WORST_CASE(?,O(n^3))
proof of input_I1YIq36w7R.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^3).

(0) CpxTRS
(1) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(2) CdtProblem
(3) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
(4) CdtProblem
(5) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) CdtProblem
(7) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
(8) CdtProblem
(9) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 105 ms]
(10) CdtProblem
(11) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 96 ms]
(12) CdtProblem
(13) CdtRuleRemovalProof [UPPER BOUND(ADD(n^3)), 734 ms]
(14) CdtProblem
(15) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
(16) BOUNDS(1, 1)


----------------------------------------

(0)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^3).


The TRS R consists of the following rules:

   minus(0, y) -> 0
   minus(s(x), 0) -> s(x)
   minus(s(x), s(y)) -> minus(x, y)
   le(0, y) -> true
   le(s(x), 0) -> false
   le(s(x), s(y)) -> le(x, y)
   if(true, x, y) -> x
   if(false, x, y) -> y
   perfectp(0) -> false
   perfectp(s(x)) -> f(x, s(0), s(x), s(x))
   f(0, y, 0, u) -> true
   f(0, y, s(z), u) -> false
   f(s(x), 0, z, u) -> f(x, u, minus(z, s(x)), u)
   f(s(x), s(y), z, u) -> if(le(x, y), f(s(x), minus(y, x), z, u), f(x, u, z, u))

S is empty.
Rewrite Strategy: PARALLEL_INNERMOST
----------------------------------------

(1) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
----------------------------------------

(2)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(0, z0) -> 0
   minus(s(z0), 0) -> s(z0)
   minus(s(z0), s(z1)) -> minus(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   perfectp(0) -> false
   perfectp(s(z0)) -> f(z0, s(0), s(z0), s(z0))
   f(0, z0, 0, z1) -> true
   f(0, z0, s(z1), z2) -> false
   f(s(z0), 0, z1, z2) -> f(z0, z2, minus(z1, s(z0)), z2)
   f(s(z0), s(z1), z2, z3) -> if(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3))
Tuples:
   MINUS(0, z0) -> c
   MINUS(s(z0), 0) -> c1
   MINUS(s(z0), s(z1)) -> c2(MINUS(z0, z1))
   LE(0, z0) -> c3
   LE(s(z0), 0) -> c4
   LE(s(z0), s(z1)) -> c5(LE(z0, z1))
   IF(true, z0, z1) -> c6
   IF(false, z0, z1) -> c7
   PERFECTP(0) -> c8
   PERFECTP(s(z0)) -> c9(F(z0, s(0), s(z0), s(z0)))
   F(0, z0, 0, z1) -> c10
   F(0, z0, s(z1), z2) -> c11
   F(s(z0), 0, z1, z2) -> c12(F(z0, z2, minus(z1, s(z0)), z2), MINUS(z1, s(z0)))
   F(s(z0), s(z1), z2, z3) -> c13(IF(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3)), LE(z0, z1))
   F(s(z0), s(z1), z2, z3) -> c14(IF(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3)), F(s(z0), minus(z1, z0), z2, z3), MINUS(z1, z0))
   F(s(z0), s(z1), z2, z3) -> c15(IF(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3)), F(z0, z3, z2, z3))
S tuples:
   MINUS(0, z0) -> c
   MINUS(s(z0), 0) -> c1
   MINUS(s(z0), s(z1)) -> c2(MINUS(z0, z1))
   LE(0, z0) -> c3
   LE(s(z0), 0) -> c4
   LE(s(z0), s(z1)) -> c5(LE(z0, z1))
   IF(true, z0, z1) -> c6
   IF(false, z0, z1) -> c7
   PERFECTP(0) -> c8
   PERFECTP(s(z0)) -> c9(F(z0, s(0), s(z0), s(z0)))
   F(0, z0, 0, z1) -> c10
   F(0, z0, s(z1), z2) -> c11
   F(s(z0), 0, z1, z2) -> c12(F(z0, z2, minus(z1, s(z0)), z2), MINUS(z1, s(z0)))
   F(s(z0), s(z1), z2, z3) -> c13(IF(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3)), LE(z0, z1))
   F(s(z0), s(z1), z2, z3) -> c14(IF(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3)), F(s(z0), minus(z1, z0), z2, z3), MINUS(z1, z0))
   F(s(z0), s(z1), z2, z3) -> c15(IF(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3)), F(z0, z3, z2, z3))
K tuples:none
Defined Rule Symbols:   minus_2, le_2, if_3, perfectp_1, f_4

