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

(0) Obligation:

Clauses:

normal(F, N) :- ','(rewrite(F, F1), normal(F1, N)).
normal(F, F).
rewrite(op(op(A, B), C), op(A, op(B, C))).
rewrite(op(A, op(B, C)), op(A, L)) :- rewrite(op(B, C), L).

Query: normal(g,a)

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph ICLP10.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: normal(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: normal(T1, T2), SCOPE: 1, CLAUSE: 0 | normal(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: ','(rewrite(T5, X5), normal(X5, T7)) | normal(T5, T2), SCOPE: 1, CLAUSE: 1\n\nT5 is ground\nX5 is free", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: ','(rewrite(T5, X5), normal(X5, T7)), SCOPE: 2, CLAUSE: 2 | ','(rewrite(T5, X5), normal(X5, T7)), SCOPE: 2, CLAUSE: 3 | ?_2 | normal(T5, T2), SCOPE: 1, CLAUSE: 1\n\nT5 is ground\nX5 is free", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: ','(rewrite(T5, X5), normal(X5, T7)), SCOPE: 2, CLAUSE: 2\n\nT5 is ground\nX5 is free", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ','(rewrite(T5, X5), normal(X5, T7)), SCOPE: 2, CLAUSE: 3 | ?_2 | normal(T5, T2), SCOPE: 1, CLAUSE: 1\n\nT5 is ground\nX5 is free", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: normal(op(T20, op(T21, T22)), T7)\n\nT20 is ground\nT21 is ground\nT22 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: ','(rewrite(T5, X5), normal(X5, T7)), SCOPE: 2, CLAUSE: 3\n\nT5 is ground\nX5 is free", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: ?_2 | normal(T5, T2), SCOPE: 1, CLAUSE: 1\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(rewrite(op(T40, T41), X48), normal(op(T39, X48), T7))\n\nT39 is ground\nT40 is ground\nT41 is ground\nX48 is free", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: rewrite(op(T40, T41), X48)\n\nT40 is ground\nT41 is ground\nX48 is free", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: normal(op(T39, T48), T7)\n\nT39 is ground\nT48 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: rewrite(op(T40, T41), X48), SCOPE: 3, CLAUSE: 2 | rewrite(op(T40, T41), X48), SCOPE: 3, CLAUSE: 3\n\nT40 is ground\nT41 is ground\nX48 is free", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: rewrite(op(T40, T41), X48), SCOPE: 3, CLAUSE: 2\n\nT40 is ground\nT41 is ground\nX48 is free", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: rewrite(op(T40, T41), X48), SCOPE: 3, CLAUSE: 3\n\nT40 is ground\nT41 is ground\nX48 is free", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: rewrite(op(T77, T78), X90)\n\nT77 is ground\nT78 is ground\nX90 is free", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: normal(T5, T2), SCOPE: 1, CLAUSE: 1\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 27 [arrowhead = none ];
27 -> 2;

28 [label="ONLY EVAL with clause\nnormal(X3, X4) :- ','(rewrite(X3, X5), normal(X5, X4)).\nand substitutionT1 -> T5,\nX3 -> T5,\nT2 -> T7,\nX4 -> T7,\nT6 -> T7", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 28 [arrowhead = none ];
28 -> 3;

29 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 29 [arrowhead = none ];
29 -> 4;

30 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
4 -> 30 [arrowhead = none ];
30 -> 5;

31 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
4 -> 31 [arrowhead = none ];
31 -> 6;

32 [label="EVAL with clause\nrewrite(op(op(X18, X19), X20), op(X18, op(X19, X20))).\nand substitutionX18 -> T20,\nX19 -> T21,\nX20 -> T22,\nT5 -> op(op(T20, T21), T22),\nX5 -> op(T20, op(T21, T22))", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 32 [arrowhead = none ];
32 -> 7;

33 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 33 [arrowhead = none ];
33 -> 8;

34 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
6 -> 34 [arrowhead = none ];
34 -> 9;

35 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
6 -> 35 [arrowhead = none ];
35 -> 10;

