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 (4ms).
 | – Problem 2 was processed with processor ForwardNarrowing (2ms).
 |    | – Problem 3 was processed with processor ForwardNarrowing (3ms).
 |    |    | – Problem 4 was processed with processor ForwardNarrowing (2ms).
 |    |    |    | – Problem 5 was processed with processor ForwardNarrowing (3ms).
 |    |    |    |    | – Problem 6 was processed with processor ForwardNarrowing (4ms).
 |    |    |    |    |    | – Problem 7 was processed with processor ForwardNarrowing (1ms).
 |    |    |    |    |    |    | – Problem 8 was processed with processor ForwardNarrowing (1ms).
 |    |    |    |    |    |    |    | – Problem 9 was processed with processor ForwardNarrowing (1ms).
 |    |    |    |    |    |    |    |    | – Problem 10 was processed with processor ForwardNarrowing (2ms).
 |    |    |    |    |    |    |    |    |    | – Problem 11 was processed with processor ForwardNarrowing (2ms).
 |    |    |    |    |    |    |    |    |    |    | – Problem 12 was processed with processor ForwardNarrowing (5ms).
 |    |    |    |    |    |    |    |    |    |    |    | – Problem 13 was processed with processor ForwardNarrowing (3ms).
 |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 14 was processed with processor ForwardNarrowing (4ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 15 was processed with processor ForwardNarrowing (2ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 16 was processed with processor ForwardNarrowing (1ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 17 was processed with processor ForwardNarrowing (4ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 18 was processed with processor ForwardNarrowing (3ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 19 was processed with processor ForwardNarrowing (3ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 20 was processed with processor ForwardNarrowing (3ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 21 was processed with processor ForwardNarrowing (7ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 22 was processed with processor ForwardNarrowing (13ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 23 was processed with processor ForwardNarrowing (30ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 24 was processed with processor ForwardNarrowing (25ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 25 was processed with processor ForwardNarrowing (49ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 26 was processed with processor ForwardNarrowing (85ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 27 was processed with processor ForwardNarrowing (18ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 28 was processed with processor ForwardNarrowing (34ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 29 was processed with processor ForwardNarrowing (58ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 30 was processed with processor ForwardNarrowing (66ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 31 was processed with processor ForwardNarrowing (102ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 32 remains open; application of the following processors failed [ForwardNarrowing (161ms), ForwardNarrowing (118ms), ForwardNarrowing (102ms), ForwardNarrowing (160ms), ForwardNarrowing (108ms), ForwardNarrowing (123ms), ForwardNarrowing (170ms), ForwardNarrowing (109ms), ForwardNarrowing (110ms), ForwardNarrowing (124ms), ForwardNarrowing (133ms), ForwardNarrowing (108ms), ForwardNarrowing (88ms), ForwardNarrowing (90ms), ForwardNarrowing (103ms), ForwardNarrowing (128ms), ForwardNarrowing (123ms), ForwardNarrowing (122ms), ForwardNarrowing (123ms), ForwardNarrowing (153ms), ForwardNarrowing (129ms), ForwardNarrowing (129ms), ForwardNarrowing (158ms), ForwardNarrowing (111ms), ForwardNarrowing (102ms), ForwardNarrowing (99ms), ForwardNarrowing (124ms), ForwardNarrowing (76ms), ForwardNarrowing (65ms), ForwardNarrowing (80ms), ForwardNarrowing (68ms), ForwardNarrowing (80ms), ForwardNarrowing (65ms), ForwardNarrowing (79ms), ForwardNarrowing (86ms), ForwardNarrowing (85ms), ForwardNarrowing (71ms), ForwardNarrowing (69ms), ForwardNarrowing (72ms), ForwardNarrowing (81ms), ForwardNarrowing (84ms), ForwardNarrowing (70ms), ForwardNarrowing (88ms), ForwardNarrowing (74ms), ForwardNarrowing (89ms), ForwardNarrowing (timeout)].

The following open problems remain:



Open Dependency Pair Problem 1

Dependency Pairs

app#(app(sumwith, f), app(app(cons, x), xs))app#(sumwith, f)app#(app(sumwith, f), app(app(cons, x), xs))app#(app(sumwith, f), xs)
app#(app(sumwith, f), app(app(cons, x), xs))app#(f, x)app#(app(sumwith, f), app(app(cons, x), xs))app#(app(plus, app(f, x)), app(app(sumwith, f), xs))
app#(app(plus, app(s, x)), y)app#(s, app(app(plus, x), y))app#(app(sumwith, f), app(app(cons, x), xs))app#(plus, app(f, x))
app#(app(plus, app(s, x)), y)app#(plus, x)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(app(sumwith, f), nil)nilapp(app(sumwith, f), app(app(cons, x), xs))app(app(plus, app(f, x)), app(app(sumwith, f), xs))

Original Signature

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


Problem 1: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

app#(app(sumwith, f), app(app(cons, x), xs))app#(sumwith, f)app#(app(sumwith, f), app(app(cons, x), xs))app#(app(plus, app(f, x)), app(app(sumwith, f), xs))
app#(app(sumwith, f), app(app(cons, x), xs))app#(f, x)app#(app(sumwith, f), app(app(cons, x), xs))app#(app(sumwith, f), xs)
app#(app(plus, app(s, x)), y)app#(s, app(app(plus, x), y))app#(app(sumwith, f), app(app(cons, x), xs))app#(plus, app(f, x))
app#(app(plus, app(s, x)), y)app#(plus, x)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(app(sumwith, f), nil)nilapp(app(sumwith, f), app(app(cons, x), xs))app(app(plus, app(f, x)), app(app(sumwith, f), xs))

Original Signature

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

Strategy


The right-hand side of the rule app#(app(sumwith, f), app(app(cons, x), xs)) → app#(sumwith, 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(sumwith, f), app(app(cons, x), xs)) → app#(sumwith, f) is deleted.

Problem 2: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

app#(app(sumwith, f), app(app(cons, x), xs))app#(app(sumwith, f), xs)app#(app(sumwith, f), app(app(cons, x), xs))app#(f, x)
app#(app(sumwith, f), app(app(cons, x), xs))app#(app(plus, app(f, x)), app(app(sumwith, f), xs))app#(app(plus, app(s, x)), y)app#(s, app(app(plus, x), y))
app#(app(sumwith, f), app(app(cons, x), xs))app#(plus, app(f, x))app#(app(plus, app(s, x)), y)app#(plus, x)
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(app(sumwith, f), nil)nilapp(app(sumwith, f), app(app(cons, x), xs))app(app(plus, app(f, x)), app(app(sumwith, f), xs))

Original Signature

Termination of terms over the following signature is verified: plus, app, 0, s, sumwith, 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(sumwith, f), app(app(cons, x), xs))app#(app(plus, app(f, x)), app(app(sumwith, f), xs))
app#(app(sumwith, f), app(app(cons, x), xs))app#(f, x)app#(app(sumwith, f), app(app(cons, x), xs))app#(app(sumwith, f), xs)
app#(app(sumwith, f), app(app(cons, x), xs))app#(plus, app(f, x))app#(app(plus, app(s, x)), y)app#(plus, x)
app#(app(plus, app(s, x)), y)app#(app(plus, x), y)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(app(sumwith, f), nil)nilapp(app(sumwith, f), app(app(cons, x), xs))app(app(plus, app(f, x)), app(app(sumwith, f), xs))

Original Signature

Termination of terms over the following signature is verified: app, plus, 0, s, sumwith, 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(sumwith, f), app(app(cons, x), xs))app#(app(sumwith, f), xs)app#(app(sumwith, f), app(app(cons, x), xs))app#(f, x)
app#(app(sumwith, f), app(app(cons, x), xs))app#(app(plus, app(f, x)), app(app(sumwith, f), xs))app#(app(plus, app(s, app(s, 0))), _x61)app#(s, app(s, _x61))
app#(app(sumwith, f), app(app(cons, x), xs))app#(plus, app(f, x))app#(app(plus, app(s, x)), y)app#(plus, x)
app#(app(plus, app(s, x)), y)app#(app(plus, x), y)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)

Rewrite Rules

app(app(plus, 0), y)yapp(app(plus, app(s, x)), y)app(s, app(app(plus, x), y))
app(app(sumwith, f), nil)nilapp(app(sumwith, f), app(app(cons, x), xs))app(app(plus, app(f, x)), app(app(sumwith, f), xs))

