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

(0) Obligation:

Clauses:

p(d(e(t)), const(1)).
p(d(e(const(A))), const(0)).
p(d(e(+(X, Y))), +(DX, DY)) :- ','(p(d(e(X)), DX), p(d(e(Y)), DY)).
p(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) :- ','(p(d(e(X)), DX), p(d(e(Y)), DY)).
p(d(d(X)), DDX) :- ','(p(d(X), DX), p(d(e(DX)), DDX)).

Query: p(g,a)

(1) PrologToTRSTransformerProof (SOUND transformation)

Transformed Prolog program to TRS.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
2 [label="2: p(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
5 [label="5: p(T1, T2), SCOPE: 1, CLAUSE: 0 | p(T1, T2), SCOPE: 1, CLAUSE: 1 | p(T1, T2), SCOPE: 1, CLAUSE: 2 | p(T1, T2), SCOPE: 1, CLAUSE: 3 | p(T1, T2), SCOPE: 1, CLAUSE: 4\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: p(T1, T2), SCOPE: 1, CLAUSE: 0\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: p(T1, T2), SCOPE: 1, CLAUSE: 1 | p(T1, T2), SCOPE: 1, CLAUSE: 2 | p(T1, T2), SCOPE: 1, CLAUSE: 3 | p(T1, T2), SCOPE: 1, CLAUSE: 4\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: p(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: p(T1, T2), SCOPE: 1, CLAUSE: 2 | p(T1, T2), SCOPE: 1, CLAUSE: 3 | p(T1, T2), SCOPE: 1, CLAUSE: 4\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: p(T1, T2), SCOPE: 1, CLAUSE: 2\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: p(T1, T2), SCOPE: 1, CLAUSE: 3 | p(T1, T2), SCOPE: 1, CLAUSE: 4\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: ','(p(d(e(T24)), T28), p(d(e(T25)), T29))\n\nT24 is ground\nT25 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: p(d(e(T24)), T28)\n\nT24 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: p(d(e(T25)), T30)\n\nT25 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: p(T1, T2), SCOPE: 1, CLAUSE: 3\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: p(T1, T2), SCOPE: 1, CLAUSE: 4\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: ','(p(d(e(T47)), T51), p(d(e(T48)), T52))\n\nT47 is ground\nT48 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
81 [label="81: ','(p(d(T57), X53), p(d(e(X53)), T59))\n\nT57 is ground\nX53 is free", fontsize=16, style=filled, fillcolor=lightyellow];
82 [label="82: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
85 [label="85: p(d(T57), X53)\n\nT57 is ground\nX53 is free", fontsize=16, style=filled, fillcolor=lightyellow];
86 [label="86: p(d(e(T60)), T59)\n\nT60 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
87 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
2 -> 87 [arrowhead = none ];
87 -> 5;

88 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
5 -> 88 [arrowhead = none ];
88 -> 7;

89 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
5 -> 89 [arrowhead = none ];
89 -> 8;

90 [label="EVAL with clause\np(d(e(t)), const(1)).\nand substitutionT1 -> d(e(t)),\nT2 -> const(1)", fontsize=14, style = filled, fillcolor = lightblue];
7 -> 90 [arrowhead = none ];
90 -> 9;

91 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 91 [arrowhead = none ];
91 -> 11;

92 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
8 -> 92 [arrowhead = none ];
92 -> 20;

93 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
8 -> 93 [arrowhead = none ];
93 -> 21;

94 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 94 [arrowhead = none ];
94 -> 15;

95 [label="EVAL with clause\np(d(e(const(X5))), const(0)).\nand substitutionX5 -> T7,\nT1 -> d(e(const(T7))),\nT2 -> const(0)", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 95 [arrowhead = none ];
95 -> 26;

96 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
20 -> 96 [arrowhead = none ];
96 -> 29;

97 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
21 -> 97 [arrowhead = none ];
97 -> 32;

98 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
21 -> 98 [arrowhead = none ];
98 -> 33;

99 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
26 -> 99 [arrowhead = none ];
99 -> 30;