36 [label="INSTANCE with matching:\nT1 -> op(T20, op(T21, T22))\nT2 -> T7", fontsize=14, style = filled, fillcolor = lightgrey];
7 -> 36 [arrowhead = none , style=dashed];
36 -> 1[style=dashed];

37 [label="EVAL with clause\nrewrite(op(X44, op(X45, X46)), op(X44, X47)) :- rewrite(op(X45, X46), X47).\nand substitutionX44 -> T39,\nX45 -> T40,\nX46 -> T41,\nT5 -> op(T39, op(T40, T41)),\nX47 -> X48,\nX5 -> op(T39, X48)", fontsize=14, style = filled, fillcolor = lightblue];
9 -> 37 [arrowhead = none ];
37 -> 11;

38 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 38 [arrowhead = none ];
38 -> 12;

39 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 39 [arrowhead = none ];
39 -> 23;

40 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
11 -> 40 [arrowhead = none ];
40 -> 13;

41 [label="SPLIT 2\nnew knowledge:\nT40 is ground\nT41 is ground\nT48 is ground\nreplacements:X48 -> T48", fontsize=14, style = filled, fillcolor = violet];
11 -> 41 [arrowhead = none ];
41 -> 14;

42 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
13 -> 42 [arrowhead = none ];
42 -> 15;

43 [label="INSTANCE with matching:\nT1 -> op(T39, T48)\nT2 -> T7", fontsize=14, style = filled, fillcolor = lightgrey];
14 -> 43 [arrowhead = none , style=dashed];
43 -> 1[style=dashed];

44 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
15 -> 44 [arrowhead = none ];
44 -> 16;

45 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
15 -> 45 [arrowhead = none ];
45 -> 17;

46 [label="EVAL with clause\nrewrite(op(op(X73, X74), X75), op(X73, op(X74, X75))).\nand substitutionX73 -> T67,\nX74 -> T68,\nT40 -> op(T67, T68),\nT41 -> T69,\nX75 -> T69,\nX48 -> op(T67, op(T68, T69))", fontsize=14, style = filled, fillcolor = lightblue];
16 -> 46 [arrowhead = none ];
46 -> 18;

47 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
16 -> 47 [arrowhead = none ];
47 -> 19;

48 [label="EVAL with clause\nrewrite(op(X86, op(X87, X88)), op(X86, X89)) :- rewrite(op(X87, X88), X89).\nand substitutionT40 -> T76,\nX86 -> T76,\nX87 -> T77,\nX88 -> T78,\nT41 -> op(T77, T78),\nX89 -> X90,\nX48 -> op(T76, X90)", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 48 [arrowhead = none ];
48 -> 21;

49 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
17 -> 49 [arrowhead = none ];
49 -> 22;

50 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
18 -> 50 [arrowhead = none ];
50 -> 20;

51 [label="INSTANCE with matching:\nT40 -> T77\nT41 -> T78\nX48 -> X90", fontsize=14, style = filled, fillcolor = lightgrey];
21 -> 51 [arrowhead = none , style=dashed];
51 -> 13[style=dashed];

52 [label="EVAL with clause\nnormal(X102, X102).\nand substitutionT5 -> T90,\nX102 -> T90,\nT2 -> T90", fontsize=14, style = filled, fillcolor = lightblue];
23 -> 52 [arrowhead = none ];
52 -> 24;

53 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
23 -> 53 [arrowhead = none ];
53 -> 25;

54 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
24 -> 54 [arrowhead = none ];
54 -> 26;

}

(2) Obligation:

Clauses:

rewriteA(op(T67, T68), T69, op(T67, op(T68, T69))).
rewriteA(T76, op(T77, T78), op(T76, X90)) :- rewriteA(T77, T78, X90).
normalB(op(op(T20, T21), T22), T7) :- normalB(op(T20, op(T21, T22)), T7).
normalB(op(T39, op(T40, T41)), T7) :- rewriteA(T40, T41, X48).
normalB(op(T39, op(T40, T41)), T7) :- ','(rewriteA(T40, T41, T48), normalB(op(T39, T48), T7)).
normalB(T90, T90).

