TIMEOUT

The TRS could not be proven terminating. The proof attempt took 60000 ms.

The following DP Processors were used


Problem 1 was processed with processor ForwardNarrowing (3ms).
 | – Problem 2 was processed with processor ForwardNarrowing (4ms).
 |    | – Problem 3 was processed with processor ForwardNarrowing (5ms).
 |    |    | – Problem 4 was processed with processor ForwardNarrowing (2ms).
 |    |    |    | – Problem 5 was processed with processor ForwardNarrowing (1ms).
 |    |    |    |    | – Problem 6 was processed with processor ForwardNarrowing (1ms).
 |    |    |    |    |    | – Problem 7 was processed with processor ForwardNarrowing (6ms).
 |    |    |    |    |    |    | – Problem 8 was processed with processor ForwardNarrowing (2ms).
 |    |    |    |    |    |    |    | – Problem 9 was processed with processor ForwardNarrowing (4ms).
 |    |    |    |    |    |    |    |    | – Problem 10 was processed with processor ForwardNarrowing (3ms).
 |    |    |    |    |    |    |    |    |    | – Problem 11 was processed with processor ForwardNarrowing (3ms).
 |    |    |    |    |    |    |    |    |    |    | – Problem 12 was processed with processor ForwardNarrowing (3ms).
 |    |    |    |    |    |    |    |    |    |    |    | – Problem 13 was processed with processor ForwardNarrowing (3ms).
 |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 14 was processed with processor ForwardNarrowing (2ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 15 was processed with processor ForwardNarrowing (5ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 16 was processed with processor ForwardNarrowing (4ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 17 was processed with processor ForwardNarrowing (2ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 18 was processed with processor ForwardNarrowing (2ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 19 was processed with processor ForwardNarrowing (4ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 20 was processed with processor ForwardNarrowing (6ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 21 was processed with processor ForwardNarrowing (10ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 22 was processed with processor ForwardNarrowing (127ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 23 was processed with processor ForwardNarrowing (147ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 24 was processed with processor ForwardNarrowing (222ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 25 remains open; application of the following processors failed [ForwardNarrowing (111ms), ForwardNarrowing (134ms), ForwardNarrowing (224ms), ForwardNarrowing (137ms), ForwardNarrowing (216ms), ForwardNarrowing (145ms), ForwardNarrowing (303ms), ForwardNarrowing (196ms), ForwardNarrowing (173ms), ForwardNarrowing (177ms), ForwardNarrowing (235ms), ForwardNarrowing (258ms), ForwardNarrowing (304ms), ForwardNarrowing (228ms), ForwardNarrowing (209ms), ForwardNarrowing (219ms), ForwardNarrowing (262ms), ForwardNarrowing (258ms), ForwardNarrowing (316ms), ForwardNarrowing (249ms), ForwardNarrowing (283ms), ForwardNarrowing (87ms), ForwardNarrowing (126ms), ForwardNarrowing (237ms), ForwardNarrowing (83ms), ForwardNarrowing (189ms), ForwardNarrowing (274ms), ForwardNarrowing (185ms), ForwardNarrowing (193ms), ForwardNarrowing (262ms), ForwardNarrowing (217ms), ForwardNarrowing (268ms), ForwardNarrowing (207ms), ForwardNarrowing (230ms), ForwardNarrowing (211ms), ForwardNarrowing (223ms), ForwardNarrowing (217ms), ForwardNarrowing (timeout)].

The following open problems remain:



Open Dependency Pair Problem 1

Dependency Pairs

app#(inc, xs)app#(map, app(plus, app(s, 0)))app#(app(map, f), app(app(cons, x), xs))app#(f, x)
app#(app(map, f), app(app(cons, x), xs))app#(app(cons, app(f, x)), app(app(map, f), xs))app#(app(plus, app(s, x)), y)app#(s, app(app(plus, x), y))
app#(inc, xs)app#(s, 0)app#(app(plus, app(s, x)), y)app#(plus, x)
app#(app(map, f), app(app(cons, x), xs))app#(app(map, f), xs)app#(app(plus, app(s, x)), y)app#(app(plus, x), y)
app#(app(map, f), app(app(cons, x), xs))app#(map, f)app#(app(map, f), app(app(cons, x), xs))app#(cons, app(f, x))
app#(inc, xs)app#(app(map, app(plus, app(s, 0))), xs)app#(inc, xs)app#(plus, app(s, 0))

Rewrite Rules

app(app(plus, 0), y)yapp(app(plus, app(s, x)), y)app(s, app(app(plus, x), y))
app(inc, xs)app(app(map, app(plus, app(s, 0))), xs)app(app(map, f), nil)nil
app(app(map, f), app(app(cons, x), xs))app(app(cons, app(f, x)), app(app(map, f), xs))

Original Signature

Termination of terms over the following signature is verified: plus, app, 0, inc, s, map, cons, nil


Problem 1: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

app#(inc, xs)app#(map, app(plus, app(s, 0)))app#(app(map, f), app(app(cons, x), xs))app#(app(cons, app(f, x)), app(app(map, f), xs))
app#(app(map, f), app(app(cons, x), xs))app#(f, x)app#(app(plus, app(s, x)), y)app#(s, app(app(plus, x), y))
app#(inc, xs)app#(s, 0)app#(app(plus, app(s, x)), y)app#(plus, x)
app#(app(map, f), app(app(cons, x), xs))app#(app(map, f), xs)app#(app(plus, app(s, x)), y)app#(app(plus, x), y)
app#(inc, xs)app#(app(map, app(plus, app(s, 0))), xs)app#(app(map, f), app(app(cons, x), xs))app#(cons, app(f, x))
app#(app(map, f), app(app(cons, x), xs))app#(map, f)app#(inc, xs)app#(plus, app(s, 0))

Rewrite Rules

app(app(plus, 0), y)yapp(app(plus, app(s, x)), y)app(s, app(app(plus, x), y))
app(inc, xs)app(app(map, app(plus, app(s, 0))), xs)app(app(map, f), nil)nil
app(app(map, f), app(app(cons, x), xs))app(app(cons, app(f, x)), app(app(map, f), xs))

Original Signature

Termination of terms over the following signature is verified: app, plus, 0, s, inc, map, nil, cons

Strategy


The right-hand side of the rule app#(inc, xs) → app#(map, app(plus, app(s, 0))) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
Thus, the rule app#(inc, xs) → app#(map, app(plus, app(s, 0))) is deleted.

Problem 2: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

app#(app(map, f), app(app(cons, x), xs))app#(f, x)app#(app(map, f), app(app(cons, x), xs))app#(app(cons, app(f, x)), app(app(map, f), xs))
app#(app(plus, app(s, x)), y)app#(s, app(app(plus, x), y))app#(inc, xs)app#(s, 0)
app#(app(plus, app(s, x)), y)app#(plus, x)app#(app(map, f), app(app(cons, x), xs))app#(app(map, f), xs)
app#(app(plus, app(s, x)), y)app#(app(plus, x), y)app#(app(map, f), app(app(cons, x), xs))app#(map, f)
app#(app(map, f), app(app(cons, x), xs))app#(cons, app(f, x))app#(inc, xs)app#(app(map, app(plus, app(s, 0))), xs)
app#(inc, xs)app#(plus, app(s, 0))

Rewrite Rules

app(app(plus, 0), y)yapp(app(plus, app(s, x)), y)app(s, app(app(plus, x), y))
app(inc, xs)app(app(map, app(plus, app(s, 0))), xs)app(app(map, f), nil)nil
app(app(map, f), app(app(cons, x), xs))app(app(cons, app(f, x)), app(app(map, f), xs))

Original Signature

Termination of terms over the following signature is verified: plus, app, 0, inc, s, map, cons, nil

Strategy


The right-hand side of the rule app#(app(plus, app(s, x)), y) → app#(s, app(app(plus, x), y)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
app#(s, _x31) 
app#(s, app(s, app(app(plus, _x32), _x31))) 
Thus, the rule app#(app(plus, app(s, x)), y) → app#(s, app(app(plus, x), y)) is replaced by the following rules:
app#(app(plus, app(s, app(s, _x32))), _x31) → app#(s, app(s, app(app(plus, _x32), _x31)))app#(app(plus, app(s, 0)), _x31) → app#(s, _x31)

Problem 3: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

app#(app(plus, app(s, app(s, _x32))), _x31)app#(s, app(s, app(app(plus, _x32), _x31)))app#(app(map, f), app(app(cons, x), xs))app#(app(cons, app(f, x)), app(app(map, f), xs))
app#(app(map, f), app(app(cons, x), xs))app#(f, x)app#(inc, xs)app#(s, 0)
app#(app(plus, app(s, x)), y)app#(plus, x)app#(app(map, f), app(app(cons, x), xs))app#(app(map, f), xs)
app#(app(plus, app(s, x)), y)app#(app(plus, x), y)app#(inc, xs)app#(app(map, app(plus, app(s, 0))), xs)
app#(app(map, f), app(app(cons, x), xs))app#(cons, app(f, x))app#(app(map, f), app(app(cons, x), xs))app#(map, f)
app#(inc, xs)app#(plus, app(s, 0))app#(app(plus, app(s, 0)), _x31)app#(s, _x31)

Rewrite Rules

app(app(plus, 0), y)yapp(app(plus, app(s, x)), y)app(s, app(app(plus, x), y))
app(inc, xs)app(app(map, app(plus, app(s, 0))), xs)app(app(map, f), nil)nil
app(app(map, f), app(app(cons, x), xs))app(app(cons, app(f, x)), app(app(map, f), xs))

Original Signature

Termination of terms over the following signature is verified: app, plus, 0, s, inc, map, nil, cons

Strategy


The right-hand side of the rule app#(app(plus, app(s, app(s, _x32))), _x31) → app#(s, app(s, app(app(plus, _x32), _x31))) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
app#(s, app(s, _x61)) 
app#(s, app(s, app(s, app(app(plus, _x62), _x61)))) 
Thus, the rule app#(app(plus, app(s, app(s, _x32))), _x31) → app#(s, app(s, app(app(plus, _x32), _x31))) is replaced by the following rules:
app#(app(plus, app(s, app(s, 0))), _x61) → app#(s, app(s, _x61))app#(app(plus, app(s, app(s, app(s, _x62)))), _x61) → app#(s, app(s, app(s, app(app(plus, _x62), _x61))))

Problem 4: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

app#(app(map, f), app(app(cons, x), xs))app#(f, x)app#(app(map, f), app(app(cons, x), xs))app#(app(cons, app(f, x)), app(app(map, f), xs))
app#(app(plus, app(s, app(s, 0))), _x61)app#(s, app(s, _x61))app#(app(plus, app(s, x)), y)app#(plus, x)
app#(app(map, f), app(app(cons, x), xs))app#(app(map, f), xs)app#(inc, xs)app#(app(map, app(plus, app(s, 0))), xs)
app#(app(plus, app(s, app(s, app(s, _x62)))), _x61)app#(s, app(s, app(s, app(app(plus, _x62), _x61))))app#(app(plus, app(s, 0)), _x31)app#(s, _x31)
app#(inc, xs)app#(plus, app(s, 0))app#(inc, xs)app#(s, 0)
app#(app(plus, app(s, x)), y)app#(app(plus, x), y)app#(app(map, f), app(app(cons, x), xs))app#(map, f)
app#(app(map, f), app(app(cons, x), xs))app#(cons, app(f, x))

Rewrite Rules

app(app(plus, 0), y)yapp(app(plus, app(s, x)), y)app(s, app(app(plus, x), y))
app(inc, xs)app(app(map, app(plus, app(s, 0))), xs)app(app(map, f), nil)nil
app(app(map, f), app(app(cons, x), xs))app(app(cons, app(f, x)), app(app(map, f), xs))

Original Signature

Termination of terms over the following signature is verified: plus, app, 0, inc, s, map, cons, nil

Strategy


The right-hand side of the rule app#(app(plus, app(s, app(s, 0))), _x61) → app#(s, app(s, _x61)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
Thus, the rule app#(app(plus, app(s, app(s, 0))), _x61) → app#(s, app(s, _x61)) is deleted.

Problem 5: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

app#(app(map, f), app(app(cons, x), xs))app#(app(cons, app(f, x)), app(app(map, f), xs))app#(app(map, f), app(app(cons, x), xs))app#(f, x)
app#(inc, xs)app#(s, 0)app#(app(plus, app(s, x)), y)app#(plus, x)
app#(app(map, f), app(app(cons, x), xs))app#(app(map, f), xs)app#(app(plus, app(s, x)), y)app#(app(plus, x), y)
app#(app(map, f), app(app(cons, x), xs))app#(cons, app(f, x))app#(app(map, f), app(app(cons, x), xs))app#(map, f)
app#(app(plus, app(s, app(s, app(s, _x62)))), _x61)app#(s, app(s, app(s, app(app(plus, _x62), _x61))))app#(inc, xs)app#(app(map, app(plus, app(s, 0))), xs)
app#(inc, xs)app#(plus, app(s, 0))app#(app(plus, app(s, 0)), _x31)app#(s, _x31)

Rewrite Rules

app(app(plus, 0), y)yapp(app(plus, app(s, x)), y)app(s, app(app(plus, x), y))
app(inc, xs)app(app(map, app(plus, app(s, 0))), xs)app(app(map, f), nil)nil
app(app(map, f), app(app(cons, x), xs))app(app(cons, app(f, x)), app(app(map, f), xs))

Original Signature

Termination of terms over the following signature is verified: app, plus, 0, s, inc, map, nil, cons

Strategy


The right-hand side of the rule app#(app(plus, app(s, x)), y) → app#(plus, x) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
Thus, the rule app#(app(plus, app(s, x)), y) → app#(plus, x) is deleted.

Problem 6: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

app#(app(map, f), app(app(cons, x), xs))app#(f, x)app#(app(map, f), app(app(cons, x), xs))app#(app(cons, app(f, x)), app(app(map, f), xs))
app#(inc, xs)app#(s, 0)app#(app(map, f), app(app(cons, x), xs))app#(app(map, f), xs)
app#(app(plus, app(s, x)), y)app#(app(plus, x), y)app#(inc, xs)app#(app(map, app(plus, app(s, 0))), xs)
app#(app(plus, app(s, app(s, app(s, _x62)))), _x61)app#(s, app(s, app(s, app(app(plus, _x62), _x61))))app#(app(map, f), app(app(cons, x), xs))app#(map, f)
app#(app(map, f), app(app(cons, x), xs))app#(cons, app(f, x))app#(app(plus, app(s, 0)), _x31)app#(s, _x31)
app#(inc, xs)app#(plus, app(s, 0))

Rewrite Rules

app(app(plus, 0), y)yapp(app(plus, app(s, x)), y)app(s, app(app(plus, x), y))
app(inc, xs)app(app(map, app(plus, app(s, 0))), xs)app(app(map, f), nil)nil
app(app(map, f), app(app(cons, x), xs))app(app(cons, app(f, x)), app(app(map, f), xs))

Original Signature

Termination of terms over the following signature is verified: plus, app, 0, inc, s, map, cons, nil

Strategy


The right-hand side of the rule app#(inc, xs) → app#(s, 0) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
Thus, the rule app#(inc, xs) → app#(s, 0) is deleted.

Problem 7: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

app#(app(map, f), app(app(cons, x), xs))app#(app(cons, app(f, x)), app(app(map, f), xs))app#(app(map, f), app(app(cons, x), xs))app#(f, x)
app#(app(map, f), app(app(cons, x), xs))app#(app(map, f), xs)app#(app(plus, app(s, x)), y)app#(app(plus, x), y)
app#(app(map, f), app(app(cons, x), xs))app#(cons, app(f, x))app#(app(map, f), app(app(cons, x), xs))app#(map, f)
app#(app(plus, app(s, app(s, app(s, _x62)))), _x61)app#(s, app(s, app(s, app(app(plus, _x62), _x61))))app#(inc, xs)app#(app(map, app(plus, app(s, 0))), xs)
app#(inc, xs)app#(plus, app(s, 0))app#(app(plus, app(s, 0)), _x31)app#(s, _x31)

Rewrite Rules

app(app(plus, 0), y)yapp(app(plus, app(s, x)), y)app(s, app(app(plus, x), y))
app(inc, xs)app(app(map, app(plus, app(s, 0))), xs)app(app(map, f), nil)nil
app(app(map, f), app(app(cons, x), xs))app(app(cons, app(f, x)), app(app(map, f), xs))

Original Signature

Termination of terms over the following signature is verified: app, plus, 0, s, inc, map, nil, cons

Strategy


The right-hand side of the rule app#(app(plus, app(s, app(s, app(s, _x62)))), _x61) → app#(s, app(s, app(s, app(app(plus, _x62), _x61)))) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
app#(s, app(s, app(s, app(s, app(app(plus, _x72), _x71))))) 
app#(s, app(s, app(s, _x71))) 
Thus, the rule app#(app(plus, app(s, app(s, app(s, _x62)))), _x61) → app#(s, app(s, app(s, app(app(plus, _x62), _x61)))) is replaced by the following rules:
app#(app(plus, app(s, app(s, app(s, 0)))), _x71) → app#(s, app(s, app(s, _x71)))app#(app(plus, app(s, app(s, app(s, app(s, _x72))))), _x71) → app#(s, app(s, app(s, app(s, app(app(plus, _x72), _x71)))))

Problem 8: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

app#(app(plus, app(s, app(s, app(s, 0)))), _x71)app#(s, app(s, app(s, _x71)))app#(app(map, f), app(app(cons, x), xs))app#(f, x)
app#(app(map, f), app(app(cons, x), xs))app#(app(cons, app(f, x)), app(app(map, f), xs))app#(app(plus, app(s, app(s, app(s, app(s, _x72))))), _x71)app#(s, app(s, app(s, app(s, app(app(plus, _x72), _x71)))))
app#(app(map, f), app(app(cons, x), xs))app#(app(map, f), xs)app#(app(plus, app(s, x)), y)app#(app(plus, x), y)
app#(inc, xs)app#(app(map, app(plus, app(s, 0))), xs)app#(app(map, f), app(app(cons, x), xs))app#(map, f)
app#(app(map, f), app(app(cons, x), xs))app#(cons, app(f, x))app#(app(plus, app(s, 0)), _x31)app#(s, _x31)
app#(inc, xs)app#(plus, app(s, 0))

Rewrite Rules

app(app(plus, 0), y)yapp(app(plus, app(s, x)), y)app(s, app(app(plus, x), y))
app(inc, xs)app(app(map, app(plus, app(s, 0))), xs)app(app(map, f), nil)nil
app(app(map, f), app(app(cons, x), xs))app(app(cons, app(f, x)), app(app(map, f), xs))

Original Signature

Termination of terms over the following signature is verified: plus, app, 0, inc, s, map, cons, nil

Strategy


The right-hand side of the rule app#(app(plus, app(s, app(s, app(s, 0)))), _x71) → app#(s, app(s, app(s, _x71))) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
Thus, the rule app#(app(plus, app(s, app(s, app(s, 0)))), _x71) → app#(s, app(s, app(s, _x71))) is deleted.

