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

(0) Obligation:

Clauses:

ordered([]).
ordered(.(X1, [])).
ordered(Xs) :- ','(no(max1el_list(Xs)), ','(head(Xs, X), ','(tail(Xs, Ys), ','(head(Ys, Y), ','(tail(Ys, Zs), ','(less(X, s(Y)), ordered(.(Y, Zs)))))))).
head([], X2).
head(.(X, X3), X).
tail([], []).
tail(.(X4, Xs), Xs).
less(0, s(X5)).
less(X, Y) :- ','(no(zero(X)), ','(p(X, Px), ','(p(Y, Py), less(Px, Py)))).
p(0, 0).
p(s(X), X).
max1el_list([]).
max1el_list(.(X6, [])).
zero(0).
no(X) :- ','(X, ','(!, failure(a))).
no(X7).
failure(b).

Query: ordered(g)

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph ICLP10.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: ordered(T1)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: ordered(T1), SCOPE: 1, CLAUSE: 0 | ordered(T1), SCOPE: 1, CLAUSE: 1 | ordered(T1), SCOPE: 1, CLAUSE: 2\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: true | ordered([]), SCOPE: 1, CLAUSE: 1 | ordered([]), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: ordered(T1), SCOPE: 1, CLAUSE: 1 | ordered(T1), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nordered(T1) !=? ordered([])", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: ordered([]), SCOPE: 1, CLAUSE: 1 | ordered([]), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ordered([]), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ','(no(max1el_list([])), ','(head([], X11), ','(tail([], X12), ','(head(X12, X13), ','(tail(X12, X14), ','(less(X11, s(X13)), ordered(.(X13, X14))))))))\n\nX11 is free\nX12 is free\nX13 is free\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: ','(no(max1el_list([])), ','(head([], X11), ','(tail([], X12), ','(head(X12, X13), ','(tail(X12, X14), ','(less(X11, s(X13)), ordered(.(X13, X14)))))))), SCOPE: 2, CLAUSE: 14 | ','(no(max1el_list([])), ','(head([], X11), ','(tail([], X12), ','(head(X12, X13), ','(tail(X12, X14), ','(less(X11, s(X13)), ordered(.(X13, X14)))))))), SCOPE: 2, CLAUSE: 15\n\nX11 is free\nX12 is free\nX13 is free\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: ','(','(call(max1el_list([])), ','(!_2, failure(a))), ','(head([], X11), ','(tail([], X12), ','(head(X12, X13), ','(tail(X12, X14), ','(less(X11, s(X13)), ordered(.(X13, X14)))))))) | ','(no(max1el_list([])), ','(head([], X11), ','(tail([], X12), ','(head(X12, X13), ','(tail(X12, X14), ','(less(X11, s(X13)), ordered(.(X13, X14)))))))), SCOPE: 2, CLAUSE: 15\n\nX11 is free\nX12 is free\nX13 is free\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: ','(max1el_list([]), ','(','(!_2, failure(a)), ','(head([], X11), ','(tail([], X12), ','(head(X12, X13), ','(tail(X12, X14), ','(less(X11, s(X13)), ordered(.(X13, X14))))))))) | ?_3 | ','(no(max1el_list([])), ','(head([], X11), ','(tail([], X12), ','(head(X12, X13), ','(tail(X12, X14), ','(less(X11, s(X13)), ordered(.(X13, X14)))))))), SCOPE: 2, CLAUSE: 15\n\nX11 is free\nX12 is free\nX13 is free\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(max1el_list([]), ','(','(!_2, failure(a)), ','(head([], X11), ','(tail([], X12), ','(head(X12, X13), ','(tail(X12, X14), ','(less(X11, s(X13)), ordered(.(X13, X14))))))))), SCOPE: 4, CLAUSE: 11 | ','(max1el_list([]), ','(','(!_2, failure(a)), ','(head([], X11), ','(tail([], X12), ','(head(X12, X13), ','(tail(X12, X14), ','(less(X11, s(X13)), ordered(.(X13, X14))))))))), SCOPE: 4, CLAUSE: 12 | ?_4 | ?_3 | ','(no(max1el_list([])), ','(head([], X11), ','(tail([], X12), ','(head(X12, X13), ','(tail(X12, X14), ','(less(X11, s(X13)), ordered(.(X13, X14)))))))), SCOPE: 2, CLAUSE: 15\n\nX11 is free\nX12 is free\nX13 is free\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: ','(','(!_2, failure(a)), ','(head([], X11), ','(tail([], X12), ','(head(X12, X13), ','(tail(X12, X14), ','(less(X11, s(X13)), ordered(.(X13, X14)))))))) | ','(max1el_list([]), ','(','(!_2, failure(a)), ','(head([], X11), ','(tail([], X12), ','(head(X12, X13), ','(tail(X12, X14), ','(less(X11, s(X13)), ordered(.(X13, X14))))))))), SCOPE: 4, CLAUSE: 12 | ?_4 | ?_3 | ','(no(max1el_list([])), ','(head([], X11), ','(tail([], X12), ','(head(X12, X13), ','(tail(X12, X14), ','(less(X11, s(X13)), ordered(.(X13, X14)))))))), SCOPE: 2, CLAUSE: 15\n\nX11 is free\nX12 is free\nX13 is free\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: ','(failure(a), ','(head([], X11), ','(tail([], X12), ','(head(X12, X13), ','(tail(X12, X14), ','(less(X11, s(X13)), ordered(.(X13, X14))))))))\n\nX11 is free\nX12 is free\nX13 is free\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: ','(failure(a), ','(head([], X11), ','(tail([], X12), ','(head(X12, X13), ','(tail(X12, X14), ','(less(X11, s(X13)), ordered(.(X13, X14)))))))), SCOPE: 5, CLAUSE: 16\n\nX11 is free\nX12 is free\nX13 is free\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: true | ordered(.(T3, [])), SCOPE: 1, CLAUSE: 2\n\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: ordered(T1), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nX21 is free\nordered(T1) !=? ordered([])\nordered(T1) !=? ordered(.(X21, []))", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: ordered(.(T3, [])), SCOPE: 1, CLAUSE: 2\n\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: ','(no(max1el_list(.(T5, []))), ','(head(.(T5, []), X24), ','(tail(.(T5, []), X25), ','(head(X25, X26), ','(tail(X25, X27), ','(less(X24, s(X26)), ordered(.(X26, X27))))))))\n\nT5 is ground\nX24 is free\nX25 is free\nX26 is free\nX27 is free", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: ','(no(max1el_list(.(T5, []))), ','(head(.(T5, []), X24), ','(tail(.(T5, []), X25), ','(head(X25, X26), ','(tail(X25, X27), ','(less(X24, s(X26)), ordered(.(X26, X27)))))))), SCOPE: 6, CLAUSE: 14 | ','(no(max1el_list(.(T5, []))), ','(head(.(T5, []), X24), ','(tail(.(T5, []), X25), ','(head(X25, X26), ','(tail(X25, X27), ','(less(X24, s(X26)), ordered(.(X26, X27)))))))), SCOPE: 6, CLAUSE: 15\n\nT5 is ground\nX24 is free\nX25 is free\nX26 is free\nX27 is free", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: ','(','(call(max1el_list(.(T8, []))), ','(!_6, failure(a))), ','(head(.(T8, []), X24), ','(tail(.(T8, []), X25), ','(head(X25, X26), ','(tail(X25, X27), ','(less(X24, s(X26)), ordered(.(X26, X27)))))))) | ','(no(max1el_list(.(T8, []))), ','(head(.(T8, []), X24), ','(tail(.(T8, []), X25), ','(head(X25, X26), ','(tail(X25, X27), ','(less(X24, s(X26)), ordered(.(X26, X27)))))))), SCOPE: 6, CLAUSE: 15\n\nT8 is ground\nX24 is free\nX25 is free\nX26 is free\nX27 is free", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: ','(max1el_list(.(T8, [])), ','(','(!_6, failure(a)), ','(head(.(T8, []), X24), ','(tail(.(T8, []), X25), ','(head(X25, X26), ','(tail(X25, X27), ','(less(X24, s(X26)), ordered(.(X26, X27))))))))) | ?_7 | ','(no(max1el_list(.(T8, []))), ','(head(.(T8, []), X24), ','(tail(.(T8, []), X25), ','(head(X25, X26), ','(tail(X25, X27), ','(less(X24, s(X26)), ordered(.(X26, X27)))))))), SCOPE: 6, CLAUSE: 15\n\nT8 is ground\nX24 is free\nX25 is free\nX26 is free\nX27 is free", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: ','(max1el_list(.(T8, [])), ','(','(!_6, failure(a)), ','(head(.(T8, []), X24), ','(tail(.(T8, []), X25), ','(head(X25, X26), ','(tail(X25, X27), ','(less(X24, s(X26)), ordered(.(X26, X27))))))))), SCOPE: 8, CLAUSE: 11 | ','(max1el_list(.(T8, [])), ','(','(!_6, failure(a)), ','(head(.(T8, []), X24), ','(tail(.(T8, []), X25), ','(head(X25, X26), ','(tail(X25, X27), ','(less(X24, s(X26)), ordered(.(X26, X27))))))))), SCOPE: 8, CLAUSE: 12 | ?_8 | ?_7 | ','(no(max1el_list(.(T8, []))), ','(head(.(T8, []), X24), ','(tail(.(T8, []), X25), ','(head(X25, X26), ','(tail(X25, X27), ','(less(X24, s(X26)), ordered(.(X26, X27)))))))), SCOPE: 6, CLAUSE: 15\n\nT8 is ground\nX24 is free\nX25 is free\nX26 is free\nX27 is free", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: ','(max1el_list(.(T8, [])), ','(','(!_6, failure(a)), ','(head(.(T8, []), X24), ','(tail(.(T8, []), X25), ','(head(X25, X26), ','(tail(X25, X27), ','(less(X24, s(X26)), ordered(.(X26, X27))))))))), SCOPE: 8, CLAUSE: 12 | ?_8 | ?_7 | ','(no(max1el_list(.(T8, []))), ','(head(.(T8, []), X24), ','(tail(.(T8, []), X25), ','(head(X25, X26), ','(tail(X25, X27), ','(less(X24, s(X26)), ordered(.(X26, X27)))))))), SCOPE: 6, CLAUSE: 15\n\nT8 is ground\nX24 is free\nX25 is free\nX26 is free\nX27 is free", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: ','(','(!_6, failure(a)), ','(head(.(T11, []), X24), ','(tail(.(T11, []), X25), ','(head(X25, X26), ','(tail(X25, X27), ','(less(X24, s(X26)), ordered(.(X26, X27)))))))) | ?_8 | ?_7 | ','(no(max1el_list(.(T11, []))), ','(head(.(T11, []), X24), ','(tail(.(T11, []), X25), ','(head(X25, X26), ','(tail(X25, X27), ','(less(X24, s(X26)), ordered(.(X26, X27)))))))), SCOPE: 6, CLAUSE: 15\n\nT11 is ground\nX24 is free\nX25 is free\nX26 is free\nX27 is free", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: ','(failure(a), ','(head(.(T11, []), X24), ','(tail(.(T11, []), X25), ','(head(X25, X26), ','(tail(X25, X27), ','(less(X24, s(X26)), ordered(.(X26, X27))))))))\n\nT11 is ground\nX24 is free\nX25 is free\nX26 is free\nX27 is free", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: ','(failure(a), ','(head(.(T11, []), X24), ','(tail(.(T11, []), X25), ','(head(X25, X26), ','(tail(X25, X27), ','(less(X24, s(X26)), ordered(.(X26, X27)))))))), SCOPE: 9, CLAUSE: 16\n\nT11 is ground\nX24 is free\nX25 is free\nX26 is free\nX27 is free", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: ','(no(max1el_list(T13)), ','(head(T13, X38), ','(tail(T13, X39), ','(head(X39, X40), ','(tail(X39, X41), ','(less(X38, s(X40)), ordered(.(X40, X41))))))))\n\nT13 is ground\nX21 is free\nX38 is free\nX39 is free\nX40 is free\nX41 is free\nordered(T13) !=? ordered([])\nordered(T13) !=? ordered(.(X21, []))", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: ','(no(max1el_list(T13)), ','(head(T13, X38), ','(tail(T13, X39), ','(head(X39, X40), ','(tail(X39, X41), ','(less(X38, s(X40)), ordered(.(X40, X41)))))))), SCOPE: 10, CLAUSE: 14 | ','(no(max1el_list(T13)), ','(head(T13, X38), ','(tail(T13, X39), ','(head(X39, X40), ','(tail(X39, X41), ','(less(X38, s(X40)), ordered(.(X40, X41)))))))), SCOPE: 10, CLAUSE: 15\n\nT13 is ground\nX21 is free\nX38 is free\nX39 is free\nX40 is free\nX41 is free\nordered(T13) !=? ordered([])\nordered(T13) !=? ordered(.(X21, []))", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: ','(','(call(max1el_list(T16)), ','(!_10, failure(a))), ','(head(T16, X38), ','(tail(T16, X39), ','(head(X39, X40), ','(tail(X39, X41), ','(less(X38, s(X40)), ordered(.(X40, X41)))))))) | ','(no(max1el_list(T16)), ','(head(T16, X38), ','(tail(T16, X39), ','(head(X39, X40), ','(tail(X39, X41), ','(less(X38, s(X40)), ordered(.(X40, X41)))))))), SCOPE: 10, CLAUSE: 15\n\nT16 is ground\nX21 is free\nX38 is free\nX39 is free\nX40 is free\nX41 is free\nordered(T16) !=? ordered([])\nordered(T16) !=? ordered(.(X21, []))", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: ','(max1el_list(T16), ','(','(!_10, failure(a)), ','(head(T16, X38), ','(tail(T16, X39), ','(head(X39, X40), ','(tail(X39, X41), ','(less(X38, s(X40)), ordered(.