 [arrowhead = none ];
300 -> 133;

301 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
129 -> 301 [arrowhead = none ];
301 -> 134;

302 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
130 -> 302 [arrowhead = none ];
302 -> 132;

303 [label="INSTANCE with matching:\nT103 -> T137\nT102 -> T136\nX209 -> X285", fontsize=14, style = filled, fillcolor = lightgrey];
132 -> 303 [arrowhead = none , style=dashed];
303 -> 91[style=dashed];

304 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
133 -> 304 [arrowhead = none ];
304 -> 135;

305 [label="ONLY EVAL with clause\nreduce(X312, s(s(s(s(0)))), X313, []) :- !_9.\nand substitutionT146 -> T154,\nX312 -> T154,\nT145 -> T155,\nX313 -> T155,\nX305 -> []", fontsize=14, style = filled, fillcolor = lightblue];
135 -> 305 [arrowhead = none ];
305 -> 136;

306 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
136 -> 306 [arrowhead = none ];
306 -> 137;

307 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
137 -> 307 [arrowhead = none ];
307 -> 138;

308 [label="EVAL with clause\neq(X316, X316).\nand substitutionT7 -> T158,\nX316 -> T158,\nT18 -> T158", fontsize=14, style = filled, fillcolor = lightblue];
139 -> 308 [arrowhead = none ];
308 -> 140;

309 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
139 -> 309 [arrowhead = none ];
309 -> 141;

310 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
140 -> 310 [arrowhead = none ];
310 -> 142;

}

(2) Obligation:

Clauses:

reduceA([], T23, []).
reduceA(.(97, T33), 97, X52) :- reduceA(T33, 97, X52).
reduceA(.(101, T33), 101, X52) :- reduceA(T33, 101, X52).
reduceA(.(105, T33), 105, X52) :- reduceA(T33, 105, X52).
reduceA(.(111, T33), 111, X52) :- reduceA(T33, 111, X52).
reduceA(.(117, T33), 117, X52) :- reduceA(T33, 117, X52).
reduceA(.(104, T33), 104, X52) :- reduceA(T33, 104, X52).
reduceA(.(119, T33), 119, X52) :- reduceA(T33, 119, X52).
reduceA(.(121, T33), 121, X52) :- reduceA(T33, 121, X52).
reduceA(.(T50, T51), T50, X93) :- reduceA(T51, T50, X93).
reduceA(.(T59, T60), T61, .(T59, X113)) :- reduceB(T60, T59, X113).
reduceB([], T66, []).
reduceB(.(97, T76), 97, X148) :- reduceB(T76, 97, X148).
reduceB(.(101, T76), 101, X148) :- reduceB(T76, 101, X148).
reduceB(.(105, T76), 105, X148) :- reduceB(T76, 105, X148).
reduceB(.(111, T76), 111, X148) :- reduceB(T76, 111, X148).
reduceB(.(117, T76), 117, X148) :- reduceB(T76, 117, X148).
reduceB(.(104, T76), 104, X148) :- reduceB(T76, 104, X148).
reduceB(.(119, T76), 119, X148) :- reduceB(T76, 119, X148).
reduceB(.(121, T76), 121, X148) :- reduceB(T76, 121, X148).
reduceB(.(T93, T94), T93, X189) :- reduceB(T94, T93, X189).
reduceB(.(T102, T103), T104, .(T102, X209)) :- reduceC(T103, T102, X209).
reduceC([], T109, []).
reduceC(.(97, T119), 97, X244) :- reduceC(T119, 97, X244).
reduceC(.(101, T119), 101, X244) :- reduceC(T119, 101, X244).
reduceC(.(105, T119), 105, X244) :- reduceC(T119, 105, X244).
reduceC(.(111, T119), 111, X244) :- reduceC(T119, 111, X244).
reduceC(.(117, T119), 117, X244) :- reduceC(T119, 117, X244).
reduceC(.(104, T119), 104, X244) :- reduceC(T119, 104, X244).
reduceC(.(119, T119), 119, X244) :- reduceC(T119, 119, X244).
reduceC(.(121, T119), 121, X244) :- reduceC(T119, 121, X244).
reduceC(.(T136, T137), T136, X285) :- reduceC(T137, T136, X285).
reduceC(.(T155, T154), T147, .(T155, [])).
goalD(.(T15, T16), T7) :- reduceA(T16, T15, X12).
goalD(.(T15, T16), T158) :- reduceA(T16, T15, T158).

Query: goalD(g,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:
goalD_in: (b,f)
reduceA_in: (b,b,f)
reduceB_in: (b,b,f)
reduceC_in: (b,b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

goalD_in_ga(.(T15, T16), T7) → U30_ga(T15, T16, T7, reduceA_in_gga(T16, T15, X12))
reduceA_in_gga([], T23, []) → reduceA_out_gga([], T23, [])
reduceA_in_gga(.(97, T33), 97, X52) → U1_gga(T33, X52, reduceA_in_gga(T33, 97, X52))
reduceA_in_gga(.(101, T33), 101, X52) → U2_gga(T33, X52, reduceA_in_gga(T33, 101, X52))
reduceA_in_gga(.(105, T33), 105, X52) → U3_gga(T33, X52, reduceA_in_gga(T33, 105, X52))
reduceA_in_gga(.(111, T33), 111, X52) → U4_gga(T33, X52, reduceA_in_gga(T33, 111, X52))
reduceA_in_gga(.(117, T33), 117, X52) → U5_gga(T33, X52, reduceA_in_gga(T33, 117, X52))
reduceA_in_gga(.(104, T33), 104, X52) → U6_gga(T33, X52, reduceA_in_gga(T33, 104, X52))
reduceA_in_gga(.(119, T33), 119, X52) → U7_gga(T33, X52, reduceA_in_gga(T33, 119, X52))
reduceA_in_gga(.(121, T33), 121, X52) → U8_gga(T33, X52, reduceA_in_gga(T33, 121, X52))
reduceA_in_gga(.(T50, T51), T50, X93) → U9_gga(T50, T51, X93, reduceA_in_gga(T51, T50, X93))
reduceA_in_gga(.(T59, T60), T61, .(T59, X113)) → U10_gga(T59, T60, T61, X113, reduceB_in_gga(T60, T59, X113))
reduceB_in_gga([], T66, []) → reduceB_out_gga([], T66, [])
reduceB_in_gga(.(97, T76), 97, X148) → U11_gga(T76, X148, reduceB_in_gga(T76, 97, X148))
reduceB_in_gga(.(101, T76), 101, X148) → U12_gga(T76, X148, reduceB_in_gga(T76, 101, X148))
reduceB_in_gga(.(105, T76), 105, X148) → U13_gga(T76, X148, reduceB_in_gga(T76, 105, X148))
reduceB_in_gga(.(111, T76), 111, X148) → U14_gga(T76, X148, reduceB_in_gga(T76, 111, X148))
reduceB_in_gga(.(117, T76), 117, X148) → U15_gga(T76, X148, reduceB_in_gga(T76, 117, X148))
reduceB_in_gga(.(104, T76), 104, X148) → U16_gga(T76, X148, reduceB_in_gga(T76, 104, X148))
reduceB_in_gga(.(119, T76), 119, X148) → U17_gga(T76, X148, reduceB_in_gga(T76, 119, X148))
reduceB_in_gga(.(121, T76), 121, X148) → U18_gga(T76, X148, reduceB_in_gga(T76, 121, X148))
reduceB_in_gga(.(T93, T94), T93, X189) → U19_gga(T93, T94, X189, reduceB_in_gga(T94, T93, X189))
reduceB_in_gga(.(T102, T103), T104, .(T102, X209)) → U20_gga(T102, T103, T104, X209, reduceC_in_gga(T103, T102, X209))
reduceC_in_gga([], T109, []) → reduceC_out_gga([], T109, [])
reduceC_in_gga(.(97, T119), 97, X244) → U21_gga(T119, X244, reduceC_in_gga(T119, 97, X244))
reduceC_in_gga(.(101, T119), 101, X244) → U22_gga(T119, X244, reduceC_in_gga(T119, 101, X244))
reduceC_in_gga(.(105, T119), 105, X244) → U23_gga(T119, X244, reduceC_in_gga(T119, 105, X244))
reduceC_in_gga(.(111, T119), 111, X244) → U24_gga(T119, X244, reduceC_in_gga(T119, 111, X244))
reduceC_in_gga(.(117, T119), 117, X244) → U25_gga(T119, X244, reduceC_in_gga(T119, 117, X244))
reduceC_in_gga(.(104, T119), 104, X244) → U26_gga(T119, X244, reduceC_in_gga(T119, 104, X244))
reduceC_in_gga(.(119, T119), 119, X244) → U27_gga(T119, X244, reduceC_in_gga(T119, 119, X244))
reduceC_in_gga(.(121, T119), 121, X244) → U28_gga(T119, X244, reduceC_in_gga(T119, 121, X244))
reduceC_in_gga(.(T136, T137), T136, X285) → U29_gga(T136, T137, X285, reduceC_in_gga(T137, T136, X285))
reduceC_in_gga(.(T155, T154), T147, .(T155, [])) → reduceC_out_gga(.(T155, T154), T147, .(T155, []))
U29_gga(T136, T137, X285, reduceC_out_gga(T137, T136, X285)) → reduceC_out_gga(.(T136, T137), T136, X285)
U28_gga(T119, X244, reduceC_out_gga(T119, 121, X244)) → reduceC_out_gga(.(121, T119), 121, X244)
U27_gga(T119, X244, reduceC_out_gga(T119, 119, X244)) → reduceC_out_gga(.(119, T119), 119, X244)
U26_gga(T119, X244, reduceC_out_gga(T119, 104, X244)) → reduceC_out_gga(.(104, T119), 104, X244)
U25_gga(T119, X244, reduceC_out_gga(T119, 117, X244)) → reduceC_out_gga(.(117, T119), 117, X244)
U24_gga(T119, X244, reduceC_out_gga(T119, 111, X244)) → reduceC_out_gga(.(111, T119), 111, X244)
U23_gga(T119, X244, reduceC_out_gga(T119, 105, X244)) → reduceC_out_gga(.(105, T119), 105, X244)
U22_gga(T119, X244, reduceC_out_gga(T119, 101, X244)) → reduceC_out_gga(.(101, T119), 101, X244)
U21_gga(T119, X244, reduceC_out_gga(T119, 97, X244)) → reduceC_out_gga(.(97, T119), 97, X244)
U20_gga(T102, T103, T104, X209, reduceC_out_gga(T103, T102, X209)) → reduceB_out_gga(.(T102, T103), T104, .(T102, X209))
U19_gga(T93, T94, X189, reduceB_out_gga(T94, T93, X189)) → reduceB_out_gga(.(T93, T94), T93, X189)
U18_gga(T76, X148, reduceB_out_gga(T76, 121, X148)) → reduceB_out_gga(.(121, T76), 121, X148)
U17_gga(T76, X148, reduceB_out_gga(T76, 119, X148)) → reduceB_out_gga(.(119, T76), 119, X148)
U16_gga(T76, X148, reduceB_out_gga(T76, 104, X148)) → reduceB_out_gga(.(104, T76), 104, X148)
U15_gga(T76, X148, reduceB_out_gga(T76, 117, X148)) → reduceB_out_gga(.(117, T76), 117, X148)
U14_gga(T76, X148, reduceB_out_gga(T76, 111, X148)) → reduceB_out_gga(.(111, T76), 111, X148)
U13_gga(T76, X148, reduceB_out_gga(T76, 105, X148)) → reduceB_out_gga(.(105, T76), 105, X148)
U12_gga(T76, X148, reduceB_out_gga(T76, 101, X148)) → reduceB_out_gga(.(101, T76), 101, X148)
U11_gga(T76, X148, reduceB_out_gga(T76, 97, X148)) → reduceB_out_gga(.(97, T76), 97, X148)
U10_gga(T59, T60, T61, X113, reduceB_out_gga(T60, T59, X113)) → reduceA_out_gga(.(T59, T60), T61, .(T59, X113))
U9_gga(T50, T51, X93, reduceA_out_gga(T51, T50, X93)) → reduceA_out_gga(.(T50, T51), T50, X93)
U8_gga(T33, X52, reduceA_out_gga(T33, 121, X52)) → reduceA_out_gga(.(121, T33), 121, X52)
U7_gga(T33, X52, reduceA_out_gga(T33, 119, X52)) → reduceA_out_gga(.(119, T33), 119, X52)
U6_gga(T33, X52, reduceA_out_gga(T33, 104, X52)) → reduceA_out_gga(.(104, T33), 104, X52)
U5_gga(T33, X52, reduceA_out_gga(T33, 117, X52)) → reduceA_out_gga(.(117, T33), 117, X52)
U4_gga(T33, X52, reduceA_out_gga(T33, 111, X52)) → reduceA_out_gga(.(111, T33), 111, X52)
U3_gga(T33, X52, reduceA_out_gga(T33, 105, X52)) → reduceA_out_gga(.(105, T33), 105, X52)
U2_gga(T33, X52, reduceA_out_gga(T33, 101, X52)) → reduceA_out_gga(.(101, T33), 101, X52)
U1_gga(T33, X52, reduceA_out_gga(T33, 97, X52)) → reduceA_out_gga(.(97, T33), 97, X52)
U30_ga(T15, T16, T7, reduceA_out_gga(T16, T15, X12)) → goalD_out_ga(.(T15, T16), T7)
goalD_in_ga(.(T15, T16), T158) → U31_ga(T15, T16, T158, reduceA_in_gga(T16, T15, T158))
U31_ga(T15, T16, T158, reduceA_out_gga(T16, T15, T158)) → goalD_out_ga(.(T15, T16), T158)

The argument filtering Pi contains the following mapping:
goalD_in_ga(x1, x2)  =  goalD_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U30_ga(x1, x2, x3, x4)  =  U30_ga(x4)
reduceA_in_gga(x1, x2, x3)  =  reduceA_in_gga(x1, x2)
[]  =  []
reduceA_out_gga(x1, x2, x3)  =  reduceA_out_gga(x3)
97  =  97
U1_gga(x1, x2, x3)  =  U1_gga(x3)
101  =  101
U2_gga(x1, x2, x3)  =  U2_gga(x3)
105  =  105
U3_gga(x1, x2, x3)  =  U3_gga(x3)
111  =  111
U4_gga(x1, x2, x3)  =  U4_gga(x3)
117  =  117
U5_gga(x1, x2, x3)  =  U5_gga(x3)
104  =  104
U6_gga(x1, x2, x3)  =  U6_gga(x3)
119  =  119
U7_gga(x1, x2, x3)  =  U7_gga(x3)
121  =  121
U8_gga(x1, x2, x3)  =  U8_gga(x3)
U9_gga(x1, x2, x3, x4)  =  U9_gga(x4)
U10_gga(x1, x2, x3, x4, x5)  =  U10_gga(x1, x5)
reduceB_in_gga(x1, x2, x3)  =  reduceB_in_gga(x1, x2)
reduceB_out_gga(x1, x2, x3)  =  reduceB_out_gga(x3)
U11_gga(x1, x2, x3)  =  U11_gga(x3)
U12_gga(x1, x2, x3)  =  U12_gga(x3)
U13_gga(x1, x2, x3)  =  U13_gga(x3)
U14_gga(x1, x2, x3)  =  U14_gga(x3)
U15_gga(x1, x2, x3)  =  U15_gga(x3)
U16_gga(x1, x2, x3)  =  U16_gga(x3)
U17_gga(x1, x2, x3)  =  U17_gga(x3)
U18_gga(x1, x2, x3)  =  U18_gga(x3)
U19_gga(x1, x2, x3, x4)  =  U19_gga(x4)
U20_gga(x1, x2, x3, x4, x5)  =  U20_gga(x1, x5)
reduceC_in_gga(x1, x2, x3)  =  reduceC_in_gga(x1, x2)
reduceC_out_gga(x1, x2, x3)  =  reduceC_out_gga(x3)
U21_gga(x1, x2, x3)  =  U21_gga(x3)
U22_gga(x1, x2, x3)  =  U22_gga(x3)
U23_gga(x1, x2, x3)  =  U23_gga(x3)
U24_gga(x1, x2, x3)  =  U24_gga(x3)
U25_gga(x1, x2, x3)  =  U25_gga(x3)
U26_gga(x1, x2, x3)  =  U26_gga(x3)
U27_gga(x1, x2, x3)  =  U27_gga(x3)
U28_gga(x1, x2, x3)  =  U28_gga(x3)
U29_gga(x1, x2, x3, x4)  =  U29_gga(x4)
goalD_out_ga(x1, x2)  =  goalD_out_ga
U31_ga(x1, x2, x3, x4)  =  U31_ga(x4)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