Problem 9: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

app#(app(map, f), app(app(cons, x), xs))app#(app(cons, app(f, x)), app(app(map, f), xs))app#(app(map, f), app(app(cons, x), xs))app#(f, x)
app#(app(plus, app(s, app(s, app(s, app(s, _x72))))), _x71)app#(s, app(s, app(s, app(s, app(app(plus, _x72), _x71)))))app#(app(map, f), app(app(cons, x), xs))app#(app(map, f), xs)
app#(app(plus, app(s, x)), y)app#(app(plus, x), y)app#(app(map, f), app(app(cons, x), xs))app#(cons, app(f, x))
app#(app(map, f), app(app(cons, x), xs))app#(map, f)app#(inc, xs)app#(app(map, app(plus, app(s, 0))), xs)
app#(inc, xs)app#(plus, app(s, 0))app#(app(plus, app(s, 0)), _x31)app#(s, _x31)

Rewrite Rules

app(app(plus, 0), y)yapp(app(plus, app(s, x)), y)app(s, app(app(plus, x), y))
app(inc, xs)app(app(map, app(plus, app(s, 0))), xs)app(app(map, f), nil)nil
app(app(map, f), app(app(cons, x), xs))app(app(cons, app(f, x)), app(app(map, f), xs))

Original Signature

Termination of terms over the following signature is verified: app, plus, 0, s, inc, map, nil, cons

Strategy


The right-hand side of the rule app#(app(plus, app(s, app(s, app(s, app(s, _x72))))), _x71) → app#(s, app(s, app(s, app(s, app(app(plus, _x72), _x71))))) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
app#(s, app(s, app(s, app(s, _x101)))) 
app#(s, app(s, app(s, app(s, app(s, app(app(plus, _x102), _x101)))))) 
Thus, the rule app#(app(plus, app(s, app(s, app(s, app(s, _x72))))), _x71) → app#(s, app(s, app(s, app(s, app(app(plus, _x72), _x71))))) is replaced by the following rules:
app#(app(plus, app(s, app(s, app(s, app(s, 0))))), _x101) → app#(s, app(s, app(s, app(s, _x101))))app#(app(plus, app(s, app(s, app(s, app(s, app(s, _x102)))))), _x101) → app#(s, app(s, app(s, app(s, app(s, app(app(plus, _x102), _x101))))))

Problem 10: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

app#(app(plus, app(s, app(s, app(s, app(s, 0))))), _x101)app#(s, app(s, app(s, app(s, _x101))))app#(app(map, f), app(app(cons, x), xs))app#(f, x)
app#(app(map, f), app(app(cons, x), xs))app#(app(cons, app(f, x)), app(app(map, f), xs))app#(app(plus, app(s, app(s, app(s, app(s, app(s, _x102)))))), _x101)app#(s, app(s, app(s, app(s, app(s, app(app(plus, _x102), _x101))))))
app#(app(map, f), app(app(cons, x), xs))app#(app(map, f), xs)app#(app(plus, app(s, x)), y)app#(app(plus, x), y)
app#(inc, xs)app#(app(map, app(plus, app(s, 0))), xs)app#(app(map, f), app(app(cons, x), xs))app#(map, f)
app#(app(map, f), app(app(cons, x), xs))app#(cons, app(f, x))app#(app(plus, app(s, 0)), _x31)app#(s, _x31)
app#(inc, xs)app#(plus, app(s, 0))

Rewrite Rules

app(app(plus, 0), y)yapp(app(plus, app(s, x)), y)app(s, app(app(plus, x), y))
app(inc, xs)app(app(map, app(plus, app(s, 0))), xs)app(app(map, f), nil)nil
app(app(map, f), app(app(cons, x), xs))app(app(cons, app(f, x)), app(app(map, f), xs))

Original Signature

Termination of terms over the following signature is verified: plus, app, 0, inc, s, map, cons, nil

Strategy


The right-hand side of the rule app#(app(plus, app(s, app(s, app(s, app(s, 0))))), _x101) → app#(s, app(s, app(s, app(s, _x101)))) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
Thus, the rule app#(app(plus, app(s, app(s, app(s, app(s, 0))))), _x101) → app#(s, app(s, app(s, app(s, _x101)))) is deleted.

Problem 11: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

app#(app(map, f), app(app(cons, x), xs))app#(app(cons, app(f, x)), app(app(map, f), xs))app#(app(map, f), app(app(cons, x), xs))app#(f, x)
app#(app(plus, app(s, app(s, app(s, app(s, app(s, _x102)))))), _x101)app#(s, app(s, app(s, app(s, app(s, app(app(plus, _x102), _x101))))))app#(app(map, f), app(app(cons, x), xs))app#(app(map, f), xs)
app#(app(plus, app(s, x)), y)app#(app(plus, x), y)app#(app(map, f), app(app(cons, x), xs))app#(cons, app(f, x))
app#(app(map, f), app(app(cons, x), xs))app#(map, f)app#(inc, xs)app#(app(map, app(plus, app(s, 0))), xs)
app#(inc, xs)app#(plus, app(s, 0))app#(app(plus, app(s, 0)), _x31)app#(s, _x31)

Rewrite Rules

app(app(plus, 0), y)yapp(app(plus, app(s, x)), y)app(s, app(app(plus, x), y))
app(inc, xs)app(app(map, app(plus, app(s, 0))), xs)app(app(map, f), nil)nil
app(app(map, f), app(app(cons, x), xs))app(app(cons, app(f, x)), app(app(map, f), xs))

Original Signature

Termination of terms over the following signature is verified: app, plus, 0, s, inc, map, nil, cons

Strategy


The right-hand side of the rule app#(app(plus, app(s, app(s, app(s, app(s, app(s, _x102)))))), _x101) → app#(s, app(s, app(s, app(s, app(s, app(app(plus, _x102), _x101)))))) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
app#(s, app(s, app(s, app(s, app(s, app(s, app(app(plus, _x112), _x111))))))) 
app#(s, app(s, app(s, app(s, app(s, _x111))))) 
Thus, the rule app#(app(plus, app(s, app(s, app(s, app(s, app(s, _x102)))))), _x101) → app#(s, app(s, app(s, app(s, app(s, app(app(plus, _x102), _x101)))))) is replaced by the following rules:
app#(app(plus, app(s, app(s, app(s, app(s, app(s, 0)))))), _x111) → app#(s, app(s, app(s, app(s, app(s, _x111)))))app#(app(plus, app(s, app(s, app(s, app(s, app(s, app(s, _x112))))))), _x111) → app#(s, app(s, app(s, app(s, app(s, app(s, app(app(plus, _x112), _x111)))))))

Problem 12: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

app#(app(plus, app(s, app(s, app(s, app(s, app(s, 0)))))), _x111)app#(s, app(s, app(s, app(s, app(s, _x111)))))app#(app(map, f), app(app(cons, x), xs))app#(f, x)
app#(app(map, f), app(app(cons, x), xs))app#(app(cons, app(f, x)), app(app(map, f), xs))app#(app(map, f), app(app(cons, x), xs))app#(app(map, f), xs)
app#(app(plus, app(s, x)), y)app#(app(plus, x), y)app#(inc, xs)app#(app(map, app(plus, app(s, 0))), xs)
app#(app(map, f), app(app(cons, x), xs))app#(map, f)app#(app(map, f), app(app(cons, x), xs))app#(cons, app(f, x))
app#(app(plus, app(s, 0)), _x31)app#(s, _x31)app#(inc, xs)app#(plus, app(s, 0))
app#(app(plus, app(s, app(s, app(s, app(s, app(s, app(s, _x112))))))), _x111)app#(s, app(s, app(s, app(s, app(s, app(s, app(app(plus, _x112), _x111)))))))

Rewrite Rules

app(app(plus, 0), y)yapp(app(plus, app(s, x)), y)app(s, app(app(plus, x), y))
app(inc, xs)app(app(map, app(plus, app(s, 0))), xs)app(app(map, f), nil)nil
app(app(map, f), app(app(cons, x), xs))app(app(cons, app(f, x)), app(app(map, f), xs))

Original Signature

Termination of terms over the following signature is verified: plus, app, 0, inc, s, map, cons, nil

Strategy


The right-hand side of the rule app#(app(plus, app(s, app(s, app(s, app(s, app(s, 0)))))), _x111) → app#(s, app(s, app(s, app(s, app(s, _x111))))) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
Thus, the rule app#(app(plus, app(s, app(s, app(s, app(s, app(s, 0)))))), _x111) → app#(s, app(s, app(s, app(s, app(s, _x111))))) is deleted.

Problem 13: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

app#(app(map, f), app(app(cons, x), xs))app#(app(cons, app(f, x)), app(app(map, f), xs))app#(app(map, f), app(app(cons, x), xs))app#(f, x)
app#(app(map, f), app(app(cons, x), xs))app#(app(map, f), xs)app#(app(plus, app(s, x)), y)app#(app(plus, x), y)
app#(app(map, f), app(app(cons, x), xs))app#(cons, app(f, x))app#(app(map, f), app(app(cons, x), xs))app#(map, f)
app#(inc, xs)app#(app(map, app(plus, app(s, 0))), xs)app#(app(plus, app(s, app(s, app(s, app(s, app(s, app(s, _x112))))))), _x111)app#(s, app(s, app(s, app(s, app(s, app(s, app(app(plus, _x112), _x111)))))))
app#(inc, xs)app#(plus, app(s, 0))app#(app(plus, app(s, 0)), _x31)app#(s, _x31)

Rewrite Rules

app(app(plus, 0), y)yapp(app(plus, app(s, x)), y)app(s, app(app(plus, x), y))
app(inc, xs)app(app(map, app(plus, app(s, 0))), xs)app(app(map, f), nil)nil
app(app(map, f), app(app(cons, x), xs))app(app(cons, app(f, x)), app(app(map, f), xs))

Original Signature

Termination of terms over the following signature is verified: app, plus, 0, s, inc, map, nil, cons

Strategy


The right-hand side of the rule app#(app(map, f), app(app(cons, x), xs)) → app#(map, f) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
Thus, the rule app#(app(map, f), app(app(cons, x), xs)) → app#(map, f) is deleted.

Problem 14: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

app#(app(map, f), app(app(cons, x), xs))app#(f, x)app#(app(map, f), app(app(cons, x), xs))app#(app(cons, app(f, x)), app(app(map, f), xs))
app#(app(map, f), app(app(cons, x), xs))app#(app(map, f), xs)app#(app(plus, app(s, x)), y)app#(app(plus, x), y)
app#(inc, xs)app#(app(map, app(plus, app(s, 0))), xs)app#(app(map, f), app(app(cons, x), xs))app#(cons, app(f, x))
app#(app(plus, app(s, 0)), _x31)app#(s, _x31)app#(inc, xs)app#(plus, app(s, 0))
app#(app(plus, app(s, app(s, app(s, app(s, app(s, app(s, _x112))))))), _x111)app#(s, app(s, app(s, app(s, app(s, app(s, app(app(plus, _x112), _x111)))))))

Rewrite Rules

app(app(plus, 0), y)yapp(app(plus, app(s, x)), y)app(s, app(app(plus, x), y))
app(inc, xs)app(app(map, app(plus, app(s, 0))), xs)app(app(map, f), nil)nil
app(app(map, f), app(app(cons, x), xs))app(app(cons, app(f, x)), app(app(map, f), xs))

Original Signature

Termination of terms over the following signature is verified: plus, app, 0, inc, s, map, cons, nil

Strategy


The right-hand side of the rule app#(app(map, f), app(app(cons, x), xs)) → app#(cons, app(f, x)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
app#(cons, _x31) 
app#(cons, app(app(cons, app(_x31, _x33)), app(app(map, _x31), _x32))) 
app#(cons, app(app(map, app(plus, app(s, 0))), _x31)) 
app#(cons, app(s, app(app(plus, _x32), _x31))) 
app#(cons, nil) 
Thus, the rule app#(app(map, f), app(app(cons, x), xs)) → app#(cons, app(f, x)) is replaced by the following rules:
app#(app(map, app(plus, app(s, _x32))), app(app(cons, _x31), xs)) → app#(cons, app(s, app(app(plus, _x32), _x31)))app#(app(map, inc), app(app(cons, _x31), xs)) → app#(cons, app(app(map, app(plus, app(s, 0))), _x31))
app#(app(map, app(map, _x31)), app(app(cons, nil), xs)) → app#(cons, nil)app#(app(map, app(plus, 0)), app(app(cons, _x31), xs)) → app#(cons, _x31)
app#(app(map, app(map, _x31)), app(app(cons, app(app(cons, _x33), _x32)), xs)) → app#(cons, app(app(cons, app(_x31, _x33)), app(app(map, _x31), _x32)))

Problem 15: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

app#(app(map, f), app(app(cons, x), xs))app#(app(cons, app(f, x)), app(app(map, f), xs))app#(app(map, f), app(app(cons, x), xs))app#(f, x)
app#(app(map, app(map, _x31)), app(app(cons, app(app(cons, _x33), _x32)), xs))app#(cons, app(app(cons, app(_x31, _x33)), app(app(map, _x31), _x32)))app#(app(map, f), app(app(cons, x), xs))app#(app(map, f), xs)
app#(inc, xs)app#(app(map, app(plus, app(s, 0))), xs)app#(app(plus, app(s, app(s, app(s, app(s, app(s, app(s, _x112))))))), _x111)app#(s, app(s, app(s, app(s, app(s, app(s, app(app(plus, _x112), _x111)))))))
app#(inc, xs)app#(plus, app(s, 0))app#(app(plus, app(s, 0)), _x31)app#(s, _x31)
app#(app(map, app(plus, app(s, _x32))), app(app(cons, _x31), xs))app#(cons, app(s, app(app(plus, _x32), _x31)))app#(app(map, inc), app(app(cons, _x31), xs))app#(cons, app(app(map, app(plus, app(s, 0))), _x31))
app#(app(map, app(map, _x31)), app(app(cons, nil), xs))app#(cons, nil)app#(app(map, app(plus, 0)), app(app(cons, _x31), xs))app#(cons, _x31)
app#(app(plus, app(s, x)), y)app#(app(plus, x), y)

Rewrite Rules

app(app(plus, 0), y)yapp(app(plus, app(s, x)), y)app(s, app(app(plus, x), y))
app(inc, xs)app(app(map, app(plus, app(s, 0))), xs)app(app(map, f), nil)nil
app(app(map, f), app(app(cons, x), xs))app(app(cons, app(f, x)), app(app(map, f), xs))

Original Signature

Termination of terms over the following signature is verified: app, plus, 0, s, inc, map, nil, cons

Strategy


The right-hand side of the rule app#(app(plus, app(s, app(s, app(s, app(s, app(s, app(s, _x112))))))), _x111) → app#(s, app(s, app(s, app(s, app(s, app(s, app(app(plus, _x112), _x111))))))) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
app#(s, app(s, app(s, app(s, app(s, app(s, app(s, app(app(plus, _x142), _x141)))))))) 
app#(s, app(s, app(s, app(s, app(s, app(s, _x141)))))) 
Thus, the rule app#(app(plus, app(s, app(s, app(s, app(s, app(s, app(s, _x112))))))), _x111) → app#(s, app(s, app(s, app(s, app(s, app(s, app(app(plus, _x112), _x111))))))) is replaced by the following rules:
app#(app(plus, app(s, app(s, app(s, app(s, app(s, app(s, 0))))))), _x141) → app#(s, app(s, app(s, app(s, app(s, app(s, _x141))))))app#(app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x142)))))))), _x141) → app#(s, app(s, app(s, app(s, app(s, app(s, app(s, app(app(plus, _x142), _x141))))))))

Problem 16: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

app#(app(map, f), app(app(cons, x), xs))app#(app(cons, app(f, x)), app(app(map, f), xs))app#(app(map, f), app(app(cons, x), xs))app#(f, x)
app#(app(plus, app(s, app(s, app(s, app(s, app(s, app(s, 0))))))), _x141)app#(s, app(s, app(s, app(s, app(s, app(s, _x141))))))app#(app(map, app(map, _x31)), app(app(cons, app(app(cons, _x33), _x32)), xs))app#(cons, app(app(cons, app(_x31, _x33)), app(app(map, _x31), _x32)))
app#(app(map, f), app(app(cons, x), xs))app#(app(map, f), xs)app#(inc, xs)app#(app(map, app(plus, app(s, 0))), xs)
app#(inc, xs)app#(plus, app(s, 0))app#(app(plus, app(s, 0)), _x31)app#(s, _x31)
app#(app(map, app(plus, app(s, _x32))), app(app(cons, _x31), xs))app#(cons, app(s, app(app(plus, _x32), _x31)))app#(app(map, inc), app(app(cons, _x31), xs))app#(cons, app(app(map, app(plus, app(s, 0))), _x31))
app#(app(map, app(map, _x31)), app(app(cons, nil), xs))app#(cons, nil)app#(app(map, app(plus, 0)), app(app(cons, _x31), xs))app#(cons, _x31)
app#(app(plus, app(s, x)), y)app#(app(plus, x), y)app#(app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x142)))))))), _x141)app#(s, app(s, app(s, app(s, app(s, app(s, app(s, app(app(plus, _x142), _x141))))))))

Rewrite Rules

app(app(plus, 0), y)yapp(app(plus, app(s, x)), y)app(s, app(app(plus, x), y))
app(inc, xs)app(app(map, app(plus, app(s, 0))), xs)app(app(map, f), nil)nil
app(app(map, f), app(app(cons, x), xs))app(app(cons, app(f, x)), app(app(map, f), xs))

Original Signature

Termination of terms over the following signature is verified: plus, app, 0, inc, s, map, cons, nil

Strategy


The right-hand side of the rule app#(app(plus, app(s, app(s, app(s, app(s, app(s, app(s, 0))))))), _x141) → app#(s, app(s, app(s, app(s, app(s, app(s, _x141)))))) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
Thus, the rule app#(app(plus, app(s, app(s, app(s, app(s, app(s, app(s, 0))))))), _x141) → app#(s, app(s, app(s, app(s, app(s, app(s, _x141)))))) is deleted.

