Left Termination proof of /tmp/tmpdbIYqD/less2.pl

Left Termination of the query pattern less_in_2(g, a) w.r.t. the given Prolog program could successfully be proven:

0 Prolog

↳1 PrologToPrologProblemTransformerProof (⇐)

↳2 Prolog

↳3 PrologToPiTRSProof (⇐)

↳4 PiTRS

↳5 DependencyPairsProof (⇔)

↳6 PiDP

↳7 DependencyGraphProof (⇔)

↳8 AND

    ↳9 PiDP

        ↳10 UsableRulesProof (⇔)

            ↳11 PiDP

                ↳12 PiDPToQDPProof (⇔)

                    ↳13 QDP

                        ↳14 QDPSizeChangeProof (⇔)

                            ↳15 TRUE

    ↳16 PiDP

        ↳17 UsableRulesProof (⇔)

            ↳18 PiDP

                ↳19 PiDPToQDPProof (⇐)

                    ↳20 QDP

                        ↳21 QDPSizeChangeProof (⇔)

                            ↳22 TRUE

(0) Obligation:

Clauses:

less(0, Y) :- ','(!, =(Y, s(X2))).
less(X, Y) :- ','(p(X, X1), ','(p(Y, Y1), less(X1, Y1))).
p(0, 0).
p(s(X), X).
=(X, X).

Queries:

less(g,a).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: less(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: less(T1, T2), SCOPE: 1, CLAUSE: 0 | less(T1, T2), SCOPE: 1, CLAUSE: 1 | ?_1\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: ','(!_1, =(T4, s(X5))) | less(0, T2), SCOPE: 1, CLAUSE: 1 | ?_1\n\nX5 is free", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: less(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nX4 is free; \nless(T1, T2) !=? less(0, X4)", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: =(T4, s(X5))\n\nX5 is free", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: =(T4, s(X5)), SCOPE: 2, CLAUSE: 4\n\nX5 is free", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: \n\nX5 is free\nX7 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: ','(p(T7, X12), ','(p(T9, X13), less(X12, X13)))\n\nT7 is ground\nX4 is free; \nX12 is free; \nX13 is free; \nless(T7, T2) !=? less(0, X4)", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(p(T7, X12), ','(p(T9, X13), less(X12, X13))), SCOPE: 3, CLAUSE: 2 | ','(p(T7, X12), ','(p(T9, X13), less(X12, X13))), SCOPE: 3, CLAUSE: 3\n\nT7 is ground\nX4 is free; \nX12 is free; \nX13 is free; \nless(T7, T2) !=? less(0, X4)", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: ','(p(T7, X12), ','(p(T9, X13), less(X12, X13))), SCOPE: 3, CLAUSE: 3\n\nT7 is ground\nX4 is free; \nX12 is free; \nX13 is free; \nless(T7, T2) !=? less(0, X4)\np(T7, X12) !=? p(0, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: ','(p(T9, X13), less(T10, X13))\n\nT10 is ground\nX4 is free; \nX12 is free; \nX13 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: \n\nX4 is free\nX12 is free; \nX15 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: ','(p(T9, X13), less(T10, X13)), SCOPE: 4, CLAUSE: 2 | ','(p(T9, X13), less(T10, X13)), SCOPE: 4, CLAUSE: 3\n\nT10 is ground\nX13 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: less(T10, 0) | ','(p(T9, X13), less(T10, X13)), SCOPE: 4, CLAUSE: 3\n\nT10 is ground\nX13 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: ','(p(T9, X13), less(T10, X13)), SCOPE: 4, CLAUSE: 3\n\nT10 is ground\nX13 is free; \np(T9, X13) !=? p(0, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: less(T10, 0)\n\nT10 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: ','(p(T9, X13), less(T10, X13)), SCOPE: 4, CLAUSE: 3\n\nT10 is ground\nX13 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: less(T10, 0), SCOPE: 5, CLAUSE: 0 | less(T10, 0), SCOPE: 5, CLAUSE: 1 | ?_5\n\nT10 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: ','(!_5, =(0, s(X18))) | less(0, 0), SCOPE: 5, CLAUSE: 1 | ?_5\n\nX18 is free", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: less(T10, 0), SCOPE: 5, CLAUSE: 1\n\nT10 is ground\nX17 is free; \nless(T10, 0) !=? less(0, X17)", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: =(0, s(X18))\n\nX18 is free", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: =(0, s(X18)), SCOPE: 6, CLAUSE: 4\n\nX18 is free", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: ','(p(T11, X24), ','(p(0, X25), less(X24, X25)))\n\nT11 is ground\nX17 is free; \nX24 is free; \nX25 is free; \nless(T11, 0) !=? less(0, X17)", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: ','(p(T11, X24), ','(p(0, X25), less(X24, X25))), SCOPE: 7, CLAUSE: 2 | ','(p(T11, X24), ','(p(0, X25), less(X24, X25))), SCOPE: 7, CLAUSE: 3\n\nT11 is ground\nX17 is free; \nX24 is free; \nX25 is free; \nless(T11, 0) !=? less(0, X17)", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: ','(p(T11, X24), ','(p(0, X25), less(X24, X25))), SCOPE: 7, CLAUSE: 3\n\nT11 is ground\nX17 is free; \nX24 is free; \nX25 is free; \nless(T11, 0) !=? less(0, X17)\np(T11, X24) !=? p(0, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: ','(p(0, X25), less(T12, X25))\n\nT12 is ground\nX17 is free; \nX24 is free; \nX25 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: \n\nX17 is free\nX24 is free; \nX27 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: ','(p(0, X25), less(T12, X25)), SCOPE: 8, CLAUSE: 2 | ','(p(0, X25), less(T12, X25)), SCOPE: 8, CLAUSE: 3\n\nT12 is ground\nX25 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: less(T12, 0) | ','(p(0, X25), less(T12, X25)), SCOPE: 8, CLAUSE: 3\n\nT12 is ground\nX25 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: less(T12, 0)\n\nT12 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: ','(p(0, X25), less(T12, X25)), SCOPE: 8, CLAUSE: 3\n\nT12 is ground\nX25 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: less(T10, T14)\n\nT10 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: \n\nX13 is free\nX30 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: less(T10, T16)\n\nT10 is ground\nX13 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: \n\nX13 is free\nX32 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 40 [arrowhead = none ];
40 -> 2;

