TIMEOUT

The TRS could not be proven terminating. The proof attempt took 60001 ms.

The following DP Processors were used

Problem 1 was processed with processor DependencyGraph (76ms).
 | – Problem 2 was processed with processor SubtermCriterion (1ms).
 | – Problem 3 was processed with processor PolynomialLinearRange4iUR (525ms).
 |    | – Problem 6 was processed with processor ForwardNarrowing (2ms).
 |    |    | – Problem 8 was processed with processor ForwardNarrowing (6ms).
 |    |    |    | – Problem 9 was processed with processor ForwardNarrowing (4ms).
 |    |    |    |    | – Problem 10 was processed with processor ForwardNarrowing (3ms).
 |    |    |    |    |    | – Problem 11 was processed with processor ForwardNarrowing (5ms).
 |    |    |    |    |    |    | – Problem 12 was processed with processor ForwardNarrowing (4ms).
 |    |    |    |    |    |    |    | – Problem 13 was processed with processor ForwardNarrowing (3ms).
 |    |    |    |    |    |    |    |    | – Problem 14 was processed with processor ForwardNarrowing (7ms).
 |    |    |    |    |    |    |    |    |    | – Problem 15 was processed with processor ForwardNarrowing (8ms).
 |    |    |    |    |    |    |    |    |    |    | – Problem 16 was processed with processor ForwardNarrowing (5ms).
 |    |    |    |    |    |    |    |    |    |    |    | – Problem 17 was processed with processor ForwardNarrowing (5ms).
 |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 18 was processed with processor ForwardNarrowing (5ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 19 was processed with processor ForwardNarrowing (4ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 20 was processed with processor ForwardNarrowing (10ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 21 was processed with processor ForwardNarrowing (6ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 22 was processed with processor ForwardNarrowing (4ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 23 was processed with processor ForwardNarrowing (4ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 24 was processed with processor ForwardNarrowing (3ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 25 was processed with processor ForwardNarrowing (5ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 26 was processed with processor ForwardNarrowing (5ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 27 was processed with processor ForwardNarrowing (14ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 28 was processed with processor ForwardNarrowing (8ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 29 was processed with processor ForwardNarrowing (21ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 30 was processed with processor ForwardNarrowing (30ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 31 was processed with processor ForwardNarrowing (20ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 32 was processed with processor ForwardNarrowing (29ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 33 was processed with processor ForwardNarrowing (103ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 34 was processed with processor ForwardNarrowing (83ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 35 was processed with processor ForwardNarrowing (224ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 36 was processed with processor ForwardNarrowing (233ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 37 remains open; application of the following processors failed [ForwardNarrowing (264ms), ForwardNarrowing (349ms), ForwardNarrowing (207ms), ForwardNarrowing (333ms), ForwardNarrowing (265ms), ForwardNarrowing (251ms), ForwardNarrowing (385ms), ForwardNarrowing (412ms), ForwardNarrowing (434ms), ForwardNarrowing (443ms), ForwardNarrowing (473ms), ForwardNarrowing (485ms), ForwardNarrowing (64ms), ForwardNarrowing (136ms), ForwardNarrowing (597ms), ForwardNarrowing (87ms), ForwardNarrowing (529ms), ForwardNarrowing (421ms), ForwardNarrowing (55ms), ForwardNarrowing (147ms), ForwardNarrowing (71ms), ForwardNarrowing (457ms), ForwardNarrowing (478ms), ForwardNarrowing (332ms), ForwardNarrowing (508ms), ForwardNarrowing (336ms), ForwardNarrowing (624ms), ForwardNarrowing (687ms), ForwardNarrowing (348ms), ForwardNarrowing (437ms), ForwardNarrowing (511ms), ForwardNarrowing (589ms), ForwardNarrowing (500ms), ForwardNarrowing (143ms), ForwardNarrowing (272ms), ForwardNarrowing (99ms), ForwardNarrowing (220ms), ForwardNarrowing (289ms), ForwardNarrowing (223ms), ForwardNarrowing (198ms), ForwardNarrowing (timeout)].
 | – Problem 4 was processed with processor PolynomialLinearRange4iUR (331ms).
 |    | – Problem 7 remains open; application of the following processors failed [DependencyGraph (1ms), PolynomialLinearRange4iUR (289ms), DependencyGraph (1ms), PolynomialLinearRange8NegiUR (2626ms), DependencyGraph (0ms), ReductionPairSAT (550ms), DependencyGraph (0ms), SizeChangePrinciple (20ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (1ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (1ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (2ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (2ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (1ms), ForwardNarrowing (1ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (1ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (3ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (1ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (1ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (1ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (3ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (1ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (4ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (6ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (4ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (1ms), ForwardNarrowing (1ms), ForwardNarrowing (1ms), ForwardNarrowing (1ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (1ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (1ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms)].
 | – Problem 5 remains open; application of the following processors failed [SubtermCriterion (1ms), DependencyGraph (1ms), PolynomialLinearRange4iUR (276ms), DependencyGraph (1ms), PolynomialLinearRange4iUR (273ms), DependencyGraph (1ms), PolynomialLinearRange8NegiUR (1343ms), DependencyGraph (1ms), ReductionPairSAT (569ms), DependencyGraph (0ms), SizeChangePrinciple (30ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (5ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing 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ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (33ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (1ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (1ms), 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ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (1ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (1ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (1ms), ForwardNarrowing (1ms), ForwardNarrowing (1ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (1ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms), ForwardNarrowing (0ms)].

The following open problems remain:

Open Dependency Pair Problem 5

Dependency Pairs

plus^#(s(x), y)

→

plus^#(y, minus(s(x), s(0)))

Rewrite Rules

minus(x, 0)	→	x	minus(0, y)	→	0
minus(s(x), s(y))	→	minus(p(s(x)), p(s(y)))	minus(x, plus(y, z))	→	minus(minus(x, y), z)
p(s(s(x)))	→	s(p(s(x)))	p(0)	→	s(s(0))
div(s(x), s(y))	→	s(div(minus(x, y), s(y)))	div(plus(x, y), z)	→	plus(div(x, z), div(y, z))
plus(0, y)	→	y	plus(s(x), y)	→	s(plus(y, minus(s(x), s(0))))

Original Signature

Termination of terms over the following signature is verified: plus, 0, minus, s, p, div

Open Dependency Pair Problem 6

Dependency Pairs

minus^#(s(x), s(y))

→

minus^#(p(s(x)), p(s(y)))

Rewrite Rules

minus(x, 0)	→	x	minus(0, y)	→	0
minus(s(x), s(y))	→	minus(p(s(x)), p(s(y)))	minus(x, plus(y, z))	→	minus(minus(x, y), z)
p(s(s(x)))	→	s(p(s(x)))	p(0)	→	s(s(0))
div(s(x), s(y))	→	s(div(minus(x, y), s(y)))	div(plus(x, y), z)	→	plus(div(x, z), div(y, z))
plus(0, y)	→	y	plus(s(x), y)	→	s(plus(y, minus(s(x), s(0))))

Original Signature

Termination of terms over the following signature is verified: plus, minus, 0, s, p, div

Open Dependency Pair Problem 7

Dependency Pairs

div^#(s(x), s(y))

→

div^#(minus(x, y), s(y))

Rewrite Rules

minus(x, 0)	→	x	minus(0, y)	→	0
minus(s(x), s(y))	→	minus(p(s(x)), p(s(y)))	minus(x, plus(y, z))	→	minus(minus(x, y), z)
p(s(s(x)))	→	s(p(s(x)))	p(0)	→	s(s(0))
div(s(x), s(y))	→	s(div(minus(x, y), s(y)))	div(plus(x, y), z)	→	plus(div(x, z), div(y, z))
plus(0, y)	→	y	plus(s(x), y)	→	s(plus(y, minus(s(x), s(0))))

Original Signature

Termination of terms over the following signature is verified: plus, minus, 0, s, p, div

Problem 1: DependencyGraph

Dependency Pair Problem

Dependency Pairs

div^#(plus(x, y), z)	→	div^#(x, z)	minus^#(s(x), s(y))	→	p^#(s(x))
minus^#(x, plus(y, z))	→	minus^#(minus(x, y), z)	minus^#(s(x), s(y))	→	minus^#(p(s(x)), p(s(y)))
plus^#(s(x), y)	→	plus^#(y, minus(s(x), s(0)))	minus^#(s(x), s(y))	→	p^#(s(y))
div^#(s(x), s(y))	→	minus^#(x, y)	div^#(plus(x, y), z)	→	div^#(y, z)
div^#(plus(x, y), z)	→	plus^#(div(x, z), div(y, z))	plus^#(s(x), y)	→	minus^#(s(x), s(0))
p^#(s(s(x)))	→	p^#(s(x))	div^#(s(x), s(y))	→	div^#(minus(x, y), s(y))
minus^#(x, plus(y, z))	→	minus^#(x, y)

Rewrite Rules

minus(x, 0)	→	x	minus(0, y)	→	0
minus(s(x), s(y))	→	minus(p(s(x)), p(s(y)))	minus(x, plus(y, z))	→	minus(minus(x, y), z)
p(s(s(x)))	→	s(p(s(x)))	p(0)	→	s(s(0))
div(s(x), s(y))	→	s(div(minus(x, y), s(y)))	div(plus(x, y), z)	→	plus(div(x, z), div(y, z))
plus(0, y)	→	y	plus(s(x), y)	→	s(plus(y, minus(s(x), s(0))))

Original Signature

Termination of terms over the following signature is verified: plus, minus, 0, s, p, div

Strategy

The following SCCs where found

div^#(plus(x, y), z) → div^#(x, z)	div^#(plus(x, y), z) → div^#(y, z)
div^#(s(x), s(y)) → div^#(minus(x, y), s(y))

minus^#(x, plus(y, z)) → minus^#(minus(x, y), z)	minus^#(s(x), s(y)) → minus^#(p(s(x)), p(s(y)))
minus^#(x, plus(y, z)) → minus^#(x, y)

p^#(s(s(x))) → p^#(s(x))

plus^#(s(x), y) → plus^#(y, minus(s(x), s(0)))

Problem 2: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

p^#(s(s(x)))

→

p^#(s(x))

Rewrite Rules

minus(x, 0)	→	x	minus(0, y)	→	0
minus(s(x), s(y))	→	minus(p(s(x)), p(s(y)))	minus(x, plus(y, z))	→	minus(minus(x, y), z)
p(s(s(x)))	→	s(p(s(x)))	p(0)	→	s(s(0))
div(s(x), s(y))	→	s(div(minus(x, y), s(y)))	div(plus(x, y), z)	→	plus(div(x, z), div(y, z))
plus(0, y)	→	y	plus(s(x), y)	→	s(plus(y, minus(s(x), s(0))))

Original Signature

Termination of terms over the following signature is verified: plus, minus, 0, s, p, div

Strategy

Projection

The following projection was used:

π (p#): 1

Thus, the following dependency pairs are removed:

p^#(s(s(x)))

→

p^#(s(x))

Problem 3: PolynomialLinearRange4iUR

Dependency Pair Problem

Dependency Pairs

minus^#(x, plus(y, z))	→	minus^#(minus(x, y), z)		minus^#(s(x), s(y))	→	minus^#(p(s(x)), p(s(y)))
minus^#(x, plus(y, z))	→	minus^#(x, y)

Rewrite Rules

minus(x, 0)	→	x	minus(0, y)	→	0
minus(s(x), s(y))	→	minus(p(s(x)), p(s(y)))	minus(x, plus(y, z))	→	minus(minus(x, y), z)
p(s(s(x)))	→	s(p(s(x)))	p(0)	→	s(s(0))
div(s(x), s(y))	→	s(div(minus(x, y), s(y)))	div(plus(x, y), z)	→	plus(div(x, z), div(y, z))
plus(0, y)	→	y	plus(s(x), y)	→	s(plus(y, minus(s(x), s(0))))

Original Signature

Termination of terms over the following signature is verified: plus, minus, 0, s, p, div

Strategy

Polynomial Interpretation

0: 3
div(x,y): 0
minus(x,y): 0
minus#(x,y): y
p(x): 0
plus(x,y): y + x + 2
s(x): 0

Improved Usable rules

p(s(s(x)))

→

s(p(s(x)))

p(0)

→

s(s(0))

The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:

minus^#(x, plus(y, z))

→

minus^#(minus(x, y), z)

minus^#(x, plus(y, z))

→

minus^#(x, y)

Problem 6: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

minus^#(s(x), s(y))

→

minus^#(p(s(x)), p(s(y)))

Rewrite Rules

minus(x, 0)	→	x	minus(0, y)	→	0
minus(s(x), s(y))	→	minus(p(s(x)), p(s(y)))	minus(x, plus(y, z))	→	minus(minus(x, y), z)
p(s(s(x)))	→	s(p(s(x)))	p(0)	→	s(s(0))
div(s(x), s(y))	→	s(div(minus(x, y), s(y)))	div(plus(x, y), z)	→	plus(div(x, z), div(y, z))
plus(0, y)	→	y	plus(s(x), y)	→	s(plus(y, minus(s(x), s(0))))

Original Signature

Termination of terms over the following signature is verified: plus, 0, minus, s, p, div

Strategy

The right-hand side of the rule minus^#(s(x), s(y)) → minus^#(p(s(x)), p(s(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 Terms	Irrelevant Terms
minus^#(p(s(x)), s(p(s(_x31))))
minus^#(s(p(s(_x21))), p(s(y)))

Thus, the rule minus^#(s(x), s(y)) → minus^#(p(s(x)), p(s(y))) is replaced by the following rules:

minus^#(s(s(_x21)), s(y)) → minus^#(s(p(s(_x21))), p(s(y)))

minus^#(s(x), s(s(_x31))) → minus^#(p(s(x)), s(p(s(_x31))))

Problem 8: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

minus^#(s(s(_x21)), s(y))

→

minus^#(s(p(s(_x21))), p(s(y)))

minus^#(s(x), s(s(_x31)))

→

minus^#(p(s(x)), s(p(s(_x31))))

Rewrite Rules

minus(x, 0)	→	x	minus(0, y)	→	0
minus(s(x), s(y))	→	minus(p(s(x)), p(s(y)))	minus(x, plus(y, z))	→	minus(minus(x, y), z)
p(s(s(x)))	→	s(p(s(x)))	p(0)	→	s(s(0))
div(s(x), s(y))	→	s(div(minus(x, y), s(y)))	div(plus(x, y), z)	→	plus(div(x, z), div(y, z))
plus(0, y)	→	y	plus(s(x), y)	→	s(plus(y, minus(s(x), s(0))))

Original Signature

Termination of terms over the following signature is verified: plus, minus, 0, s, p, div

Strategy

The right-hand side of the rule minus^#(s(s(_x21)), s(y)) → minus^#(s(p(s(_x21))), p(s(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 Terms	Irrelevant Terms
minus^#(s(s(p(s(_x41)))), p(s(y)))
minus^#(s(p(s(_x21))), s(p(s(_x31))))

Thus, the rule minus^#(s(s(_x21)), s(y)) → minus^#(s(p(s(_x21))), p(s(y))) is replaced by the following rules:

minus^#(s(s(s(_x41))), s(y)) → minus^#(s(s(p(s(_x41)))), p(s(y)))

minus^#(s(s(_x21)), s(s(_x31))) → minus^#(s(p(s(_x21))), s(p(s(_x31))))

Problem 9: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

minus^#(s(s(s(_x41))), s(y))	→	minus^#(s(s(p(s(_x41)))), p(s(y)))		minus^#(s(s(_x21)), s(s(_x31)))	→	minus^#(s(p(s(_x21))), s(p(s(_x31))))
minus^#(s(x), s(s(_x31)))	→	minus^#(p(s(x)), s(p(s(_x31))))

Rewrite Rules

minus(x, 0)	→	x	minus(0, y)	→	0
minus(s(x), s(y))	→	minus(p(s(x)), p(s(y)))	minus(x, plus(y, z))	→	minus(minus(x, y), z)
p(s(s(x)))	→	s(p(s(x)))	p(0)	→	s(s(0))
div(s(x), s(y))	→	s(div(minus(x, y), s(y)))	div(plus(x, y), z)	→	plus(div(x, z), div(y, z))
plus(0, y)	→	y	plus(s(x), y)	→	s(plus(y, minus(s(x), s(0))))

Original Signature

Termination of terms over the following signature is verified: plus, 0, minus, s, p, div

Strategy

The right-hand side of the rule minus^#(s(s(s(_x41))), s(y)) → minus^#(s(s(p(s(_x41)))), p(s(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 Terms	Irrelevant Terms
minus^#(s(s(s(p(s(_x51))))), p(s(y)))
minus^#(s(s(p(s(_x41)))), s(p(s(_x31))))

Thus, the rule minus^#(s(s(s(_x41))), s(y)) → minus^#(s(s(p(s(_x41)))), p(s(y))) is replaced by the following rules:

minus^#(s(s(s(s(_x51)))), s(y)) → minus^#(s(s(s(p(s(_x51))))), p(s(y)))

minus^#(s(s(s(_x41))), s(s(_x31))) → minus^#(s(s(p(s(_x41)))), s(p(s(_x31))))

Problem 10: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

minus^#(s(s(s(s(_x51)))), s(y))	→	minus^#(s(s(s(p(s(_x51))))), p(s(y)))		minus^#(s(s(s(_x41))), s(s(_x31)))	→	minus^#(s(s(p(s(_x41)))), s(p(s(_x31))))
minus^#(s(s(_x21)), s(s(_x31)))	→	minus^#(s(p(s(_x21))), s(p(s(_x31))))		minus^#(s(x), s(s(_x31)))	→	minus^#(p(s(x)), s(p(s(_x31))))

Rewrite Rules

minus(x, 0)	→	x	minus(0, y)	→	0
minus(s(x), s(y))	→	minus(p(s(x)), p(s(y)))	minus(x, plus(y, z))	→	minus(minus(x, y), z)
p(s(s(x)))	→	s(p(s(x)))	p(0)	→	s(s(0))
div(s(x), s(y))	→	s(div(minus(x, y), s(y)))	div(plus(x, y), z)	→	plus(div(x, z), div(y, z))
plus(0, y)	→	y	plus(s(x), y)	→	s(plus(y, minus(s(x), s(0))))

Original Signature

Termination of terms over the following signature is verified: plus, minus, 0, s, p, div

Strategy

The right-hand side of the rule minus^#(s(s(s(_x41))), s(s(_x31))) → minus^#(s(s(p(s(_x41)))), s(p(s(_x31)))) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
minus^#(s(s(s(p(s(_x51))))), s(p(s(_x31))))
minus^#(s(s(p(s(_x41)))), s(s(p(s(_x51)))))

Thus, the rule minus^#(s(s(s(_x41))), s(s(_x31))) → minus^#(s(s(p(s(_x41)))), s(p(s(_x31)))) is replaced by the following rules:

minus^#(s(s(s(s(_x51)))), s(s(_x31))) → minus^#(s(s(s(p(s(_x51))))), s(p(s(_x31))))

minus^#(s(s(s(_x41))), s(s(s(_x51)))) → minus^#(s(s(p(s(_x41)))), s(s(p(s(_x51)))))

