Left Termination proof of /tmp/tmpDkhzyF/add3.pl

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

0 Prolog

↳1 PrologToPiTRSViaGraphTransformerProof (⇒, 53 ms)

↳2 PiTRS

↳3 DependencyPairsProof (⇔, 0 ms)

↳4 PiDP

↳5 DependencyGraphProof (⇔, 3 ms)

↳6 PiDP

↳7 UsableRulesProof (⇔, 0 ms)

↳8 PiDP

↳9 PiDPToQDPProof (⇒, 0 ms)

↳10 QDP

↳11 QDPSizeChangeProof (⇔, 0 ms)

↳12 YES

(0) Obligation:

Clauses:

add(X, 0, X).
add(X, Y, s(Z)) :- ','(no(zero(Y)), ','(p(Y, P), add(X, P, Z))).
p(0, 0).
p(s(X), X).
zero(0).
no(X) :- ','(X, ','(!, failure(a))).
no(X).
failure(b).

Query: add(a,g,a)

(1) PrologToPiTRSViaGraphTransformerProof (SOUND transformation)

Transformed Prolog program to (Pi-)TRS.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
2 [label="A: add(T1, T2, T3)\n\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
6 [label="6: add(T1, T2, T3), SCOPE: 1, CLAUSE: 0 | add(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: true | add(T1, 0, T3), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: add(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\nT2 is ground\nX2 is free\nadd(T1, T2, T3) !=? add(X2, 0, X2)", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: add(T1, 0, T3), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: ','(no(zero(0)), ','(p(0, X9), add(T10, X9, T11)))\n\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: ','(no(zero(0)), ','(p(0, X9), add(T10, X9, T11))), SCOPE: 2, CLAUSE: 5 | ','(no(zero(0)), ','(p(0, X9), add(T10, X9, T11))), SCOPE: 2, CLAUSE: 6\n\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: ','(','(call(zero(0)), ','(!_2, failure(a))), ','(p(0, X9), add(T10, X9, T11))) | ','(no(zero(0)), ','(p(0, X9), add(T10, X9, T11))), SCOPE: 2, CLAUSE: 6\n\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: ','(zero(0), ','(','(!_2, failure(a)), ','(p(0, X9), add(T10, X9, T11)))) | ?_3 | ','(no(zero(0)), ','(p(0, X9), add(T10, X9, T11))), SCOPE: 2, CLAUSE: 6\n\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: ','(zero(0), ','(','(!_2, failure(a)), ','(p(0, X9), add(T10, X9, T11)))), SCOPE: 4, CLAUSE: 4 | ?_4 | ?_3 | ','(no(zero(0)), ','(p(0, X9), add(T10, X9, T11))), SCOPE: 2, CLAUSE: 6\n\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: ','(','(!_2, failure(a)), ','(p(0, X9), add(T10, X9, T11))) | ?_4 | ?_3 | ','(no(zero(0)), ','(p(0, X9), add(T10, X9, T11))), SCOPE: 2, CLAUSE: 6\n\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: ','(failure(a), ','(p(0, X9), add(T10, X9, T11)))\n\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: ','(failure(a), ','(p(0, X9), add(T10, X9, T11))), SCOPE: 5, CLAUSE: 7\n\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: ','(no(zero(T18)), ','(p(T18, X21), add(T20, X21, T21)))\n\nT18 is ground\nX2 is free\nX21 is free\nadd(T1, T18, T3) !=? add(X2, 0, X2)", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: ','(no(zero(T18)), ','(p(T18, X21), add(T20, X21, T21))), SCOPE: 6, CLAUSE: 5 | ','(no(zero(T18)), ','(p(T18, X21), add(T20, X21, T21))), SCOPE: 6, CLAUSE: 6\n\nT18 is ground\nX2 is free\nX21 is free\nadd(T1, T18, T3) !=? add(X2, 0, X2)", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: ','(','(call(zero(T24)), ','(!_6, failure(a))), ','(p(T24, X21), add(T20, X21, T21))) | ','(no(zero(T24)), ','(p(T24, X21), add(T20, X21, T21))), SCOPE: 6, CLAUSE: 6\n\nT24 is ground\nX2 is free\nX21 is free\nadd(T1, T24, T3) !