WORST_CASE(?,O(n^1))
proof of input_wm35vJnyWU.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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 [BOTH BOUNDS(ID, ID), 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) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
(10) CpxRelTRS
(11) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(12) CpxTRS
(13) CpxTrsMatchBoundsTAProof [FINISHED, 3 ms]
(14) BOUNDS(1, n^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:

   -(0, y) -> 0
   -(x, 0) -> x
   -(x, s(y)) -> if(greater(x, s(y)), s(-(x, p(s(y)))), 0)
   p(0) -> 0
   p(s(x)) -> x

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:
   -(0, z0) -> 0
   -(z0, 0) -> z0
   -(z0, s(z1)) -> if(greater(z0, s(z1)), s(-(z0, p(s(z1)))), 0)
   p(0) -> 0
   p(s(z0)) -> z0
Tuples:
   -'(0, z0) -> c
   -'(z0, 0) -> c1
   -'(z0, s(z1)) -> c2(-'(z0, p(s(z1))), P(s(z1)))
   P(0) -> c3
   P(s(z0)) -> c4
S tuples:
   -'(0, z0) -> c
   -'(z0, 0) -> c1
   -'(z0, s(z1)) -> c2(-'(z0, p(s(z1))), P(s(z1)))
   P(0) -> c3
   P(s(z0)) -> c4
K tuples:none
Defined Rule Symbols:   -_2, p_1

Defined Pair Symbols:   -'_2, P_1

Compound Symbols:   c, c1, c2_2, c3, c4


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

(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 4 trailing nodes:
   P(0) -> c3
   -'(0, z0) -> c
   P(s(z0)) -> c4
   -'(z0, 0) -> c1

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

(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(0, z0) -> 0
   -(z0, 0) -> z0
   -(z0, s(z1)) -> if(greater(z0, s(z1)), s(-(z0, p(s(z1)))), 0)
   p(0) -> 0
   p(s(z0)) -> z0
Tuples:
   -'(z0, s(z1)) -> c2(-'(z0, p(s(z1))), P(s(z1)))
S tuples:
   -'(z0, s(z1)) -> c2(-'(z0, p(s(z1))), P(s(z1)))
K tuples:none
Defined Rule Symbols:   -_2, p_1

Defined Pair Symbols:   -'_2

Compound Symbols:   c2_2


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

(5) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing tuple parts
----------------------------------------

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(0, z0) -> 0
   -(z0, 0) -> z0
   -(z0, s(z1)) -> if(greater(z0, s(z1)), s(-(z0, p(s(z1)))), 0)
   p(0) -> 0
   p(s(z0)) -> z0
Tuples:
   -'(z0, s(z1)) -> c2(-'(z0, p(s(z1))))
S tuples:
   -'(z0, s(z1)) -> c2(-'(z0, p(s(z1))))
K tuples:none
Defined Rule Symbols:   -_2, p_1

Defined Pair Symbols:   -'_2

Compound Symbols:   c2_1


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

(7) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   -(0, z0) -> 0
   -(z0, 0) -> z0
   -(z0, s(z1)) -> if(greater(z0, s(z1)), s(-(z0, p(s(z1)))), 0)
   p(0) -> 0

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

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(s(z0)) -> z0
Tuples:
   -'(z0, s(z1)) -> c2(-'(z0, p(s(z1))))
S tuples:
   -'(z0, s(z1)) -> c2(-'(z0, p(s(z1))))
K tuples:none
Defined Rule Symbols:   p_1

Defined Pair Symbols:   -'_2

Compound Symbols:   c2_1


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

(9) CdtToCpxRelTrsProof (BOTH BOUNDS(ID, ID))
Converted S to standard rules, and D \ S as well as R to relative rules.
----------------------------------------

(10)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   -'(z0, s(z1)) -> c2(-'(z0, p(s(z1))))

The (relative) TRS S consists of the following rules:

   p(s(z0)) -> z0

Rewrite Strategy: INNERMOST
----------------------------------------

(11) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
----------------------------------------

(12)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   -'(z0, s(z1)) -> c2(-'(z0, p(s(z1))))
   p(s(z0)) -> z0

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

(13) CpxTrsMatchBoundsTAProof (FINISHED)
A linear upper bound on the runtime complexity of the TRS R could be shown with a Match-Bound[TAB_LEFTLINEAR,TAB_NONLEFTLINEAR] (for contructor-based start-terms) of 1. 

The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
final states : [1, 2]
transitions: 
s0(0) -> 0
c20(0) -> 0
-'0(0, 0) -> 1
p0(0) -> 2
s1(0) -> 5
p1(5) -> 4
-'1(0, 4) -> 3
c21(3) -> 1
c21(3) -> 3
0 -> 2
0 -> 4

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

(14)
BOUNDS(1, n^1)