100 [label="EVAL with clause\np(d(e(+(X22, X23))), +(X24, X25)) :- ','(p(d(e(X22)), X24), p(d(e(X23)), X25)).\nand substitutionX22 -> T24,\nX23 -> T25,\nT1 -> d(e(+(T24, T25))),\nX24 -> T28,\nX25 -> T29,\nT2 -> +(T28, T29),\nT26 -> T28,\nT27 -> T29", fontsize=14, style = filled, fillcolor = lightblue];
32 -> 100 [arrowhead = none ];
100 -> 44;

101 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
32 -> 101 [arrowhead = none ];
101 -> 46;

102 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
33 -> 102 [arrowhead = none ];
102 -> 61;

103 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
33 -> 103 [arrowhead = none ];
103 -> 62;

104 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
44 -> 104 [arrowhead = none ];
104 -> 55;

105 [label="SPLIT 2\nnew knowledge:\nT24 is ground\nT28 is ground\nreplacements:T29 -> T30", fontsize=14, style = filled, fillcolor = violet];
44 -> 105 [arrowhead = none ];
105 -> 56;

106 [label="INSTANCE with matching:\nT1 -> d(e(T24))\nT2 -> T28", fontsize=14, style = filled, fillcolor = lightgrey];
55 -> 106 [arrowhead = none , style=dashed];
106 -> 2[style=dashed];

107 [label="INSTANCE with matching:\nT1 -> d(e(T25))\nT2 -> T30", fontsize=14, style = filled, fillcolor = lightgrey];
56 -> 107 [arrowhead = none , style=dashed];
107 -> 2[style=dashed];

108 [label="EVAL with clause\np(d(e(*(X42, X43))), +(*(X42, X44), *(X43, X45))) :- ','(p(d(e(X42)), X45), p(d(e(X43)), X44)).\nand substitutionX42 -> T47,\nX43 -> T48,\nT1 -> d(e(*(T47, T48))),\nX44 -> T52,\nX45 -> T51,\nT2 -> +(*(T47, T52), *(T48, T51)),\nT50 -> T51,\nT49 -> T52", fontsize=14, style = filled, fillcolor = lightblue];
61 -> 108 [arrowhead = none ];
108 -> 72;

109 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
61 -> 109 [arrowhead = none ];
109 -> 74;

110 [label="EVAL with clause\np(d(d(X51)), X52) :- ','(p(d(X51), X53), p(d(e(X53)), X52)).\nand substitutionX51 -> T57,\nT1 -> d(d(T57)),\nT2 -> T59,\nX52 -> T59,\nT58 -> T59", fontsize=14, style = filled, fillcolor = lightblue];
62 -> 110 [arrowhead = none ];
110 -> 81;

111 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
62 -> 111 [arrowhead = none ];
111 -> 82;

112 [label="INSTANCE with matching:\nT24 -> T47\nT28 -> T51\nT25 -> T48\nT29 -> T52", fontsize=14, style = filled, fillcolor = lightgrey];
72 -> 112 [arrowhead = none , style=dashed];
112 -> 44[style=dashed];

113 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
81 -> 113 [arrowhead = none ];
113 -> 85;

114 [label="SPLIT 2\nnew knowledge:\nT57 is ground\nT60 is ground\nreplacements:X53 -> T60", fontsize=14, style = filled, fillcolor = violet];
81 -> 114 [arrowhead = none ];
114 -> 86;

115 [label="INSTANCE with matching:\nT1 -> d(T57)\nT2 -> X53", fontsize=14, style = filled, fillcolor = lightgrey];
85 -> 115 [arrowhead = none , style=dashed];
115 -> 2[style=dashed];

116 [label="INSTANCE with matching:\nT1 -> d(e(T60))\nT2 -> T59", fontsize=14, style = filled, fillcolor = lightgrey];
86 -> 116 [arrowhead = none , style=dashed];
116 -> 2[style=dashed];