41 [label="EVAL with clause\nless(0, X4) :- ','(!, =(X4, s(X5))).\nand substitution\nT1 -> 0\nT2 -> T4\nX4 -> T4\nT3 -> T4", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 41 [arrowhead = none ];
41 -> 3;

42 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 42 [arrowhead = none ];
42 -> 4;

43 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 43 [arrowhead = none ];
43 -> 5;

44 [label="EVAL with clause\nless(X10, X11) :- ','(p(X10, X12), ','(p(X11, X13), less(X12, X13))).\nand substitution\nT1 -> T7\nX10 -> T7\nT2 -> T9\nX11 -> T9\nT8 -> T9", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 44 [arrowhead = none ];
44 -> 10;

45 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
5 -> 45 [arrowhead = none ];
45 -> 6;

46 [label="EVAL with clause\n=(X7, X7).\nand substitution\nT4 -> s(T6)\nX7 -> s(T6)\nX5 -> T6\nT5 -> s(T6)", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 46 [arrowhead = none ];
46 -> 7;

47 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 47 [arrowhead = none ];
47 -> 8;

48 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 48 [arrowhead = none ];
48 -> 9;

49 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
10 -> 49 [arrowhead = none ];
49 -> 11;

50 [label="BACKTRACK\nfor clause: p(0, 0)\nwith clash: (less(T7, T2), less(0, X4))", fontsize=14, style = filled, fillcolor = white];
11 -> 50 [arrowhead = none ];
50 -> 12;