goalD_in_ga(.(T15, T16), T7) → U30_ga(T15, T16, T7, reduceA_in_gga(T16, T15, X12))
reduceA_in_gga([], T23, []) → reduceA_out_gga([], T23, [])
reduceA_in_gga(.(97, T33), 97, X52) → U1_gga(T33, X52, reduceA_in_gga(T33, 97, X52))
reduceA_in_gga(.(101, T33), 101, X52) → U2_gga(T33, X52, reduceA_in_gga(T33, 101, X52))
reduceA_in_gga(.(105, T33), 105, X52) → U3_gga(T33, X52, reduceA_in_gga(T33, 105, X52))
reduceA_in_gga(.(111, T33), 111, X52) → U4_gga(T33, X52, reduceA_in_gga(T33, 111, X52))
reduceA_in_gga(.(117, T33), 117, X52) → U5_gga(T33, X52, reduceA_in_gga(T33, 117, X52))
reduceA_in_gga(.(104, T33), 104, X52) → U6_gga(T33, X52, reduceA_in_gga(T33, 104, X52))
reduceA_in_gga(.(119, T33), 119, X52) → U7_gga(T33, X52, reduceA_in_gga(T33, 119, X52))
reduceA_in_gga(.(121, T33), 121, X52) → U8_gga(T33, X52, reduceA_in_gga(T33, 121, X52))
reduceA_in_gga(.(T50, T51), T50, X93) → U9_gga(T50, T51, X93, reduceA_in_gga(T51, T50, X93))
reduceA_in_gga(.(T59, T60), T61, .(T59, X113)) → U10_gga(T59, T60, T61, X113, reduceB_in_gga(T60, T59, X113))
reduceB_in_gga([], T66, []) → reduceB_out_gga([], T66, [])
reduceB_in_gga(.(97, T76), 97, X148) → U11_gga(T76, X148, reduceB_in_gga(T76, 97, X148))
reduceB_in_gga(.(101, T76), 101, X148) → U12_gga(T76, X148, reduceB_in_gga(T76, 101, X148))
reduceB_in_gga(.(105, T76), 105, X148) → U13_gga(T76, X148, reduceB_in_gga(T76, 105, X148))
reduceB_in_gga(.(111, T76), 111, X148) → U14_gga(T76, X148, reduceB_in_gga(T76, 111, X148))
reduceB_in_gga(.(117, T76), 117, X148) → U15_gga(T76, X148, reduceB_in_gga(T76, 117, X148))
reduceB_in_gga(.(104, T76), 104, X148) → U16_gga(T76, X148, reduceB_in_gga(T76, 104, X148))
reduceB_in_gga(.(119, T76), 119, X148) → U17_gga(T76, X148, reduceB_in_gga(T76, 119, X148))
reduceB_in_gga(.(121, T76), 121, X148) → U18_gga(T76, X148, reduceB_in_gga(T76, 121, X148))
reduceB_in_gga(.(T93, T94), T93, X189) → U19_gga(T93, T94, X189, reduceB_in_gga(T94, T93, X189))
reduceB_in_gga(.(T102, T103), T104, .(T102, X209)) → U20_gga(T102, T103, T104, X209, reduceC_in_gga(T103, T102, X209))
reduceC_in_gga([], T109, []) → reduceC_out_gga([], T109, [])
reduceC_in_gga(.(97, T119), 97, X244) → U21_gga(T119, X244, reduceC_in_gga(T119, 97, X244))
reduceC_in_gga(.(101, T119), 101, X244) → U22_gga(T119, X244, reduceC_in_gga(T119, 101, X244))
reduceC_in_gga(.(105, T119), 105, X244) → U23_gga(T119, X244, reduceC_in_gga(T119, 105, X244))
reduceC_in_gga(.(111, T119), 111, X244) → U24_gga(T119, X244, reduceC_in_gga(T119, 111, X244))
reduceC_in_gga(.(117, T119), 117, X244) → U25_gga(T119, X244, reduceC_in_gga(T119, 117, X244))
reduceC_in_gga(.(104, T119), 104, X244) → U26_gga(T119, X244, reduceC_in_gga(T119, 104, X244))
reduceC_in_gga(.(119, T119), 119, X244) → U27_gga(T119, X244, reduceC_in_gga(T119, 119, X244))
reduceC_in_gga(.(121, T119), 121, X244) → U28_gga(T119, X244, reduceC_in_gga(T119, 121, X244))
reduceC_in_gga(.(T136, T137), T136, X285) → U29_gga(T136, T137, X285, reduceC_in_gga(T137, T136, X285))
reduceC_in_gga(.(T155, T154), T147, .(T155, [])) → reduceC_out_gga(.(T155, T154), T147, .(T155, []))
U29_gga(T136, T137, X285, reduceC_out_gga(T137, T136, X285)) → reduceC_out_gga(.(T136, T137), T136, X285)
U28_gga(T119, X244, reduceC_out_gga(T119, 121, X244)) → reduceC_out_gga(.(121, T119), 121, X244)
U27_gga(T119, X244, reduceC_out_gga(T119, 119, X244)) → reduceC_out_gga(.(119, T119), 119, X244)
U26_gga(T119, X244, reduceC_out_gga(T119, 104, X244)) → reduceC_out_gga(.(104, T119), 104, X244)
U25_gga(T119, X244, reduceC_out_gga(T119, 117, X244)) → reduceC_out_gga(.(117, T119), 117, X244)
U24_gga(T119, X244, reduceC_out_gga(T119, 111, X244)) → reduceC_out_gga(.(111, T119), 111, X244)
U23_gga(T119, X244, reduceC_out_gga(T119, 105, X244)) → reduceC_out_gga(.(105, T119), 105, X244)
U22_gga(T119, X244, reduceC_out_gga(T119, 101, X244)) → reduceC_out_gga(.(101, T119), 101, X244)
U21_gga(T119, X244, reduceC_out_gga(T119, 97, X244)) → reduceC_out_gga(.(97, T119), 97, X244)
U20_gga(T102, T103, T104, X209, reduceC_out_gga(T103, T102, X209)) → reduceB_out_gga(.(T102, T103), T104, .(T102, X209))
U19_gga(T93, T94, X189, reduceB_out_gga(T94, T93, X189)) → reduceB_out_gga(.(T93, T94), T93, X189)
U18_gga(T76, X148, reduceB_out_gga(T76, 121, X148)) → reduceB_out_gga(.(121, T76), 121, X148)
U17_gga(T76, X148, reduceB_out_gga(T76, 119, X148)) → reduceB_out_gga(.(119, T76), 119, X148)
U16_gga(T76, X148, reduceB_out_gga(T76, 104, X148)) → reduceB_out_gga(.(104, T76), 104, X148)
U15_gga(T76, X148, reduceB_out_gga(T76, 117, X148)) → reduceB_out_gga(.(117, T76), 117, X148)
U14_gga(T76, X148, reduceB_out_gga(T76, 111, X148)) → reduceB_out_gga(.(111, T76), 111, X148)
U13_gga(T76, X148, reduceB_out_gga(T76, 105, X148)) → reduceB_out_gga(.(105, T76), 105, X148)
U12_gga(T76, X148, reduceB_out_gga(T76, 101, X148)) → reduceB_out_gga(.(101, T76), 101, X148)
U11_gga(T76, X148, reduceB_out_gga(T76, 97, X148)) → reduceB_out_gga(.(97, T76), 97, X148)
U10_gga(T59, T60, T61, X113, reduceB_out_gga(T60, T59, X113)) → reduceA_out_gga(.(T59, T60), T61, .(T59, X113))
U9_gga(T50, T51, X93, reduceA_out_gga(T51, T50, X93)) → reduceA_out_gga(.(T50, T51), T50, X93)
U8_gga(T33, X52, reduceA_out_gga(T33, 121, X52)) → reduceA_out_gga(.(121, T33), 121, X52)
U7_gga(T33, X52, reduceA_out_gga(T33, 119, X52)) → reduceA_out_gga(.(119, T33), 119, X52)
U6_gga(T33, X52, reduceA_out_gga(T33, 104, X52)) → reduceA_out_gga(.(104, T33), 104, X52)
U5_gga(T33, X52, reduceA_out_gga(T33, 117, X52)) → reduceA_out_gga(.(117, T33), 117, X52)
U4_gga(T33, X52, reduceA_out_gga(T33, 111, X52)) → reduceA_out_gga(.(111, T33), 111, X52)
U3_gga(T33, X52, reduceA_out_gga(T33, 105, X52)) → reduceA_out_gga(.(105, T33), 105, X52)
U2_gga(T33, X52, reduceA_out_gga(T33, 101, X52)) → reduceA_out_gga(.(101, T33), 101, X52)
U1_gga(T33, X52, reduceA_out_gga(T33, 97, X52)) → reduceA_out_gga(.(97, T33), 97, X52)
U30_ga(T15, T16, T7, reduceA_out_gga(T16, T15, X12)) → goalD_out_ga(.(T15, T16), T7)
goalD_in_ga(.(T15, T16), T158) → U31_ga(T15, T16, T158, reduceA_in_gga(T16, T15, T158))
U31_ga(T15, T16, T158, reduceA_out_gga(T16, T15, T158)) → goalD_out_ga(.(T15, T16), T158)

The argument filtering Pi contains the following mapping:
goalD_in_ga(x1, x2)  =  goalD_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U30_ga(x1, x2, x3, x4)  =  U30_ga(x4)
reduceA_in_gga(x1, x2, x3)  =  reduceA_in_gga(x1, x2)
[]  =  []
reduceA_out_gga(x1, x2, x3)  =  reduceA_out_gga(x3)
97  =  97
U1_gga(x1, x2, x3)  =  U1_gga(x3)
101  =  101
U2_gga(x1, x2, x3)  =  U2_gga(x3)
105  =  105
U3_gga(x1, x2, x3)  =  U3_gga(x3)
111  =  111
U4_gga(x1, x2, x3)  =  U4_gga(x3)
117  =  117
U5_gga(x1, x2, x3)  =  U5_gga(x3)
104  =  104
U6_gga(x1, x2, x3)  =  U6_gga(x3)
119  =  119
U7_gga(x1, x2, x3)  =  U7_gga(x3)
121  =  121
U8_gga(x1, x2, x3)  =  U8_gga(x3)
U9_gga(x1, x2, x3, x4)  =  U9_gga(x4)
U10_gga(x1, x2, x3, x4, x5)  =  U10_gga(x1, x5)
reduceB_in_gga(x1, x2, x3)  =  reduceB_in_gga(x1, x2)
reduceB_out_gga(x1, x2, x3)  =  reduceB_out_gga(x3)
U11_gga(x1, x2, x3)  =  U11_gga(x3)
U12_gga(x1, x2, x3)  =  U12_gga(x3)
U13_gga(x1, x2, x3)  =  U13_gga(x3)
U14_gga(x1, x2, x3)  =  U14_gga(x3)
U15_gga(x1, x2, x3)  =  U15_gga(x3)
U16_gga(x1, x2, x3)  =  U16_gga(x3)
U17_gga(x1, x2, x3)  =  U17_gga(x3)
U18_gga(x1, x2, x3)  =  U18_gga(x3)
U19_gga(x1, x2, x3, x4)  =  U19_gga(x4)
U20_gga(x1, x2, x3, x4, x5)  =  U20_gga(x1, x5)
reduceC_in_gga(x1, x2, x3)  =  reduceC_in_gga(x1, x2)
reduceC_out_gga(x1, x2, x3)  =  reduceC_out_gga(x3)
U21_gga(x1, x2, x3)  =  U21_gga(x3)
U22_gga(x1, x2, x3)  =  U22_gga(x3)
U23_gga(x1, x2, x3)  =  U23_gga(x3)
U24_gga(x1, x2, x3)  =  U24_gga(x3)
U25_gga(x1, x2, x3)  =  U25_gga(x3)
U26_gga(x1, x2, x3)  =  U26_gga(x3)
U27_gga(x1, x2, x3)  =  U27_gga(x3)
U28_gga(x1, x2, x3)  =  U28_gga(x3)
U29_gga(x1, x2, x3, x4)  =  U29_gga(x4)
goalD_out_ga(x1, x2)  =  goalD_out_ga
U31_ga(x1, x2, x3, x4)  =  U31_ga(x4)

(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:

GOALD_IN_GA(.(T15, T16), T7) → U30_GA(T15, T16, T7, reduceA_in_gga(T16, T15, X12))
GOALD_IN_GA(.(T15, T16), T7) → REDUCEA_IN_GGA(T16, T15, X12)
REDUCEA_IN_GGA(.(97, T33), 97, X52) → U1_GGA(T33, X52, reduceA_in_gga(T33, 97, X52))
REDUCEA_IN_GGA(.(97, T33), 97, X52) → REDUCEA_IN_GGA(T33, 97, X52)
REDUCEA_IN_GGA(.(101, T33), 101, X52) → U2_GGA(T33, X52, reduceA_in_gga(T33, 101, X52))
REDUCEA_IN_GGA(.(101, T33), 101, X52) → REDUCEA_IN_GGA(T33, 101, X52)
REDUCEA_IN_GGA(.(105, T33), 105, X52) → U3_GGA(T33, X52, reduceA_in_gga(T33, 105, X52))
REDUCEA_IN_GGA(.(105, T33), 105, X52) → REDUCEA_IN_GGA(T33, 105, X52)
REDUCEA_IN_GGA(.(111, T33), 111, X52) → U4_GGA(T33, X52, reduceA_in_gga(T33, 111, X52))
REDUCEA_IN_GGA(.(111, T33), 111, X52) → REDUCEA_IN_GGA(T33, 111, X52)
REDUCEA_IN_GGA(.(117, T33), 117, X52) → U5_GGA(T33, X52, reduceA_in_gga(T33, 117, X52))
REDUCEA_IN_GGA(.(117, T33), 117, X52) → REDUCEA_IN_GGA(T33, 117, X52)
REDUCEA_IN_GGA(.(104, T33), 104, X52) → U6_GGA(T33, X52, reduceA_in_gga(T33, 104, X52))
REDUCEA_IN_GGA(.(104, T33), 104, X52) → REDUCEA_IN_GGA(T33, 104, X52)
REDUCEA_IN_GGA(.(119, T33), 119, X52) → U7_GGA(T33, X52, reduceA_in_gga(T33, 119, X52))
REDUCEA_IN_GGA(.(119, T33), 119, X52) → REDUCEA_IN_GGA(T33, 119, X52)
REDUCEA_IN_GGA(.(121, T33), 121, X52) → U8_GGA(T33, X52, reduceA_in_gga(T33, 121, X52))
REDUCEA_IN_GGA(.(121, T33), 121, X52) → REDUCEA_IN_GGA(T33, 121, X52)
REDUCEA_IN_GGA(.(T50, T51), T50, X93) → U9_GGA(T50, T51, X93, reduceA_in_gga(T51, T50, X93))
REDUCEA_IN_GGA(.(T50, T51), T50, X93) → REDUCEA_IN_GGA(T51, T50, X93)
REDUCEA_IN_GGA(.(T59, T60), T61, .(T59, X113)) → U10_GGA(T59, T60, T61, X113, reduceB_in_gga(T60, T59, X113))
REDUCEA_IN_GGA(.(T59, T60), T61, .(T59, X113)) → REDUCEB_IN_GGA(T60, T59, X113)
REDUCEB_IN_GGA(.(97, T76), 97, X148) → U11_GGA(T76, X148, reduceB_in_gga(T76, 97, X148))
REDUCEB_IN_GGA(.(97, T76), 97, X148) → REDUCEB_IN_GGA(T76, 97, X148)
REDUCEB_IN_GGA(.(101, T76), 101, X148) → U12_GGA(T76, X148, reduceB_in_gga(T76, 101, X148))
REDUCEB_IN_GGA(.(101, T76), 101, X148) → REDUCEB_IN_GGA(T76, 101, X148)
REDUCEB_IN_GGA(.(105, T76), 105, X148) → U13_GGA(T76, X148, reduceB_in_gga(T76, 105, X148))
REDUCEB_IN_GGA(.(105, T76), 105, X148) → REDUCEB_IN_GGA(T76, 105, X148)
REDUCEB_IN_GGA(.(111, T76), 111, X148) → U14_GGA(T76, X148, reduceB_in_gga(T76, 111, X148))
REDUCEB_IN_GGA(.(111, T76), 111, X148) → REDUCEB_IN_GGA(T76, 111, X148)
REDUCEB_IN_GGA(.(117, T76), 117, X148) → U15_GGA(T76, X148, reduceB_in_gga(T76, 117, X148))
REDUCEB_IN_GGA(.(117, T76), 117, X148) → REDUCEB_IN_GGA(T76, 117, X148)
REDUCEB_IN_GGA(.(104, T76), 104, X148) → U16_GGA(T76, X148, reduceB_in_gga(T76, 104, X148))
REDUCEB_IN_GGA(.(104, T76), 104, X148) → REDUCEB_IN_GGA(T76, 104, X148)
REDUCEB_IN_GGA(.(119, T76), 119, X148) → U17_GGA(T76, X148, reduceB_in_gga(T76, 119, X148))
REDUCEB_IN_GGA(.(119, T76), 119, X148) → REDUCEB_IN_GGA(T76, 119, X148)
REDUCEB_IN_GGA(.(121, T76), 121, X148) → U18_GGA(T76, X148, reduceB_in_gga(T76, 121, X148))
REDUCEB_IN_GGA(.(121, T76), 121, X148) → REDUCEB_IN_GGA(T76, 121, X148)
REDUCEB_IN_GGA(.(T93, T94), T93, X189) → U19_GGA(T93, T94, X189, reduceB_in_gga(T94, T93, X189))
REDUCEB_IN_GGA(.(T93, T94), T93, X189) → REDUCEB_IN_GGA(T94, T93, X189)
REDUCEB_IN_GGA(.(T102, T103), T104, .(T102, X209)) → U20_GGA(T102, T103, T104, X209, reduceC_in_gga(T103, T102, X209))
REDUCEB_IN_GGA(.(T102, T103), T104, .(T102, X209)) → REDUCEC_IN_GGA(T103, T102, X209)
REDUCEC_IN_GGA(.(97, T119), 97, X244) → U21_GGA(T119, X244, reduceC_in_gga(T119, 97, X244))
REDUCEC_IN_GGA(.(97, T119), 97, X244) → REDUCEC_IN_GGA(T119, 97, X244)
REDUCEC_IN_GGA(.(101, T119), 101, X244) → U22_GGA(T119, X244, reduceC_in_gga(T119, 101, X244))
REDUCEC_IN_GGA(.(101, T119), 101, X244) → REDUCEC_IN_GGA(T119, 101, X244)
REDUCEC_IN_GGA(.(105, T119), 105, X244) → U23_GGA(T119, X244, reduceC_in_gga(T119, 105, X244))
REDUCEC_IN_GGA(.(105, T119), 105, X244) → REDUCEC_IN_GGA(T119, 105, X244)
REDUCEC_IN_GGA(.(111, T119), 111, X244) → U24_GGA(T119, X244, reduceC_in_gga(T119, 111, X244))
REDUCEC_IN_GGA(.(111, T119), 111, X244) → REDUCEC_IN_GGA(T119, 111, X244)
REDUCEC_IN_GGA(.(117, T119), 117, X244) → U25_GGA(T119, X244, reduceC_in_gga(T119, 117, X244))
REDUCEC_IN_GGA(.(117, T119), 117, X244) → REDUCEC_IN_GGA(T119, 117, X244)
REDUCEC_IN_GGA(.(104, T119), 104, X244) → U26_GGA(T119, X244, reduceC_in_gga(T119, 104, X244))
REDUCEC_IN_GGA(.(104, T119), 104, X244) → REDUCEC_IN_GGA(T119, 104, X244)
REDUCEC_IN_GGA(.(119, T119), 119, X244) → U27_GGA(T119, X244, reduceC_in_gga(T119, 119, X244))
REDUCEC_IN_GGA(.(119, T119), 119, X244) → REDUCEC_IN_GGA(T119, 119, X244)
REDUCEC_IN_GGA(.(121, T119), 121, X244) → U28_GGA(T119, X244, reduceC_in_gga(T119, 121, X244))
REDUCEC_IN_GGA(.(121, T119), 121, X244) → REDUCEC_IN_GGA(T119, 121, X244)
REDUCEC_IN_GGA(.(T136, T137), T136, X285) → U29_GGA(T136, T137, X285, reduceC_in_gga(T137, T136, X285))
REDUCEC_IN_GGA(.(T136, T137), T136, X285) → REDUCEC_IN_GGA(T137, T136, X285)
GOALD_IN_GA(.(T15, T16), T158) → U31_GA(T15, T16, T158, reduceA_in_gga(T16, T15, T158))
GOALD_IN_GA(.(T15, T16), T158) → REDUCEA_IN_GGA(T16, T15, T158)

The TRS R consists of the following rules:

goalD_in_ga(.(T15, T16), T7) → U30_ga(T15, T16, T7, reduceA_in_gga(T16, T15, X12))
reduceA_in_gga([], T23, []) → reduceA_out_gga([], T23, [])
reduceA_in_gga(.(97, T33), 97, X52) → U1_gga(T33, X52, reduceA_in_gga(T33, 97, X52))
reduceA_in_gga(.(101, T33), 101, X52) → U2_gga(T33, X52, reduceA_in_gga(T33, 101, X52))
reduceA_in_gga(.(105, T33), 105, X52) → U3_gga(T33, X52, reduceA_in_gga(T33, 105, X52))
reduceA_in_gga(.(111, T33), 111, X52) → U4_gga(T33, X52, reduceA_in_gga(T33, 111, X52))
reduceA_in_gga(.(117, T33), 117, X52) → U5_gga(T33, X52, reduceA_in_gga(T33, 117, X52))
reduceA_in_gga(.(104, T33), 104, X52) → U6_gga(T33, X52, reduceA_in_gga(T33, 104, X52))
reduceA_in_gga(.(119, T33), 119, X52) → U7_gga(T33, X52, reduceA_in_gga(T33, 119, X52))
reduceA_in_gga(.(121, T33), 121, X52) → U8_gga(T33, X52, reduceA_in_gga(T33, 121, X52))
reduceA_in_gga(.(T50, T51), T50, X93) → U9_gga(T50, T51, X93, reduceA_in_gga(T51, T50, X93))
reduceA_in_gga(.(T59, T60), T61, .(T59, X113)) → U10_gga(T59, T60, T61, X113, reduceB_in_gga(T60, T59, X113))
reduceB_in_gga([], T66, []) → reduceB_out_gga([], T66, [])
reduceB_in_gga(.(97, T76), 97, X148) → U11_gga(T76, X148, reduceB_in_gga(T76, 97, X148))
reduceB_in_gga(.(101, T76), 101, X148) → U12_gga(T76, X148, reduceB_in_gga(T76, 101, X148))
reduceB_in_gga(.(105, T76), 105, X148) → U13_gga(T76, X148, reduceB_in_gga(T76, 105, X148))
reduceB_in_gga(.(111, T76), 111, X148) → U14_gga(T76, X148, reduceB_in_gga(T76, 111, X148))
reduceB_in_gga(.(117, T76), 117, X148) → U15_gga(T76, X148, reduceB_in_gga(T76, 117, X148))
reduceB_in_gga(.(104, T76), 104, X148) → U16_gga(T76, X148, reduceB_in_gga(T76, 104, X148))
reduceB_in_gga(.(119, T76), 119, X148) → U17_gga(T76, X148, reduceB_in_gga(T76, 119, X148))
reduceB_in_gga(.(121, T76), 121, X148) → U18_gga(T76, X148, reduceB_in_gga(T76, 121, X148))
reduceB_in_gga(.(T93, T94), T93, X189) → U19_gga(T93, T94, X189, reduceB_in_gga(T94, T93, X189))
reduceB_in_gga(.(T102, T103), T104, .(T102, X209)) → U20_gga(T102, T103, T104, X209, reduceC_in_gga(T103, T102, X209))
reduceC_in_gga([], T109, []) → reduceC_out_gga([], T109, [])
reduceC_in_gga(.(97, T119), 97, X244) → U21_gga(T119, X244, reduceC_in_gga(T119, 97, X244))
reduceC_in_gga(.(101, T119), 101, X244) → U22_gga(T119, X244, reduceC_in_gga(T119, 101, X244))
reduceC_in_gga(.(105, T119), 105, X244) → U23_gga(T119, X244, reduceC_in_gga(T119, 105, X244))
reduceC_in_gga(.(111, T119), 111, X244) → U24_gga(T119, X244, reduceC_in_gga(T119, 111, X244))
reduceC_in_gga(.(117, T119), 117, X244) → U25_gga(T119, X244, reduceC_in_gga(T119, 117, X244))
reduceC_in_gga(.(104, T119), 104, X244) → U26_gga(T119, X244, reduceC_in_gga(T119, 104, X244))
reduceC_in_gga(.(119, T119), 119, X244) → U27_gga(T119, X244, reduceC_in_gga(T119, 119, X244))
reduceC_in_gga(.(121, T119), 121, X244) → U28_gga(T119, X244, reduceC_in_gga(T119, 121, X244))
reduceC_in_gga(.(T136, T137), T136, X285) → U29_gga(T136, T137, X285, reduceC_in_gga(T137, T136, X285))
reduceC_in_gga(.(T155, T154), T147, .(T155, [])) → reduceC_out_gga(.(T155, T154), T147, .(T155, []))
U29_gga(T136, T137, X285, reduceC_out_gga(T137, T136, X285)) → reduceC_out_gga(.(T136, T137), T136, X285)
U28_gga(T119, X244, reduceC_out_gga(T119, 121, X244)) → reduceC_out_gga(.(121, T119), 121, X244)
U27_gga(T119, X244, reduceC_out_gga(T119, 119, X244)) → reduceC_out_gga(.(119, T119), 119, X244)
U26_gga(T119, X244, reduceC_out_gga(T119, 104, X244)) → reduceC_out_gga(.(104, T119), 104, X244)
U25_gga(T119, X244, reduceC_out_gga(T119, 117, X244)) → reduceC_out_gga(.(117, T119), 117, X244)
U24_gga(T119, X244, reduceC_out_gga(T119, 111, X244)) → reduceC_out_gga(.(111, T119), 111, X244)
U23_gga(T119, X244, reduceC_out_gga(T119, 105, X244)) → reduceC_out_gga(.(105, T119), 105, X244)
U22_gga(T119, X244, reduceC_out_gga(T119, 101, X244)) → reduceC_out_gga(.(101, T119), 101, X244)
U21_gga(T119, X244, reduceC_out_gga(T119, 97, X244)) → reduceC_out_gga(.(97, T119), 97, X244)
U20_gga(T102, T103, T104, X209, reduceC_out_gga(T103, T102, X209)) → reduceB_out_gga(.(T102, T103), T104, .(T102, X209))
U19_gga(T93, T94, X189, reduceB_out_gga(T94, T93, X189)) → reduceB_out_gga(.(T93, T94), T93, X189)
U18_gga(T76, X148, reduceB_out_gga(T76, 121, X148)) → reduceB_out_gga(.(121, T76), 121, X148)
U17_gga(T76, X148, reduceB_out_gga(T76, 119, X148)) → reduceB_out_gga(.(119, T76), 119, X148)
U16_gga(T76, X148, reduceB_out_gga(T76, 104, X148)) → reduceB_out_gga(.(104, T76), 104, X148)
U15_gga(T76, X148, reduceB_out_gga(T76, 117, X148)) → reduceB_out_gga(.(117, T76), 117, X148)
U14_gga(T76, X148, reduceB_out_gga(T76, 111, X148)) → reduceB_out_gga(.(111, T76), 111, X148)
U13_gga(T76, X148, reduceB_out_gga(T76, 105, X148)) → reduceB_out_gga(.(105, T76), 105, X148)
U12_gga(T76, X148, reduceB_out_gga(T76, 101, X148)) → reduceB_out_gga(.(101, T76), 101, X148)
U11_gga(T76, X148, reduceB_out_gga(T76, 97, X148)) → reduceB_out_gga(.(97, T76), 97, X148)
U10_gga(T59, T60, T61, X113, reduceB_out_gga(T60, T59, X113)) → reduceA_out_gga(.(T59, T60), T61, .(T59, X113))
U9_gga(T50, T51, X93, reduceA_out_gga(T51, T50, X93)) → reduceA_out_gga(.(T50, T51), T50, X93)
U8_gga(T33, X52, reduceA_out_gga(T33, 121, X52)) → reduceA_out_gga(.(121, T33), 121, X52)
U7_gga(T33, X52, reduceA_out_gga(T33, 119, X52)) → reduceA_out_gga(.(119, T33), 119, X52)
U6_gga(T33, X52, reduceA_out_gga(T33, 104, X52)) → reduceA_out_gga(.(104, T33), 104, X52)
U5_gga(T33, X52, reduceA_out_gga(T33, 117, X52)) → reduceA_out_gga(.(117, T33), 117, X52)
U4_gga(T33, X52, reduceA_out_gga(T33, 111, X52)) → reduceA_out_gga(.(111, T33), 111, X52)
U3_gga(T33, X52, reduceA_out_gga(T33, 105, X52)) → reduceA_out_gga(.(105, T33), 105, X52)
U2_gga(T33, X52, reduceA_out_gga(T33, 101, X52)) → reduceA_out_gga(.(101, T33), 101, X52)
U1_gga(T33, X52, reduceA_out_gga(T33, 97, X52)) → reduceA_out_gga(.(97, T33), 97, X52)
U30_ga(T15, T16, T7, reduceA_out_gga(T16, T15, X12)) → goalD_out_ga(.(T15, T16), T7)
goalD_in_ga(.(T15, T16), T158) → U31_ga(T15, T16, T158, reduceA_in_gga(T16, T15, T158))
U31_ga(T15, T16, T158, reduceA_out_gga(T16, T15, T158)) → goalD_out_ga(.(T15, T16), T158)