}

(2) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

f2_in(d(e(t))) → f2_out1(const(1))
f2_in(d(e(const(T7)))) → f2_out1(const(0))
f2_in(d(e(+(T24, T25)))) → U1(f44_in(T24, T25), d(e(+(T24, T25))))
U1(f44_out1(T28, T29), d(e(+(T24, T25)))) → f2_out1(+(T28, T29))
f2_in(d(e(*(T47, T48)))) → U2(f44_in(T47, T48), d(e(*(T47, T48))))
U2(f44_out1(T51, T52), d(e(*(T47, T48)))) → f2_out1(+(*(T47, T52), *(T48, T51)))
f2_in(d(d(T57))) → U3(f81_in(T57), d(d(T57)))
U3(f81_out1(X53, T59), d(d(T57))) → f2_out1(T59)
f44_in(T24, T25) → U4(f2_in(d(e(T24))), T24, T25)
U4(f2_out1(T28), T24, T25) → U5(f2_in(d(e(T25))), T24, T25, T28)
U5(f2_out1(T30), T24, T25, T28) → f44_out1(T28, T30)
f81_in(T57) → U6(f2_in(d(T57)), T57)
U6(f2_out1(T60), T57) → U7(f2_in(d(e(T60))), T57, T60)
U7(f2_out1(T59), T57, T60) → f81_out1(T60, T59)

Q is empty.

(3) Overlay + Local Confluence (EQUIVALENT transformation)

The TRS is overlay and locally confluent. By [NOC] we can switch to innermost.

(4) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

f2_in(d(e(t))) → f2_out1(const(1))
f2_in(d(e(const(T7)))) → f2_out1(const(0))
f2_in(d(e(+(T24, T25)))) → U1(f44_in(T24, T25), d(e(+(T24, T25))))
U1(f44_out1(T28, T29), d(e(+(T24, T25)))) → f2_out1(+(T28, T29))
f2_in(d(e(*(T47, T48)))) → U2(f44_in(T47, T48), d(e(*(T47, T48))))
U2(f44_out1(T51, T52), d(e(*(T47, T48)))) → f2_out1(+(*(T47, T52), *(T48, T51)))
f2_in(d(d(T57))) → U3(f81_in(T57), d(d(T57)))
U3(f81_out1(X53, T59), d(d(T57))) → f2_out1(T59)
f44_in(T24, T25) → U4(f2_in(d(e(T24))), T24, T25)
U4(f2_out1(T28), T24, T25) → U5(f2_in(d(e(T25))), T24, T25, T28)
U5(f2_out1(T30), T24, T25, T28) → f44_out1(T28, T30)
f81_in(T57) → U6(f2_in(d(T57)), T57)
U6(f2_out1(T60), T57) → U7(f2_in(d(e(T60))), T57, T60)
U7(f2_out1(T59), T57, T60) → f81_out1(T60, T59)

The set Q consists of the following terms:

f2_in(d(e(t)))
f2_in(d(e(const(x0))))
f2_in(d(e(+(x0, x1))))
U1(f44_out1(x0, x1), d(e(+(x2, x3))))
f2_in(d(e(*(x0, x1))))
U2(f44_out1(x0, x1), d(e(*(x2, x3))))
f2_in(d(d(x0)))
U3(f81_out1(x0, x1), d(d(x2)))
f44_in(x0, x1)
U4(f2_out1(x0), x1, x2)
U5(f2_out1(x0), x1, x2, x3)
f81_in(x0)
U6(f2_out1(x0), x1)
U7(f2_out1(x0), x1, x2)

(5) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.

(6) Obligation:

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

F2_IN(d(e(+(T24, T25)))) → U1¹(f44_in(T24, T25), d(e(+(T24, T25))))
F2_IN(d(e(+(T24, T25)))) → F44_IN(T24, T25)
F2_IN(d(e(*(T47, T48)))) → U2¹(f44_in(T47, T48), d(e(*(T47, T48))))
F2_IN(d(e(*(T47, T48)))) → F44_IN(T47, T48)
F2_IN(d(d(T57))) → U3¹(f81_in(T57), d(d(T57)))
F2_IN(d(d(T57))) → F81_IN(T57)
F44_IN(T24, T25) → U4¹(f2_in(d(e(T24))), T24, T25)
F44_IN(T24, T25) → F2_IN(d(e(T24)))
U4¹(f2_out1(T28), T24, T25) → U5¹(f2_in(d(e(T25))), T24, T25, T28)
U4¹(f2_out1(T28), T24, T25) → F2_IN(d(e(T25)))
F81_IN(T57) → U6¹(f2_in(d(T57)), T57)
F81_IN(T57) → F2_IN(d(T57))
U6¹(f2_out1(T60), T57) → U7¹(f2_in(d(e(T60))), T57, T60)
U6¹(f2_out1(T60), T57) → F2_IN(d(e(T60)))