Original Signature

Termination of terms over the following signature is verified: plus, app, 0, s, sumwith, 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(sumwith, f), app(app(cons, x), xs))app#(app(plus, app(f, x)), app(app(sumwith, f), xs))app#(app(sumwith, f), app(app(cons, x), xs))app#(f, x)
app#(app(sumwith, f), app(app(cons, x), xs))app#(app(sumwith, f), xs)app#(app(sumwith, f), app(app(cons, x), xs))app#(plus, app(f, x))
app#(app(plus, app(s, x)), y)app#(plus, x)app#(app(plus, app(s, x)), y)app#(app(plus, x), y)
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)

Rewrite Rules

app(app(plus, 0), y)yapp(app(plus, app(s, x)), y)app(s, app(app(plus, x), y))
app(app(sumwith, f), nil)nilapp(app(sumwith, f), app(app(cons, x), xs))app(app(plus, app(f, x)), app(app(sumwith, f), xs))

Original Signature

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

Strategy


The right-hand side of the rule app#(app(sumwith, f), app(app(cons, x), xs)) → app#(plus, 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#(plus, app(app(plus, app(_x31, _x33)), app(app(sumwith, _x31), _x32))) 
app#(plus, _x31) 
app#(plus, app(s, app(app(plus, _x32), _x31))) 
app#(plus, nil) 
Thus, the rule app#(app(sumwith, f), app(app(cons, x), xs)) → app#(plus, app(f, x)) is replaced by the following rules:
app#(app(sumwith, app(plus, 0)), app(app(cons, _x31), xs)) → app#(plus, _x31)app#(app(sumwith, app(plus, app(s, _x32))), app(app(cons, _x31), xs)) → app#(plus, app(s, app(app(plus, _x32), _x31)))
app#(app(sumwith, app(sumwith, _x31)), app(app(cons, app(app(cons, _x33), _x32)), xs)) → app#(plus, app(app(plus, app(_x31, _x33)), app(app(sumwith, _x31), _x32)))app#(app(sumwith, app(sumwith, _x31)), app(app(cons, nil), xs)) → app#(plus, nil)

Problem 6: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

app#(app(sumwith, app(plus, 0)), app(app(cons, _x31), xs))app#(plus, _x31)app#(app(sumwith, app(plus, app(s, _x32))), app(app(cons, _x31), xs))app#(plus, app(s, app(app(plus, _x32), _x31)))
app#(app(sumwith, app(sumwith, _x31)), app(app(cons, app(app(cons, _x33), _x32)), xs))app#(plus, app(app(plus, app(_x31, _x33)), app(app(sumwith, _x31), _x32)))app#(app(sumwith, f), app(app(cons, x), xs))app#(app(sumwith, f), xs)
app#(app(sumwith, f), app(app(cons, x), xs))app#(f, x)app#(app(sumwith, f), app(app(cons, x), xs))app#(app(plus, app(f, x)), app(app(sumwith, f), xs))
app#(app(sumwith, app(sumwith, _x31)), app(app(cons, nil), xs))app#(plus, nil)app#(app(plus, app(s, x)), y)app#(plus, x)
app#(app(plus, app(s, x)), y)app#(app(plus, x), y)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)

Rewrite Rules

app(app(plus, 0), y)yapp(app(plus, app(s, x)), y)app(s, app(app(plus, x), y))
app(app(sumwith, f), nil)nilapp(app(sumwith, f), app(app(cons, x), xs))app(app(plus, app(f, x)), app(app(sumwith, f), xs))

Original Signature

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

Strategy


The right-hand side of the rule app#(app(sumwith, app(plus, app(s, _x32))), app(app(cons, _x31), xs)) → app#(plus, 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#(plus, app(s, app(s, app(app(plus, _x62), _x61)))) 
app#(plus, app(s, _x61)) 
Thus, the rule app#(app(sumwith, app(plus, app(s, _x32))), app(app(cons, _x31), xs)) → app#(plus, app(s, app(app(plus, _x32), _x31))) is replaced by the following rules:
app#(app(sumwith, app(plus, app(s, app(s, _x62)))), app(app(cons, _x61), xs)) → app#(plus, app(s, app(s, app(app(plus, _x62), _x61))))app#(app(sumwith, app(plus, app(s, 0))), app(app(cons, _x61), xs)) → app#(plus, app(s, _x61))

Problem 7: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

app#(app(sumwith, app(sumwith, _x31)), app(app(cons, app(app(cons, _x33), _x32)), xs))app#(plus, app(app(plus, app(_x31, _x33)), app(app(sumwith, _x31), _x32)))app#(app(sumwith, app(plus, 0)), app(app(cons, _x31), xs))app#(plus, _x31)
app#(app(sumwith, f), app(app(cons, x), xs))app#(app(plus, app(f, x)), app(app(sumwith, f), xs))app#(app(sumwith, f), app(app(cons, x), xs))app#(f, x)
app#(app(sumwith, f), app(app(cons, x), xs))app#(app(sumwith, f), xs)app#(app(sumwith, app(sumwith, _x31)), app(app(cons, nil), xs))app#(plus, nil)
app#(app(sumwith, app(plus, app(s, app(s, _x62)))), app(app(cons, _x61), xs))app#(plus, app(s, app(s, app(app(plus, _x62), _x61))))app#(app(sumwith, app(plus, app(s, 0))), app(app(cons, _x61), xs))app#(plus, app(s, _x61))
app#(app(plus, app(s, x)), y)app#(plus, x)app#(app(plus, app(s, x)), y)app#(app(plus, x), y)
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)

Rewrite Rules

app(app(plus, 0), y)yapp(app(plus, app(s, x)), y)app(s, app(app(plus, x), y))
app(app(sumwith, f), nil)nilapp(app(sumwith, f), app(app(cons, x), xs))app(app(plus, app(f, x)), app(app(sumwith, f), xs))

Original Signature

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

Strategy


The right-hand side of the rule app#(app(sumwith, app(plus, 0)), app(app(cons, _x31), xs)) → app#(plus, _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(sumwith, app(plus, 0)), app(app(cons, _x31), xs)) → app#(plus, _x31) is deleted.

Problem 8: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

app#(app(sumwith, app(sumwith, _x31)), app(app(cons, app(app(cons, _x33), _x32)), xs))app#(plus, app(app(plus, app(_x31, _x33)), app(app(sumwith, _x31), _x32)))app#(app(sumwith, f), app(app(cons, x), xs))app#(app(sumwith, f), xs)
app#(app(sumwith, f), app(app(cons, x), xs))app#(f, x)app#(app(sumwith, f), app(app(cons, x), xs))app#(app(plus, app(f, x)), app(app(sumwith, f), xs))
app#(app(sumwith, app(plus, app(s, 0))), app(app(cons, _x61), xs))app#(plus, app(s, _x61))app#(app(sumwith, app(plus, app(s, app(s, _x62)))), app(app(cons, _x61), xs))app#(plus, app(s, app(s, app(app(plus, _x62), _x61))))
app#(app(sumwith, app(sumwith, _x31)), app(app(cons, nil), xs))app#(plus, nil)app#(app(plus, app(s, x)), y)app#(plus, x)
app#(app(plus, app(s, x)), y)app#(app(plus, x), y)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)

Rewrite Rules

app(app(plus, 0), y)yapp(app(plus, app(s, x)), y)app(s, app(app(plus, x), y))
app(app(sumwith, f), nil)nilapp(app(sumwith, f), app(app(cons, x), xs))app(app(plus, app(f, x)), app(app(sumwith, f), xs))

Original Signature

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

Strategy


The right-hand side of the rule app#(app(sumwith, app(sumwith, _x31)), app(app(cons, nil), xs)) → app#(plus, 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
Thus, the rule app#(app(sumwith, app(sumwith, _x31)), app(app(cons, nil), xs)) → app#(plus, nil) is deleted.