Problem 11: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

minus^#(s(s(s(s(_x51)))), s(s(_x31)))	→	minus^#(s(s(s(p(s(_x51))))), s(p(s(_x31))))	minus^#(s(s(s(s(_x51)))), s(y))	→	minus^#(s(s(s(p(s(_x51))))), p(s(y)))
minus^#(s(s(s(_x41))), s(s(s(_x51))))	→	minus^#(s(s(p(s(_x41)))), s(s(p(s(_x51)))))	minus^#(s(s(_x21)), s(s(_x31)))	→	minus^#(s(p(s(_x21))), s(p(s(_x31))))
minus^#(s(x), s(s(_x31)))	→	minus^#(p(s(x)), s(p(s(_x31))))

Rewrite Rules

minus(x, 0)	→	x	minus(0, y)	→	0
minus(s(x), s(y))	→	minus(p(s(x)), p(s(y)))	minus(x, plus(y, z))	→	minus(minus(x, y), z)
p(s(s(x)))	→	s(p(s(x)))	p(0)	→	s(s(0))
div(s(x), s(y))	→	s(div(minus(x, y), s(y)))	div(plus(x, y), z)	→	plus(div(x, z), div(y, z))
plus(0, y)	→	y	plus(s(x), y)	→	s(plus(y, minus(s(x), s(0))))

Original Signature

Termination of terms over the following signature is verified: plus, 0, minus, s, p, div

Strategy

The right-hand side of the rule minus^#(s(s(s(s(_x51)))), s(s(_x31))) → minus^#(s(s(s(p(s(_x51))))), s(p(s(_x31)))) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
minus^#(s(s(s(p(s(_x51))))), s(s(p(s(_x51)))))
minus^#(s(s(s(s(p(s(_x61)))))), s(p(s(_x31))))

Thus, the rule minus^#(s(s(s(s(_x51)))), s(s(_x31))) → minus^#(s(s(s(p(s(_x51))))), s(p(s(_x31)))) is replaced by the following rules:

minus^#(s(s(s(s(_x51)))), s(s(s(_x51)))) → minus^#(s(s(s(p(s(_x51))))), s(s(p(s(_x51)))))

minus^#(s(s(s(s(s(_x61))))), s(s(_x31))) → minus^#(s(s(s(s(p(s(_x61)))))), s(p(s(_x31))))

Problem 12: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

minus^#(s(s(s(s(_x51)))), s(s(s(_x51))))	→	minus^#(s(s(s(p(s(_x51))))), s(s(p(s(_x51)))))	minus^#(s(s(s(s(s(_x61))))), s(s(_x31)))	→	minus^#(s(s(s(s(p(s(_x61)))))), s(p(s(_x31))))
minus^#(s(s(s(s(_x51)))), s(y))	→	minus^#(s(s(s(p(s(_x51))))), p(s(y)))	minus^#(s(s(s(_x41))), s(s(s(_x51))))	→	minus^#(s(s(p(s(_x41)))), s(s(p(s(_x51)))))
minus^#(s(s(_x21)), s(s(_x31)))	→	minus^#(s(p(s(_x21))), s(p(s(_x31))))	minus^#(s(x), s(s(_x31)))	→	minus^#(p(s(x)), s(p(s(_x31))))

Rewrite Rules

minus(x, 0)	→	x	minus(0, y)	→	0
minus(s(x), s(y))	→	minus(p(s(x)), p(s(y)))	minus(x, plus(y, z))	→	minus(minus(x, y), z)
p(s(s(x)))	→	s(p(s(x)))	p(0)	→	s(s(0))
div(s(x), s(y))	→	s(div(minus(x, y), s(y)))	div(plus(x, y), z)	→	plus(div(x, z), div(y, z))
plus(0, y)	→	y	plus(s(x), y)	→	s(plus(y, minus(s(x), s(0))))

Original Signature

Termination of terms over the following signature is verified: plus, minus, 0, s, p, div

Strategy

The right-hand side of the rule minus^#(s(s(s(s(_x51)))), s(y)) → minus^#(s(s(s(p(s(_x51))))), p(s(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 Terms	Irrelevant Terms
minus^#(s(s(s(p(s(_x51))))), s(p(s(_x31))))
minus^#(s(s(s(s(p(s(_x61)))))), p(s(y)))

Thus, the rule minus^#(s(s(s(s(_x51)))), s(y)) → minus^#(s(s(s(p(s(_x51))))), p(s(y))) is replaced by the following rules:

minus^#(s(s(s(s(s(_x61))))), s(y)) → minus^#(s(s(s(s(p(s(_x61)))))), p(s(y)))

minus^#(s(s(s(s(_x51)))), s(s(_x31))) → minus^#(s(s(s(p(s(_x51))))), s(p(s(_x31))))

Problem 13: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

minus^#(s(s(s(s(s(_x61))))), s(y))	→	minus^#(s(s(s(s(p(s(_x61)))))), p(s(y)))	minus^#(s(s(s(s(_x51)))), s(s(s(_x51))))	→	minus^#(s(s(s(p(s(_x51))))), s(s(p(s(_x51)))))
minus^#(s(s(s(s(s(_x61))))), s(s(_x31)))	→	minus^#(s(s(s(s(p(s(_x61)))))), s(p(s(_x31))))	minus^#(s(s(s(s(_x51)))), s(s(_x31)))	→	minus^#(s(s(s(p(s(_x51))))), s(p(s(_x31))))
minus^#(s(s(s(_x41))), s(s(s(_x51))))	→	minus^#(s(s(p(s(_x41)))), s(s(p(s(_x51)))))	minus^#(s(s(_x21)), s(s(_x31)))	→	minus^#(s(p(s(_x21))), s(p(s(_x31))))
minus^#(s(x), s(s(_x31)))	→	minus^#(p(s(x)), s(p(s(_x31))))

Rewrite Rules

minus(x, 0)	→	x	minus(0, y)	→	0
minus(s(x), s(y))	→	minus(p(s(x)), p(s(y)))	minus(x, plus(y, z))	→	minus(minus(x, y), z)
p(s(s(x)))	→	s(p(s(x)))	p(0)	→	s(s(0))
div(s(x), s(y))	→	s(div(minus(x, y), s(y)))	div(plus(x, y), z)	→	plus(div(x, z), div(y, z))
plus(0, y)	→	y	plus(s(x), y)	→	s(plus(y, minus(s(x), s(0))))

Original Signature

Termination of terms over the following signature is verified: plus, 0, minus, s, p, div

Strategy

The right-hand side of the rule minus^#(s(s(s(s(s(_x61))))), s(y)) → minus^#(s(s(s(s(p(s(_x61)))))), p(s(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 Terms	Irrelevant Terms
minus^#(s(s(s(s(p(s(_x61)))))), s(p(s(_x31))))
minus^#(s(s(s(s(s(p(s(_x71))))))), p(s(y)))

Thus, the rule minus^#(s(s(s(s(s(_x61))))), s(y)) → minus^#(s(s(s(s(p(s(_x61)))))), p(s(y))) is replaced by the following rules:

minus^#(s(s(s(s(s(s(_x71)))))), s(y)) → minus^#(s(s(s(s(s(p(s(_x71))))))), p(s(y)))

minus^#(s(s(s(s(s(_x61))))), s(s(_x31))) → minus^#(s(s(s(s(p(s(_x61)))))), s(p(s(_x31))))

Problem 14: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

minus^#(s(s(s(s(s(s(_x71)))))), s(y))	→	minus^#(s(s(s(s(s(p(s(_x71))))))), p(s(y)))	minus^#(s(s(s(s(_x51)))), s(s(s(_x51))))	→	minus^#(s(s(s(p(s(_x51))))), s(s(p(s(_x51)))))
minus^#(s(s(s(s(s(_x61))))), s(s(_x31)))	→	minus^#(s(s(s(s(p(s(_x61)))))), s(p(s(_x31))))	minus^#(s(s(s(s(_x51)))), s(s(_x31)))	→	minus^#(s(s(s(p(s(_x51))))), s(p(s(_x31))))
minus^#(s(s(s(_x41))), s(s(s(_x51))))	→	minus^#(s(s(p(s(_x41)))), s(s(p(s(_x51)))))	minus^#(s(s(_x21)), s(s(_x31)))	→	minus^#(s(p(s(_x21))), s(p(s(_x31))))
minus^#(s(x), s(s(_x31)))	→	minus^#(p(s(x)), s(p(s(_x31))))

Rewrite Rules

minus(x, 0)	→	x	minus(0, y)	→	0
minus(s(x), s(y))	→	minus(p(s(x)), p(s(y)))	minus(x, plus(y, z))	→	minus(minus(x, y), z)
p(s(s(x)))	→	s(p(s(x)))	p(0)	→	s(s(0))
div(s(x), s(y))	→	s(div(minus(x, y), s(y)))	div(plus(x, y), z)	→	plus(div(x, z), div(y, z))
plus(0, y)	→	y	plus(s(x), y)	→	s(plus(y, minus(s(x), s(0))))

Original Signature

Termination of terms over the following signature is verified: plus, minus, 0, s, p, div

Strategy

The right-hand side of the rule minus^#(s(s(s(s(s(s(_x71)))))), s(y)) → minus^#(s(s(s(s(s(p(s(_x71))))))), p(s(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 Terms	Irrelevant Terms
minus^#(s(s(s(s(s(s(p(s(_x81)))))))), p(s(y)))
minus^#(s(s(s(s(s(p(s(_x71))))))), s(p(s(_x31))))

Thus, the rule minus^#(s(s(s(s(s(s(_x71)))))), s(y)) → minus^#(s(s(s(s(s(p(s(_x71))))))), p(s(y))) is replaced by the following rules:

minus^#(s(s(s(s(s(s(_x71)))))), s(s(_x31))) → minus^#(s(s(s(s(s(p(s(_x71))))))), s(p(s(_x31))))

minus^#(s(s(s(s(s(s(s(_x81))))))), s(y)) → minus^#(s(s(s(s(s(s(p(s(_x81)))))))), p(s(y)))

Problem 15: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

minus^#(s(s(s(s(s(s(_x71)))))), s(s(_x31)))	→	minus^#(s(s(s(s(s(p(s(_x71))))))), s(p(s(_x31))))	minus^#(s(s(s(s(_x51)))), s(s(s(_x51))))	→	minus^#(s(s(s(p(s(_x51))))), s(s(p(s(_x51)))))
minus^#(s(s(s(s(s(s(s(_x81))))))), s(y))	→	minus^#(s(s(s(s(s(s(p(s(_x81)))))))), p(s(y)))	minus^#(s(s(s(s(s(_x61))))), s(s(_x31)))	→	minus^#(s(s(s(s(p(s(_x61)))))), s(p(s(_x31))))
minus^#(s(s(s(s(_x51)))), s(s(_x31)))	→	minus^#(s(s(s(p(s(_x51))))), s(p(s(_x31))))	minus^#(s(s(s(_x41))), s(s(s(_x51))))	→	minus^#(s(s(p(s(_x41)))), s(s(p(s(_x51)))))
minus^#(s(s(_x21)), s(s(_x31)))	→	minus^#(s(p(s(_x21))), s(p(s(_x31))))	minus^#(s(x), s(s(_x31)))	→	minus^#(p(s(x)), s(p(s(_x31))))

Rewrite Rules

minus(x, 0)	→	x	minus(0, y)	→	0
minus(s(x), s(y))	→	minus(p(s(x)), p(s(y)))	minus(x, plus(y, z))	→	minus(minus(x, y), z)
p(s(s(x)))	→	s(p(s(x)))	p(0)	→	s(s(0))
div(s(x), s(y))	→	s(div(minus(x, y), s(y)))	div(plus(x, y), z)	→	plus(div(x, z), div(y, z))
plus(0, y)	→	y	plus(s(x), y)	→	s(plus(y, minus(s(x), s(0))))

Original Signature

Termination of terms over the following signature is verified: plus, 0, minus, s, p, div

Strategy

The right-hand side of the rule minus^#(s(s(s(s(s(s(_x71)))))), s(s(_x31))) → minus^#(s(s(s(s(s(p(s(_x71))))))), s(p(s(_x31)))) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
minus^#(s(s(s(s(s(p(s(_x71))))))), s(s(p(s(_x51)))))
minus^#(s(s(s(s(s(s(p(s(_x81)))))))), s(p(s(_x31))))

Thus, the rule minus^#(s(s(s(s(s(s(_x71)))))), s(s(_x31))) → minus^#(s(s(s(s(s(p(s(_x71))))))), s(p(s(_x31)))) is replaced by the following rules:

minus^#(s(s(s(s(s(s(s(_x81))))))), s(s(_x31))) → minus^#(s(s(s(s(s(s(p(s(_x81)))))))), s(p(s(_x31))))

minus^#(s(s(s(s(s(s(_x71)))))), s(s(s(_x51)))) → minus^#(s(s(s(s(s(p(s(_x71))))))), s(s(p(s(_x51)))))

Problem 16: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

minus^#(s(s(s(s(s(s(s(_x81))))))), s(s(_x31)))	→	minus^#(s(s(s(s(s(s(p(s(_x81)))))))), s(p(s(_x31))))	minus^#(s(s(s(s(_x51)))), s(s(s(_x51))))	→	minus^#(s(s(s(p(s(_x51))))), s(s(p(s(_x51)))))
minus^#(s(s(s(s(s(s(s(_x81))))))), s(y))	→	minus^#(s(s(s(s(s(s(p(s(_x81)))))))), p(s(y)))	minus^#(s(s(s(s(s(_x61))))), s(s(_x31)))	→	minus^#(s(s(s(s(p(s(_x61)))))), s(p(s(_x31))))
minus^#(s(s(s(s(_x51)))), s(s(_x31)))	→	minus^#(s(s(s(p(s(_x51))))), s(p(s(_x31))))	minus^#(s(s(s(_x41))), s(s(s(_x51))))	→	minus^#(s(s(p(s(_x41)))), s(s(p(s(_x51)))))
minus^#(s(s(_x21)), s(s(_x31)))	→	minus^#(s(p(s(_x21))), s(p(s(_x31))))	minus^#(s(s(s(s(s(s(_x71)))))), s(s(s(_x51))))	→	minus^#(s(s(s(s(s(p(s(_x71))))))), s(s(p(s(_x51)))))
minus^#(s(x), s(s(_x31)))	→	minus^#(p(s(x)), s(p(s(_x31))))

Rewrite Rules

minus(x, 0)	→	x	minus(0, y)	→	0
minus(s(x), s(y))	→	minus(p(s(x)), p(s(y)))	minus(x, plus(y, z))	→	minus(minus(x, y), z)
p(s(s(x)))	→	s(p(s(x)))	p(0)	→	s(s(0))
div(s(x), s(y))	→	s(div(minus(x, y), s(y)))	div(plus(x, y), z)	→	plus(div(x, z), div(y, z))
plus(0, y)	→	y	plus(s(x), y)	→	s(plus(y, minus(s(x), s(0))))

Original Signature

Termination of terms over the following signature is verified: plus, minus, 0, s, p, div

Strategy

The right-hand side of the rule minus^#(s(s(s(s(s(s(s(_x81))))))), s(s(_x31))) → minus^#(s(s(s(s(s(s(p(s(_x81)))))))), s(p(s(_x31)))) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
minus^#(s(s(s(s(s(s(p(s(_x81)))))))), s(s(p(s(_x51)))))
minus^#(s(s(s(s(s(s(s(p(s(_x91))))))))), s(p(s(_x31))))

Thus, the rule minus^#(s(s(s(s(s(s(s(_x81))))))), s(s(_x31))) → minus^#(s(s(s(s(s(s(p(s(_x81)))))))), s(p(s(_x31)))) is replaced by the following rules:

minus^#(s(s(s(s(s(s(s(_x81))))))), s(s(s(_x51)))) → minus^#(s(s(s(s(s(s(p(s(_x81)))))))), s(s(p(s(_x51)))))

minus^#(s(s(s(s(s(s(s(s(_x91)))))))), s(s(_x31))) → minus^#(s(s(s(s(s(s(s(p(s(_x91))))))))), s(p(s(_x31))))

Problem 17: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

minus^#(s(s(s(s(_x51)))), s(s(s(_x51))))	→	minus^#(s(s(s(p(s(_x51))))), s(s(p(s(_x51)))))	minus^#(s(s(s(s(s(s(s(_x81))))))), s(y))	→	minus^#(s(s(s(s(s(s(p(s(_x81)))))))), p(s(y)))
minus^#(s(s(s(s(s(_x61))))), s(s(_x31)))	→	minus^#(s(s(s(s(p(s(_x61)))))), s(p(s(_x31))))	minus^#(s(s(s(s(_x51)))), s(s(_x31)))	→	minus^#(s(s(s(p(s(_x51))))), s(p(s(_x31))))
minus^#(s(s(s(_x41))), s(s(s(_x51))))	→	minus^#(s(s(p(s(_x41)))), s(s(p(s(_x51)))))	minus^#(s(s(_x21)), s(s(_x31)))	→	minus^#(s(p(s(_x21))), s(p(s(_x31))))
minus^#(s(s(s(s(s(s(s(_x81))))))), s(s(s(_x51))))	→	minus^#(s(s(s(s(s(s(p(s(_x81)))))))), s(s(p(s(_x51)))))	minus^#(s(s(s(s(s(s(s(s(_x91)))))))), s(s(_x31)))	→	minus^#(s(s(s(s(s(s(s(p(s(_x91))))))))), s(p(s(_x31))))
minus^#(s(s(s(s(s(s(_x71)))))), s(s(s(_x51))))	→	minus^#(s(s(s(s(s(p(s(_x71))))))), s(s(p(s(_x51)))))	minus^#(s(x), s(s(_x31)))	→	minus^#(p(s(x)), s(p(s(_x31))))

Rewrite Rules

minus(x, 0)	→	x	minus(0, y)	→	0
minus(s(x), s(y))	→	minus(p(s(x)), p(s(y)))	minus(x, plus(y, z))	→	minus(minus(x, y), z)
p(s(s(x)))	→	s(p(s(x)))	p(0)	→	s(s(0))
div(s(x), s(y))	→	s(div(minus(x, y), s(y)))	div(plus(x, y), z)	→	plus(div(x, z), div(y, z))
plus(0, y)	→	y	plus(s(x), y)	→	s(plus(y, minus(s(x), s(0))))

Original Signature

Termination of terms over the following signature is verified: plus, 0, minus, s, p, div

Strategy

The right-hand side of the rule minus^#(s(s(s(s(s(s(s(_x81))))))), s(y)) → minus^#(s(s(s(s(s(s(p(s(_x81)))))))), p(s(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 Terms	Irrelevant Terms
minus^#(s(s(s(s(s(s(s(p(s(_x91))))))))), p(s(y)))
minus^#(s(s(s(s(s(s(p(s(_x81)))))))), s(p(s(_x31))))

Thus, the rule minus^#(s(s(s(s(s(s(s(_x81))))))), s(y)) → minus^#(s(s(s(s(s(s(p(s(_x81)))))))), p(s(y))) is replaced by the following rules:

minus^#(s(s(s(s(s(s(s(_x81))))))), s(s(_x31))) → minus^#(s(s(s(s(s(s(p(s(_x81)))))))), s(p(s(_x31))))

minus^#(s(s(s(s(s(s(s(s(_x91)))))))), s(y)) → minus^#(s(s(s(s(s(s(s(p(s(_x91))))))))), p(s(y)))