The argument filtering Pi contains the following mapping:
goalD_in_ga(x1, x2)  =  goalD_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U30_ga(x1, x2, x3, x4)  =  U30_ga(x4)
reduceA_in_gga(x1, x2, x3)  =  reduceA_in_gga(x1, x2)
[]  =  []
reduceA_out_gga(x1, x2, x3)  =  reduceA_out_gga(x3)
97  =  97
U1_gga(x1, x2, x3)  =  U1_gga(x3)
101  =  101
U2_gga(x1, x2, x3)  =  U2_gga(x3)
105  =  105
U3_gga(x1, x2, x3)  =  U3_gga(x3)
111  =  111
U4_gga(x1, x2, x3)  =  U4_gga(x3)
117  =  117
U5_gga(x1, x2, x3)  =  U5_gga(x3)
104  =  104
U6_gga(x1, x2, x3)  =  U6_gga(x3)
119  =  119
U7_gga(x1, x2, x3)  =  U7_gga(x3)
121  =  121
U8_gga(x1, x2, x3)  =  U8_gga(x3)
U9_gga(x1, x2, x3, x4)  =  U9_gga(x4)
U10_gga(x1, x2, x3, x4, x5)  =  U10_gga(x1, x5)
reduceB_in_gga(x1, x2, x3)  =  reduceB_in_gga(x1, x2)
reduceB_out_gga(x1, x2, x3)  =  reduceB_out_gga(x3)
U11_gga(x1, x2, x3)  =  U11_gga(x3)
U12_gga(x1, x2, x3)  =  U12_gga(x3)
U13_gga(x1, x2, x3)  =  U13_gga(x3)
U14_gga(x1, x2, x3)  =  U14_gga(x3)
U15_gga(x1, x2, x3)  =  U15_gga(x3)
U16_gga(x1, x2, x3)  =  U16_gga(x3)
U17_gga(x1, x2, x3)  =  U17_gga(x3)
U18_gga(x1, x2, x3)  =  U18_gga(x3)
U19_gga(x1, x2, x3, x4)  =  U19_gga(x4)
U20_gga(x1, x2, x3, x4, x5)  =  U20_gga(x1, x5)
reduceC_in_gga(x1, x2, x3)  =  reduceC_in_gga(x1, x2)
reduceC_out_gga(x1, x2, x3)  =  reduceC_out_gga(x3)
U21_gga(x1, x2, x3)  =  U21_gga(x3)
U22_gga(x1, x2, x3)  =  U22_gga(x3)
U23_gga(x1, x2, x3)  =  U23_gga(x3)
U24_gga(x1, x2, x3)  =  U24_gga(x3)
U25_gga(x1, x2, x3)  =  U25_gga(x3)
U26_gga(x1, x2, x3)  =  U26_gga(x3)
U27_gga(x1, x2, x3)  =  U27_gga(x3)
U28_gga(x1, x2, x3)  =  U28_gga(x3)
U29_gga(x1, x2, x3, x4)  =  U29_gga(x4)
goalD_out_ga(x1, x2)  =  goalD_out_ga
U31_ga(x1, x2, x3, x4)  =  U31_ga(x4)
GOALD_IN_GA(x1, x2)  =  GOALD_IN_GA(x1)
U30_GA(x1, x2, x3, x4)  =  U30_GA(x4)
REDUCEA_IN_GGA(x1, x2, x3)  =  REDUCEA_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3)  =  U1_GGA(x3)
U2_GGA(x1, x2, x3)  =  U2_GGA(x3)
U3_GGA(x1, x2, x3)  =  U3_GGA(x3)
U4_GGA(x1, x2, x3)  =  U4_GGA(x3)
U5_GGA(x1, x2, x3)  =  U5_GGA(x3)
U6_GGA(x1, x2, x3)  =  U6_GGA(x3)
U7_GGA(x1, x2, x3)  =  U7_GGA(x3)
U8_GGA(x1, x2, x3)  =  U8_GGA(x3)
U9_GGA(x1, x2, x3, x4)  =  U9_GGA(x4)
U10_GGA(x1, x2, x3, x4, x5)  =  U10_GGA(x1, x5)
REDUCEB_IN_GGA(x1, x2, x3)  =  REDUCEB_IN_GGA(x1, x2)
U11_GGA(x1, x2, x3)  =  U11_GGA(x3)
U12_GGA(x1, x2, x3)  =  U12_GGA(x3)
U13_GGA(x1, x2, x3)  =  U13_GGA(x3)
U14_GGA(x1, x2, x3)  =  U14_GGA(x3)
U15_GGA(x1, x2, x3)  =  U15_GGA(x3)
U16_GGA(x1, x2, x3)  =  U16_GGA(x3)
U17_GGA(x1, x2, x3)  =  U17_GGA(x3)
U18_GGA(x1, x2, x3)  =  U18_GGA(x3)
U19_GGA(x1, x2, x3, x4)  =  U19_GGA(x4)
U20_GGA(x1, x2, x3, x4, x5)  =  U20_GGA(x1, x5)
REDUCEC_IN_GGA(x1, x2, x3)  =  REDUCEC_IN_GGA(x1, x2)
U21_GGA(x1, x2, x3)  =  U21_GGA(x3)
U22_GGA(x1, x2, x3)  =  U22_GGA(x3)
U23_GGA(x1, x2, x3)  =  U23_GGA(x3)
U24_GGA(x1, x2, x3)  =  U24_GGA(x3)
U25_GGA(x1, x2, x3)  =  U25_GGA(x3)
U26_GGA(x1, x2, x3)  =  U26_GGA(x3)
U27_GGA(x1, x2, x3)  =  U27_GGA(x3)
U28_GGA(x1, x2, x3)  =  U28_GGA(x3)
U29_GGA(x1, x2, x3, x4)  =  U29_GGA(x4)
U31_GA(x1, x2, x3, x4)  =  U31_GA(x4)

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

(6) Obligation:

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

GOALD_IN_GA(.(T15, T16), T7) → U30_GA(T15, T16, T7, reduceA_in_gga(T16, T15, X12))
GOALD_IN_GA(.(T15, T16), T7) → REDUCEA_IN_GGA(T16, T15, X12)
REDUCEA_IN_GGA(.(97, T33), 97, X52) → U1_GGA(T33, X52, reduceA_in_gga(T33, 97, X52))
REDUCEA_IN_GGA(.(97, T33), 97, X52) → REDUCEA_IN_GGA(T33, 97, X52)
REDUCEA_IN_GGA(.(101, T33), 101, X52) → U2_GGA(T33, X52, reduceA_in_gga(T33, 101, X52))
REDUCEA_IN_GGA(.(101, T33), 101, X52) → REDUCEA_IN_GGA(T33, 101, X52)
REDUCEA_IN_GGA(.(105, T33), 105, X52) → U3_GGA(T33, X52, reduceA_in_gga(T33, 105, X52))
REDUCEA_IN_GGA(.(105, T33), 105, X52) → REDUCEA_IN_GGA(T33, 105, X52)
REDUCEA_IN_GGA(.(111, T33), 111, X52) → U4_GGA(T33, X52, reduceA_in_gga(T33, 111, X52))
REDUCEA_IN_GGA(.(111, T33), 111, X52) → REDUCEA_IN_GGA(T33, 111, X52)
REDUCEA_IN_GGA(.(117, T33), 117, X52) → U5_GGA(T33, X52, reduceA_in_gga(T33, 117, X52))
REDUCEA_IN_GGA(.(117, T33), 117, X52) → REDUCEA_IN_GGA(T33, 117, X52)
REDUCEA_IN_GGA(.(104, T33), 104, X52) → U6_GGA(T33, X52, reduceA_in_gga(T33, 104, X52))
REDUCEA_IN_GGA(.(104, T33), 104, X52) → REDUCEA_IN_GGA(T33, 104, X52)
REDUCEA_IN_GGA(.(119, T33), 119, X52) → U7_GGA(T33, X52, reduceA_in_gga(T33, 119, X52))
REDUCEA_IN_GGA(.(119, T33), 119, X52) → REDUCEA_IN_GGA(T33, 119, X52)
REDUCEA_IN_GGA(.(121, T33), 121, X52) → U8_GGA(T33, X52, reduceA_in_gga(T33, 121, X52))
REDUCEA_IN_GGA(.(121, T33), 121, X52) → REDUCEA_IN_GGA(T33, 121, X52)
REDUCEA_IN_GGA(.(T50, T51), T50, X93) → U9_GGA(T50, T51, X93, reduceA_in_gga(T51, T50, X93))
REDUCEA_IN_GGA(.(T50, T51), T50, X93) → REDUCEA_IN_GGA(T51, T50, X93)
REDUCEA_IN_GGA(.(T59, T60), T61, .(T59, X113)) → U10_GGA(T59, T60, T61, X113, reduceB_in_gga(T60, T59, X113))
REDUCEA_IN_GGA(.(T59, T60), T61, .(T59, X113)) → REDUCEB_IN_GGA(T60, T59, X113)
REDUCEB_IN_GGA(.(97, T76), 97, X148) → U11_GGA(T76, X148, reduceB_in_gga(T76, 97, X148))
REDUCEB_IN_GGA(.(97, T76), 97, X148) → REDUCEB_IN_GGA(T76, 97, X148)
REDUCEB_IN_GGA(.(101, T76), 101, X148) → U12_GGA(T76, X148, reduceB_in_gga(T76, 101, X148))
REDUCEB_IN_GGA(.(101, T76), 101, X148) → REDUCEB_IN_GGA(T76, 101, X148)
REDUCEB_IN_GGA(.(105, T76), 105, X148) → U13_GGA(T76, X148, reduceB_in_gga(T76, 105, X148))
REDUCEB_IN_GGA(.(105, T76), 105, X148) → REDUCEB_IN_GGA(T76, 105, X148)
REDUCEB_IN_GGA(.(111, T76), 111, X148) → U14_GGA(T76, X148, reduceB_in_gga(T76, 111, X148))
REDUCEB_IN_GGA(.(111, T76), 111, X148) → REDUCEB_IN_GGA(T76, 111, X148)
REDUCEB_IN_GGA(.(117, T76), 117, X148) → U15_GGA(T76, X148, reduceB_in_gga(T76, 117, X148))
REDUCEB_IN_GGA(.(117, T76), 117, X148) → REDUCEB_IN_GGA(T76, 117, X148)
REDUCEB_IN_GGA(.(104, T76), 104, X148) → U16_GGA(T76, X148, reduceB_in_gga(T76, 104, X148))
REDUCEB_IN_GGA(.(104, T76), 104, X148) → REDUCEB_IN_GGA(T76, 104, X148)
REDUCEB_IN_GGA(.(119, T76), 119, X148) → U17_GGA(T76, X148, reduceB_in_gga(T76, 119, X148))
REDUCEB_IN_GGA(.(119, T76), 119, X148) → REDUCEB_IN_GGA(T76, 119, X148)
REDUCEB_IN_GGA(.(121, T76), 121, X148) → U18_GGA(T76, X148, reduceB_in_gga(T76, 121, X148))
REDUCEB_IN_GGA(.(121, T76), 121, X148) → REDUCEB_IN_GGA(T76, 121, X148)
REDUCEB_IN_GGA(.(T93, T94), T93, X189) → U19_GGA(T93, T94, X189, reduceB_in_gga(T94, T93, X189))
REDUCEB_IN_GGA(.(T93, T94), T93, X189) → REDUCEB_IN_GGA(T94, T93, X189)
REDUCEB_IN_GGA(.(T102, T103), T104, .(T102, X209)) → U20_GGA(T102, T103, T104, X209, reduceC_in_gga(T103, T102, X209))
REDUCEB_IN_GGA(.(T102, T103), T104, .(T102, X209)) → REDUCEC_IN_GGA(T103, T102, X209)
REDUCEC_IN_GGA(.(97, T119), 97, X244) → U21_GGA(T119, X244, reduceC_in_gga(T119, 97, X244))
REDUCEC_IN_GGA(.(97, T119), 97, X244) → REDUCEC_IN_GGA(T119, 97, X244)
REDUCEC_IN_GGA(.(101, T119), 101, X244) → U22_GGA(T119, X244, reduceC_in_gga(T119, 101, X244))
REDUCEC_IN_GGA(.(101, T119), 101, X244) → REDUCEC_IN_GGA(T119, 101, X244)
REDUCEC_IN_GGA(.(105, T119), 105, X244) → U23_GGA(T119, X244, reduceC_in_gga(T119, 105, X244))
REDUCEC_IN_GGA(.(105, T119), 105, X244) → REDUCEC_IN_GGA(T119, 105, X244)
REDUCEC_IN_GGA(.(111, T119), 111, X244) → U24_GGA(T119, X244, reduceC_in_gga(T119, 111, X244))
REDUCEC_IN_GGA(.(111, T119), 111, X244) → REDUCEC_IN_GGA(T119, 111, X244)
REDUCEC_IN_GGA(.(117, T119), 117, X244) → U25_GGA(T119, X244, reduceC_in_gga(T119, 117, X244))
REDUCEC_IN_GGA(.(117, T119), 117, X244) → REDUCEC_IN_GGA(T119, 117, X244)
REDUCEC_IN_GGA(.(104, T119), 104, X244) → U26_GGA(T119, X244, reduceC_in_gga(T119, 104, X244))
REDUCEC_IN_GGA(.(104, T119), 104, X244) → REDUCEC_IN_GGA(T119, 104, X244)
REDUCEC_IN_GGA(.(119, T119), 119, X244) → U27_GGA(T119, X244, reduceC_in_gga(T119, 119, X244))
REDUCEC_IN_GGA(.(119, T119), 119, X244) → REDUCEC_IN_GGA(T119, 119, X244)
REDUCEC_IN_GGA(.(121, T119), 121, X244) → U28_GGA(T119, X244, reduceC_in_gga(T119, 121, X244))
REDUCEC_IN_GGA(.(121, T119), 121, X244) → REDUCEC_IN_GGA(T119, 121, X244)
REDUCEC_IN_GGA(.(T136, T137), T136, X285) → U29_GGA(T136, T137, X285, reduceC_in_gga(T137, T136, X285))
REDUCEC_IN_GGA(.(T136, T137), T136, X285) → REDUCEC_IN_GGA(T137, T136, X285)
GOALD_IN_GA(.(T15, T16), T158) → U31_GA(T15, T16, T158, reduceA_in_gga(T16, T15, T158))
GOALD_IN_GA(.(T15, T16), T158) → REDUCEA_IN_GGA(T16, T15, T158)

The TRS R consists of the following rules:

goalD_in_ga(.(T15, T16), T7) → U30_ga(T15, T16, T7, reduceA_in_gga(T16, T15, X12))
reduceA_in_gga([], T23, []) → reduceA_out_gga([], T23, [])
reduceA_in_gga(.(97, T33), 97, X52) → U1_gga(T33, X52, reduceA_in_gga(T33, 97, X52))
reduceA_in_gga(.(101, T33), 101, X52) → U2_gga(T33, X52, reduceA_in_gga(T33, 101, X52))
reduceA_in_gga(.(105, T33), 105, X52) → U3_gga(T33, X52, reduceA_in_gga(T33, 105, X52))
reduceA_in_gga(.(111, T33), 111, X52) → U4_gga(T33, X52, reduceA_in_gga(T33, 111, X52))
reduceA_in_gga(.(117, T33), 117, X52) → U5_gga(T33, X52, reduceA_in_gga(T33, 117, X52))
reduceA_in_gga(.(104, T33), 104, X52) → U6_gga(T33, X52, reduceA_in_gga(T33, 104, X52))
reduceA_in_gga(.(119, T33), 119, X52) → U7_gga(T33, X52, reduceA_in_gga(T33, 119, X52))
reduceA_in_gga(.(121, T33), 121, X52) → U8_gga(T33, X52, reduceA_in_gga(T33, 121, X52))
reduceA_in_gga(.(T50, T51), T50, X93) → U9_gga(T50, T51, X93, reduceA_in_gga(T51, T50, X93))
reduceA_in_gga(.(T59, T60), T61, .(T59, X113)) → U10_gga(T59, T60, T61, X113, reduceB_in_gga(T60, T59, X113))
reduceB_in_gga([], T66, []) → reduceB_out_gga([], T66, [])
reduceB_in_gga(.(97, T76), 97, X148) → U11_gga(T76, X148, reduceB_in_gga(T76, 97, X148))
reduceB_in_gga(.(101, T76), 101, X148) → U12_gga(T76, X148, reduceB_in_gga(T76, 101, X148))
reduceB_in_gga(.(105, T76), 105, X148) → U13_gga(T76, X148, reduceB_in_gga(T76, 105, X148))
reduceB_in_gga(.(111, T76), 111, X148) → U14_gga(T76, X148, reduceB_in_gga(T76, 111, X148))
reduceB_in_gga(.(117, T76), 117, X148) → U15_gga(T76, X148, reduceB_in_gga(T76, 117, X148))
reduceB_in_gga(.(104, T76), 104, X148) → U16_gga(T76, X148, reduceB_in_gga(T76, 104, X148))
reduceB_in_gga(.(119, T76), 119, X148) → U17_gga(T76, X148, reduceB_in_gga(T76, 119, X148))
reduceB_in_gga(.(121, T76), 121, X148) → U18_gga(T76, X148, reduceB_in_gga(T76, 121, X148))
reduceB_in_gga(.(T93, T94), T93, X189) → U19_gga(T93, T94, X189, reduceB_in_gga(T94, T93, X189))
reduceB_in_gga(.(T102, T103), T104, .(T102, X209)) → U20_gga(T102, T103, T104, X209, reduceC_in_gga(T103, T102, X209))
reduceC_in_gga([], T109, []) → reduceC_out_gga([], T109, [])
reduceC_in_gga(.(97, T119), 97, X244) → U21_gga(T119, X244, reduceC_in_gga(T119, 97, X244))
reduceC_in_gga(.(101, T119), 101, X244) → U22_gga(T119, X244, reduceC_in_gga(T119, 101, X244))
reduceC_in_gga(.(105, T119), 105, X244) → U23_gga(T119, X244, reduceC_in_gga(T119, 105, X244))
reduceC_in_gga(.(111, T119), 111, X244) → U24_gga(T119, X244, reduceC_in_gga(T119, 111, X244))
reduceC_in_gga(.(117, T119), 117, X244) → U25_gga(T119, X244, reduceC_in_gga(T119, 117, X244))
reduceC_in_gga(.(104, T119), 104, X244) → U26_gga(T119, X244, reduceC_in_gga(T119, 104, X244))
reduceC_in_gga(.(119, T119), 119, X244) → U27_gga(T119, X244, reduceC_in_gga(T119, 119, X244))
reduceC_in_gga(.(121, T119), 121, X244) → U28_gga(T119, X244, reduceC_in_gga(T119, 121, X244))
reduceC_in_gga(.(T136, T137), T136, X285) → U29_gga(T136, T137, X285, reduceC_in_gga(T137, T136, X285))
reduceC_in_gga(.(T155, T154), T147, .(T155, [])) → reduceC_out_gga(.(T155, T154), T147, .(T155, []))
U29_gga(T136, T137, X285, reduceC_out_gga(T137, T136, X285)) → reduceC_out_gga(.(T136, T137), T136, X285)
U28_gga(T119, X244, reduceC_out_gga(T119, 121, X244)) → reduceC_out_gga(.(121, T119), 121, X244)
U27_gga(T119, X244, reduceC_out_gga(T119, 119, X244)) → reduceC_out_gga(.(119, T119), 119, X244)
U26_gga(T119, X244, reduceC_out_gga(T119, 104, X244)) → reduceC_out_gga(.(104, T119), 104, X244)
U25_gga(T119, X244, reduceC_out_gga(T119, 117, X244)) → reduceC_out_gga(.(117, T119), 117, X244)
U24_gga(T119, X244, reduceC_out_gga(T119, 111, X244)) → reduceC_out_gga(.(111, T119), 111, X244)
U23_gga(T119, X244, reduceC_out_gga(T119, 105, X244)) → reduceC_out_gga(.(105, T119), 105, X244)
U22_gga(T119, X244, reduceC_out_gga(T119, 101, X244)) → reduceC_out_gga(.(101, T119), 101, X244)
U21_gga(T119, X244, reduceC_out_gga(T119, 97, X244)) → reduceC_out_gga(.(97, T119), 97, X244)
U20_gga(T102, T103, T104, X209, reduceC_out_gga(T103, T102, X209)) → reduceB_out_gga(.(T102, T103), T104, .(T102, X209))
U19_gga(T93, T94, X189, reduceB_out_gga(T94, T93, X189)) → reduceB_out_gga(.(T93, T94), T93, X189)
U18_gga(T76, X148, reduceB_out_gga(T76, 121, X148)) → reduceB_out_gga(.(121, T76), 121, X148)
U17_gga(T76, X148, reduceB_out_gga(T76, 119, X148)) → reduceB_out_gga(.(119, T76), 119, X148)
U16_gga(T76, X148, reduceB_out_gga(T76, 104, X148)) → reduceB_out_gga(.(104, T76), 104, X148)
U15_gga(T76, X148, reduceB_out_gga(T76, 117, X148)) → reduceB_out_gga(.(117, T76), 117, X148)
U14_gga(T76, X148, reduceB_out_gga(T76, 111, X148)) → reduceB_out_gga(.(111, T76), 111, X148)
U13_gga(T76, X148, reduceB_out_gga(T76, 105, X148)) → reduceB_out_gga(.(105, T76), 105, X148)
U12_gga(T76, X148, reduceB_out_gga(T76, 101, X148)) → reduceB_out_gga(.(101, T76), 101, X148)
U11_gga(T76, X148, reduceB_out_gga(T76, 97, X148)) → reduceB_out_gga(.(97, T76), 97, X148)
U10_gga(T59, T60, T61, X113, reduceB_out_gga(T60, T59, X113)) → reduceA_out_gga(.(T59, T60), T61, .(T59, X113))
U9_gga(T50, T51, X93, reduceA_out_gga(T51, T50, X93)) → reduceA_out_gga(.(T50, T51), T50, X93)
U8_gga(T33, X52, reduceA_out_gga(T33, 121, X52)) → reduceA_out_gga(.(121, T33), 121, X52)
U7_gga(T33, X52, reduceA_out_gga(T33, 119, X52)) → reduceA_out_gga(.(119, T33), 119, X52)
U6_gga(T33, X52, reduceA_out_gga(T33, 104, X52)) → reduceA_out_gga(.(104, T33), 104, X52)
U5_gga(T33, X52, reduceA_out_gga(T33, 117, X52)) → reduceA_out_gga(.(117, T33), 117, X52)
U4_gga(T33, X52, reduceA_out_gga(T33, 111, X52)) → reduceA_out_gga(.(111, T33), 111, X52)
U3_gga(T33, X52, reduceA_out_gga(T33, 105, X52)) → reduceA_out_gga(.(105, T33), 105, X52)
U2_gga(T33, X52, reduceA_out_gga(T33, 101, X52)) → reduceA_out_gga(.(101, T33), 101, X52)
U1_gga(T33, X52, reduceA_out_gga(T33, 97, X52)) → reduceA_out_gga(.(97, T33), 97, X52)
U30_ga(T15, T16, T7, reduceA_out_gga(T16, T15, X12)) → goalD_out_ga(.(T15, T16), T7)
goalD_in_ga(.(T15, T16), T158) → U31_ga(T15, T16, T158, reduceA_in_gga(T16, T15, T158))
U31_ga(T15, T16, T158, reduceA_out_gga(T16, T15, T158)) → goalD_out_ga(.(T15, T16), T158)

The argument filtering Pi contains the following mapping:
goalD_in_ga(x1, x2)  =  goalD_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U30_ga(x1, x2, x3, x4)  =  U30_ga(x4)
reduceA_in_gga(x1, x2, x3)  =  reduceA_in_gga(x1, x2)
[]  =  []
reduceA_out_gga(x1, x2, x3)  =  reduceA_out_gga(x3)
97  =  97
U1_gga(x1, x2, x3)  =  U1_gga(x3)
101  =  101
U2_gga(x1, x2, x3)  =  U2_gga(x3)
105  =  105
U3_gga(x1, x2, x3)  =  U3_gga(x3)
111  =  111
U4_gga(x1, x2, x3)  =  U4_gga(x3)
117  =  117
U5_gga(x1, x2, x3)  =  U5_gga(x3)
104  =  104
U6_gga(x1, x2, x3)  =  U6_gga(x3)
119  =  119
U7_gga(x1, x2, x3)  =  U7_gga(x3)
121  =  121
U8_gga(x1, x2, x3)  =  U8_gga(x3)
U9_gga(x1, x2, x3, x4)  =  U9_gga(x4)
U10_gga(x1, x2, x3, x4, x5)  =  U10_gga(x1, x5)
reduceB_in_gga(x1, x2, x3)  =  reduceB_in_gga(x1, x2)
reduceB_out_gga(x1, x2, x3)  =  reduceB_out_gga(x3)
U11_gga(x1, x2, x3)  =  U11_gga(x3)
U12_gga(x1, x2, x3)  =  U12_gga(x3)
U13_gga(x1, x2, x3)  =  U13_gga(x3)
U14_gga(x1, x2, x3)  =  U14_gga(x3)
U15_gga(x1, x2, x3)  =  U15_gga(x3)
U16_gga(x1, x2, x3)  =  U16_gga(x3)
U17_gga(x1, x2, x3)  =  U17_gga(x3)
U18_gga(x1, x2, x3)  =  U18_gga(x3)
U19_gga(x1, x2, x3, x4)  =  U19_gga(x4)
U20_gga(x1, x2, x3, x4, x5)  =  U20_gga(x1, x5)
reduceC_in_gga(x1, x2, x3)  =  reduceC_in_gga(x1, x2)
reduceC_out_gga(x1, x2, x3)  =  reduceC_out_gga(x3)
U21_gga(x1, x2, x3)  =  U21_gga(x3)
U22_gga(x1, x2, x3)  =  U22_gga(x3)
U23_gga(x1, x2, x3)  =  U23_gga(x3)
U24_gga(x1, x2, x3)  =  U24_gga(x3)
U25_gga(x1, x2, x3)  =  U25_gga(x3)
U26_gga(x1, x2, x3)  =  U26_gga(x3)
U27_gga(x1, x2, x3)  =  U27_gga(x3)
U28_gga(x1, x2, x3)  =  U28_gga(x3)
U29_gga(x1, x2, x3, x4)  =  U29_gga(x4)
goalD_out_ga(x1, x2)  =  goalD_out_ga
U31_ga(x1, x2, x3, x4)  =  U31_ga(x4)
GOALD_IN_GA(x1, x2)  =  GOALD_IN_GA(x1)
U30_GA(x1, x2, x3, x4)  =  U30_GA(x4)
REDUCEA_IN_GGA(x1, x2, x3)  =  REDUCEA_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3)  =  U1_GGA(x3)
U2_GGA(x1, x2, x3)  =  U2_GGA(x3)
U3_GGA(x1, x2, x3)  =  U3_GGA(x3)
U4_GGA(x1, x2, x3)  =  U4_GGA(x3)
U5_GGA(x1, x2, x3)  =  U5_GGA(x3)
U6_GGA(x1, x2, x3)  =  U6_GGA(x3)
U7_GGA(x1, x2, x3)  =  U7_GGA(x3)
U8_GGA(x1, x2, x3)  =  U8_GGA(x3)
U9_GGA(x1, x2, x3, x4)  =  U9_GGA(x4)
U10_GGA(x1, x2, x3, x4, x5)  =  U10_GGA(x1, x5)
REDUCEB_IN_GGA(x1, x2, x3)  =  REDUCEB_IN_GGA(x1, x2)
U11_GGA(x1, x2, x3)  =  U11_GGA(x3)
U12_GGA(x1, x2, x3)  =  U12_GGA(x3)
U13_GGA(x1, x2, x3)  =  U13_GGA(x3)
U14_GGA(x1, x2, x3)  =  U14_GGA(x3)
U15_GGA(x1, x2, x3)  =  U15_GGA(x3)
U16_GGA(x1, x2, x3)  =  U16_GGA(x3)
U17_GGA(x1, x2, x3)  =  U17_GGA(x3)
U18_GGA(x1, x2, x3)  =  U18_GGA(x3)
U19_GGA(x1, x2, x3, x4)  =  U19_GGA(x4)
U20_GGA(x1, x2, x3, x4, x5)  =  U20_GGA(x1, x5)
REDUCEC_IN_GGA(x1, x2, x3)  =  REDUCEC_IN_GGA(x1, x2)
U21_GGA(x1, x2, x3)  =  U21_GGA(x3)
U22_GGA(x1, x2, x3)  =  U22_GGA(x3)
U23_GGA(x1, x2, x3)  =  U23_GGA(x3)
U24_GGA(x1, x2, x3)  =  U24_GGA(x3)
U25_GGA(x1, x2, x3)  =  U25_GGA(x3)
U26_GGA(x1, x2, x3)  =  U26_GGA(x3)
U27_GGA(x1, x2, x3)  =  U27_GGA(x3)
U28_GGA(x1, x2, x3)  =  U28_GGA(x3)
U29_GGA(x1, x2, x3, x4)  =  U29_GGA(x4)
U31_GA(x1, x2, x3, x4)  =  U31_GA(x4)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 35 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

REDUCEC_IN_GGA(.(T136, T137), T136, X285) → REDUCEC_IN_GGA(T137, T136, X285)
REDUCEC_IN_GGA(.(97, T119), 97, X244) → REDUCEC_IN_GGA(T119, 97, X244)
REDUCEC_IN_GGA(.(101, T119), 101, X244) → REDUCEC_IN_GGA(T119, 101, X244)
REDUCEC_IN_GGA(.(105, T119), 105, X244) → REDUCEC_IN_GGA(T119, 105, X244)
REDUCEC_IN_GGA(.(111, T119), 111, X244) → REDUCEC_IN_GGA(T119, 111, X244)
REDUCEC_IN_GGA(.(117, T119), 117, X244) → REDUCEC_IN_GGA(T119, 117, X244)
REDUCEC_IN_GGA(.(104, T119), 104, X244) → REDUCEC_IN_GGA(T119, 104, X244)
REDUCEC_IN_GGA(.(119, T119), 119, X244) → REDUCEC_IN_GGA(T119, 119, X244)
REDUCEC_IN_GGA(.(121, T119), 121, X244) → REDUCEC_IN_GGA(T119, 121, X244)

The TRS R consists of the following rules:

goalD_in_ga(.(T15, T16), T7) → U30_ga(T15, T16, T7, reduceA_in_gga(T16, T15, X12))
reduceA_in_gga([], T23, []) → reduceA_out_gga([], T23, [])
reduceA_in_gga(.(97, T33), 97, X52) → U1_gga(T33, X52, reduceA_in_gga(T33, 97, X52))
reduceA_in_gga(.(101, T33), 101, X52) → U2_gga(T33, X52, reduceA_in_gga(T33, 101, X52))
reduceA_in_gga(.(105, T33), 105, X52) → U3_gga(T33, X52, reduceA_in_gga(T33, 105, X52))
reduceA_in_gga(.(111, T33), 111, X52) → U4_gga(T33, X52, reduceA_in_gga(T33, 111, X52))
reduceA_in_gga(.(117, T33), 117, X52) → U5_gga(T33, X52, reduceA_in_gga(T33, 117, X52))
reduceA_in_gga(.(104, T33), 104, X52) → U6_gga(T33, X52, reduceA_in_gga(T33, 104, X52))
reduceA_in_gga(.(119, T33), 119, X52) → U7_gga(T33, X52, reduceA_in_gga(T33, 119, X52))
reduceA_in_gga(.(121, T33), 121, X52) → U8_gga(T33, X52, reduceA_in_gga(T33, 121, X52))
reduceA_in_gga(.(T50, T51), T50, X93) → U9_gga(T50, T51, X93, reduceA_in_gga(T51, T50, X93))
reduceA_in_gga(.(T59, T60), T61, .(T59, X113)) → U10_gga(T59, T60, T61, X113, reduceB_in_gga(T60, T59, X113))
reduceB_in_gga([], T66, []) → reduceB_out_gga([], T66, [])
reduceB_in_gga(.(97, T76), 97, X148) → U11_gga(T76, X148, reduceB_in_gga(T76, 97, X148))
reduceB_in_gga(.(101, T76), 101, X148) → U12_gga(T76, X148, reduceB_in_gga(T76, 101, X148))
reduceB_in_gga(.(105, T76), 105, X148) → U13_gga(T76, X148, reduceB_in_gga(T76, 105, X148))
reduceB_in_gga(.(111, T76), 111, X148) → U14_gga(T76, X148, reduceB_in_gga(T76, 111, X148))
reduceB_in_gga(.(117, T76), 117, X148) → U15_gga(T76, X148, reduceB_in_gga(T76, 117, X148))
reduceB_in_gga(.(104, T76), 104, X148) → U16_gga(T76, X148, reduceB_in_gga(T76, 104, X148))
reduceB_in_gga(.(119, T76), 119, X148) → U17_gga(T76, X148, reduceB_in_gga(T76, 119, X148))
reduceB_in_gga(.(121, T76), 121, X148) → U18_gga(T76, X148, reduceB_in_gga(T76, 121, X148))
reduceB_in_gga(.(T93, T94), T93, X189) → U19_gga(T93, T94, X189, reduceB_in_gga(T94, T93, X189))
reduceB_in_gga(.(T102, T103), T104, .(T102, X209)) → U20_gga(T102, T103, T104, X209, reduceC_in_gga(T103, T102, X209))
reduceC_in_gga([], T109, []) → reduceC_out_gga([], T109, [])
reduceC_in_gga(.(97, T119), 97, X244) → U21_gga(T119, X244, reduceC_in_gga(T119, 97, X244))
reduceC_in_gga(.(101, T119), 101, X244) → U22_gga(T119, X244, reduceC_in_gga(T119, 101, X244))
reduceC_in_gga(.(105, T119), 105, X244) → U23_gga(T119, X244, reduceC_in_gga(T119, 105, X244))
reduceC_in_gga(.(111, T119), 111, X244) → U24_gga(T119, X244, reduceC_in_gga(T119, 111, X244))
reduceC_in_gga(.(117, T119), 117, X244) → U25_gga(T119, X244, reduceC_in_gga(T119, 117, X244))
reduceC_in_gga(.(104, T119), 104, X244) → U26_gga(T119, X244, reduceC_in_gga(T119, 104, X244))
reduceC_in_gga(.(119, T119), 119, X244) → U27_gga(T119, X244, reduceC_in_gga(T119, 119, X244))
reduceC_in_gga(.(121, T119), 121, X244) → U28_gga(T119, X244, reduceC_in_gga(T119, 121, X244))
reduceC_in_gga(.(T136, T137), T136, X285) → U29_gga(T136, T137, X285, reduceC_in_gga(T137, T136, X285))
reduceC_in_gga(.(T155, T154), T147, .(T155, [])) → reduceC_out_gga(.(T155, T154), T147, .(T155, []))
U29_gga(T136, T137, X285, reduceC_out_gga(T137, T136, X285)) → reduceC_out_gga(.(T136, T137), T136, X285)
U28_gga(T119, X244, reduceC_out_gga(T119, 121, X244)) → reduceC_out_gga(.(121, T119), 121, X244)
U27_gga(T119, X244, reduceC_out_gga(T119, 119, X244)) → reduceC_out_gga(.(119, T119), 119, X244)
U26_gga(T119, X244, reduceC_out_gga(T119, 104, X244)) → reduceC_out_gga(.(104, T119), 104, X244)
U25_gga(T119, X244, reduceC_out_gga(T119, 117, X244)) → reduceC_out_gga(.(117, T119), 117, X244)
U24_gga(T119, X244, reduceC_out_gga(T119, 111, X244)) → reduceC_out_gga(.(111, T119), 111, X244)
U23_gga(T119, X244, reduceC_out_gga(T119, 105, X244)) → reduceC_out_gga(.(105, T119), 105, X244)
U22_gga(T119, X244, reduceC_out_gga(T119, 101, X244)) → reduceC_out_gga(.(101, T119), 101, X244)
U21_gga(T119, X244, reduceC_out_gga(T119, 97, X244)) → reduceC_out_gga(.(97, T119), 97, X244)
U20_gga(T102, T103, T104, X209, reduceC_out_gga(T103, T102, X209)) → reduceB_out_gga(.(T102, T103), T104, .(T102, X209))
U19_gga(T93, T94, X189, reduceB_out_gga(T94, T93, X189)) → reduceB_out_gga(.(T93, T94), T93, X189)
U18_gga(T76, X148, reduceB_out_gga(T76, 121, X148)) → reduceB_out_gga(.(121, T76), 121, X148)
U17_gga(T76, X148, reduceB_out_gga(T76, 119, X148)) → reduceB_out_gga(.(119, T76), 119, X148)
U16_gga(T76, X148, reduceB_out_gga(T76, 104, X148)) → reduceB_out_gga(.(104, T76), 104, X148)
U15_gga(T76, X148, reduceB_out_gga(T76, 117, X148)) → reduceB_out_gga(.(117, T76), 117, X148)
U14_gga(T76, X148, reduceB_out_gga(T76, 111, X148)) → reduceB_out_gga(.(111, T76), 111, X148)
U13_gga(T76, X148, reduceB_out_gga(T76, 105, X148)) → reduceB_out_gga(.(105, T76), 105, X148)
U12_gga(T76, X148, reduceB_out_gga(T76, 101, X148)) → reduceB_out_gga(.(101, T76), 101, X148)
U11_gga(T76, X148, reduceB_out_gga(T76, 97, X148)) → reduceB_out_gga(.(97, T76), 97, X148)
U10_gga(T59, T60, T61, X113, reduceB_out_gga(T60, T59, X113)) → reduceA_out_gga(.(T59, T60), T61, .(T59, X113))
U9_gga(T50, T51, X93, reduceA_out_gga(T51, T50, X93)) → reduceA_out_gga(.(T50, T51), T50, X93)
U8_gga(T33, X52, reduceA_out_gga(T33, 121, X52)) → reduceA_out_gga(.(121, T33), 121, X52)
U7_gga(T33, X52, reduceA_out_gga(T33, 119, X52)) → reduceA_out_gga(.(119, T33), 119, X52)
U6_gga(T33, X52, reduceA_out_gga(T33, 104, X52)) → reduceA_out_gga(.(104, T33), 104, X52)
U5_gga(T33, X52, reduceA_out_gga(T33, 117, X52)) → reduceA_out_gga(.(117, T33), 117, X52)
U4_gga(T33, X52, reduceA_out_gga(T33, 111, X52)) → reduceA_out_gga(.(111, T33), 111, X52)
U3_gga(T33, X52, reduceA_out_gga(T33, 105, X52)) → reduceA_out_gga(.(105, T33), 105, X52)
U2_gga(T33, X52, reduceA_out_gga(T33, 101, X52)) → reduceA_out_gga(.(101, T33), 101, X52)
U1_gga(T33, X52, reduceA_out_gga(T33, 97, X52)) → reduceA_out_gga(.(97, T33), 97, X52)
U30_ga(T15, T16, T7, reduceA_out_gga(T16, T15, X12)) → goalD_out_ga(.(T15, T16), T7)
goalD_in_ga(.(T15, T16), T158) → U31_ga(T15, T16, T158, reduceA_in_gga(T16, T15, T158))
U31_ga(T15, T16, T158, reduceA_out_gga(T16, T15, T158)) → goalD_out_ga(.(T15, T16), T158)

The argument filtering Pi contains the following mapping:
goalD_in_ga(x1, x2)  =  goalD_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U30_ga(x1, x2, x3, x4)  =  U30_ga(x4)
reduceA_in_gga(x1, x2, x3)  =  reduceA_in_gga(x1, x2)
[]  =  []
reduceA_out_gga(x1, x2, x3)  =  reduceA_out_gga(x3)
97  =  97
U1_gga(x1, x2, x3)  =  U1_gga(x3)
101  =  101
U2_gga(x1, x2, x3)  =  U2_gga(x3)
105  =  105
U3_gga(x1, x2, x3)  =  U3_gga(x3)
111  =  111
U4_gga(x1, x2, x3)  =  U4_gga(x3)
117  =  117
U5_gga(x1, x2, x3)  =  U5_gga(x3)
104  =  104
U6_gga(x1, x2, x3)  =  U6_gga(x3)
119  =  119
U7_gga(x1, x2, x3)  =  U7_gga(x3)
121  =  121
U8_gga(x1, x2, x3)  =  U8_gga(x3)
U9_gga(x1, x2, x3, x4)  =  U9_gga(x4)
U10_gga(x1, x2, x3, x4, x5)  =  U10_gga(x1, x5)
reduceB_in_gga(x1, x2, x3)  =  reduceB_in_gga(x1, x2)
reduceB_out_gga(x1, x2, x3)  =  reduceB_out_gga(x3)
U11_gga(x1, x2, x3)  =  U11_gga(x3)
U12_gga(x1, x2, x3)  =  U12_gga(x3)
U13_gga(x1, x2, x3)  =  U13_gga(x3)
U14_gga(x1, x2, x3)  =  U14_gga(x3)
U15_gga(x1, x2, x3)  =  U15_gga(x3)
U16_gga(x1, x2, x3)  =  U16_gga(x3)
U17_gga(x1, x2, x3)  =  U17_gga(x3)
U18_gga(x1, x2, x3)  =  U18_gga(x3)
U19_gga(x1, x2, x3, x4)  =  U19_gga(x4)
U20_gga(x1, x2, x3, x4, x5)  =  U20_gga(x1, x5)
reduceC_in_gga(x1, x2, x3)  =  reduceC_in_gga(x1, x2)
reduceC_out_gga(x1, x2, x3)  =  reduceC_out_gga(x3)
U21_gga(x1, x2, x3)  =  U21_gga(x3)
U22_gga(x1, x2, x3)  =  U22_gga(x3)
U23_gga(x1, x2, x3)  =  U23_gga(x3)
U24_gga(x1, x2, x3)  =  U24_gga(x3)
U25_gga(x1, x2, x3)  =  U25_gga(x3)
U26_gga(x1, x2, x3)  =  U26_gga(x3)
U27_gga(x1, x2, x3)  =  U27_gga(x3)
U28_gga(x1, x2, x3)  =  U28_gga(x3)
U29_gga(x1, x2, x3, x4)  =  U29_gga(x4)
goalD_out_ga(x1, x2)  =  goalD_out_ga
U31_ga(x1, x2, x3, x4)  =  U31_ga(x4)
REDUCEC_IN_GGA(x1, x2, x3)  =  REDUCEC_IN_GGA(x1, x2)

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

(10) UsableRulesProof (EQUIVALENT transformation)

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

(11) Obligation:

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

REDUCEC_IN_GGA(.(T136, T137), T136, X285) → REDUCEC_IN_GGA(T137, T136, X285)
REDUCEC_IN_GGA(.(97, T119), 97, X244) → REDUCEC_IN_GGA(T119, 97, X244)
REDUCEC_IN_GGA(.(101, T119), 101, X244) → REDUCEC_IN_GGA(T119, 101, X244)
REDUCEC_IN_GGA(.(105, T119), 105, X244) → REDUCEC_IN_GGA(T119, 105, X244)
REDUCEC_IN_GGA(.(111, T119), 111, X244) → REDUCEC_IN_GGA(T119, 111, X244)
REDUCEC_IN_GGA(.(117, T119), 117, X244) → REDUCEC_IN_GGA(T119, 117, X244)
REDUCEC_IN_GGA(.(104, T119), 104, X244) → REDUCEC_IN_GGA(T119, 104, X244)
REDUCEC_IN_GGA(.(119, T119), 119, X244) → REDUCEC_IN_GGA(T119, 119, X244)
REDUCEC_IN_GGA(.(121, T119), 121, X244) → REDUCEC_IN_GGA(T119, 121, X244)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
97  =  97
101  =  101
105  =  105
111  =  111
117  =  117
104  =  104
119  =  119
121  =  121
REDUCEC_IN_GGA(x1, x2, x3)  =  REDUCEC_IN_GGA(x1, x2)

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

(12) PiDPToQDPProof (SOUND transformation)

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

(13) Obligation:

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

REDUCEC_IN_GGA(.(T136, T137), T136) → REDUCEC_IN_GGA(T137, T136)
REDUCEC_IN_GGA(.(97, T119), 97) → REDUCEC_IN_GGA(T119, 97)
REDUCEC_IN_GGA(.(101, T119), 101) → REDUCEC_IN_GGA(T119, 101)
REDUCEC_IN_GGA(.(105, T119), 105) → REDUCEC_IN_GGA(T119, 105)
REDUCEC_IN_GGA(.(111, T119), 111) → REDUCEC_IN_GGA(T119, 111)
REDUCEC_IN_GGA(.(117, T119), 117) → REDUCEC_IN_GGA(T119, 117)
REDUCEC_IN_GGA(.(104, T119), 104) → REDUCEC_IN_GGA(T119, 104)
REDUCEC_IN_GGA(.(119, T119), 119) → REDUCEC_IN_GGA(T119, 119)
REDUCEC_IN_GGA(.(121, T119), 121) → REDUCEC_IN_GGA(T119, 121)

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

(14) 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:

• REDUCEC_IN_GGA(.(T136, T137), T136) → REDUCEC_IN_GGA(T137, T136)
  The graph contains the following edges 1 > 1, 1 > 2, 2 >= 2

• REDUCEC_IN_GGA(.(97, T119), 97) → REDUCEC_IN_GGA(T119, 97)
  The graph contains the following edges 1 > 1, 1 > 2, 2 >= 2

• REDUCEC_IN_GGA(.(101, T119), 101) → REDUCEC_IN_GGA(T119, 101)
  The graph contains the following edges 1 > 1, 1 > 2, 2 >= 2

• REDUCEC_IN_GGA(.(105, T119), 105) → REDUCEC_IN_GGA(T119, 105)
  The graph contains the following edges 1 > 1, 1 > 2, 2 >= 2

• REDUCEC_IN_GGA(.(111, T119), 111) → REDUCEC_IN_GGA(T119, 111)
  The graph contains the following edges 1 > 1, 1 > 2, 2 >= 2

• REDUCEC_IN_GGA(.(117, T119), 117) → REDUCEC_IN_GGA(T119, 117)
  The graph contains the following edges 1 > 1, 1 > 2, 2 >= 2

• REDUCEC_IN_GGA(.(104, T119), 104) → REDUCEC_IN_GGA(T119, 104)
  The graph contains the following edges 1 > 1, 1 > 2, 2 >= 2

• REDUCEC_IN_GGA(.(119, T119), 119) → REDUCEC_IN_GGA(T119, 119)
  The graph contains the following edges 1 > 1, 1 > 2, 2 >= 2

• REDUCEC_IN_GGA(.(121, T119), 121) → REDUCEC_IN_GGA(T119, 121)
  The graph contains the following edges 1 > 1, 1 > 2, 2 >= 2

(15) YES

(16) Obligation:

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

REDUCEB_IN_GGA(.(T93, T94), T93, X189) → REDUCEB_IN_GGA(T94, T93, X189)
REDUCEB_IN_GGA(.(97, T76), 97, X148) → REDUCEB_IN_GGA(T76, 97, X148)
REDUCEB_IN_GGA(.(101, T76), 101, X148) → REDUCEB_IN_GGA(T76, 101, X148)
REDUCEB_IN_GGA(.(105, T76), 105, X148) → REDUCEB_IN_GGA(T76, 105, X148)
REDUCEB_IN_GGA(.(111, T76), 111, X148) → REDUCEB_IN_GGA(T76, 111, X148)
REDUCEB_IN_GGA(.(117, T76), 117, X148) → REDUCEB_IN_GGA(T76, 117, X148)
REDUCEB_IN_GGA(.(104, T76), 104, X148) → REDUCEB_IN_GGA(T76, 104, X148)
REDUCEB_IN_GGA(.(119, T76), 119, X148) → REDUCEB_IN_GGA(T76, 119, X148)
REDUCEB_IN_GGA(.(121, T76), 121, X148) → REDUCEB_IN_GGA(T76, 121, X148)