51 [label="EVAL with clause\np(s(X15), X15).\nand substitution\nX15 -> T10\nT7 -> s(T10)\nX12 -> T10", fontsize=14, style = filled, fillcolor = lightblue];
12 -> 51 [arrowhead = none ];
51 -> 13;

52 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
12 -> 52 [arrowhead = none ];
52 -> 14;

53 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
13 -> 53 [arrowhead = none ];
53 -> 15;

54 [label="EVAL with clause\np(0, 0).\nand substitution\nT9 -> 0\nX13 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
15 -> 54 [arrowhead = none ];
54 -> 16;

55 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
15 -> 55 [arrowhead = none ];
55 -> 17;

56 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
16 -> 56 [arrowhead = none ];
56 -> 18;

57 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
16 -> 57 [arrowhead = none ];
57 -> 19;

58 [label="EVAL with clause\np(s(X32), X32).\nand substitution\nX32 -> T16\nT9 -> s(T16)\nX13 -> T16\nT15 -> T16", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 58 [arrowhead = none ];
58 -> 38;

59 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 59 [arrowhead = none ];
59 -> 39;

60 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
18 -> 60 [arrowhead = none ];
60 -> 20;

61 [label="EVAL with clause\np(s(X30), X30).\nand substitution\nX30 -> T14\nT9 -> s(T14)\nX13 -> T14\nT13 -> T14", fontsize=14, style = filled, fillcolor = lightblue];
19 -> 61 [arrowhead = none ];
61 -> 36;

62 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
19 -> 62 [arrowhead = none ];
62 -> 37;

63 [label="EVAL with clause\nless(0, X17) :- ','(!, =(X17, s(X18))).\nand substitution\nT10 -> 0\nX17 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 63 [arrowhead = none ];
63 -> 21;

64 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 64 [arrowhead = none ];
64 -> 22;

65 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
21 -> 65 [arrowhead = none ];
65 -> 23;

66 [label="EVAL with clause\nless(X22, X23) :- ','(p(X22, X24), ','(p(X23, X25), less(X24, X25))).\nand substitution\nT10 -> T11\nX22 -> T11\nX23 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
22 -> 66 [arrowhead = none ];
66 -> 26;

67 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
23 -> 67 [arrowhead = none ];
67 -> 24;

68 [label="BACKTRACK\nfor clause: =(X, X)because of non-unification", fontsize=14, style = filled, fillcolor = white];
24 -> 68 [arrowhead = none ];
68 -> 25;

69 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
26 -> 69 [arrowhead = none ];
69 -> 27;

70 [label="BACKTRACK\nfor clause: p(0, 0)\nwith clash: (less(T11, 0), less(0, X17))", fontsize=14, style = filled, fillcolor = white];
27 -> 70 [arrowhead = none ];
70 -> 28;

71 [label="EVAL with clause\np(s(X27), X27).\nand substitution\nX27 -> T12\nT11 -> s(T12)\nX24 -> T12", fontsize=14, style = filled, fillcolor = lightblue];
28 -> 71 [arrowhead = none ];
71 -> 29;

72 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
28 -> 72 [arrowhead = none ];
72 -> 30;

73 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
29 -> 73 [arrowhead = none ];
73 -> 31;

74 [label="EVAL with clause\np(0, 0).\nand substitution\nX25 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
31 -> 74 [arrowhead = none ];
74 -> 32;

75 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
32 -> 75 [arrowhead = none ];
75 -> 33;

76 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
32 -> 76 [arrowhead = none ];
76 -> 34;

77 [label="INSTANCE with matching:\nT10 -> T12", fontsize=14, style = filled, fillcolor = lightgrey];
33 -> 77 [arrowhead = none , style=dashed];
77 -> 18[style=dashed];