The TRS R consists of the following rules:

f2_in(d(e(t))) → f2_out1(const(1))
f2_in(d(e(const(T7)))) → f2_out1(const(0))
f2_in(d(e(+(T24, T25)))) → U1(f44_in(T24, T25), d(e(+(T24, T25))))
U1(f44_out1(T28, T29), d(e(+(T24, T25)))) → f2_out1(+(T28, T29))
f2_in(d(e(*(T47, T48)))) → U2(f44_in(T47, T48), d(e(*(T47, T48))))
U2(f44_out1(T51, T52), d(e(*(T47, T48)))) → f2_out1(+(*(T47, T52), *(T48, T51)))
f2_in(d(d(T57))) → U3(f81_in(T57), d(d(T57)))
U3(f81_out1(X53, T59), d(d(T57))) → f2_out1(T59)
f44_in(T24, T25) → U4(f2_in(d(e(T24))), T24, T25)
U4(f2_out1(T28), T24, T25) → U5(f2_in(d(e(T25))), T24, T25, T28)
U5(f2_out1(T30), T24, T25, T28) → f44_out1(T28, T30)
f81_in(T57) → U6(f2_in(d(T57)), T57)
U6(f2_out1(T60), T57) → U7(f2_in(d(e(T60))), T57, T60)
U7(f2_out1(T59), T57, T60) → f81_out1(T60, T59)

The set Q consists of the following terms:

f2_in(d(e(t)))
f2_in(d(e(const(x0))))
f2_in(d(e(+(x0, x1))))
U1(f44_out1(x0, x1), d(e(+(x2, x3))))
f2_in(d(e(*(x0, x1))))
U2(f44_out1(x0, x1), d(e(*(x2, x3))))
f2_in(d(d(x0)))
U3(f81_out1(x0, x1), d(d(x2)))
f44_in(x0, x1)
U4(f2_out1(x0), x1, x2)
U5(f2_out1(x0), x1, x2, x3)
f81_in(x0)
U6(f2_out1(x0), x1)
U7(f2_out1(x0), x1, x2)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 7 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

F2_IN(d(e(+(T24, T25)))) → F44_IN(T24, T25)
F44_IN(T24, T25) → U4¹(f2_in(d(e(T24))), T24, T25)
U4¹(f2_out1(T28), T24, T25) → F2_IN(d(e(T25)))
F2_IN(d(e(*(T47, T48)))) → F44_IN(T47, T48)
F44_IN(T24, T25) → F2_IN(d(e(T24)))

The TRS R consists of the following rules:

f2_in(d(e(t))) → f2_out1(const(1))
f2_in(d(e(const(T7)))) → f2_out1(const(0))
f2_in(d(e(+(T24, T25)))) → U1(f44_in(T24, T25), d(e(+(T24, T25))))
U1(f44_out1(T28, T29), d(e(+(T24, T25)))) → f2_out1(+(T28, T29))
f2_in(d(e(*(T47, T48)))) → U2(f44_in(T47, T48), d(e(*(T47, T48))))
U2(f44_out1(T51, T52), d(e(*(T47, T48)))) → f2_out1(+(*(T47, T52), *(T48, T51)))
f2_in(d(d(T57))) → U3(f81_in(T57), d(d(T57)))
U3(f81_out1(X53, T59), d(d(T57))) → f2_out1(T59)
f44_in(T24, T25) → U4(f2_in(d(e(T24))), T24, T25)
U4(f2_out1(T28), T24, T25) → U5(f2_in(d(e(T25))), T24, T25, T28)
U5(f2_out1(T30), T24, T25, T28) → f44_out1(T28, T30)
f81_in(T57) → U6(f2_in(d(T57)), T57)
U6(f2_out1(T60), T57) → U7(f2_in(d(e(T60))), T57, T60)
U7(f2_out1(T59), T57, T60) → f81_out1(T60, T59)