The TRS R consists of the following rules:

goalD_in_ga(.(T15, T16), T7) → U30_ga(T15, T16, T7, reduceA_in_gga(T16, T15, X12))
reduceA_in_gga([], T23, []) → reduceA_out_gga([], T23, [])
reduceA_in_gga(.(97, T33), 97, X52) → U1_gga(T33, X52, reduceA_in_gga(T33, 97, X52))
reduceA_in_gga(.(101, T33), 101, X52) → U2_gga(T33, X52, reduceA_in_gga(T33, 101, X52))
reduceA_in_gga(.(105, T33), 105, X52) → U3_gga(T33, X52, reduceA_in_gga(T33, 105, X52))
reduceA_in_gga(.(111, T33), 111, X52) → U4_gga(T33, X52, reduceA_in_gga(T33, 111, X52))
reduceA_in_gga(.(117, T33), 117, X52) → U5_gga(T33, X52, reduceA_in_gga(T33, 117, X52))
reduceA_in_gga(.(104, T33), 104, X52) → U6_gga(T33, X52, reduceA_in_gga(T33, 104, X52))
reduceA_in_gga(.(119, T33), 119, X52) → U7_gga(T33, X52, reduceA_in_gga(T33, 119, X52))
reduceA_in_gga(.(121, T33), 121, X52) → U8_gga(T33, X52, reduceA_in_gga(T33, 121, X52))
reduceA_in_gga(.(T50, T51), T50, X93) → U9_gga(T50, T51, X93, reduceA_in_gga(T51, T50, X93))
reduceA_in_gga(.(T59, T60), T61, .(T59, X113)) → U10_gga(T59, T60, T61, X113, reduceB_in_gga(T60, T59, X113))
reduceB_in_gga([], T66, []) → reduceB_out_gga([], T66, [])
reduceB_in_gga(.(97, T76), 97, X148) → U11_gga(T76, X148, reduceB_in_gga(T76, 97, X148))
reduceB_in_gga(.(101, T76), 101, X148) → U12_gga(T76, X148, reduceB_in_gga(T76, 101, X148))
reduceB_in_gga(.(105, T76), 105, X148) → U13_gga(T76, X148, reduceB_in_gga(T76, 105, X148))
reduceB_in_gga(.(111, T76), 111, X148) → U14_gga(T76, X148, reduceB_in_gga(T76, 111, X148))
reduceB_in_gga(.(117, T76), 117, X148) → U15_gga(T76, X148, reduceB_in_gga(T76, 117, X148))
reduceB_in_gga(.(104, T76), 104, X148) → U16_gga(T76, X148, reduceB_in_gga(T76, 104, X148))
reduceB_in_gga(.(119, T76), 119, X148) → U17_gga(T76, X148, reduceB_in_gga(T76, 119, X148))
reduceB_in_gga(.(121, T76), 121, X148) → U18_gga(T76, X148, reduceB_in_gga(T76, 121, X148))
reduceB_in_gga(.(T93, T94), T93, X189) → U19_gga(T93, T94, X189, reduceB_in_gga(T94, T93, X189))
reduceB_in_gga(.(T102, T103), T104, .(T102, X209)) → U20_gga(T102, T103, T104, X209, reduceC_in_gga(T103, T102, X209))
reduceC_in_gga([], T109, []) → reduceC_out_gga([], T109, [])
reduceC_in_gga(.(97, T119), 97, X244) → U21_gga(T119, X244, reduceC_in_gga(T119, 97, X244))
reduceC_in_gga(.(101, T119), 101, X244) → U22_gga(T119, X244, reduceC_in_gga(T119, 101, X244))
reduceC_in_gga(.(105, T119), 105, X244) → U23_gga(T119, X244, reduceC_in_gga(T119, 105, X244))
reduceC_in_gga(.(111, T119), 111, X244) → U24_gga(T119, X244, reduceC_in_gga(T119, 111, X244))
reduceC_in_gga(.(117, T119), 117, X244) → U25_gga(T119, X244, reduceC_in_gga(T119, 117, X244))
reduceC_in_gga(.(104, T119), 104, X244) → U26_gga(T119, X244, reduceC_in_gga(T119, 104, X244))
reduceC_in_gga(.(119, T119), 119, X244) → U27_gga(T119, X244, reduceC_in_gga(T119, 119, X244))
reduceC_in_gga(.(121, T119), 121, X244) → U28_gga(T119, X244, reduceC_in_gga(T119, 121, X244))
reduceC_in_gga(.(T136, T137), T136, X285) → U29_gga(T136, T137, X285, reduceC_in_gga(T137, T136, X285))
reduceC_in_gga(.(T155, T154), T147, .(T155, [])) → reduceC_out_gga(.(T155, T154), T147, .(T155, []))
U29_gga(T136, T137, X285, reduceC_out_gga(T137, T136, X285)) → reduceC_out_gga(.(T136, T137), T136, X285)
U28_gga(T119, X244, reduceC_out_gga(T119, 121, X244)) → reduceC_out_gga(.(121, T119), 121, X244)
U27_gga(T119, X244, reduceC_out_gga(T119, 119, X244)) → reduceC_out_gga(.(119, T119), 119, X244)
U26_gga(T119, X244, reduceC_out_gga(T119, 104, X244)) → reduceC_out_gga(.(104, T119), 104, X244)
U25_gga(T119, X244, reduceC_out_gga(T119, 117, X244)) → reduceC_out_gga(.(117, T119), 117, X244)
U24_gga(T119, X244, reduceC_out_gga(T119, 111, X244)) → reduceC_out_gga(.(111, T119), 111, X244)
U23_gga(T119, X244, reduceC_out_gga(T119, 105, X244)) → reduceC_out_gga(.(105, T119), 105, X244)
U22_gga(T119, X244, reduceC_out_gga(T119, 101, X244)) → reduceC_out_gga(.(101, T119), 101, X244)
U21_gga(T119, X244, reduceC_out_gga(T119, 97, X244)) → reduceC_out_gga(.(97, T119), 97, X244)
U20_gga(T102, T103, T104, X209, reduceC_out_gga(T103, T102, X209)) → reduceB_out_gga(.(T102, T103), T104, .(T102, X209))
U19_gga(T93, T94, X189, reduceB_out_gga(T94, T93, X189)) → reduceB_out_gga(.(T93, T94), T93, X189)
U18_gga(T76, X148, reduceB_out_gga(T76, 121, X148)) → reduceB_out_gga(.(121, T76), 121, X148)
U17_gga(T76, X148, reduceB_out_gga(T76, 119, X148)) → reduceB_out_gga(.(119, T76), 119, X148)
U16_gga(T76, X148, reduceB_out_gga(T76, 104, X148)) → reduceB_out_gga(.(104, T76), 104, X148)
U15_gga(T76, X148, reduceB_out_gga(T76, 117, X148)) → reduceB_out_gga(.(117, T76), 117, X148)
U14_gga(T76, X148, reduceB_out_gga(T76, 111, X148)) → reduceB_out_gga(.(111, T76), 111, X148)
U13_gga(T76, X148, reduceB_out_gga(T76, 105, X148)) → reduceB_out_gga(.(105, T76), 105, X148)
U12_gga(T76, X148, reduceB_out_gga(T76, 101, X148)) → reduceB_out_gga(.(101, T76), 101, X148)
U11_gga(T76, X148, reduceB_out_gga(T76, 97, X148)) → reduceB_out_gga(.(97, T76), 97, X148)
U10_gga(T59, T60, T61, X113, reduceB_out_gga(T60, T59, X113)) → reduceA_out_gga(.(T59, T60), T61, .(T59, X113))
U9_gga(T50, T51, X93, reduceA_out_gga(T51, T50, X93)) → reduceA_out_gga(.(T50, T51), T50, X93)
U8_gga(T33, X52, reduceA_out_gga(T33, 121, X52)) → reduceA_out_gga(.(121, T33), 121, X52)
U7_gga(T33, X52, reduceA_out_gga(T33, 119, X52)) → reduceA_out_gga(.(119, T33), 119, X52)
U6_gga(T33, X52, reduceA_out_gga(T33, 104, X52)) → reduceA_out_gga(.(104, T33), 104, X52)
U5_gga(T33, X52, reduceA_out_gga(T33, 117, X52)) → reduceA_out_gga(.(117, T33), 117, X52)
U4_gga(T33, X52, reduceA_out_gga(T33, 111, X52)) → reduceA_out_gga(.(111, T33), 111, X52)
U3_gga(T33, X52, reduceA_out_gga(T33, 105, X52)) → reduceA_out_gga(.(105, T33), 105, X52)
U2_gga(T33, X52, reduceA_out_gga(T33, 101, X52)) → reduceA_out_gga(.(101, T33), 101, X52)
U1_gga(T33, X52, reduceA_out_gga(T33, 97, X52)) → reduceA_out_gga(.(97, T33), 97, X52)
U30_ga(T15, T16, T7, reduceA_out_gga(T16, T15, X12)) → goalD_out_ga(.(T15, T16), T7)
goalD_in_ga(.(T15, T16), T158) → U31_ga(T15, T16, T158, reduceA_in_gga(T16, T15, T158))
U31_ga(T15, T16, T158, reduceA_out_gga(T16, T15, T158)) → goalD_out_ga(.(T15, T16), T158)

The argument filtering Pi contains the following mapping:
goalD_in_ga(x1, x2)  =  goalD_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U30_ga(x1, x2, x3, x4)  =  U30_ga(x4)
reduceA_in_gga(x1, x2, x3)  =  reduceA_in_gga(x1, x2)
[]  =  []
reduceA_out_gga(x1, x2, x3)  =  reduceA_out_gga(x3)
97  =  97
U1_gga(x1, x2, x3)  =  U1_gga(x3)
101  =  101
U2_gga(x1, x2, x3)  =  U2_gga(x3)
105  =  105
U3_gga(x1, x2, x3)  =  U3_gga(x3)
111  =  111
U4_gga(x1, x2, x3)  =  U4_gga(x3)
117  =  117
U5_gga(x1, x2, x3)  =  U5_gga(x3)
104  =  104
U6_gga(x1, x2, x3)  =  U6_gga(x3)
119  =  119
U7_gga(x1, x2, x3)  =  U7_gga(x3)
121  =  121
U8_gga(x1, x2, x3)  =  U8_gga(x3)
U9_gga(x1, x2, x3, x4)  =  U9_gga(x4)
U10_gga(x1, x2, x3, x4, x5)  =  U10_gga(x1, x5)
reduceB_in_gga(x1, x2, x3)  =  reduceB_in_gga(x1, x2)
reduceB_out_gga(x1, x2, x3)  =  reduceB_out_gga(x3)
U11_gga(x1, x2, x3)  =  U11_gga(x3)
U12_gga(x1, x2, x3)  =  U12_gga(x3)
U13_gga(x1, x2, x3)  =  U13_gga(x3)
U14_gga(x1, x2, x3)  =  U14_gga(x3)
U15_gga(x1, x2, x3)  =  U15_gga(x3)
U16_gga(x1, x2, x3)  =  U16_gga(x3)
U17_gga(x1, x2, x3)  =  U17_gga(x3)
U18_gga(x1, x2, x3)  =  U18_gga(x3)
U19_gga(x1, x2, x3, x4)  =  U19_gga(x4)
U20_gga(x1, x2, x3, x4, x5)  =  U20_gga(x1, x5)
reduceC_in_gga(x1, x2, x3)  =  reduceC_in_gga(x1, x2)
reduceC_out_gga(x1, x2, x3)  =  reduceC_out_gga(x3)
U21_gga(x1, x2, x3)  =  U21_gga(x3)
U22_gga(x1, x2, x3)  =  U22_gga(x3)
U23_gga(x1, x2, x3)  =  U23_gga(x3)
U24_gga(x1, x2, x3)  =  U24_gga(x3)
U25_gga(x1, x2, x3)  =  U25_gga(x3)
U26_gga(x1, x2, x3)  =  U26_gga(x3)
U27_gga(x1, x2, x3)  =  U27_gga(x3)
U28_gga(x1, x2, x3)  =  U28_gga(x3)
U29_gga(x1, x2, x3, x4)  =  U29_gga(x4)
goalD_out_ga(x1, x2)  =  goalD_out_ga
U31_ga(x1, x2, x3, x4)  =  U31_ga(x4)
REDUCEB_IN_GGA(x1, x2, x3)  =  REDUCEB_IN_GGA(x1, x2)

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

(17) UsableRulesProof (EQUIVALENT transformation)

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

(18) Obligation:

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

REDUCEB_IN_GGA(.(T93, T94), T93, X189) → REDUCEB_IN_GGA(T94, T93, X189)
REDUCEB_IN_GGA(.(97, T76), 97, X148) → REDUCEB_IN_GGA(T76, 97, X148)
REDUCEB_IN_GGA(.(101, T76), 101, X148) → REDUCEB_IN_GGA(T76, 101, X148)
REDUCEB_IN_GGA(.(105, T76), 105, X148) → REDUCEB_IN_GGA(T76, 105, X148)
REDUCEB_IN_GGA(.(111, T76), 111, X148) → REDUCEB_IN_GGA(T76, 111, X148)
REDUCEB_IN_GGA(.(117, T76), 117, X148) → REDUCEB_IN_GGA(T76, 117, X148)
REDUCEB_IN_GGA(.(104, T76), 104, X148) → REDUCEB_IN_GGA(T76, 104, X148)
REDUCEB_IN_GGA(.(119, T76), 119, X148) → REDUCEB_IN_GGA(T76, 119, X148)
REDUCEB_IN_GGA(.(121, T76), 121, X148) → REDUCEB_IN_GGA(T76, 121, X148)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
97  =  97
101  =  101
105  =  105
111  =  111
117  =  117
104  =  104
119  =  119
121  =  121
REDUCEB_IN_GGA(x1, x2, x3)  =  REDUCEB_IN_GGA(x1, x2)

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

(19) PiDPToQDPProof (SOUND transformation)

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

(20) Obligation:

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

REDUCEB_IN_GGA(.(T93, T94), T93) → REDUCEB_IN_GGA(T94, T93)
REDUCEB_IN_GGA(.(97, T76), 97) → REDUCEB_IN_GGA(T76, 97)
REDUCEB_IN_GGA(.(101, T76), 101) → REDUCEB_IN_GGA(T76, 101)
REDUCEB_IN_GGA(.(105, T76), 105) → REDUCEB_IN_GGA(T76, 105)
REDUCEB_IN_GGA(.(111, T76), 111) → REDUCEB_IN_GGA(T76, 111)
REDUCEB_IN_GGA(.(117, T76), 117) → REDUCEB_IN_GGA(T76, 117)
REDUCEB_IN_GGA(.(104, T76), 104) → REDUCEB_IN_GGA(T76, 104)
REDUCEB_IN_GGA(.(119, T76), 119) → REDUCEB_IN_GGA(T76, 119)
REDUCEB_IN_GGA(.(121, T76), 121) → REDUCEB_IN_GGA(T76, 121)

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

(21) 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:

• REDUCEB_IN_GGA(.(T93, T94), T93) → REDUCEB_IN_GGA(T94, T93)
  The graph contains the following edges 1 > 1, 1 > 2, 2 >= 2

• REDUCEB_IN_GGA(.(97, T76), 97) → REDUCEB_IN_GGA(T76, 97)
  The graph contains the following edges 1 > 1, 1 > 2, 2 >= 2

• REDUCEB_IN_GGA(.(101, T76), 101) → REDUCEB_IN_GGA(T76, 101)
  The graph contains the following edges 1 > 1, 1 > 2, 2 >= 2

• REDUCEB_IN_GGA(.(105, T76), 105) → REDUCEB_IN_GGA(T76, 105)
  The graph contains the following edges 1 > 1, 1 > 2, 2 >= 2

• REDUCEB_IN_GGA(.(111, T76), 111) → REDUCEB_IN_GGA(T76, 111)
  The graph contains the following edges 1 > 1, 1 > 2, 2 >= 2

• REDUCEB_IN_GGA(.(117, T76), 117) → REDUCEB_IN_GGA(T76, 117)
  The graph contains the following edges 1 > 1, 1 > 2, 2 >= 2

• REDUCEB_IN_GGA(.(104, T76), 104) → REDUCEB_IN_GGA(T76, 104)
  The graph contains the following edges 1 > 1, 1 > 2, 2 >= 2

• REDUCEB_IN_GGA(.(119, T76), 119) → REDUCEB_IN_GGA(T76, 119)
  The graph contains the following edges 1 > 1, 1 > 2, 2 >= 2

• REDUCEB_IN_GGA(.(121, T76), 121) → REDUCEB_IN_GGA(T76, 121)
  The graph contains the following edges 1 > 1, 1 > 2, 2 >= 2

(22) YES

(23) Obligation:

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

REDUCEA_IN_GGA(.(T50, T51), T50, X93) → REDUCEA_IN_GGA(T51, T50, X93)
REDUCEA_IN_GGA(.(97, T33), 97, X52) → REDUCEA_IN_GGA(T33, 97, X52)
REDUCEA_IN_GGA(.(101, T33), 101, X52) → REDUCEA_IN_GGA(T33, 101, X52)
REDUCEA_IN_GGA(.(105, T33), 105, X52) → REDUCEA_IN_GGA(T33, 105, X52)
REDUCEA_IN_GGA(.(111, T33), 111, X52) → REDUCEA_IN_GGA(T33, 111, X52)
REDUCEA_IN_GGA(.(117, T33), 117, X52) → REDUCEA_IN_GGA(T33, 117, X52)
REDUCEA_IN_GGA(.(104, T33), 104, X52) → REDUCEA_IN_GGA(T33, 104, X52)
REDUCEA_IN_GGA(.(119, T33), 119, X52) → REDUCEA_IN_GGA(T33, 119, X52)
REDUCEA_IN_GGA(.(121, T33), 121, X52) → REDUCEA_IN_GGA(T33, 121, X52)