78 [label="BACKTRACK\nfor clause: p(s(X), X)because of non-unification", fontsize=14, style = filled, fillcolor = white];
34 -> 78 [arrowhead = none ];
78 -> 35;

79 [label="INSTANCE with matching:\nT1 -> T10\nT2 -> T14", fontsize=14, style = filled, fillcolor = lightgrey];
36 -> 79 [arrowhead = none , style=dashed];
79 -> 1[style=dashed];

80 [label="INSTANCE with matching:\nT1 -> T10\nT2 -> T16", fontsize=14, style = filled, fillcolor = lightgrey];
38 -> 80 [arrowhead = none , style=dashed];
80 -> 1[style=dashed];

}

(2) Obligation:

Clauses:

less18(s(T12)) :- less18(T12).
less1(0, s(T6)).
less1(s(s(T12)), 0) :- less18(T12).
less1(s(T10), s(T14)) :- less1(T10, T14).
less1(s(T10), s(T16)) :- less1(T10, T16).

Queries:

less1(g,a).

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
less1_in: (b,f)
less18_in: (b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

less1_in_ga(0, s(T6)) → less1_out_ga(0, s(T6))
less1_in_ga(s(s(T12)), 0) → U2_ga(T12, less18_in_g(T12))
less18_in_g(s(T12)) → U1_g(T12, less18_in_g(T12))
U1_g(T12, less18_out_g(T12)) → less18_out_g(s(T12))
U2_ga(T12, less18_out_g(T12)) → less1_out_ga(s(s(T12)), 0)
less1_in_ga(s(T10), s(T14)) → U3_ga(T10, T14, less1_in_ga(T10, T14))
less1_in_ga(s(T10), s(T16)) → U4_ga(T10, T16, less1_in_ga(T10, T16))
U4_ga(T10, T16, less1_out_ga(T10, T16)) → less1_out_ga(s(T10), s(T16))
U3_ga(T10, T14, less1_out_ga(T10, T14)) → less1_out_ga(s(T10), s(T14))

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

less1_in_ga(0, s(T6)) → less1_out_ga(0, s(T6))
less1_in_ga(s(s(T12)), 0) → U2_ga(T12, less18_in_g(T12))
less18_in_g(s(T12)) → U1_g(T12, less18_in_g(T12))
U1_g(T12, less18_out_g(T12)) → less18_out_g(s(T12))
U2_ga(T12, less18_out_g(T12)) → less1_out_ga(s(s(T12)), 0)
less1_in_ga(s(T10), s(T14)) → U3_ga(T10, T14, less1_in_ga(T10, T14))
less1_in_ga(s(T10), s(T16)) → U4_ga(T10, T16, less1_in_ga(T10, T16))
U4_ga(T10, T16, less1_out_ga(T10, T16)) → less1_out_ga(s(T10), s(T16))
U3_ga(T10, T14, less1_out_ga(T10, T14)) → less1_out_ga(s(T10), s(T14))

(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:

LESS1_IN_GA(s(s(T12)), 0) → U2_GA(T12, less18_in_g(T12))
LESS1_IN_GA(s(s(T12)), 0) → LESS18_IN_G(T12)
LESS18_IN_G(s(T12)) → U1_G(T12, less18_in_g(T12))
LESS18_IN_G(s(T12)) → LESS18_IN_G(T12)
LESS1_IN_GA(s(T10), s(T14)) → U3_GA(T10, T14, less1_in_ga(T10, T14))
LESS1_IN_GA(s(T10), s(T14)) → LESS1_IN_GA(T10, T14)
LESS1_IN_GA(s(T10), s(T16)) → U4_GA(T10, T16, less1_in_ga(T10, T16))