Query: normalB(g,a)

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
normalB_in: (b,f)
rewriteA_in: (b,b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

normalB_in_ga(op(op(T20, T21), T22), T7) → U2_ga(T20, T21, T22, T7, normalB_in_ga(op(T20, op(T21, T22)), T7))
normalB_in_ga(op(T39, op(T40, T41)), T7) → U3_ga(T39, T40, T41, T7, rewriteA_in_gga(T40, T41, X48))
rewriteA_in_gga(op(T67, T68), T69, op(T67, op(T68, T69))) → rewriteA_out_gga(op(T67, T68), T69, op(T67, op(T68, T69)))
rewriteA_in_gga(T76, op(T77, T78), op(T76, X90)) → U1_gga(T76, T77, T78, X90, rewriteA_in_gga(T77, T78, X90))
U1_gga(T76, T77, T78, X90, rewriteA_out_gga(T77, T78, X90)) → rewriteA_out_gga(T76, op(T77, T78), op(T76, X90))
U3_ga(T39, T40, T41, T7, rewriteA_out_gga(T40, T41, X48)) → normalB_out_ga(op(T39, op(T40, T41)), T7)
normalB_in_ga(op(T39, op(T40, T41)), T7) → U4_ga(T39, T40, T41, T7, rewriteA_in_gga(T40, T41, T48))
U4_ga(T39, T40, T41, T7, rewriteA_out_gga(T40, T41, T48)) → U5_ga(T39, T40, T41, T7, normalB_in_ga(op(T39, T48), T7))
normalB_in_ga(T90, T90) → normalB_out_ga(T90, T90)
U5_ga(T39, T40, T41, T7, normalB_out_ga(op(T39, T48), T7)) → normalB_out_ga(op(T39, op(T40, T41)), T7)
U2_ga(T20, T21, T22, T7, normalB_out_ga(op(T20, op(T21, T22)), T7)) → normalB_out_ga(op(op(T20, T21), T22), T7)

The argument filtering Pi contains the following mapping:
normalB_in_ga(x1, x2)  =  normalB_in_ga(x1)
op(x1, x2)  =  op(x1, x2)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x5)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x5)
rewriteA_in_gga(x1, x2, x3)  =  rewriteA_in_gga(x1, x2)
rewriteA_out_gga(x1, x2, x3)  =  rewriteA_out_gga(x3)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x1, x5)
normalB_out_ga(x1, x2)  =  normalB_out_ga
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x1, x5)
U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x5)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

normalB_in_ga(op(op(T20, T21), T22), T7) → U2_ga(T20, T21, T22, T7, normalB_in_ga(op(T20, op(T21, T22)), T7))
normalB_in_ga(op(T39, op(T40, T41)), T7) → U3_ga(T39, T40, T41, T7, rewriteA_in_gga(T40, T41, X48))
rewriteA_in_gga(op(T67, T68), T69, op(T67, op(T68, T69))) → rewriteA_out_gga(op(T67, T68), T69, op(T67, op(T68, T69)))
rewriteA_in_gga(T76, op(T77, T78), op(T76, X90)) → U1_gga(T76, T77, T78, X90, rewriteA_in_gga(T77, T78, X90))
U1_gga(T76, T77, T78, X90, rewriteA_out_gga(T77, T78, X90)) → rewriteA_out_gga(T76, op(T77, T78), op(T76, X90))
U3_ga(T39, T40, T41, T7, rewriteA_out_gga(T40, T41, X48)) → normalB_out_ga(op(T39, op(T40, T41)), T7)
normalB_in_ga(op(T39, op(T40, T41)), T7) → U4_ga(T39, T40, T41, T7, rewriteA_in_gga(T40, T41, T48))
U4_ga(T39, T40, T41, T7, rewriteA_out_gga(T40, T41, T48)) → U5_ga(T39, T40, T41, T7, normalB_in_ga(op(T39, T48), T7))
normalB_in_ga(T90, T90) → normalB_out_ga(T90, T90)
U5_ga(T39, T40, T41, T7, normalB_out_ga(op(T39, T48), T7)) → normalB_out_ga(op(T39, op(T40, T41)), T7)
U2_ga(T20, T21, T22, T7, normalB_out_ga(op(T20, op(T21, T22)), T7)) → normalB_out_ga(op(op(T20, T21), T22), T7)