Problem 9: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

app#(app(sumwith, app(sumwith, _x31)), app(app(cons, app(app(cons, _x33), _x32)), xs))app#(plus, app(app(plus, app(_x31, _x33)), app(app(sumwith, _x31), _x32)))app#(app(sumwith, f), app(app(cons, x), xs))app#(app(plus, app(f, x)), app(app(sumwith, f), xs))
app#(app(sumwith, f), app(app(cons, x), xs))app#(f, x)app#(app(sumwith, f), app(app(cons, x), xs))app#(app(sumwith, f), xs)
app#(app(sumwith, app(plus, app(s, app(s, _x62)))), app(app(cons, _x61), xs))app#(plus, app(s, app(s, app(app(plus, _x62), _x61))))app#(app(sumwith, app(plus, app(s, 0))), app(app(cons, _x61), xs))app#(plus, app(s, _x61))
app#(app(plus, app(s, x)), y)app#(plus, x)app#(app(plus, app(s, x)), y)app#(app(plus, x), y)
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)

Rewrite Rules

app(app(plus, 0), y)yapp(app(plus, app(s, x)), y)app(s, app(app(plus, x), y))
app(app(sumwith, f), nil)nilapp(app(sumwith, f), app(app(cons, x), xs))app(app(plus, app(f, x)), app(app(sumwith, f), xs))

Original Signature

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

Strategy


The right-hand side of the rule app#(app(sumwith, app(plus, app(s, 0))), app(app(cons, _x61), xs)) → app#(plus, 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(sumwith, app(plus, app(s, 0))), app(app(cons, _x61), xs)) → app#(plus, app(s, _x61)) is deleted.

Problem 10: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

app#(app(sumwith, app(sumwith, _x31)), app(app(cons, app(app(cons, _x33), _x32)), xs))app#(plus, app(app(plus, app(_x31, _x33)), app(app(sumwith, _x31), _x32)))app#(app(sumwith, f), app(app(cons, x), xs))app#(app(sumwith, f), xs)
app#(app(sumwith, f), app(app(cons, x), xs))app#(f, x)app#(app(sumwith, f), app(app(cons, x), xs))app#(app(plus, app(f, x)), app(app(sumwith, f), xs))
app#(app(sumwith, app(plus, app(s, app(s, _x62)))), app(app(cons, _x61), xs))app#(plus, app(s, app(s, app(app(plus, _x62), _x61))))app#(app(plus, app(s, x)), y)app#(plus, x)
app#(app(plus, app(s, x)), y)app#(app(plus, x), y)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)

Rewrite Rules

app(app(plus, 0), y)yapp(app(plus, app(s, x)), y)app(s, app(app(plus, x), y))
app(app(sumwith, f), nil)nilapp(app(sumwith, f), app(app(cons, x), xs))app(app(plus, app(f, x)), app(app(sumwith, f), xs))

Original Signature

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

Strategy


The right-hand side of the rule app#(app(sumwith, app(plus, app(s, app(s, _x62)))), app(app(cons, _x61), xs)) → app#(plus, 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#(plus, app(s, app(s, _x71))) 
app#(plus, app(s, app(s, app(s, app(app(plus, _x72), _x71))))) 
Thus, the rule app#(app(sumwith, app(plus, app(s, app(s, _x62)))), app(app(cons, _x61), xs)) → app#(plus, app(s, app(s, app(app(plus, _x62), _x61)))) is replaced by the following rules:
app#(app(sumwith, app(plus, app(s, app(s, 0)))), app(app(cons, _x71), xs)) → app#(plus, app(s, app(s, _x71)))app#(app(sumwith, app(plus, app(s, app(s, app(s, _x72))))), app(app(cons, _x71), xs)) → app#(plus, app(s, app(s, app(s, app(app(plus, _x72), _x71)))))

Problem 11: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

app#(app(sumwith, app(sumwith, _x31)), app(app(cons, app(app(cons, _x33), _x32)), xs))app#(plus, app(app(plus, app(_x31, _x33)), app(app(sumwith, _x31), _x32)))app#(app(sumwith, app(plus, app(s, app(s, 0)))), app(app(cons, _x71), xs))app#(plus, app(s, app(s, _x71)))
app#(app(sumwith, f), app(app(cons, x), xs))app#(app(plus, app(f, x)), app(app(sumwith, f), xs))app#(app(sumwith, f), app(app(cons, x), xs))app#(f, x)
app#(app(sumwith, f), app(app(cons, x), xs))app#(app(sumwith, f), xs)app#(app(sumwith, app(plus, app(s, app(s, app(s, _x72))))), app(app(cons, _x71), xs))app#(plus, app(s, app(s, app(s, app(app(plus, _x72), _x71)))))
app#(app(plus, app(s, x)), y)app#(plus, x)app#(app(plus, app(s, x)), y)app#(app(plus, x), y)
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)

Rewrite Rules

app(app(plus, 0), y)yapp(app(plus, app(s, x)), y)app(s, app(app(plus, x), y))
app(app(sumwith, f), nil)nilapp(app(sumwith, f), app(app(cons, x), xs))app(app(plus, app(f, x)), app(app(sumwith, f), xs))

Original Signature

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

Strategy


The right-hand side of the rule app#(app(sumwith, app(plus, app(s, app(s, 0)))), app(app(cons, _x71), xs)) → app#(plus, 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(sumwith, app(plus, app(s, app(s, 0)))), app(app(cons, _x71), xs)) → app#(plus, app(s, app(s, _x71))) is deleted.

Problem 12: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

app#(app(sumwith, app(sumwith, _x31)), app(app(cons, app(app(cons, _x33), _x32)), xs))app#(plus, app(app(plus, app(_x31, _x33)), app(app(sumwith, _x31), _x32)))app#(app(sumwith, f), app(app(cons, x), xs))app#(app(sumwith, f), xs)
app#(app(sumwith, f), app(app(cons, x), xs))app#(f, x)app#(app(sumwith, f), app(app(cons, x), xs))app#(app(plus, app(f, x)), app(app(sumwith, f), xs))
app#(app(sumwith, app(plus, app(s, app(s, app(s, _x72))))), app(app(cons, _x71), xs))app#(plus, app(s, app(s, app(s, app(app(plus, _x72), _x71)))))app#(app(plus, app(s, x)), y)app#(plus, x)
app#(app(plus, app(s, x)), y)app#(app(plus, x), y)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)

Rewrite Rules

app(app(plus, 0), y)yapp(app(plus, app(s, x)), y)app(s, app(app(plus, x), y))
app(app(sumwith, f), nil)nilapp(app(sumwith, f), app(app(cons, x), xs))app(app(plus, app(f, x)), app(app(sumwith, f), xs))

Original Signature

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

Strategy


The right-hand side of the rule app#(app(sumwith, app(plus, app(s, app(s, app(s, _x72))))), app(app(cons, _x71), xs)) → app#(plus, 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#(plus, app(s, app(s, app(s, app(s, app(app(plus, _x102), _x101)))))) 
app#(plus, app(s, app(s, app(s, _x101)))) 
Thus, the rule app#(app(sumwith, app(plus, app(s, app(s, app(s, _x72))))), app(app(cons, _x71), xs)) → app#(plus, app(s, app(s, app(s, app(app(plus, _x72), _x71))))) is replaced by the following rules:
app#(app(sumwith, app(plus, app(s, app(s, app(s, app(s, _x102)))))), app(app(cons, _x101), xs)) → app#(plus, app(s, app(s, app(s, app(s, app(app(plus, _x102), _x101))))))app#(app(sumwith, app(plus, app(s, app(s, app(s, 0))))), app(app(cons, _x101), xs)) → app#(plus, app(s, app(s, app(s, _x101))))

Problem 13: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

app#(app(sumwith, app(plus, app(s, app(s, app(s, app(s, _x102)))))), app(app(cons, _x101), xs))app#(plus, app(s, app(s, app(s, app(s, app(app(plus, _x102), _x101))))))app#(app(sumwith, app(sumwith, _x31)), app(app(cons, app(app(cons, _x33), _x32)), xs))app#(plus, app(app(plus, app(_x31, _x33)), app(app(sumwith, _x31), _x32)))
app#(app(sumwith, f), app(app(cons, x), xs))app#(app(plus, app(f, x)), app(app(sumwith, f), xs))app#(app(sumwith, f), app(app(cons, x), xs))app#(f, x)
app#(app(sumwith, f), app(app(cons, x), xs))app#(app(sumwith, f), xs)app#(app(sumwith, app(plus, app(s, app(s, app(s, 0))))), app(app(cons, _x101), xs))app#(plus, app(s, app(s, app(s, _x101))))
app#(app(plus, app(s, x)), y)app#(plus, x)app#(app(plus, app(s, x)), y)app#(app(plus, x), y)
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)