The set Q consists of the following terms:

f2_in(d(e(t)))
f2_in(d(e(const(x0))))
f2_in(d(e(+(x0, x1))))
U1(f44_out1(x0, x1), d(e(+(x2, x3))))
f2_in(d(e(*(x0, x1))))
U2(f44_out1(x0, x1), d(e(*(x2, x3))))
f2_in(d(d(x0)))
U3(f81_out1(x0, x1), d(d(x2)))
f44_in(x0, x1)
U4(f2_out1(x0), x1, x2)
U5(f2_out1(x0), x1, x2, x3)
f81_in(x0)
U6(f2_out1(x0), x1)
U7(f2_out1(x0), x1, x2)

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

(10) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

(11) Obligation:

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

F2_IN(d(e(+(T24, T25)))) → F44_IN(T24, T25)
F44_IN(T24, T25) → U4¹(f2_in(d(e(T24))), T24, T25)
U4¹(f2_out1(T28), T24, T25) → F2_IN(d(e(T25)))
F2_IN(d(e(*(T47, T48)))) → F44_IN(T47, T48)
F44_IN(T24, T25) → F2_IN(d(e(T24)))

The TRS R consists of the following rules:

f2_in(d(e(t))) → f2_out1(const(1))
f2_in(d(e(const(T7)))) → f2_out1(const(0))
f2_in(d(e(+(T24, T25)))) → U1(f44_in(T24, T25), d(e(+(T24, T25))))
f2_in(d(e(*(T47, T48)))) → U2(f44_in(T47, T48), d(e(*(T47, T48))))
f44_in(T24, T25) → U4(f2_in(d(e(T24))), T24, T25)
U2(f44_out1(T51, T52), d(e(*(T47, T48)))) → f2_out1(+(*(T47, T52), *(T48, T51)))
U4(f2_out1(T28), T24, T25) → U5(f2_in(d(e(T25))), T24, T25, T28)
U5(f2_out1(T30), T24, T25, T28) → f44_out1(T28, T30)
U1(f44_out1(T28, T29), d(e(+(T24, T25)))) → f2_out1(+(T28, T29))

The set Q consists of the following terms:

f2_in(d(e(t)))
f2_in(d(e(const(x0))))
f2_in(d(e(+(x0, x1))))
U1(f44_out1(x0, x1), d(e(+(x2, x3))))
f2_in(d(e(*(x0, x1))))
U2(f44_out1(x0, x1), d(e(*(x2, x3))))
f2_in(d(d(x0)))
U3(f81_out1(x0, x1), d(d(x2)))
f44_in(x0, x1)
U4(f2_out1(x0), x1, x2)
U5(f2_out1(x0), x1, x2, x3)
f81_in(x0)
U6(f2_out1(x0), x1)
U7(f2_out1(x0), x1, x2)

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

(12) QReductionProof (EQUIVALENT transformation)

We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

U3(f81_out1(x0, x1), d(d(x2)))
f81_in(x0)
U6(f2_out1(x0), x1)
U7(f2_out1(x0), x1, x2)

(13) Obligation:

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

F2_IN(d(e(+(T24, T25)))) → F44_IN(T24, T25)
F44_IN(T24, T25) → U4¹(f2_in(d(e(T24))), T24, T25)
U4¹(f2_out1(T28), T24, T25) → F2_IN(d(e(T25)))
F2_IN(d(e(*(T47, T48)))) → F44_IN(T47, T48)
F44_IN(T24, T25) → F2_IN(d(e(T24)))

The TRS R consists of the following rules:

f2_in(d(e(t))) → f2_out1(const(1))
f2_in(d(e(const(T7)))) → f2_out1(const(0))
f2_in(d(e(+(T24, T25)))) → U1(f44_in(T24, T25), d(e(+(T24, T25))))
f2_in(d(e(*(T47, T48)))) → U2(f44_in(T47, T48), d(e(*(T47, T48))))
f44_in(T24, T25) → U4(f2_in(d(e(T24))), T24, T25)
U2(f44_out1(T51, T52), d(e(*(T47, T48)))) → f2_out1(+(*(T47, T52), *(T48, T51)))
U4(f2_out1(T28), T24, T25) → U5(f2_in(d(e(T25))), T24, T25, T28)
U5(f2_out1(T30), T24, T25, T28) → f44_out1(T28, T30)
U1(f44_out1(T28, T29), d(e(+(T24, T25)))) → f2_out1(+(T28, T29))