Problem 18: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

minus^#(s(s(s(s(s(s(s(_x81))))))), s(s(_x31)))	→	minus^#(s(s(s(s(s(s(p(s(_x81)))))))), s(p(s(_x31))))	minus^#(s(s(s(s(_x51)))), s(s(s(_x51))))	→	minus^#(s(s(s(p(s(_x51))))), s(s(p(s(_x51)))))
minus^#(s(s(s(s(s(_x61))))), s(s(_x31)))	→	minus^#(s(s(s(s(p(s(_x61)))))), s(p(s(_x31))))	minus^#(s(s(s(s(_x51)))), s(s(_x31)))	→	minus^#(s(s(s(p(s(_x51))))), s(p(s(_x31))))
minus^#(s(s(s(s(s(s(s(s(_x91)))))))), s(y))	→	minus^#(s(s(s(s(s(s(s(p(s(_x91))))))))), p(s(y)))	minus^#(s(s(s(_x41))), s(s(s(_x51))))	→	minus^#(s(s(p(s(_x41)))), s(s(p(s(_x51)))))
minus^#(s(s(_x21)), s(s(_x31)))	→	minus^#(s(p(s(_x21))), s(p(s(_x31))))	minus^#(s(s(s(s(s(s(s(s(_x91)))))))), s(s(_x31)))	→	minus^#(s(s(s(s(s(s(s(p(s(_x91))))))))), s(p(s(_x31))))
minus^#(s(s(s(s(s(s(s(_x81))))))), s(s(s(_x51))))	→	minus^#(s(s(s(s(s(s(p(s(_x81)))))))), s(s(p(s(_x51)))))	minus^#(s(s(s(s(s(s(_x71)))))), s(s(s(_x51))))	→	minus^#(s(s(s(s(s(p(s(_x71))))))), s(s(p(s(_x51)))))
minus^#(s(x), s(s(_x31)))	→	minus^#(p(s(x)), s(p(s(_x31))))

Rewrite Rules

minus(x, 0)	→	x	minus(0, y)	→	0
minus(s(x), s(y))	→	minus(p(s(x)), p(s(y)))	minus(x, plus(y, z))	→	minus(minus(x, y), z)
p(s(s(x)))	→	s(p(s(x)))	p(0)	→	s(s(0))
div(s(x), s(y))	→	s(div(minus(x, y), s(y)))	div(plus(x, y), z)	→	plus(div(x, z), div(y, z))
plus(0, y)	→	y	plus(s(x), y)	→	s(plus(y, minus(s(x), s(0))))

Original Signature

Termination of terms over the following signature is verified: plus, minus, 0, s, p, div

Strategy

Relevant Terms	Irrelevant Terms
minus^#(s(s(s(s(s(s(p(s(_x81)))))))), s(s(p(s(_x51)))))
minus^#(s(s(s(s(s(s(s(p(s(_x91))))))))), s(p(s(_x31))))

Thus, the rule minus^#(s(s(s(s(s(s(s(_x81))))))), s(s(_x31))) → minus^#(s(s(s(s(s(s(p(s(_x81)))))))), s(p(s(_x31)))) is replaced by the following rules:

minus^#(s(s(s(s(s(s(s(_x81))))))), s(s(s(_x51)))) → minus^#(s(s(s(s(s(s(p(s(_x81)))))))), s(s(p(s(_x51)))))

minus^#(s(s(s(s(s(s(s(s(_x91)))))))), s(s(_x31))) → minus^#(s(s(s(s(s(s(s(p(s(_x91))))))))), s(p(s(_x31))))

Problem 19: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

minus^#(s(s(s(s(_x51)))), s(s(s(_x51))))	→	minus^#(s(s(s(p(s(_x51))))), s(s(p(s(_x51)))))	minus^#(s(s(s(s(s(_x61))))), s(s(_x31)))	→	minus^#(s(s(s(s(p(s(_x61)))))), s(p(s(_x31))))
minus^#(s(s(s(s(_x51)))), s(s(_x31)))	→	minus^#(s(s(s(p(s(_x51))))), s(p(s(_x31))))	minus^#(s(s(s(s(s(s(s(s(_x91)))))))), s(y))	→	minus^#(s(s(s(s(s(s(s(p(s(_x91))))))))), p(s(y)))
minus^#(s(s(s(_x41))), s(s(s(_x51))))	→	minus^#(s(s(p(s(_x41)))), s(s(p(s(_x51)))))	minus^#(s(s(_x21)), s(s(_x31)))	→	minus^#(s(p(s(_x21))), s(p(s(_x31))))
minus^#(s(s(s(s(s(s(s(_x81))))))), s(s(s(_x51))))	→	minus^#(s(s(s(s(s(s(p(s(_x81)))))))), s(s(p(s(_x51)))))	minus^#(s(s(s(s(s(s(s(s(_x91)))))))), s(s(_x31)))	→	minus^#(s(s(s(s(s(s(s(p(s(_x91))))))))), s(p(s(_x31))))
minus^#(s(s(s(s(s(s(_x71)))))), s(s(s(_x51))))	→	minus^#(s(s(s(s(s(p(s(_x71))))))), s(s(p(s(_x51)))))	minus^#(s(x), s(s(_x31)))	→	minus^#(p(s(x)), s(p(s(_x31))))

Rewrite Rules

minus(x, 0)	→	x	minus(0, y)	→	0
minus(s(x), s(y))	→	minus(p(s(x)), p(s(y)))	minus(x, plus(y, z))	→	minus(minus(x, y), z)
p(s(s(x)))	→	s(p(s(x)))	p(0)	→	s(s(0))
div(s(x), s(y))	→	s(div(minus(x, y), s(y)))	div(plus(x, y), z)	→	plus(div(x, z), div(y, z))
plus(0, y)	→	y	plus(s(x), y)	→	s(plus(y, minus(s(x), s(0))))

Original Signature

Termination of terms over the following signature is verified: plus, 0, minus, s, p, div

Strategy

The right-hand side of the rule minus^#(s(s(s(s(s(_x61))))), s(s(_x31))) → minus^#(s(s(s(s(p(s(_x61)))))), s(p(s(_x31)))) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
minus^#(s(s(s(s(s(p(s(_x71))))))), s(p(s(_x31))))
minus^#(s(s(s(s(p(s(_x61)))))), s(s(p(s(_x51)))))

Thus, the rule minus^#(s(s(s(s(s(_x61))))), s(s(_x31))) → minus^#(s(s(s(s(p(s(_x61)))))), s(p(s(_x31)))) is replaced by the following rules:

minus^#(s(s(s(s(s(s(_x71)))))), s(s(_x31))) → minus^#(s(s(s(s(s(p(s(_x71))))))), s(p(s(_x31))))

minus^#(s(s(s(s(s(_x61))))), s(s(s(_x51)))) → minus^#(s(s(s(s(p(s(_x61)))))), s(s(p(s(_x51)))))

Problem 20: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

minus^#(s(s(s(s(s(s(_x71)))))), s(s(_x31)))	→	minus^#(s(s(s(s(s(p(s(_x71))))))), s(p(s(_x31))))	minus^#(s(s(s(s(_x51)))), s(s(s(_x51))))	→	minus^#(s(s(s(p(s(_x51))))), s(s(p(s(_x51)))))
minus^#(s(s(s(s(s(_x61))))), s(s(s(_x51))))	→	minus^#(s(s(s(s(p(s(_x61)))))), s(s(p(s(_x51)))))	minus^#(s(s(s(s(_x51)))), s(s(_x31)))	→	minus^#(s(s(s(p(s(_x51))))), s(p(s(_x31))))
minus^#(s(s(s(s(s(s(s(s(_x91)))))))), s(y))	→	minus^#(s(s(s(s(s(s(s(p(s(_x91))))))))), p(s(y)))	minus^#(s(s(s(_x41))), s(s(s(_x51))))	→	minus^#(s(s(p(s(_x41)))), s(s(p(s(_x51)))))
minus^#(s(s(_x21)), s(s(_x31)))	→	minus^#(s(p(s(_x21))), s(p(s(_x31))))	minus^#(s(s(s(s(s(s(s(s(_x91)))))))), s(s(_x31)))	→	minus^#(s(s(s(s(s(s(s(p(s(_x91))))))))), s(p(s(_x31))))
minus^#(s(s(s(s(s(s(s(_x81))))))), s(s(s(_x51))))	→	minus^#(s(s(s(s(s(s(p(s(_x81)))))))), s(s(p(s(_x51)))))	minus^#(s(s(s(s(s(s(_x71)))))), s(s(s(_x51))))	→	minus^#(s(s(s(s(s(p(s(_x71))))))), s(s(p(s(_x51)))))
minus^#(s(x), s(s(_x31)))	→	minus^#(p(s(x)), s(p(s(_x31))))

Rewrite Rules

minus(x, 0)	→	x	minus(0, y)	→	0
minus(s(x), s(y))	→	minus(p(s(x)), p(s(y)))	minus(x, plus(y, z))	→	minus(minus(x, y), z)
p(s(s(x)))	→	s(p(s(x)))	p(0)	→	s(s(0))
div(s(x), s(y))	→	s(div(minus(x, y), s(y)))	div(plus(x, y), z)	→	plus(div(x, z), div(y, z))
plus(0, y)	→	y	plus(s(x), y)	→	s(plus(y, minus(s(x), s(0))))

Original Signature

Termination of terms over the following signature is verified: plus, minus, 0, s, p, div

Strategy

Relevant Terms	Irrelevant Terms
minus^#(s(s(s(s(s(p(s(_x71))))))), s(s(p(s(_x51)))))
minus^#(s(s(s(s(s(s(p(s(_x81)))))))), s(p(s(_x31))))

Thus, the rule minus^#(s(s(s(s(s(s(_x71)))))), s(s(_x31))) → minus^#(s(s(s(s(s(p(s(_x71))))))), s(p(s(_x31)))) is replaced by the following rules:

minus^#(s(s(s(s(s(s(s(_x81))))))), s(s(_x31))) → minus^#(s(s(s(s(s(s(p(s(_x81)))))))), s(p(s(_x31))))

minus^#(s(s(s(s(s(s(_x71)))))), s(s(s(_x51)))) → minus^#(s(s(s(s(s(p(s(_x71))))))), s(s(p(s(_x51)))))

Problem 21: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

minus^#(s(s(s(s(s(s(s(_x81))))))), s(s(_x31)))	→	minus^#(s(s(s(s(s(s(p(s(_x81)))))))), s(p(s(_x31))))	minus^#(s(s(s(s(_x51)))), s(s(s(_x51))))	→	minus^#(s(s(s(p(s(_x51))))), s(s(p(s(_x51)))))
minus^#(s(s(s(s(s(_x61))))), s(s(s(_x51))))	→	minus^#(s(s(s(s(p(s(_x61)))))), s(s(p(s(_x51)))))	minus^#(s(s(s(s(_x51)))), s(s(_x31)))	→	minus^#(s(s(s(p(s(_x51))))), s(p(s(_x31))))
minus^#(s(s(s(s(s(s(s(s(_x91)))))))), s(y))	→	minus^#(s(s(s(s(s(s(s(p(s(_x91))))))))), p(s(y)))	minus^#(s(s(s(_x41))), s(s(s(_x51))))	→	minus^#(s(s(p(s(_x41)))), s(s(p(s(_x51)))))
minus^#(s(s(_x21)), s(s(_x31)))	→	minus^#(s(p(s(_x21))), s(p(s(_x31))))	minus^#(s(s(s(s(s(s(s(_x81))))))), s(s(s(_x51))))	→	minus^#(s(s(s(s(s(s(p(s(_x81)))))))), s(s(p(s(_x51)))))
minus^#(s(s(s(s(s(s(s(s(_x91)))))))), s(s(_x31)))	→	minus^#(s(s(s(s(s(s(s(p(s(_x91))))))))), s(p(s(_x31))))	minus^#(s(s(s(s(s(s(_x71)))))), s(s(s(_x51))))	→	minus^#(s(s(s(s(s(p(s(_x71))))))), s(s(p(s(_x51)))))
minus^#(s(x), s(s(_x31)))	→	minus^#(p(s(x)), s(p(s(_x31))))

Rewrite Rules

minus(x, 0)	→	x	minus(0, y)	→	0
minus(s(x), s(y))	→	minus(p(s(x)), p(s(y)))	minus(x, plus(y, z))	→	minus(minus(x, y), z)
p(s(s(x)))	→	s(p(s(x)))	p(0)	→	s(s(0))
div(s(x), s(y))	→	s(div(minus(x, y), s(y)))	div(plus(x, y), z)	→	plus(div(x, z), div(y, z))
plus(0, y)	→	y	plus(s(x), y)	→	s(plus(y, minus(s(x), s(0))))

Original Signature

Termination of terms over the following signature is verified: plus, 0, minus, s, p, div

Strategy

Relevant Terms	Irrelevant Terms
minus^#(s(s(s(s(s(s(p(s(_x81)))))))), s(s(p(s(_x51)))))
minus^#(s(s(s(s(s(s(s(p(s(_x91))))))))), s(p(s(_x31))))

Thus, the rule minus^#(s(s(s(s(s(s(s(_x81))))))), s(s(_x31))) → minus^#(s(s(s(s(s(s(p(s(_x81)))))))), s(p(s(_x31)))) is replaced by the following rules:

minus^#(s(s(s(s(s(s(s(_x81))))))), s(s(s(_x51)))) → minus^#(s(s(s(s(s(s(p(s(_x81)))))))), s(s(p(s(_x51)))))

minus^#(s(s(s(s(s(s(s(s(_x91)))))))), s(s(_x31))) → minus^#(s(s(s(s(s(s(s(p(s(_x91))))))))), s(p(s(_x31))))

Problem 22: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

minus^#(s(s(s(s(_x51)))), s(s(s(_x51))))	→	minus^#(s(s(s(p(s(_x51))))), s(s(p(s(_x51)))))	minus^#(s(s(s(s(s(_x61))))), s(s(s(_x51))))	→	minus^#(s(s(s(s(p(s(_x61)))))), s(s(p(s(_x51)))))
minus^#(s(s(s(s(_x51)))), s(s(_x31)))	→	minus^#(s(s(s(p(s(_x51))))), s(p(s(_x31))))	minus^#(s(s(s(s(s(s(s(s(_x91)))))))), s(y))	→	minus^#(s(s(s(s(s(s(s(p(s(_x91))))))))), p(s(y)))
minus^#(s(s(s(_x41))), s(s(s(_x51))))	→	minus^#(s(s(p(s(_x41)))), s(s(p(s(_x51)))))	minus^#(s(s(_x21)), s(s(_x31)))	→	minus^#(s(p(s(_x21))), s(p(s(_x31))))
minus^#(s(s(s(s(s(s(s(s(_x91)))))))), s(s(_x31)))	→	minus^#(s(s(s(s(s(s(s(p(s(_x91))))))))), s(p(s(_x31))))	minus^#(s(s(s(s(s(s(s(_x81))))))), s(s(s(_x51))))	→	minus^#(s(s(s(s(s(s(p(s(_x81)))))))), s(s(p(s(_x51)))))
minus^#(s(s(s(s(s(s(_x71)))))), s(s(s(_x51))))	→	minus^#(s(s(s(s(s(p(s(_x71))))))), s(s(p(s(_x51)))))	minus^#(s(x), s(s(_x31)))	→	minus^#(p(s(x)), s(p(s(_x31))))

Rewrite Rules

minus(x, 0)	→	x	minus(0, y)	→	0
minus(s(x), s(y))	→	minus(p(s(x)), p(s(y)))	minus(x, plus(y, z))	→	minus(minus(x, y), z)
p(s(s(x)))	→	s(p(s(x)))	p(0)	→	s(s(0))
div(s(x), s(y))	→	s(div(minus(x, y), s(y)))	div(plus(x, y), z)	→	plus(div(x, z), div(y, z))
plus(0, y)	→	y	plus(s(x), y)	→	s(plus(y, minus(s(x), s(0))))

Original Signature

Termination of terms over the following signature is verified: plus, minus, 0, s, p, div

Strategy

Relevant Terms	Irrelevant Terms
minus^#(s(s(s(p(s(_x51))))), s(s(p(s(_x51)))))
minus^#(s(s(s(s(p(s(_x61)))))), s(p(s(_x31))))

Thus, the rule minus^#(s(s(s(s(_x51)))), s(s(_x31))) → minus^#(s(s(s(p(s(_x51))))), s(p(s(_x31)))) is replaced by the following rules:

minus^#(s(s(s(s(_x51)))), s(s(s(_x51)))) → minus^#(s(s(s(p(s(_x51))))), s(s(p(s(_x51)))))

minus^#(s(s(s(s(s(_x61))))), s(s(_x31))) → minus^#(s(s(s(s(p(s(_x61)))))), s(p(s(_x31))))

Problem 23: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

minus^#(s(s(s(s(_x51)))), s(s(s(_x51))))	→	minus^#(s(s(s(p(s(_x51))))), s(s(p(s(_x51)))))	minus^#(s(s(s(s(s(_x61))))), s(s(s(_x51))))	→	minus^#(s(s(s(s(p(s(_x61)))))), s(s(p(s(_x51)))))
minus^#(s(s(s(s(s(_x61))))), s(s(_x31)))	→	minus^#(s(s(s(s(p(s(_x61)))))), s(p(s(_x31))))	minus^#(s(s(s(s(s(s(s(s(_x91)))))))), s(y))	→	minus^#(s(s(s(s(s(s(s(p(s(_x91))))))))), p(s(y)))
minus^#(s(s(s(_x41))), s(s(s(_x51))))	→	minus^#(s(s(p(s(_x41)))), s(s(p(s(_x51)))))	minus^#(s(s(_x21)), s(s(_x31)))	→	minus^#(s(p(s(_x21))), s(p(s(_x31))))
minus^#(s(s(s(s(s(s(s(_x81))))))), s(s(s(_x51))))	→	minus^#(s(s(s(s(s(s(p(s(_x81)))))))), s(s(p(s(_x51)))))	minus^#(s(s(s(s(s(s(s(s(_x91)))))))), s(s(_x31)))	→	minus^#(s(s(s(s(s(s(s(p(s(_x91))))))))), s(p(s(_x31))))
minus^#(s(s(s(s(s(s(_x71)))))), s(s(s(_x51))))	→	minus^#(s(s(s(s(s(p(s(_x71))))))), s(s(p(s(_x51)))))	minus^#(s(x), s(s(_x31)))	→	minus^#(p(s(x)), s(p(s(_x31))))

Rewrite Rules

minus(x, 0)	→	x	minus(0, y)	→	0
minus(s(x), s(y))	→	minus(p(s(x)), p(s(y)))	minus(x, plus(y, z))	→	minus(minus(x, y), z)
p(s(s(x)))	→	s(p(s(x)))	p(0)	→	s(s(0))
div(s(x), s(y))	→	s(div(minus(x, y), s(y)))	div(plus(x, y), z)	→	plus(div(x, z), div(y, z))
plus(0, y)	→	y	plus(s(x), y)	→	s(plus(y, minus(s(x), s(0))))