Defined Pair Symbols:   MINUS_2, LE_2, IF_3, PERFECTP_1, F_4

Compound Symbols:   c, c1, c2_1, c3, c4, c5_1, c6, c7, c8, c9_1, c10, c11, c12_2, c13_2, c14_3, c15_2


----------------------------------------

(3) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   PERFECTP(s(z0)) -> c9(F(z0, s(0), s(z0), s(z0)))
Removed 9 trailing nodes:
   F(0, z0, 0, z1) -> c10
   MINUS(s(z0), 0) -> c1
   F(0, z0, s(z1), z2) -> c11
   IF(false, z0, z1) -> c7
   PERFECTP(0) -> c8
   IF(true, z0, z1) -> c6
   LE(s(z0), 0) -> c4
   MINUS(0, z0) -> c
   LE(0, z0) -> c3

----------------------------------------

(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(0, z0) -> 0
   minus(s(z0), 0) -> s(z0)
   minus(s(z0), s(z1)) -> minus(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   perfectp(0) -> false
   perfectp(s(z0)) -> f(z0, s(0), s(z0), s(z0))
   f(0, z0, 0, z1) -> true
   f(0, z0, s(z1), z2) -> false
   f(s(z0), 0, z1, z2) -> f(z0, z2, minus(z1, s(z0)), z2)
   f(s(z0), s(z1), z2, z3) -> if(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3))
Tuples:
   MINUS(s(z0), s(z1)) -> c2(MINUS(z0, z1))
   LE(s(z0), s(z1)) -> c5(LE(z0, z1))
   F(s(z0), 0, z1, z2) -> c12(F(z0, z2, minus(z1, s(z0)), z2), MINUS(z1, s(z0)))
   F(s(z0), s(z1), z2, z3) -> c13(IF(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3)), LE(z0, z1))
   F(s(z0), s(z1), z2, z3) -> c14(IF(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3)), F(s(z0), minus(z1, z0), z2, z3), MINUS(z1, z0))
   F(s(z0), s(z1), z2, z3) -> c15(IF(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3)), F(z0, z3, z2, z3))
S tuples:
   MINUS(s(z0), s(z1)) -> c2(MINUS(z0, z1))
   LE(s(z0), s(z1)) -> c5(LE(z0, z1))
   F(s(z0), 0, z1, z2) -> c12(F(z0, z2, minus(z1, s(z0)), z2), MINUS(z1, s(z0)))
   F(s(z0), s(z1), z2, z3) -> c13(IF(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3)), LE(z0, z1))
   F(s(z0), s(z1), z2, z3) -> c14(IF(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3)), F(s(z0), minus(z1, z0), z2, z3), MINUS(z1, z0))
   F(s(z0), s(z1), z2, z3) -> c15(IF(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3)), F(z0, z3, z2, z3))
K tuples:none
Defined Rule Symbols:   minus_2, le_2, if_3, perfectp_1, f_4

Defined Pair Symbols:   MINUS_2, LE_2, F_4

Compound Symbols:   c2_1, c5_1, c12_2, c13_2, c14_3, c15_2


----------------------------------------

(5) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 3 trailing tuple parts
----------------------------------------

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(0, z0) -> 0
   minus(s(z0), 0) -> s(z0)
   minus(s(z0), s(z1)) -> minus(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   perfectp(0) -> false
   perfectp(s(z0)) -> f(z0, s(0), s(z0), s(z0))
   f(0, z0, 0, z1) -> true
   f(0, z0, s(z1), z2) -> false
   f(s(z0), 0, z1, z2) -> f(z0, z2, minus(z1, s(z0)), z2)
   f(s(z0), s(z1), z2, z3) -> if(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3))
Tuples:
   MINUS(s(z0), s(z1)) -> c2(MINUS(z0, z1))
   LE(s(z0), s(z1)) -> c5(LE(z0, z1))
   F(s(z0), 0, z1, z2) -> c12(F(z0, z2, minus(z1, s(z0)), z2), MINUS(z1, s(z0)))
   F(s(z0), s(z1), z2, z3) -> c13(LE(z0, z1))
   F(s(z0), s(z1), z2, z3) -> c14(F(s(z0), minus(z1, z0), z2, z3), MINUS(z1, z0))
   F(s(z0), s(z1), z2, z3) -> c15(F(z0, z3, z2, z3))
S tuples:
   MINUS(s(z0), s(z1)) -> c2(MINUS(z0, z1))
   LE(s(z0), s(z1)) -> c5(LE(z0, z1))
   F(s(z0), 0, z1, z2) -> c12(F(z0, z2, minus(z1, s(z0)), z2), MINUS(z1, s(z0)))
   F(s(z0), s(z1), z2, z3) -> c13(LE(z0, z1))
   F(s(z0), s(z1), z2, z3) -> c14(F(s(z0), minus(z1, z0), z2, z3), MINUS(z1, z0))
   F(s(z0), s(z1), z2, z3) -> c15(F(z0, z3, z2, z3))
K tuples:none
Defined Rule Symbols:   minus_2, le_2, if_3, perfectp_1, f_4