The TRS R consists of the following rules:

less1_in_ga(0, s(T6)) → less1_out_ga(0, s(T6))
less1_in_ga(s(s(T12)), 0) → U2_ga(T12, less18_in_g(T12))
less18_in_g(s(T12)) → U1_g(T12, less18_in_g(T12))
U1_g(T12, less18_out_g(T12)) → less18_out_g(s(T12))
U2_ga(T12, less18_out_g(T12)) → less1_out_ga(s(s(T12)), 0)
less1_in_ga(s(T10), s(T14)) → U3_ga(T10, T14, less1_in_ga(T10, T14))
less1_in_ga(s(T10), s(T16)) → U4_ga(T10, T16, less1_in_ga(T10, T16))
U4_ga(T10, T16, less1_out_ga(T10, T16)) → less1_out_ga(s(T10), s(T16))
U3_ga(T10, T14, less1_out_ga(T10, T14)) → less1_out_ga(s(T10), s(T14))

The argument filtering Pi contains the following mapping:
less1_in_ga(x1, x2) = less1_in_ga(x1)
0 = 0
less1_out_ga(x1, x2) = less1_out_ga
s(x1) = s(x1)
U2_ga(x1, x2) = U2_ga(x2)
less18_in_g(x1) = less18_in_g(x1)
U1_g(x1, x2) = U1_g(x2)
less18_out_g(x1) = less18_out_g
U3_ga(x1, x2, x3) = U3_ga(x3)
U4_ga(x1, x2, x3) = U4_ga(x3)
LESS1_IN_GA(x1, x2) = LESS1_IN_GA(x1)
U2_GA(x1, x2) = U2_GA(x2)
LESS18_IN_G(x1) = LESS18_IN_G(x1)
U1_G(x1, x2) = U1_G(x2)
U3_GA(x1, x2, x3) = U3_GA(x3)
U4_GA(x1, x2, x3) = U4_GA(x3)

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

(6) Obligation:

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

LESS1_IN_GA(s(s(T12)), 0) → U2_GA(T12, less18_in_g(T12))
LESS1_IN_GA(s(s(T12)), 0) → LESS18_IN_G(T12)
LESS18_IN_G(s(T12)) → U1_G(T12, less18_in_g(T12))
LESS18_IN_G(s(T12)) → LESS18_IN_G(T12)
LESS1_IN_GA(s(T10), s(T14)) → U3_GA(T10, T14, less1_in_ga(T10, T14))
LESS1_IN_GA(s(T10), s(T14)) → LESS1_IN_GA(T10, T14)
LESS1_IN_GA(s(T10), s(T16)) → U4_GA(T10, T16, less1_in_ga(T10, T16))

The TRS R consists of the following rules:

less1_in_ga(0, s(T6)) → less1_out_ga(0, s(T6))
less1_in_ga(s(s(T12)), 0) → U2_ga(T12, less18_in_g(T12))
less18_in_g(s(T12)) → U1_g(T12, less18_in_g(T12))
U1_g(T12, less18_out_g(T12)) → less18_out_g(s(T12))
U2_ga(T12, less18_out_g(T12)) → less1_out_ga(s(s(T12)), 0)
less1_in_ga(s(T10), s(T14)) → U3_ga(T10, T14, less1_in_ga(T10, T14))
less1_in_ga(s(T10), s(T16)) → U4_ga(T10, T16, less1_in_ga(T10, T16))
U4_ga(T10, T16, less1_out_ga(T10, T16)) → less1_out_ga(s(T10), s(T16))
U3_ga(T10, T14, less1_out_ga(T10, T14)) → less1_out_ga(s(T10), s(T14))

(7) DependencyGraphProof (EQUIVALENT transformation)

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

(8) Complex Obligation (AND)

(9) Obligation:

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

LESS18_IN_G(s(T12)) → LESS18_IN_G(T12)