The set Q consists of the following terms:

f2_in(d(e(t)))
f2_in(d(e(const(x0))))
f2_in(d(e(+(x0, x1))))
U1(f44_out1(x0, x1), d(e(+(x2, x3))))
f2_in(d(e(*(x0, x1))))
U2(f44_out1(x0, x1), d(e(*(x2, x3))))
f2_in(d(d(x0)))
f44_in(x0, x1)
U4(f2_out1(x0), x1, x2)
U5(f2_out1(x0), x1, x2, x3)

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

(14) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04,JAR06].

The following pairs can be oriented strictly and are deleted.

F44_IN(T24, T25) → U4¹(f2_in(d(e(T24))), T24, T25)
F44_IN(T24, T25) → F2_IN(d(e(T24)))

The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(*(x₁, x₂)) = 1 + x₁ + x₂   
POL(+(x₁, x₂)) = 1 + x₁ + x₂   
POL(0) = 0   
POL(1) = 0   
POL(F2_IN(x₁)) = x₁   
POL(F44_IN(x₁, x₂)) = 1 + x₁ + x₂   
POL(U1(x₁, x₂)) = 0   
POL(U2(x₁, x₂)) = 0   
POL(U4(x₁, x₂, x₃)) = 0   
POL(U4¹(x₁, x₂, x₃)) = x₃   
POL(U5(x₁, x₂, x₃, x₄)) = 0   
POL(const(x₁)) = 0   
POL(d(x₁)) = x₁   
POL(e(x₁)) = x₁   
POL(f2_in(x₁)) = 0   
POL(f2_out1(x₁)) = 0   
POL(f44_in(x₁, x₂)) = 0   
POL(f44_out1(x₁, x₂)) = 0   
POL(t) = 0   

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
none

(15) Obligation:

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

F2_IN(d(e(+(T24, T25)))) → F44_IN(T24, T25)
U4¹(f2_out1(T28), T24, T25) → F2_IN(d(e(T25)))
F2_IN(d(e(*(T47, T48)))) → F44_IN(T47, T48)

The TRS R consists of the following rules:

f2_in(d(e(t))) → f2_out1(const(1))
f2_in(d(e(const(T7)))) → f2_out1(const(0))
f2_in(d(e(+(T24, T25)))) → U1(f44_in(T24, T25), d(e(+(T24, T25))))
f2_in(d(e(*(T47, T48)))) → U2(f44_in(T47, T48), d(e(*(T47, T48))))
f44_in(T24, T25) → U4(f2_in(d(e(T24))), T24, T25)
U2(f44_out1(T51, T52), d(e(*(T47, T48)))) → f2_out1(+(*(T47, T52), *(T48, T51)))
U4(f2_out1(T28), T24, T25) → U5(f2_in(d(e(T25))), T24, T25, T28)
U5(f2_out1(T30), T24, T25, T28) → f44_out1(T28, T30)
U1(f44_out1(T28, T29), d(e(+(T24, T25)))) → f2_out1(+(T28, T29))

The set Q consists of the following terms:

f2_in(d(e(t)))
f2_in(d(e(const(x0))))
f2_in(d(e(+(x0, x1))))
U1(f44_out1(x0, x1), d(e(+(x2, x3))))
f2_in(d(e(*(x0, x1))))
U2(f44_out1(x0, x1), d(e(*(x2, x3))))
f2_in(d(d(x0)))
f44_in(x0, x1)
U4(f2_out1(x0), x1, x2)
U5(f2_out1(x0), x1, x2, x3)

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

(16) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes.