Defined Pair Symbols:   MINUS_2, LE_2, F_4

Compound Symbols:   c2_1, c5_1, c12_2, c13_1, c14_2, c15_1


----------------------------------------

(7) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
   perfectp(0) -> false
   perfectp(s(z0)) -> f(z0, s(0), s(z0), s(z0))
   f(0, z0, 0, z1) -> true
   f(0, z0, s(z1), z2) -> false
   f(s(z0), 0, z1, z2) -> f(z0, z2, minus(z1, s(z0)), z2)
   f(s(z0), s(z1), z2, z3) -> if(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3))

----------------------------------------

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(0, z0) -> 0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   minus(s(z0), 0) -> s(z0)
Tuples:
   MINUS(s(z0), s(z1)) -> c2(MINUS(z0, z1))
   LE(s(z0), s(z1)) -> c5(LE(z0, z1))
   F(s(z0), 0, z1, z2) -> c12(F(z0, z2, minus(z1, s(z0)), z2), MINUS(z1, s(z0)))
   F(s(z0), s(z1), z2, z3) -> c13(LE(z0, z1))
   F(s(z0), s(z1), z2, z3) -> c14(F(s(z0), minus(z1, z0), z2, z3), MINUS(z1, z0))
   F(s(z0), s(z1), z2, z3) -> c15(F(z0, z3, z2, z3))
S tuples:
   MINUS(s(z0), s(z1)) -> c2(MINUS(z0, z1))
   LE(s(z0), s(z1)) -> c5(LE(z0, z1))
   F(s(z0), 0, z1, z2) -> c12(F(z0, z2, minus(z1, s(z0)), z2), MINUS(z1, s(z0)))
   F(s(z0), s(z1), z2, z3) -> c13(LE(z0, z1))
   F(s(z0), s(z1), z2, z3) -> c14(F(s(z0), minus(z1, z0), z2, z3), MINUS(z1, z0))
   F(s(z0), s(z1), z2, z3) -> c15(F(z0, z3, z2, z3))
K tuples:none
Defined Rule Symbols:   minus_2

Defined Pair Symbols:   MINUS_2, LE_2, F_4

Compound Symbols:   c2_1, c5_1, c12_2, c13_1, c14_2, c15_1


----------------------------------------

(9) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   LE(s(z0), s(z1)) -> c5(LE(z0, z1))
   F(s(z0), 0, z1, z2) -> c12(F(z0, z2, minus(z1, s(z0)), z2), MINUS(z1, s(z0)))
   F(s(z0), s(z1), z2, z3) -> c13(LE(z0, z1))
   F(s(z0), s(z1), z2, z3) -> c15(F(z0, z3, z2, z3))
We considered the (Usable) Rules:none
And the Tuples:
   MINUS(s(z0), s(z1)) -> c2(MINUS(z0, z1))
   LE(s(z0), s(z1)) -> c5(LE(z0, z1))
   F(s(z0), 0, z1, z2) -> c12(F(z0, z2, minus(z1, s(z0)), z2), MINUS(z1, s(z0)))
   F(s(z0), s(z1), z2, z3) -> c13(LE(z0, z1))
   F(s(z0), s(z1), z2, z3) -> c14(F(s(z0), minus(z1, z0), z2, z3), MINUS(z1, z0))
   F(s(z0), s(z1), z2, z3) -> c15(F(z0, z3, z2, z3))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = [1]
   POL(F(x_1, x_2, x_3, x_4)) = [1] + x_1
   POL(LE(x_1, x_2)) = [1] + x_1
   POL(MINUS(x_1, x_2)) = 0
   POL(c12(x_1, x_2)) = x_1 + x_2
   POL(c13(x_1)) = x_1
   POL(c14(x_1, x_2)) = x_1 + x_2
   POL(c15(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(minus(x_1, x_2)) = [1] + x_1 + x_2
   POL(s(x_1)) = [1] + x_1