(X40, X41))))))))) | ?_11 | ','(no(max1el_list(T16)), ','(head(T16, X38), ','(tail(T16, X39), ','(head(X39, X40), ','(tail(X39, X41), ','(less(X38, s(X40)), ordered(.(X40, X41)))))))), SCOPE: 10, CLAUSE: 15\n\nT16 is ground\nX21 is free\nX38 is free\nX39 is free\nX40 is free\nX41 is free\nordered(T16) !=? ordered([])\nordered(T16) !=? ordered(.(X21, []))", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: ','(max1el_list(T16), ','(','(!_10, failure(a)), ','(head(T16, X38), ','(tail(T16, X39), ','(head(X39, X40), ','(tail(X39, X41), ','(less(X38, s(X40)), ordered(.(X40, X41))))))))), SCOPE: 12, CLAUSE: 11 | ','(max1el_list(T16), ','(','(!_10, failure(a)), ','(head(T16, X38), ','(tail(T16, X39), ','(head(X39, X40), ','(tail(X39, X41), ','(less(X38, s(X40)), ordered(.(X40, X41))))))))), SCOPE: 12, CLAUSE: 12 | ?_12 | ?_11 | ','(no(max1el_list(T16)), ','(head(T16, X38), ','(tail(T16, X39), ','(head(X39, X40), ','(tail(X39, X41), ','(less(X38, s(X40)), ordered(.(X40, X41)))))))), SCOPE: 10, CLAUSE: 15\n\nT16 is ground\nX21 is free\nX38 is free\nX39 is free\nX40 is free\nX41 is free\nordered(T16) !=? ordered([])\nordered(T16) !=? ordered(.(X21, []))", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: ','(max1el_list(T16), ','(','(!_10, failure(a)), ','(head(T16, X38), ','(tail(T16, X39), ','(head(X39, X40), ','(tail(X39, X41), ','(less(X38, s(X40)), ordered(.(X40, X41))))))))), SCOPE: 12, CLAUSE: 12 | ?_12 | ?_11 | ','(no(max1el_list(T16)), ','(head(T16, X38), ','(tail(T16, X39), ','(head(X39, X40), ','(tail(X39, X41), ','(less(X38, s(X40)), ordered(.(X40, X41)))))))), SCOPE: 10, CLAUSE: 15\n\nT16 is ground\nX21 is free\nX38 is free\nX39 is free\nX40 is free\nX41 is free\nordered(T16) !=? ordered([])\nordered(T16) !=? ordered(.(X21, []))", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: ?_12 | ?_11 | ','(no(max1el_list(T16)), ','(head(T16, X38), ','(tail(T16, X39), ','(head(X39, X40), ','(tail(X39, X41), ','(less(X38, s(X40)), ordered(.(X40, X41)))))))), SCOPE: 10, CLAUSE: 15\n\nT16 is ground\nX21 is free\nX38 is free\nX39 is free\nX40 is free\nX41 is free\nordered(T16) !=? ordered([])\nordered(T16) !=? ordered(.(X21, []))", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: ?_11 | ','(no(max1el_list(T16)), ','(head(T16, X38), ','(tail(T16, X39), ','(head(X39, X40), ','(tail(X39, X41), ','(less(X38, s(X40)), ordered(.(X40, X41)))))))), SCOPE: 10, CLAUSE: 15\n\nT16 is ground\nX21 is free\nX38 is free\nX39 is free\nX40 is free\nX41 is free\nordered(T16) !=? ordered([])\nordered(T16) !=? ordered(.(X21, []))", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: ','(no(max1el_list(T16)), ','(head(T16, X38), ','(tail(T16, X39), ','(head(X39, X40), ','(tail(X39, X41), ','(less(X38, s(X40)), ordered(.(X40, X41)))))))), SCOPE: 10, CLAUSE: 15\n\nT16 is ground\nX21 is free\nX38 is free\nX39 is free\nX40 is free\nX41 is free\nordered(T16) !=? ordered([])\nordered(T16) !=? ordered(.(X21, []))", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: ','(head(T20, X38), ','(tail(T20, X39), ','(head(X39, X40), ','(tail(X39, X41), ','(less(X38, s(X40)), ordered(.(X40, X41)))))))\n\nT20 is ground\nX21 is free\nX38 is free\nX39 is free\nX40 is free\nX41 is free\nordered(T20) !=? ordered([])\nordered(T20) !=? ordered(.(X21, []))", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: ','(head(T20, X38), ','(tail(T20, X39), ','(head(X39, X40), ','(tail(X39, X41), ','(less(X38, s(X40)), ordered(.(X40, X41))))))), SCOPE: 13, CLAUSE: 3 | ','(head(T20, X38), ','(tail(T20, X39), ','(head(X39, X40), ','(tail(X39, X41), ','(less(X38, s(X40)), ordered(.(X40, X41))))))), SCOPE: 13, CLAUSE: 4\n\nT20 is ground\nX21 is free\nX38 is free\nX39 is free\nX40 is free\nX41 is free\nordered(T20) !=? ordered([])\nordered(T20) !=? ordered(.(X21, []))", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: ','(head(T20, X38), ','(tail(T20, X39), ','(head(X39, X40), ','(tail(X39, X41), ','(less(X38, s(X40)), ordered(.(X40, X41))))))), SCOPE: 13, CLAUSE: 4\n\nT20 is ground\nX21 is free\nX38 is free\nX39 is free\nX40 is free\nX41 is free\nordered(T20) !=? ordered([])\nordered(T20) !=? ordered(.(X21, []))", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: ','(tail(.(T25, T26), X39), ','(head(X39, X40), ','(tail(X39, X41), ','(less(T25, s(X40)), ordered(.(X40, X41))))))\n\nT25 is ground\nT26 is ground\nX21 is free\nX39 is free\nX40 is free\nX41 is free\nordered(.(T25, T26)) !=? ordered(.(X21, []))", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: ','(tail(.(T25, T26), X39), ','(head(X39, X40), ','(tail(X39, X41), ','(less(T25, s(X40)), ordered(.(X40, X41)))))), SCOPE: 14, CLAUSE: 5 | ','(tail(.(T25, T26), X39), ','(head(X39, X40), ','(tail(X39, X41), ','(less(T25, s(X40)), ordered(.(X40, X41)))))), SCOPE: 14, CLAUSE: 6\n\nT25 is ground\nT26 is ground\nX21 is free\nX39 is free\nX40 is free\nX41 is free\nordered(.(T25, T26)) !=? ordered(.(X21, []))", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: ','(tail(.(T25, T26), X39), ','(head(X39, X40), ','(tail(X39, X41), ','(less(T25, s(X40)), ordered(.(X40, X41)))))), SCOPE: 14, CLAUSE: 6\n\nT25 is ground\nT26 is ground\nX21 is free\nX39 is free\nX40 is free\nX41 is free\nordered(.(T25, T26)) !=? ordered(.(X21, []))", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: ','(head(T32, X40), ','(tail(T32, X41), ','(less(T31, s(X40)), ordered(.(X40, X41)))))\n\nT31 is ground\nT32 is ground\nX21 is free\nX40 is free\nX41 is free\nordered(.(T31, T32)) !=? ordered(.(X21, []))", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: ','(head(T32, X40), ','(tail(T32, X41), ','(less(T31, s(X40)), ordered(.(X40, X41))))), SCOPE: 15, CLAUSE: 3 | ','(head(T32, X40), ','(tail(T32, X41), ','(less(T31, s(X40)), ordered(.(X40, X41))))), SCOPE: 15, CLAUSE: 4\n\nT31 is ground\nT32 is ground\nX21 is free\nX40 is free\nX41 is free\nordered(.(T31, T32)) !=? ordered(.(X21, []))", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: ','(head(T32, X40), ','(tail(T32, X41), ','(less(T31, s(X40)), ordered(.(X40, X41))))), SCOPE: 15, CLAUSE: 4\n\nT31 is ground\nT32 is ground\nX21 is free\nX40 is free\nX41 is free\nordered(.(T31, T32)) !=? ordered(.(X21, []))", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: ','(tail(.(T37, T38), X41), ','(less(T31, s(T37)), ordered(.(T37, X41))))\n\nT31 is ground\nT37 is ground\nT38 is ground\nX41 is free", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: ','(tail(.(T37, T38), X41), ','(less(T31, s(T37)), ordered(.(T37, X41)))), SCOPE: 16, CLAUSE: 5 | ','(tail(.(T37, T38), X41), ','(less(T31, s(T37)), ordered(.(T37, X41)))), SCOPE: 16, CLAUSE: 6\n\nT31 is ground\nT37 is ground\nT38 is ground\nX41 is free", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: ','(tail(.(T37, T38), X41), ','(less(T31, s(T37)), ordered(.(T37, X41)))), SCOPE: 16, CLAUSE: 6\n\nT31 is ground\nT37 is ground\nT38 is ground\nX41 is free", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: ','(less(T31, s(T47)), ordered(.(T47, T48)))\n\nT31 is ground\nT47 is ground\nT48 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: less(T31, s(T47))\n\nT31 is ground\nT47 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: ordered(.(T47, T48))\n\nT47 is ground\nT48 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: less(T31, s(T47)), SCOPE: 17, CLAUSE: 7 | less(T31, s(T47)), SCOPE: 17, CLAUSE: 8\n\nT31 is ground\nT47 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: less(T31, s(T47)), SCOPE: 17, CLAUSE: 7\n\nT31 is ground\nT47 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: less(T31, s(T47)), SCOPE: 17, CLAUSE: 8\n\nT31 is ground\nT47 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: ','(no(zero(T62)), ','(p(T62, X116), ','(p(s(T63), X117), less(X116, X117))))\n\nT62 is ground\nT63 is ground\nX116 is free\nX117 is free", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: ','(no(zero(T62)), ','(p(T62, X116), ','(p(s(T63), X117), less(X116, X117)))), SCOPE: 18, CLAUSE: 14 | ','(no(zero(T62)), ','(p(T62, X116), ','(p(s(T63), X117), less(X116, X117)))), SCOPE: 18, CLAUSE: 15\n\nT62 is ground\nT63 is ground\nX116 is free\nX117 is free", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: ','(','(call(zero(T66)), ','(!_18, failure(a))), ','(p(T66, X116), ','(p(s(T63), X117), less(X116, X117)))) | ','(no(zero(T66)), ','(p(T66, X116), ','(p(s(T63), X117), less(X116, X117)))), SCOPE: 18, CLAUSE: 15\n\nT63 is ground\nT66 is ground\nX116 is free\nX117 is free", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: ','(zero(T66), ','(','(!_18, failure(a)), ','(p(T66, X116), ','(p(s(T63), X117), less(X116, X117))))) | ?_19 | ','(no(zero(T66)), ','(p(T66, X116), ','(p(s(T63), X117), less(X116, X117)))), SCOPE: 18, CLAUSE: 15\n\nT63 is ground\nT66 is ground\nX116 is free\nX117 is free", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: ','(zero(T66), ','(','(!_18, failure(a)), ','(p(T66, X116), ','(p(s(T63), X117), less(X116, X117))))), SCOPE: 20, CLAUSE: 13 | ?_20 | ?_19 | ','(no(zero(T66)), ','(p(T66, X116), ','(p(s(T63), X117), less(X116, X117)))), SCOPE: 18, CLAUSE: 15\n\nT63 is ground\nT66 is ground\nX116 is free\nX117 is free", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: ','(','(!_18, failure(a)), ','(p(0, X116), ','(p(s(T63), X117), less(X116, X117)))) | ?_20 | ?_19 | ','(no(zero(0)), ','(p(0, X116), ','(p(s(T63), X117), less(X116, X117)))), SCOPE: 18, CLAUSE: 15\n\nT63 is ground\nX116 is free\nX117 is free", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: ?_20 | ?_19 | ','(no(zero(T66)), ','(p(T66, X116), ','(p(s(T63), X117), less(X116, X117)))), SCOPE: 18, CLAUSE: 15\n\nT63 is ground\nT66 is ground\nX116 is free\nX117 is free\nzero(T66) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: ','(failure(a), ','(p(0, X116), ','(p(s(T63), X117), less(X116, X117))))\n\nT63 is ground\nX116 is free\nX117 is free", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: ','(failure(a), ','(p(0, X116), ','(p(s(T63), X117), less(X116, X117)))), SCOPE: 21, CLAUSE: 16\n\nT63 is ground\nX116 is free\nX117 is free", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: ?_19 | ','(no(zero(T66)), ','(p(T66, X116), ','(p(s(T63), X117), less(X116, X117)))), SCOPE: 18, CLAUSE: 15\n\nT63 is ground\nT66 is ground\nX116 is free\nX117 is free\nzero(T66) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: ','(no(zero(T66)), ','(p(T66, X116), ','(p(s(T63), X117), less(X116, X117)))), SCOPE: 18, CLAUSE: 15\n\nT63 is ground\nT66 is ground\nX116 is free\nX117 is free\nzero(T66) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: ','(p(T69, X116), ','(p(s(T63), X117), less(X116, X117)))\n\nT63 is ground\nT69 is ground\nX116 is free\nX117 is free\nzero(T69) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: ','(p(T69, X116), ','(p(s(T63), X117), less(X116, X117))), SCOPE: 22, CLAUSE: 9 | ','(p(T69, X116), ','(p(s(T63), X117), less(X116, X117))), SCOPE: 22, CLAUSE: 10\n\nT63 is ground\nT69 is ground\nX116 is free\nX117 is free\nzero(T69) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: ','(p(T69, X116), ','(p(s(T63), X117), less(X116, X117))), SCOPE: 22, CLAUSE: 10\n\nT63 is ground\nT69 is ground\nX116 is free\nX117 is free\nzero(T69) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: ','(p(s(T63), X117), less(T72, X117))\n\nT63 is ground\nT72 is ground\nX117 is free", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="77: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="78: ','(p(s(T63), X117), less(T72, X117)), SCOPE: 23, CLAUSE: 9 | ','(p(s(T63), X117), less(T72, X117)), SCOPE: 23, CLAUSE: 10\n\nT63 is ground\nT72 is ground\nX117 is free", fontsize=16, style=filled, fillcolor=lightyellow];