Problem 17: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

app#(app(map, f), app(app(cons, x), xs))app#(app(cons, app(f, x)), app(app(map, f), xs))app#(app(map, f), app(app(cons, x), xs))app#(f, x)
app#(app(map, app(map, _x31)), app(app(cons, app(app(cons, _x33), _x32)), xs))app#(cons, app(app(cons, app(_x31, _x33)), app(app(map, _x31), _x32)))app#(app(map, f), app(app(cons, x), xs))app#(app(map, f), xs)
app#(inc, xs)app#(app(map, app(plus, app(s, 0))), xs)app#(inc, xs)app#(plus, app(s, 0))
app#(app(plus, app(s, 0)), _x31)app#(s, _x31)app#(app(map, app(plus, app(s, _x32))), app(app(cons, _x31), xs))app#(cons, app(s, app(app(plus, _x32), _x31)))
app#(app(map, inc), app(app(cons, _x31), xs))app#(cons, app(app(map, app(plus, app(s, 0))), _x31))app#(app(map, app(map, _x31)), app(app(cons, nil), xs))app#(cons, nil)
app#(app(map, app(plus, 0)), app(app(cons, _x31), xs))app#(cons, _x31)app#(app(plus, app(s, x)), y)app#(app(plus, x), y)
app#(app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x142)))))))), _x141)app#(s, app(s, app(s, app(s, app(s, app(s, app(s, app(app(plus, _x142), _x141))))))))

Rewrite Rules

app(app(plus, 0), y)yapp(app(plus, app(s, x)), y)app(s, app(app(plus, x), y))
app(inc, xs)app(app(map, app(plus, app(s, 0))), xs)app(app(map, f), nil)nil
app(app(map, f), app(app(cons, x), xs))app(app(cons, app(f, x)), app(app(map, f), xs))

Original Signature

Termination of terms over the following signature is verified: app, plus, 0, s, inc, map, nil, cons

Strategy


The right-hand side of the rule app#(inc, xs) → app#(plus, app(s, 0)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
Thus, the rule app#(inc, xs) → app#(plus, app(s, 0)) is deleted.

Problem 18: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

app#(app(map, inc), app(app(cons, _x31), xs))app#(cons, app(app(map, app(plus, app(s, 0))), _x31))app#(app(map, app(plus, app(s, _x32))), app(app(cons, _x31), xs))app#(cons, app(s, app(app(plus, _x32), _x31)))
app#(app(map, f), app(app(cons, x), xs))app#(f, x)app#(app(map, f), app(app(cons, x), xs))app#(app(cons, app(f, x)), app(app(map, f), xs))
app#(app(map, app(map, _x31)), app(app(cons, nil), xs))app#(cons, nil)app#(app(map, app(plus, 0)), app(app(cons, _x31), xs))app#(cons, _x31)
app#(app(map, app(map, _x31)), app(app(cons, app(app(cons, _x33), _x32)), xs))app#(cons, app(app(cons, app(_x31, _x33)), app(app(map, _x31), _x32)))app#(app(map, f), app(app(cons, x), xs))app#(app(map, f), xs)
app#(app(plus, app(s, x)), y)app#(app(plus, x), y)app#(inc, xs)app#(app(map, app(plus, app(s, 0))), xs)
app#(app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x142)))))))), _x141)app#(s, app(s, app(s, app(s, app(s, app(s, app(s, app(app(plus, _x142), _x141))))))))app#(app(plus, app(s, 0)), _x31)app#(s, _x31)

Rewrite Rules

app(app(plus, 0), y)yapp(app(plus, app(s, x)), y)app(s, app(app(plus, x), y))
app(inc, xs)app(app(map, app(plus, app(s, 0))), xs)app(app(map, f), nil)nil
app(app(map, f), app(app(cons, x), xs))app(app(cons, app(f, x)), app(app(map, f), xs))

Original Signature

Termination of terms over the following signature is verified: plus, app, 0, inc, s, map, cons, nil

Strategy


The right-hand side of the rule app#(app(plus, app(s, 0)), _x31) → app#(s, _x31) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
Thus, the rule app#(app(plus, app(s, 0)), _x31) → app#(s, _x31) is deleted.

Problem 19: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

app#(app(map, app(plus, app(s, _x32))), app(app(cons, _x31), xs))app#(cons, app(s, app(app(plus, _x32), _x31)))app#(app(map, inc), app(app(cons, _x31), xs))app#(cons, app(app(map, app(plus, app(s, 0))), _x31))
app#(app(map, f), app(app(cons, x), xs))app#(app(cons, app(f, x)), app(app(map, f), xs))app#(app(map, f), app(app(cons, x), xs))app#(f, x)
app#(app(map, app(map, _x31)), app(app(cons, nil), xs))app#(cons, nil)app#(app(map, app(plus, 0)), app(app(cons, _x31), xs))app#(cons, _x31)
app#(app(map, app(map, _x31)), app(app(cons, app(app(cons, _x33), _x32)), xs))app#(cons, app(app(cons, app(_x31, _x33)), app(app(map, _x31), _x32)))app#(app(map, f), app(app(cons, x), xs))app#(app(map, f), xs)
app#(app(plus, app(s, x)), y)app#(app(plus, x), y)app#(inc, xs)app#(app(map, app(plus, app(s, 0))), xs)
app#(app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x142)))))))), _x141)app#(s, app(s, app(s, app(s, app(s, app(s, app(s, app(app(plus, _x142), _x141))))))))

Rewrite Rules

app(app(plus, 0), y)yapp(app(plus, app(s, x)), y)app(s, app(app(plus, x), y))
app(inc, xs)app(app(map, app(plus, app(s, 0))), xs)app(app(map, f), nil)nil
app(app(map, f), app(app(cons, x), xs))app(app(cons, app(f, x)), app(app(map, f), xs))

Original Signature

Termination of terms over the following signature is verified: app, plus, 0, s, inc, map, nil, cons

Strategy


The right-hand side of the rule app#(app(map, inc), app(app(cons, _x31), xs)) → app#(cons, app(app(map, app(plus, app(s, 0))), _x31)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
app#(cons, app(app(cons, app(app(plus, app(s, 0)), _x43)), app(app(map, app(plus, app(s, 0))), _x42))) 
app#(cons, nil) 
Thus, the rule app#(app(map, inc), app(app(cons, _x31), xs)) → app#(cons, app(app(map, app(plus, app(s, 0))), _x31)) is replaced by the following rules:
app#(app(map, inc), app(app(cons, nil), xs)) → app#(cons, nil)app#(app(map, inc), app(app(cons, app(app(cons, _x43), _x42)), xs)) → app#(cons, app(app(cons, app(app(plus, app(s, 0)), _x43)), app(app(map, app(plus, app(s, 0))), _x42)))

Problem 20: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

app#(app(map, app(plus, app(s, _x32))), app(app(cons, _x31), xs))app#(cons, app(s, app(app(plus, _x32), _x31)))app#(app(map, f), app(app(cons, x), xs))app#(f, x)
app#(app(map, f), app(app(cons, x), xs))app#(app(cons, app(f, x)), app(app(map, f), xs))app#(app(map, inc), app(app(cons, nil), xs))app#(cons, nil)
app#(app(map, app(map, _x31)), app(app(cons, nil), xs))app#(cons, nil)app#(app(map, inc), app(app(cons, app(app(cons, _x43), _x42)), xs))app#(cons, app(app(cons, app(app(plus, app(s, 0)), _x43)), app(app(map, app(plus, app(s, 0))), _x42)))
app#(app(map, app(plus, 0)), app(app(cons, _x31), xs))app#(cons, _x31)app#(app(map, app(map, _x31)), app(app(cons, app(app(cons, _x33), _x32)), xs))app#(cons, app(app(cons, app(_x31, _x33)), app(app(map, _x31), _x32)))
app#(app(map, f), app(app(cons, x), xs))app#(app(map, f), xs)app#(app(plus, app(s, x)), y)app#(app(plus, x), y)
app#(inc, xs)app#(app(map, app(plus, app(s, 0))), xs)app#(app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x142)))))))), _x141)app#(s, app(s, app(s, app(s, app(s, app(s, app(s, app(app(plus, _x142), _x141))))))))

Rewrite Rules

app(app(plus, 0), y)yapp(app(plus, app(s, x)), y)app(s, app(app(plus, x), y))
app(inc, xs)app(app(map, app(plus, app(s, 0))), xs)app(app(map, f), nil)nil
app(app(map, f), app(app(cons, x), xs))app(app(cons, app(f, x)), app(app(map, f), xs))

Original Signature

Termination of terms over the following signature is verified: plus, app, 0, inc, s, map, cons, nil

Strategy


The right-hand side of the rule app#(app(map, app(plus, app(s, _x32))), app(app(cons, _x31), xs)) → app#(cons, app(s, app(app(plus, _x32), _x31))) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
app#(cons, app(s, app(s, app(app(plus, _x62), _x61)))) 
app#(cons, app(s, _x61)) 
Thus, the rule app#(app(map, app(plus, app(s, _x32))), app(app(cons, _x31), xs)) → app#(cons, app(s, app(app(plus, _x32), _x31))) is replaced by the following rules:
app#(app(map, app(plus, app(s, 0))), app(app(cons, _x61), xs)) → app#(cons, app(s, _x61))app#(app(map, app(plus, app(s, app(s, _x62)))), app(app(cons, _x61), xs)) → app#(cons, app(s, app(s, app(app(plus, _x62), _x61))))

Problem 21: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

app#(app(map, f), app(app(cons, x), xs))app#(f, x)app#(app(map, f), app(app(cons, x), xs))app#(app(cons, app(f, x)), app(app(map, f), xs))
app#(app(map, inc), app(app(cons, app(app(cons, _x81), app(app(cons, _x91), nil))), xs))app#(cons, app(app(cons, app(s, _x81)), app(app(cons, app(s, app(app(plus, 0), _x91))), nil)))app#(app(map, inc), app(app(cons, app(app(cons, _x43), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x111), app(app(cons, _x93), _x92))))), xs))app#(cons, app(app(cons, app(app(plus, app(s, 0)), _x43)), app(app(cons, app(s, app(app(plus, 0), _x91))), app(app(cons, app(s, app(app(plus, 0), _x111))), app(app(cons, app(app(plus, app(s, 0)), _x93)), app(app(map, app(plus, app(s, 0))), _x92))))))
app#(app(map, app(map, _x31)), app(app(cons, app(app(cons, _x33), _x32)), xs))app#(cons, app(app(cons, app(_x31, _x33)), app(app(map, _x31), _x32)))app#(app(map, inc), app(app(cons, app(app(cons, _x43), app(app(cons, _x101), app(app(cons, _x111), app(app(cons, _x131), nil))))), xs))app#(cons, app(app(cons, app(app(plus, app(s, 0)), _x43)), app(app(cons, app(s, _x101)), app(app(cons, app(s, app(app(plus, 0), _x111))), app(app(cons, app(s, app(app(plus, 0), _x131))), nil)))))
app#(app(map, inc), app(app(cons, app(app(cons, _x43), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x111), nil)))), xs))app#(cons, app(app(cons, app(app(plus, app(s, 0)), _x43)), app(app(cons, app(s, app(app(plus, 0), _x91))), app(app(cons, app(s, app(app(plus, 0), _x111))), nil))))app#(app(map, inc), app(app(cons, app(app(cons, _x43), app(app(cons, _x101), app(app(cons, _x83), app(app(cons, _x93), _x92))))), xs))app#(cons, app(app(cons, app(app(plus, app(s, 0)), _x43)), app(app(cons, app(s, _x101)), app(app(cons, app(app(plus, app(s, 0)), _x83)), app(app(cons, app(app(plus, app(s, 0)), _x93)), app(app(map, app(plus, app(s, 0))), _x92))))))
app#(app(map, inc), app(app(cons, app(app(cons, _x43), app(app(cons, _x101), app(app(cons, _x111), app(app(cons, _x131), _x92))))), xs))app#(cons, app(app(cons, app(app(plus, app(s, 0)), _x43)), app(app(cons, app(s, _x101)), app(app(cons, app(s, app(app(plus, 0), _x111))), app(app(cons, app(s, app(app(plus, 0), _x131))), app(app(map, app(plus, app(s, 0))), _x92))))))app#(app(map, inc), app(app(cons, app(app(cons, _x43), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x83), app(app(cons, _x93), _x92))))), xs))app#(cons, app(app(cons, app(app(plus, app(s, 0)), _x43)), app(app(cons, app(s, app(app(plus, 0), _x91))), app(app(cons, app(app(plus, app(s, 0)), _x83)), app(app(cons, app(app(plus, app(s, 0)), _x93)), app(app(map, app(plus, app(s, 0))), _x92))))))
app#(app(map, inc), app(app(cons, app(app(cons, _x71), app(app(cons, _x101), app(app(cons, _x111), _x82)))), xs))app#(cons, app(app(cons, app(s, app(app(plus, 0), _x71))), app(app(cons, app(s, _x101)), app(app(cons, app(s, app(app(plus, 0), _x111))), app(app(map, app(plus, app(s, 0))), _x82)))))app#(app(map, inc), app(app(cons, app(app(cons, _x71), app(app(cons, _x101), app(app(cons, _x111), app(app(cons, _x93), _x92))))), xs))app#(cons, app(app(cons, app(s, app(app(plus, 0), _x71))), app(app(cons, app(s, _x101)), app(app(cons, app(s, app(app(plus, 0), _x111))), app(app(cons, app(app(plus, app(s, 0)), _x93)), app(app(map, app(plus, app(s, 0))), _x92))))))
app#(app(map, inc), app(app(cons, app(app(cons, _x43), app(app(cons, _x101), app(app(cons, _x83), nil)))), xs))app#(cons, app(app(cons, app(app(plus, app(s, 0)), _x43)), app(app(cons, app(s, _x101)), app(app(cons, app(app(plus, app(s, 0)), _x83)), nil))))app#(app(map, app(plus, 0)), app(app(cons, _x31), xs))app#(cons, _x31)
app#(app(map, inc), app(app(cons, app(app(cons, _x71), app(app(cons, _x101), app(app(cons, _x121), _x82)))), xs))app#(cons, app(app(cons, app(s, app(app(plus, 0), _x71))), app(app(cons, app(s, _x101)), app(app(cons, app(s, _x121)), app(app(map, app(plus, app(s, 0))), _x82)))))app#(app(map, app(plus, app(s, app(s, _x62)))), app(app(cons, _x61), xs))app#(cons, app(s, app(s, app(app(plus, _x62), _x61))))
app#(app(map, inc), app(app(cons, app(app(cons, _x71), app(app(cons, _x53), _x52))), xs))app#(cons, app(app(cons, app(s, app(app(plus, 0), _x71))), app(app(cons, app(app(plus, app(s, 0)), _x53)), app(app(map, app(plus, app(s, 0))), _x52))))app#(app(map, inc), app(app(cons, app(app(cons, _x43), app(app(cons, _x101), app(app(cons, _x121), _x82)))), xs))app#(cons, app(app(cons, app(app(plus, app(s, 0)), _x43)), app(app(cons, app(s, _x101)), app(app(cons, app(s, _x121)), app(app(map, app(plus, app(s, 0))), _x82)))))
app#(app(map, inc), app(app(cons, app(app(cons, _x71), app(app(cons, _x101), app(app(cons, _x83), _x82)))), xs))app#(cons, app(app(cons, app(s, app(app(plus, 0), _x71))), app(app(cons, app(s, _x101)), app(app(cons, app(app(plus, app(s, 0)), _x83)), app(app(map, app(plus, app(s, 0))), _x82)))))app#(app(map, inc), app(app(cons, app(app(cons, _x43), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x83), nil)))), xs))app#(cons, app(app(cons, app(app(plus, app(s, 0)), _x43)), app(app(cons, app(s, app(app(plus, 0), _x91))), app(app(cons, app(app(plus, app(s, 0)), _x83)), nil))))
app#(app(map, inc), app(app(cons, app(app(cons, _x43), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x121), nil)))), xs))app#(cons, app(app(cons, app(app(plus, app(s, 0)), _x43)), app(app(cons, app(s, app(app(plus, 0), _x91))), app(app(cons, app(s, _x121)), nil))))app#(app(map, inc), app(app(cons, app(app(cons, _x71), app(app(cons, _x101), nil))), xs))app#(cons, app(app(cons, app(s, app(app(plus, 0), _x71))), app(app(cons, app(s, _x101)), nil)))
app#(app(map, inc), app(app(cons, app(app(cons, _x71), app(app(cons, _x91), _x52))), xs))app#(cons, app(app(cons, app(s, app(app(plus, 0), _x71))), app(app(cons, app(s, app(app(plus, 0), _x91))), app(app(map, app(plus, app(s, 0))), _x52))))app#(app(map, inc), app(app(cons, app(app(cons, _x71), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x121), app(app(cons, _x93), _x92))))), xs))app#(cons, app(app(cons, app(s, app(app(plus, 0), _x71))), app(app(cons, app(s, app(app(plus, 0), _x91))), app(app(cons, app(s, _x121)), app(app(cons, app(app(plus, app(s, 0)), _x93)), app(app(map, app(plus, app(s, 0))), _x92))))))
app#(app(map, inc), app(app(cons, app(app(cons, _x43), app(app(cons, _x101), app(app(cons, _x111), app(app(cons, _x93), app(app(cons, _x123), _x122)))))), xs))app#(cons, app(app(cons, app(app(plus, app(s, 0)), _x43)), app(app(cons, app(s, _x101)), app(app(cons, app(s, app(app(plus, 0), _x111))), app(app(cons, app(app(plus, app(s, 0)), _x93)), app(app(cons, app(app(plus, app(s, 0)), _x123)), app(app(map, app(plus, app(s, 0))), _x122)))))))app#(app(map, inc), app(app(cons, app(app(cons, _x43), app(app(cons, _x101), app(app(cons, _x121), app(app(cons, _x93), nil))))), xs))app#(cons, app(app(cons, app(app(plus, app(s, 0)), _x43)), app(app(cons, app(s, _x101)), app(app(cons, app(s, _x121)), app(app(cons, app(app(plus, app(s, 0)), _x93)), nil)))))
app#(app(map, f), app(app(cons, x), xs))app#(app(map, f), xs)app#(app(map, inc), app(app(cons, app(app(cons, _x43), app(app(cons, _x101), app(app(cons, _x121), app(app(cons, _x93), _x92))))), xs))app#(cons, app(app(cons, app(app(plus, app(s, 0)), _x43)), app(app(cons, app(s, _x101)), app(app(cons, app(s, _x121)), app(app(cons, app(app(plus, app(s, 0)), _x93)), app(app(map, app(plus, app(s, 0))), _x92))))))
app#(app(map, inc), app(app(cons, app(app(cons, _x71), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x121), nil)))), xs))app#(cons, app(app(cons, app(s, app(app(plus, 0), _x71))), app(app(cons, app(s, app(app(plus, 0), _x91))), app(app(cons, app(s, _x121)), nil))))app#(inc, xs)app#(app(map, app(plus, app(s, 0))), xs)
app#(app(map, inc), app(app(cons, app(app(cons, _x43), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x121), app(app(cons, _x93), _x92))))), xs))app#(cons, app(app(cons, app(app(plus, app(s, 0)), _x43)), app(app(cons, app(s, app(app(plus, 0), _x91))), app(app(cons, app(s, _x121)), app(app(cons, app(app(plus, app(s, 0)), _x93)), app(app(map, app(plus, app(s, 0))), _x92))))))app#(app(map, inc), app(app(cons, app(app(cons, _x71), app(app(cons, _x91), app(app(cons, _x111), _x82)))), xs))app#(cons, app(app(cons, app(s, app(app(plus, 0), _x71))), app(app(cons, app(s, app(app(plus, 0), _x91))), app(app(cons, app(s, app(app(plus, 0), _x111))), app(app(map, app(plus, app(s, 0))), _x82)))))
app#(app(map, inc), app(app(cons, app(app(cons, _x43), app(app(cons, _x101), nil))), xs))app#(cons, app(app(cons, app(app(plus, app(s, 0)), _x43)), app(app(cons, app(s, _x101)), nil)))app#(app(map, inc), app(app(cons, app(app(cons, _x71), app(app(cons, _x53), nil))), xs))app#(cons, app(app(cons, app(s, app(app(plus, 0), _x71))), app(app(cons, app(app(plus, app(s, 0)), _x53)), nil)))
app#(app(map, inc), app(app(cons, app(app(cons, _x43), app(app(cons, _x101), app(app(cons, _x111), nil)))), xs))app#(cons, app(app(cons, app(app(plus, app(s, 0)), _x43)), app(app(cons, app(s, _x101)), app(app(cons, app(s, app(app(plus, 0), _x111))), nil))))app#(app(map, inc), app(app(cons, app(app(cons, _x81), _x42)), xs))app#(cons, app(app(cons, app(s, _x81)), app(app(map, app(plus, app(s, 0))), _x42)))
app#(app(map, inc), app(app(cons, app(app(cons, _x43), app(app(cons, _x53), app(app(cons, _x83), _x82)))), xs))app#(cons, app(app(cons, app(app(plus, app(s, 0)), _x43)), app(app(cons, app(app(plus, app(s, 0)), _x53)), app(app(cons, app(app(plus, app(s, 0)), _x83)), app(app(map, app(plus, app(s, 0))), _x82)))))app#(app(map, inc), app(app(cons, app(app(cons, _x43), app(app(cons, _x121), _x52))), xs))app#(cons, app(app(cons, app(app(plus, app(s, 0)), _x43)), app(app(cons, app(s, _x121)), app(app(map, app(plus, app(s, 0))), _x52))))
app#(app(map, inc), app(app(cons, app(app(cons, _x71), app(app(cons, _x101), app(app(cons, _x111), app(app(cons, _x93), nil))))), xs))app#(cons, app(app(cons, app(s, app(app(plus, 0), _x71))), app(app(cons, app(s, _x101)), app(app(cons, app(s, app(app(plus, 0), _x111))), app(app(cons, app(app(plus, app(s, 0)), _x93)), nil)))))app#(app(map, inc), app(app(cons, app(app(cons, _x71), app(app(cons, _x91), app(app(cons, _x83), _x82)))), xs))app#(cons, app(app(cons, app(s, app(app(plus, 0), _x71))), app(app(cons, app(s, app(app(plus, 0), _x91))), app(app(cons, app(app(plus, app(s, 0)), _x83)), app(app(map, app(plus, app(s, 0))), _x82)))))
app#(app(plus, app(s, x)), y)app#(app(plus, x), y)app#(app(map, inc), app(app(cons, app(app(cons, _x81), app(app(cons, _x91), app(app(cons, _x121), _x82)))), xs))app#(cons, app(app(cons, app(s, _x81)), app(app(cons, app(s, app(app(plus, 0), _x91))), app(app(cons, app(s, _x121)), app(app(map, app(plus, app(s, 0))), _x82)))))
app#(app(map, inc), app(app(cons, app(app(cons, _x81), app(app(cons, _x101), nil))), xs))app#(cons, app(app(cons, app(s, _x81)), app(app(cons, app(s, _x101)), nil)))app#(app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x142)))))))), _x141)app#(s, app(s, app(s, app(s, app(s, app(s, app(s, app(app(plus, _x142), _x141))))))))