----------------------------------------

(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(0, z0) -> 0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   minus(s(z0), 0) -> s(z0)
Tuples:
   MINUS(s(z0), s(z1)) -> c2(MINUS(z0, z1))
   LE(s(z0), s(z1)) -> c5(LE(z0, z1))
   F(s(z0), 0, z1, z2) -> c12(F(z0, z2, minus(z1, s(z0)), z2), MINUS(z1, s(z0)))
   F(s(z0), s(z1), z2, z3) -> c13(LE(z0, z1))
   F(s(z0), s(z1), z2, z3) -> c14(F(s(z0), minus(z1, z0), z2, z3), MINUS(z1, z0))
   F(s(z0), s(z1), z2, z3) -> c15(F(z0, z3, z2, z3))
S tuples:
   MINUS(s(z0), s(z1)) -> c2(MINUS(z0, z1))
   F(s(z0), s(z1), z2, z3) -> c14(F(s(z0), minus(z1, z0), z2, z3), MINUS(z1, z0))
K tuples:
   LE(s(z0), s(z1)) -> c5(LE(z0, z1))
   F(s(z0), 0, z1, z2) -> c12(F(z0, z2, minus(z1, s(z0)), z2), MINUS(z1, s(z0)))
   F(s(z0), s(z1), z2, z3) -> c13(LE(z0, z1))
   F(s(z0), s(z1), z2, z3) -> c15(F(z0, z3, z2, z3))
Defined Rule Symbols:   minus_2

Defined Pair Symbols:   MINUS_2, LE_2, F_4

Compound Symbols:   c2_1, c5_1, c12_2, c13_1, c14_2, c15_1


----------------------------------------

(11) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   F(s(z0), s(z1), z2, z3) -> c14(F(s(z0), minus(z1, z0), z2, z3), MINUS(z1, z0))
We considered the (Usable) Rules:
   minus(s(z0), 0) -> s(z0)
   minus(s(z0), s(z1)) -> minus(z0, z1)
   minus(0, z0) -> 0
And the Tuples:
   MINUS(s(z0), s(z1)) -> c2(MINUS(z0, z1))
   LE(s(z0), s(z1)) -> c5(LE(z0, z1))
   F(s(z0), 0, z1, z2) -> c12(F(z0, z2, minus(z1, s(z0)), z2), MINUS(z1, s(z0)))
   F(s(z0), s(z1), z2, z3) -> c13(LE(z0, z1))
   F(s(z0), s(z1), z2, z3) -> c14(F(s(z0), minus(z1, z0), z2, z3), MINUS(z1, z0))
   F(s(z0), s(z1), z2, z3) -> c15(F(z0, z3, z2, z3))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = [2]
   POL(F(x_1, x_2, x_3, x_4)) = [1] + x_2 + x_1*x_4
   POL(LE(x_1, x_2)) = [2] + x_2
   POL(MINUS(x_1, x_2)) = 0
   POL(c12(x_1, x_2)) = x_1 + x_2
   POL(c13(x_1)) = x_1
   POL(c14(x_1, x_2)) = x_1 + x_2
   POL(c15(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(minus(x_1, x_2)) = x_1
   POL(s(x_1)) = [2] + x_1

----------------------------------------

(12)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(0, z0) -> 0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   minus(s(z0), 0) -> s(z0)
Tuples:
   MINUS(s(z0), s(z1)) -> c2(MINUS(z0, z1))
   LE(s(z0), s(z1)) -> c5(LE(z0, z1))
   F(s(z0), 0, z1, z2) -> c12(F(z0, z2, minus(z1, s(z0)), z2), MINUS(z1, s(z0)))
   F(s(z0), s(z1), z2, z3) -> c13(LE(z0, z1))
   F(s(z0), s(z1), z2, z3) -> c14(F(s(z0), minus(z1, z0), z2, z3), MINUS(z1, z0))
   F(s(z0), s(z1), z2, z3) -> c15(F(z0, z3, z2, z3))
S tuples:
   MINUS(s(z0), s(z1)) -> c2(MINUS(z0, z1))
K tuples:
   LE(s(z0), s(z1)) -> c5(LE(z0, z1))
   F(s(z0), 0, z1, z2) -> c12(F(z0, z2, minus(z1, s(z0)), z2), MINUS(z1, s(z0)))
   F(s(z0), s(z1), z2, z3) -> c13(LE(z0, z1))
   F(s(z0), s(z1), z2, z3) -> c15(F(z0, z3, z2, z3))
   F(s(z0), s(z1), z2, z3) -> c14(F(s(z0), minus(z1, z0), z2, z3), MINUS(z1, z0))
Defined Rule Symbols:   minus_2