The TRS R consists of the following rules:

goalD_in_ga(.(T15, T16), T7) → U30_ga(T15, T16, T7, reduceA_in_gga(T16, T15, X12))
reduceA_in_gga([], T23, []) → reduceA_out_gga([], T23, [])
reduceA_in_gga(.(97, T33), 97, X52) → U1_gga(T33, X52, reduceA_in_gga(T33, 97, X52))
reduceA_in_gga(.(101, T33), 101, X52) → U2_gga(T33, X52, reduceA_in_gga(T33, 101, X52))
reduceA_in_gga(.(105, T33), 105, X52) → U3_gga(T33, X52, reduceA_in_gga(T33, 105, X52))
reduceA_in_gga(.(111, T33), 111, X52) → U4_gga(T33, X52, reduceA_in_gga(T33, 111, X52))
reduceA_in_gga(.(117, T33), 117, X52) → U5_gga(T33, X52, reduceA_in_gga(T33, 117, X52))
reduceA_in_gga(.(104, T33), 104, X52) → U6_gga(T33, X52, reduceA_in_gga(T33, 104, X52))
reduceA_in_gga(.(119, T33), 119, X52) → U7_gga(T33, X52, reduceA_in_gga(T33, 119, X52))
reduceA_in_gga(.(121, T33), 121, X52) → U8_gga(T33, X52, reduceA_in_gga(T33, 121, X52))
reduceA_in_gga(.(T50, T51), T50, X93) → U9_gga(T50, T51, X93, reduceA_in_gga(T51, T50, X93))
reduceA_in_gga(.(T59, T60), T61, .(T59, X113)) → U10_gga(T59, T60, T61, X113, reduceB_in_gga(T60, T59, X113))
reduceB_in_gga([], T66, []) → reduceB_out_gga([], T66, [])
reduceB_in_gga(.(97, T76), 97, X148) → U11_gga(T76, X148, reduceB_in_gga(T76, 97, X148))
reduceB_in_gga(.(101, T76), 101, X148) → U12_gga(T76, X148, reduceB_in_gga(T76, 101, X148))
reduceB_in_gga(.(105, T76), 105, X148) → U13_gga(T76, X148, reduceB_in_gga(T76, 105, X148))
reduceB_in_gga(.(111, T76), 111, X148) → U14_gga(T76, X148, reduceB_in_gga(T76, 111, X148))
reduceB_in_gga(.(117, T76), 117, X148) → U15_gga(T76, X148, reduceB_in_gga(T76, 117, X148))
reduceB_in_gga(.(104, T76), 104, X148) → U16_gga(T76, X148, reduceB_in_gga(T76, 104, X148))
reduceB_in_gga(.(119, T76), 119, X148) → U17_gga(T76, X148, reduceB_in_gga(T76, 119, X148))
reduceB_in_gga(.(121, T76), 121, X148) → U18_gga(T76, X148, reduceB_in_gga(T76, 121, X148))
reduceB_in_gga(.(T93, T94), T93, X189) → U19_gga(T93, T94, X189, reduceB_in_gga(T94, T93, X189))
reduceB_in_gga(.(T102, T103), T104, .(T102, X209)) → U20_gga(T102, T103, T104, X209, reduceC_in_gga(T103, T102, X209))
reduceC_in_gga([], T109, []) → reduceC_out_gga([], T109, [])
reduceC_in_gga(.(97, T119), 97, X244) → U21_gga(T119, X244, reduceC_in_gga(T119, 97, X244))
reduceC_in_gga(.(101, T119), 101, X244) → U22_gga(T119, X244, reduceC_in_gga(T119, 101, X244))
reduceC_in_gga(.(105, T119), 105, X244) → U23_gga(T119, X244, reduceC_in_gga(T119, 105, X244))
reduceC_in_gga(.(111, T119), 111, X244) → U24_gga(T119, X244, reduceC_in_gga(T119, 111, X244))
reduceC_in_gga(.(117, T119), 117, X244) → U25_gga(T119, X244, reduceC_in_gga(T119, 117, X244))
reduceC_in_gga(.(104, T119), 104, X244) → U26_gga(T119, X244, reduceC_in_gga(T119, 104, X244))
reduceC_in_gga(.(119, T119), 119, X244) → U27_gga(T119, X244, reduceC_in_gga(T119, 119, X244))
reduceC_in_gga(.(121, T119), 121, X244) → U28_gga(T119, X244, reduceC_in_gga(T119, 121, X244))
reduceC_in_gga(.(T136, T137), T136, X285) → U29_gga(T136, T137, X285, reduceC_in_gga(T137, T136, X285))
reduceC_in_gga(.(T155, T154), T147, .(T155, [])) → reduceC_out_gga(.(T155, T154), T147, .(T155, []))
U29_gga(T136, T137, X285, reduceC_out_gga(T137, T136, X285)) → reduceC_out_gga(.(T136, T137), T136, X285)
U28_gga(T119, X244, reduceC_out_gga(T119, 121, X244)) → reduceC_out_gga(.(121, T119), 121, X244)
U27_gga(T119, X244, reduceC_out_gga(T119, 119, X244)) → reduceC_out_gga(.(119, T119), 119, X244)
U26_gga(T119, X244, reduceC_out_gga(T119, 104, X244)) → reduceC_out_gga(.(104, T119), 104, X244)
U25_gga(T119, X244, reduceC_out_gga(T119, 117, X244)) → reduceC_out_gga(.(117, T119), 117, X244)
U24_gga(T119, X244, reduceC_out_gga(T119, 111, X244)) → reduceC_out_gga(.(111, T119), 111, X244)
U23_gga(T119, X244, reduceC_out_gga(T119, 105, X244)) → reduceC_out_gga(.(105, T119), 105, X244)
U22_gga(T119, X244, reduceC_out_gga(T119, 101, X244)) → reduceC_out_gga(.(101, T119), 101, X244)
U21_gga(T119, X244, reduceC_out_gga(T119, 97, X244)) → reduceC_out_gga(.(97, T119), 97, X244)
U20_gga(T102, T103, T104, X209, reduceC_out_gga(T103, T102, X209)) → reduceB_out_gga(.(T102, T103), T104, .(T102, X209))
U19_gga(T93, T94, X189, reduceB_out_gga(T94, T93, X189)) → reduceB_out_gga(.(T93, T94), T93, X189)
U18_gga(T76, X148, reduceB_out_gga(T76, 121, X148)) → reduceB_out_gga(.(121, T76), 121, X148)
U17_gga(T76, X148, reduceB_out_gga(T76, 119, X148)) → reduceB_out_gga(.(119, T76), 119, X148)
U16_gga(T76, X148, reduceB_out_gga(T76, 104, X148)) → reduceB_out_gga(.(104, T76), 104, X148)
U15_gga(T76, X148, reduceB_out_gga(T76, 117, X148)) → reduceB_out_gga(.(117, T76), 117, X148)
U14_gga(T76, X148, reduceB_out_gga(T76, 111, X148)) → reduceB_out_gga(.(111, T76), 111, X148)
U13_gga(T76, X148, reduceB_out_gga(T76, 105, X148)) → reduceB_out_gga(.(105, T76), 105, X148)
U12_gga(T76, X148, reduceB_out_gga(T76, 101, X148)) → reduceB_out_gga(.(101, T76), 101, X148)
U11_gga(T76, X148, reduceB_out_gga(T76, 97, X148)) → reduceB_out_gga(.(97, T76), 97, X148)
U10_gga(T59, T60, T61, X113, reduceB_out_gga(T60, T59, X113)) → reduceA_out_gga(.(T59, T60), T61, .(T59, X113))
U9_gga(T50, T51, X93, reduceA_out_gga(T51, T50, X93)) → reduceA_out_gga(.(T50, T51), T50, X93)
U8_gga(T33, X52, reduceA_out_gga(T33, 121, X52)) → reduceA_out_gga(.(121, T33), 121, X52)
U7_gga(T33, X52, reduceA_out_gga(T33, 119, X52)) → reduceA_out_gga(.(119, T33), 119, X52)
U6_gga(T33, X52, reduceA_out_gga(T33, 104, X52)) → reduceA_out_gga(.(104, T33), 104, X52)
U5_gga(T33, X52, reduceA_out_gga(T33, 117, X52)) → reduceA_out_gga(.(117, T33), 117, X52)
U4_gga(T33, X52, reduceA_out_gga(T33, 111, X52)) → reduceA_out_gga(.(111, T33), 111, X52)
U3_gga(T33, X52, reduceA_out_gga(T33, 105, X52)) → reduceA_out_gga(.(105, T33), 105, X52)
U2_gga(T33, X52, reduceA_out_gga(T33, 101, X52)) → reduceA_out_gga(.(101, T33), 101, X52)
U1_gga(T33, X52, reduceA_out_gga(T33, 97, X52)) → reduceA_out_gga(.(97, T33), 97, X52)
U30_ga(T15, T16, T7, reduceA_out_gga(T16, T15, X12)) → goalD_out_ga(.(T15, T16), T7)
goalD_in_ga(.(T15, T16), T158) → U31_ga(T15, T16, T158, reduceA_in_gga(T16, T15, T158))
U31_ga(T15, T16, T158, reduceA_out_gga(T16, T15, T158)) → goalD_out_ga(.(T15, T16), T158)

The argument filtering Pi contains the following mapping:
goalD_in_ga(x1, x2)  =  goalD_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U30_ga(x1, x2, x3, x4)  =  U30_ga(x4)
reduceA_in_gga(x1, x2, x3)  =  reduceA_in_gga(x1, x2)
[]  =  []
reduceA_out_gga(x1, x2, x3)  =  reduceA_out_gga(x3)
97  =  97
U1_gga(x1, x2, x3)  =  U1_gga(x3)
101  =  101
U2_gga(x1, x2, x3)  =  U2_gga(x3)
105  =  105
U3_gga(x1, x2, x3)  =  U3_gga(x3)
111  =  111
U4_gga(x1, x2, x3)  =  U4_gga(x3)
117  =  117
U5_gga(x1, x2, x3)  =  U5_gga(x3)
104  =  104
U6_gga(x1, x2, x3)  =  U6_gga(x3)
119  =  119
U7_gga(x1, x2, x3)  =  U7_gga(x3)
121  =  121
U8_gga(x1, x2, x3)  =  U8_gga(x3)
U9_gga(x1, x2, x3, x4)  =  U9_gga(x4)
U10_gga(x1, x2, x3, x4, x5)  =  U10_gga(x1, x5)
reduceB_in_gga(x1, x2, x3)  =  reduceB_in_gga(x1, x2)
reduceB_out_gga(x1, x2, x3)  =  reduceB_out_gga(x3)
U11_gga(x1, x2, x3)  =  U11_gga(x3)
U12_gga(x1, x2, x3)  =  U12_gga(x3)
U13_gga(x1, x2, x3)  =  U13_gga(x3)
U14_gga(x1, x2, x3)  =  U14_gga(x3)
U15_gga(x1, x2, x3)  =  U15_gga(x3)
U16_gga(x1, x2, x3)  =  U16_gga(x3)
U17_gga(x1, x2, x3)  =  U17_gga(x3)
U18_gga(x1, x2, x3)  =  U18_gga(x3)
U19_gga(x1, x2, x3, x4)  =  U19_gga(x4)
U20_gga(x1, x2, x3, x4, x5)  =  U20_gga(x1, x5)
reduceC_in_gga(x1, x2, x3)  =  reduceC_in_gga(x1, x2)
reduceC_out_gga(x1, x2, x3)  =  reduceC_out_gga(x3)
U21_gga(x1, x2, x3)  =  U21_gga(x3)
U22_gga(x1, x2, x3)  =  U22_gga(x3)
U23_gga(x1, x2, x3)  =  U23_gga(x3)
U24_gga(x1, x2, x3)  =  U24_gga(x3)
U25_gga(x1, x2, x3)  =  U25_gga(x3)
U26_gga(x1, x2, x3)  =  U26_gga(x3)
U27_gga(x1, x2, x3)  =  U27_gga(x3)
U28_gga(x1, x2, x3)  =  U28_gga(x3)
U29_gga(x1, x2, x3, x4)  =  U29_gga(x4)
goalD_out_ga(x1, x2)  =  goalD_out_ga
U31_ga(x1, x2, x3, x4)  =  U31_ga(x4)
REDUCEA_IN_GGA(x1, x2, x3)  =  REDUCEA_IN_GGA(x1, x2)

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

(24) UsableRulesProof (EQUIVALENT transformation)

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

(25) Obligation:

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

REDUCEA_IN_GGA(.(T50, T51), T50, X93) → REDUCEA_IN_GGA(T51, T50, X93)
REDUCEA_IN_GGA(.(97, T33), 97, X52) → REDUCEA_IN_GGA(T33, 97, X52)
REDUCEA_IN_GGA(.(101, T33), 101, X52) → REDUCEA_IN_GGA(T33, 101, X52)
REDUCEA_IN_GGA(.(105, T33), 105, X52) → REDUCEA_IN_GGA(T33, 105, X52)
REDUCEA_IN_GGA(.(111, T33), 111, X52) → REDUCEA_IN_GGA(T33, 111, X52)
REDUCEA_IN_GGA(.(117, T33), 117, X52) → REDUCEA_IN_GGA(T33, 117, X52)
REDUCEA_IN_GGA(.(104, T33), 104, X52) → REDUCEA_IN_GGA(T33, 104, X52)
REDUCEA_IN_GGA(.(119, T33), 119, X52) → REDUCEA_IN_GGA(T33, 119, X52)
REDUCEA_IN_GGA(.(121, T33), 121, X52) → REDUCEA_IN_GGA(T33, 121, X52)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
97  =  97
101  =  101
105  =  105
111  =  111
117  =  117
104  =  104
119  =  119
121  =  121
REDUCEA_IN_GGA(x1, x2, x3)  =  REDUCEA_IN_GGA(x1, x2)

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

(26) PiDPToQDPProof (SOUND transformation)

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

(27) Obligation:

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

REDUCEA_IN_GGA(.(T50, T51), T50) → REDUCEA_IN_GGA(T51, T50)
REDUCEA_IN_GGA(.(97, T33), 97) → REDUCEA_IN_GGA(T33, 97)
REDUCEA_IN_GGA(.(101, T33), 101) → REDUCEA_IN_GGA(T33, 101)
REDUCEA_IN_GGA(.(105, T33), 105) → REDUCEA_IN_GGA(T33, 105)
REDUCEA_IN_GGA(.(111, T33), 111) → REDUCEA_IN_GGA(T33, 111)
REDUCEA_IN_GGA(.(117, T33), 117) → REDUCEA_IN_GGA(T33, 117)
REDUCEA_IN_GGA(.(104, T33), 104) → REDUCEA_IN_GGA(T33, 104)
REDUCEA_IN_GGA(.(119, T33), 119) → REDUCEA_IN_GGA(T33, 119)
REDUCEA_IN_GGA(.(121, T33), 121) → REDUCEA_IN_GGA(T33, 121)

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

(28) 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:

• REDUCEA_IN_GGA(.(T50, T51), T50) → REDUCEA_IN_GGA(T51, T50)
  The graph contains the following edges 1 > 1, 1 > 2, 2 >= 2

• REDUCEA_IN_GGA(.(97, T33), 97) → REDUCEA_IN_GGA(T33, 97)
  The graph contains the following edges 1 > 1, 1 > 2, 2 >= 2

• REDUCEA_IN_GGA(.(101, T33), 101) → REDUCEA_IN_GGA(T33, 101)
  The graph contains the following edges 1 > 1, 1 > 2, 2 >= 2

• REDUCEA_IN_GGA(.(105, T33), 105) → REDUCEA_IN_GGA(T33, 105)
  The graph contains the following edges 1 > 1, 1 > 2, 2 >= 2

• REDUCEA_IN_GGA(.(111, T33), 111) → REDUCEA_IN_GGA(T33, 111)
  The graph contains the following edges 1 > 1, 1 > 2, 2 >= 2

• REDUCEA_IN_GGA(.(117, T33), 117) → REDUCEA_IN_GGA(T33, 117)
  The graph contains the following edges 1 > 1, 1 > 2, 2 >= 2

• REDUCEA_IN_GGA(.(104, T33), 104) → REDUCEA_IN_GGA(T33, 104)
  The graph contains the following edges 1 > 1, 1 > 2, 2 >= 2

• REDUCEA_IN_GGA(.(119, T33), 119) → REDUCEA_IN_GGA(T33, 119)
  The graph contains the following edges 1 > 1, 1 > 2, 2 >= 2

• REDUCEA_IN_GGA(.(121, T33), 121) → REDUCEA_IN_GGA(T33, 121)
  The graph contains the following edges 1 > 1, 1 > 2, 2 >= 2

(29) YES