Rewrite Rules

app(app(plus, 0), y)yapp(app(plus, app(s, x)), y)app(s, app(app(plus, x), y))
app(inc, xs)app(app(map, app(plus, app(s, 0))), xs)app(app(map, f), nil)nil
app(app(map, f), app(app(cons, x), xs))app(app(cons, app(f, x)), app(app(map, f), xs))

Original Signature

Termination of terms over the following signature is verified: app, plus, 0, s, inc, map, nil, cons

Strategy


The right-hand side of the rule app#(app(map, inc), app(app(cons, app(app(cons, _x81), app(app(cons, _x91), nil))), xs)) → app#(cons, app(app(cons, app(s, _x81)), app(app(cons, app(s, app(app(plus, 0), _x91))), nil))) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
app#(cons, app(app(cons, app(s, _x81)), app(app(cons, app(s, _x101)), nil))) 
Thus, the rule app#(app(map, inc), app(app(cons, app(app(cons, _x81), app(app(cons, _x91), nil))), xs)) → app#(cons, app(app(cons, app(s, _x81)), app(app(cons, app(s, app(app(plus, 0), _x91))), nil))) is replaced by the following rules:
app#(app(map, inc), app(app(cons, app(app(cons, _x81), app(app(cons, _x101), nil))), xs)) → app#(cons, app(app(cons, app(s, _x81)), app(app(cons, app(s, _x101)), nil)))