79 [label="79: ','(p(s(T63), X117), less(T72, X117)), SCOPE: 23, CLAUSE: 10\n\nT63 is ground\nT72 is ground\nX117 is free", fontsize=16, style=filled, fillcolor=lightyellow];
80 [label="80: less(T72, T77)\n\nT72 is ground\nT77 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
81 [label="81: less(T72, T77), SCOPE: 24, CLAUSE: 7 | less(T72, T77), SCOPE: 24, CLAUSE: 8\n\nT72 is ground\nT77 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
82 [label="82: less(T72, T77), SCOPE: 24, CLAUSE: 7\n\nT72 is ground\nT77 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
83 [label="83: less(T72, T77), SCOPE: 24, CLAUSE: 8\n\nT72 is ground\nT77 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
84 [label="84: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
85 [label="85: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
86 [label="86: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
87 [label="87: ','(no(zero(T89)), ','(p(T89, X161), ','(p(T90, X162), less(X161, X162))))\n\nT89 is ground\nT90 is ground\nX161 is free\nX162 is free", fontsize=16, style=filled, fillcolor=lightyellow];
88 [label="88: ','(no(zero(T89)), ','(p(T89, X161), ','(p(T90, X162), less(X161, X162)))), SCOPE: 25, CLAUSE: 14 | ','(no(zero(T89)), ','(p(T89, X161), ','(p(T90, X162), less(X161, X162)))), SCOPE: 25, CLAUSE: 15\n\nT89 is ground\nT90 is ground\nX161 is free\nX162 is free", fontsize=16, style=filled, fillcolor=lightyellow];
89 [label="89: ','(','(call(zero(T93)), ','(!_25, failure(a))), ','(p(T93, X161), ','(p(T90, X162), less(X161, X162)))) | ','(no(zero(T93)), ','(p(T93, X161), ','(p(T90, X162), less(X161, X162)))), SCOPE: 25, CLAUSE: 15\n\nT90 is ground\nT93 is ground\nX161 is free\nX162 is free", fontsize=16, style=filled, fillcolor=lightyellow];
90 [label="90: ','(zero(T93), ','(','(!_25, failure(a)), ','(p(T93, X161), ','(p(T90, X162), less(X161, X162))))) | ?_26 | ','(no(zero(T93)), ','(p(T93, X161), ','(p(T90, X162), less(X161, X162)))), SCOPE: 25, CLAUSE: 15\n\nT90 is ground\nT93 is ground\nX161 is free\nX162 is free", fontsize=16, style=filled, fillcolor=lightyellow];
91 [label="91: ','(zero(T93), ','(','(!_25, failure(a)), ','(p(T93, X161), ','(p(T90, X162), less(X161, X162))))), SCOPE: 27, CLAUSE: 13 | ?_27 | ?_26 | ','(no(zero(T93)), ','(p(T93, X161), ','(p(T90, X162), less(X161, X162)))), SCOPE: 25, CLAUSE: 15\n\nT90 is ground\nT93 is ground\nX161 is free\nX162 is free", fontsize=16, style=filled, fillcolor=lightyellow];
92 [label="92: ','(','(!_25, failure(a)), ','(p(0, X161), ','(p(T90, X162), less(X161, X162)))) | ?_27 | ?_26 | ','(no(zero(0)), ','(p(0, X161), ','(p(T90, X162), less(X161, X162)))), SCOPE: 25, CLAUSE: 15\n\nT90 is ground\nX161 is free\nX162 is free", fontsize=16, style=filled, fillcolor=lightyellow];
93 [label="93: ?_27 | ?_26 | ','(no(zero(T93)), ','(p(T93, X161), ','(p(T90, X162), less(X161, X162)))), SCOPE: 25, CLAUSE: 15\n\nT90 is ground\nT93 is ground\nX161 is free\nX162 is free\nzero(T93) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
94 [label="94: ','(failure(a), ','(p(0, X161), ','(p(T90, X162), less(X161, X162))))\n\nT90 is ground\nX161 is free\nX162 is free", fontsize=16, style=filled, fillcolor=lightyellow];
95 [label="95: ','(failure(a), ','(p(0, X161), ','(p(T90, X162), less(X161, X162)))), SCOPE: 28, CLAUSE: 16\n\nT90 is ground\nX161 is free\nX162 is free", fontsize=16, style=filled, fillcolor=lightyellow];
96 [label="96: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
97 [label="97: ?_26 | ','(no(zero(T93)), ','(p(T93, X161), ','(p(T90, X162), less(X161, X162)))), SCOPE: 25, CLAUSE: 15\n\nT90 is ground\nT93 is ground\nX161 is free\nX162 is free\nzero(T93) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
98 [label="98: ','(no(zero(T93)), ','(p(T93, X161), ','(p(T90, X162), less(X161, X162)))), SCOPE: 25, CLAUSE: 15\n\nT90 is ground\nT93 is ground\nX161 is free\nX162 is free\nzero(T93) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
99 [label="99: ','(p(T96, X161), ','(p(T90, X162), less(X161, X162)))\n\nT90 is ground\nT96 is ground\nX161 is free\nX162 is free\nzero(T96) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
100 [label="100: ','(p(T96, X161), ','(p(T90, X162), less(X161, X162))), SCOPE: 29, CLAUSE: 9 | ','(p(T96, X161), ','(p(T90, X162), less(X161, X162))), SCOPE: 29, CLAUSE: 10\n\nT90 is ground\nT96 is ground\nX161 is free\nX162 is free\nzero(T96) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
101 [label="101: ','(p(T96, X161), ','(p(T90, X162), less(X161, X162))), SCOPE: 29, CLAUSE: 10\n\nT90 is ground\nT96 is ground\nX161 is free\nX162 is free\nzero(T96) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
102 [label="102: ','(p(T90, X162), less(T99, X162))\n\nT90 is ground\nT99 is ground\nX162 is free", fontsize=16, style=filled, fillcolor=lightyellow];
103 [label="103: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
104 [label="104: ','(p(T90, X162), less(T99, X162)), SCOPE: 30, CLAUSE: 9 | ','(p(T90, X162), less(T99, X162)), SCOPE: 30, CLAUSE: 10\n\nT90 is ground\nT99 is ground\nX162 is free", fontsize=16, style=filled, fillcolor=lightyellow];
105 [label="105: ','(p(T90, X162), less(T99, X162)), SCOPE: 30, CLAUSE: 9\n\nT90 is ground\nT99 is ground\nX162 is free", fontsize=16, style=filled, fillcolor=lightyellow];
106 [label="106: ','(p(T90, X162), less(T99, X162)), SCOPE: 30, CLAUSE: 10\n\nT90 is ground\nT99 is ground\nX162 is free", fontsize=16, style=filled, fillcolor=lightyellow];
107 [label="107: less(T99, 0)\n\nT99 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
108 [label="108: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
109 [label="109: less(T99, 0), SCOPE: 31, CLAUSE: 7 | less(T99, 0), SCOPE: 31, CLAUSE: 8\n\nT99 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
110 [label="110: less(T99, 0), SCOPE: 31, CLAUSE: 8\n\nT99 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
111 [label="111: ','(no(zero(T102)), ','(p(T102, X197), ','(p(0, X198), less(X197, X198))))\n\nT102 is ground\nX197 is free\nX198 is free", fontsize=16, style=filled, fillcolor=lightyellow];
112 [label="112: ','(no(zero(T102)), ','(p(T102, X197), ','(p(0, X198), less(X197, X198)))), SCOPE: 32, CLAUSE: 14 | ','(no(zero(T102)), ','(p(T102, X197), ','(p(0, X198), less(X197, X198)))), SCOPE: 32, CLAUSE: 15\n\nT102 is ground\nX197 is free\nX198 is free", fontsize=16, style=filled, fillcolor=lightyellow];
113 [label="113: ','(','(call(zero(T105)), ','(!_32, failure(a))), ','(p(T105, X197), ','(p(0, X198), less(X197, X198)))) | ','(no(zero(T105)), ','(p(T105, X197), ','(p(0, X198), less(X197, X198)))), SCOPE: 32, CLAUSE: 15\n\nT105 is ground\nX197 is free\nX198 is free", fontsize=16, style=filled, fillcolor=lightyellow];
114 [label="114: ','(zero(T105), ','(','(!_32, failure(a)), ','(p(T105, X197), ','(p(0, X198), less(X197, X198))))) | ?_33 | ','(no(zero(T105)), ','(p(T105, X197), ','(p(0, X198), less(X197, X198)))), SCOPE: 32, CLAUSE: 15\n\nT105 is ground\nX197 is free\nX198 is free", fontsize=16, style=filled, fillcolor=lightyellow];
115 [label="115: ','(zero(T105), ','(','(!_32, failure(a)), ','(p(T105, X197), ','(p(0, X198), less(X197, X198))))), SCOPE: 34, CLAUSE: 13 | ?_34 | ?_33 | ','(no(zero(T105)), ','(p(T105, X197), ','(p(0, X198), less(X197, X198)))), SCOPE: 32, CLAUSE: 15\n\nT105 is ground\nX197 is free\nX198 is free", fontsize=16, style=filled, fillcolor=lightyellow];
116 [label="116: ','(','(!_32, failure(a)), ','(p(0, X197), ','(p(0, X198), less(X197, X198)))) | ?_34 | ?_33 | ','(no(zero(0)), ','(p(0, X197), ','(p(0, X198), less(X197, X198)))), SCOPE: 32, CLAUSE: 15\n\nX197 is free\nX198 is free", fontsize=16, style=filled, fillcolor=lightyellow];
117 [label="117: ?_34 | ?_33 | ','(no(zero(T105)), ','(p(T105, X197), ','(p(0, X198), less(X197, X198)))), SCOPE: 32, CLAUSE: 15\n\nT105 is ground\nX197 is free\nX198 is free\nzero(T105) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
118 [label="118: ','(failure(a), ','(p(0, X197), ','(p(0, X198), less(X197, X198))))\n\nX197 is free\nX198 is free", fontsize=16, style=filled, fillcolor=lightyellow];
119 [label="119: ','(failure(a), ','(p(0, X197), ','(p(0, X198), less(X197, X198)))), SCOPE: 35, CLAUSE: 16\n\nX197 is free\nX198 is free", fontsize=16, style=filled, fillcolor=lightyellow];
120 [label="120: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
121 [label="121: ?_33 | ','(no(zero(T105)), ','(p(T105, X197), ','(p(0, X198), less(X197, X198)))), SCOPE: 32, CLAUSE: 15\n\nT105 is ground\nX197 is free\nX198 is free\nzero(T105) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
122 [label="122: ','(no(zero(T105)), ','(p(T105, X197), ','(p(0, X198), less(X197, X198)))), SCOPE: 32, CLAUSE: 15\n\nT105 is ground\nX197 is free\nX198 is free\nzero(T105) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
123 [label="123: ','(p(T108, X197), ','(p(0, X198), less(X197, X198)))\n\nT108 is ground\nX197 is free\nX198 is free\nzero(T108) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
124 [label="124: ','(p(T108, X197), ','(p(0, X198), less(X197, X198))), SCOPE: 36, CLAUSE: 9 | ','(p(T108, X197), ','(p(0, X198), less(X197, X198))), SCOPE: 36, CLAUSE: 10\n\nT108 is ground\nX197 is free\nX198 is free\nzero(T108) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
125 [label="125: ','(p(T108, X197), ','(p(0, X198), less(X197, X198))), SCOPE: 36, CLAUSE: 10\n\nT108 is ground\nX197 is free\nX198 is free\nzero(T108) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
126 [label="126: ','(p(0, X198), less(T111, X198))\n\nT111 is ground\nX198 is free", fontsize=16, style=filled, fillcolor=lightyellow];
127 [label="127: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
128 [label="128: ','(p(0, X198), less(T111, X198)), SCOPE: 37, CLAUSE: 9 | ','(p(0, X198), less(T111, X198)), SCOPE: 37, CLAUSE: 10\n\nT111 is ground\nX198 is free", fontsize=16, style=filled, fillcolor=lightyellow];
129 [label="129: ','(p(0, X198), less(T111, X198)), SCOPE: 37, CLAUSE: 9\n\nT111 is ground\nX198 is free", fontsize=16, style=filled, fillcolor=lightyellow];
130 [label="130: ','(p(0, X198), less(T111, X198)), SCOPE: 37, CLAUSE: 10\n\nT111 is ground\nX198 is free", fontsize=16, style=filled, fillcolor=lightyellow];
131 [label="131: less(T111, 0)\n\nT111 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
132 [label="132: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
133 [label="133: less(T99, T114)\n\nT99 is ground\nT114 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
134 [label="134: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
135 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 135 [arrowhead = none ];
135 -> 2;