Rewrite Rules

app(app(plus, 0), y)yapp(app(plus, app(s, x)), y)app(s, app(app(plus, x), y))
app(app(sumwith, f), nil)nilapp(app(sumwith, f), app(app(cons, x), xs))app(app(plus, app(f, x)), app(app(sumwith, f), xs))

Original Signature

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

Strategy


The right-hand side of the rule app#(app(sumwith, app(plus, app(s, app(s, app(s, app(s, _x102)))))), app(app(cons, _x101), xs)) → app#(plus, 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#(plus, app(s, app(s, app(s, app(s, app(s, app(app(plus, _x112), _x111))))))) 
app#(plus, app(s, app(s, app(s, app(s, _x111))))) 
Thus, the rule app#(app(sumwith, app(plus, app(s, app(s, app(s, app(s, _x102)))))), app(app(cons, _x101), xs)) → app#(plus, app(s, app(s, app(s, app(s, app(app(plus, _x102), _x101)))))) is replaced by the following rules:
app#(app(sumwith, app(plus, app(s, app(s, app(s, app(s, 0)))))), app(app(cons, _x111), xs)) → app#(plus, app(s, app(s, app(s, app(s, _x111)))))app#(app(sumwith, app(plus, app(s, app(s, app(s, app(s, app(s, _x112))))))), app(app(cons, _x111), xs)) → app#(plus, app(s, app(s, app(s, app(s, app(s, app(app(plus, _x112), _x111)))))))

Problem 14: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

app#(app(sumwith, app(sumwith, _x31)), app(app(cons, app(app(cons, _x33), _x32)), xs))app#(plus, app(app(plus, app(_x31, _x33)), app(app(sumwith, _x31), _x32)))app#(app(sumwith, f), app(app(cons, x), xs))app#(app(sumwith, f), xs)
app#(app(sumwith, f), app(app(cons, x), xs))app#(f, x)app#(app(sumwith, f), app(app(cons, x), xs))app#(app(plus, app(f, x)), app(app(sumwith, f), xs))
app#(app(sumwith, app(plus, app(s, app(s, app(s, app(s, 0)))))), app(app(cons, _x111), xs))app#(plus, app(s, app(s, app(s, app(s, _x111)))))app#(app(sumwith, app(plus, app(s, app(s, app(s, 0))))), app(app(cons, _x101), xs))app#(plus, app(s, app(s, app(s, _x101))))
app#(app(plus, app(s, x)), y)app#(plus, x)app#(app(sumwith, app(plus, app(s, app(s, app(s, app(s, app(s, _x112))))))), app(app(cons, _x111), xs))app#(plus, app(s, app(s, app(s, app(s, app(s, app(app(plus, _x112), _x111)))))))
app#(app(plus, app(s, x)), y)app#(app(plus, x), y)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)

Rewrite Rules

app(app(plus, 0), y)yapp(app(plus, app(s, x)), y)app(s, app(app(plus, x), y))
app(app(sumwith, f), nil)nilapp(app(sumwith, f), app(app(cons, x), xs))app(app(plus, app(f, x)), app(app(sumwith, f), xs))

Original Signature

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

Strategy


The right-hand side of the rule app#(app(sumwith, app(plus, app(s, app(s, app(s, app(s, 0)))))), app(app(cons, _x111), xs)) → app#(plus, 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(sumwith, app(plus, app(s, app(s, app(s, app(s, 0)))))), app(app(cons, _x111), xs)) → app#(plus, app(s, app(s, app(s, app(s, _x111))))) is deleted.

Problem 15: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

app#(app(sumwith, app(sumwith, _x31)), app(app(cons, app(app(cons, _x33), _x32)), xs))app#(plus, app(app(plus, app(_x31, _x33)), app(app(sumwith, _x31), _x32)))app#(app(sumwith, f), app(app(cons, x), xs))app#(app(plus, app(f, x)), app(app(sumwith, f), xs))
app#(app(sumwith, f), app(app(cons, x), xs))app#(f, x)app#(app(sumwith, f), app(app(cons, x), xs))app#(app(sumwith, f), xs)
app#(app(sumwith, app(plus, app(s, app(s, app(s, 0))))), app(app(cons, _x101), xs))app#(plus, app(s, app(s, app(s, _x101))))app#(app(plus, app(s, x)), y)app#(plus, x)
app#(app(sumwith, app(plus, app(s, app(s, app(s, app(s, app(s, _x112))))))), app(app(cons, _x111), xs))app#(plus, app(s, app(s, app(s, app(s, app(s, app(app(plus, _x112), _x111)))))))app#(app(plus, app(s, x)), y)app#(app(plus, x), y)
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)

Rewrite Rules

app(app(plus, 0), y)yapp(app(plus, app(s, x)), y)app(s, app(app(plus, x), y))
app(app(sumwith, f), nil)nilapp(app(sumwith, f), app(app(cons, x), xs))app(app(plus, app(f, x)), app(app(sumwith, f), xs))

Original Signature

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

Strategy


The right-hand side of the rule app#(app(sumwith, app(plus, app(s, app(s, app(s, 0))))), app(app(cons, _x101), xs)) → app#(plus, 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(sumwith, app(plus, app(s, app(s, app(s, 0))))), app(app(cons, _x101), xs)) → app#(plus, app(s, app(s, app(s, _x101)))) is deleted.

Problem 16: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

app#(app(sumwith, app(sumwith, _x31)), app(app(cons, app(app(cons, _x33), _x32)), xs))app#(plus, app(app(plus, app(_x31, _x33)), app(app(sumwith, _x31), _x32)))app#(app(sumwith, f), app(app(cons, x), xs))app#(app(sumwith, f), xs)
app#(app(sumwith, f), app(app(cons, x), xs))app#(f, x)app#(app(sumwith, f), app(app(cons, x), xs))app#(app(plus, app(f, x)), app(app(sumwith, f), xs))
app#(app(plus, app(s, x)), y)app#(plus, x)app#(app(sumwith, app(plus, app(s, app(s, app(s, app(s, app(s, _x112))))))), app(app(cons, _x111), xs))app#(plus, app(s, app(s, app(s, app(s, app(s, app(app(plus, _x112), _x111)))))))
app#(app(plus, app(s, x)), y)app#(app(plus, x), y)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)

Rewrite Rules

app(app(plus, 0), y)yapp(app(plus, app(s, x)), y)app(s, app(app(plus, x), y))
app(app(sumwith, f), nil)nilapp(app(sumwith, f), app(app(cons, x), xs))app(app(plus, app(f, x)), app(app(sumwith, f), xs))

Original Signature

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

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 17: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

app#(app(sumwith, app(sumwith, _x31)), app(app(cons, app(app(cons, _x33), _x32)), xs))app#(plus, app(app(plus, app(_x31, _x33)), app(app(sumwith, _x31), _x32)))app#(app(sumwith, f), app(app(cons, x), xs))app#(app(plus, app(f, x)), app(app(sumwith, f), xs))
app#(app(sumwith, f), app(app(cons, x), xs))app#(f, x)app#(app(sumwith, f), app(app(cons, x), xs))app#(app(sumwith, f), xs)
app#(app(sumwith, app(plus, app(s, app(s, app(s, app(s, app(s, _x112))))))), app(app(cons, _x111), xs))app#(plus, app(s, app(s, app(s, app(s, app(s, app(app(plus, _x112), _x111)))))))app#(app(plus, app(s, x)), y)app#(app(plus, x), y)
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)

Rewrite Rules

app(app(plus, 0), y)yapp(app(plus, app(s, x)), y)app(s, app(app(plus, x), y))
app(app(sumwith, f), nil)nilapp(app(sumwith, f), app(app(cons, x), xs))app(app(plus, app(f, x)), app(app(sumwith, f), xs))