Defined Pair Symbols:   MINUS_2, LE_2, F_4

Compound Symbols:   c2_1, c5_1, c12_2, c13_1, c14_2, c15_1


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(13) CdtRuleRemovalProof (UPPER BOUND(ADD(n^3)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   MINUS(s(z0), s(z1)) -> c2(MINUS(z0, z1))
We considered the (Usable) Rules:
   minus(s(z0), 0) -> s(z0)
   minus(s(z0), s(z1)) -> minus(z0, z1)
   minus(0, z0) -> 0
And the Tuples:
   MINUS(s(z0), s(z1)) -> c2(MINUS(z0, z1))
   LE(s(z0), s(z1)) -> c5(LE(z0, z1))
   F(s(z0), 0, z1, z2) -> c12(F(z0, z2, minus(z1, s(z0)), z2), MINUS(z1, s(z0)))
   F(s(z0), s(z1), z2, z3) -> c13(LE(z0, z1))
   F(s(z0), s(z1), z2, z3) -> c14(F(s(z0), minus(z1, z0), z2, z3), MINUS(z1, z0))
   F(s(z0), s(z1), z2, z3) -> c15(F(z0, z3, z2, z3))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(F(x_1, x_2, x_3, x_4)) = [1] + x_2 + x_3 + x_2*x_4 + x_1^2 + x_1*x_2 + x_1*x_3 + x_2*x_3 + x_1^2*x_4 + x_1*x_4^2 + x_1*x_3*x_4 + x_1*x_3^2
   POL(LE(x_1, x_2)) = x_1 + x_2 + x_1*x_2 + x_1^2
   POL(MINUS(x_1, x_2)) = x_2
   POL(c12(x_1, x_2)) = x_1 + x_2
   POL(c13(x_1)) = x_1
   POL(c14(x_1, x_2)) = x_1 + x_2
   POL(c15(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(minus(x_1, x_2)) = x_1
   POL(s(x_1)) = [1] + x_1

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(14)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(0, z0) -> 0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   minus(s(z0), 0) -> s(z0)
Tuples:
   MINUS(s(z0), s(z1)) -> c2(MINUS(z0, z1))
   LE(s(z0), s(z1)) -> c5(LE(z0, z1))
   F(s(z0), 0, z1, z2) -> c12(F(z0, z2, minus(z1, s(z0)), z2), MINUS(z1, s(z0)))
   F(s(z0), s(z1), z2, z3) -> c13(LE(z0, z1))
   F(s(z0), s(z1), z2, z3) -> c14(F(s(z0), minus(z1, z0), z2, z3), MINUS(z1, z0))
   F(s(z0), s(z1), z2, z3) -> c15(F(z0, z3, z2, z3))
S tuples:none
K tuples:
   LE(s(z0), s(z1)) -> c5(LE(z0, z1))
   F(s(z0), 0, z1, z2) -> c12(F(z0, z2, minus(z1, s(z0)), z2), MINUS(z1, s(z0)))
   F(s(z0), s(z1), z2, z3) -> c13(LE(z0, z1))
   F(s(z0), s(z1), z2, z3) -> c15(F(z0, z3, z2, z3))
   F(s(z0), s(z1), z2, z3) -> c14(F(s(z0), minus(z1, z0), z2, z3), MINUS(z1, z0))
   MINUS(s(z0), s(z1)) -> c2(MINUS(z0, z1))
Defined Rule Symbols:   minus_2

Defined Pair Symbols:   MINUS_2, LE_2, F_4

Compound Symbols:   c2_1, c5_1, c12_2, c13_1, c14_2, c15_1


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(15) SIsEmptyProof (BOTH BOUNDS(ID, ID))
The set S is empty
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(16)
BOUNDS(1, 1)