Problem 22: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

app#(app(map, inc), app(app(cons, app(app(cons, _x71), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x121), app(app(cons, _x131), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x191), app(app(cons, _x191), _x162)))))))), xs))app#(cons, app(app(cons, app(s, app(app(plus, 0), _x71))), app(app(cons, app(s, _x101)), app(app(cons, app(s, _x121)), app(app(cons, app(s, app(app(plus, 0), _x131))), app(app(cons, app(s, app(app(plus, 0), _x161))), app(app(cons, app(s, app(app(plus, 0), _x191))), app(app(cons, app(s, app(app(plus, 0), _x191))), app(app(map, app(plus, app(s, 0))), _x162)))))))))app#(app(map, f), app(app(cons, x), xs))app#(app(map, f), xs)
app#(app(map, inc), app(app(cons, app(app(cons, _x81), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x83), app(app(cons, _x93), app(app(cons, _x123), _x122)))))), xs))app#(cons, app(app(cons, app(s, _x81)), app(app(cons, app(s, app(app(plus, 0), _x91))), app(app(cons, app(app(plus, app(s, 0)), _x83)), app(app(cons, app(app(plus, app(s, 0)), _x93)), app(app(cons, app(app(plus, app(s, 0)), _x123)), app(app(map, app(plus, app(s, 0))), _x122)))))))app#(app(map, inc), app(app(cons, app(app(cons, _x81), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x121), app(app(cons, _x141), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x191), app(app(cons, _x163), app(app(cons, _x173), _x172))))))))), xs))app#(cons, app(app(cons, app(s, _x81)), app(app(cons, app(s, _x101)), app(app(cons, app(s, _x121)), app(app(cons, app(s, _x141)), app(app(cons, app(s, app(app(plus, 0), _x161))), app(app(cons, app(s, app(app(plus, 0), _x191))), app(app(cons, app(app(plus, app(s, 0)), _x163)), app(app(cons, app(app(plus, app(s, 0)), _x173)), app(app(map, app(plus, app(s, 0))), _x172))))))))))
app#(app(map, inc), app(app(cons, app(app(cons, _x81), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x111), app(app(cons, _x161), _x92))))), xs))app#(cons, app(app(cons, app(s, _x81)), app(app(cons, app(s, app(app(plus, 0), _x91))), app(app(cons, app(s, app(app(plus, 0), _x111))), app(app(cons, app(s, _x161)), app(app(map, app(plus, app(s, 0))), _x92))))))app#(app(map, inc), app(app(cons, app(app(cons, _x43), app(app(cons, _x101), app(app(cons, _x111), app(app(cons, _x93), _x92))))), xs))app#(cons, app(app(cons, app(app(plus, app(s, 0)), _x43)), app(app(cons, app(s, _x101)), app(app(cons, app(s, app(app(plus, 0), _x111))), app(app(cons, app(app(plus, app(s, 0)), _x93)), app(app(map, app(plus, app(s, 0))), _x92))))))
app#(app(map, inc), app(app(cons, app(app(cons, _x71), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x83), app(app(cons, _x131), _x92))))), xs))app#(cons, app(app(cons, app(s, app(app(plus, 0), _x71))), app(app(cons, app(s, _x101)), app(app(cons, app(app(plus, app(s, 0)), _x83)), app(app(cons, app(s, app(app(plus, 0), _x131))), app(app(map, app(plus, app(s, 0))), _x92))))))app#(app(map, inc), app(app(cons, app(app(cons, _x71), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x111), app(app(cons, _x93), app(app(cons, _x161), app(app(cons, _x191), _x152))))))), xs))app#(cons, app(app(cons, app(s, app(app(plus, 0), _x71))), app(app(cons, app(s, app(app(plus, 0), _x91))), app(app(cons, app(s, app(app(plus, 0), _x111))), app(app(cons, app(app(plus, app(s, 0)), _x93)), app(app(cons, app(s, app(app(plus, 0), _x161))), app(app(cons, app(s, app(app(plus, 0), _x191))), app(app(map, app(plus, app(s, 0))), _x152))))))))
app#(app(map, inc), app(app(cons, app(app(cons, _x43), app(app(cons, _x101), app(app(cons, _x111), app(app(cons, _x131), _x92))))), xs))app#(cons, app(app(cons, app(app(plus, app(s, 0)), _x43)), app(app(cons, app(s, _x101)), app(app(cons, app(s, app(app(plus, 0), _x111))), app(app(cons, app(s, app(app(plus, 0), _x131))), app(app(map, app(plus, app(s, 0))), _x92))))))app#(app(map, inc), app(app(cons, app(app(cons, _x71), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x121), app(app(cons, _x131), app(app(cons, _x161), app(app(cons, _x191), _x152))))))), xs))app#(cons, app(app(cons, app(s, app(app(plus, 0), _x71))), app(app(cons, app(s, app(app(plus, 0), _x91))), app(app(cons, app(s, _x121)), app(app(cons, app(s, app(app(plus, 0), _x131))), app(app(cons, app(s, app(app(plus, 0), _x161))), app(app(cons, app(s, app(app(plus, 0), _x191))), app(app(map, app(plus, app(s, 0))), _x152))))))))
app#(app(map, inc), app(app(cons, app(app(cons, _x43), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x83), app(app(cons, _x93), nil))))), xs))app#(cons, app(app(cons, app(app(plus, app(s, 0)), _x43)), app(app(cons, app(s, _x101)), app(app(cons, app(app(plus, app(s, 0)), _x83)), app(app(cons, app(app(plus, app(s, 0)), _x93)), nil)))))app#(app(map, inc), app(app(cons, app(app(cons, _x71), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x121), app(app(cons, _x141), app(app(cons, _x123), app(app(cons, _x191), app(app(cons, _x163), _x162)))))))), xs))app#(cons, app(app(cons, app(s, app(app(plus, 0), _x71))), app(app(cons, app(s, _x101)), app(app(cons, app(s, _x121)), app(app(cons, app(s, _x141)), app(app(cons, app(app(plus, app(s, 0)), _x123)), app(app(cons, app(s, app(app(plus, 0), _x191))), app(app(cons, app(app(plus, app(s, 0)), _x163)), app(app(map, app(plus, app(s, 0))), _x162)))))))))
app#(app(map, inc), app(app(cons, app(app(cons, _x81), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x121), app(app(cons, _x141), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x181), app(app(cons, _x163), nil)))))))), xs))app#(cons, app(app(cons, app(s, _x81)), app(app(cons, app(s, _x101)), app(app(cons, app(s, _x121)), app(app(cons, app(s, _x141)), app(app(cons, app(s, _x171)), app(app(cons, app(s, _x181)), app(app(cons, app(app(plus, app(s, 0)), _x163)), nil))))))))app#(app(map, inc), app(app(cons, app(app(cons, _x71), app(app(cons, _x101), app(app(cons, _x121), app(app(cons, _x93), _x92))))), xs))app#(cons, app(app(cons, app(s, app(app(plus, 0), _x71))), app(app(cons, app(s, _x101)), app(app(cons, app(s, _x121)), app(app(cons, app(app(plus, app(s, 0)), _x93)), app(app(map, app(plus, app(s, 0))), _x92))))))
app#(app(map, inc), app(app(cons, app(app(cons, _x71), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x111), app(app(cons, _x93), app(app(cons, _x161), _x122)))))), xs))app#(cons, app(app(cons, app(s, app(app(plus, 0), _x71))), app(app(cons, app(s, app(app(plus, 0), _x91))), app(app(cons, app(s, app(app(plus, 0), _x111))), app(app(cons, app(app(plus, app(s, 0)), _x93)), app(app(cons, app(s, app(app(plus, 0), _x161))), app(app(map, app(plus, app(s, 0))), _x122)))))))app#(app(map, inc), app(app(cons, app(app(cons, _x71), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x121), app(app(cons, _x131), app(app(cons, _x123), app(app(cons, _x191), nil))))))), xs))app#(cons, app(app(cons, app(s, app(app(plus, 0), _x71))), app(app(cons, app(s, _x101)), app(app(cons, app(s, _x121)), app(app(cons, app(s, app(app(plus, 0), _x131))), app(app(cons, app(app(plus, app(s, 0)), _x123)), app(app(cons, app(s, app(app(plus, 0), _x191))), nil)))))))
app#(app(map, inc), app(app(cons, app(app(cons, _x71), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x121), app(app(cons, _x131), app(app(cons, _x151), app(app(cons, _x191), app(app(cons, _x163), _x162)))))))), xs))app#(cons, app(app(cons, app(s, app(app(plus, 0), _x71))), app(app(cons, app(s, _x101)), app(app(cons, app(s, _x121)), app(app(cons, app(s, app(app(plus, 0), _x131))), app(app(cons, app(s, app(app(plus, 0), _x151))), app(app(cons, app(s, app(app(plus, 0), _x191))), app(app(cons, app(app(plus, app(s, 0)), _x163)), app(app(map, app(plus, app(s, 0))), _x162)))))))))app#(app(map, inc), app(app(cons, app(app(cons, _x71), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x111), app(app(cons, _x93), app(app(cons, _x161), app(app(cons, _x153), _x152))))))), xs))app#(cons, app(app(cons, app(s, app(app(plus, 0), _x71))), app(app(cons, app(s, app(app(plus, 0), _x91))), app(app(cons, app(s, app(app(plus, 0), _x111))), app(app(cons, app(app(plus, app(s, 0)), _x93)), app(app(cons, app(s, app(app(plus, 0), _x161))), app(app(cons, app(app(plus, app(s, 0)), _x153)), app(app(map, app(plus, app(s, 0))), _x152))))))))
app#(app(map, inc), app(app(cons, app(app(cons, _x71), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x121), app(app(cons, _x141), app(app(cons, _x123), app(app(cons, _x191), _x152))))))), xs))app#(cons, app(app(cons, app(s, app(app(plus, 0), _x71))), app(app(cons, app(s, _x101)), app(app(cons, app(s, _x121)), app(app(cons, app(s, _x141)), app(app(cons, app(app(plus, app(s, 0)), _x123)), app(app(cons, app(s, app(app(plus, 0), _x191))), app(app(map, app(plus, app(s, 0))), _x152))))))))app#(app(map, inc), app(app(cons, app(app(cons, _x43), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x111), app(app(cons, _x93), app(app(cons, _x123), nil)))))), xs))app#(cons, app(app(cons, app(app(plus, app(s, 0)), _x43)), app(app(cons, app(s, app(app(plus, 0), _x91))), app(app(cons, app(s, app(app(plus, 0), _x111))), app(app(cons, app(app(plus, app(s, 0)), _x93)), app(app(cons, app(app(plus, app(s, 0)), _x123)), nil))))))
app#(app(map, inc), app(app(cons, app(app(cons, _x71), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x121), app(app(cons, _x93), app(app(cons, _x123), app(app(cons, _x191), nil))))))), xs))app#(cons, app(app(cons, app(s, app(app(plus, 0), _x71))), app(app(cons, app(s, app(app(plus, 0), _x91))), app(app(cons, app(s, _x121)), app(app(cons, app(app(plus, app(s, 0)), _x93)), app(app(cons, app(app(plus, app(s, 0)), _x123)), app(app(cons, app(s, app(app(plus, 0), _x191))), nil)))))))app#(app(map, inc), app(app(cons, app(app(cons, _x71), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x83), app(app(cons, _x141), app(app(cons, _x123), _x122)))))), xs))app#(cons, app(app(cons, app(s, app(app(plus, 0), _x71))), app(app(cons, app(s, app(app(plus, 0), _x91))), app(app(cons, app(app(plus, app(s, 0)), _x83)), app(app(cons, app(s, _x141)), app(app(cons, app(app(plus, app(s, 0)), _x123)), app(app(map, app(plus, app(s, 0))), _x122)))))))
app#(app(map, inc), app(app(cons, app(app(cons, _x43), app(app(cons, _x101), app(app(cons, _x121), _x82)))), xs))app#(cons, app(app(cons, app(app(plus, app(s, 0)), _x43)), app(app(cons, app(s, _x101)), app(app(cons, app(s, _x121)), app(app(map, app(plus, app(s, 0))), _x82)))))app#(app(map, inc), app(app(cons, app(app(cons, _x71), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x121), app(app(cons, _x93), app(app(cons, _x123), _x122)))))), xs))app#(cons, app(app(cons, app(s, app(app(plus, 0), _x71))), app(app(cons, app(s, app(app(plus, 0), _x91))), app(app(cons, app(s, _x121)), app(app(cons, app(app(plus, app(s, 0)), _x93)), app(app(cons, app(app(plus, app(s, 0)), _x123)), app(app(map, app(plus, app(s, 0))), _x122)))))))
app#(app(map, inc), app(app(cons, app(app(cons, _x81), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x111), app(app(cons, _x131), app(app(cons, _x123), app(app(cons, _x191), _x152))))))), xs))app#(cons, app(app(cons, app(s, _x81)), app(app(cons, app(s, _x101)), app(app(cons, app(s, app(app(plus, 0), _x111))), app(app(cons, app(s, app(app(plus, 0), _x131))), app(app(cons, app(app(plus, app(s, 0)), _x123)), app(app(cons, app(s, app(app(plus, 0), _x191))), app(app(map, app(plus, app(s, 0))), _x152))))))))app#(app(map, inc), app(app(cons, app(app(cons, _x43), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x121), nil)))), xs))app#(cons, app(app(cons, app(app(plus, app(s, 0)), _x43)), app(app(cons, app(s, app(app(plus, 0), _x91))), app(app(cons, app(s, _x121)), nil))))
app#(app(map, inc), app(app(cons, app(app(cons, _x43), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x111), app(app(cons, _x93), app(app(cons, _x123), app(app(cons, _x153), _x152))))))), xs))app#(cons, app(app(cons, app(app(plus, app(s, 0)), _x43)), app(app(cons, app(s, app(app(plus, 0), _x91))), app(app(cons, app(s, app(app(plus, 0), _x111))), app(app(cons, app(app(plus, app(s, 0)), _x93)), app(app(cons, app(app(plus, app(s, 0)), _x123)), app(app(cons, app(app(plus, app(s, 0)), _x153)), app(app(map, app(plus, app(s, 0))), _x152))))))))app#(app(map, inc), app(app(cons, app(app(cons, _x81), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x121), app(app(cons, _x141), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x181), app(app(cons, _x191), nil)))))))), xs))app#(cons, app(app(cons, app(s, _x81)), app(app(cons, app(s, _x101)), app(app(cons, app(s, _x121)), app(app(cons, app(s, _x141)), app(app(cons, app(s, app(app(plus, 0), _x161))), app(app(cons, app(s, _x181)), app(app(cons, app(s, app(app(plus, 0), _x191))), nil))))))))
app#(app(map, inc), app(app(cons, app(app(cons, _x43), app(app(cons, _x101), app(app(cons, _x111), app(app(cons, _x141), nil))))), xs))app#(cons, app(app(cons, app(app(plus, app(s, 0)), _x43)), app(app(cons, app(s, _x101)), app(app(cons, app(s, app(app(plus, 0), _x111))), app(app(cons, app(s, _x141)), nil)))))app#(app(map, inc), app(app(cons, app(app(cons, _x81), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x83), app(app(cons, _x131), app(app(cons, _x123), _x122)))))), xs))app#(cons, app(app(cons, app(s, _x81)), app(app(cons, app(s, app(app(plus, 0), _x91))), app(app(cons, app(app(plus, app(s, 0)), _x83)), app(app(cons, app(s, app(app(plus, 0), _x131))), app(app(cons, app(app(plus, app(s, 0)), _x123)), app(app(map, app(plus, app(s, 0))), _x122)))))))
app#(app(map, inc), app(app(cons, app(app(cons, _x71), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x121), app(app(cons, _x93), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x191), nil))))))), xs))app#(cons, app(app(cons, app(s, app(app(plus, 0), _x71))), app(app(cons, app(s, app(app(plus, 0), _x91))), app(app(cons, app(s, _x121)), app(app(cons, app(app(plus, app(s, 0)), _x93)), app(app(cons, app(s, app(app(plus, 0), _x161))), app(app(cons, app(s, app(app(plus, 0), _x191))), nil)))))))app#(app(map, inc), app(app(cons, app(app(cons, _x81), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x111), app(app(cons, _x93), app(app(cons, _x123), app(app(cons, _x191), _x152))))))), xs))app#(cons, app(app(cons, app(s, _x81)), app(app(cons, app(s, app(app(plus, 0), _x91))), app(app(cons, app(s, app(app(plus, 0), _x111))), app(app(cons, app(app(plus, app(s, 0)), _x93)), app(app(cons, app(app(plus, app(s, 0)), _x123)), app(app(cons, app(s, app(app(plus, 0), _x191))), app(app(map, app(plus, app(s, 0))), _x152))))))))
app#(app(map, inc), app(app(cons, app(app(cons, _x43), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x121), app(app(cons, _x93), _x92))))), xs))app#(cons, app(app(cons, app(app(plus, app(s, 0)), _x43)), app(app(cons, app(s, app(app(plus, 0), _x91))), app(app(cons, app(s, _x121)), app(app(cons, app(app(plus, app(s, 0)), _x93)), app(app(map, app(plus, app(s, 0))), _x92))))))app#(app(map, inc), app(app(cons, app(app(cons, _x71), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x111), app(app(cons, _x93), app(app(cons, _x123), app(app(cons, _x153), nil))))))), xs))app#(cons, app(app(cons, app(s, app(app(plus, 0), _x71))), app(app(cons, app(s, app(app(plus, 0), _x91))), app(app(cons, app(s, app(app(plus, 0), _x111))), app(app(cons, app(app(plus, app(s, 0)), _x93)), app(app(cons, app(app(plus, app(s, 0)), _x123)), app(app(cons, app(app(plus, app(s, 0)), _x153)), nil)))))))
app#(app(map, inc), app(app(cons, app(app(cons, _x71), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x111), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x123), app(app(cons, _x133), _x132))))))), xs))app#(cons, app(app(cons, app(s, app(app(plus, 0), _x71))), app(app(cons, app(s, app(app(plus, 0), _x91))), app(app(cons, app(s, app(app(plus, 0), _x111))), app(app(cons, app(s, app(app(plus, 0), _x131))), app(app(cons, app(app(plus, app(s, 0)), _x123)), app(app(cons, app(app(plus, app(s, 0)), _x133)), app(app(map, app(plus, app(s, 0))), _x132))))))))app#(app(map, inc), app(app(cons, app(app(cons, _x81), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x121), app(app(cons, _x141), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x191), nil))))))), xs))app#(cons, app(app(cons, app(s, _x81)), app(app(cons, app(s, _x101)), app(app(cons, app(s, _x121)), app(app(cons, app(s, _x141)), app(app(cons, app(s, app(app(plus, 0), _x161))), app(app(cons, app(s, app(app(plus, 0), _x191))), nil)))))))
app#(app(map, inc), app(app(cons, app(app(cons, _x71), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x121), app(app(cons, _x131), app(app(cons, _x123), app(app(cons, _x181), _x152))))))), xs))app#(cons, app(app(cons, app(s, app(app(plus, 0), _x71))), app(app(cons, app(s, _x101)), app(app(cons, app(s, _x121)), app(app(cons, app(s, app(app(plus, 0), _x131))), app(app(cons, app(app(plus, app(s, 0)), _x123)), app(app(cons, app(s, _x181)), app(app(map, app(plus, app(s, 0))), _x152))))))))app#(app(map, inc), app(app(cons, app(app(cons, _x71), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x121), app(app(cons, _x93), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x191), app(app(cons, _x163), _x162)))))))), xs))app#(cons, app(app(cons, app(s, app(app(plus, 0), _x71))), app(app(cons, app(s, _x101)), app(app(cons, app(s, _x121)), app(app(cons, app(app(plus, app(s, 0)), _x93)), app(app(cons, app(s, app(app(plus, 0), _x161))), app(app(cons, app(s, app(app(plus, 0), _x191))), app(app(cons, app(app(plus, app(s, 0)), _x163)), app(app(map, app(plus, app(s, 0))), _x162)))))))))
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app#(app(map, inc), app(app(cons, app(app(cons, _x43), app(app(cons, _x121), _x52))), xs))app#(cons, app(app(cons, app(app(plus, app(s, 0)), _x43)), app(app(cons, app(s, _x121)), app(app(map, app(plus, app(s, 0))), _x52))))app#(app(map, inc), app(app(cons, app(app(cons, _x43), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x83), app(app(cons, _x93), app(app(cons, _x123), _x122)))))), xs))app#(cons, app(app(cons, app(app(plus, app(s, 0)), _x43)), app(app(cons, app(s, app(app(plus, 0), _x91))), app(app(cons, app(app(plus, app(s, 0)), _x83)), app(app(cons, app(app(plus, app(s, 0)), _x93)), app(app(cons, app(app(plus, app(s, 0)), _x123)), app(app(map, app(plus, app(s, 0))), _x122)))))))
app#(app(map, inc), app(app(cons, app(app(cons, _x81), app(app(cons, _x91), app(app(cons, _x121), _x82)))), xs))app#(cons, app(app(cons, app(s, _x81)), app(app(cons, app(s, app(app(plus, 0), _x91))), app(app(cons, app(s, _x121)), app(app(map, app(plus, app(s, 0))), _x82)))))app#(app(map, inc), app(app(cons, app(app(cons, _x81), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x121), app(app(cons, _x131), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x191), app(app(cons, _x163), app(app(cons, _x173), _x172))))))))), xs))app#(cons, app(app(cons, app(s, _x81)), app(app(cons, app(s, _x101)), app(app(cons, app(s, _x121)), app(app(cons, app(s, app(app(plus, 0), _x131))), app(app(cons, app(s, app(app(plus, 0), _x161))), app(app(cons, app(s, app(app(plus, 0), _x191))), app(app(cons, app(app(plus, app(s, 0)), _x163)), app(app(cons, app(app(plus, app(s, 0)), _x173)), app(app(map, app(plus, app(s, 0))), _x172))))))))))
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app#(app(map, inc), app(app(cons, app(app(cons, _x71), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x121), app(app(cons, _x131), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x191), app(app(cons, _x163), nil)))))))), xs))app#(cons, app(app(cons, app(s, app(app(plus, 0), _x71))), app(app(cons, app(s, _x101)), app(app(cons, app(s, _x121)), app(app(cons, app(s, app(app(plus, 0), _x131))), app(app(cons, app(s, app(app(plus, 0), _x161))), app(app(cons, app(s, app(app(plus, 0), _x191))), app(app(cons, app(app(plus, app(s, 0)), _x163)), nil))))))))app#(app(map, inc), app(app(cons, app(app(cons, _x71), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x111), app(app(cons, _x131), app(app(cons, _x123), app(app(cons, _x181), _x152))))))), xs))app#(cons, app(app(cons, app(s, app(app(plus, 0), _x71))), app(app(cons, app(s, _x101)), app(app(cons, app(s, app(app(plus, 0), _x111))), app(app(cons, app(s, app(app(plus, 0), _x131))), app(app(cons, app(app(plus, app(s, 0)), _x123)), app(app(cons, app(s, _x181)), app(app(map, app(plus, app(s, 0))), _x152))))))))
app#(app(map, inc), app(app(cons, app(app(cons, _x71), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x111), app(app(cons, _x131), app(app(cons, _x123), app(app(cons, _x153), _x152))))))), xs))app#(cons, app(app(cons, app(s, app(app(plus, 0), _x71))), app(app(cons, app(s, app(app(plus, 0), _x91))), app(app(cons, app(s, app(app(plus, 0), _x111))), app(app(cons, app(s, app(app(plus, 0), _x131))), app(app(cons, app(app(plus, app(s, 0)), _x123)), app(app(cons, app(app(plus, app(s, 0)), _x153)), app(app(map, app(plus, app(s, 0))), _x152))))))))app#(app(map, inc), app(app(cons, app(app(cons, _x71), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x121), app(app(cons, _x93), app(app(cons, _x123), app(app(cons, _x191), app(app(cons, _x183), _x182)))))))), xs))app#(cons, app(app(cons, app(s, app(app(plus, 0), _x71))), app(app(cons, app(s, app(app(plus, 0), _x91))), app(app(cons, app(s, _x121)), app(app(cons, app(app(plus, app(s, 0)), _x93)), app(app(cons, app(app(plus, app(s, 0)), _x123)), app(app(cons, app(s, app(app(plus, 0), _x191))), app(app(cons, app(app(plus, app(s, 0)), _x183)), app(app(map, app(plus, app(s, 0))), _x182)))))))))
app#(app(map, inc), app(app(cons, app(app(cons, _x43), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x111), app(app(cons, _x93), app(app(cons, _x123), _x122)))))), xs))app#(cons, app(app(cons, app(app(plus, app(s, 0)), _x43)), app(app(cons, app(s, _x101)), app(app(cons, app(s, app(app(plus, 0), _x111))), app(app(cons, app(app(plus, app(s, 0)), _x93)), app(app(cons, app(app(plus, app(s, 0)), _x123)), app(app(map, app(plus, app(s, 0))), _x122)))))))app#(app(map, inc), app(app(cons, app(app(cons, _x71), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x121), app(app(cons, _x93), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x191), app(app(cons, _x163), _x162)))))))), xs))app#(cons, app(app(cons, app(s, app(app(plus, 0), _x71))), app(app(cons, app(s, app(app(plus, 0), _x91))), app(app(cons, app(s, _x121)), app(app(cons, app(app(plus, app(s, 0)), _x93)), app(app(cons, app(s, app(app(plus, 0), _x161))), app(app(cons, app(s, app(app(plus, 0), _x191))), app(app(cons, app(app(plus, app(s, 0)), _x163)), app(app(map, app(plus, app(s, 0))), _x162)))))))))
app#(app(map, inc), app(app(cons, app(app(cons, _x81), _x42)), xs))app#(cons, app(app(cons, app(s, _x81)), app(app(map, app(plus, app(s, 0))), _x42)))app#(app(map, inc), app(app(cons, app(app(cons, _x71), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x121), app(app(cons, _x131), app(app(cons, _x161), app(app(cons, _x181), _x152))))))), xs))app#(cons, app(app(cons, app(s, app(app(plus, 0), _x71))), app(app(cons, app(s, _x101)), app(app(cons, app(s, _x121)), app(app(cons, app(s, app(app(plus, 0), _x131))), app(app(cons, app(s, app(app(plus, 0), _x161))), app(app(cons, app(s, _x181)), app(app(map, app(plus, app(s, 0))), _x152))))))))
app#(app(map, inc), app(app(cons, app(app(cons, _x71), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x111), app(app(cons, _x141), app(app(cons, _x123), _x122)))))), xs))app#(cons, app(app(cons, app(s, app(app(plus, 0), _x71))), app(app(cons, app(s, app(app(plus, 0), _x91))), app(app(cons, app(s, app(app(plus, 0), _x111))), app(app(cons, app(s, _x141)), app(app(cons, app(app(plus, app(s, 0)), _x123)), app(app(map, app(plus, app(s, 0))), _x122)))))))app#(app(map, inc), app(app(cons, app(app(cons, _x81), app(app(cons, _x101), nil))), xs))app#(cons, app(app(cons, app(s, _x81)), app(app(cons, app(s, _x101)), nil)))
app#(app(map, inc), app(app(cons, app(app(cons, _x71), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x83), app(app(cons, _x161), app(app(cons, _x123), _x122)))))), xs))app#(cons, app(app(cons, app(s, app(app(plus, 0), _x71))), app(app(cons, app(s, app(app(plus, 0), _x91))), app(app(cons, app(app(plus, app(s, 0)), _x83)), app(app(cons, app(s, _x161)), app(app(cons, app(app(plus, app(s, 0)), _x123)), app(app(map, app(plus, app(s, 0))), _x122)))))))

Rewrite Rules

app(app(plus, 0), y)yapp(app(plus, app(s, x)), y)app(s, app(app(plus, x), y))
app(inc, xs)app(app(map, app(plus, app(s, 0))), xs)app(app(map, f), nil)nil
app(app(map, f), app(app(cons, x), xs))app(app(cons, app(f, x)), app(app(map, f), xs))

Original Signature

Termination of terms over the following signature is verified: app, plus, 0, s, inc, map, nil, cons

Strategy


The right-hand side of the rule app#(app(map, inc), app(app(cons, app(app(cons, _x81), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x83), app(app(cons, _x93), app(app(cons, _x123), _x122)))))), xs)) → app#(cons, app(app(cons, app(s, _x81)), app(app(cons, app(s, app(app(plus, 0), _x91))), app(app(cons, app(app(plus, app(s, 0)), _x83)), app(app(cons, app(app(plus, app(s, 0)), _x93)), app(app(cons, app(app(plus, app(s, 0)), _x123)), app(app(map, app(plus, app(s, 0))), _x122))))))) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
app#(cons, app(app(cons, app(s, _x81)), app(app(cons, app(s, app(app(plus, 0), _x91))), app(app(cons, app(app(plus, app(s, 0)), _x83)), app(app(cons, app(app(plus, app(s, 0)), _x93)), app(app(cons, app(app(plus, app(s, 0)), _x123)), nil)))))) 
app#(cons, app(app(cons, app(s, _x81)), app(app(cons, app(s, _x101)), app(app(cons, app(app(plus, app(s, 0)), _x83)), app(app(cons, app(app(plus, app(s, 0)), _x93)), app(app(cons, app(app(plus, app(s, 0)), _x123)), app(app(map, app(plus, app(s, 0))), _x122))))))) 
app#(cons, app(app(cons, app(s, _x81)), app(app(cons, app(s, app(app(plus, 0), _x91))), app(app(cons, app(app(plus, app(s, 0)), _x83)), app(app(cons, app(app(plus, app(s, 0)), _x93)), app(app(cons, app(app(plus, app(s, 0)), _x123)), app(app(cons, app(app(plus, app(s, 0)), _x153)), app(app(map, app(plus, app(s, 0))), _x152)))))))) 
app#(cons, app(app(cons, app(s, _x81)), app(app(cons, app(s, app(app(plus, 0), _x91))), app(app(cons, app(app(plus, app(s, 0)), _x83)), app(app(cons, app(s, app(app(plus, 0), _x131))), app(app(cons, app(app(plus, app(s, 0)), _x123)), app(app(map, app(plus, app(s, 0))), _x122))))))) 
app#(cons, app(app(cons, app(s, _x81)), app(app(cons, app(s, app(app(plus, 0), _x91))), app(app(cons, app(s, app(app(plus, 0), _x111))), app(app(cons, app(app(plus, app(s, 0)), _x93)), app(app(cons, app(app(plus, app(s, 0)), _x123)), app(app(map, app(plus, app(s, 0))), _x122))))))) 
app#(cons, app(app(cons, app(s, _x81)), app(app(cons, app(s, app(app(plus, 0), _x91))), app(app(cons, app(app(plus, app(s, 0)), _x83)), app(app(cons, app(app(plus, app(s, 0)), _x93)), app(app(cons, app(s, app(app(plus, 0), _x161))), app(app(map, app(plus, app(s, 0))), _x122))))))) 
Thus, the rule app#(app(map, inc), app(app(cons, app(app(cons, _x81), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x83), app(app(cons, _x93), app(app(cons, _x123), _x122)))))), xs)) → app#(cons, app(app(cons, app(s, _x81)), app(app(cons, app(s, app(app(plus, 0), _x91))), app(app(cons, app(app(plus, app(s, 0)), _x83)), app(app(cons, app(app(plus, app(s, 0)), _x93)), app(app(cons, app(app(plus, app(s, 0)), _x123)), app(app(map, app(plus, app(s, 0))), _x122))))))) is replaced by the following rules:
app#(app(map, inc), app(app(cons, app(app(cons, _x81), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x83), app(app(cons, _x93), app(app(cons, _x123), nil)))))), xs)) → app#(cons, app(app(cons, app(s, _x81)), app(app(cons, app(s, app(app(plus, 0), _x91))), app(app(cons, app(app(plus, app(s, 0)), _x83)), app(app(cons, app(app(plus, app(s, 0)), _x93)), app(app(cons, app(app(plus, app(s, 0)), _x123)), nil))))))app#(app(map, inc), app(app(cons, app(app(cons, _x81), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x83), app(app(cons, _x93), app(app(cons, _x123), _x122)))))), xs)) → app#(cons, app(app(cons, app(s, _x81)), app(app(cons, app(s, _x101)), app(app(cons, app(app(plus, app(s, 0)), _x83)), app(app(cons, app(app(plus, app(s, 0)), _x93)), app(app(cons, app(app(plus, app(s, 0)), _x123)), app(app(map, app(plus, app(s, 0))), _x122)))))))
app#(app(map, inc), app(app(cons, app(app(cons, _x81), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x83), app(app(cons, _x93), app(app(cons, _x161), _x122)))))), xs)) → app#(cons, app(app(cons, app(s, _x81)), app(app(cons, app(s, app(app(plus, 0), _x91))), app(app(cons, app(app(plus, app(s, 0)), _x83)), app(app(cons, app(app(plus, app(s, 0)), _x93)), app(app(cons, app(s, app(app(plus, 0), _x161))), app(app(map, app(plus, app(s, 0))), _x122)))))))app#(app(map, inc), app(app(cons, app(app(cons, _x81), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x83), app(app(cons, _x93), app(app(cons, _x123), app(app(cons, _x153), _x152))))))), xs)) → app#(cons, app(app(cons, app(s, _x81)), app(app(cons, app(s, app(app(plus, 0), _x91))), app(app(cons, app(app(plus, app(s, 0)), _x83)), app(app(cons, app(app(plus, app(s, 0)), _x93)), app(app(cons, app(app(plus, app(s, 0)), _x123)), app(app(cons, app(app(plus, app(s, 0)), _x153)), app(app(map, app(plus, app(s, 0))), _x152))))))))
app#(app(map, inc), app(app(cons, app(app(cons, _x81), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x111), app(app(cons, _x93), app(app(cons, _x123), _x122)))))), xs)) → app#(cons, app(app(cons, app(s, _x81)), app(app(cons, app(s, app(app(plus, 0), _x91))), app(app(cons, app(s, app(app(plus, 0), _x111))), app(app(cons, app(app(plus, app(s, 0)), _x93)), app(app(cons, app(app(plus, app(s, 0)), _x123)), app(app(map, app(plus, app(s, 0))), _x122)))))))app#(app(map, inc), app(app(cons, app(app(cons, _x81), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x83), app(app(cons, _x131), app(app(cons, _x123), _x122)))))), xs)) → app#(cons, app(app(cons, app(s, _x81)), app(app(cons, app(s, app(app(plus, 0), _x91))), app(app(cons, app(app(plus, app(s, 0)), _x83)), app(app(cons, app(s, app(app(plus, 0), _x131))), app(app(cons, app(app(plus, app(s, 0)), _x123)), app(app(map, app(plus, app(s, 0))), _x122)))))))