=? add(X2, 0, X2)", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: ','(zero(T24), ','(','(!_6, failure(a)), ','(p(T24, X21), add(T20, X21, T21)))) | ?_7 | ','(no(zero(T24)), ','(p(T24, X21), add(T20, X21, T21))), SCOPE: 6, CLAUSE: 6\n\nT24 is ground\nX2 is free\nX21 is free\nadd(T1, T24, T3) !=? add(X2, 0, X2)", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: ','(zero(T24), ','(','(!_6, failure(a)), ','(p(T24, X21), add(T20, X21, T21)))), SCOPE: 8, CLAUSE: 4 | ?_8 | ?_7 | ','(no(zero(T24)), ','(p(T24, X21), add(T20, X21, T21))), SCOPE: 6, CLAUSE: 6\n\nT24 is ground\nX2 is free\nX21 is free\nadd(T1, T24, T3) !=? add(X2, 0, X2)", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="78: ','(','(!_6, failure(a)), ','(p(0, X21), add(T20, X21, T21))) | ?_8 | ?_7 | ','(no(zero(0)), ','(p(0, X21), add(T20, X21, T21))), SCOPE: 6, CLAUSE: 6\n\nX21 is free", fontsize=16, style=filled, fillcolor=lightyellow];
79 [label="79: ?_8 | ?_7 | ','(no(zero(T24)), ','(p(T24, X21), add(T20, X21, T21))), SCOPE: 6, CLAUSE: 6\n\nT24 is ground\nX2 is free\nX21 is free\nadd(T1, T24, T3) !=? add(X2, 0, X2)\nzero(T24) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
80 [label="80: ','(failure(a), ','(p(0, X21), add(T20, X21, T21)))\n\nX21 is free", fontsize=16, style=filled, fillcolor=lightyellow];
83 [label="83: ','(failure(a), ','(p(0, X21), add(T20, X21, T21))), SCOPE: 9, CLAUSE: 7\n\nX21 is free", fontsize=16, style=filled, fillcolor=lightyellow];
84 [label="84: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
85 [label="85: ?_7 | ','(no(zero(T24)), ','(p(T24, X21), add(T20, X21, T21))), SCOPE: 6, CLAUSE: 6\n\nT24 is ground\nX2 is free\nX21 is free\nadd(T1, T24, T3) !=? add(X2, 0, X2)\nzero(T24) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
86 [label="86: ','(no(zero(T24)), ','(p(T24, X21), add(T20, X21, T21))), SCOPE: 6, CLAUSE: 6\n\nT24 is ground\nX2 is free\nX21 is free\nadd(T1, T24, T3) !=? add(X2, 0, X2)\nzero(T24) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
87 [label="87: ','(p(T31, X21), add(T20, X21, T21))\n\nT31 is ground\nX2 is free\nX21 is free\nadd(T1, T31, T3) !=? add(X2, 0, X2)\nzero(T31) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
88 [label="88: ','(p(T31, X21), add(T20, X21, T21)), SCOPE: 10, CLAUSE: 2 | ','(p(T31, X21), add(T20, X21, T21)), SCOPE: 10, CLAUSE: 3\n\nT31 is ground\nX2 is free\nX21 is free\nadd(T1, T31, T3) !=? add(X2, 0, X2)\nzero(T31) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
89 [label="89: ','(p(T31, X21), add(T20, X21, T21)), SCOPE: 10, CLAUSE: 3\n\nT31 is ground\nX2 is free\nX21 is free\nadd(T1, T31, T3) !=? add(X2, 0, X2)\nzero(T31) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
90 [label="90: add(T20, T36, T21)\n\nT36 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
91 [label="91: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
92 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
2 -> 92 [arrowhead = none ];
92 -> 6;

93 [label="EVAL with clause\nadd(X2, 0, X2).\nand substitutionT1 -> T5,\nX2 -> T5,\nT2 -> 0,\nT3 -> T5", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 93 [arrowhead = none ];
93 -> 10;

94 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 94 [arrowhead = none ];
94 -> 14;

95 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 95 [arrowhead = none ];
95 -> 17;