136 [label="EVAL with clause\nordered([]).\nand substitutionT1 -> []", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 136 [arrowhead = none ];
136 -> 3;

137 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
2 -> 137 [arrowhead = none ];
137 -> 4;

138 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 138 [arrowhead = none ];
138 -> 5;

139 [label="EVAL with clause\nordered(.(X21, [])).\nand substitutionX21 -> T3,\nT1 -> .(T3, [])", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 139 [arrowhead = none ];
139 -> 16;

140 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 140 [arrowhead = none ];
140 -> 17;

141 [label="BACKTRACK\nfor clause: ordered(.(X1, []))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 141 [arrowhead = none ];
141 -> 6;

142 [label="ONLY EVAL with clause\nordered(X10) :- ','(no(max1el_list(X10)), ','(head(X10, X11), ','(tail(X10, X12), ','(head(X12, X13), ','(tail(X12, X14), ','(less(X11, s(X13)), ordered(.(X13, X14)))))))).\nand substitutionX10 -> []", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 142 [arrowhead = none ];
142 -> 7;

143 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
7 -> 143 [arrowhead = none ];
143 -> 8;

144 [label="ONLY EVAL with clause\nno(X17) :- ','(call(X17), ','(!_2, failure(a))).\nand substitutionX17 -> max1el_list([])", fontsize=14, style = filled, fillcolor = lightblue];
8 -> 144 [arrowhead = none ];
144 -> 9;

145 [label="CALL", fontsize=14, style = filled, fillcolor = lightcyan];
9 -> 145 [arrowhead = none ];
145 -> 10;

146 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
10 -> 146 [arrowhead = none ];
146 -> 11;

147 [label="ONLY EVAL with clause\nmax1el_list([]).\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 147 [arrowhead = none ];
147 -> 12;

148 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
12 -> 148 [arrowhead = none ];
148 -> 13;

149 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
13 -> 149 [arrowhead = none ];
149 -> 14;

150 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
14 -> 150 [arrowhead = none ];
150 -> 15;

151 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
16 -> 151 [arrowhead = none ];
151 -> 18;

152 [label="ONLY EVAL with clause\nordered(X37) :- ','(no(max1el_list(X37)), ','(head(X37, X38), ','(tail(X37, X39), ','(head(X39, X40), ','(tail(X39, X41), ','(less(X38, s(X40)), ordered(.(X40, X41)))))))).\nand substitutionT1 -> T13,\nX37 -> T13", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 152 [arrowhead = none ];
152 -> 29;

153 [label="ONLY EVAL with clause\nordered(X23) :- ','(no(max1el_list(X23)), ','(head(X23, X24), ','(tail(X23, X25), ','(head(X25, X26), ','(tail(X25, X27), ','(less(X24, s(X26)), ordered(.(X26, X27)))))))).\nand substitutionT3 -> T5,\nX23 -> .(T5, [])", fontsize=14, style = filled, fillcolor = lightblue];
18 -> 153 [arrowhead = none ];
153 -> 19;

154 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
19 -> 154 [arrowhead = none ];
154 -> 20;

155 [label="ONLY EVAL with clause\nno(X30) :- ','(call(X30), ','(!_6, failure(a))).\nand substitutionT5 -> T8,\nX30 -> max1el_list(.(T8, []))", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 155 [arrowhead = none ];
155 -> 21;

156 [label="CALL", fontsize=14, style = filled, fillcolor = lightcyan];
21 -> 156 [arrowhead = none ];
156 -> 22;

157 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
22 -> 157 [arrowhead = none ];
157 -> 23;

158 [label="BACKTRACK\nfor clause: max1el_list([])because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
23 -> 158 [arrowhead = none ];
158 -> 24;

159 [label="ONLY EVAL with clause\nmax1el_list(.(X33, [])).\nand substitutionT8 -> T11,\nX33 -> T11", fontsize=14, style = filled, fillcolor = lightblue];
24 -> 159 [arrowhead = none ];
159 -> 25;

160 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
25 -> 160 [arrowhead = none ];
160 -> 26;

161 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
26 -> 161 [arrowhead = none ];
161 -> 27;

162 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
27 -> 162 [arrowhead = none ];
162 -> 28;

163 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
29 -> 163 [arrowhead = none ];
163 -> 30;

164 [label="ONLY EVAL with clause\nno(X44) :- ','(call(X44), ','(!_10, failure(a))).\nand substitutionT13 -> T16,\nX44 -> max1el_list(T16)", fontsize=14, style = filled, fillcolor = lightblue];
30 -> 164 [arrowhead = none ];
164 -> 31;

165 [label="CALL", fontsize=14, style = filled, fillcolor = lightcyan];
31 -> 165 [arrowhead = none ];
165 -> 32;

166 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
32 -> 166 [arrowhead = none ];
166 -> 33;

167 [label="BACKTRACK\nfor clause: max1el_list([])\nwith clash: (ordered(T16), ordered([]))", fontsize=14, style = filled, fillcolor = lightgreen];
33 -> 167 [arrowhead = none ];
167 -> 34;

168 [label="BACKTRACK\nfor clause: max1el_list(.(X6, []))\nwith clash: (ordered(T16), ordered(.(X21, [])))", fontsize=14, style = filled, fillcolor = lightgreen];
34 -> 168 [arrowhead = none ];
168 -> 35;

169 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
35 -> 169 [arrowhead = none ];
169 -> 36;

170 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
36 -> 170 [arrowhead = none ];
170 -> 37;

171 [label="ONLY EVAL with clause\nno(X50).\nand substitutionT16 -> T20,\nX50 -> max1el_list(T20)", fontsize=14, style = filled, fillcolor = lightblue];
37 -> 171 [arrowhead = none ];
171 -> 38;

172 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
38 -> 172 [arrowhead = none ];
172 -> 39;