The TRS R consists of the following rules:

less1_in_ga(0, s(T6)) → less1_out_ga(0, s(T6))
less1_in_ga(s(s(T12)), 0) → U2_ga(T12, less18_in_g(T12))
less18_in_g(s(T12)) → U1_g(T12, less18_in_g(T12))
U1_g(T12, less18_out_g(T12)) → less18_out_g(s(T12))
U2_ga(T12, less18_out_g(T12)) → less1_out_ga(s(s(T12)), 0)
less1_in_ga(s(T10), s(T14)) → U3_ga(T10, T14, less1_in_ga(T10, T14))
less1_in_ga(s(T10), s(T16)) → U4_ga(T10, T16, less1_in_ga(T10, T16))
U4_ga(T10, T16, less1_out_ga(T10, T16)) → less1_out_ga(s(T10), s(T16))
U3_ga(T10, T14, less1_out_ga(T10, T14)) → less1_out_ga(s(T10), s(T14))

The argument filtering Pi contains the following mapping:
less1_in_ga(x1, x2) = less1_in_ga(x1)
0 = 0
less1_out_ga(x1, x2) = less1_out_ga
s(x1) = s(x1)
U2_ga(x1, x2) = U2_ga(x2)
less18_in_g(x1) = less18_in_g(x1)
U1_g(x1, x2) = U1_g(x2)
less18_out_g(x1) = less18_out_g
U3_ga(x1, x2, x3) = U3_ga(x3)
U4_ga(x1, x2, x3) = U4_ga(x3)
LESS18_IN_G(x1) = LESS18_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:

LESS18_IN_G(s(T12)) → LESS18_IN_G(T12)

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:

LESS18_IN_G(s(T12)) → LESS18_IN_G(T12)

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:

LESS18_IN_G(s(T12)) → LESS18_IN_G(T12)
The graph contains the following edges 1 > 1

(15) TRUE

(16) Obligation:

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

LESS1_IN_GA(s(T10), s(T14)) → LESS1_IN_GA(T10, T14)

The TRS R consists of the following rules:

less1_in_ga(0, s(T6)) → less1_out_ga(0, s(T6))
less1_in_ga(s(s(T12)), 0) → U2_ga(T12, less18_in_g(T12))
less18_in_g(s(T12)) → U1_g(T12, less18_in_g(T12))
U1_g(T12, less18_out_g(T12)) → less18_out_g(s(T12))
U2_ga(T12, less18_out_g(T12)) → less1_out_ga(s(s(T12)), 0)
less1_in_ga(s(T10), s(T14)) → U3_ga(T10, T14, less1_in_ga(T10, T14))
less1_in_ga(s(T10), s(T16)) → U4_ga(T10, T16, less1_in_ga(T10, T16))
U4_ga(T10, T16, less1_out_ga(T10, T16)) → less1_out_ga(s(T10), s(T16))
U3_ga(T10, T14, less1_out_ga(T10, T14)) → less1_out_ga(s(T10), s(T14))

(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:

LESS1_IN_GA(s(T10), s(T14)) → LESS1_IN_GA(T10, T14)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
LESS1_IN_GA(x1, x2) = LESS1_IN_GA(x1)

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:

LESS1_IN_GA(s(T10)) → LESS1_IN_GA(T10)

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:

LESS1_IN_GA(s(T10)) → LESS1_IN_GA(T10)
The graph contains the following edges 1 > 1

(0) Obligation:

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

(2) Obligation:

(3) PrologToPiTRSProof (SOUND transformation)

(4) Obligation:

(5) DependencyPairsProof (EQUIVALENT transformation)

(6) Obligation:

(7) DependencyGraphProof (EQUIVALENT transformation)

(8) Complex Obligation (AND)

(9) Obligation:

(10) UsableRulesProof (EQUIVALENT transformation)

(11) Obligation:

(12) PiDPToQDPProof (EQUIVALENT transformation)

(13) Obligation:

(14) QDPSizeChangeProof (EQUIVALENT transformation)

(15) TRUE

(16) Obligation:

(17) UsableRulesProof (EQUIVALENT transformation)

(18) Obligation:

(19) PiDPToQDPProof (SOUND transformation)

(20) Obligation:

(21) QDPSizeChangeProof (EQUIVALENT transformation)

(22) TRUE