WORST_CASE(?,O(n^1))
proof of input_6dgiKFKuj5.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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^1).

(0) CpxTRS
(1) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(2) CdtProblem
(3) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
(4) CdtProblem
(5) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) CdtProblem
(7) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 49 ms]
(8) CdtProblem
(9) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 0 ms]
(10) CdtProblem
(11) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
(12) BOUNDS(1, 1)


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

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


The TRS R consists of the following rules:

   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:
   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:
   PERFECTP(0) -> c
   PERFECTP(s(z0)) -> c1(F(z0, s(0), s(z0), s(z0)))
   F(0, z0, 0, z1) -> c2
   F(0, z0, s(z1), z2) -> c3
   F(s(z0), 0, z1, z2) -> c4(F(z0, z2, minus(z1, s(z0)), z2))
   F(s(z0), s(z1), z2, z3) -> c5(F(s(z0), minus(z1, z0), z2, z3))
   F(s(z0), s(z1), z2, z3) -> c6(F(z0, z3, z2, z3))
S tuples:
   PERFECTP(0) -> c
   PERFECTP(s(z0)) -> c1(F(z0, s(0), s(z0), s(z0)))
   F(0, z0, 0, z1) -> c2
   F(0, z0, s(z1), z2) -> c3
   F(s(z0), 0, z1, z2) -> c4(F(z0, z2, minus(z1, s(z0)), z2))
   F(s(z0), s(z1), z2, z3) -> c5(F(s(z0), minus(z1, z0), z2, z3))
   F(s(z0), s(z1), z2, z3) -> c6(F(z0, z3, z2, z3))
K tuples:none
Defined Rule Symbols:   perfectp_1, f_4

Defined Pair Symbols:   PERFECTP_1, F_4

Compound Symbols:   c, c1_1, c2, c3, c4_1, c5_1, c6_1


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

(3) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   PERFECTP(s(z0)) -> c1(F(z0, s(0), s(z0), s(z0)))
Removed 4 trailing nodes:
   F(0, z0, 0, z1) -> c2
   F(0, z0, s(z1), z2) -> c3
   PERFECTP(0) -> c
   F(s(z0), s(z1), z2, z3) -> c5(F(s(z0), minus(z1, z0), z2, z3))

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

(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   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:
   F(s(z0), 0, z1, z2) -> c4(F(z0, z2, minus(z1, s(z0)), z2))
   F(s(z0), s(z1), z2, z3) -> c6(F(z0, z3, z2, z3))
S tuples:
   F(s(z0), 0, z1, z2) -> c4(F(z0, z2, minus(z1, s(z0)), z2))
   F(s(z0), s(z1), z2, z3) -> c6(F(z0, z3, z2, z3))
K tuples:none
Defined Rule Symbols:   perfectp_1, f_4

Defined Pair Symbols:   F_4

Compound Symbols:   c4_1, c6_1


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

(5) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   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))

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

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   F(s(z0), 0, z1, z2) -> c4(F(z0, z2, minus(z1, s(z0)), z2))
   F(s(z0), s(z1), z2, z3) -> c6(F(z0, z3, z2, z3))
S tuples:
   F(s(z0), 0, z1, z2) -> c4(F(z0, z2, minus(z1, s(z0)), z2))
   F(s(z0), s(z1), z2, z3) -> c6(F(z0, z3, z2, z3))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   F_4

Compound Symbols:   c4_1, c6_1


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

(7) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   F(s(z0), s(z1), z2, z3) -> c6(F(z0, z3, z2, z3))
We considered the (Usable) Rules:none
And the Tuples:
   F(s(z0), 0, z1, z2) -> c4(F(z0, z2, minus(z1, s(z0)), z2))
   F(s(z0), s(z1), z2, z3) -> c6(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)) = x_1 + x_3
   POL(c4(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(minus(x_1, x_2)) = [1]
   POL(s(x_1)) = [1] + x_1

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

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   F(s(z0), 0, z1, z2) -> c4(F(z0, z2, minus(z1, s(z0)), z2))
   F(s(z0), s(z1), z2, z3) -> c6(F(z0, z3, z2, z3))
S tuples:
   F(s(z0), 0, z1, z2) -> c4(F(z0, z2, minus(z1, s(z0)), z2))
K tuples:
   F(s(z0), s(z1), z2, z3) -> c6(F(z0, z3, z2, z3))
Defined Rule Symbols:none

Defined Pair Symbols:   F_4

Compound Symbols:   c4_1, c6_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.
   F(s(z0), 0, z1, z2) -> c4(F(z0, z2, minus(z1, s(z0)), z2))
We considered the (Usable) Rules:none
And the Tuples:
   F(s(z0), 0, z1, z2) -> c4(F(z0, z2, minus(z1, s(z0)), z2))
   F(s(z0), s(z1), z2, z3) -> c6(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)) = x_1 + x_3
   POL(c4(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(minus(x_1, x_2)) = x_1
   POL(s(x_1)) = [1] + x_1

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

(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   F(s(z0), 0, z1, z2) -> c4(F(z0, z2, minus(z1, s(z0)), z2))
   F(s(z0), s(z1), z2, z3) -> c6(F(z0, z3, z2, z3))
S tuples:none
K tuples:
   F(s(z0), s(z1), z2, z3) -> c6(F(z0, z3, z2, z3))
   F(s(z0), 0, z1, z2) -> c4(F(z0, z2, minus(z1, s(z0)), z2))
Defined Rule Symbols:none

Defined Pair Symbols:   F_4

Compound Symbols:   c4_1, c6_1


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

(11) SIsEmptyProof (BOTH BOUNDS(ID, ID))
The set S is empty
----------------------------------------

(12)
BOUNDS(1, 1)