Problem 23: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

app#(app(map, inc), app(app(cons, app(app(cons, _x71), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x121), app(app(cons, _x131), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x191), app(app(cons, _x191), _x162)))))))), xs))app#(cons, app(app(cons, app(s, app(app(plus, 0), _x71))), app(app(cons, app(s, _x101)), app(app(cons, app(s, _x121)), app(app(cons, app(s, app(app(plus, 0), _x131))), app(app(cons, app(s, app(app(plus, 0), _x161))), app(app(cons, app(s, app(app(plus, 0), _x191))), app(app(cons, app(s, app(app(plus, 0), _x191))), app(app(map, app(plus, app(s, 0))), _x162)))))))))app#(app(map, f), app(app(cons, x), xs))app#(app(map, f), xs)
app#(app(map, inc), app(app(cons, app(app(cons, _x71), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x121), app(app(cons, _x141), app(app(cons, _x123), app(app(cons, _x191), _x152))))))), xs))app#(cons, app(app(cons, app(s, app(app(plus, 0), _x71))), app(app(cons, app(s, _x101)), app(app(cons, app(s, _x121)), app(app(cons, app(s, _x141)), app(app(cons, app(app(plus, app(s, 0)), _x123)), app(app(cons, app(s, app(app(plus, 0), _x191))), app(app(map, app(plus, app(s, 0))), _x152))))))))app#(app(map, inc), app(app(cons, app(app(cons, _x71), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x111), app(app(cons, _x131), app(app(cons, _x161), app(app(cons, _x191), _x152))))))), xs))app#(cons, app(app(cons, app(s, app(app(plus, 0), _x71))), app(app(cons, app(s, app(app(plus, 0), _x91))), app(app(cons, app(s, app(app(plus, 0), _x111))), app(app(cons, app(s, app(app(plus, 0), _x131))), app(app(cons, app(s, app(app(plus, 0), _x161))), app(app(cons, app(s, app(app(plus, 0), _x191))), app(app(map, app(plus, app(s, 0))), _x152))))))))
app#(app(map, inc), app(app(cons, app(app(cons, _x81), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x121), app(app(cons, _x131), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x181), app(app(cons, _x163), nil)))))))), xs))app#(cons, app(app(cons, app(s, _x81)), app(app(cons, app(s, app(app(plus, 0), _x91))), app(app(cons, app(s, _x121)), app(app(cons, app(s, app(app(plus, 0), _x131))), app(app(cons, app(s, _x171)), app(app(cons, app(s, _x181)), app(app(cons, app(app(plus, app(s, 0)), _x163)), nil))))))))app#(app(map, inc), app(app(cons, app(app(cons, _x71), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x83), app(app(cons, _x141), app(app(cons, _x123), _x122)))))), xs))app#(cons, app(app(cons, app(s, app(app(plus, 0), _x71))), app(app(cons, app(s, app(app(plus, 0), _x91))), app(app(cons, app(app(plus, app(s, 0)), _x83)), app(app(cons, app(s, _x141)), app(app(cons, app(app(plus, app(s, 0)), _x123)), app(app(map, app(plus, app(s, 0))), _x122)))))))
app#(app(map, inc), app(app(cons, app(app(cons, _x71), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x121), app(app(cons, _x93), app(app(cons, _x123), _x122)))))), xs))app#(cons, app(app(cons, app(s, app(app(plus, 0), _x71))), app(app(cons, app(s, app(app(plus, 0), _x91))), app(app(cons, app(s, _x121)), app(app(cons, app(app(plus, app(s, 0)), _x93)), app(app(cons, app(app(plus, app(s, 0)), _x123)), app(app(map, app(plus, app(s, 0))), _x122)))))))app#(app(map, inc), app(app(cons, app(app(cons, _x43), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x121), nil)))), xs))app#(cons, app(app(cons, app(app(plus, app(s, 0)), _x43)), app(app(cons, app(s, app(app(plus, 0), _x91))), app(app(cons, app(s, _x121)), nil))))
app#(app(map, inc), app(app(cons, app(app(cons, _x43), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x111), app(app(cons, _x93), app(app(cons, _x123), app(app(cons, _x153), _x152))))))), xs))app#(cons, app(app(cons, app(app(plus, app(s, 0)), _x43)), app(app(cons, app(s, app(app(plus, 0), _x91))), app(app(cons, app(s, app(app(plus, 0), _x111))), app(app(cons, app(app(plus, app(s, 0)), _x93)), app(app(cons, app(app(plus, app(s, 0)), _x123)), app(app(cons, app(app(plus, app(s, 0)), _x153)), app(app(map, app(plus, app(s, 0))), _x152))))))))app#(app(map, inc), app(app(cons, app(app(cons, _x43), app(app(cons, _x101), app(app(cons, _x111), app(app(cons, _x141), nil))))), xs))app#(cons, app(app(cons, app(app(plus, app(s, 0)), _x43)), app(app(cons, app(s, _x101)), app(app(cons, app(s, app(app(plus, 0), _x111))), app(app(cons, app(s, _x141)), nil)))))
app#(app(map, inc), app(app(cons, app(app(cons, _x81), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x83), app(app(cons, _x131), app(app(cons, _x123), _x122)))))), xs))app#(cons, app(app(cons, app(s, _x81)), app(app(cons, app(s, app(app(plus, 0), _x91))), app(app(cons, app(app(plus, app(s, 0)), _x83)), app(app(cons, app(s, app(app(plus, 0), _x131))), app(app(cons, app(app(plus, app(s, 0)), _x123)), app(app(map, app(plus, app(s, 0))), _x122)))))))app#(app(map, inc), app(app(cons, app(app(cons, _x81), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x121), app(app(cons, _x141), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x191), app(app(cons, _x163), app(app(cons, _x173), _x172))))))))), xs))app#(cons, app(app(cons, app(s, _x81)), app(app(cons, app(s, _x101)), app(app(cons, app(s, _x121)), app(app(cons, app(s, _x141)), app(app(cons, app(s, _x171)), app(app(cons, app(s, app(app(plus, 0), _x191))), app(app(cons, app(app(plus, app(s, 0)), _x163)), app(app(cons, app(app(plus, app(s, 0)), _x173)), app(app(map, app(plus, app(s, 0))), _x172))))))))))
app#(app(map, inc), app(app(cons, app(app(cons, _x81), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x121), app(app(cons, _x131), app(app(cons, _x151), app(app(cons, _x181), nil))))))), xs))app#(cons, app(app(cons, app(s, _x81)), app(app(cons, app(s, _x101)), app(app(cons, app(s, _x121)), app(app(cons, app(s, app(app(plus, 0), _x131))), app(app(cons, app(s, app(app(plus, 0), _x151))), app(app(cons, app(s, _x181)), nil)))))))app#(app(map, inc), app(app(cons, app(app(cons, _x81), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x83), app(app(cons, _x93), app(app(cons, _x161), nil)))))), xs))app#(cons, app(app(cons, app(s, _x81)), app(app(cons, app(s, _x101)), app(app(cons, app(app(plus, app(s, 0)), _x83)), app(app(cons, app(app(plus, app(s, 0)), _x93)), app(app(cons, app(s, app(app(plus, 0), _x161))), nil))))))
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app#(app(map, inc), app(app(cons, app(app(cons, _x71), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x83), app(app(cons, _x93), nil))))), xs))app#(cons, app(app(cons, app(s, app(app(plus, 0), _x71))), app(app(cons, app(s, _x101)), app(app(cons, app(app(plus, app(s, 0)), _x83)), app(app(cons, app(app(plus, app(s, 0)), _x93)), nil)))))app#(app(map, inc), app(app(cons, app(app(cons, _x81), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x111), app(app(cons, _x93), app(app(cons, _x171), app(app(cons, _x153), _x152))))))), xs))app#(cons, app(app(cons, app(s, _x81)), app(app(cons, app(s, app(app(plus, 0), _x91))), app(app(cons, app(s, app(app(plus, 0), _x111))), app(app(cons, app(app(plus, app(s, 0)), _x93)), app(app(cons, app(s, _x171)), app(app(cons, app(app(plus, app(s, 0)), _x153)), app(app(map, app(plus, app(s, 0))), _x152))))))))
app#(app(map, inc), app(app(cons, app(app(cons, _x43), app(app(cons, _x101), app(app(cons, _x111), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x123), app(app(cons, _x171), _x132))))))), xs))app#(cons, app(app(cons, app(app(plus, app(s, 0)), _x43)), app(app(cons, app(s, _x101)), app(app(cons, app(s, app(app(plus, 0), _x111))), app(app(cons, app(s, app(app(plus, 0), _x131))), app(app(cons, app(app(plus, app(s, 0)), _x123)), app(app(cons, app(s, app(app(plus, 0), _x171))), app(app(map, app(plus, app(s, 0))), _x132))))))))app#(app(map, inc), app(app(cons, app(app(cons, _x81), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x111), app(app(cons, _x131), _x92))))), xs))app#(cons, app(app(cons, app(s, _x81)), app(app(cons, app(s, _x101)), app(app(cons, app(s, app(app(plus, 0), _x111))), app(app(cons, app(s, app(app(plus, 0), _x131))), app(app(map, app(plus, app(s, 0))), _x92))))))
app#(app(map, inc), app(app(cons, app(app(cons, _x81), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x121), app(app(cons, _x131), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x181), app(app(cons, _x163), _x162)))))))), xs))app#(cons, app(app(cons, app(s, _x81)), app(app(cons, app(s, _x101)), app(app(cons, app(s, _x121)), app(app(cons, app(s, app(app(plus, 0), _x131))), app(app(cons, app(s, app(app(plus, 0), _x161))), app(app(cons, app(s, _x181)), app(app(cons, app(app(plus, app(s, 0)), _x163)), app(app(map, app(plus, app(s, 0))), _x162)))))))))app#(app(map, inc), app(app(cons, app(app(cons, _x71), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x121), app(app(cons, _x131), app(app(cons, _x171), app(app(cons, _x191), _x152))))))), xs))app#(cons, app(app(cons, app(s, app(app(plus, 0), _x71))), app(app(cons, app(s, app(app(plus, 0), _x91))), app(app(cons, app(s, _x121)), app(app(cons, app(s, app(app(plus, 0), _x131))), app(app(cons, app(s, _x171)), app(app(cons, app(s, app(app(plus, 0), _x191))), app(app(map, app(plus, app(s, 0))), _x152))))))))
app#(app(map, inc), app(app(cons, app(app(cons, _x71), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x121), app(app(cons, _x93), app(app(cons, _x171), app(app(cons, _x191), nil))))))), xs))app#(cons, app(app(cons, app(s, app(app(plus, 0), _x71))), app(app(cons, app(s, _x101)), app(app(cons, app(s, _x121)), app(app(cons, app(app(plus, app(s, 0)), _x93)), app(app(cons, app(s, _x171)), app(app(cons, app(s, app(app(plus, 0), _x191))), nil)))))))app#(app(map, inc), app(app(cons, app(app(cons, _x81), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x121), app(app(cons, _x131), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x191), app(app(cons, _x191), _x162)))))))), xs))app#(cons, app(app(cons, app(s, _x81)), app(app(cons, app(s, _x101)), app(app(cons, app(s, _x121)), app(app(cons, app(s, app(app(plus, 0), _x131))), app(app(cons, app(s, app(app(plus, 0), _x161))), app(app(cons, app(s, app(app(plus, 0), _x191))), app(app(cons, app(s, app(app(plus, 0), _x191))), app(app(map, app(plus, app(s, 0))), _x162)))))))))
app#(app(map, inc), app(app(cons, app(app(cons, _x81), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x121), app(app(cons, _x141), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x191), app(app(cons, _x191), app(app(cons, _x173), _x172))))))))), xs))app#(cons, app(app(cons, app(s, _x81)), app(app(cons, app(s, _x101)), app(app(cons, app(s, _x121)), app(app(cons, app(s, _x141)), app(app(cons, app(s, app(app(plus, 0), _x161))), app(app(cons, app(s, app(app(plus, 0), _x191))), app(app(cons, app(s, app(app(plus, 0), _x191))), app(app(cons, app(app(plus, app(s, 0)), _x173)), app(app(map, app(plus, app(s, 0))), _x172))))))))))app#(app(map, inc), app(app(cons, app(app(cons, _x71), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x111), app(app(cons, _x93), app(app(cons, _x123), app(app(cons, _x191), nil))))))), xs))app#(cons, app(app(cons, app(s, app(app(plus, 0), _x71))), app(app(cons, app(s, app(app(plus, 0), _x91))), app(app(cons, app(s, app(app(plus, 0), _x111))), app(app(cons, app(app(plus, app(s, 0)), _x93)), app(app(cons, app(app(plus, app(s, 0)), _x123)), app(app(cons, app(s, app(app(plus, 0), _x191))), nil)))))))
app#(app(map, inc), app(app(cons, app(app(cons, _x43), app(app(cons, _x101), app(app(cons, _x121), app(app(cons, _x131), app(app(cons, _x123), _x122)))))), xs))app#(cons, app(app(cons, app(app(plus, app(s, 0)), _x43)), app(app(cons, app(s, _x101)), app(app(cons, app(s, _x121)), app(app(cons, app(s, app(app(plus, 0), _x131))), app(app(cons, app(app(plus, app(s, 0)), _x123)), app(app(map, app(plus, app(s, 0))), _x122)))))))app#(app(map, inc), app(app(cons, app(app(cons, _x81), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x121), app(app(cons, _x141), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x191), app(app(cons, _x191), _x162)))))))), xs))app#(cons, app(app(cons, app(s, _x81)), app(app(cons, app(s, _x101)), app(app(cons, app(s, _x121)), app(app(cons, app(s, _x141)), app(app(cons, app(s, app(app(plus, 0), _x161))), app(app(cons, app(s, app(app(plus, 0), _x191))), app(app(cons, app(s, app(app(plus, 0), _x191))), app(app(map, app(plus, app(s, 0))), _x162)))))))))
app#(app(map, inc), app(app(cons, app(app(cons, _x81), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x121), app(app(cons, _x141), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x181), nil))))))), xs))app#(cons, app(app(cons, app(s, _x81)), app(app(cons, app(s, app(app(plus, 0), _x91))), app(app(cons, app(s, _x121)), app(app(cons, app(s, _x141)), app(app(cons, app(s, app(app(plus, 0), _x161))), app(app(cons, app(s, _x181)), nil)))))))app#(app(map, inc), app(app(cons, app(app(cons, _x71), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x121), app(app(cons, _x131), app(app(cons, _x123), app(app(cons, _x181), app(app(cons, _x163), _x162)))))))), xs))app#(cons, app(app(cons, app(s, app(app(plus, 0), _x71))), app(app(cons, app(s, _x101)), app(app(cons, app(s, _x121)), app(app(cons, app(s, app(app(plus, 0), _x131))), app(app(cons, app(app(plus, app(s, 0)), _x123)), app(app(cons, app(s, _x181)), app(app(cons, app(app(plus, app(s, 0)), _x163)), app(app(map, app(plus, app(s, 0))), _x162)))))))))
app#(app(map, inc), app(app(cons, app(app(cons, _x71), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x121), app(app(cons, _x141), app(app(cons, _x151), app(app(cons, _x191), app(app(cons, _x191), app(app(cons, _x211), _x172))))))))), xs))app#(cons, app(app(cons, app(s, app(app(plus, 0), _x71))), app(app(cons, app(s, _x101)), app(app(cons, app(s, _x121)), app(app(cons, app(s, _x141)), app(app(cons, app(s, app(app(plus, 0), _x151))), app(app(cons, app(s, app(app(plus, 0), _x191))), app(app(cons, app(s, app(app(plus, 0), _x191))), app(app(cons, app(s, app(app(plus, 0), _x211))), app(app(map, app(plus, app(s, 0))), _x172))))))))))app#(app(map, inc), app(app(cons, app(app(cons, _x81), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x121), app(app(cons, _x131), app(app(cons, _x123), app(app(cons, _x181), nil))))))), xs))app#(cons, app(app(cons, app(s, _x81)), app(app(cons, app(s, _x101)), app(app(cons, app(s, _x121)), app(app(cons, app(s, app(app(plus, 0), _x131))), app(app(cons, app(app(plus, app(s, 0)), _x123)), app(app(cons, app(s, _x181)), nil)))))))
app#(app(map, inc), app(app(cons, app(app(cons, _x81), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x83), app(app(cons, _x93), app(app(cons, _x191), _x122)))))), xs))app#(cons, app(app(cons, app(s, _x81)), app(app(cons, app(s, app(app(plus, 0), _x91))), app(app(cons, app(app(plus, app(s, 0)), _x83)), app(app(cons, app(app(plus, app(s, 0)), _x93)), app(app(cons, app(s, _x191)), app(app(map, app(plus, app(s, 0))), _x122)))))))app#(app(map, inc), app(app(cons, app(app(cons, _x81), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x83), app(app(cons, _x93), app(app(cons, _x171), app(app(cons, _x153), _x152))))))), xs))app#(cons, app(app(cons, app(s, _x81)), app(app(cons, app(s, _x101)), app(app(cons, app(app(plus, app(s, 0)), _x83)), app(app(cons, app(app(plus, app(s, 0)), _x93)), app(app(cons, app(s, _x171)), app(app(cons, app(app(plus, app(s, 0)), _x153)), app(app(map, app(plus, app(s, 0))), _x152))))))))
app#(app(map, inc), app(app(cons, app(app(cons, _x81), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x83), app(app(cons, _x93), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x153), app(app(cons, _x163), app(app(cons, _x193), _x192))))))))), xs))app#(cons, app(app(cons, app(s, _x81)), app(app(cons, app(s, app(app(plus, 0), _x91))), app(app(cons, app(app(plus, app(s, 0)), _x83)), app(app(cons, app(app(plus, app(s, 0)), _x93)), app(app(cons, app(s, app(app(plus, 0), _x161))), app(app(cons, app(app(plus, app(s, 0)), _x153)), app(app(cons, app(app(plus, app(s, 0)), _x163)), app(app(cons, app(app(plus, app(s, 0)), _x193)), app(app(map, app(plus, app(s, 0))), _x192))))))))))app#(app(map, inc), app(app(cons, app(app(cons, _x71), app(app(cons, _x101), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x93), app(app(cons, _x123), _x122)))))), xs))app#(cons, app(app(cons, app(s, app(app(plus, 0), _x71))), app(app(cons, app(s, _x101)), app(app(cons, app(s, _x121)), app(app(cons, app(app(plus, app(s, 0)), _x93)), app(app(cons, app(app(plus, app(s, 0)), _x123)), app(app(map, app(plus, app(s, 0))), _x122)))))))
app#(app(map, inc), app(app(cons, app(app(cons, _x43), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x111), app(app(cons, _x93), app(app(cons, _x123), _x122)))))), xs))app#(cons, app(app(cons, app(app(plus, app(s, 0)), _x43)), app(app(cons, app(s, _x101)), app(app(cons, app(s, app(app(plus, 0), _x111))), app(app(cons, app(app(plus, app(s, 0)), _x93)), app(app(cons, app(app(plus, app(s, 0)), _x123)), app(app(map, app(plus, app(s, 0))), _x122)))))))app#(app(map, inc), app(app(cons, app(app(cons, _x81), _x42)), xs))app#(cons, app(app(cons, app(s, _x81)), app(app(map, app(plus, app(s, 0))), _x42)))
app#(app(map, inc), app(app(cons, app(app(cons, _x71), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x121), app(app(cons, _x131), app(app(cons, _x161), app(app(cons, _x181), _x152))))))), xs))app#(cons, app(app(cons, app(s, app(app(plus, 0), _x71))), app(app(cons, app(s, _x101)), app(app(cons, app(s, _x121)), app(app(cons, app(s, app(app(plus, 0), _x131))), app(app(cons, app(s, app(app(plus, 0), _x161))), app(app(cons, app(s, _x181)), app(app(map, app(plus, app(s, 0))), _x152))))))))app#(app(map, inc), app(app(cons, app(app(cons, _x71), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x111), app(app(cons, _x141), app(app(cons, _x123), _x122)))))), xs))app#(cons, app(app(cons, app(s, app(app(plus, 0), _x71))), app(app(cons, app(s, app(app(plus, 0), _x91))), app(app(cons, app(s, app(app(plus, 0), _x111))), app(app(cons, app(s, _x141)), app(app(cons, app(app(plus, app(s, 0)), _x123)), app(app(map, app(plus, app(s, 0))), _x122)))))))
app#(app(map, inc), app(app(cons, app(app(cons, _x71), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x83), app(app(cons, _x161), app(app(cons, _x123), _x122)))))), xs))app#(cons, app(app(cons, app(s, app(app(plus, 0), _x71))), app(app(cons, app(s, app(app(plus, 0), _x91))), app(app(cons, app(app(plus, app(s, 0)), _x83)), app(app(cons, app(s, _x161)), app(app(cons, app(app(plus, app(s, 0)), _x123)), app(app(map, app(plus, app(s, 0))), _x122)))))))