Original Signature

Termination of terms over the following signature is verified: plus, 0, minus, s, p, div

Strategy

Relevant Terms	Irrelevant Terms
minus^#(s(s(s(s(s(p(s(_x71))))))), s(p(s(_x31))))
minus^#(s(s(s(s(p(s(_x61)))))), s(s(p(s(_x51)))))

Thus, the rule minus^#(s(s(s(s(s(_x61))))), s(s(_x31))) → minus^#(s(s(s(s(p(s(_x61)))))), s(p(s(_x31)))) is replaced by the following rules:

minus^#(s(s(s(s(s(s(_x71)))))), s(s(_x31))) → minus^#(s(s(s(s(s(p(s(_x71))))))), s(p(s(_x31))))

minus^#(s(s(s(s(s(_x61))))), s(s(s(_x51)))) → minus^#(s(s(s(s(p(s(_x61)))))), s(s(p(s(_x51)))))

Problem 24: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

minus^#(s(s(s(s(s(s(_x71)))))), s(s(_x31)))	→	minus^#(s(s(s(s(s(p(s(_x71))))))), s(p(s(_x31))))	minus^#(s(s(s(s(_x51)))), s(s(s(_x51))))	→	minus^#(s(s(s(p(s(_x51))))), s(s(p(s(_x51)))))
minus^#(s(s(s(s(s(_x61))))), s(s(s(_x51))))	→	minus^#(s(s(s(s(p(s(_x61)))))), s(s(p(s(_x51)))))	minus^#(s(s(s(s(s(s(s(s(_x91)))))))), s(y))	→	minus^#(s(s(s(s(s(s(s(p(s(_x91))))))))), p(s(y)))
minus^#(s(s(s(_x41))), s(s(s(_x51))))	→	minus^#(s(s(p(s(_x41)))), s(s(p(s(_x51)))))	minus^#(s(s(_x21)), s(s(_x31)))	→	minus^#(s(p(s(_x21))), s(p(s(_x31))))
minus^#(s(s(s(s(s(s(s(s(_x91)))))))), s(s(_x31)))	→	minus^#(s(s(s(s(s(s(s(p(s(_x91))))))))), s(p(s(_x31))))	minus^#(s(s(s(s(s(s(s(_x81))))))), s(s(s(_x51))))	→	minus^#(s(s(s(s(s(s(p(s(_x81)))))))), s(s(p(s(_x51)))))
minus^#(s(s(s(s(s(s(_x71)))))), s(s(s(_x51))))	→	minus^#(s(s(s(s(s(p(s(_x71))))))), s(s(p(s(_x51)))))	minus^#(s(x), s(s(_x31)))	→	minus^#(p(s(x)), s(p(s(_x31))))

Rewrite Rules

minus(x, 0)	→	x	minus(0, y)	→	0
minus(s(x), s(y))	→	minus(p(s(x)), p(s(y)))	minus(x, plus(y, z))	→	minus(minus(x, y), z)
p(s(s(x)))	→	s(p(s(x)))	p(0)	→	s(s(0))
div(s(x), s(y))	→	s(div(minus(x, y), s(y)))	div(plus(x, y), z)	→	plus(div(x, z), div(y, z))
plus(0, y)	→	y	plus(s(x), y)	→	s(plus(y, minus(s(x), s(0))))

Original Signature

Termination of terms over the following signature is verified: plus, minus, 0, s, p, div

Strategy

Relevant Terms	Irrelevant Terms
minus^#(s(s(s(s(s(p(s(_x71))))))), s(s(p(s(_x51)))))
minus^#(s(s(s(s(s(s(p(s(_x81)))))))), s(p(s(_x31))))

Thus, the rule minus^#(s(s(s(s(s(s(_x71)))))), s(s(_x31))) → minus^#(s(s(s(s(s(p(s(_x71))))))), s(p(s(_x31)))) is replaced by the following rules:

minus^#(s(s(s(s(s(s(s(_x81))))))), s(s(_x31))) → minus^#(s(s(s(s(s(s(p(s(_x81)))))))), s(p(s(_x31))))

minus^#(s(s(s(s(s(s(_x71)))))), s(s(s(_x51)))) → minus^#(s(s(s(s(s(p(s(_x71))))))), s(s(p(s(_x51)))))

Problem 25: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

minus^#(s(s(s(s(s(s(s(_x81))))))), s(s(_x31)))	→	minus^#(s(s(s(s(s(s(p(s(_x81)))))))), s(p(s(_x31))))	minus^#(s(s(s(s(_x51)))), s(s(s(_x51))))	→	minus^#(s(s(s(p(s(_x51))))), s(s(p(s(_x51)))))
minus^#(s(s(s(s(s(_x61))))), s(s(s(_x51))))	→	minus^#(s(s(s(s(p(s(_x61)))))), s(s(p(s(_x51)))))	minus^#(s(s(s(s(s(s(s(s(_x91)))))))), s(y))	→	minus^#(s(s(s(s(s(s(s(p(s(_x91))))))))), p(s(y)))
minus^#(s(s(s(_x41))), s(s(s(_x51))))	→	minus^#(s(s(p(s(_x41)))), s(s(p(s(_x51)))))	minus^#(s(s(_x21)), s(s(_x31)))	→	minus^#(s(p(s(_x21))), s(p(s(_x31))))
minus^#(s(s(s(s(s(s(s(_x81))))))), s(s(s(_x51))))	→	minus^#(s(s(s(s(s(s(p(s(_x81)))))))), s(s(p(s(_x51)))))	minus^#(s(s(s(s(s(s(s(s(_x91)))))))), s(s(_x31)))	→	minus^#(s(s(s(s(s(s(s(p(s(_x91))))))))), s(p(s(_x31))))
minus^#(s(s(s(s(s(s(_x71)))))), s(s(s(_x51))))	→	minus^#(s(s(s(s(s(p(s(_x71))))))), s(s(p(s(_x51)))))	minus^#(s(x), s(s(_x31)))	→	minus^#(p(s(x)), s(p(s(_x31))))

Rewrite Rules

minus(x, 0)	→	x	minus(0, y)	→	0
minus(s(x), s(y))	→	minus(p(s(x)), p(s(y)))	minus(x, plus(y, z))	→	minus(minus(x, y), z)
p(s(s(x)))	→	s(p(s(x)))	p(0)	→	s(s(0))
div(s(x), s(y))	→	s(div(minus(x, y), s(y)))	div(plus(x, y), z)	→	plus(div(x, z), div(y, z))
plus(0, y)	→	y	plus(s(x), y)	→	s(plus(y, minus(s(x), s(0))))

Original Signature

Termination of terms over the following signature is verified: plus, 0, minus, s, p, div

Strategy

Relevant Terms	Irrelevant Terms
minus^#(s(s(s(s(s(s(p(s(_x81)))))))), s(s(p(s(_x51)))))
minus^#(s(s(s(s(s(s(s(p(s(_x91))))))))), s(p(s(_x31))))

Thus, the rule minus^#(s(s(s(s(s(s(s(_x81))))))), s(s(_x31))) → minus^#(s(s(s(s(s(s(p(s(_x81)))))))), s(p(s(_x31)))) is replaced by the following rules:

minus^#(s(s(s(s(s(s(s(s(_x91)))))))), s(s(_x31))) → minus^#(s(s(s(s(s(s(s(p(s(_x91))))))))), s(p(s(_x31))))

minus^#(s(s(s(s(s(s(s(_x81))))))), s(s(s(_x51)))) → minus^#(s(s(s(s(s(s(p(s(_x81)))))))), s(s(p(s(_x51)))))

Problem 26: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

minus^#(s(s(s(s(_x51)))), s(s(s(_x51))))	→	minus^#(s(s(s(p(s(_x51))))), s(s(p(s(_x51)))))	minus^#(s(s(s(s(s(_x61))))), s(s(s(_x51))))	→	minus^#(s(s(s(s(p(s(_x61)))))), s(s(p(s(_x51)))))
minus^#(s(s(s(s(s(s(s(s(_x91)))))))), s(y))	→	minus^#(s(s(s(s(s(s(s(p(s(_x91))))))))), p(s(y)))	minus^#(s(s(s(_x41))), s(s(s(_x51))))	→	minus^#(s(s(p(s(_x41)))), s(s(p(s(_x51)))))
minus^#(s(s(_x21)), s(s(_x31)))	→	minus^#(s(p(s(_x21))), s(p(s(_x31))))	minus^#(s(s(s(s(s(s(s(s(_x91)))))))), s(s(_x31)))	→	minus^#(s(s(s(s(s(s(s(p(s(_x91))))))))), s(p(s(_x31))))
minus^#(s(s(s(s(s(s(s(_x81))))))), s(s(s(_x51))))	→	minus^#(s(s(s(s(s(s(p(s(_x81)))))))), s(s(p(s(_x51)))))	minus^#(s(s(s(s(s(s(_x71)))))), s(s(s(_x51))))	→	minus^#(s(s(s(s(s(p(s(_x71))))))), s(s(p(s(_x51)))))
minus^#(s(x), s(s(_x31)))	→	minus^#(p(s(x)), s(p(s(_x31))))

Rewrite Rules

minus(x, 0)	→	x	minus(0, y)	→	0
minus(s(x), s(y))	→	minus(p(s(x)), p(s(y)))	minus(x, plus(y, z))	→	minus(minus(x, y), z)
p(s(s(x)))	→	s(p(s(x)))	p(0)	→	s(s(0))
div(s(x), s(y))	→	s(div(minus(x, y), s(y)))	div(plus(x, y), z)	→	plus(div(x, z), div(y, z))
plus(0, y)	→	y	plus(s(x), y)	→	s(plus(y, minus(s(x), s(0))))

Original Signature

Termination of terms over the following signature is verified: plus, minus, 0, s, p, div

Strategy

The right-hand side of the rule minus^#(s(s(s(s(s(s(s(s(_x91)))))))), s(y)) → minus^#(s(s(s(s(s(s(s(p(s(_x91))))))))), p(s(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 Terms	Irrelevant Terms
minus^#(s(s(s(s(s(s(s(s(p(s(_x101)))))))))), p(s(y)))
minus^#(s(s(s(s(s(s(s(p(s(_x91))))))))), s(p(s(_x31))))

Thus, the rule minus^#(s(s(s(s(s(s(s(s(_x91)))))))), s(y)) → minus^#(s(s(s(s(s(s(s(p(s(_x91))))))))), p(s(y))) is replaced by the following rules:

minus^#(s(s(s(s(s(s(s(s(_x91)))))))), s(s(_x31))) → minus^#(s(s(s(s(s(s(s(p(s(_x91))))))))), s(p(s(_x31))))

minus^#(s(s(s(s(s(s(s(s(s(_x101))))))))), s(y)) → minus^#(s(s(s(s(s(s(s(s(p(s(_x101)))))))))), p(s(y)))

Problem 27: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x231))))))))))))))))))))), s(s(_x31)))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x231)))))))))))))))))))))), s(p(s(_x31))))	minus^#(s(s(s(s(s(_x61))))), s(s(s(_x51))))	→	minus^#(s(s(s(s(p(s(_x61)))))), s(s(p(s(_x51)))))
minus^#(s(s(s(s(s(s(s(s(s(s(s(_x121))))))))))), s(s(s(_x51))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(p(s(_x121)))))))))))), s(s(p(s(_x51)))))	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x241)))))))))))))))))))))), s(y))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x241))))))))))))))))))))))), p(s(y)))
minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x191)))))))))))))))))), s(s(_x31)))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x191))))))))))))))))))), s(p(s(_x31))))	minus^#(s(s(s(s(s(s(s(_x81))))))), s(s(s(_x51))))	→	minus^#(s(s(s(s(s(s(p(s(_x81)))))))), s(s(p(s(_x51)))))
minus^#(s(s(_x21)), s(s(s(_x51))))	→	minus^#(s(p(s(_x21))), s(s(p(s(_x51)))))	minus^#(s(s(s(s(s(s(_x71)))))), s(s(s(_x51))))	→	minus^#(s(s(s(s(s(p(s(_x71))))))), s(s(p(s(_x51)))))
minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x181))))))))))))))))), s(s(s(_x51))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x181)))))))))))))))))), s(s(p(s(_x51)))))	minus^#(s(s(s(s(_x51)))), s(s(s(_x51))))	→	minus^#(s(s(s(p(s(_x51))))), s(s(p(s(_x51)))))
minus^#(s(s(s(s(s(s(s(s(s(_x101))))))))), s(s(s(_x51))))	→	minus^#(s(s(s(s(s(s(s(s(p(s(_x101)))))))))), s(s(p(s(_x51)))))	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(_x141))))))))))))), s(s(_x31)))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x141)))))))))))))), s(p(s(_x31))))
minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x171)))))))))))))))), s(s(s(_x51))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x171))))))))))))))))), s(s(p(s(_x51)))))	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x221)))))))))))))))))))), s(s(_x31)))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x221))))))))))))))))))))), s(p(s(_x31))))
minus^#(s(s(s(s(s(s(s(s(s(s(s(s(_x131)))))))))))), s(s(s(_x51))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(p(s(_x131))))))))))))), s(s(p(s(_x51)))))	minus^#(s(s(s(_x41))), s(s(s(_x51))))	→	minus^#(s(s(p(s(_x41)))), s(s(p(s(_x51)))))
minus^#(s(s(s(s(s(s(s(s(_x91)))))))), s(s(s(_x51))))	→	minus^#(s(s(s(s(s(s(s(p(s(_x91))))))))), s(s(p(s(_x51)))))	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x151)))))))))))))), s(s(s(_x51))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x151))))))))))))))), s(s(p(s(_x51)))))
minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x161))))))))))))))), s(s(_x31)))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x161)))))))))))))))), s(p(s(_x31))))	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x191)))))))))))))))))), s(s(s(_x51))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x191))))))))))))))))))), s(s(p(s(_x51)))))
minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x201))))))))))))))))))), s(s(_x31)))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x201)))))))))))))))))))), s(p(s(_x31))))	minus^#(s(s(s(s(s(s(s(s(s(s(_x111)))))))))), s(s(_x31)))	→	minus^#(s(s(s(s(s(s(s(s(s(p(s(_x111))))))))))), s(p(s(_x31))))
minus^#(s(x), s(s(_x31)))	→	minus^#(p(s(x)), s(p(s(_x31))))

Rewrite Rules

minus(x, 0)	→	x	minus(0, y)	→	0
minus(s(x), s(y))	→	minus(p(s(x)), p(s(y)))	minus(x, plus(y, z))	→	minus(minus(x, y), z)
p(s(s(x)))	→	s(p(s(x)))	p(0)	→	s(s(0))
div(s(x), s(y))	→	s(div(minus(x, y), s(y)))	div(plus(x, y), z)	→	plus(div(x, z), div(y, z))
plus(0, y)	→	y	plus(s(x), y)	→	s(plus(y, minus(s(x), s(0))))

Original Signature

Termination of terms over the following signature is verified: plus, 0, minus, s, p, div

Strategy

The right-hand side of the rule minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x231))))))))))))))))))))), s(s(_x31))) → minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x231)))))))))))))))))))))), s(p(s(_x31)))) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).

Relevant Terms	Irrelevant Terms
minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x231)))))))))))))))))))))), s(s(p(s(_x51)))))
minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x241))))))))))))))))))))))), s(p(s(_x31))))

Thus, the rule minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x231))))))))))))))))))))), s(s(_x31))) → minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x231)))))))))))))))))))))), s(p(s(_x31)))) is replaced by the following rules:

minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x241)))))))))))))))))))))), s(s(_x31))) → minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x241))))))))))))))))))))))), s(p(s(_x31))))

minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x231))))))))))))))))))))), s(s(s(_x51)))) → minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x231)))))))))))))))))))))), s(s(p(s(_x51)))))

Problem 28: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

minus^#(s(s(s(s(s(s(s(_x81))))))), s(s(s(s(_x61)))))	→	minus^#(s(s(s(s(s(s(p(s(_x81)))))))), s(s(s(p(s(_x61))))))	minus^#(s(s(s(s(s(_x61))))), s(s(s(s(_x61)))))	→	minus^#(s(s(s(s(p(s(_x61)))))), s(s(s(p(s(_x61))))))
minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x181))))))))))))))))), s(s(s(_x51))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x181)))))))))))))))))), s(s(p(s(_x51)))))	minus^#(s(s(s(_x41))), s(s(s(s(s(s(_x81)))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(p(s(_x81))))))))
minus^#(s(s(_x21)), s(s(s(s(s(_x71))))))	→	minus^#(s(p(s(_x21))), s(s(s(s(p(s(_x71)))))))	minus^#(s(s(s(s(_x51)))), s(s(s(_x51))))	→	minus^#(s(s(s(p(s(_x51))))), s(s(p(s(_x51)))))
minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x221)))))))))))))))))))), s(s(s(_x51))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x221))))))))))))))))))))), s(s(p(s(_x51)))))	minus^#(s(s(s(s(s(s(_x71)))))), s(s(s(s(_x61)))))	→	minus^#(s(s(s(s(s(p(s(_x71))))))), s(s(s(p(s(_x61))))))
minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x231))))))))))))))))))))), s(s(s(_x51))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x231)))))))))))))))))))))), s(s(p(s(_x51)))))	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x171)))))))))))))))), s(s(s(_x51))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x171))))))))))))))))), s(s(p(s(_x51)))))
minus^#(s(s(s(s(s(s(s(s(_x91)))))))), s(s(s(s(_x61)))))	→	minus^#(s(s(s(s(s(s(s(p(s(_x91))))))))), s(s(s(p(s(_x61))))))	minus^#(s(s(s(_x41))), s(s(s(s(s(_x71))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(p(s(_x71)))))))
minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x151)))))))))))))), s(s(s(_x51))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x151))))))))))))))), s(s(p(s(_x51)))))	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x191)))))))))))))))))), s(s(s(_x51))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x191))))))))))))))))))), s(s(p(s(_x51)))))
minus^#(s(s(s(_x41))), s(s(s(s(_x61)))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(p(s(_x61))))))	minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(_x91))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(p(s(_x91)))))))))
minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x161))))))))))))))), s(s(s(_x51))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x161)))))))))))))))), s(s(p(s(_x51)))))	minus^#(s(s(s(s(s(s(s(s(s(s(s(_x121))))))))))), s(s(s(_x51))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(p(s(_x121)))))))))))), s(s(p(s(_x51)))))
minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x241)))))))))))))))))))))), s(s(s(_x51))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x241))))))))))))))))))))))), s(s(p(s(_x51)))))	minus^#(s(s(s(s(s(s(s(s(s(_x101))))))))), s(s(s(_x51))))	→	minus^#(s(s(s(s(s(s(s(s(p(s(_x101)))))))))), s(s(p(s(_x51)))))
minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x251))))))))))))))))))))))), s(s(_x31)))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x251)))))))))))))))))))))))), s(p(s(_x31))))	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(_x131)))))))))))), s(s(s(_x51))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(p(s(_x131))))))))))))), s(s(p(s(_x51)))))
minus^#(s(s(s(s(s(s(s(s(s(s(_x111)))))))))), s(s(s(_x51))))	→	minus^#(s(s(s(s(s(s(s(s(s(p(s(_x111))))))))))), s(s(p(s(_x51)))))	minus^#(s(s(s(_x41))), s(s(s(_x51))))	→	minus^#(s(s(p(s(_x41)))), s(s(p(s(_x51)))))
minus^#(s(x), s(s(s(s(s(s(s(_x91))))))))	→	minus^#(p(s(x)), s(s(s(s(s(s(p(s(_x91)))))))))	minus^#(s(s(s(s(s(s(s(s(_x91)))))))), s(s(_x31)))	→	minus^#(s(s(s(s(s(s(s(p(s(_x91))))))))), s(p(s(_x31))))
minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x201))))))))))))))))))), s(s(_x31)))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x201)))))))))))))))))))), s(p(s(_x31))))	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x251))))))))))))))))))))))), s(y))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x251)))))))))))))))))))))))), p(s(y)))
minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(_x141))))))))))))), s(s(s(_x51))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x141)))))))))))))), s(s(p(s(_x51)))))	minus^#(s(s(_x21)), s(s(s(s(s(s(s(s(_x101)))))))))	→	minus^#(s(p(s(_x21))), s(s(s(s(s(s(s(p(s(_x101))))))))))