173 [label="BACKTRACK\nfor clause: head([], X2)\nwith clash: (ordered(T20), ordered([]))", fontsize=14, style = filled, fillcolor = lightgreen];
39 -> 173 [arrowhead = none ];
173 -> 40;

174 [label="EVAL with clause\nhead(.(X59, X60), X59).\nand substitutionX59 -> T25,\nX60 -> T26,\nT20 -> .(T25, T26),\nX38 -> T25", fontsize=14, style = filled, fillcolor = lightblue];
40 -> 174 [arrowhead = none ];
174 -> 41;

175 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
40 -> 175 [arrowhead = none ];
175 -> 42;

176 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
41 -> 176 [arrowhead = none ];
176 -> 43;

177 [label="BACKTRACK\nfor clause: tail([], [])because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
43 -> 177 [arrowhead = none ];
177 -> 44;

178 [label="ONLY EVAL with clause\ntail(.(X73, X74), X74).\nand substitutionT25 -> T31,\nX73 -> T31,\nT26 -> T32,\nX74 -> T32,\nX39 -> T32", fontsize=14, style = filled, fillcolor = lightblue];
44 -> 178 [arrowhead = none ];
178 -> 45;

179 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
45 -> 179 [arrowhead = none ];
179 -> 46;

180 [label="BACKTRACK\nfor clause: head([], X2)\nwith clash: (ordered(.(T31, T32)), ordered(.(X21, [])))", fontsize=14, style = filled, fillcolor = lightgreen];
46 -> 180 [arrowhead = none ];
180 -> 47;

181 [label="EVAL with clause\nhead(.(X85, X86), X85).\nand substitutionX85 -> T37,\nX86 -> T38,\nT32 -> .(T37, T38),\nX40 -> T37", fontsize=14, style = filled, fillcolor = lightblue];
47 -> 181 [arrowhead = none ];
181 -> 48;

182 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
47 -> 182 [arrowhead = none ];
182 -> 49;

183 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
48 -> 183 [arrowhead = none ];
183 -> 50;

184 [label="BACKTRACK\nfor clause: tail([], [])because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
50 -> 184 [arrowhead = none ];
184 -> 51;

185 [label="ONLY EVAL with clause\ntail(.(X95, X96), X96).\nand substitutionT37 -> T47,\nX95 -> T47,\nT38 -> T48,\nX96 -> T48,\nX41 -> T48", fontsize=14, style = filled, fillcolor = lightblue];
51 -> 185 [arrowhead = none ];
185 -> 52;

186 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
52 -> 186 [arrowhead = none ];
186 -> 53;

187 [label="SPLIT 2\nnew knowledge:\nT31 is ground\nT47 is ground", fontsize=14, style = filled, fillcolor = violet];
52 -> 187 [arrowhead = none ];
187 -> 54;

188 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
53 -> 188 [arrowhead = none ];
188 -> 55;

189 [label="INSTANCE with matching:\nT1 -> .(T47, T48)", fontsize=14, style = filled, fillcolor = lightgrey];
54 -> 189 [arrowhead = none , style=dashed];
189 -> 1[style=dashed];

190 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
55 -> 190 [arrowhead = none ];
190 -> 56;

191 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
55 -> 191 [arrowhead = none ];
191 -> 57;

192 [label="EVAL with clause\nless(0, s(X105)).\nand substitutionT31 -> 0,\nT47 -> T57,\nX105 -> T57", fontsize=14, style = filled, fillcolor = lightblue];
56 -> 192 [arrowhead = none ];
192 -> 58;

193 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
56 -> 193 [arrowhead = none ];
193 -> 59;

194 [label="ONLY EVAL with clause\nless(X114, X115) :- ','(no(zero(X114)), ','(p(X114, X116), ','(p(X115, X117), less(X116, X117)))).\nand substitutionT31 -> T62,\nX114 -> T62,\nT47 -> T63,\nX115 -> s(T63)", fontsize=14, style = filled, fillcolor = lightblue];
57 -> 194 [arrowhead = none ];
194 -> 61;

195 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
58 -> 195 [arrowhead = none ];
195 -> 60;

196 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
61 -> 196 [arrowhead = none ];
196 -> 62;

197 [label="ONLY EVAL with clause\nno(X122) :- ','(call(X122), ','(!_18, failure(a))).\nand substitutionT62 -> T66,\nX122 -> zero(T66)", fontsize=14, style = filled, fillcolor = lightblue];
62 -> 197 [arrowhead = none ];
197 -> 63;

198 [label="CALL", fontsize=14, style = filled, fillcolor = lightcyan];
63 -> 198 [arrowhead = none ];
198 -> 64;

199 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
64 -> 199 [arrowhead = none ];
199 -> 65;

200 [label="EVAL with clause\nzero(0).\nand substitutionT66 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
65 -> 200 [arrowhead = none ];
200 -> 66;

201 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
65 -> 201 [arrowhead = none ];
201 -> 67;

202 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
66 -> 202 [arrowhead = none ];
202 -> 68;

203 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
67 -> 203 [arrowhead = none ];
203 -> 71;

204 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
68 -> 204 [arrowhead = none ];
204 -> 69;

205 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
69 -> 205 [arrowhead = none ];
205 -> 70;

206 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
71 -> 206 [arrowhead = none ];
206 -> 72;

207 [label="ONLY EVAL with clause\nno(X129).\nand substitutionT66 -> T69,\nX129 -> zero(T69)", fontsize=14, style = filled, fillcolor = lightblue];
72 -> 207 [arrowhead = none ];
207 -> 73;

208 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
73 -> 208 [arrowhead = none ];
208 -> 74;

209 [label="BACKTRACK\nfor clause: p(0, 0)\nwith clash: (zero(T69), zero(0))", fontsize=14, style = filled, fillcolor = lightgreen];
74 -> 209 [arrowhead = none ];
209 -> 75;

210 [label="EVAL with clause\np(s(X134), X134).\nand substitutionX134 -> T72,\nT69 -> s(T72),\nX116 -> T72", fontsize=14, style = filled, fillcolor = lightblue];
75 -> 210 [arrowhead = none ];
210 -> 76;

211 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
75 -> 211 [arrowhead = none ];
211 -> 77;

212 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
76 -> 212 [arrowhead = none ];
212 -> 78;

213 [label="BACKTRACK\nfor clause: p(0, 0)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
78 -> 213 [arrowhead = none ];
213 -> 79;

214 [label="ONLY EVAL with clause\np(s(X143), X143).\nand substitutionT63 -> T77,\nX143 -> T77,\nX117 -> T77", fontsize=14, style = filled, fillcolor = lightblue];
79 -> 214 [arrowhead = none ];
214 -> 80;

215 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
80 -> 215 [arrowhead = none ];
215 -> 81;

216 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
81 -> 216 [arrowhead = none ];
216 -> 82;

217 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
81 -> 217 [arrowhead = none ];
217 -> 83;

218 [label="EVAL with clause\nless(0, s(X150)).\nand substitutionT72 -> 0,\nX150 -> T84,\nT77 -> s(T84)", fontsize=14, style = filled, fillcolor = lightblue];
82 -> 218 [arrowhead = none ];
218 -> 84;

219 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
82 -> 219 [arrowhead = none ];
219 -> 85;

220 [label="ONLY EVAL with clause\nless(X159, X160) :- ','(no(zero(X159)), ','(p(X159, X161), ','(p(X160, X162), less(X161, X162)))).\nand substitutionT72 -> T89,\nX159 -> T89,\nT77 -> T90,\nX160 -> T90", fontsize=14, style = filled, fillcolor = lightblue];
83 -> 220 [arrowhead = none ];
220 -> 87;

221 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
84 -> 221 [arrowhead = none ];
221 -> 86;

222 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
87 -> 222 [arrowhead = none ];
222 -> 88;

223 [label="ONLY EVAL with clause\nno(X167) :- ','(call(X167), ','(!_25, failure(a))).\nand substitutionT89 -> T93,\nX167 -> zero(T93)", fontsize=14, style = filled, fillcolor = lightblue];
88 -> 223 [arrowhead = none ];
223 -> 89;

224 [label="CALL", fontsize=14, style = filled, fillcolor = lightcyan];
89 -> 224 [arrowhead = none ];
224 -> 90;

225 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
90 -> 225 [arrowhead = none ];
225 -> 91;

226 [label="EVAL with clause\nzero(0).\nand substitutionT93 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
91 -> 226 [arrowhead = none ];
226 -> 92;

227 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
91 -> 227 [arrowhead = none ];
227 -> 93;

228 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
92 -> 228 [arrowhead = none ];
228 -> 94;

229 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
93 -> 229 [arrowhead = none ];
229 -> 97;

230 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
94 -> 230 [arrowhead = none ];
230 -> 95;

231 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
95 -> 231 [arrowhead = none ];
231 -> 96;

232 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
97 -> 232 [arrowhead = none ];
232 -> 98;

233 [label="ONLY EVAL with clause\nno(X174).\nand substitutionT93 -> T96,\nX174 -> zero(T96)", fontsize=14, style = filled, fillcolor = lightblue];
98 -> 233 [arrowhead = none ];
233 -> 99;

234 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
99 -> 234 [arrowhead = none ];
234 -> 100;

235 [label="BACKTRACK\nfor clause: p(0, 0)\nwith clash: (zero(T96), zero(0))", fontsize=14, style = filled, fillcolor = lightgreen];
100 -> 235 [arrowhead = none ];
235 -> 101;

236 [label="EVAL with clause\np(s(X179), X179).\nand substitutionX179 -> T99,\nT96 -> s(T99),\nX161 -> T99", fontsize=14, style = filled, fillcolor = lightblue];
101 -> 236 [arrowhead = none ];
236 -> 102;

237 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
101 -> 237 [arrowhead = none ];
237 -> 103;

238 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
102 -> 238 [arrowhead = none ];
238 -> 104;

239 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
104 -> 239 [arrowhead = none ];
239 -> 105;

240 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
104 -> 240 [arrowhead = none ];
240 -> 106;

241 [label="EVAL with clause\np(0, 0).\nand substitutionT90 -> 0,\nX162 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
105 -> 241 [arrowhead = none ];
241 -> 107;

242 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
105 -> 242 [arrowhead = none ];
242 -> 108;

243 [label="EVAL with clause\np(s(X229), X229).\nand substitutionX229 -> T114,\nT90 -> s(T114),\nX162 -> T114", fontsize=14, style = filled, fillcolor = lightblue];
106 -> 243 [arrowhead = none ];
243 -> 133;

244 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
106 -> 244 [arrowhead = none ];
244 -> 134;

245 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
107 -> 245 [arrowhead = none ];
245 -> 109;

246 [label="BACKTRACK\nfor clause: less(0, s(X5))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
109 -> 246 [arrowhead = none ];
246 -> 110;

247 [label="ONLY EVAL with clause\nless(X195, X196) :- ','(no(zero(X195)), ','(p(X195, X197), ','(p(X196, X198), less(X197, X198)))).\nand substitutionT99 -> T102,\nX195 -> T102,\nX196 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
110 -> 247 [arrowhead = none ];
247 -> 111;

248 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
111 -> 248 [arrowhead = none ];
248 -> 112;

249 [label="ONLY EVAL with clause\nno(X203) :- ','(call(X203), ','(!_32, failure(a))).\nand substitutionT102 -> T105,\nX203 -> zero(T105)", fontsize=14, style = filled, fillcolor = lightblue];
112 -> 249 [arrowhead = none ];
249 -> 113;

250 [label="CALL", fontsize=14, style = filled, fillcolor = lightcyan];
113 -> 250 [arrowhead = none ];
250 -> 114;

251 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
114 -> 251 [arrowhead = none ];
251 -> 115;