Rewrite Rules

app(app(plus, 0), y)yapp(app(plus, app(s, x)), y)app(s, app(app(plus, x), y))
app(inc, xs)app(app(map, app(plus, app(s, 0))), xs)app(app(map, f), nil)nil
app(app(map, f), app(app(cons, x), xs))app(app(cons, app(f, x)), app(app(map, f), xs))

Original Signature

Termination of terms over the following signature is verified: app, plus, 0, s, inc, map, nil, cons

Strategy


The right-hand side of the rule app#(app(map, inc), app(app(cons, app(app(cons, _x71), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x121), app(app(cons, _x141), app(app(cons, _x123), app(app(cons, _x191), _x152))))))), xs)) → app#(cons, app(app(cons, app(s, app(app(plus, 0), _x71))), app(app(cons, app(s, _x101)), app(app(cons, app(s, _x121)), app(app(cons, app(s, _x141)), app(app(cons, app(app(plus, app(s, 0)), _x123)), app(app(cons, app(s, app(app(plus, 0), _x191))), app(app(map, app(plus, app(s, 0))), _x152)))))))) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
app#(cons, app(app(cons, app(s, app(app(plus, 0), _x71))), app(app(cons, app(s, _x101)), app(app(cons, app(s, _x121)), app(app(cons, app(s, _x141)), app(app(cons, app(app(plus, app(s, 0)), _x123)), app(app(cons, app(s, _x181)), app(app(map, app(plus, app(s, 0))), _x152)))))))) 
app#(cons, app(app(cons, app(s, app(app(plus, 0), _x71))), app(app(cons, app(s, _x101)), app(app(cons, app(s, _x121)), app(app(cons, app(s, _x141)), app(app(cons, app(s, app(app(plus, 0), _x151))), app(app(cons, app(s, app(app(plus, 0), _x191))), app(app(map, app(plus, app(s, 0))), _x152)))))))) 
app#(cons, app(app(cons, app(s, _x81)), app(app(cons, app(s, _x101)), app(app(cons, app(s, _x121)), app(app(cons, app(s, _x141)), app(app(cons, app(app(plus, app(s, 0)), _x123)), app(app(cons, app(s, app(app(plus, 0), _x191))), app(app(map, app(plus, app(s, 0))), _x152)))))))) 
app#(cons, app(app(cons, app(s, app(app(plus, 0), _x71))), app(app(cons, app(s, _x101)), app(app(cons, app(s, _x121)), app(app(cons, app(s, _x141)), app(app(cons, app(app(plus, app(s, 0)), _x123)), app(app(cons, app(s, app(app(plus, 0), _x191))), app(app(cons, app(app(plus, app(s, 0)), _x163)), app(app(map, app(plus, app(s, 0))), _x162))))))))) 
app#(cons, app(app(cons, app(s, app(app(plus, 0), _x71))), app(app(cons, app(s, _x101)), app(app(cons, app(s, _x121)), app(app(cons, app(s, _x141)), app(app(cons, app(app(plus, app(s, 0)), _x123)), app(app(cons, app(s, app(app(plus, 0), _x191))), nil))))))) 
Thus, the rule app#(app(map, inc), app(app(cons, app(app(cons, _x71), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x121), app(app(cons, _x141), app(app(cons, _x123), app(app(cons, _x191), _x152))))))), xs)) → app#(cons, app(app(cons, app(s, app(app(plus, 0), _x71))), app(app(cons, app(s, _x101)), app(app(cons, app(s, _x121)), app(app(cons, app(s, _x141)), app(app(cons, app(app(plus, app(s, 0)), _x123)), app(app(cons, app(s, app(app(plus, 0), _x191))), app(app(map, app(plus, app(s, 0))), _x152)))))))) is replaced by the following rules:
app#(app(map, inc), app(app(cons, app(app(cons, _x71), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x121), app(app(cons, _x141), app(app(cons, _x123), app(app(cons, _x191), app(app(cons, _x163), _x162)))))))), xs)) → app#(cons, app(app(cons, app(s, app(app(plus, 0), _x71))), app(app(cons, app(s, _x101)), app(app(cons, app(s, _x121)), app(app(cons, app(s, _x141)), app(app(cons, app(app(plus, app(s, 0)), _x123)), app(app(cons, app(s, app(app(plus, 0), _x191))), app(app(cons, app(app(plus, app(s, 0)), _x163)), app(app(map, app(plus, app(s, 0))), _x162)))))))))app#(app(map, inc), app(app(cons, app(app(cons, _x81), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x121), app(app(cons, _x141), app(app(cons, _x123), app(app(cons, _x191), _x152))))))), xs)) → app#(cons, app(app(cons, app(s, _x81)), app(app(cons, app(s, _x101)), app(app(cons, app(s, _x121)), app(app(cons, app(s, _x141)), app(app(cons, app(app(plus, app(s, 0)), _x123)), app(app(cons, app(s, app(app(plus, 0), _x191))), app(app(map, app(plus, app(s, 0))), _x152))))))))
app#(app(map, inc), app(app(cons, app(app(cons, _x71), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x121), app(app(cons, _x141), app(app(cons, _x123), app(app(cons, _x191), nil))))))), xs)) → app#(cons, app(app(cons, app(s, app(app(plus, 0), _x71))), app(app(cons, app(s, _x101)), app(app(cons, app(s, _x121)), app(app(cons, app(s, _x141)), app(app(cons, app(app(plus, app(s, 0)), _x123)), app(app(cons, app(s, app(app(plus, 0), _x191))), nil)))))))app#(app(map, inc), app(app(cons, app(app(cons, _x71), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x121), app(app(cons, _x141), app(app(cons, _x123), app(app(cons, _x181), _x152))))))), xs)) → app#(cons, app(app(cons, app(s, app(app(plus, 0), _x71))), app(app(cons, app(s, _x101)), app(app(cons, app(s, _x121)), app(app(cons, app(s, _x141)), app(app(cons, app(app(plus, app(s, 0)), _x123)), app(app(cons, app(s, _x181)), app(app(map, app(plus, app(s, 0))), _x152))))))))
app#(app(map, inc), app(app(cons, app(app(cons, _x71), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x121), app(app(cons, _x141), app(app(cons, _x151), app(app(cons, _x191), 0))))))), xs)) → app#(cons, app(app(cons, app(s, app(app(plus, 0), _x71))), app(app(cons, app(s, _x101)), app(app(cons, app(s, _x121)), app(app(cons, app(s, _x141)), app(app(cons, app(s, app(app(plus, 0), _x151))), app(app(cons, app(s, app(app(plus, 0), _x191))), app(app(map, app(plus, app(s, 0))), _x152))))))))