Rewrite Rules

minus(x, 0)	→	x	minus(0, y)	→	0
minus(s(x), s(y))	→	minus(p(s(x)), p(s(y)))	minus(x, plus(y, z))	→	minus(minus(x, y), z)
p(s(s(x)))	→	s(p(s(x)))	p(0)	→	s(s(0))
div(s(x), s(y))	→	s(div(minus(x, y), s(y)))	div(plus(x, y), z)	→	plus(div(x, z), div(y, z))
plus(0, y)	→	y	plus(s(x), y)	→	s(plus(y, minus(s(x), s(0))))

Original Signature

Termination of terms over the following signature is verified: plus, 0, minus, s, p, div

Strategy

The right-hand side of the rule minus^#(s(s(_x21)), s(s(s(s(s(_x71)))))) → minus^#(s(p(s(_x21))), s(s(s(s(p(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 Terms	Irrelevant Terms
minus^#(s(p(s(_x21))), s(s(s(s(s(p(s(_x81))))))))
minus^#(s(s(p(s(_x41)))), s(s(s(s(p(s(_x71)))))))

Thus, the rule minus^#(s(s(_x21)), s(s(s(s(s(_x71)))))) → minus^#(s(p(s(_x21))), s(s(s(s(p(s(_x71))))))) is replaced by the following rules:

minus^#(s(s(_x21)), s(s(s(s(s(s(_x81))))))) → minus^#(s(p(s(_x21))), s(s(s(s(s(p(s(_x81))))))))

minus^#(s(s(s(_x41))), s(s(s(s(s(_x71)))))) → minus^#(s(s(p(s(_x41)))), s(s(s(s(p(s(_x71)))))))

Problem 29: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(s(_x141)))))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(s(p(s(_x141))))))))))))))	minus^#(s(s(s(s(s(s(s(s(s(_x101))))))))), s(s(s(s(s(_x71))))))	→	minus^#(s(s(s(s(s(s(s(s(p(s(_x101)))))))))), s(s(s(s(p(s(_x71)))))))
minus^#(s(s(s(s(s(s(_x71)))))), s(s(s(s(s(_x71))))))	→	minus^#(s(s(s(s(s(p(s(_x71))))))), s(s(s(s(p(s(_x71)))))))	minus^#(s(s(s(s(s(_x61))))), s(s(s(s(_x61)))))	→	minus^#(s(s(s(s(p(s(_x61)))))), s(s(s(p(s(_x61))))))
minus^#(s(s(s(s(s(s(s(s(s(_x101))))))))), s(s(s(s(_x61)))))	→	minus^#(s(s(s(s(s(s(s(s(p(s(_x101)))))))))), s(s(s(p(s(_x61))))))	minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(_x111))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(p(s(_x111)))))))))))
minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x181))))))))))))))))), s(s(s(_x51))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x181)))))))))))))))))), s(s(p(s(_x51)))))	minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(_x121)))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(p(s(_x121))))))))))))
minus^#(s(s(s(s(s(_x61))))), s(s(s(s(s(s(s(_x91))))))))	→	minus^#(s(s(s(s(p(s(_x61)))))), s(s(s(s(s(s(p(s(_x91)))))))))	minus^#(s(s(s(s(_x51)))), s(s(s(_x51))))	→	minus^#(s(s(s(p(s(_x51))))), s(s(p(s(_x51)))))
minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x221)))))))))))))))))))), s(s(s(_x51))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x221))))))))))))))))))))), s(s(p(s(_x51)))))	minus^#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(_x151))))))))))))))	→	minus^#(s(p(s(_x21))), s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x151)))))))))))))))
minus^#(s(s(s(s(s(_x61))))), s(s(s(s(s(s(_x81)))))))	→	minus^#(s(s(s(s(p(s(_x61)))))), s(s(s(s(s(p(s(_x81))))))))	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x231))))))))))))))))))))), s(s(s(_x51))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x231)))))))))))))))))))))), s(s(p(s(_x51)))))
minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x171)))))))))))))))), s(s(s(_x51))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x171))))))))))))))))), s(s(p(s(_x51)))))	minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x161)))))))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x161))))))))))))))))
minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x151)))))))))))))), s(s(s(_x51))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x151))))))))))))))), s(s(p(s(_x51)))))	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x191)))))))))))))))))), s(s(s(_x51))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x191))))))))))))))))))), s(s(p(s(_x51)))))
minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x161))))))))))))))), s(s(s(_x51))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x161)))))))))))))))), s(s(p(s(_x51)))))	minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(_x131))))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(p(s(_x131)))))))))))))
minus^#(s(s(s(s(s(s(s(s(s(s(s(_x121))))))))))), s(s(s(_x51))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(p(s(_x121)))))))))))), s(s(p(s(_x51)))))	minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x171))))))))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x171)))))))))))))))))
minus^#(s(s(s(s(s(s(s(s(_x91)))))))), s(s(s(s(s(s(_x81)))))))	→	minus^#(s(s(s(s(s(s(s(p(s(_x91))))))))), s(s(s(s(s(p(s(_x81))))))))	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x241)))))))))))))))))))))), s(s(s(_x51))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x241))))))))))))))))))))))), s(s(p(s(_x51)))))
minus^#(s(s(s(s(_x51)))), s(s(s(s(s(s(s(_x91))))))))	→	minus^#(s(s(s(p(s(_x51))))), s(s(s(s(s(s(p(s(_x91)))))))))	minus^#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x181)))))))))))))))))	→	minus^#(s(p(s(_x21))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x181))))))))))))))))))
minus^#(s(s(s(s(s(s(s(s(s(_x101))))))))), s(s(s(_x51))))	→	minus^#(s(s(s(s(s(s(s(s(p(s(_x101)))))))))), s(s(p(s(_x51)))))	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x251))))))))))))))))))))))), s(s(s(_x51))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x251)))))))))))))))))))))))), s(s(p(s(_x51)))))
minus^#(s(s(s(s(s(s(s(_x81))))))), s(s(s(s(s(s(_x81)))))))	→	minus^#(s(s(s(s(s(s(p(s(_x81)))))))), s(s(s(s(s(p(s(_x81))))))))	minus^#(s(s(_x21)), s(s(s(s(s(s(s(s(s(_x111))))))))))	→	minus^#(s(p(s(_x21))), s(s(s(s(s(s(s(s(p(s(_x111)))))))))))
minus^#(s(s(s(s(s(s(s(s(s(s(_x111)))))))))), s(s(s(_x51))))	→	minus^#(s(s(s(s(s(s(s(s(s(p(s(_x111))))))))))), s(s(p(s(_x51)))))	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(_x131)))))))))))), s(s(s(_x51))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(p(s(_x131))))))))))))), s(s(p(s(_x51)))))
minus^#(s(s(s(s(s(s(s(s(_x91)))))))), s(s(_x31)))	→	minus^#(s(s(s(s(s(s(s(p(s(_x91))))))))), s(p(s(_x31))))	minus^#(s(s(s(s(_x51)))), s(s(s(s(s(_x71))))))	→	minus^#(s(s(s(p(s(_x51))))), s(s(s(s(p(s(_x71)))))))
minus^#(s(s(s(s(_x51)))), s(s(s(s(s(s(s(s(_x101)))))))))	→	minus^#(s(s(s(p(s(_x51))))), s(s(s(s(s(s(s(p(s(_x101))))))))))	minus^#(s(x), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x181)))))))))))))))))	→	minus^#(p(s(x)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x181))))))))))))))))))
minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x201))))))))))))))))))), s(s(_x31)))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x201)))))))))))))))))))), s(p(s(_x31))))	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x251))))))))))))))))))))))), s(y))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x251)))))))))))))))))))))))), p(s(y)))
minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(_x141))))))))))))), s(s(s(_x51))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x141)))))))))))))), s(s(p(s(_x51)))))	minus^#(s(s(_x21)), s(s(s(s(s(s(s(s(_x101)))))))))	→	minus^#(s(p(s(_x21))), s(s(s(s(s(s(s(p(s(_x101))))))))))
minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x261)))))))))))))))))))))))), s(s(_x31)))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x261))))))))))))))))))))))))), s(p(s(_x31))))	minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(s(s(_x151))))))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x151)))))))))))))))

Rewrite Rules

minus(x, 0)	→	x	minus(0, y)	→	0
minus(s(x), s(y))	→	minus(p(s(x)), p(s(y)))	minus(x, plus(y, z))	→	minus(minus(x, y), z)
p(s(s(x)))	→	s(p(s(x)))	p(0)	→	s(s(0))
div(s(x), s(y))	→	s(div(minus(x, y), s(y)))	div(plus(x, y), z)	→	plus(div(x, z), div(y, z))
plus(0, y)	→	y	plus(s(x), y)	→	s(plus(y, minus(s(x), s(0))))

Original Signature

Termination of terms over the following signature is verified: plus, 0, minus, s, p, div

Strategy

The right-hand side of the rule minus^#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(_x151)))))))))))))) → minus^#(s(p(s(_x21))), s(s(s(s(s(s(s(s(s(s(s(s(p(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 Terms	Irrelevant Terms
minus^#(s(p(s(_x21))), s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x161))))))))))))))))
minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x151)))))))))))))))

Thus, the rule minus^#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(_x151)))))))))))))) → minus^#(s(p(s(_x21))), s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x151))))))))))))))) is replaced by the following rules:

minus^#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x161))))))))))))))) → minus^#(s(p(s(_x21))), s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x161))))))))))))))))

minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(s(s(_x151)))))))))))))) → minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x151)))))))))))))))

Problem 30: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(s(_x141)))))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(s(p(s(_x141))))))))))))))	minus^#(s(s(s(s(s(s(_x71)))))), s(s(s(s(s(_x71))))))	→	minus^#(s(s(s(s(s(p(s(_x71))))))), s(s(s(s(p(s(_x71)))))))
minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x171)))))))))))))))), s(s(s(s(_x61)))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x171))))))))))))))))), s(s(s(p(s(_x61))))))	minus^#(s(s(s(s(s(_x61))))), s(s(s(s(_x61)))))	→	minus^#(s(s(s(s(p(s(_x61)))))), s(s(s(p(s(_x61))))))
minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(_x111))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(p(s(_x111)))))))))))	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x151)))))))))))))), s(s(s(s(_x61)))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x151))))))))))))))), s(s(s(p(s(_x61))))))
minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(_x121)))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(p(s(_x121))))))))))))	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x181))))))))))))))))), s(s(s(s(_x61)))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x181)))))))))))))))))), s(s(s(p(s(_x61))))))
minus^#(s(s(s(s(s(s(s(s(s(s(s(s(_x131)))))))))))), s(s(s(s(_x61)))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(p(s(_x131))))))))))))), s(s(s(p(s(_x61))))))	minus^#(s(s(s(s(s(_x61))))), s(s(s(s(s(s(s(_x91))))))))	→	minus^#(s(s(s(s(p(s(_x61)))))), s(s(s(s(s(s(p(s(_x91)))))))))
minus^#(s(s(s(s(_x51)))), s(s(s(_x51))))	→	minus^#(s(s(s(p(s(_x51))))), s(s(p(s(_x51)))))	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x221)))))))))))))))))))), s(s(s(_x51))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x221))))))))))))))))))))), s(s(p(s(_x51)))))
minus^#(s(s(s(s(s(_x61))))), s(s(s(s(s(s(_x81)))))))	→	minus^#(s(s(s(s(p(s(_x61)))))), s(s(s(s(s(p(s(_x81))))))))	minus^#(s(s(s(s(s(s(s(s(s(s(s(_x121))))))))))), s(s(s(s(_x61)))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(p(s(_x121)))))))))))), s(s(s(p(s(_x61))))))
minus^#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x191))))))))))))))))))	→	minus^#(s(p(s(_x21))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x191)))))))))))))))))))	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x231))))))))))))))))))))), s(s(s(_x51))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x231)))))))))))))))))))))), s(s(p(s(_x51)))))
minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x161)))))))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x161))))))))))))))))	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x161))))))))))))))), s(s(s(s(_x61)))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x161)))))))))))))))), s(s(s(p(s(_x61))))))
minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x201))))))))))))))))))), s(s(s(s(_x61)))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x201)))))))))))))))))))), s(s(s(p(s(_x61))))))	minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x181)))))))))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x181))))))))))))))))))
minus^#(s(s(s(s(s(s(s(s(s(s(s(_x121))))))))))), s(s(s(_x51))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(p(s(_x121)))))))))))), s(s(p(s(_x51)))))	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x191)))))))))))))))))), s(s(s(s(_x61)))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x191))))))))))))))))))), s(s(s(p(s(_x61))))))
minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(_x131))))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(p(s(_x131)))))))))))))	minus^#(s(s(s(s(s(s(s(s(_x91)))))))), s(s(s(s(s(s(_x81)))))))	→	minus^#(s(s(s(s(s(s(s(p(s(_x91))))))))), s(s(s(s(s(p(s(_x81))))))))
minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x171))))))))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x171)))))))))))))))))	minus^#(s(s(s(s(s(s(s(s(s(s(_x111)))))))))), s(s(s(s(s(s(_x81)))))))	→	minus^#(s(s(s(s(s(s(s(s(s(p(s(_x111))))))))))), s(s(s(s(s(p(s(_x81))))))))
minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x241)))))))))))))))))))))), s(s(s(_x51))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x241))))))))))))))))))))))), s(s(p(s(_x51)))))	minus^#(s(s(s(s(s(s(s(s(s(s(s(_x121))))))))))), s(s(s(s(s(_x71))))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(p(s(_x121)))))))))))), s(s(s(s(p(s(_x71)))))))
minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x251))))))))))))))))))))))), s(s(s(_x51))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x251)))))))))))))))))))))))), s(s(p(s(_x51)))))	minus^#(s(s(s(s(s(s(s(_x81))))))), s(s(s(s(s(s(_x81)))))))	→	minus^#(s(s(s(s(s(s(p(s(_x81)))))))), s(s(s(s(s(p(s(_x81))))))))
minus^#(s(s(s(s(s(s(s(s(s(_x101))))))))), s(s(s(s(s(s(_x81)))))))	→	minus^#(s(s(s(s(s(s(s(s(p(s(_x101)))))))))), s(s(s(s(s(p(s(_x81))))))))	minus^#(s(s(s(s(s(s(s(s(s(s(_x111)))))))))), s(s(s(s(_x61)))))	→	minus^#(s(s(s(s(s(s(s(s(s(p(s(_x111))))))))))), s(s(s(p(s(_x61))))))
minus^#(s(s(s(s(_x51)))), s(s(s(s(s(_x71))))))	→	minus^#(s(s(s(p(s(_x51))))), s(s(s(s(p(s(_x71)))))))	minus^#(s(s(s(s(_x51)))), s(s(s(s(s(s(s(s(_x101)))))))))	→	minus^#(s(s(s(p(s(_x51))))), s(s(s(s(s(s(s(p(s(_x101))))))))))
minus^#(s(x), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x181)))))))))))))))))	→	minus^#(p(s(x)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x181))))))))))))))))))	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x201))))))))))))))))))), s(s(_x31)))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x201)))))))))))))))))))), s(p(s(_x31))))
minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x251))))))))))))))))))))))), s(y))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x251)))))))))))))))))))))))), p(s(y)))	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(_x141))))))))))))), s(s(s(_x51))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x141)))))))))))))), s(s(p(s(_x51)))))
minus^#(s(s(_x21)), s(s(s(s(s(s(s(s(_x101)))))))))	→	minus^#(s(p(s(_x21))), s(s(s(s(s(s(s(p(s(_x101))))))))))	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x261)))))))))))))))))))))))), s(s(_x31)))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x261))))))))))))))))))))))))), s(p(s(_x31))))
minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(s(s(_x151))))))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x151)))))))))))))))

Rewrite Rules

minus(x, 0)	→	x	minus(0, y)	→	0
minus(s(x), s(y))	→	minus(p(s(x)), p(s(y)))	minus(x, plus(y, z))	→	minus(minus(x, y), z)
p(s(s(x)))	→	s(p(s(x)))	p(0)	→	s(s(0))
div(s(x), s(y))	→	s(div(minus(x, y), s(y)))	div(plus(x, y), z)	→	plus(div(x, z), div(y, z))
plus(0, y)	→	y	plus(s(x), y)	→	s(plus(y, minus(s(x), s(0))))

Original Signature

Termination of terms over the following signature is verified: plus, 0, minus, s, p, div

Strategy

The right-hand side of the rule minus^#(s(s(s(s(s(s(s(s(s(s(s(_x121))))))))))), s(s(s(s(_x61))))) → minus^#(s(s(s(s(s(s(s(s(s(s(p(s(_x121)))))))))))), s(s(s(p(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 Terms	Irrelevant Terms
minus^#(s(s(s(s(s(s(s(s(s(s(s(p(s(_x131))))))))))))), s(s(s(p(s(_x61))))))
minus^#(s(s(s(s(s(s(s(s(s(s(p(s(_x121)))))))))))), s(s(s(s(p(s(_x71)))))))

Thus, the rule minus^#(s(s(s(s(s(s(s(s(s(s(s(_x121))))))))))), s(s(s(s(_x61))))) → minus^#(s(s(s(s(s(s(s(s(s(s(p(s(_x121)))))))))))), s(s(s(p(s(_x61)))))) is replaced by the following rules:

minus^#(s(s(s(s(s(s(s(s(s(s(s(s(_x131)))))))))))), s(s(s(s(_x61))))) → minus^#(s(s(s(s(s(s(s(s(s(s(s(p(s(_x131))))))))))))), s(s(s(p(s(_x61))))))

minus^#(s(s(s(s(s(s(s(s(s(s(s(_x121))))))))))), s(s(s(s(s(_x71)))))) → minus^#(s(s(s(s(s(s(s(s(s(s(p(s(_x121)))))))))))), s(s(s(s(p(s(_x71)))))))