The argument filtering Pi contains the following mapping:
normalB_in_ga(x1, x2)  =  normalB_in_ga(x1)
op(x1, x2)  =  op(x1, x2)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x5)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x5)
rewriteA_in_gga(x1, x2, x3)  =  rewriteA_in_gga(x1, x2)
rewriteA_out_gga(x1, x2, x3)  =  rewriteA_out_gga(x3)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x1, x5)
normalB_out_ga(x1, x2)  =  normalB_out_ga
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x1, x5)
U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x5)

(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:

NORMALB_IN_GA(op(op(T20, T21), T22), T7) → U2_GA(T20, T21, T22, T7, normalB_in_ga(op(T20, op(T21, T22)), T7))
NORMALB_IN_GA(op(op(T20, T21), T22), T7) → NORMALB_IN_GA(op(T20, op(T21, T22)), T7)
NORMALB_IN_GA(op(T39, op(T40, T41)), T7) → U3_GA(T39, T40, T41, T7, rewriteA_in_gga(T40, T41, X48))
NORMALB_IN_GA(op(T39, op(T40, T41)), T7) → REWRITEA_IN_GGA(T40, T41, X48)
REWRITEA_IN_GGA(T76, op(T77, T78), op(T76, X90)) → U1_GGA(T76, T77, T78, X90, rewriteA_in_gga(T77, T78, X90))
REWRITEA_IN_GGA(T76, op(T77, T78), op(T76, X90)) → REWRITEA_IN_GGA(T77, T78, X90)
NORMALB_IN_GA(op(T39, op(T40, T41)), T7) → U4_GA(T39, T40, T41, T7, rewriteA_in_gga(T40, T41, T48))
U4_GA(T39, T40, T41, T7, rewriteA_out_gga(T40, T41, T48)) → U5_GA(T39, T40, T41, T7, normalB_in_ga(op(T39, T48), T7))
U4_GA(T39, T40, T41, T7, rewriteA_out_gga(T40, T41, T48)) → NORMALB_IN_GA(op(T39, T48), T7)

The TRS R consists of the following rules:

normalB_in_ga(op(op(T20, T21), T22), T7) → U2_ga(T20, T21, T22, T7, normalB_in_ga(op(T20, op(T21, T22)), T7))
normalB_in_ga(op(T39, op(T40, T41)), T7) → U3_ga(T39, T40, T41, T7, rewriteA_in_gga(T40, T41, X48))
rewriteA_in_gga(op(T67, T68), T69, op(T67, op(T68, T69))) → rewriteA_out_gga(op(T67, T68), T69, op(T67, op(T68, T69)))
rewriteA_in_gga(T76, op(T77, T78), op(T76, X90)) → U1_gga(T76, T77, T78, X90, rewriteA_in_gga(T77, T78, X90))
U1_gga(T76, T77, T78, X90, rewriteA_out_gga(T77, T78, X90)) → rewriteA_out_gga(T76, op(T77, T78), op(T76, X90))
U3_ga(T39, T40, T41, T7, rewriteA_out_gga(T40, T41, X48)) → normalB_out_ga(op(T39, op(T40, T41)), T7)
normalB_in_ga(op(T39, op(T40, T41)), T7) → U4_ga(T39, T40, T41, T7, rewriteA_in_gga(T40, T41, T48))
U4_ga(T39, T40, T41, T7, rewriteA_out_gga(T40, T41, T48)) → U5_ga(T39, T40, T41, T7, normalB_in_ga(op(T39, T48), T7))
normalB_in_ga(T90, T90) → normalB_out_ga(T90, T90)
U5_ga(T39, T40, T41, T7, normalB_out_ga(op(T39, T48), T7)) → normalB_out_ga(op(T39, op(T40, T41)), T7)
U2_ga(T20, T21, T22, T7, normalB_out_ga(op(T20, op(T21, T22)), T7)) → normalB_out_ga(op(op(T20, T21), T22), T7)