Problem 24: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

app#(app(map, inc), app(app(cons, app(app(cons, _x71), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x121), app(app(cons, _x131), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x191), app(app(cons, _x191), _x162)))))))), xs))app#(cons, app(app(cons, app(s, app(app(plus, 0), _x71))), app(app(cons, app(s, _x101)), app(app(cons, app(s, _x121)), app(app(cons, app(s, app(app(plus, 0), _x131))), app(app(cons, app(s, app(app(plus, 0), _x161))), app(app(cons, app(s, app(app(plus, 0), _x191))), app(app(cons, app(s, app(app(plus, 0), _x191))), app(app(map, app(plus, app(s, 0))), _x162)))))))))app#(app(map, f), app(app(cons, x), xs))app#(app(map, f), xs)
app#(app(map, inc), app(app(cons, app(app(cons, _x71), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x121), app(app(cons, _x141), app(app(cons, _x151), app(app(cons, _x181), nil))))))), xs))app#(cons, app(app(cons, app(s, app(app(plus, 0), _x71))), app(app(cons, app(s, _x101)), app(app(cons, app(s, _x121)), app(app(cons, app(s, _x141)), app(app(cons, app(s, app(app(plus, 0), _x151))), app(app(cons, app(s, _x181)), nil)))))))app#(app(map, inc), app(app(cons, app(app(cons, _x81), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x121), app(app(cons, _x131), app(app(cons, _x151), app(app(cons, _x181), nil))))))), xs))app#(cons, app(app(cons, app(s, _x81)), app(app(cons, app(s, _x101)), app(app(cons, app(s, _x121)), app(app(cons, app(s, app(app(plus, 0), _x131))), app(app(cons, app(s, app(app(plus, 0), _x151))), app(app(cons, app(s, _x181)), nil)))))))
app#(app(map, inc), app(app(cons, app(app(cons, _x81), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x121), app(app(cons, _x93), app(app(cons, _x123), _x122)))))), xs))app#(cons, app(app(cons, app(s, _x81)), app(app(cons, app(s, app(app(plus, 0), _x91))), app(app(cons, app(s, _x121)), app(app(cons, app(app(plus, app(s, 0)), _x93)), app(app(cons, app(app(plus, app(s, 0)), _x123)), app(app(map, app(plus, app(s, 0))), _x122)))))))app#(app(map, inc), app(app(cons, app(app(cons, _x81), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x121), app(app(cons, _x141), app(app(cons, _x151), app(app(cons, _x191), app(app(cons, _x163), app(app(cons, _x211), _x172))))))))), xs))app#(cons, app(app(cons, app(s, _x81)), app(app(cons, app(s, _x101)), app(app(cons, app(s, _x121)), app(app(cons, app(s, _x141)), app(app(cons, app(s, app(app(plus, 0), _x151))), app(app(cons, app(s, app(app(plus, 0), _x191))), app(app(cons, app(app(plus, app(s, 0)), _x163)), app(app(cons, app(s, app(app(plus, 0), _x211))), app(app(map, app(plus, app(s, 0))), _x172))))))))))
app#(app(map, inc), app(app(cons, app(app(cons, _x81), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x121), app(app(cons, _x141), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x191), app(app(cons, _x191), app(app(cons, _x173), _x172))))))))), xs))app#(cons, app(app(cons, app(s, _x81)), app(app(cons, app(s, _x101)), app(app(cons, app(s, _x121)), app(app(cons, app(s, _x141)), app(app(cons, app(s, _x171)), app(app(cons, app(s, app(app(plus, 0), _x191))), app(app(cons, app(s, app(app(plus, 0), _x191))), app(app(cons, app(app(plus, app(s, 0)), _x173)), app(app(map, app(plus, app(s, 0))), _x172))))))))))app#(app(map, inc), app(app(cons, app(app(cons, _x81), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x121), app(app(cons, _x93), app(app(cons, _x123), app(app(cons, _x153), nil))))))), xs))app#(cons, app(app(cons, app(s, _x81)), app(app(cons, app(s, _x101)), app(app(cons, app(s, _x121)), app(app(cons, app(app(plus, app(s, 0)), _x93)), app(app(cons, app(app(plus, app(s, 0)), _x123)), app(app(cons, app(app(plus, app(s, 0)), _x153)), nil)))))))
app#(app(map, inc), app(app(cons, app(app(cons, _x81), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x83), app(app(cons, _x93), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x153), app(app(cons, _x163), nil)))))))), xs))app#(cons, app(app(cons, app(s, _x81)), app(app(cons, app(s, app(app(plus, 0), _x91))), app(app(cons, app(app(plus, app(s, 0)), _x83)), app(app(cons, app(app(plus, app(s, 0)), _x93)), app(app(cons, app(s, app(app(plus, 0), _x161))), app(app(cons, app(app(plus, app(s, 0)), _x153)), app(app(cons, app(app(plus, app(s, 0)), _x163)), nil))))))))app#(app(map, inc), app(app(cons, app(app(cons, _x81), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x121), app(app(cons, _x131), app(app(cons, _x151), app(app(cons, _x191), app(app(cons, _x163), app(app(cons, _x211), app(app(cons, _x203), _x202)))))))))), xs))app#(cons, app(app(cons, app(s, _x81)), app(app(cons, app(s, _x101)), app(app(cons, app(s, _x121)), app(app(cons, app(s, app(app(plus, 0), _x131))), app(app(cons, app(s, app(app(plus, 0), _x151))), app(app(cons, app(s, app(app(plus, 0), _x191))), app(app(cons, app(app(plus, app(s, 0)), _x163)), app(app(cons, app(s, app(app(plus, 0), _x211))), app(app(cons, app(app(plus, app(s, 0)), _x203)), app(app(map, app(plus, app(s, 0))), _x202)))))))))))
app#(app(map, inc), app(app(cons, app(app(cons, _x43), app(app(cons, _x101), app(app(cons, _x121), app(app(cons, _x131), nil))))), xs))app#(cons, app(app(cons, app(app(plus, app(s, 0)), _x43)), app(app(cons, app(s, _x101)), app(app(cons, app(s, _x121)), app(app(cons, app(s, app(app(plus, 0), _x131))), nil)))))app#(app(map, inc), app(app(cons, app(app(cons, _x71), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x121), app(app(cons, _x131), app(app(cons, _x151), app(app(cons, _x181), app(app(cons, _x163), app(app(cons, _x173), _x172))))))))), xs))app#(cons, app(app(cons, app(s, app(app(plus, 0), _x71))), app(app(cons, app(s, _x101)), app(app(cons, app(s, _x121)), app(app(cons, app(s, app(app(plus, 0), _x131))), app(app(cons, app(s, app(app(plus, 0), _x151))), app(app(cons, app(s, _x181)), app(app(cons, app(app(plus, app(s, 0)), _x163)), app(app(cons, app(app(plus, app(s, 0)), _x173)), app(app(map, app(plus, app(s, 0))), _x172))))))))))
app#(app(map, inc), app(app(cons, app(app(cons, _x71), app(app(cons, _x101), app(app(cons, _x121), _x82)))), xs))app#(cons, app(app(cons, app(s, app(app(plus, 0), _x71))), app(app(cons, app(s, _x101)), app(app(cons, app(s, _x121)), app(app(map, app(plus, app(s, 0))), _x82)))))app#(app(map, inc), app(app(cons, app(app(cons, _x81), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x121), app(app(cons, _x93), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x191), app(app(cons, _x163), _x162)))))))), xs))app#(cons, app(app(cons, app(s, _x81)), app(app(cons, app(s, _x101)), app(app(cons, app(s, _x121)), app(app(cons, app(app(plus, app(s, 0)), _x93)), app(app(cons, app(s, app(app(plus, 0), _x161))), app(app(cons, app(s, app(app(plus, 0), _x191))), app(app(cons, app(app(plus, app(s, 0)), _x163)), app(app(map, app(plus, app(s, 0))), _x162)))))))))
app#(app(map, inc), app(app(cons, app(app(cons, _x71), app(app(cons, app(plus, app(s, 0))), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x93), app(app(cons, _x123), _x122)))))), xs))app#(cons, app(app(cons, app(s, app(app(plus, 0), _x71))), app(app(cons, app(s, app(app(plus, 0), _x91))), app(app(cons, app(s, _x121)), app(app(cons, app(app(plus, app(s, 0)), _x93)), app(app(cons, app(app(plus, app(s, 0)), _x123)), app(app(map, app(plus, app(s, 0))), _x122)))))))app#(app(map, inc), app(app(cons, app(app(cons, _x81), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x83), app(app(cons, _x131), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x153), app(app(cons, _x163), _x162)))))))), xs))app#(cons, app(app(cons, app(s, _x81)), app(app(cons, app(s, app(app(plus, 0), _x91))), app(app(cons, app(app(plus, app(s, 0)), _x83)), app(app(cons, app(s, app(app(plus, 0), _x131))), app(app(cons, app(s, app(app(plus, 0), _x161))), app(app(cons, app(app(plus, app(s, 0)), _x153)), app(app(cons, app(app(plus, app(s, 0)), _x163)), app(app(map, app(plus, app(s, 0))), _x162)))))))))
app#(app(map, inc), app(app(cons, app(app(cons, _x71), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x83), app(app(cons, _x141), app(app(cons, _x123), app(app(cons, _x133), _x132))))))), xs))app#(cons, app(app(cons, app(s, app(app(plus, 0), _x71))), app(app(cons, app(s, _x101)), app(app(cons, app(app(plus, app(s, 0)), _x83)), app(app(cons, app(s, _x141)), app(app(cons, app(app(plus, app(s, 0)), _x123)), app(app(cons, app(app(plus, app(s, 0)), _x133)), app(app(map, app(plus, app(s, 0))), _x132))))))))app#(app(map, inc), app(app(cons, app(app(cons, _x71), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x121), app(app(cons, _x141), app(app(cons, _x151), app(app(cons, _x181), app(app(cons, _x163), app(app(cons, _x211), _x172))))))))), xs))app#(cons, app(app(cons, app(s, app(app(plus, 0), _x71))), app(app(cons, app(s, _x101)), app(app(cons, app(s, _x121)), app(app(cons, app(s, _x141)), app(app(cons, app(s, app(app(plus, 0), _x151))), app(app(cons, app(s, _x181)), app(app(cons, app(app(plus, app(s, 0)), _x163)), app(app(cons, app(s, app(app(plus, 0), _x211))), app(app(map, app(plus, app(s, 0))), _x172))))))))))
app#(app(map, inc), app(app(cons, app(app(cons, _x43), app(app(cons, _x101), app(app(cons, _x111), app(app(cons, _x141), app(app(cons, _x123), _x122)))))), xs))app#(cons, app(app(cons, app(app(plus, app(s, 0)), _x43)), app(app(cons, app(s, _x101)), app(app(cons, app(s, app(app(plus, 0), _x111))), app(app(cons, app(s, _x141)), app(app(cons, app(app(plus, app(s, 0)), _x123)), app(app(map, app(plus, app(s, 0))), _x122)))))))app#(app(map, inc), app(app(cons, app(app(cons, _x81), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x111), app(app(cons, _x131), app(app(cons, _x123), app(app(cons, _x191), _x152))))))), xs))app#(cons, app(app(cons, app(s, _x81)), app(app(cons, app(s, app(app(plus, 0), _x91))), app(app(cons, app(s, app(app(plus, 0), _x111))), app(app(cons, app(s, app(app(plus, 0), _x131))), app(app(cons, app(app(plus, app(s, 0)), _x123)), app(app(cons, app(s, app(app(plus, 0), _x191))), app(app(map, app(plus, app(s, 0))), _x152))))))))
app#(app(map, inc), app(app(cons, app(app(cons, _x81), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x111), app(app(cons, _x93), app(app(cons, _x123), app(app(cons, _x153), _x152))))))), xs))app#(cons, app(app(cons, app(s, _x81)), app(app(cons, app(s, _x101)), app(app(cons, app(s, app(app(plus, 0), _x111))), app(app(cons, app(app(plus, app(s, 0)), _x93)), app(app(cons, app(app(plus, app(s, 0)), _x123)), app(app(cons, app(app(plus, app(s, 0)), _x153)), app(app(map, app(plus, app(s, 0))), _x152))))))))app#(app(map, inc), app(app(cons, app(app(cons, _x71), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x121), app(app(cons, _x131), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x181), app(app(cons, _x163), app(app(cons, _x173), _x172))))))))), xs))app#(cons, app(app(cons, app(s, app(app(plus, 0), _x71))), app(app(cons, app(s, app(app(plus, 0), _x91))), app(app(cons, app(s, _x121)), app(app(cons, app(s, app(app(plus, 0), _x131))), app(app(cons, app(s, app(app(plus, 0), _x161))), app(app(cons, app(s, _x181)), app(app(cons, app(app(plus, app(s, 0)), _x163)), app(app(cons, app(app(plus, app(s, 0)), _x173)), app(app(map, app(plus, app(s, 0))), _x172))))))))))
app#(app(map, inc), app(app(cons, app(app(cons, _x81), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x83), app(app(cons, _x131), nil))))), xs))app#(cons, app(app(cons, app(s, _x81)), app(app(cons, app(s, _x101)), app(app(cons, app(app(plus, app(s, 0)), _x83)), app(app(cons, app(s, app(app(plus, 0), _x131))), nil)))))app#(app(map, inc), app(app(cons, app(app(cons, _x71), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x121), app(app(cons, _x131), app(app(cons, _x151), app(app(cons, _x191), nil))))))), xs))app#(cons, app(app(cons, app(s, app(app(plus, 0), _x71))), app(app(cons, app(s, _x101)), app(app(cons, app(s, _x121)), app(app(cons, app(s, app(app(plus, 0), _x131))), app(app(cons, app(s, app(app(plus, 0), _x151))), app(app(cons, app(s, app(app(plus, 0), _x191))), nil)))))))
app#(app(map, inc), app(app(cons, app(app(cons, _x71), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x121), app(app(cons, _x131), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x191), app(app(cons, _x163), _x162)))))))), xs))app#(cons, app(app(cons, app(s, app(app(plus, 0), _x71))), app(app(cons, app(s, _x101)), app(app(cons, app(s, _x121)), app(app(cons, app(s, app(app(plus, 0), _x131))), app(app(cons, app(s, _x171)), app(app(cons, app(s, app(app(plus, 0), _x191))), app(app(cons, app(app(plus, app(s, 0)), _x163)), app(app(map, app(plus, app(s, 0))), _x162)))))))))app#(app(map, inc), app(app(cons, app(app(cons, _x43), app(app(cons, _x101), app(app(cons, _x111), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x123), app(app(cons, _x133), app(app(cons, _x163), _x162)))))))), xs))app#(cons, app(app(cons, app(app(plus, app(s, 0)), _x43)), app(app(cons, app(s, _x101)), app(app(cons, app(s, app(app(plus, 0), _x111))), app(app(cons, app(s, _x141)), app(app(cons, app(app(plus, app(s, 0)), _x123)), app(app(cons, app(app(plus, app(s, 0)), _x133)), app(app(cons, app(app(plus, app(s, 0)), _x163)), app(app(map, app(plus, app(s, 0))), _x162)))))))))
app#(app(map, inc), app(app(cons, app(app(cons, _x71), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x83), app(app(cons, _x131), app(app(cons, _x123), _x122)))))), xs))app#(cons, app(app(cons, app(s, app(app(plus, 0), _x71))), app(app(cons, app(s, _x101)), app(app(cons, app(app(plus, app(s, 0)), _x83)), app(app(cons, app(s, app(app(plus, 0), _x131))), app(app(cons, app(app(plus, app(s, 0)), _x123)), app(app(map, app(plus, app(s, 0))), _x122)))))))app#(app(map, inc), app(app(cons, app(app(cons, _x71), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x121), app(app(cons, _x131), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x181), app(app(cons, _x163), _x162)))))))), xs))app#(cons, app(app(cons, app(s, app(app(plus, 0), _x71))), app(app(cons, app(s, _x101)), app(app(cons, app(s, _x121)), app(app(cons, app(s, app(app(plus, 0), _x131))), app(app(cons, app(s, app(app(plus, 0), _x161))), app(app(cons, app(s, _x181)), app(app(cons, app(app(plus, app(s, 0)), _x163)), app(app(map, app(plus, app(s, 0))), _x162)))))))))
app#(app(map, inc), app(app(cons, app(app(cons, _x71), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x121), app(app(cons, _x141), app(app(cons, _x151), app(app(cons, _x191), app(app(cons, _x163), _x162)))))))), xs))app#(cons, app(app(cons, app(s, app(app(plus, 0), _x71))), app(app(cons, app(s, _x101)), app(app(cons, app(s, _x121)), app(app(cons, app(s, _x141)), app(app(cons, app(s, app(app(plus, 0), _x151))), app(app(cons, app(s, app(app(plus, 0), _x191))), app(app(cons, app(app(plus, app(s, 0)), _x163)), app(app(map, app(plus, app(s, 0))), _x162)))))))))app#(app(map, inc), app(app(cons, app(app(cons, _x71), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x111), app(app(cons, _x93), app(app(cons, _x161), nil)))))), xs))app#(cons, app(app(cons, app(s, app(app(plus, 0), _x71))), app(app(cons, app(s, app(app(plus, 0), _x91))), app(app(cons, app(s, app(app(plus, 0), _x111))), app(app(cons, app(app(plus, app(s, 0)), _x93)), app(app(cons, app(s, app(app(plus, 0), _x161))), nil))))))
app#(app(map, inc), app(app(cons, app(app(cons, _x71), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x121), app(app(cons, _x131), app(app(cons, _x151), app(app(cons, _x191), 0))))))), xs))app#(cons, app(app(cons, app(s, app(app(plus, 0), _x71))), app(app(cons, app(s, _x101)), app(app(cons, app(s, _x121)), app(app(cons, app(s, app(app(plus, 0), _x131))), app(app(cons, app(s, app(app(plus, 0), _x151))), app(app(cons, app(s, app(app(plus, 0), _x191))), app(app(map, app(plus, app(s, 0))), _x152))))))))app#(app(map, inc), app(app(cons, app(app(cons, _x71), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x121), app(app(cons, _x131), app(app(cons, _x161), app(app(cons, _x191), app(app(cons, _x163), app(app(cons, _x211), nil))))))))), xs))app#(cons, app(app(cons, app(s, app(app(plus, 0), _x71))), app(app(cons, app(s, _x101)), app(app(cons, app(s, _x121)), app(app(cons, app(s, app(app(plus, 0), _x131))), app(app(cons, app(s, _x161)), app(app(cons, app(s, app(app(plus, 0), _x191))), app(app(cons, app(app(plus, app(s, 0)), _x163)), app(app(cons, app(s, app(app(plus, 0), _x211))), nil)))))))))
app#(app(map, inc), app(app(cons, app(app(cons, _x43), app(app(cons, _x121), _x52))), xs))app#(cons, app(app(cons, app(app(plus, app(s, 0)), _x43)), app(app(cons, app(s, _x121)), app(app(map, app(plus, app(s, 0))), _x52))))app#(app(map, inc), app(app(cons, app(app(cons, _x81), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x121), app(app(cons, _x141), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x181), app(app(cons, _x163), _x162)))))))), xs))app#(cons, app(app(cons, app(s, _x81)), app(app(cons, app(s, app(app(plus, 0), _x91))), app(app(cons, app(s, _x121)), app(app(cons, app(s, _x141)), app(app(cons, app(s, _x171)), app(app(cons, app(s, _x181)), app(app(cons, app(app(plus, app(s, 0)), _x163)), app(app(map, app(plus, app(s, 0))), _x162)))))))))
app#(app(map, inc), app(app(cons, app(app(cons, _x71), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x121), app(app(cons, _x93), app(app(cons, _x171), app(app(cons, _x191), _x152))))))), xs))app#(cons, app(app(cons, app(s, app(app(plus, 0), _x71))), app(app(cons, app(s, app(app(plus, 0), _x91))), app(app(cons, app(s, _x121)), app(app(cons, app(app(plus, app(s, 0)), _x93)), app(app(cons, app(s, _x171)), app(app(cons, app(s, app(app(plus, 0), _x191))), app(app(map, app(plus, app(s, 0))), _x152))))))))app#(app(map, f), app(app(cons, x), xs))app#(app(cons, app(f, x)), app(app(map, f), xs))
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app#(app(map, inc), app(app(cons, app(app(cons, _x71), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x111), app(app(cons, _x131), app(app(cons, _x123), app(app(cons, _x153), nil))))))), xs))app#(cons, app(app(cons, app(s, app(app(plus, 0), _x71))), app(app(cons, app(s, app(app(plus, 0), _x91))), app(app(cons, app(s, app(app(plus, 0), _x111))), app(app(cons, app(s, app(app(plus, 0), _x131))), app(app(cons, app(app(plus, app(s, 0)), _x123)), app(app(cons, app(app(plus, app(s, 0)), _x153)), nil)))))))app#(app(map, inc), app(app(cons, app(app(cons, _x71), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x111), app(app(cons, _x131), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x181), nil))))))), xs))app#(cons, app(app(cons, app(s, app(app(plus, 0), _x71))), app(app(cons, app(s, app(app(plus, 0), _x91))), app(app(cons, app(s, app(app(plus, 0), _x111))), app(app(cons, app(s, app(app(plus, 0), _x131))), app(app(cons, app(s, app(app(plus, 0), _x161))), app(app(cons, app(s, _x181)), nil)))))))
app#(app(map, inc), app(app(cons, app(app(cons, _x81), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x111), app(app(cons, _x93), app(app(cons, _x123), app(app(cons, _x191), nil))))))), xs))app#(cons, app(app(cons, app(s, _x81)), app(app(cons, app(s, app(app(plus, 0), _x91))), app(app(cons, app(s, app(app(plus, 0), _x111))), app(app(cons, app(app(plus, app(s, 0)), _x93)), app(app(cons, app(app(plus, app(s, 0)), _x123)), app(app(cons, app(s, app(app(plus, 0), _x191))), nil)))))))app#(app(map, inc), app(app(cons, app(app(cons, _x81), _x42)), xs))app#(cons, app(app(cons, app(s, _x81)), app(app(map, app(plus, app(s, 0))), _x42)))
app#(app(map, inc), app(app(cons, app(app(cons, _x71), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x121), app(app(cons, _x131), app(app(cons, _x161), app(app(cons, _x181), _x152))))))), xs))app#(cons, app(app(cons, app(s, app(app(plus, 0), _x71))), app(app(cons, app(s, _x101)), app(app(cons, app(s, _x121)), app(app(cons, app(s, app(app(plus, 0), _x131))), app(app(cons, app(s, app(app(plus, 0), _x161))), app(app(cons, app(s, _x181)), app(app(map, app(plus, app(s, 0))), _x152))))))))app#(app(map, inc), app(app(cons, app(app(cons, _x71), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x111), app(app(cons, _x141), app(app(cons, _x123), _x122)))))), xs))app#(cons, app(app(cons, app(s, app(app(plus, 0), _x71))), app(app(cons, app(s, app(app(plus, 0), _x91))), app(app(cons, app(s, app(app(plus, 0), _x111))), app(app(cons, app(s, _x141)), app(app(cons, app(app(plus, app(s, 0)), _x123)), app(app(map, app(plus, app(s, 0))), _x122)))))))
app#(app(map, inc), app(app(cons, app(app(cons, _x71), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x83), app(app(cons, _x161), app(app(cons, _x123), _x122)))))), xs))app#(cons, app(app(cons, app(s, app(app(plus, 0), _x71))), app(app(cons, app(s, app(app(plus, 0), _x91))), app(app(cons, app(app(plus, app(s, 0)), _x83)), app(app(cons, app(s, _x161)), app(app(cons, app(app(plus, app(s, 0)), _x123)), app(app(map, app(plus, app(s, 0))), _x122)))))))

Rewrite Rules

app(app(plus, 0), y)yapp(app(plus, app(s, x)), y)app(s, app(app(plus, x), y))
app(inc, xs)app(app(map, app(plus, app(s, 0))), xs)app(app(map, f), nil)nil
app(app(map, f), app(app(cons, x), xs))app(app(cons, app(f, x)), app(app(map, f), xs))

Original Signature

Termination of terms over the following signature is verified: app, plus, 0, s, inc, map, nil, cons

Strategy


The right-hand side of the rule app#(app(map, inc), app(app(cons, app(app(cons, _x71), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x121), app(app(cons, _x141), app(app(cons, _x151), app(app(cons, _x181), nil))))))), xs)) → app#(cons, app(app(cons, app(s, app(app(plus, 0), _x71))), app(app(cons, app(s, _x101)), app(app(cons, app(s, _x121)), app(app(cons, app(s, _x141)), app(app(cons, app(s, app(app(plus, 0), _x151))), app(app(cons, app(s, _x181)), nil))))))) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
app#(cons, app(app(cons, app(s, app(app(plus, 0), _x71))), app(app(cons, app(s, _x101)), app(app(cons, app(s, _x121)), app(app(cons, app(s, _x141)), app(app(cons, app(s, _x161)), app(app(cons, app(s, _x181)), nil))))))) 
app#(cons, app(app(cons, app(s, _x81)), app(app(cons, app(s, _x101)), app(app(cons, app(s, _x121)), app(app(cons, app(s, _x141)), app(app(cons, app(s, app(app(plus, 0), _x151))), app(app(cons, app(s, _x181)), nil))))))) 
Thus, the rule app#(app(map, inc), app(app(cons, app(app(cons, _x71), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x121), app(app(cons, _x141), app(app(cons, _x151), app(app(cons, _x181), nil))))))), xs)) → app#(cons, app(app(cons, app(s, app(app(plus, 0), _x71))), app(app(cons, app(s, _x101)), app(app(cons, app(s, _x121)), app(app(cons, app(s, _x141)), app(app(cons, app(s, app(app(plus, 0), _x151))), app(app(cons, app(s, _x181)), nil))))))) is replaced by the following rules:
app#(app(map, inc), app(app(cons, app(app(cons, _x71), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x121), app(app(cons, _x141), app(app(cons, _x161), app(app(cons, _x181), nil))))))), xs)) → app#(cons, app(app(cons, app(s, app(app(plus, 0), _x71))), app(app(cons, app(s, _x101)), app(app(cons, app(s, _x121)), app(app(cons, app(s, _x141)), app(app(cons, app(s, _x161)), app(app(cons, app(s, _x181)), nil)))))))app#(app(map, inc), app(app(cons, app(app(cons, _x81), app(app(cons, app(plus, app(s, 0))), app(app(cons, _x121), app(app(cons, _x141), app(app(cons, _x151), app(app(cons, _x181), nil))))))), xs)) → app#(cons, app(app(cons, app(s, _x81)), app(app(cons, app(s, _x101)), app(app(cons, app(s, _x121)), app(app(cons, app(s, _x141)), app(app(cons, app(s, app(app(plus, 0), _x151))), app(app(cons, app(s, _x181)), nil)))))))