Original Signature

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

Strategy


The right-hand side of the rule app#(app(sumwith, app(plus, app(s, app(s, app(s, app(s, app(s, _x112))))))), app(app(cons, _x111), xs)) → app#(plus, 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#(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(app(plus, _x142), _x141)))))))) 
app#(plus, app(s, app(s, app(s, app(s, app(s, _x141)))))) 
Thus, the rule app#(app(sumwith, app(plus, app(s, app(s, app(s, app(s, app(s, _x112))))))), app(app(cons, _x111), xs)) → app#(plus, app(s, app(s, app(s, app(s, app(s, app(app(plus, _x112), _x111))))))) is replaced by the following rules:
app#(app(sumwith, app(plus, app(s, app(s, app(s, app(s, app(s, app(s, _x142)))))))), app(app(cons, _x141), xs)) → app#(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(app(plus, _x142), _x141))))))))app#(app(sumwith, app(plus, app(s, app(s, app(s, app(s, app(s, 0))))))), app(app(cons, _x141), xs)) → app#(plus, app(s, app(s, app(s, app(s, app(s, _x141))))))

Problem 18: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

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

Rewrite Rules

app(app(plus, 0), y)yapp(app(plus, app(s, x)), y)app(s, app(app(plus, x), y))
app(app(sumwith, f), nil)nilapp(app(sumwith, f), app(app(cons, x), xs))app(app(plus, app(f, x)), app(app(sumwith, f), xs))

Original Signature

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

Strategy


The right-hand side of the rule app#(app(sumwith, app(plus, app(s, app(s, app(s, app(s, app(s, app(s, _x142)))))))), app(app(cons, _x141), xs)) → app#(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(app(plus, _x142), _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
app#(plus, app(s, app(s, app(s, app(s, app(s, app(s, _x151))))))) 
app#(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(app(plus, _x152), _x151))))))))) 
Thus, the rule app#(app(sumwith, app(plus, app(s, app(s, app(s, app(s, app(s, app(s, _x142)))))))), app(app(cons, _x141), xs)) → app#(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(app(plus, _x142), _x141)))))))) is replaced by the following rules:
app#(app(sumwith, app(plus, app(s, app(s, app(s, app(s, app(s, app(s, 0)))))))), app(app(cons, _x151), xs)) → app#(plus, app(s, app(s, app(s, app(s, app(s, app(s, _x151)))))))app#(app(sumwith, app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x152))))))))), app(app(cons, _x151), xs)) → app#(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(app(plus, _x152), _x151)))))))))

Problem 19: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

app#(app(sumwith, app(sumwith, _x31)), app(app(cons, app(app(cons, _x33), _x32)), xs))app#(plus, app(app(plus, app(_x31, _x33)), app(app(sumwith, _x31), _x32)))app#(app(sumwith, app(plus, app(s, app(s, app(s, app(s, app(s, app(s, 0)))))))), app(app(cons, _x151), xs))app#(plus, app(s, app(s, app(s, app(s, app(s, app(s, _x151)))))))
app#(app(sumwith, f), app(app(cons, x), xs))app#(app(plus, app(f, x)), app(app(sumwith, f), xs))app#(app(sumwith, f), app(app(cons, x), xs))app#(f, x)
app#(app(sumwith, f), app(app(cons, x), xs))app#(app(sumwith, f), xs)app#(app(sumwith, app(plus, app(s, app(s, app(s, app(s, app(s, 0))))))), app(app(cons, _x141), xs))app#(plus, app(s, app(s, app(s, app(s, app(s, _x141))))))
app#(app(plus, app(s, x)), y)app#(app(plus, x), y)app#(app(plus, app(s, app(s, app(s, _x62)))), _x61)app#(s, app(s, app(s, app(app(plus, _x62), _x61))))
app#(app(sumwith, app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x152))))))))), app(app(cons, _x151), xs))app#(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(app(plus, _x152), _x151)))))))))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(app(sumwith, f), nil)nilapp(app(sumwith, f), app(app(cons, x), xs))app(app(plus, app(f, x)), app(app(sumwith, f), xs))

Original Signature

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

Strategy


The right-hand side of the rule app#(app(sumwith, app(plus, app(s, app(s, app(s, app(s, app(s, app(s, 0)))))))), app(app(cons, _x151), xs)) → app#(plus, app(s, app(s, app(s, app(s, app(s, app(s, _x151))))))) 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(sumwith, app(plus, app(s, app(s, app(s, app(s, app(s, app(s, 0)))))))), app(app(cons, _x151), xs)) → app#(plus, app(s, app(s, app(s, app(s, app(s, app(s, _x151))))))) is deleted.

Problem 20: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

app#(app(sumwith, app(sumwith, _x31)), app(app(cons, app(app(cons, _x33), _x32)), xs))app#(plus, app(app(plus, app(_x31, _x33)), app(app(sumwith, _x31), _x32)))app#(app(sumwith, f), app(app(cons, x), xs))app#(app(sumwith, f), xs)
app#(app(sumwith, f), app(app(cons, x), xs))app#(f, x)app#(app(sumwith, f), app(app(cons, x), xs))app#(app(plus, app(f, x)), app(app(sumwith, f), xs))
app#(app(sumwith, app(plus, app(s, app(s, app(s, app(s, app(s, 0))))))), app(app(cons, _x141), xs))app#(plus, app(s, app(s, app(s, app(s, app(s, _x141))))))app#(app(plus, app(s, x)), y)app#(app(plus, x), y)
app#(app(sumwith, app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x152))))))))), app(app(cons, _x151), xs))app#(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(app(plus, _x152), _x151)))))))))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)

Rewrite Rules

app(app(plus, 0), y)yapp(app(plus, app(s, x)), y)app(s, app(app(plus, x), y))
app(app(sumwith, f), nil)nilapp(app(sumwith, f), app(app(cons, x), xs))app(app(plus, app(f, x)), app(app(sumwith, f), xs))

Original Signature

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

Strategy


The right-hand side of the rule app#(app(sumwith, app(plus, app(s, app(s, app(s, app(s, app(s, 0))))))), app(app(cons, _x141), xs)) → app#(plus, 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(sumwith, app(plus, app(s, app(s, app(s, app(s, app(s, 0))))))), app(app(cons, _x141), xs)) → app#(plus, app(s, app(s, app(s, app(s, app(s, _x141)))))) is deleted.

Problem 21: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

app#(app(sumwith, app(sumwith, _x31)), app(app(cons, app(app(cons, _x33), _x32)), xs))app#(plus, app(app(plus, app(_x31, _x33)), app(app(sumwith, _x31), _x32)))app#(app(sumwith, f), app(app(cons, x), xs))app#(app(plus, app(f, x)), app(app(sumwith, f), xs))
app#(app(sumwith, f), app(app(cons, x), xs))app#(f, x)app#(app(sumwith, f), app(app(cons, x), xs))app#(app(sumwith, f), xs)
app#(app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x332))))))))))))))))), _x331)app#(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(app(plus, _x332), _x331)))))))))))))))))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, app(s, app(s, app(s, 0))))))))))), _x221)app#(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x221))))))))))app#(app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, 0)))))))))))))))), _x331)app#(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x331)))))))))))))))
app#(app(plus, app(s, 0)), _x31)app#(s, _x31)app#(app(sumwith, app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x222)))))))))))), app(app(cons, _x221), xs))app#(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(app(plus, _x222), _x221))))))))))))

Rewrite Rules

app(app(plus, 0), y)yapp(app(plus, app(s, x)), y)app(s, app(app(plus, x), y))
app(app(sumwith, f), nil)nilapp(app(sumwith, f), app(app(cons, x), xs))app(app(plus, app(f, x)), app(app(sumwith, f), xs))

Original Signature

Termination of terms over the following signature is verified: app, plus, 0, s, sumwith, 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, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x332))))))))))))))))), _x331) → app#(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(app(plus, _x332), _x331))))))))))))))))) 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(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(app(plus, _x342), _x341)))))))))))))))))) 
app#(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x341)))))))))))))))) 
Thus, the rule app#(app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x332))))))))))))))))), _x331) → app#(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(app(plus, _x332), _x331))))))))))))))))) is replaced by the following rules:
app#(app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x342)))))))))))))))))), _x341) → app#(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(app(plus, _x342), _x341))))))))))))))))))app#(app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, 0))))))))))))))))), _x341) → app#(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x341))))))))))))))))