252 [label="EVAL with clause\nzero(0).\nand substitutionT105 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
115 -> 252 [arrowhead = none ];
252 -> 116;

253 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
115 -> 253 [arrowhead = none ];
253 -> 117;

254 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
116 -> 254 [arrowhead = none ];
254 -> 118;

255 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
117 -> 255 [arrowhead = none ];
255 -> 121;

256 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
118 -> 256 [arrowhead = none ];
256 -> 119;

257 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
119 -> 257 [arrowhead = none ];
257 -> 120;

258 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
121 -> 258 [arrowhead = none ];
258 -> 122;

259 [label="ONLY EVAL with clause\nno(X210).\nand substitutionT105 -> T108,\nX210 -> zero(T108)", fontsize=14, style = filled, fillcolor = lightblue];
122 -> 259 [arrowhead = none ];
259 -> 123;

260 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
123 -> 260 [arrowhead = none ];
260 -> 124;

261 [label="BACKTRACK\nfor clause: p(0, 0)\nwith clash: (zero(T108), zero(0))", fontsize=14, style = filled, fillcolor = lightgreen];
124 -> 261 [arrowhead = none ];
261 -> 125;

262 [label="EVAL with clause\np(s(X215), X215).\nand substitutionX215 -> T111,\nT108 -> s(T111),\nX197 -> T111", fontsize=14, style = filled, fillcolor = lightblue];
125 -> 262 [arrowhead = none ];
262 -> 126;

263 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
125 -> 263 [arrowhead = none ];
263 -> 127;

264 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
126 -> 264 [arrowhead = none ];
264 -> 128;

265 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
128 -> 265 [arrowhead = none ];
265 -> 129;

266 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
128 -> 266 [arrowhead = none ];
266 -> 130;

267 [label="ONLY EVAL with clause\np(0, 0).\nand substitutionX198 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
129 -> 267 [arrowhead = none ];
267 -> 131;

268 [label="BACKTRACK\nfor clause: p(s(X), X)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
130 -> 268 [arrowhead = none ];
268 -> 132;

269 [label="INSTANCE with matching:\nT99 -> T111", fontsize=14, style = filled, fillcolor = lightgrey];
131 -> 269 [arrowhead = none , style=dashed];
269 -> 107[style=dashed];

270 [label="INSTANCE with matching:\nT72 -> T99\nT77 -> T114", fontsize=14, style = filled, fillcolor = lightgrey];
133 -> 270 [arrowhead = none , style=dashed];
270 -> 80[style=dashed];

}

(2) Obligation:

Clauses:

lessA(s(T111)) :- lessA(T111).
lessB(0, s(T84)).
lessB(s(T99), 0) :- lessA(T99).
lessB(s(T99), s(T114)) :- lessB(T99, T114).
lessC(0, T57).
lessC(s(T72), T77) :- lessB(T72, T77).
orderedD([]).
orderedD(.(T3, [])).
orderedD(.(T31, .(T47, T48))) :- lessC(T31, T47).
orderedD(.(T31, .(T47, T48))) :- ','(lessC(T31, T47), orderedD(.(T47, T48))).

Query: orderedD(g)

(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:
orderedD_in: (b)
lessC_in: (b,b)
lessB_in: (b,b)
lessA_in: (b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

orderedD_in_g([]) → orderedD_out_g([])
orderedD_in_g(.(T3, [])) → orderedD_out_g(.(T3, []))
orderedD_in_g(.(T31, .(T47, T48))) → U5_g(T31, T47, T48, lessC_in_gg(T31, T47))
lessC_in_gg(0, T57) → lessC_out_gg(0, T57)
lessC_in_gg(s(T72), T77) → U4_gg(T72, T77, lessB_in_gg(T72, T77))
lessB_in_gg(0, s(T84)) → lessB_out_gg(0, s(T84))
lessB_in_gg(s(T99), 0) → U2_gg(T99, lessA_in_g(T99))
lessA_in_g(s(T111)) → U1_g(T111, lessA_in_g(T111))
U1_g(T111, lessA_out_g(T111)) → lessA_out_g(s(T111))
U2_gg(T99, lessA_out_g(T99)) → lessB_out_gg(s(T99), 0)
lessB_in_gg(s(T99), s(T114)) → U3_gg(T99, T114, lessB_in_gg(T99, T114))
U3_gg(T99, T114, lessB_out_gg(T99, T114)) → lessB_out_gg(s(T99), s(T114))
U4_gg(T72, T77, lessB_out_gg(T72, T77)) → lessC_out_gg(s(T72), T77)
U5_g(T31, T47, T48, lessC_out_gg(T31, T47)) → orderedD_out_g(.(T31, .(T47, T48)))
U5_g(T31, T47, T48, lessC_out_gg(T31, T47)) → U6_g(T31, T47, T48, orderedD_in_g(.(T47, T48)))
U6_g(T31, T47, T48, orderedD_out_g(.(T47, T48))) → orderedD_out_g(.(T31, .(T47, T48)))

The argument filtering Pi contains the following mapping:
orderedD_in_g(x1)  =  orderedD_in_g(x1)
[]  =  []
orderedD_out_g(x1)  =  orderedD_out_g
.(x1, x2)  =  .(x1, x2)
U5_g(x1, x2, x3, x4)  =  U5_g(x2, x3, x4)
lessC_in_gg(x1, x2)  =  lessC_in_gg(x1, x2)
0  =  0
lessC_out_gg(x1, x2)  =  lessC_out_gg
s(x1)  =  s(x1)
U4_gg(x1, x2, x3)  =  U4_gg(x3)
lessB_in_gg(x1, x2)  =  lessB_in_gg(x1, x2)
lessB_out_gg(x1, x2)  =  lessB_out_gg
U2_gg(x1, x2)  =  U2_gg(x2)
lessA_in_g(x1)  =  lessA_in_g(x1)
U1_g(x1, x2)  =  U1_g(x2)
lessA_out_g(x1)  =  lessA_out_g
U3_gg(x1, x2, x3)  =  U3_gg(x3)
U6_g(x1, x2, x3, x4)  =  U6_g(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:

orderedD_in_g([]) → orderedD_out_g([])
orderedD_in_g(.(T3, [])) → orderedD_out_g(.(T3, []))
orderedD_in_g(.(T31, .(T47, T48))) → U5_g(T31, T47, T48, lessC_in_gg(T31, T47))
lessC_in_gg(0, T57) → lessC_out_gg(0, T57)
lessC_in_gg(s(T72), T77) → U4_gg(T72, T77, lessB_in_gg(T72, T77))
lessB_in_gg(0, s(T84)) → lessB_out_gg(0, s(T84))
lessB_in_gg(s(T99), 0) → U2_gg(T99, lessA_in_g(T99))
lessA_in_g(s(T111)) → U1_g(T111, lessA_in_g(T111))
U1_g(T111, lessA_out_g(T111)) → lessA_out_g(s(T111))
U2_gg(T99, lessA_out_g(T99)) → lessB_out_gg(s(T99), 0)
lessB_in_gg(s(T99), s(T114)) → U3_gg(T99, T114, lessB_in_gg(T99, T114))
U3_gg(T99, T114, lessB_out_gg(T99, T114)) → lessB_out_gg(s(T99), s(T114))
U4_gg(T72, T77, lessB_out_gg(T72, T77)) → lessC_out_gg(s(T72), T77)
U5_g(T31, T47, T48, lessC_out_gg(T31, T47)) → orderedD_out_g(.(T31, .(T47, T48)))
U5_g(T31, T47, T48, lessC_out_gg(T31, T47)) → U6_g(T31, T47, T48, orderedD_in_g(.(T47, T48)))
U6_g(T31, T47, T48, orderedD_out_g(.(T47, T48))) → orderedD_out_g(.(T31, .(T47, T48)))

The argument filtering Pi contains the following mapping:
orderedD_in_g(x1)  =  orderedD_in_g(x1)
[]  =  []
orderedD_out_g(x1)  =  orderedD_out_g
.(x1, x2)  =  .(x1, x2)
U5_g(x1, x2, x3, x4)  =  U5_g(x2, x3, x4)
lessC_in_gg(x1, x2)  =  lessC_in_gg(x1, x2)
0  =  0
lessC_out_gg(x1, x2)  =  lessC_out_gg
s(x1)  =  s(x1)
U4_gg(x1, x2, x3)  =  U4_gg(x3)
lessB_in_gg(x1, x2)  =  lessB_in_gg(x1, x2)
lessB_out_gg(x1, x2)  =  lessB_out_gg
U2_gg(x1, x2)  =  U2_gg(x2)
lessA_in_g(x1)  =  lessA_in_g(x1)
U1_g(x1, x2)  =  U1_g(x2)
lessA_out_g(x1)  =  lessA_out_g
U3_gg(x1, x2, x3)  =  U3_gg(x3)
U6_g(x1, x2, x3, x4)  =  U6_g(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:

ORDEREDD_IN_G(.(T31, .(T47, T48))) → U5_G(T31, T47, T48, lessC_in_gg(T31, T47))
ORDEREDD_IN_G(.(T31, .(T47, T48))) → LESSC_IN_GG(T31, T47)
LESSC_IN_GG(s(T72), T77) → U4_GG(T72, T77, lessB_in_gg(T72, T77))
LESSC_IN_GG(s(T72), T77) → LESSB_IN_GG(T72, T77)
LESSB_IN_GG(s(T99), 0) → U2_GG(T99, lessA_in_g(T99))
LESSB_IN_GG(s(T99), 0) → LESSA_IN_G(T99)
LESSA_IN_G(s(T111)) → U1_G(T111, lessA_in_g(T111))
LESSA_IN_G(s(T111)) → LESSA_IN_G(T111)
LESSB_IN_GG(s(T99), s(T114)) → U3_GG(T99, T114, lessB_in_gg(T99, T114))
LESSB_IN_GG(s(T99), s(T114)) → LESSB_IN_GG(T99, T114)
U5_G(T31, T47, T48, lessC_out_gg(T31, T47)) → U6_G(T31, T47, T48, orderedD_in_g(.(T47, T48)))
U5_G(T31, T47, T48, lessC_out_gg(T31, T47)) → ORDEREDD_IN_G(.(T47, T48))

The TRS R consists of the following rules:

orderedD_in_g([]) → orderedD_out_g([])
orderedD_in_g(.(T3, [])) → orderedD_out_g(.(T3, []))
orderedD_in_g(.(T31, .(T47, T48))) → U5_g(T31, T47, T48, lessC_in_gg(T31, T47))
lessC_in_gg(0, T57) → lessC_out_gg(0, T57)
lessC_in_gg(s(T72), T77) → U4_gg(T72, T77, lessB_in_gg(T72, T77))
lessB_in_gg(0, s(T84)) → lessB_out_gg(0, s(T84))
lessB_in_gg(s(T99), 0) → U2_gg(T99, lessA_in_g(T99))
lessA_in_g(s(T111)) → U1_g(T111, lessA_in_g(T111))
U1_g(T111, lessA_out_g(T111)) → lessA_out_g(s(T111))
U2_gg(T99, lessA_out_g(T99)) → lessB_out_gg(s(T99), 0)
lessB_in_gg(s(T99), s(T114)) → U3_gg(T99, T114, lessB_in_gg(T99, T114))
U3_gg(T99, T114, lessB_out_gg(T99, T114)) → lessB_out_gg(s(T99), s(T114))
U4_gg(T72, T77, lessB_out_gg(T72, T77)) → lessC_out_gg(s(T72), T77)
U5_g(T31, T47, T48, lessC_out_gg(T31, T47)) → orderedD_out_g(.(T31, .(T47, T48)))
U5_g(T31, T47, T48, lessC_out_gg(T31, T47)) → U6_g(T31, T47, T48, orderedD_in_g(.(T47, T48)))
U6_g(T31, T47, T48, orderedD_out_g(.(T47, T48))) → orderedD_out_g(.(T31, .(T47, T48)))