The argument filtering Pi contains the following mapping:
normalB_in_ga(x1, x2)  =  normalB_in_ga(x1)
op(x1, x2)  =  op(x1, x2)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x5)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x5)
rewriteA_in_gga(x1, x2, x3)  =  rewriteA_in_gga(x1, x2)
rewriteA_out_gga(x1, x2, x3)  =  rewriteA_out_gga(x3)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x1, x5)
normalB_out_ga(x1, x2)  =  normalB_out_ga
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x1, x5)
U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x5)
NORMALB_IN_GA(x1, x2)  =  NORMALB_IN_GA(x1)
U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x5)
U3_GA(x1, x2, x3, x4, x5)  =  U3_GA(x5)
REWRITEA_IN_GGA(x1, x2, x3)  =  REWRITEA_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4, x5)  =  U1_GGA(x1, x5)
U4_GA(x1, x2, x3, x4, x5)  =  U4_GA(x1, x5)
U5_GA(x1, x2, x3, x4, x5)  =  U5_GA(x5)

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

(6) Obligation:

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

NORMALB_IN_GA(op(op(T20, T21), T22), T7) → U2_GA(T20, T21, T22, T7, normalB_in_ga(op(T20, op(T21, T22)), T7))
NORMALB_IN_GA(op(op(T20, T21), T22), T7) → NORMALB_IN_GA(op(T20, op(T21, T22)), T7)
NORMALB_IN_GA(op(T39, op(T40, T41)), T7) → U3_GA(T39, T40, T41, T7, rewriteA_in_gga(T40, T41, X48))
NORMALB_IN_GA(op(T39, op(T40, T41)), T7) → REWRITEA_IN_GGA(T40, T41, X48)
REWRITEA_IN_GGA(T76, op(T77, T78), op(T76, X90)) → U1_GGA(T76, T77, T78, X90, rewriteA_in_gga(T77, T78, X90))
REWRITEA_IN_GGA(T76, op(T77, T78), op(T76, X90)) → REWRITEA_IN_GGA(T77, T78, X90)
NORMALB_IN_GA(op(T39, op(T40, T41)), T7) → U4_GA(T39, T40, T41, T7, rewriteA_in_gga(T40, T41, T48))
U4_GA(T39, T40, T41, T7, rewriteA_out_gga(T40, T41, T48)) → U5_GA(T39, T40, T41, T7, normalB_in_ga(op(T39, T48), T7))
U4_GA(T39, T40, T41, T7, rewriteA_out_gga(T40, T41, T48)) → NORMALB_IN_GA(op(T39, T48), T7)

The TRS R consists of the following rules:

normalB_in_ga(op(op(T20, T21), T22), T7) → U2_ga(T20, T21, T22, T7, normalB_in_ga(op(T20, op(T21, T22)), T7))
normalB_in_ga(op(T39, op(T40, T41)), T7) → U3_ga(T39, T40, T41, T7, rewriteA_in_gga(T40, T41, X48))
rewriteA_in_gga(op(T67, T68), T69, op(T67, op(T68, T69))) → rewriteA_out_gga(op(T67, T68), T69, op(T67, op(T68, T69)))
rewriteA_in_gga(T76, op(T77, T78), op(T76, X90)) → U1_gga(T76, T77, T78, X90, rewriteA_in_gga(T77, T78, X90))
U1_gga(T76, T77, T78, X90, rewriteA_out_gga(T77, T78, X90)) → rewriteA_out_gga(T76, op(T77, T78), op(T76, X90))
U3_ga(T39, T40, T41, T7, rewriteA_out_gga(T40, T41, X48)) → normalB_out_ga(op(T39, op(T40, T41)), T7)
normalB_in_ga(op(T39, op(T40, T41)), T7) → U4_ga(T39, T40, T41, T7, rewriteA_in_gga(T40, T41, T48))
U4_ga(T39, T40, T41, T7, rewriteA_out_gga(T40, T41, T48)) → U5_ga(T39, T40, T41, T7, normalB_in_ga(op(T39, T48), T7))
normalB_in_ga(T90, T90) → normalB_out_ga(T90, T90)
U5_ga(T39, T40, T41, T7, normalB_out_ga(op(T39, T48), T7)) → normalB_out_ga(op(T39, op(T40, T41)), T7)
U2_ga(T20, T21, T22, T7, normalB_out_ga(op(T20, op(T21, T22)), T7)) → normalB_out_ga(op(op(T20, T21), T22), T7)