Problem 22: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

app#(app(sumwith, app(sumwith, _x31)), app(app(cons, app(app(cons, _x33), _x32)), xs))app#(plus, app(app(plus, app(_x31, _x33)), app(app(sumwith, _x31), _x32)))app#(app(sumwith, f), app(app(cons, x), xs))app#(app(plus, app(f, x)), app(app(sumwith, f), xs))
app#(app(sumwith, f), app(app(cons, x), xs))app#(f, x)app#(app(sumwith, f), app(app(cons, x), xs))app#(app(sumwith, f), xs)
app#(app(sumwith, app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, 0)))))))))))))))))))))))))))), app(app(cons, _x571), xs))app#(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x571)))))))))))))))))))))))))))app#(app(sumwith, app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x572))))))))))))))))))))))))))))), app(app(cons, _x571), xs))app#(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(app(plus, _x572), _x571)))))))))))))))))))))))))))))
app#(app(plus, app(s, x)), y)app#(app(plus, x), y)app#(app(sumwith, app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, 0))))))))))))))))))))))))))), app(app(cons, _x541), xs))app#(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x541))))))))))))))))))))))))))
app#(app(sumwith, app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, 0))))))))))))))))))))))))), app(app(cons, _x501), xs))app#(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x501))))))))))))))))))))))))app#(app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x472))))))))))))))))))))))))), _x471)app#(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(app(plus, _x472), _x471)))))))))))))))))))))))))

Rewrite Rules

app(app(plus, 0), y)yapp(app(plus, app(s, x)), y)app(s, app(app(plus, x), y))
app(app(sumwith, f), nil)nilapp(app(sumwith, f), app(app(cons, x), xs))app(app(plus, app(f, x)), app(app(sumwith, f), xs))

Original Signature

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

Strategy


The right-hand side of the rule app#(app(sumwith, app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x572))))))))))))))))))))))))))))), app(app(cons, _x571), xs)) → app#(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(app(plus, _x572), _x571))))))))))))))))))))))))))))) 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#(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(app(plus, _x582), _x581)))))))))))))))))))))))))))))) 
app#(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x581)))))))))))))))))))))))))))) 
Thus, the rule app#(app(sumwith, app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x572))))))))))))))))))))))))))))), app(app(cons, _x571), xs)) → app#(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(app(plus, _x572), _x571))))))))))))))))))))))))))))) is replaced by the following rules:
app#(app(sumwith, app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, 0))))))))))))))))))))))))))))), app(app(cons, _x581), xs)) → app#(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x581))))))))))))))))))))))))))))app#(app(sumwith, app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x582)))))))))))))))))))))))))))))), app(app(cons, _x581), xs)) → app#(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(app(plus, _x582), _x581))))))))))))))))))))))))))))))

Problem 23: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

app#(app(sumwith, f), app(app(cons, x), xs))app#(app(sumwith, f), xs)app#(app(sumwith, f), app(app(cons, x), xs))app#(f, x)
app#(app(sumwith, f), app(app(cons, x), xs))app#(app(plus, app(f, x)), app(app(sumwith, f), xs))app#(app(sumwith, app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x1102)))))))))))))))))))))))))))))))))))))))))))))))))))))))), app(app(cons, _x1101), xs))app#(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(app(plus, _x1102), _x1101))))))))))))))))))))))))))))))))))))))))))))))))))))))))
app#(app(sumwith, app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, 0))))))))))))))))))))))))))))))))))))))))))))))))))))))), app(app(cons, _x1101), xs))app#(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x1101))))))))))))))))))))))))))))))))))))))))))))))))))))))app#(app(sumwith, app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, 0))))))))))))))))))))))))))))))))))))))))))))))))))))), app(app(cons, _x1061), xs))app#(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x1061))))))))))))))))))))))))))))))))))))))))))))))))))))
app#(app(sumwith, app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, 0))))))))))))))))))))))))))))))))))))))))))), app(app(cons, _x861), xs))app#(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x861))))))))))))))))))))))))))))))))))))))))))app#(app(sumwith, app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, 0)))))))))))))))))))))))))))))))))))))), app(app(cons, _x771), xs))app#(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x771)))))))))))))))))))))))))))))))))))))
app#(app(sumwith, app(sumwith, _x31)), app(app(cons, app(app(cons, _x33), _x32)), xs))app#(plus, app(app(plus, app(_x31, _x33)), app(app(sumwith, _x31), _x32)))app#(app(sumwith, app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, 0)))))))))))))))))))))))))))))))))))))))))))))))))))))), app(app(cons, _x1071), xs))app#(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x1071)))))))))))))))))))))))))))))))))))))))))))))))))))))
app#(app(plus, app(s, x)), y)app#(app(plus, x), y)app#(app(sumwith, app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, 0)))))))))))))))))))))))))))))))))))))))))))))))))), app(app(cons, _x1011), xs))app#(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x1011)))))))))))))))))))))))))))))))))))))))))))))))))
app#(app(sumwith, app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, 0)))))))))))))))))))))))))))))))))))))))))))))), app(app(cons, _x931), xs))app#(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x931)))))))))))))))))))))))))))))))))))))))))))))app#(app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x472))))))))))))))))))))))))), _x471)app#(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(app(plus, _x472), _x471)))))))))))))))))))))))))

Rewrite Rules

app(app(plus, 0), y)yapp(app(plus, app(s, x)), y)app(s, app(app(plus, x), y))
app(app(sumwith, f), nil)nilapp(app(sumwith, f), app(app(cons, x), xs))app(app(plus, app(f, x)), app(app(sumwith, f), xs))

Original Signature

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

Strategy


The right-hand side of the rule app#(app(sumwith, app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x1102)))))))))))))))))))))))))))))))))))))))))))))))))))))))), app(app(cons, _x1101), xs)) → app#(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(app(plus, _x1102), _x1101)))))))))))))))))))))))))))))))))))))))))))))))))))))))) 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#(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x1131))))))))))))))))))))))))))))))))))))))))))))))))))))))) 
app#(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(app(plus, _x1132), _x1131))))))))))))))))))))))))))))))))))))))))))))))))))))))))) 
Thus, the rule app#(app(sumwith, app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x1102)))))))))))))))))))))))))))))))))))))))))))))))))))))))), app(app(cons, _x1101), xs)) → app#(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(app(plus, _x1102), _x1101)))))))))))))))))))))))))))))))))))))))))))))))))))))))) is replaced by the following rules:
app#(app(sumwith, app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x1132))))))))))))))))))))))))))))))))))))))))))))))))))))))))), app(app(cons, _x1131), xs)) → app#(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(app(plus, _x1132), _x1131)))))))))))))))))))))))))))))))))))))))))))))))))))))))))app#(app(sumwith, app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, 0)))))))))))))))))))))))))))))))))))))))))))))))))))))))), app(app(cons, _x1131), xs)) → app#(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x1131)))))))))))))))))))))))))))))))))))))))))))))))))))))))

Problem 24: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

app#(app(sumwith, app(sumwith, _x31)), app(app(cons, app(app(cons, _x33), _x32)), xs))app#(plus, app(app(plus, app(_x31, _x33)), app(app(sumwith, _x31), _x32)))app#(app(sumwith, f), app(app(cons, x), xs))app#(app(plus, app(f, x)), app(app(sumwith, f), xs))
app#(app(sumwith, f), app(app(cons, x), xs))app#(f, x)app#(app(sumwith, f), app(app(cons, x), xs))app#(app(sumwith, f), xs)
app#(app(plus, app(s, x)), y)app#(app(plus, x), y)app#(app(sumwith, app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, 0))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), app(app(cons, _x1181), xs))app#(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x1181))))))))))))))))))))))))))))))))))))))))))))))))))))))))))
app#(app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x472))))))))))))))))))))))))), _x471)app#(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(app(plus, _x472), _x471)))))))))))))))))))))))))app#(app(sumwith, app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, 0)))))))))))))))))))))))))))))))))))))))))))))))))))))))), app(app(cons, _x1131), xs))app#(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x1131)))))))))))))))))))))))))))))))))))))))))))))))))))))))
app#(app(sumwith, app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, 0)))))))))))))))))))))))))))))))))))))))))))))), app(app(cons, _x931), xs))app#(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x931)))))))))))))))))))))))))))))))))))))))))))))app#(app(sumwith, app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x1572))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), app(app(cons, _x1571), xs))app#(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(app(plus, _x1572), _x1571)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))

Rewrite Rules

app(app(plus, 0), y)yapp(app(plus, app(s, x)), y)app(s, app(app(plus, x), y))
app(app(sumwith, f), nil)nilapp(app(sumwith, f), app(app(cons, x), xs))app(app(plus, app(f, x)), app(app(sumwith, f), xs))

Original Signature

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

Strategy


The right-hand side of the rule app#(app(sumwith, app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, 0))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), app(app(cons, _x1181), xs)) → app#(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x1181)))))))))))))))))))))))))))))))))))))))))))))))))))))))))) 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(sumwith, app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, 0))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), app(app(cons, _x1181), xs)) → app#(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x1181)))))))))))))))))))))))))))))))))))))))))))))))))))))))))) is deleted.