(17) TRUE

(18) Obligation:

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

F2_IN(d(d(T57))) → F81_IN(T57)
F81_IN(T57) → F2_IN(d(T57))

The TRS R consists of the following rules:

f2_in(d(e(t))) → f2_out1(const(1))
f2_in(d(e(const(T7)))) → f2_out1(const(0))
f2_in(d(e(+(T24, T25)))) → U1(f44_in(T24, T25), d(e(+(T24, T25))))
U1(f44_out1(T28, T29), d(e(+(T24, T25)))) → f2_out1(+(T28, T29))
f2_in(d(e(*(T47, T48)))) → U2(f44_in(T47, T48), d(e(*(T47, T48))))
U2(f44_out1(T51, T52), d(e(*(T47, T48)))) → f2_out1(+(*(T47, T52), *(T48, T51)))
f2_in(d(d(T57))) → U3(f81_in(T57), d(d(T57)))
U3(f81_out1(X53, T59), d(d(T57))) → f2_out1(T59)
f44_in(T24, T25) → U4(f2_in(d(e(T24))), T24, T25)
U4(f2_out1(T28), T24, T25) → U5(f2_in(d(e(T25))), T24, T25, T28)
U5(f2_out1(T30), T24, T25, T28) → f44_out1(T28, T30)
f81_in(T57) → U6(f2_in(d(T57)), T57)
U6(f2_out1(T60), T57) → U7(f2_in(d(e(T60))), T57, T60)
U7(f2_out1(T59), T57, T60) → f81_out1(T60, T59)

The set Q consists of the following terms:

f2_in(d(e(t)))
f2_in(d(e(const(x0))))
f2_in(d(e(+(x0, x1))))
U1(f44_out1(x0, x1), d(e(+(x2, x3))))
f2_in(d(e(*(x0, x1))))
U2(f44_out1(x0, x1), d(e(*(x2, x3))))
f2_in(d(d(x0)))
U3(f81_out1(x0, x1), d(d(x2)))
f44_in(x0, x1)
U4(f2_out1(x0), x1, x2)
U5(f2_out1(x0), x1, x2, x3)
f81_in(x0)
U6(f2_out1(x0), x1)
U7(f2_out1(x0), x1, x2)

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

(19) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

(20) Obligation:

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

F2_IN(d(d(T57))) → F81_IN(T57)
F81_IN(T57) → F2_IN(d(T57))

R is empty.
The set Q consists of the following terms:

f2_in(d(e(t)))
f2_in(d(e(const(x0))))
f2_in(d(e(+(x0, x1))))
U1(f44_out1(x0, x1), d(e(+(x2, x3))))
f2_in(d(e(*(x0, x1))))
U2(f44_out1(x0, x1), d(e(*(x2, x3))))
f2_in(d(d(x0)))
U3(f81_out1(x0, x1), d(d(x2)))
f44_in(x0, x1)
U4(f2_out1(x0), x1, x2)
U5(f2_out1(x0), x1, x2, x3)
f81_in(x0)
U6(f2_out1(x0), x1)
U7(f2_out1(x0), x1, x2)

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

(21) QReductionProof (EQUIVALENT transformation)

We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

f2_in(d(e(t)))
f2_in(d(e(const(x0))))
f2_in(d(e(+(x0, x1))))
U1(f44_out1(x0, x1), d(e(+(x2, x3))))
f2_in(d(e(*(x0, x1))))
U2(f44_out1(x0, x1), d(e(*(x2, x3))))
f2_in(d(d(x0)))
U3(f81_out1(x0, x1), d(d(x2)))
f44_in(x0, x1)
U4(f2_out1(x0), x1, x2)
U5(f2_out1(x0), x1, x2, x3)
f81_in(x0)
U6(f2_out1(x0), x1)
U7(f2_out1(x0), x1, x2)

(22) Obligation:

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

F2_IN(d(d(T57))) → F81_IN(T57)
F81_IN(T57) → F2_IN(d(T57))

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

(23) MRRProof (EQUIVALENT transformation)

By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented dependency pairs:

F2_IN(d(d(T57))) → F81_IN(T57)
F81_IN(T57) → F2_IN(d(T57))

Used ordering: Knuth-Bendix order [KBO] with precedence:

F81_IN1 > d₁ > F2_IN1

and weight map:

F2_IN_1=1
d_1=1
F81_IN_1=2

The variable weight is 1

(24) Obligation:

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

(25) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(26) YES