Problem 31: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

minus^#(s(s(s(s(s(s(s(s(s(_x101))))))))), s(s(s(s(s(s(s(s(_x101)))))))))	→	minus^#(s(s(s(s(s(s(s(s(p(s(_x101)))))))))), s(s(s(s(s(s(s(p(s(_x101))))))))))	minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(s(_x141)))))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(s(p(s(_x141))))))))))))))
minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x151)))))))))))))), s(s(s(s(s(_x71))))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x151))))))))))))))), s(s(s(s(p(s(_x71)))))))	minus^#(s(s(s(s(s(s(_x71)))))), s(s(s(s(s(_x71))))))	→	minus^#(s(s(s(s(s(p(s(_x71))))))), s(s(s(s(p(s(_x71)))))))
minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x171)))))))))))))))), s(s(s(s(_x61)))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x171))))))))))))))))), s(s(s(p(s(_x61))))))	minus^#(s(s(s(s(s(_x61))))), s(s(s(s(_x61)))))	→	minus^#(s(s(s(s(p(s(_x61)))))), s(s(s(p(s(_x61))))))
minus^#(s(s(s(s(s(s(s(s(s(s(s(_x121))))))))))), s(s(s(s(s(s(_x81)))))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(p(s(_x121)))))))))))), s(s(s(s(s(p(s(_x81))))))))	minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(_x111))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(p(s(_x111)))))))))))
minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x151)))))))))))))), s(s(s(s(_x61)))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x151))))))))))))))), s(s(s(p(s(_x61))))))	minus^#(s(s(s(s(s(s(s(s(s(s(s(_x121))))))))))), s(s(s(s(s(s(s(_x91))))))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(p(s(_x121)))))))))))), s(s(s(s(s(s(p(s(_x91)))))))))
minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(_x121)))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(p(s(_x121))))))))))))	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x181))))))))))))))))), s(s(s(s(_x61)))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x181)))))))))))))))))), s(s(s(p(s(_x61))))))
minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x221)))))))))))))))))))), s(s(s(_x51))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x221))))))))))))))))))))), s(s(p(s(_x51)))))	minus^#(s(s(s(s(_x51)))), s(s(s(_x51))))	→	minus^#(s(s(s(p(s(_x51))))), s(s(p(s(_x51)))))
minus^#(s(s(s(s(s(s(_x71)))))), s(s(s(s(s(s(s(s(s(_x111))))))))))	→	minus^#(s(s(s(s(s(p(s(_x71))))))), s(s(s(s(s(s(s(s(p(s(_x111)))))))))))	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x231))))))))))))))))))))), s(s(s(_x51))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x231)))))))))))))))))))))), s(s(p(s(_x51)))))
minus^#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x191))))))))))))))))))	→	minus^#(s(p(s(_x21))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x191)))))))))))))))))))	minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x161)))))))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x161))))))))))))))))
minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x161))))))))))))))), s(s(s(s(_x61)))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x161)))))))))))))))), s(s(s(p(s(_x61))))))	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x201))))))))))))))))))), s(s(s(s(_x61)))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x201)))))))))))))))))))), s(s(s(p(s(_x61))))))
minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x181)))))))))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x181))))))))))))))))))	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x191)))))))))))))))))), s(s(s(s(_x61)))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x191))))))))))))))))))), s(s(s(p(s(_x61))))))
minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(_x131))))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(p(s(_x131)))))))))))))	minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x171))))))))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x171)))))))))))))))))
minus^#(s(s(s(s(s(_x61))))), s(s(s(s(s(s(s(s(s(_x111))))))))))	→	minus^#(s(s(s(s(p(s(_x61)))))), s(s(s(s(s(s(s(s(p(s(_x111)))))))))))	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x151)))))))))))))), s(s(s(s(s(s(_x81)))))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x151))))))))))))))), s(s(s(s(s(p(s(_x81))))))))
minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(_x141))))))))))))), s(s(s(s(s(s(s(_x91))))))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x141)))))))))))))), s(s(s(s(s(s(p(s(_x91)))))))))	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x241)))))))))))))))))))))), s(s(s(_x51))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x241))))))))))))))))))))))), s(s(p(s(_x51)))))
minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x251))))))))))))))))))))))), s(s(s(_x51))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x251)))))))))))))))))))))))), s(s(p(s(_x51)))))	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(_x131)))))))))))), s(s(s(s(s(s(s(_x91))))))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(p(s(_x131))))))))))))), s(s(s(s(s(s(p(s(_x91)))))))))
minus^#(s(s(s(s(s(s(s(_x81))))))), s(s(s(s(s(s(_x81)))))))	→	minus^#(s(s(s(s(s(s(p(s(_x81)))))))), s(s(s(s(s(p(s(_x81))))))))	minus^#(s(s(s(s(s(s(s(s(s(s(_x111)))))))))), s(s(s(s(s(s(s(_x91))))))))	→	minus^#(s(s(s(s(s(s(s(s(s(p(s(_x111))))))))))), s(s(s(s(s(s(p(s(_x91)))))))))
minus^#(s(s(s(s(s(s(s(s(_x91)))))))), s(s(s(s(s(s(s(_x91))))))))	→	minus^#(s(s(s(s(s(s(s(p(s(_x91))))))))), s(s(s(s(s(s(p(s(_x91)))))))))	minus^#(s(s(s(s(_x51)))), s(s(s(s(s(s(s(s(_x101)))))))))	→	minus^#(s(s(s(p(s(_x51))))), s(s(s(s(s(s(s(p(s(_x101))))))))))
minus^#(s(x), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x181)))))))))))))))))	→	minus^#(p(s(x)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x181))))))))))))))))))	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x201))))))))))))))))))), s(s(_x31)))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x201)))))))))))))))))))), s(p(s(_x31))))
minus^#(s(s(s(s(s(s(s(_x81))))))), s(s(s(s(s(s(s(s(_x101)))))))))	→	minus^#(s(s(s(s(s(s(p(s(_x81)))))))), s(s(s(s(s(s(s(p(s(_x101))))))))))	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x251))))))))))))))))))))))), s(y))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x251)))))))))))))))))))))))), p(s(y)))
minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(_x141))))))))))))), s(s(s(_x51))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x141)))))))))))))), s(s(p(s(_x51)))))	minus^#(s(s(_x21)), s(s(s(s(s(s(s(s(_x101)))))))))	→	minus^#(s(p(s(_x21))), s(s(s(s(s(s(s(p(s(_x101))))))))))
minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x261)))))))))))))))))))))))), s(s(_x31)))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x261))))))))))))))))))))))))), s(p(s(_x31))))	minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(s(s(_x151))))))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x151)))))))))))))))

Rewrite Rules

minus(x, 0)	→	x	minus(0, y)	→	0
minus(s(x), s(y))	→	minus(p(s(x)), p(s(y)))	minus(x, plus(y, z))	→	minus(minus(x, y), z)
p(s(s(x)))	→	s(p(s(x)))	p(0)	→	s(s(0))
div(s(x), s(y))	→	s(div(minus(x, y), s(y)))	div(plus(x, y), z)	→	plus(div(x, z), div(y, z))
plus(0, y)	→	y	plus(s(x), y)	→	s(plus(y, minus(s(x), s(0))))

Original Signature

Termination of terms over the following signature is verified: plus, 0, minus, s, p, div

Strategy

The right-hand side of the rule minus^#(s(s(s(s(s(s(s(s(s(s(s(_x121))))))))))), s(s(s(s(s(s(_x81))))))) → minus^#(s(s(s(s(s(s(s(s(s(s(p(s(_x121)))))))))))), s(s(s(s(s(p(s(_x81)))))))) 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 Terms	Irrelevant Terms
minus^#(s(s(s(s(s(s(s(s(s(s(s(p(s(_x131))))))))))))), s(s(s(s(s(p(s(_x81))))))))
minus^#(s(s(s(s(s(s(s(s(s(s(p(s(_x121)))))))))))), s(s(s(s(s(s(p(s(_x91)))))))))

Thus, the rule minus^#(s(s(s(s(s(s(s(s(s(s(s(_x121))))))))))), s(s(s(s(s(s(_x81))))))) → minus^#(s(s(s(s(s(s(s(s(s(s(p(s(_x121)))))))))))), s(s(s(s(s(p(s(_x81)))))))) is replaced by the following rules:

minus^#(s(s(s(s(s(s(s(s(s(s(s(s(_x131)))))))))))), s(s(s(s(s(s(_x81))))))) → minus^#(s(s(s(s(s(s(s(s(s(s(s(p(s(_x131))))))))))))), s(s(s(s(s(p(s(_x81))))))))

minus^#(s(s(s(s(s(s(s(s(s(s(s(_x121))))))))))), s(s(s(s(s(s(s(_x91)))))))) → minus^#(s(s(s(s(s(s(s(s(s(s(p(s(_x121)))))))))))), s(s(s(s(s(s(p(s(_x91)))))))))

Problem 32: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x151)))))))))))))), s(s(s(s(s(_x71))))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x151))))))))))))))), s(s(s(s(p(s(_x71)))))))	minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(s(_x141)))))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(s(p(s(_x141))))))))))))))
minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x251))))))))))))))))))))))), s(s(s(s(_x61)))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x251)))))))))))))))))))))))), s(s(s(p(s(_x61))))))	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x171)))))))))))))))), s(s(s(s(_x61)))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x171))))))))))))))))), s(s(s(p(s(_x61))))))
minus^#(s(s(s(s(s(_x61))))), s(s(s(s(_x61)))))	→	minus^#(s(s(s(s(p(s(_x61)))))), s(s(s(p(s(_x61))))))	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x241)))))))))))))))))))))), s(s(s(s(_x61)))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x241))))))))))))))))))))))), s(s(s(p(s(_x61))))))
minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(_x111))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(p(s(_x111)))))))))))	minus^#(s(s(s(s(_x51)))), s(s(s(_x51))))	→	minus^#(s(s(s(p(s(_x51))))), s(s(p(s(_x51)))))
minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x161)))))))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x161))))))))))))))))	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x151)))))))))))))), s(s(s(_x51))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x151))))))))))))))), s(s(p(s(_x51)))))
minus^#(s(s(s(s(s(s(s(s(s(s(s(s(_x131)))))))))))), s(s(s(s(s(s(s(s(s(_x111))))))))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(p(s(_x131))))))))))))), s(s(s(s(s(s(s(s(p(s(_x111)))))))))))	minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x201)))))))))))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x201))))))))))))))))))))
minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(_x141))))))))))))), s(s(s(s(_x61)))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x141)))))))))))))), s(s(s(p(s(_x61))))))	minus^#(s(s(s(s(s(s(s(_x81))))))), s(s(s(s(s(s(_x81)))))))	→	minus^#(s(s(s(s(s(s(p(s(_x81)))))))), s(s(s(s(s(p(s(_x81))))))))
minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x251))))))))))))))))))))))), s(y))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x251)))))))))))))))))))))))), p(s(y)))	minus^#(s(s(s(s(s(s(s(s(s(_x101))))))))), s(s(s(s(s(s(s(s(_x101)))))))))	→	minus^#(s(s(s(s(s(s(s(s(p(s(_x101)))))))))), s(s(s(s(s(s(s(p(s(_x101))))))))))
minus^#(s(s(s(s(s(s(_x71)))))), s(s(s(s(s(_x71))))))	→	minus^#(s(s(s(s(s(p(s(_x71))))))), s(s(s(s(p(s(_x71)))))))	minus^#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x221)))))))))))))))))))))	→	minus^#(s(p(s(_x21))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x221))))))))))))))))))))))
minus^#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x201)))))))))))))))))))	→	minus^#(s(p(s(_x21))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x201))))))))))))))))))))	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x151)))))))))))))), s(s(s(s(_x61)))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x151))))))))))))))), s(s(s(p(s(_x61))))))
minus^#(s(s(s(s(_x51)))), s(s(s(s(s(s(s(s(s(_x111))))))))))	→	minus^#(s(s(s(p(s(_x51))))), s(s(s(s(s(s(s(s(p(s(_x111)))))))))))	minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(_x121)))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(p(s(_x121))))))))))))
minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x221)))))))))))))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x221))))))))))))))))))))))	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x181))))))))))))))))), s(s(s(s(_x61)))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x181)))))))))))))))))), s(s(s(p(s(_x61))))))
minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x221)))))))))))))))))))), s(s(s(_x51))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x221))))))))))))))))))))), s(s(p(s(_x51)))))	minus^#(s(s(s(s(s(s(_x71)))))), s(s(s(s(s(s(s(s(s(_x111))))))))))	→	minus^#(s(s(s(s(s(p(s(_x71))))))), s(s(s(s(s(s(s(s(p(s(_x111)))))))))))
minus^#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x191))))))))))))))))))	→	minus^#(s(p(s(_x21))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x191)))))))))))))))))))	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x231))))))))))))))))))))), s(s(s(_x51))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x231)))))))))))))))))))))), s(s(p(s(_x51)))))
minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x151)))))))))))))), s(s(s(s(s(s(s(s(_x101)))))))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x151))))))))))))))), s(s(s(s(s(s(s(p(s(_x101))))))))))	minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x191))))))))))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x191)))))))))))))))))))
minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x161))))))))))))))), s(s(s(s(_x61)))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x161)))))))))))))))), s(s(s(p(s(_x61))))))	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(_x141))))))))))))), s(s(s(s(s(s(s(s(s(_x111))))))))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x141)))))))))))))), s(s(s(s(s(s(s(s(p(s(_x111)))))))))))
minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x201))))))))))))))))))), s(s(s(s(_x61)))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x201)))))))))))))))))))), s(s(s(p(s(_x61))))))	minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x181)))))))))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x181))))))))))))))))))
minus^#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x251))))))))))))))))))))))))	→	minus^#(s(p(s(_x21))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x251)))))))))))))))))))))))))	minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(_x131))))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(p(s(_x131)))))))))))))
minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x191)))))))))))))))))), s(s(s(s(_x61)))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x191))))))))))))))))))), s(s(s(p(s(_x61))))))	minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x171))))))))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x171)))))))))))))))))
minus^#(s(s(s(s(s(_x61))))), s(s(s(s(s(s(s(s(s(_x111))))))))))	→	minus^#(s(s(s(s(p(s(_x61)))))), s(s(s(s(s(s(s(s(p(s(_x111)))))))))))	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x151)))))))))))))), s(s(s(s(s(s(_x81)))))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x151))))))))))))))), s(s(s(s(s(p(s(_x81))))))))
minus^#(s(s(s(s(s(s(s(s(s(s(s(_x121))))))))))), s(s(s(s(s(s(s(s(s(_x111))))))))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(p(s(_x121)))))))))))), s(s(s(s(s(s(s(s(p(s(_x111)))))))))))	minus^#(s(s(s(s(s(s(s(s(s(s(_x111)))))))))), s(s(s(s(s(s(s(s(s(_x111))))))))))	→	minus^#(s(s(s(s(s(s(s(s(s(p(s(_x111))))))))))), s(s(s(s(s(s(s(s(p(s(_x111)))))))))))
minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x211))))))))))))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x211)))))))))))))))))))))	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x261)))))))))))))))))))))))), s(s(s(_x51))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x261))))))))))))))))))))))))), s(s(p(s(_x51)))))
minus^#(s(x), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x261)))))))))))))))))))))))))	→	minus^#(p(s(x)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x261))))))))))))))))))))))))))	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x151)))))))))))))), s(s(s(s(s(s(s(_x91))))))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x151))))))))))))))), s(s(s(s(s(s(p(s(_x91)))))))))
minus^#(s(s(s(s(s(s(s(s(_x91)))))))), s(s(s(s(s(s(s(_x91))))))))	→	minus^#(s(s(s(s(s(s(s(p(s(_x91))))))))), s(s(s(s(s(s(p(s(_x91)))))))))	minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x241)))))))))))))))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x241))))))))))))))))))))))))
minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x231))))))))))))))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x231)))))))))))))))))))))))	minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(s(s(_x151))))))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x151)))))))))))))))
minus^#(s(s(s(s(s(s(s(_x81))))))), s(s(s(s(s(s(s(s(_x101)))))))))	→	minus^#(s(s(s(s(s(s(p(s(_x81)))))))), s(s(s(s(s(s(s(p(s(_x101))))))))))	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x261)))))))))))))))))))))))), s(s(_x31)))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x261))))))))))))))))))))))))), s(p(s(_x31))))
minus^#(s(s(_x21)), s(s(s(s(s(s(s(s(_x101)))))))))	→	minus^#(s(p(s(_x21))), s(s(s(s(s(s(s(p(s(_x101))))))))))

Rewrite Rules

minus(x, 0)	→	x	minus(0, y)	→	0
minus(s(x), s(y))	→	minus(p(s(x)), p(s(y)))	minus(x, plus(y, z))	→	minus(minus(x, y), z)
p(s(s(x)))	→	s(p(s(x)))	p(0)	→	s(s(0))
div(s(x), s(y))	→	s(div(minus(x, y), s(y)))	div(plus(x, y), z)	→	plus(div(x, z), div(y, z))
plus(0, y)	→	y	plus(s(x), y)	→	s(plus(y, minus(s(x), s(0))))

Original Signature

Termination of terms over the following signature is verified: plus, 0, minus, s, p, div

Strategy

The right-hand side of the rule minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x151)))))))))))))), s(s(s(_x51)))) → minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x151))))))))))))))), s(s(p(s(_x51))))) 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 Terms	Irrelevant Terms
minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x151))))))))))))))), s(s(s(p(s(_x61))))))
minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x161)))))))))))))))), s(s(p(s(_x51)))))

Thus, the rule minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x151)))))))))))))), s(s(s(_x51)))) → minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x151))))))))))))))), s(s(p(s(_x51))))) is replaced by the following rules:

minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x161))))))))))))))), s(s(s(_x51)))) → minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x161)))))))))))))))), s(s(p(s(_x51)))))

minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x151)))))))))))))), s(s(s(s(_x61))))) → minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x151))))))))))))))), s(s(s(p(s(_x61))))))