Problem 25: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

app#(app(sumwith, app(sumwith, _x31)), app(app(cons, app(app(cons, _x33), _x32)), xs))app#(plus, app(app(plus, app(_x31, _x33)), app(app(sumwith, _x31), _x32)))app#(app(sumwith, f), app(app(cons, x), xs))app#(app(plus, app(f, x)), app(app(sumwith, f), xs))
app#(app(sumwith, f), app(app(cons, x), xs))app#(f, x)app#(app(sumwith, f), app(app(cons, x), xs))app#(app(sumwith, f), xs)
app#(app(sumwith, app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x1662)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), app(app(cons, _x1661), xs))app#(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(app(plus, _x1662), _x1661))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))app#(app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, 0))))))))))))))))))))))))))))))))))))))))))), _x861)app#(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x861))))))))))))))))))))))))))))))))))))))))))
app#(app(sumwith, app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, 0))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), app(app(cons, _x1581), xs))app#(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x1581))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))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, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, 0))))))))))))))))))))))))))))))))))))), _x741)app#(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x741))))))))))))))))))))))))))))))))))))app#(app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, 0)))))))))))))))))))))))))))))))), _x631)app#(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x631)))))))))))))))))))))))))))))))
app#(app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x902)))))))))))))))))))))))))))))))))))))))))))))), _x901)app#(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(app(plus, _x902), _x901))))))))))))))))))))))))))))))))))))))))))))))app#(app(sumwith, app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, 0)))))))))))))))))))))))))))))))))))))))))))))), app(app(cons, _x931), xs))app#(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x931)))))))))))))))))))))))))))))))))))))))))))))

Rewrite Rules

app(app(plus, 0), y)yapp(app(plus, app(s, x)), y)app(s, app(app(plus, x), y))
app(app(sumwith, f), nil)nilapp(app(sumwith, f), app(app(cons, x), xs))app(app(plus, app(f, x)), app(app(sumwith, f), xs))

Original Signature

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

Strategy


The right-hand side of the rule app#(app(sumwith, app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x1662)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), app(app(cons, _x1661), xs)) → app#(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(app(plus, _x1662), _x1661)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) 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#(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(app(plus, _x1672), _x1671))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) 
app#(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x1671))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) 
Thus, the rule app#(app(sumwith, app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x1662)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), app(app(cons, _x1661), xs)) → app#(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(app(plus, _x1662), _x1661)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) is replaced by the following rules:
app#(app(sumwith, app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x1672))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), app(app(cons, _x1671), xs)) → app#(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(app(plus, _x1672), _x1671)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))app#(app(sumwith, app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, 0)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), app(app(cons, _x1671), xs)) → app#(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x1671)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))

Problem 26: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

app#(app(sumwith, f), app(app(cons, x), xs))app#(app(sumwith, f), xs)app#(app(sumwith, f), app(app(cons, x), xs))app#(f, x)
app#(app(sumwith, f), app(app(cons, x), xs))app#(app(plus, app(f, x)), app(app(sumwith, f), xs))app#(app(sumwith, app(sumwith, _x31)), app(app(cons, app(app(cons, _x33), _x32)), xs))app#(plus, app(app(plus, app(_x31, _x33)), app(app(sumwith, _x31), _x32)))
app#(app(sumwith, app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, 0))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), app(app(cons, _x2181), xs))app#(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x2181))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))app#(app(sumwith, app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x2182)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), app(app(cons, _x2181), xs))app#(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(app(plus, _x2182), _x2181))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))
app#(app(sumwith, app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, 0))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), app(app(cons, _x2021), xs))app#(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x2021))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))app#(app(sumwith, app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, 0))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), app(app(cons, _x1981), xs))app#(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x1981))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))
app#(app(plus, app(s, x)), y)app#(app(plus, x), y)app#(app(sumwith, app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, 0))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), app(app(cons, _x1861), xs))app#(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x1861))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))
app#(app(sumwith, app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, 0)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), app(app(cons, _x2031), xs))app#(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x2031)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))app#(app(sumwith, app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, 0)))))))))))))))))))))))))))))))))))))))))))))), app(app(cons, _x931), xs))app#(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x931)))))))))))))))))))))))))))))))))))))))))))))
app#(app(sumwith, app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, 0))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), app(app(cons, _x2141), xs))app#(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x2141))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))app#(app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x902)))))))))))))))))))))))))))))))))))))))))))))), _x901)app#(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(app(plus, _x902), _x901))))))))))))))))))))))))))))))))))))))))))))))

Rewrite Rules

app(app(plus, 0), y)yapp(app(plus, app(s, x)), y)app(s, app(app(plus, x), y))
app(app(sumwith, f), nil)nilapp(app(sumwith, f), app(app(cons, x), xs))app(app(plus, app(f, x)), app(app(sumwith, f), xs))

Original Signature

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

Strategy


The right-hand side of the rule app#(app(sumwith, app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, 0))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), app(app(cons, _x2181), xs)) → app#(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x2181)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) 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(sumwith, app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, 0))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), app(app(cons, _x2181), xs)) → app#(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x2181)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) is deleted.

Problem 27: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

app#(app(sumwith, app(sumwith, _x31)), app(app(cons, app(app(cons, _x33), _x32)), xs))app#(plus, app(app(plus, app(_x31, _x33)), app(app(sumwith, _x31), _x32)))app#(app(sumwith, f), app(app(cons, x), xs))app#(app(plus, app(f, x)), app(app(sumwith, f), xs))
app#(app(sumwith, f), app(app(cons, x), xs))app#(f, x)app#(app(sumwith, f), app(app(cons, x), xs))app#(app(sumwith, f), xs)
app#(app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, 0)))))))))))))))))))))))))))))))))))))))))))))))), _x971)app#(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x971)))))))))))))))))))))))))))))))))))))))))))))))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, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x972))))))))))))))))))))))))))))))))))))))))))))))))), _x971)app#(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(app(plus, _x972), _x971)))))))))))))))))))))))))))))))))))))))))))))))))app#(app(sumwith, app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x2582)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), app(app(cons, _x2581), xs))app#(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(app(plus, _x2582), _x2581))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))
app#(app(sumwith, app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, 0))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), app(app(cons, _x2141), xs))app#(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x2141))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))app#(app(sumwith, app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, 0))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), app(app(cons, _x2461), xs))app#(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x2461))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))

Rewrite Rules

app(app(plus, 0), y)yapp(app(plus, app(s, x)), y)app(s, app(app(plus, x), y))
app(app(sumwith, f), nil)nilapp(app(sumwith, f), app(app(cons, x), xs))app(app(plus, app(f, x)), app(app(sumwith, f), xs))

Original Signature

Termination of terms over the following signature is verified: app, plus, 0, s, sumwith, 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, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, 0)))))))))))))))))))))))))))))))))))))))))))))))), _x971) → app#(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x971))))))))))))))))))))))))))))))))))))))))))))))) 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, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, 0)))))))))))))))))))))))))))))))))))))))))))))))), _x971) → app#(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x971))))))))))))))))))))))))))))))))))))))))))))))) is deleted.