The argument filtering Pi contains the following mapping:
normalB_in_ga(x1, x2)  =  normalB_in_ga(x1)
op(x1, x2)  =  op(x1, x2)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x5)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x5)
rewriteA_in_gga(x1, x2, x3)  =  rewriteA_in_gga(x1, x2)
rewriteA_out_gga(x1, x2, x3)  =  rewriteA_out_gga(x3)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x1, x5)
normalB_out_ga(x1, x2)  =  normalB_out_ga
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x1, x5)
U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x5)
NORMALB_IN_GA(x1, x2)  =  NORMALB_IN_GA(x1)
U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x5)
U3_GA(x1, x2, x3, x4, x5)  =  U3_GA(x5)
REWRITEA_IN_GGA(x1, x2, x3)  =  REWRITEA_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4, x5)  =  U1_GGA(x1, x5)
U4_GA(x1, x2, x3, x4, x5)  =  U4_GA(x1, x5)
U5_GA(x1, x2, x3, x4, x5)  =  U5_GA(x5)

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

(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:

REWRITEA_IN_GGA(T76, op(T77, T78), op(T76, X90)) → REWRITEA_IN_GGA(T77, T78, X90)

The TRS R consists of the following rules:

normalB_in_ga(op(op(T20, T21), T22), T7) → U2_ga(T20, T21, T22, T7, normalB_in_ga(op(T20, op(T21, T22)), T7))
normalB_in_ga(op(T39, op(T40, T41)), T7) → U3_ga(T39, T40, T41, T7, rewriteA_in_gga(T40, T41, X48))
rewriteA_in_gga(op(T67, T68), T69, op(T67, op(T68, T69))) → rewriteA_out_gga(op(T67, T68), T69, op(T67, op(T68, T69)))
rewriteA_in_gga(T76, op(T77, T78), op(T76, X90)) → U1_gga(T76, T77, T78, X90, rewriteA_in_gga(T77, T78, X90))
U1_gga(T76, T77, T78, X90, rewriteA_out_gga(T77, T78, X90)) → rewriteA_out_gga(T76, op(T77, T78), op(T76, X90))
U3_ga(T39, T40, T41, T7, rewriteA_out_gga(T40, T41, X48)) → normalB_out_ga(op(T39, op(T40, T41)), T7)
normalB_in_ga(op(T39, op(T40, T41)), T7) → U4_ga(T39, T40, T41, T7, rewriteA_in_gga(T40, T41, T48))
U4_ga(T39, T40, T41, T7, rewriteA_out_gga(T40, T41, T48)) → U5_ga(T39, T40, T41, T7, normalB_in_ga(op(T39, T48), T7))
normalB_in_ga(T90, T90) → normalB_out_ga(T90, T90)
U5_ga(T39, T40, T41, T7, normalB_out_ga(op(T39, T48), T7)) → normalB_out_ga(op(T39, op(T40, T41)), T7)
U2_ga(T20, T21, T22, T7, normalB_out_ga(op(T20, op(T21, T22)), T7)) → normalB_out_ga(op(op(T20, T21), T22), T7)

The argument filtering Pi contains the following mapping:
normalB_in_ga(x1, x2)  =  normalB_in_ga(x1)
op(x1, x2)  =  op(x1, x2)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x5)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x5)
rewriteA_in_gga(x1, x2, x3)  =  rewriteA_in_gga(x1, x2)
rewriteA_out_gga(x1, x2, x3)  =  rewriteA_out_gga(x3)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x1, x5)
normalB_out_ga(x1, x2)  =  normalB_out_ga
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x1, x5)
U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x5)
REWRITEA_IN_GGA(x1, x2, x3)  =  REWRITEA_IN_GGA(x1, x2)