Problem 33: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(s(_x141)))))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(s(p(s(_x141))))))))))))))	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x241)))))))))))))))))))))), s(s(s(s(s(_x71))))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x241))))))))))))))))))))))), s(s(s(s(p(s(_x71)))))))
minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x221)))))))))))))))))))), s(s(s(s(s(_x71))))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x221))))))))))))))))))))), s(s(s(s(p(s(_x71)))))))	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x231))))))))))))))))))))), s(s(s(s(s(_x71))))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x231)))))))))))))))))))))), s(s(s(s(p(s(_x71)))))))
minus^#(s(s(s(s(s(_x61))))), s(s(s(s(_x61)))))	→	minus^#(s(s(s(s(p(s(_x61)))))), s(s(s(p(s(_x61))))))	minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(_x111))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(p(s(_x111)))))))))))
minus^#(s(s(s(s(_x51)))), s(s(s(_x51))))	→	minus^#(s(s(s(p(s(_x51))))), s(s(p(s(_x51)))))	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x251))))))))))))))))))))))), s(s(s(s(s(_x71))))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x251)))))))))))))))))))))))), s(s(s(s(p(s(_x71)))))))
minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x161)))))))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x161))))))))))))))))	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(_x131)))))))))))), s(s(s(s(s(s(s(s(s(_x111))))))))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(p(s(_x131))))))))))))), s(s(s(s(s(s(s(s(p(s(_x111)))))))))))
minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x171)))))))))))))))), s(s(s(s(s(_x71))))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x171))))))))))))))))), s(s(s(s(p(s(_x71)))))))	minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x201)))))))))))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x201))))))))))))))))))))
minus^#(s(s(s(s(s(s(s(_x81))))))), s(s(s(s(s(s(_x81)))))))	→	minus^#(s(s(s(s(s(s(p(s(_x81)))))))), s(s(s(s(s(p(s(_x81))))))))	minus^#(s(s(s(s(s(s(s(s(s(_x101))))))))), s(s(s(s(s(s(s(s(_x101)))))))))	→	minus^#(s(s(s(s(s(s(s(s(p(s(_x101)))))))))), s(s(s(s(s(s(s(p(s(_x101))))))))))
minus^#(s(s(s(s(s(s(_x71)))))), s(s(s(s(s(_x71))))))	→	minus^#(s(s(s(s(s(p(s(_x71))))))), s(s(s(s(p(s(_x71)))))))	minus^#(s(s(s(s(_x51)))), s(s(s(s(s(s(s(s(s(_x111))))))))))	→	minus^#(s(s(s(p(s(_x51))))), s(s(s(s(s(s(s(s(p(s(_x111)))))))))))
minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(_x121)))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(p(s(_x121))))))))))))	minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x221)))))))))))))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x221))))))))))))))))))))))
minus^#(s(s(s(s(s(s(_x71)))))), s(s(s(s(s(s(s(s(s(_x111))))))))))	→	minus^#(s(s(s(s(s(p(s(_x71))))))), s(s(s(s(s(s(s(s(p(s(_x111)))))))))))	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x151)))))))))))))), s(s(s(s(s(s(s(s(_x101)))))))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x151))))))))))))))), s(s(s(s(s(s(s(p(s(_x101))))))))))
minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x191))))))))))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x191)))))))))))))))))))	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x181))))))))))))))))), s(s(s(s(s(_x71))))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x181)))))))))))))))))), s(s(s(s(p(s(_x71)))))))
minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x161))))))))))))))), s(s(s(s(_x61)))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x161)))))))))))))))), s(s(s(p(s(_x61))))))	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x261)))))))))))))))))))))))), s(s(s(s(_x61)))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x261))))))))))))))))))))))))), s(s(s(p(s(_x61))))))
minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(_x141))))))))))))), s(s(s(s(s(s(s(s(s(_x111))))))))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x141)))))))))))))), s(s(s(s(s(s(s(s(p(s(_x111)))))))))))	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x201))))))))))))))))))), s(s(s(s(_x61)))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x201)))))))))))))))))))), s(s(s(p(s(_x61))))))
minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x181)))))))))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x181))))))))))))))))))	minus^#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x251))))))))))))))))))))))))	→	minus^#(s(p(s(_x21))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x251)))))))))))))))))))))))))
minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(_x131))))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(p(s(_x131)))))))))))))	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x191)))))))))))))))))), s(s(s(s(_x61)))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x191))))))))))))))))))), s(s(s(p(s(_x61))))))
minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x171))))))))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x171)))))))))))))))))	minus^#(s(s(s(s(s(_x61))))), s(s(s(s(s(s(s(s(s(_x111))))))))))	→	minus^#(s(s(s(s(p(s(_x61)))))), s(s(s(s(s(s(s(s(p(s(_x111)))))))))))
minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x151)))))))))))))), s(s(s(s(s(s(_x81)))))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x151))))))))))))))), s(s(s(s(s(p(s(_x81))))))))	minus^#(s(s(s(s(s(s(s(s(s(s(s(_x121))))))))))), s(s(s(s(s(s(s(s(s(_x111))))))))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(p(s(_x121)))))))))))), s(s(s(s(s(s(s(s(p(s(_x111)))))))))))
minus^#(s(s(s(s(s(s(s(s(s(s(_x111)))))))))), s(s(s(s(s(s(s(s(s(_x111))))))))))	→	minus^#(s(s(s(s(s(s(s(s(s(p(s(_x111))))))))))), s(s(s(s(s(s(s(s(p(s(_x111)))))))))))	minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x211))))))))))))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x211)))))))))))))))))))))
minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x261)))))))))))))))))))))))), s(s(s(_x51))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x261))))))))))))))))))))))))), s(s(p(s(_x51)))))	minus^#(s(x), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x261)))))))))))))))))))))))))	→	minus^#(p(s(x)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x261))))))))))))))))))))))))))
minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x151)))))))))))))), s(s(s(s(s(s(s(_x91))))))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x151))))))))))))))), s(s(s(s(s(s(p(s(_x91)))))))))	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x161))))))))))))))), s(s(s(s(s(_x71))))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x161)))))))))))))))), s(s(s(s(p(s(_x71)))))))
minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(_x141))))))))))))), s(s(s(s(s(_x71))))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x141)))))))))))))), s(s(s(s(p(s(_x71)))))))	minus^#(s(s(s(s(s(s(s(s(_x91)))))))), s(s(s(s(s(s(s(_x91))))))))	→	minus^#(s(s(s(s(s(s(s(p(s(_x91))))))))), s(s(s(s(s(s(p(s(_x91)))))))))
minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x241)))))))))))))))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x241))))))))))))))))))))))))	minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x231))))))))))))))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x231)))))))))))))))))))))))
minus^#(s(s(_x21)), s(s(s(s(s(s(s(s(_x101)))))))))	→	minus^#(s(p(s(_x21))), s(s(s(s(s(s(s(p(s(_x101))))))))))	minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(s(s(_x151))))))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x151)))))))))))))))
minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x261)))))))))))))))))))))))), s(s(_x31)))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x261))))))))))))))))))))))))), s(p(s(_x31))))	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x261)))))))))))))))))))))))), s(y))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x261))))))))))))))))))))))))), p(s(y)))
minus^#(s(s(s(s(s(s(s(_x81))))))), s(s(s(s(s(s(s(s(_x101)))))))))	→	minus^#(s(s(s(s(s(s(p(s(_x81)))))))), s(s(s(s(s(s(s(p(s(_x101))))))))))

Rewrite Rules

minus(x, 0)	→	x	minus(0, y)	→	0
minus(s(x), s(y))	→	minus(p(s(x)), p(s(y)))	minus(x, plus(y, z))	→	minus(minus(x, y), z)
p(s(s(x)))	→	s(p(s(x)))	p(0)	→	s(s(0))
div(s(x), s(y))	→	s(div(minus(x, y), s(y)))	div(plus(x, y), z)	→	plus(div(x, z), div(y, z))
plus(0, y)	→	y	plus(s(x), y)	→	s(plus(y, minus(s(x), s(0))))

Original Signature

Termination of terms over the following signature is verified: plus, 0, minus, s, p, div

Strategy

The right-hand side of the rule minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x261)))))))))))))))))))))))), s(s(s(s(_x61))))) → minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x261))))))))))))))))))))))))), s(s(s(p(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 Terms	Irrelevant Terms
minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x271)))))))))))))))))))))))))), s(s(s(p(s(_x61))))))
minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x261))))))))))))))))))))))))), s(s(s(s(p(s(_x71)))))))

Thus, the rule minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x261)))))))))))))))))))))))), s(s(s(s(_x61))))) → minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x261))))))))))))))))))))))))), s(s(s(p(s(_x61)))))) is replaced by the following rules:

minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x261)))))))))))))))))))))))), s(s(s(s(s(_x71)))))) → minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x261))))))))))))))))))))))))), s(s(s(s(p(s(_x71)))))))

minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x271))))))))))))))))))))))))), s(s(s(s(_x61))))) → minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x271)))))))))))))))))))))))))), s(s(s(p(s(_x61))))))

Problem 34: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(s(_x141)))))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(s(p(s(_x141))))))))))))))	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x261)))))))))))))))))))))))), s(s(s(s(s(s(_x81)))))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x261))))))))))))))))))))))))), s(s(s(s(s(p(s(_x81))))))))
minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x271))))))))))))))))))))))))), s(s(s(_x51))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x271)))))))))))))))))))))))))), s(s(p(s(_x51)))))	minus^#(s(s(s(s(s(_x61))))), s(s(s(s(_x61)))))	→	minus^#(s(s(s(s(p(s(_x61)))))), s(s(s(p(s(_x61))))))
minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(_x111))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(p(s(_x111)))))))))))	minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x261)))))))))))))))))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x261))))))))))))))))))))))))))
minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x271))))))))))))))))))))))))), s(s(s(s(_x61)))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x271)))))))))))))))))))))))))), s(s(s(p(s(_x61))))))	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x201))))))))))))))))))), s(s(s(s(s(s(_x81)))))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x201)))))))))))))))))))), s(s(s(s(s(p(s(_x81))))))))
minus^#(s(s(s(s(_x51)))), s(s(s(_x51))))	→	minus^#(s(s(s(p(s(_x51))))), s(s(p(s(_x51)))))	minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x161)))))))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x161))))))))))))))))
minus^#(s(s(s(s(s(s(s(s(s(s(s(s(_x131)))))))))))), s(s(s(s(s(s(s(s(s(_x111))))))))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(p(s(_x131))))))))))))), s(s(s(s(s(s(s(s(p(s(_x111)))))))))))	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x241)))))))))))))))))))))), s(s(s(s(s(s(_x81)))))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x241))))))))))))))))))))))), s(s(s(s(s(p(s(_x81))))))))
minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x181))))))))))))))))), s(s(s(s(s(s(_x81)))))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x181)))))))))))))))))), s(s(s(s(s(p(s(_x81))))))))	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x271))))))))))))))))))))))))), s(s(s(s(s(_x71))))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x271)))))))))))))))))))))))))), s(s(s(s(p(s(_x71)))))))
minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x191)))))))))))))))))), s(s(s(s(s(s(_x81)))))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x191))))))))))))))))))), s(s(s(s(s(p(s(_x81))))))))	minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x201)))))))))))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x201))))))))))))))))))))
minus^#(s(s(s(s(s(s(s(_x81))))))), s(s(s(s(s(s(_x81)))))))	→	minus^#(s(s(s(s(s(s(p(s(_x81)))))))), s(s(s(s(s(p(s(_x81))))))))	minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x251))))))))))))))))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x251)))))))))))))))))))))))))
minus^#(s(s(s(s(s(s(s(s(s(_x101))))))))), s(s(s(s(s(s(s(s(_x101)))))))))	→	minus^#(s(s(s(s(s(s(s(s(p(s(_x101)))))))))), s(s(s(s(s(s(s(p(s(_x101))))))))))	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x221)))))))))))))))))))), s(s(s(s(s(s(_x81)))))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x221))))))))))))))))))))), s(s(s(s(s(p(s(_x81))))))))
minus^#(s(x), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x321))))))))))))))))))))))))))))))	→	minus^#(p(s(x)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x321)))))))))))))))))))))))))))))))	minus^#(s(s(s(s(s(s(_x71)))))), s(s(s(s(s(_x71))))))	→	minus^#(s(s(s(s(s(p(s(_x71))))))), s(s(s(s(p(s(_x71)))))))
minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(_x121)))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(p(s(_x121))))))))))))	minus^#(s(s(s(s(_x51)))), s(s(s(s(s(s(s(s(s(_x111))))))))))	→	minus^#(s(s(s(p(s(_x51))))), s(s(s(s(s(s(s(s(p(s(_x111)))))))))))
minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x221)))))))))))))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x221))))))))))))))))))))))	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x251))))))))))))))))))))))), s(s(s(s(s(s(_x81)))))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x251)))))))))))))))))))))))), s(s(s(s(s(p(s(_x81))))))))
minus^#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x281)))))))))))))))))))))))))))	→	minus^#(s(p(s(_x21))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x281))))))))))))))))))))))))))))	minus^#(s(s(s(s(s(s(_x71)))))), s(s(s(s(s(s(s(s(s(_x111))))))))))	→	minus^#(s(s(s(s(s(p(s(_x71))))))), s(s(s(s(s(s(s(s(p(s(_x111)))))))))))
minus^#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x261)))))))))))))))))))))))))	→	minus^#(s(p(s(_x21))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x261))))))))))))))))))))))))))	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x171)))))))))))))))), s(s(s(s(s(s(_x81)))))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x171))))))))))))))))), s(s(s(s(s(p(s(_x81))))))))
minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x151)))))))))))))), s(s(s(s(s(s(s(s(_x101)))))))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x151))))))))))))))), s(s(s(s(s(s(s(p(s(_x101))))))))))	minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x191))))))))))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x191)))))))))))))))))))
minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(_x141))))))))))))), s(s(s(s(s(s(s(s(s(_x111))))))))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x141)))))))))))))), s(s(s(s(s(s(s(s(p(s(_x111)))))))))))	minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x271))))))))))))))))))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x271)))))))))))))))))))))))))))
minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x181)))))))))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x181))))))))))))))))))	minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(_x131))))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(p(s(_x131)))))))))))))
minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x171))))))))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x171)))))))))))))))))	minus^#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x291))))))))))))))))))))))))))))	→	minus^#(s(p(s(_x21))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x291)))))))))))))))))))))))))))))
minus^#(s(s(s(s(s(_x61))))), s(s(s(s(s(s(s(s(s(_x111))))))))))	→	minus^#(s(s(s(s(p(s(_x61)))))), s(s(s(s(s(s(s(s(p(s(_x111)))))))))))	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x151)))))))))))))), s(s(s(s(s(s(_x81)))))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x151))))))))))))))), s(s(s(s(s(p(s(_x81))))))))
minus^#(s(s(s(s(s(s(s(s(s(s(s(_x121))))))))))), s(s(s(s(s(s(s(s(s(_x111))))))))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(p(s(_x121)))))))))))), s(s(s(s(s(s(s(s(p(s(_x111)))))))))))	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x231))))))))))))))))))))), s(s(s(s(s(s(_x81)))))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x231)))))))))))))))))))))), s(s(s(s(s(p(s(_x81))))))))
minus^#(s(s(s(s(s(s(s(s(s(s(_x111)))))))))), s(s(s(s(s(s(s(s(s(_x111))))))))))	→	minus^#(s(s(s(s(s(s(s(s(s(p(s(_x111))))))))))), s(s(s(s(s(s(s(s(p(s(_x111)))))))))))	minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x211))))))))))))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x211)))))))))))))))))))))
minus^#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x301)))))))))))))))))))))))))))))	→	minus^#(s(p(s(_x21))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x301))))))))))))))))))))))))))))))	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x151)))))))))))))), s(s(s(s(s(s(s(_x91))))))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x151))))))))))))))), s(s(s(s(s(s(p(s(_x91)))))))))
minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x161))))))))))))))), s(s(s(s(s(_x71))))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x161)))))))))))))))), s(s(s(s(p(s(_x71)))))))	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(_x141))))))))))))), s(s(s(s(s(_x71))))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x141)))))))))))))), s(s(s(s(p(s(_x71)))))))
minus^#(s(s(s(s(s(s(s(s(_x91)))))))), s(s(s(s(s(s(s(_x91))))))))	→	minus^#(s(s(s(s(s(s(s(p(s(_x91))))))))), s(s(s(s(s(s(p(s(_x91)))))))))	minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x241)))))))))))))))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x241))))))))))))))))))))))))
minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x231))))))))))))))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x231)))))))))))))))))))))))	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x261)))))))))))))))))))))))), s(s(_x31)))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x261))))))))))))))))))))))))), s(p(s(_x31))))
minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x261)))))))))))))))))))))))), s(y))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x261))))))))))))))))))))))))), p(s(y)))	minus^#(s(s(s(s(s(s(s(_x81))))))), s(s(s(s(s(s(s(s(_x101)))))))))	→	minus^#(s(s(s(s(s(s(p(s(_x81)))))))), s(s(s(s(s(s(s(p(s(_x101))))))))))
minus^#(s(s(_x21)), s(s(s(s(s(s(s(s(_x101)))))))))	→	minus^#(s(p(s(_x21))), s(s(s(s(s(s(s(p(s(_x101))))))))))	minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(s(s(_x151))))))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x151)))))))))))))))

Rewrite Rules

minus(x, 0)	→	x	minus(0, y)	→	0
minus(s(x), s(y))	→	minus(p(s(x)), p(s(y)))	minus(x, plus(y, z))	→	minus(minus(x, y), z)
p(s(s(x)))	→	s(p(s(x)))	p(0)	→	s(s(0))
div(s(x), s(y))	→	s(div(minus(x, y), s(y)))	div(plus(x, y), z)	→	plus(div(x, z), div(y, z))
plus(0, y)	→	y	plus(s(x), y)	→	s(plus(y, minus(s(x), s(0))))

Original Signature

Termination of terms over the following signature is verified: plus, 0, minus, s, p, div

Strategy

The right-hand side of the rule minus^#(s(x), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x321)))))))))))))))))))))))))))))) → minus^#(p(s(x)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x321))))))))))))))))))))))))))))))) 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 Terms	Irrelevant Terms
minus^#(s(p(s(_x21))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x321)))))))))))))))))))))))))))))))
minus^#(p(s(x)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x331))))))))))))))))))))))))))))))))

Thus, the rule minus^#(s(x), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x321)))))))))))))))))))))))))))))) → minus^#(p(s(x)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x321))))))))))))))))))))))))))))))) is replaced by the following rules:

minus^#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x321)))))))))))))))))))))))))))))) → minus^#(s(p(s(_x21))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x321)))))))))))))))))))))))))))))))

minus^#(s(x), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x331))))))))))))))))))))))))))))))) → minus^#(p(s(x)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x331))))))))))))))))))))))))))))))))