Problem 28: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

app#(app(sumwith, app(sumwith, _x31)), app(app(cons, app(app(cons, _x33), _x32)), xs))app#(plus, app(app(plus, app(_x31, _x33)), app(app(sumwith, _x31), _x32)))app#(app(sumwith, f), app(app(cons, x), xs))app#(app(plus, app(f, x)), app(app(sumwith, f), xs))
app#(app(sumwith, f), app(app(cons, x), xs))app#(f, x)app#(app(sumwith, f), app(app(cons, x), xs))app#(app(sumwith, f), xs)
app#(app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x1462)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), _x1461)app#(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(app(plus, _x1462), _x1461))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))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, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, 0))))))))))))))))))))))))))))))))))))))))))))))))))))))), _x1101)app#(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x1101))))))))))))))))))))))))))))))))))))))))))))))))))))))app#(app(sumwith, app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, 0))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), app(app(cons, _x2141), xs))app#(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x2141))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))
app#(app(sumwith, app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, 0))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), app(app(cons, _x2461), xs))app#(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x2461))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))app#(app(sumwith, app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x2582)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), app(app(cons, _x2581), xs))app#(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(app(plus, _x2582), _x2581))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))

Rewrite Rules

app(app(plus, 0), y)yapp(app(plus, app(s, x)), y)app(s, app(app(plus, x), y))
app(app(sumwith, f), nil)nilapp(app(sumwith, f), app(app(cons, x), xs))app(app(plus, app(f, x)), app(app(sumwith, f), xs))

Original Signature

Termination of terms over the following signature is verified: app, plus, 0, s, sumwith, 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, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x1462)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), _x1461) → app#(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(app(plus, _x1462), _x1461)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) 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(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x1471))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) 
app#(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(app(plus, _x1472), _x1471))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) 
Thus, the rule app#(app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x1462)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), _x1461) → app#(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(app(plus, _x1462), _x1461)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) is replaced by the following rules:
app#(app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x1472))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), _x1471) → app#(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(app(plus, _x1472), _x1471)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))app#(app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, 0)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), _x1471) → app#(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x1471)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))

Problem 29: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

app#(app(sumwith, f), app(app(cons, x), xs))app#(app(sumwith, f), xs)app#(app(sumwith, f), app(app(cons, x), xs))app#(f, x)
app#(app(sumwith, f), app(app(cons, x), xs))app#(app(plus, app(f, x)), app(app(sumwith, f), xs))app#(app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x2012))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), _x2011)app#(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(app(plus, _x2012), _x2011)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))
app#(app(sumwith, app(sumwith, _x31)), app(app(cons, app(app(cons, _x33), _x32)), xs))app#(plus, app(app(plus, app(_x31, _x33)), app(app(sumwith, _x31), _x32)))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, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, 0)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), _x1971)app#(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x1971)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))app#(app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, 0))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), _x1541)app#(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x1541))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))
app#(app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, 0)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), _x1771)app#(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x1771)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))app#(app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, 0))))))))))))))))))))))))))))))))))))))))))))))))))))))), _x1101)app#(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x1101))))))))))))))))))))))))))))))))))))))))))))))))))))))
app#(app(sumwith, app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, 0))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), app(app(cons, _x2461), xs))app#(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x2461))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))app#(app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, 0)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), _x2011)app#(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x2011)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))
app#(app(sumwith, app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, 0))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), app(app(cons, _x2141), xs))app#(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x2141))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))app#(app(sumwith, app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x2582)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), app(app(cons, _x2581), xs))app#(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(app(plus, _x2582), _x2581))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))

Rewrite Rules

app(app(plus, 0), y)yapp(app(plus, app(s, x)), y)app(s, app(app(plus, x), y))
app(app(sumwith, f), nil)nilapp(app(sumwith, f), app(app(cons, x), xs))app(app(plus, app(f, x)), app(app(sumwith, f), xs))

Original Signature

Termination of terms over the following signature is verified: app, plus, 0, s, sumwith, 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, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x2012))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), _x2011) → app#(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(app(plus, _x2012), _x2011))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) 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(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x2021)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) 
app#(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(app(plus, _x2022), _x2021)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) 
Thus, the rule app#(app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x2012))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), _x2011) → app#(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(app(plus, _x2012), _x2011))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) is replaced by the following rules:
app#(app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, 0))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), _x2021) → app#(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x2021))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))app#(app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x2022)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), _x2021) → app#(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(app(plus, _x2022), _x2021))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))

Problem 30: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

app#(app(sumwith, app(sumwith, _x31)), app(app(cons, app(app(cons, _x33), _x32)), xs))app#(plus, app(app(plus, app(_x31, _x33)), app(app(sumwith, _x31), _x32)))app#(app(sumwith, f), app(app(cons, x), xs))app#(app(plus, app(f, x)), app(app(sumwith, f), xs))
app#(app(sumwith, f), app(app(cons, x), xs))app#(f, x)app#(app(sumwith, f), app(app(cons, x), xs))app#(app(sumwith, f), xs)
app#(app(sumwith, app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, 0)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), app(app(cons, _x2611), xs))app#(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x2611)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))app#(app(sumwith, app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, 0))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), app(app(cons, _x2621), xs))app#(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x2621))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))
app#(app(plus, app(s, x)), y)app#(app(plus, x), y)app#(app(sumwith, app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x2622)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), app(app(cons, _x2621), xs))app#(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(app(plus, _x2622), _x2621))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))
app#(app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x2432))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), _x2431)app#(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(app(plus, _x2432), _x2431)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))app#(app(sumwith, app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, 0))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), app(app(cons, _x2141), xs))app#(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x2141))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))
app#(app(sumwith, app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, 0))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), app(app(cons, _x2461), xs))app#(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x2461))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))app#(app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, 0)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), _x2011)app#(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x2011)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))

Rewrite Rules

app(app(plus, 0), y)yapp(app(plus, app(s, x)), y)app(s, app(app(plus, x), y))
app(app(sumwith, f), nil)nilapp(app(sumwith, f), app(app(cons, x), xs))app(app(plus, app(f, x)), app(app(sumwith, f), xs))

Original Signature

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

Strategy


The right-hand side of the rule app#(app(sumwith, app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, 0))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), app(app(cons, _x2621), xs)) → app#(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x2621)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) 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(sumwith, app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, 0))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), app(app(cons, _x2621), xs)) → app#(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x2621)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) is deleted.

Problem 31: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

app#(app(sumwith, f), app(app(cons, x), xs))app#(app(sumwith, f), xs)app#(app(sumwith, f), app(app(cons, x), xs))app#(f, x)
app#(app(sumwith, f), app(app(cons, x), xs))app#(app(plus, app(f, x)), app(app(sumwith, f), xs))app#(app(sumwith, app(sumwith, _x31)), app(app(cons, app(app(cons, _x33), _x32)), xs))app#(plus, app(app(plus, app(_x31, _x33)), app(app(sumwith, _x31), _x32)))
app#(app(sumwith, app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, 0))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), app(app(cons, _x3121), xs))app#(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x3121))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))app#(app(sumwith, app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x3132))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), app(app(cons, _x3131), xs))app#(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(app(plus, _x3132), 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app#(app(sumwith, app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, 0))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), app(app(cons, _x2901), xs))app#(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x2901))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))app#(app(sumwith, app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, 0)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), app(app(cons, _x2831), xs))app#(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x2831)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))
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app#(app(sumwith, app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, 0))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), app(app(cons, _x2461), xs))app#(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x2461))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))app#(app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, 0))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), _x2461)app#(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x2461))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))
app#(app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, 0)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), _x2011)app#(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x2011)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))app#(app(sumwith, app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, 0))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), app(app(cons, _x2141), xs))app#(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x2141))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))

Rewrite Rules

app(app(plus, 0), y)yapp(app(plus, app(s, x)), y)app(s, app(app(plus, x), y))
app(app(sumwith, f), nil)nilapp(app(sumwith, f), app(app(cons, x), xs))app(app(plus, app(f, x)), app(app(sumwith, f), xs))

Original Signature

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

Strategy


The right-hand side of the rule app#(app(sumwith, app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, 0))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), app(app(cons, _x3121), xs)) → app#(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x3121)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) 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(sumwith, app(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, 0))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), app(app(cons, _x3121), xs)) → app#(plus, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, app(s, _x3121)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) is deleted.