The argument filtering Pi contains the following mapping:
orderedD_in_g(x1)  =  orderedD_in_g(x1)
[]  =  []
orderedD_out_g(x1)  =  orderedD_out_g
.(x1, x2)  =  .(x1, x2)
U5_g(x1, x2, x3, x4)  =  U5_g(x2, x3, x4)
lessC_in_gg(x1, x2)  =  lessC_in_gg(x1, x2)
0  =  0
lessC_out_gg(x1, x2)  =  lessC_out_gg
s(x1)  =  s(x1)
U4_gg(x1, x2, x3)  =  U4_gg(x3)
lessB_in_gg(x1, x2)  =  lessB_in_gg(x1, x2)
lessB_out_gg(x1, x2)  =  lessB_out_gg
U2_gg(x1, x2)  =  U2_gg(x2)
lessA_in_g(x1)  =  lessA_in_g(x1)
U1_g(x1, x2)  =  U1_g(x2)
lessA_out_g(x1)  =  lessA_out_g
U3_gg(x1, x2, x3)  =  U3_gg(x3)
U6_g(x1, x2, x3, x4)  =  U6_g(x4)
ORDEREDD_IN_G(x1)  =  ORDEREDD_IN_G(x1)
U5_G(x1, x2, x3, x4)  =  U5_G(x2, x3, x4)
LESSC_IN_GG(x1, x2)  =  LESSC_IN_GG(x1, x2)
U4_GG(x1, x2, x3)  =  U4_GG(x3)
LESSB_IN_GG(x1, x2)  =  LESSB_IN_GG(x1, x2)
U2_GG(x1, x2)  =  U2_GG(x2)
LESSA_IN_G(x1)  =  LESSA_IN_G(x1)
U1_G(x1, x2)  =  U1_G(x2)
U3_GG(x1, x2, x3)  =  U3_GG(x3)
U6_G(x1, x2, x3, x4)  =  U6_G(x4)

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

(6) Obligation:

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

ORDEREDD_IN_G(.(T31, .(T47, T48))) → U5_G(T31, T47, T48, lessC_in_gg(T31, T47))
ORDEREDD_IN_G(.(T31, .(T47, T48))) → LESSC_IN_GG(T31, T47)
LESSC_IN_GG(s(T72), T77) → U4_GG(T72, T77, lessB_in_gg(T72, T77))
LESSC_IN_GG(s(T72), T77) → LESSB_IN_GG(T72, T77)
LESSB_IN_GG(s(T99), 0) → U2_GG(T99, lessA_in_g(T99))
LESSB_IN_GG(s(T99), 0) → LESSA_IN_G(T99)
LESSA_IN_G(s(T111)) → U1_G(T111, lessA_in_g(T111))
LESSA_IN_G(s(T111)) → LESSA_IN_G(T111)
LESSB_IN_GG(s(T99), s(T114)) → U3_GG(T99, T114, lessB_in_gg(T99, T114))
LESSB_IN_GG(s(T99), s(T114)) → LESSB_IN_GG(T99, T114)
U5_G(T31, T47, T48, lessC_out_gg(T31, T47)) → U6_G(T31, T47, T48, orderedD_in_g(.(T47, T48)))
U5_G(T31, T47, T48, lessC_out_gg(T31, T47)) → ORDEREDD_IN_G(.(T47, T48))

The TRS R consists of the following rules:

orderedD_in_g([]) → orderedD_out_g([])
orderedD_in_g(.(T3, [])) → orderedD_out_g(.(T3, []))
orderedD_in_g(.(T31, .(T47, T48))) → U5_g(T31, T47, T48, lessC_in_gg(T31, T47))
lessC_in_gg(0, T57) → lessC_out_gg(0, T57)
lessC_in_gg(s(T72), T77) → U4_gg(T72, T77, lessB_in_gg(T72, T77))
lessB_in_gg(0, s(T84)) → lessB_out_gg(0, s(T84))
lessB_in_gg(s(T99), 0) → U2_gg(T99, lessA_in_g(T99))
lessA_in_g(s(T111)) → U1_g(T111, lessA_in_g(T111))
U1_g(T111, lessA_out_g(T111)) → lessA_out_g(s(T111))
U2_gg(T99, lessA_out_g(T99)) → lessB_out_gg(s(T99), 0)
lessB_in_gg(s(T99), s(T114)) → U3_gg(T99, T114, lessB_in_gg(T99, T114))
U3_gg(T99, T114, lessB_out_gg(T99, T114)) → lessB_out_gg(s(T99), s(T114))
U4_gg(T72, T77, lessB_out_gg(T72, T77)) → lessC_out_gg(s(T72), T77)
U5_g(T31, T47, T48, lessC_out_gg(T31, T47)) → orderedD_out_g(.(T31, .(T47, T48)))
U5_g(T31, T47, T48, lessC_out_gg(T31, T47)) → U6_g(T31, T47, T48, orderedD_in_g(.(T47, T48)))
U6_g(T31, T47, T48, orderedD_out_g(.(T47, T48))) → orderedD_out_g(.(T31, .(T47, T48)))

The argument filtering Pi contains the following mapping:
orderedD_in_g(x1)  =  orderedD_in_g(x1)
[]  =  []
orderedD_out_g(x1)  =  orderedD_out_g
.(x1, x2)  =  .(x1, x2)
U5_g(x1, x2, x3, x4)  =  U5_g(x2, x3, x4)
lessC_in_gg(x1, x2)  =  lessC_in_gg(x1, x2)
0  =  0
lessC_out_gg(x1, x2)  =  lessC_out_gg
s(x1)  =  s(x1)
U4_gg(x1, x2, x3)  =  U4_gg(x3)
lessB_in_gg(x1, x2)  =  lessB_in_gg(x1, x2)
lessB_out_gg(x1, x2)  =  lessB_out_gg
U2_gg(x1, x2)  =  U2_gg(x2)
lessA_in_g(x1)  =  lessA_in_g(x1)
U1_g(x1, x2)  =  U1_g(x2)
lessA_out_g(x1)  =  lessA_out_g
U3_gg(x1, x2, x3)  =  U3_gg(x3)
U6_g(x1, x2, x3, x4)  =  U6_g(x4)
ORDEREDD_IN_G(x1)  =  ORDEREDD_IN_G(x1)
U5_G(x1, x2, x3, x4)  =  U5_G(x2, x3, x4)
LESSC_IN_GG(x1, x2)  =  LESSC_IN_GG(x1, x2)
U4_GG(x1, x2, x3)  =  U4_GG(x3)
LESSB_IN_GG(x1, x2)  =  LESSB_IN_GG(x1, x2)
U2_GG(x1, x2)  =  U2_GG(x2)
LESSA_IN_G(x1)  =  LESSA_IN_G(x1)
U1_G(x1, x2)  =  U1_G(x2)
U3_GG(x1, x2, x3)  =  U3_GG(x3)
U6_G(x1, x2, x3, x4)  =  U6_G(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 8 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

LESSA_IN_G(s(T111)) → LESSA_IN_G(T111)

The TRS R consists of the following rules:

orderedD_in_g([]) → orderedD_out_g([])
orderedD_in_g(.(T3, [])) → orderedD_out_g(.(T3, []))
orderedD_in_g(.(T31, .(T47, T48))) → U5_g(T31, T47, T48, lessC_in_gg(T31, T47))
lessC_in_gg(0, T57) → lessC_out_gg(0, T57)
lessC_in_gg(s(T72), T77) → U4_gg(T72, T77, lessB_in_gg(T72, T77))
lessB_in_gg(0, s(T84)) → lessB_out_gg(0, s(T84))
lessB_in_gg(s(T99), 0) → U2_gg(T99, lessA_in_g(T99))
lessA_in_g(s(T111)) → U1_g(T111, lessA_in_g(T111))
U1_g(T111, lessA_out_g(T111)) → lessA_out_g(s(T111))
U2_gg(T99, lessA_out_g(T99)) → lessB_out_gg(s(T99), 0)
lessB_in_gg(s(T99), s(T114)) → U3_gg(T99, T114, lessB_in_gg(T99, T114))
U3_gg(T99, T114, lessB_out_gg(T99, T114)) → lessB_out_gg(s(T99), s(T114))
U4_gg(T72, T77, lessB_out_gg(T72, T77)) → lessC_out_gg(s(T72), T77)
U5_g(T31, T47, T48, lessC_out_gg(T31, T47)) → orderedD_out_g(.(T31, .(T47, T48)))
U5_g(T31, T47, T48, lessC_out_gg(T31, T47)) → U6_g(T31, T47, T48, orderedD_in_g(.(T47, T48)))
U6_g(T31, T47, T48, orderedD_out_g(.(T47, T48))) → orderedD_out_g(.(T31, .(T47, T48)))

The argument filtering Pi contains the following mapping:
orderedD_in_g(x1)  =  orderedD_in_g(x1)
[]  =  []
orderedD_out_g(x1)  =  orderedD_out_g
.(x1, x2)  =  .(x1, x2)
U5_g(x1, x2, x3, x4)  =  U5_g(x2, x3, x4)
lessC_in_gg(x1, x2)  =  lessC_in_gg(x1, x2)
0  =  0
lessC_out_gg(x1, x2)  =  lessC_out_gg
s(x1)  =  s(x1)
U4_gg(x1, x2, x3)  =  U4_gg(x3)
lessB_in_gg(x1, x2)  =  lessB_in_gg(x1, x2)
lessB_out_gg(x1, x2)  =  lessB_out_gg
U2_gg(x1, x2)  =  U2_gg(x2)
lessA_in_g(x1)  =  lessA_in_g(x1)
U1_g(x1, x2)  =  U1_g(x2)
lessA_out_g(x1)  =  lessA_out_g
U3_gg(x1, x2, x3)  =  U3_gg(x3)
U6_g(x1, x2, x3, x4)  =  U6_g(x4)
LESSA_IN_G(x1)  =  LESSA_IN_G(x1)

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:

LESSA_IN_G(s(T111)) → LESSA_IN_G(T111)

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

(12) PiDPToQDPProof (EQUIVALENT 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:

LESSA_IN_G(s(T111)) → LESSA_IN_G(T111)

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:

• LESSA_IN_G(s(T111)) → LESSA_IN_G(T111)
  The graph contains the following edges 1 > 1

(15) YES

(16) Obligation:

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

LESSB_IN_GG(s(T99), s(T114)) → LESSB_IN_GG(T99, T114)

The TRS R consists of the following rules:

orderedD_in_g([]) → orderedD_out_g([])
orderedD_in_g(.(T3, [])) → orderedD_out_g(.(T3, []))
orderedD_in_g(.(T31, .(T47, T48))) → U5_g(T31, T47, T48, lessC_in_gg(T31, T47))
lessC_in_gg(0, T57) → lessC_out_gg(0, T57)
lessC_in_gg(s(T72), T77) → U4_gg(T72, T77, lessB_in_gg(T72, T77))
lessB_in_gg(0, s(T84)) → lessB_out_gg(0, s(T84))
lessB_in_gg(s(T99), 0) → U2_gg(T99, lessA_in_g(T99))
lessA_in_g(s(T111)) → U1_g(T111, lessA_in_g(T111))
U1_g(T111, lessA_out_g(T111)) → lessA_out_g(s(T111))
U2_gg(T99, lessA_out_g(T99)) → lessB_out_gg(s(T99), 0)
lessB_in_gg(s(T99), s(T114)) → U3_gg(T99, T114, lessB_in_gg(T99, T114))
U3_gg(T99, T114, lessB_out_gg(T99, T114)) → lessB_out_gg(s(T99), s(T114))
U4_gg(T72, T77, lessB_out_gg(T72, T77)) → lessC_out_gg(s(T72), T77)
U5_g(T31, T47, T48, lessC_out_gg(T31, T47)) → orderedD_out_g(.(T31, .(T47, T48)))
U5_g(T31, T47, T48, lessC_out_gg(T31, T47)) → U6_g(T31, T47, T48, orderedD_in_g(.(T47, T48)))
U6_g(T31, T47, T48, orderedD_out_g(.(T47, T48))) → orderedD_out_g(.(T31, .(T47, T48)))