Problem 35: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(s(_x141)))))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(s(p(s(_x141))))))))))))))	minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x401))))))))))))))))))))))))))))))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x401)))))))))))))))))))))))))))))))))))))))
minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x341))))))))))))))))))))))))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x341)))))))))))))))))))))))))))))))))	minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x381))))))))))))))))))))))))))))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x381)))))))))))))))))))))))))))))))))))))
minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x261)))))))))))))))))))))))), s(s(s(s(s(s(_x81)))))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x261))))))))))))))))))))))))), s(s(s(s(s(p(s(_x81))))))))	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x271))))))))))))))))))))))))), s(s(s(_x51))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x271)))))))))))))))))))))))))), s(s(p(s(_x51)))))
minus^#(s(s(s(s(s(_x61))))), s(s(s(s(_x61)))))	→	minus^#(s(s(s(s(p(s(_x61)))))), s(s(s(p(s(_x61))))))	minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(_x111))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(p(s(_x111)))))))))))
minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x261)))))))))))))))))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x261))))))))))))))))))))))))))	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x271))))))))))))))))))))))))), s(s(s(s(_x61)))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x271)))))))))))))))))))))))))), s(s(s(p(s(_x61))))))
minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x201))))))))))))))))))), s(s(s(s(s(s(_x81)))))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x201)))))))))))))))))))), s(s(s(s(s(p(s(_x81))))))))	minus^#(s(s(s(s(_x51)))), s(s(s(_x51))))	→	minus^#(s(s(s(p(s(_x51))))), s(s(p(s(_x51)))))
minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x161)))))))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x161))))))))))))))))	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(_x131)))))))))))), s(s(s(s(s(s(s(s(s(_x111))))))))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(p(s(_x131))))))))))))), s(s(s(s(s(s(s(s(p(s(_x111)))))))))))
minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x371)))))))))))))))))))))))))))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x371))))))))))))))))))))))))))))))))))))	minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x361))))))))))))))))))))))))))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x361)))))))))))))))))))))))))))))))))))
minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x351)))))))))))))))))))))))))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x351))))))))))))))))))))))))))))))))))	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x241)))))))))))))))))))))), s(s(s(s(s(s(_x81)))))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x241))))))))))))))))))))))), s(s(s(s(s(p(s(_x81))))))))
minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x421))))))))))))))))))))))))))))))))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x421)))))))))))))))))))))))))))))))))))))))))	minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x411)))))))))))))))))))))))))))))))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x411))))))))))))))))))))))))))))))))))))))))
minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x181))))))))))))))))), s(s(s(s(s(s(_x81)))))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x181)))))))))))))))))), s(s(s(s(s(p(s(_x81))))))))	minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x431)))))))))))))))))))))))))))))))))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x431))))))))))))))))))))))))))))))))))))))))))
minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x271))))))))))))))))))))))))), s(s(s(s(s(_x71))))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x271)))))))))))))))))))))))))), s(s(s(s(p(s(_x71)))))))	minus^#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x451)))))))))))))))))))))))))))))))))))))))))))	→	minus^#(s(p(s(_x21))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x451))))))))))))))))))))))))))))))))))))))))))))
minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x191)))))))))))))))))), s(s(s(s(s(s(_x81)))))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x191))))))))))))))))))), s(s(s(s(s(p(s(_x81))))))))	minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x201)))))))))))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x201))))))))))))))))))))
minus^#(s(s(s(s(s(s(s(_x81))))))), s(s(s(s(s(s(_x81)))))))	→	minus^#(s(s(s(s(s(s(p(s(_x81)))))))), s(s(s(s(s(p(s(_x81))))))))	minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x391)))))))))))))))))))))))))))))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x391))))))))))))))))))))))))))))))))))))))
minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x251))))))))))))))))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x251)))))))))))))))))))))))))	minus^#(s(s(s(s(s(s(s(s(s(_x101))))))))), s(s(s(s(s(s(s(s(_x101)))))))))	→	minus^#(s(s(s(s(s(s(s(s(p(s(_x101)))))))))), s(s(s(s(s(s(s(p(s(_x101))))))))))
minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x221)))))))))))))))))))), s(s(s(s(s(s(_x81)))))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x221))))))))))))))))))))), s(s(s(s(s(p(s(_x81))))))))	minus^#(s(s(s(s(s(s(_x71)))))), s(s(s(s(s(_x71))))))	→	minus^#(s(s(s(s(s(p(s(_x71))))))), s(s(s(s(p(s(_x71)))))))
minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x221)))))))))))))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x221))))))))))))))))))))))	minus^#(s(s(s(s(_x51)))), s(s(s(s(s(s(s(s(s(_x111))))))))))	→	minus^#(s(s(s(p(s(_x51))))), s(s(s(s(s(s(s(s(p(s(_x111)))))))))))
minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(_x121)))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(p(s(_x121))))))))))))	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x251))))))))))))))))))))))), s(s(s(s(s(s(_x81)))))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x251)))))))))))))))))))))))), s(s(s(s(s(p(s(_x81))))))))
minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x441))))))))))))))))))))))))))))))))))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x441)))))))))))))))))))))))))))))))))))))))))))	minus^#(s(s(s(s(s(s(_x71)))))), s(s(s(s(s(s(s(s(s(_x111))))))))))	→	minus^#(s(s(s(s(s(p(s(_x71))))))), s(s(s(s(s(s(s(s(p(s(_x111)))))))))))
minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x171)))))))))))))))), s(s(s(s(s(s(_x81)))))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x171))))))))))))))))), s(s(s(s(s(p(s(_x81))))))))	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x151)))))))))))))), s(s(s(s(s(s(s(s(_x101)))))))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x151))))))))))))))), s(s(s(s(s(s(s(p(s(_x101))))))))))
minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x191))))))))))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x191)))))))))))))))))))	minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x281)))))))))))))))))))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x281))))))))))))))))))))))))))))
minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(_x141))))))))))))), s(s(s(s(s(s(s(s(s(_x111))))))))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x141)))))))))))))), s(s(s(s(s(s(s(s(p(s(_x111)))))))))))	minus^#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x431)))))))))))))))))))))))))))))))))))))))))	→	minus^#(s(p(s(_x21))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x431))))))))))))))))))))))))))))))))))))))))))
minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x271))))))))))))))))))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x271)))))))))))))))))))))))))))	minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x181)))))))))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x181))))))))))))))))))
minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(_x131))))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(p(s(_x131)))))))))))))	minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x171))))))))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x171)))))))))))))))))
minus^#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x291))))))))))))))))))))))))))))	→	minus^#(s(p(s(_x21))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x291)))))))))))))))))))))))))))))	minus^#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x331)))))))))))))))))))))))))))))))	→	minus^#(s(p(s(_x21))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x331))))))))))))))))))))))))))))))))
minus^#(s(s(s(s(s(_x61))))), s(s(s(s(s(s(s(s(s(_x111))))))))))	→	minus^#(s(s(s(s(p(s(_x61)))))), s(s(s(s(s(s(s(s(p(s(_x111)))))))))))	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x151)))))))))))))), s(s(s(s(s(s(_x81)))))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x151))))))))))))))), s(s(s(s(s(p(s(_x81))))))))
minus^#(s(s(s(s(s(s(s(s(s(s(s(_x121))))))))))), s(s(s(s(s(s(s(s(s(_x111))))))))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(p(s(_x121)))))))))))), s(s(s(s(s(s(s(s(p(s(_x111)))))))))))	minus^#(s(s(s(s(s(s(s(s(s(s(_x111)))))))))), s(s(s(s(s(s(s(s(s(_x111))))))))))	→	minus^#(s(s(s(s(s(s(s(s(s(p(s(_x111))))))))))), s(s(s(s(s(s(s(s(p(s(_x111)))))))))))
minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x231))))))))))))))))))))), s(s(s(s(s(s(_x81)))))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x231)))))))))))))))))))))), s(s(s(s(s(p(s(_x81))))))))	minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x211))))))))))))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x211)))))))))))))))))))))
minus^#(s(x), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x441))))))))))))))))))))))))))))))))))))))))))	→	minus^#(p(s(x)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x441)))))))))))))))))))))))))))))))))))))))))))	minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x321))))))))))))))))))))))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x321)))))))))))))))))))))))))))))))
minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x151)))))))))))))), s(s(s(s(s(s(s(_x91))))))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x151))))))))))))))), s(s(s(s(s(s(p(s(_x91)))))))))	minus^#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x301)))))))))))))))))))))))))))))	→	minus^#(s(p(s(_x21))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x301))))))))))))))))))))))))))))))
minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(_x141))))))))))))), s(s(s(s(s(_x71))))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x141)))))))))))))), s(s(s(s(p(s(_x71)))))))	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x161))))))))))))))), s(s(s(s(s(_x71))))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x161)))))))))))))))), s(s(s(s(p(s(_x71)))))))
minus^#(s(s(s(s(s(s(s(s(_x91)))))))), s(s(s(s(s(s(s(_x91))))))))	→	minus^#(s(s(s(s(s(s(s(p(s(_x91))))))))), s(s(s(s(s(s(p(s(_x91)))))))))	minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x241)))))))))))))))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x241))))))))))))))))))))))))
minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x231))))))))))))))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x231)))))))))))))))))))))))	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x261)))))))))))))))))))))))), s(s(_x31)))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x261))))))))))))))))))))))))), s(p(s(_x31))))
minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x261)))))))))))))))))))))))), s(y))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x261))))))))))))))))))))))))), p(s(y)))	minus^#(s(s(s(s(s(s(s(_x81))))))), s(s(s(s(s(s(s(s(_x101)))))))))	→	minus^#(s(s(s(s(s(s(p(s(_x81)))))))), s(s(s(s(s(s(s(p(s(_x101))))))))))
minus^#(s(s(_x21)), s(s(s(s(s(s(s(s(_x101)))))))))	→	minus^#(s(p(s(_x21))), s(s(s(s(s(s(s(p(s(_x101))))))))))	minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(s(s(_x151))))))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x151)))))))))))))))

Rewrite Rules

minus(x, 0)	→	x	minus(0, y)	→	0
minus(s(x), s(y))	→	minus(p(s(x)), p(s(y)))	minus(x, plus(y, z))	→	minus(minus(x, y), z)
p(s(s(x)))	→	s(p(s(x)))	p(0)	→	s(s(0))
div(s(x), s(y))	→	s(div(minus(x, y), s(y)))	div(plus(x, y), z)	→	plus(div(x, z), div(y, z))
plus(0, y)	→	y	plus(s(x), y)	→	s(plus(y, minus(s(x), s(0))))

Original Signature

Termination of terms over the following signature is verified: plus, 0, minus, s, p, div

Strategy

The right-hand side of the rule minus^#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x431))))))))))))))))))))))))))))))))))))))))) → minus^#(s(p(s(_x21))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x431)))))))))))))))))))))))))))))))))))))))))) 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 Terms	Irrelevant Terms
minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x431))))))))))))))))))))))))))))))))))))))))))
minus^#(s(p(s(_x21))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x441)))))))))))))))))))))))))))))))))))))))))))

Thus, the rule minus^#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x431))))))))))))))))))))))))))))))))))))))))) → minus^#(s(p(s(_x21))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x431)))))))))))))))))))))))))))))))))))))))))) is replaced by the following rules:

minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x431))))))))))))))))))))))))))))))))))))))))) → minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x431))))))))))))))))))))))))))))))))))))))))))

minus^#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x441)))))))))))))))))))))))))))))))))))))))))) → minus^#(s(p(s(_x21))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x441)))))))))))))))))))))))))))))))))))))))))))

Problem 36: ForwardNarrowing

Dependency Pair Problem

Dependency Pairs

minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(s(_x141)))))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(s(p(s(_x141))))))))))))))	minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x401))))))))))))))))))))))))))))))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x401)))))))))))))))))))))))))))))))))))))))
minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x341))))))))))))))))))))))))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x341)))))))))))))))))))))))))))))))))	minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x381))))))))))))))))))))))))))))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x381)))))))))))))))))))))))))))))))))))))
minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x261)))))))))))))))))))))))), s(s(s(s(s(s(_x81)))))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x261))))))))))))))))))))))))), s(s(s(s(s(p(s(_x81))))))))	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x271))))))))))))))))))))))))), s(s(s(_x51))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x271)))))))))))))))))))))))))), s(s(p(s(_x51)))))
minus^#(s(s(s(s(s(_x61))))), s(s(s(s(_x61)))))	→	minus^#(s(s(s(s(p(s(_x61)))))), s(s(s(p(s(_x61))))))	minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(_x111))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(p(s(_x111)))))))))))
minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x261)))))))))))))))))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x261))))))))))))))))))))))))))	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x271))))))))))))))))))))))))), s(s(s(s(_x61)))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x271)))))))))))))))))))))))))), s(s(s(p(s(_x61))))))
minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x481))))))))))))))))))))))))))))))))))))))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x481)))))))))))))))))))))))))))))))))))))))))))))))	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x201))))))))))))))))))), s(s(s(s(s(s(_x81)))))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x201)))))))))))))))))))), s(s(s(s(s(p(s(_x81))))))))
minus^#(s(s(s(s(_x51)))), s(s(s(_x51))))	→	minus^#(s(s(s(p(s(_x51))))), s(s(p(s(_x51)))))	minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x511)))))))))))))))))))))))))))))))))))))))))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x511))))))))))))))))))))))))))))))))))))))))))))))))))
minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x161)))))))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x161))))))))))))))))	minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x551)))))))))))))))))))))))))))))))))))))))))))))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x551))))))))))))))))))))))))))))))))))))))))))))))))))))))
minus^#(s(s(s(s(s(s(s(s(s(s(s(s(_x131)))))))))))), s(s(s(s(s(s(s(s(s(_x111))))))))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(p(s(_x131))))))))))))), s(s(s(s(s(s(s(s(p(s(_x111)))))))))))	minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x371)))))))))))))))))))))))))))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x371))))))))))))))))))))))))))))))))))))
minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x361))))))))))))))))))))))))))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x361)))))))))))))))))))))))))))))))))))	minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x351)))))))))))))))))))))))))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x351))))))))))))))))))))))))))))))))))
minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x241)))))))))))))))))))))), s(s(s(s(s(s(_x81)))))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x241))))))))))))))))))))))), s(s(s(s(s(p(s(_x81))))))))	minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x421))))))))))))))))))))))))))))))))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x421)))))))))))))))))))))))))))))))))))))))))
minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x411)))))))))))))))))))))))))))))))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x411))))))))))))))))))))))))))))))))))))))))	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x181))))))))))))))))), s(s(s(s(s(s(_x81)))))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x181)))))))))))))))))), s(s(s(s(s(p(s(_x81))))))))
minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x431)))))))))))))))))))))))))))))))))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x431))))))))))))))))))))))))))))))))))))))))))	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x271))))))))))))))))))))))))), s(s(s(s(s(_x71))))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x271)))))))))))))))))))))))))), s(s(s(s(p(s(_x71)))))))
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minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x151)))))))))))))), s(s(s(s(s(s(s(_x91))))))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x151))))))))))))))), s(s(s(s(s(s(p(s(_x91)))))))))	minus^#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x301)))))))))))))))))))))))))))))	→	minus^#(s(p(s(_x21))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x301))))))))))))))))))))))))))))))
minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(_x141))))))))))))), s(s(s(s(s(_x71))))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x141)))))))))))))), s(s(s(s(p(s(_x71)))))))	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x161))))))))))))))), s(s(s(s(s(_x71))))))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x161)))))))))))))))), s(s(s(s(p(s(_x71)))))))
minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x471)))))))))))))))))))))))))))))))))))))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x471))))))))))))))))))))))))))))))))))))))))))))))	minus^#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x541))))))))))))))))))))))))))))))))))))))))))))))))))))	→	minus^#(s(p(s(_x21))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x541)))))))))))))))))))))))))))))))))))))))))))))))))))))
minus^#(s(s(s(s(s(s(s(s(_x91)))))))), s(s(s(s(s(s(s(_x91))))))))	→	minus^#(s(s(s(s(s(s(s(p(s(_x91))))))))), s(s(s(s(s(s(p(s(_x91)))))))))	minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x241)))))))))))))))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x241))))))))))))))))))))))))
minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x231))))))))))))))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x231)))))))))))))))))))))))	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x261)))))))))))))))))))))))), s(s(_x31)))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x261))))))))))))))))))))))))), s(p(s(_x31))))
minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x261)))))))))))))))))))))))), s(y))	→	minus^#(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x261))))))))))))))))))))))))), p(s(y)))	minus^#(s(s(s(s(s(s(s(_x81))))))), s(s(s(s(s(s(s(s(_x101)))))))))	→	minus^#(s(s(s(s(s(s(p(s(_x81)))))))), s(s(s(s(s(s(s(p(s(_x101))))))))))
minus^#(s(s(_x21)), s(s(s(s(s(s(s(s(_x101)))))))))	→	minus^#(s(p(s(_x21))), s(s(s(s(s(s(s(p(s(_x101))))))))))	minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(s(s(_x151))))))))))))))	→	minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x151)))))))))))))))

Rewrite Rules

minus(x, 0)	→	x	minus(0, y)	→	0
minus(s(x), s(y))	→	minus(p(s(x)), p(s(y)))	minus(x, plus(y, z))	→	minus(minus(x, y), z)
p(s(s(x)))	→	s(p(s(x)))	p(0)	→	s(s(0))
div(s(x), s(y))	→	s(div(minus(x, y), s(y)))	div(plus(x, y), z)	→	plus(div(x, z), div(y, z))
plus(0, y)	→	y	plus(s(x), y)	→	s(plus(y, minus(s(x), s(0))))

Original Signature

Termination of terms over the following signature is verified: plus, 0, minus, s, p, div

Strategy

The right-hand side of the rule minus^#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x561)))))))))))))))))))))))))))))))))))))))))))))))))))))) → minus^#(s(p(s(_x21))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x561))))))))))))))))))))))))))))))))))))))))))))))))))))))) 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 Terms	Irrelevant Terms
minus^#(s(p(s(_x21))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x571))))))))))))))))))))))))))))))))))))))))))))))))))))))))
minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x561)))))))))))))))))))))))))))))))))))))))))))))))))))))))

Thus, the rule minus^#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x561)))))))))))))))))))))))))))))))))))))))))))))))))))))) → minus^#(s(p(s(_x21))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x561))))))))))))))))))))))))))))))))))))))))))))))))))))))) is replaced by the following rules:

minus^#(s(s(_x21)), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x571))))))))))))))))))))))))))))))))))))))))))))))))))))))) → minus^#(s(p(s(_x21))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x571))))))))))))))))))))))))))))))))))))))))))))))))))))))))

minus^#(s(s(s(_x41))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(_x561)))))))))))))))))))))))))))))))))))))))))))))))))))))) → minus^#(s(s(p(s(_x41)))), s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(s(_x561)))))))))))))))))))))))))))))))))))))))))))))))))))))))

Problem 4: PolynomialLinearRange4iUR

Dependency Pair Problem

Dependency Pairs

div^#(plus(x, y), z)	→	div^#(x, z)		div^#(plus(x, y), z)	→	div^#(y, z)
div^#(s(x), s(y))	→	div^#(minus(x, y), s(y))

Rewrite Rules

minus(x, 0)	→	x	minus(0, y)	→	0
minus(s(x), s(y))	→	minus(p(s(x)), p(s(y)))	minus(x, plus(y, z))	→	minus(minus(x, y), z)
p(s(s(x)))	→	s(p(s(x)))	p(0)	→	s(s(0))
div(s(x), s(y))	→	s(div(minus(x, y), s(y)))	div(plus(x, y), z)	→	plus(div(x, z), div(y, z))
plus(0, y)	→	y	plus(s(x), y)	→	s(plus(y, minus(s(x), s(0))))

Original Signature

Termination of terms over the following signature is verified: plus, minus, 0, s, p, div

Strategy

Polynomial Interpretation

0: 0
div(x,y): 0
div#(x,y): 2x
minus(x,y): x
p(x): 0
plus(x,y): 2y + 2x + 1
s(x): 2x

Improved Usable rules

p(s(s(x)))	→	s(p(s(x)))	minus(x, plus(y, z))	→	minus(minus(x, y), z)
minus(0, y)	→	0	minus(s(x), s(y))	→	minus(p(s(x)), p(s(y)))
p(0)	→	s(s(0))	minus(x, 0)	→	x

The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:

div^#(plus(x, y), z)

→

div^#(x, z)

div^#(plus(x, y), z)

→

div^#(y, z)