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

(10) UsableRulesProof (EQUIVALENT transformation)

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

(11) Obligation:

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

REWRITEA_IN_GGA(T76, op(T77, T78), op(T76, X90)) → REWRITEA_IN_GGA(T77, T78, X90)

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

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

(12) PiDPToQDPProof (SOUND transformation)

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

(13) Obligation:

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

REWRITEA_IN_GGA(T76, op(T77, T78)) → REWRITEA_IN_GGA(T77, T78)

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:

• REWRITEA_IN_GGA(T76, op(T77, T78)) → REWRITEA_IN_GGA(T77, T78)
  The graph contains the following edges 2 > 1, 2 > 2

(15) YES

(16) Obligation:

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

NORMALB_IN_GA(op(T39, op(T40, T41)), T7) → U4_GA(T39, T40, T41, T7, rewriteA_in_gga(T40, T41, T48))
U4_GA(T39, T40, T41, T7, rewriteA_out_gga(T40, T41, T48)) → NORMALB_IN_GA(op(T39, T48), T7)
NORMALB_IN_GA(op(op(T20, T21), T22), T7) → NORMALB_IN_GA(op(T20, op(T21, T22)), T7)

The TRS R consists of the following rules:

normalB_in_ga(op(op(T20, T21), T22), T7) → U2_ga(T20, T21, T22, T7, normalB_in_ga(op(T20, op(T21, T22)), T7))
normalB_in_ga(op(T39, op(T40, T41)), T7) → U3_ga(T39, T40, T41, T7, rewriteA_in_gga(T40, T41, X48))
rewriteA_in_gga(op(T67, T68), T69, op(T67, op(T68, T69))) → rewriteA_out_gga(op(T67, T68), T69, op(T67, op(T68, T69)))
rewriteA_in_gga(T76, op(T77, T78), op(T76, X90)) → U1_gga(T76, T77, T78, X90, rewriteA_in_gga(T77, T78, X90))
U1_gga(T76, T77, T78, X90, rewriteA_out_gga(T77, T78, X90)) → rewriteA_out_gga(T76, op(T77, T78), op(T76, X90))
U3_ga(T39, T40, T41, T7, rewriteA_out_gga(T40, T41, X48)) → normalB_out_ga(op(T39, op(T40, T41)), T7)
normalB_in_ga(op(T39, op(T40, T41)), T7) → U4_ga(T39, T40, T41, T7, rewriteA_in_gga(T40, T41, T48))
U4_ga(T39, T40, T41, T7, rewriteA_out_gga(T40, T41, T48)) → U5_ga(T39, T40, T41, T7, normalB_in_ga(op(T39, T48), T7))
normalB_in_ga(T90, T90) → normalB_out_ga(T90, T90)
U5_ga(T39, T40, T41, T7, normalB_out_ga(op(T39, T48), T7)) → normalB_out_ga(op(T39, op(T40, T41)), T7)
U2_ga(T20, T21, T22, T7, normalB_out_ga(op(T20, op(T21, T22)), T7)) → normalB_out_ga(op(op(T20, T21), T22), T7)

The argument filtering Pi contains the following mapping:
normalB_in_ga(x1, x2)  =  normalB_in_ga(x1)
op(x1, x2)  =  op(x1, x2)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x5)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x5)
rewriteA_in_gga(x1, x2, x3)  =  rewriteA_in_gga(x1, x2)
rewriteA_out_gga(x1, x2, x3)  =  rewriteA_out_gga(x3)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x1, x5)
normalB_out_ga(x1, x2)  =  normalB_out_ga
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x1, x5)
U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x5)
NORMALB_IN_GA(x1, x2)  =  NORMALB_IN_GA(x1)
U4_GA(x1, x2, x3, x4, x5)  =  U4_GA(x1, x5)

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:

NORMALB_IN_GA(op(T39, op(T40, T41)), T7) → U4_GA(T39, T40, T41, T7, rewriteA_in_gga(T40, T41, T48))
U4_GA(T39, T40, T41, T7, rewriteA_out_gga(T40, T41, T48)) → NORMALB_IN_GA(op(T39, T48), T7)
NORMALB_IN_GA(op(op(T20, T21), T22), T7) → NORMALB_IN_GA(op(T20, op(T21, T22)), T7)

The TRS R consists of the following rules:

rewriteA_in_gga(op(T67, T68), T69, op(T67, op(T68, T69))) → rewriteA_out_gga(op(T67, T68), T69, op(T67, op(T68, T69)))
rewriteA_in_gga(T76, op(T77, T78), op(T76, X90)) → U1_gga(T76, T77, T78, X90, rewriteA_in_gga(T77, T78, X90))
U1_gga(T76, T77, T78, X90, rewriteA_out_gga(T77, T78, X90)) → rewriteA_out_gga(T76, op(T77, T78), op(T76, X90))

The argument filtering Pi contains the following mapping:
op(x1, x2)  =  op(x1, x2)
rewriteA_in_gga(x1, x2, x3)  =  rewriteA_in_gga(x1, x2)
rewriteA_out_gga(x1, x2, x3)  =  rewriteA_out_gga(x3)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x1, x5)
NORMALB_IN_GA(x1, x2)  =  NORMALB_IN_GA(x1)
U4_GA(x1, x2, x3, x4, x5)  =  U4_GA(x1, x5)

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:

NORMALB_IN_GA(op(T39, op(T40, T41))) → U4_GA(T39, rewriteA_in_gga(T40, T41))
U4_GA(T39, rewriteA_out_gga(T48)) → NORMALB_IN_GA(op(T39, T48))
NORMALB_IN_GA(op(op(T20, T21), T22)) → NORMALB_IN_GA(op(T20, op(T21, T22)))

The TRS R consists of the following rules:

rewriteA_in_gga(op(T67, T68), T69) → rewriteA_out_gga(op(T67, op(T68, T69)))
rewriteA_in_gga(T76, op(T77, T78)) → U1_gga(T76, rewriteA_in_gga(T77, T78))
U1_gga(T76, rewriteA_out_gga(X90)) → rewriteA_out_gga(op(T76, X90))

The set Q consists of the following terms:

rewriteA_in_gga(x0, x1)
U1_gga(x0, x1)

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

(21) 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:

NORMALB_IN_GA(op(T39, op(T40, T41))) → U4_GA(T39, rewriteA_in_gga(T40, T41))
NORMALB_IN_GA(op(op(T20, T21), T22)) → NORMALB_IN_GA(op(T20, op(T21, T22)))

Used ordering: Polynomial interpretation [POLO]:

POL(NORMALB_IN_GA(x₁)) = x₁   
POL(U1_gga(x₁, x₂)) = 2 + 2·x₁ + x₂   
POL(U4_GA(x₁, x₂)) = 2 + 2·x₁ + x₂   
POL(op(x₁, x₂)) = 2 + 2·x₁ + x₂   
POL(rewriteA_in_gga(x₁, x₂)) = 2·x₁ + x₂   
POL(rewriteA_out_gga(x₁)) = x₁   

(22) Obligation:

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

U4_GA(T39, rewriteA_out_gga(T48)) → NORMALB_IN_GA(op(T39, T48))

The TRS R consists of the following rules:

rewriteA_in_gga(op(T67, T68), T69) → rewriteA_out_gga(op(T67, op(T68, T69)))
rewriteA_in_gga(T76, op(T77, T78)) → U1_gga(T76, rewriteA_in_gga(T77, T78))
U1_gga(T76, rewriteA_out_gga(X90)) → rewriteA_out_gga(op(T76, X90))

The set Q consists of the following terms:

rewriteA_in_gga(x0, x1)
U1_gga(x0, x1)

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

(23) DependencyGraphProof (EQUIVALENT transformation)

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

(24) TRUE