The argument filtering Pi contains the following mapping:
orderedD_in_g(x1)  =  orderedD_in_g(x1)
[]  =  []
orderedD_out_g(x1)  =  orderedD_out_g
.(x1, x2)  =  .(x1, x2)
U5_g(x1, x2, x3, x4)  =  U5_g(x2, x3, x4)
lessC_in_gg(x1, x2)  =  lessC_in_gg(x1, x2)
0  =  0
lessC_out_gg(x1, x2)  =  lessC_out_gg
s(x1)  =  s(x1)
U4_gg(x1, x2, x3)  =  U4_gg(x3)
lessB_in_gg(x1, x2)  =  lessB_in_gg(x1, x2)
lessB_out_gg(x1, x2)  =  lessB_out_gg
U2_gg(x1, x2)  =  U2_gg(x2)
lessA_in_g(x1)  =  lessA_in_g(x1)
U1_g(x1, x2)  =  U1_g(x2)
lessA_out_g(x1)  =  lessA_out_g
U3_gg(x1, x2, x3)  =  U3_gg(x3)
U6_g(x1, x2, x3, x4)  =  U6_g(x4)
LESSB_IN_GG(x1, x2)  =  LESSB_IN_GG(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:

LESSB_IN_GG(s(T99), s(T114)) → LESSB_IN_GG(T99, T114)

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

(19) PiDPToQDPProof (EQUIVALENT 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:

LESSB_IN_GG(s(T99), s(T114)) → LESSB_IN_GG(T99, T114)

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:

• LESSB_IN_GG(s(T99), s(T114)) → LESSB_IN_GG(T99, T114)
  The graph contains the following edges 1 > 1, 2 > 2

(22) YES

(23) Obligation:

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

U5_G(T31, T47, T48, lessC_out_gg(T31, T47)) → ORDEREDD_IN_G(.(T47, T48))
ORDEREDD_IN_G(.(T31, .(T47, T48))) → U5_G(T31, T47, T48, lessC_in_gg(T31, T47))

The TRS R consists of the following rules:

orderedD_in_g([]) → orderedD_out_g([])
orderedD_in_g(.(T3, [])) → orderedD_out_g(.(T3, []))
orderedD_in_g(.(T31, .(T47, T48))) → U5_g(T31, T47, T48, lessC_in_gg(T31, T47))
lessC_in_gg(0, T57) → lessC_out_gg(0, T57)
lessC_in_gg(s(T72), T77) → U4_gg(T72, T77, lessB_in_gg(T72, T77))
lessB_in_gg(0, s(T84)) → lessB_out_gg(0, s(T84))
lessB_in_gg(s(T99), 0) → U2_gg(T99, lessA_in_g(T99))
lessA_in_g(s(T111)) → U1_g(T111, lessA_in_g(T111))
U1_g(T111, lessA_out_g(T111)) → lessA_out_g(s(T111))
U2_gg(T99, lessA_out_g(T99)) → lessB_out_gg(s(T99), 0)
lessB_in_gg(s(T99), s(T114)) → U3_gg(T99, T114, lessB_in_gg(T99, T114))
U3_gg(T99, T114, lessB_out_gg(T99, T114)) → lessB_out_gg(s(T99), s(T114))
U4_gg(T72, T77, lessB_out_gg(T72, T77)) → lessC_out_gg(s(T72), T77)
U5_g(T31, T47, T48, lessC_out_gg(T31, T47)) → orderedD_out_g(.(T31, .(T47, T48)))
U5_g(T31, T47, T48, lessC_out_gg(T31, T47)) → U6_g(T31, T47, T48, orderedD_in_g(.(T47, T48)))
U6_g(T31, T47, T48, orderedD_out_g(.(T47, T48))) → orderedD_out_g(.(T31, .(T47, T48)))

The argument filtering Pi contains the following mapping:
orderedD_in_g(x1)  =  orderedD_in_g(x1)
[]  =  []
orderedD_out_g(x1)  =  orderedD_out_g
.(x1, x2)  =  .(x1, x2)
U5_g(x1, x2, x3, x4)  =  U5_g(x2, x3, x4)
lessC_in_gg(x1, x2)  =  lessC_in_gg(x1, x2)
0  =  0
lessC_out_gg(x1, x2)  =  lessC_out_gg
s(x1)  =  s(x1)
U4_gg(x1, x2, x3)  =  U4_gg(x3)
lessB_in_gg(x1, x2)  =  lessB_in_gg(x1, x2)
lessB_out_gg(x1, x2)  =  lessB_out_gg
U2_gg(x1, x2)  =  U2_gg(x2)
lessA_in_g(x1)  =  lessA_in_g(x1)
U1_g(x1, x2)  =  U1_g(x2)
lessA_out_g(x1)  =  lessA_out_g
U3_gg(x1, x2, x3)  =  U3_gg(x3)
U6_g(x1, x2, x3, x4)  =  U6_g(x4)
ORDEREDD_IN_G(x1)  =  ORDEREDD_IN_G(x1)
U5_G(x1, x2, x3, x4)  =  U5_G(x2, x3, x4)

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:

U5_G(T31, T47, T48, lessC_out_gg(T31, T47)) → ORDEREDD_IN_G(.(T47, T48))
ORDEREDD_IN_G(.(T31, .(T47, T48))) → U5_G(T31, T47, T48, lessC_in_gg(T31, T47))

The TRS R consists of the following rules:

lessC_in_gg(0, T57) → lessC_out_gg(0, T57)
lessC_in_gg(s(T72), T77) → U4_gg(T72, T77, lessB_in_gg(T72, T77))
U4_gg(T72, T77, lessB_out_gg(T72, T77)) → lessC_out_gg(s(T72), T77)
lessB_in_gg(0, s(T84)) → lessB_out_gg(0, s(T84))
lessB_in_gg(s(T99), 0) → U2_gg(T99, lessA_in_g(T99))
lessB_in_gg(s(T99), s(T114)) → U3_gg(T99, T114, lessB_in_gg(T99, T114))
U2_gg(T99, lessA_out_g(T99)) → lessB_out_gg(s(T99), 0)
U3_gg(T99, T114, lessB_out_gg(T99, T114)) → lessB_out_gg(s(T99), s(T114))
lessA_in_g(s(T111)) → U1_g(T111, lessA_in_g(T111))
U1_g(T111, lessA_out_g(T111)) → lessA_out_g(s(T111))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
lessC_in_gg(x1, x2)  =  lessC_in_gg(x1, x2)
0  =  0
lessC_out_gg(x1, x2)  =  lessC_out_gg
s(x1)  =  s(x1)
U4_gg(x1, x2, x3)  =  U4_gg(x3)
lessB_in_gg(x1, x2)  =  lessB_in_gg(x1, x2)
lessB_out_gg(x1, x2)  =  lessB_out_gg
U2_gg(x1, x2)  =  U2_gg(x2)
lessA_in_g(x1)  =  lessA_in_g(x1)
U1_g(x1, x2)  =  U1_g(x2)
lessA_out_g(x1)  =  lessA_out_g
U3_gg(x1, x2, x3)  =  U3_gg(x3)
ORDEREDD_IN_G(x1)  =  ORDEREDD_IN_G(x1)
U5_G(x1, x2, x3, x4)  =  U5_G(x2, x3, x4)

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:

U5_G(T47, T48, lessC_out_gg) → ORDEREDD_IN_G(.(T47, T48))
ORDEREDD_IN_G(.(T31, .(T47, T48))) → U5_G(T47, T48, lessC_in_gg(T31, T47))

The TRS R consists of the following rules:

lessC_in_gg(0, T57) → lessC_out_gg
lessC_in_gg(s(T72), T77) → U4_gg(lessB_in_gg(T72, T77))
U4_gg(lessB_out_gg) → lessC_out_gg
lessB_in_gg(0, s(T84)) → lessB_out_gg
lessB_in_gg(s(T99), 0) → U2_gg(lessA_in_g(T99))
lessB_in_gg(s(T99), s(T114)) → U3_gg(lessB_in_gg(T99, T114))
U2_gg(lessA_out_g) → lessB_out_gg
U3_gg(lessB_out_gg) → lessB_out_gg
lessA_in_g(s(T111)) → U1_g(lessA_in_g(T111))
U1_g(lessA_out_g) → lessA_out_g

The set Q consists of the following terms:

lessC_in_gg(x0, x1)
U4_gg(x0)
lessB_in_gg(x0, x1)
U2_gg(x0)
U3_gg(x0)
lessA_in_g(x0)
U1_g(x0)

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

(28) UsableRulesReductionPairsProof (EQUIVALENT transformation)

By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.

The following dependency pairs can be deleted:

ORDEREDD_IN_G(.(T31, .(T47, T48))) → U5_G(T47, T48, lessC_in_gg(T31, T47))

The following rules are removed from R:

lessC_in_gg(0, T57) → lessC_out_gg
lessC_in_gg(s(T72), T77) → U4_gg(lessB_in_gg(T72, T77))
U4_gg(lessB_out_gg) → lessC_out_gg
lessB_in_gg(0, s(T84)) → lessB_out_gg
lessB_in_gg(s(T99), 0) → U2_gg(lessA_in_g(T99))
lessB_in_gg(s(T99), s(T114)) → U3_gg(lessB_in_gg(T99, T114))
U2_gg(lessA_out_g) → lessB_out_gg
U3_gg(lessB_out_gg) → lessB_out_gg
lessA_in_g(s(T111)) → U1_g(lessA_in_g(T111))

Used ordering: POLO with Polynomial interpretation [POLO]:

POL(.(x₁, x₂)) = 1 + x₁ + 2·x₂   
POL(0) = 0   
POL(ORDEREDD_IN_G(x₁)) = x₁   
POL(U1_g(x₁)) = x₁   
POL(U2_gg(x₁)) = 1 + 2·x₁   
POL(U3_gg(x₁)) = 2 + 2·x₁   
POL(U4_gg(x₁)) = 1 + x₁   
POL(U5_G(x₁, x₂, x₃)) = 1 + x₁ + 2·x₂ + x₃   
POL(lessA_in_g(x₁)) = 1 + x₁   
POL(lessA_out_g) = 2   
POL(lessB_in_gg(x₁, x₂)) = 1 + 2·x₁ + x₂   
POL(lessB_out_gg) = 1   
POL(lessC_in_gg(x₁, x₂)) = 1 + x₁ + x₂   
POL(lessC_out_gg) = 0   
POL(s(x₁)) = 1 + 2·x₁   

(29) Obligation:

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

U5_G(T47, T48, lessC_out_gg) → ORDEREDD_IN_G(.(T47, T48))

The TRS R consists of the following rules:

U1_g(lessA_out_g) → lessA_out_g

The set Q consists of the following terms:

lessC_in_gg(x0, x1)
U4_gg(x0)
lessB_in_gg(x0, x1)
U2_gg(x0)
U3_gg(x0)
lessA_in_g(x0)
U1_g(x0)

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

(30) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.

(31) TRUE