96 [label="EVAL with clause\nadd(X18, X19, s(X20)) :- ','(no(zero(X19)), ','(p(X19, X21), add(X18, X21, X20))).\nand substitutionT1 -> T20,\nX18 -> T20,\nT2 -> T18,\nX19 -> T18,\nX20 -> T21,\nT3 -> s(T21),\nT17 -> T20,\nT19 -> T21", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 96 [arrowhead = none ];
96 -> 55;

97 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
14 -> 97 [arrowhead = none ];
97 -> 69;

98 [label="EVAL with clause\nadd(X6, X7, s(X8)) :- ','(no(zero(X7)), ','(p(X7, X9), add(X6, X9, X8))).\nand substitutionT1 -> T10,\nX6 -> T10,\nX7 -> 0,\nX8 -> T11,\nT3 -> s(T11),\nT8 -> T10,\nT9 -> T11", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 98 [arrowhead = none ];
98 -> 19;

99 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
17 -> 99 [arrowhead = none ];
99 -> 21;

100 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
19 -> 100 [arrowhead = none ];
100 -> 23;

101 [label="ONLY EVAL with clause\nno(X12) :- ','(call(X12), ','(!_2, failure(a))).\nand substitutionX12 -> zero(0)", fontsize=14, style = filled, fillcolor = lightblue];
23 -> 101 [arrowhead = none ];
101 -> 27;

102 [label="CALL", fontsize=14, style = filled, fillcolor = lightcyan];
27 -> 102 [arrowhead = none ];
102 -> 30;

103 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
30 -> 103 [arrowhead = none ];
103 -> 33;

104 [label="ONLY EVAL with clause\nzero(0).\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
33 -> 104 [arrowhead = none ];
104 -> 38;

105 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
38 -> 105 [arrowhead = none ];
105 -> 39;

106 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
39 -> 106 [arrowhead = none ];
106 -> 48;

107 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
48 -> 107 [arrowhead = none ];
107 -> 49;

108 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
55 -> 108 [arrowhead = none ];
108 -> 71;

109 [label="ONLY EVAL with clause\nno(X24) :- ','(call(X24), ','(!_6, failure(a))).\nand substitutionT18 -> T24,\nX24 -> zero(T24)", fontsize=14, style = filled, fillcolor = lightblue];
71 -> 109 [arrowhead = none ];
109 -> 72;

110 [label="CALL", fontsize=14, style = filled, fillcolor = lightcyan];
72 -> 110 [arrowhead = none ];
110 -> 74;

111 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
74 -> 111 [arrowhead = none ];
111 -> 75;

112 [label="EVAL with clause\nzero(0).\nand substitutionT24 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
75 -> 112 [arrowhead = none ];
112 -> 78;

113 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
75 -> 113 [arrowhead = none ];
113 -> 79;

114 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
78 -> 114 [arrowhead = none ];
114 -> 80;

115 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
79 -> 115 [arrowhead = none ];
115 -> 85;

116 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
80 -> 116 [arrowhead = none ];
116 -> 83;

117 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
83 -> 117 [arrowhead = none ];
117 -> 84;

118 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
85 -> 118 [arrowhead = none ];
118 -> 86;

119 [label="ONLY EVAL with clause\nno(X31).\nand substitutionT24 -> T31,\nX31 -> zero(T31)", fontsize=14, style = filled, fillcolor = lightblue];
86 -> 119 [arrowhead = none ];
119 -> 87;

120 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
87 -> 120 [arrowhead = none ];
120 -> 88;

121 [label="BACKTRACK\nfor clause: p(0, 0)\nwith clash: (zero(T31), zero(0))", fontsize=14, style = filled, fillcolor = lightgreen];
88 -> 121 [arrowhead = none ];
121 -> 89;

122 [label="EVAL with clause\np(s(X36), X36).\nand substitutionX36 -> T36,\nT31 -> s(T36),\nX21 -> T36", fontsize=14, style = filled, fillcolor = lightblue];
89 -> 122 [arrowhead = none ];
122 -> 90;

123 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
89 -> 123 [arrowhead = none ];
123 -> 91;

124 [label="INSTANCE with matching:\nT1 -> T20\nT2 -> T36\nT3 -> T21", fontsize=14, style = filled, fillcolor = lightgrey];
90 -> 124 [arrowhead = none , style=dashed];
124 -> 2[style=dashed];

}

(2) Obligation:

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

addA_in_aga(T5, 0, T5) → addA_out_aga(T5, 0, T5)
addA_in_aga(T20, s(T36), s(T21)) → U1_aga(T20, T36, T21, addA_in_aga(T20, T36, T21))
U1_aga(T20, T36, T21, addA_out_aga(T20, T36, T21)) → addA_out_aga(T20, s(T36), s(T21))

(3) DependencyPairsProof (EQUIVALENT transformation)

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

ADDA_IN_AGA(T20, s(T36), s(T21)) → U1_AGA(T20, T36, T21, addA_in_aga(T20, T36, T21))
ADDA_IN_AGA(T20, s(T36), s(T21)) → ADDA_IN_AGA(T20, T36, T21)

The TRS R consists of the following rules:

addA_in_aga(T5, 0, T5) → addA_out_aga(T5, 0, T5)
addA_in_aga(T20, s(T36), s(T21)) → U1_aga(T20, T36, T21, addA_in_aga(T20, T36, T21))
U1_aga(T20, T36, T21, addA_out_aga(T20, T36, T21)) → addA_out_aga(T20, s(T36), s(T21))

The argument filtering Pi contains the following mapping:
addA_in_aga(x1, x2, x3) = addA_in_aga(x2)
0 = 0
addA_out_aga(x1, x2, x3) = addA_out_aga(x2)
s(x1) = s(x1)
U1_aga(x1, x2, x3, x4) = U1_aga(x2, x4)
ADDA_IN_AGA(x1, x2, x3) = ADDA_IN_AGA(x2)
U1_AGA(x1, x2, x3, x4) = U1_AGA(x2, x4)

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

(4) Obligation:

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

ADDA_IN_AGA(T20, s(T36), s(T21)) → U1_AGA(T20, T36, T21, addA_in_aga(T20, T36, T21))
ADDA_IN_AGA(T20, s(T36), s(T21)) → ADDA_IN_AGA(T20, T36, T21)

The TRS R consists of the following rules:

addA_in_aga(T5, 0, T5) → addA_out_aga(T5, 0, T5)
addA_in_aga(T20, s(T36), s(T21)) → U1_aga(T20, T36, T21, addA_in_aga(T20, T36, T21))
U1_aga(T20, T36, T21, addA_out_aga(T20, T36, T21)) → addA_out_aga(T20, s(T36), s(T21))

(5) DependencyGraphProof (EQUIVALENT transformation)

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

(6) Obligation:

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

ADDA_IN_AGA(T20, s(T36), s(T21)) → ADDA_IN_AGA(T20, T36, T21)

The TRS R consists of the following rules:

addA_in_aga(T5, 0, T5) → addA_out_aga(T5, 0, T5)
addA_in_aga(T20, s(T36), s(T21)) → U1_aga(T20, T36, T21, addA_in_aga(T20, T36, T21))
U1_aga(T20, T36, T21, addA_out_aga(T20, T36, T21)) → addA_out_aga(T20, s(T36), s(T21))

(7) UsableRulesProof (EQUIVALENT transformation)

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

(8) Obligation:

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

ADDA_IN_AGA(T20, s(T36), s(T21)) → ADDA_IN_AGA(T20, T36, T21)

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

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

(9) PiDPToQDPProof (SOUND transformation)

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

(10) Obligation:

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

ADDA_IN_AGA(s(T36)) → ADDA_IN_AGA(T36)

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

(11) 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:

ADDA_IN_AGA(s(T36)) → ADDA_IN_AGA(T36)
The graph contains the following edges 1 > 1

(0) Obligation:

(1) PrologToPiTRSViaGraphTransformerProof (SOUND transformation)

(2) Obligation:

(3) DependencyPairsProof (EQUIVALENT transformation)

(4) Obligation:

(5) DependencyGraphProof (EQUIVALENT transformation)

(6) Obligation:

(7) UsableRulesProof (EQUIVALENT transformation)

(8) Obligation:

(9) PiDPToQDPProof (SOUND transformation)

(10) Obligation:

(11) QDPSizeChangeProof (EQUIVALENT transformation)

(12) YES