WORST_CASE(?,O(n^1))
proof of input_lpFhQiZbA8.trs
# AProVE Commit ID: 5b976082cb74a395683ed8cc7acf94bd611ab29f fuhs 20230524 unpublished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) CpxTRS
(11) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(12) CpxTRS
(13) CpxTrsMatchBoundsTAProof [FINISHED, 0 ms]
(14) BOUNDS(1, n^1)


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(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:

   f(0) -> true
   f(1) -> false
   f(s(x)) -> f(x)
   if(true, s(x), s(y)) -> s(x)
   if(false, s(x), s(y)) -> s(y)
   g(x, c(y)) -> c(g(x, y))
   g(x, c(y)) -> g(x, if(f(x), c(g(s(x), y)), c(y)))

S is empty.
Rewrite Strategy: PARALLEL_INNERMOST
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(1) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
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(2)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(0) -> true
   f(1) -> false
   f(s(z0)) -> f(z0)
   if(true, s(z0), s(z1)) -> s(z0)
   if(false, s(z0), s(z1)) -> s(z1)
   g(z0, c(z1)) -> c(g(z0, z1))
   g(z0, c(z1)) -> g(z0, if(f(z0), c(g(s(z0), z1)), c(z1)))
Tuples:
   F(0) -> c1
   F(1) -> c2
   F(s(z0)) -> c3(F(z0))
   IF(true, s(z0), s(z1)) -> c4
   IF(false, s(z0), s(z1)) -> c5
   G(z0, c(z1)) -> c6(G(z0, z1))
   G(z0, c(z1)) -> c7(G(z0, if(f(z0), c(g(s(z0), z1)), c(z1))), IF(f(z0), c(g(s(z0), z1)), c(z1)), F(z0))
   G(z0, c(z1)) -> c8(G(z0, if(f(z0), c(g(s(z0), z1)), c(z1))), IF(f(z0), c(g(s(z0), z1)), c(z1)), G(s(z0), z1))
S tuples:
   F(0) -> c1
   F(1) -> c2
   F(s(z0)) -> c3(F(z0))
   IF(true, s(z0), s(z1)) -> c4
   IF(false, s(z0), s(z1)) -> c5
   G(z0, c(z1)) -> c6(G(z0, z1))
   G(z0, c(z1)) -> c7(G(z0, if(f(z0), c(g(s(z0), z1)), c(z1))), IF(f(z0), c(g(s(z0), z1)), c(z1)), F(z0))
   G(z0, c(z1)) -> c8(G(z0, if(f(z0), c(g(s(z0), z1)), c(z1))), IF(f(z0), c(g(s(z0), z1)), c(z1)), G(s(z0), z1))
K tuples:none
Defined Rule Symbols:   f_1, if_3, g_2

Defined Pair Symbols:   F_1, IF_3, G_2

Compound Symbols:   c1, c2, c3_1, c4, c5, c6_1, c7_3, c8_3


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(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 4 trailing nodes:
   IF(false, s(z0), s(z1)) -> c5
   F(1) -> c2
   F(0) -> c1
   IF(true, s(z0), s(z1)) -> c4

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(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(0) -> true
   f(1) -> false
   f(s(z0)) -> f(z0)
   if(true, s(z0), s(z1)) -> s(z0)
   if(false, s(z0), s(z1)) -> s(z1)
   g(z0, c(z1)) -> c(g(z0, z1))
   g(z0, c(z1)) -> g(z0, if(f(z0), c(g(s(z0), z1)), c(z1)))
Tuples:
   F(s(z0)) -> c3(F(z0))
   G(z0, c(z1)) -> c6(G(z0, z1))
   G(z0, c(z1)) -> c7(G(z0, if(f(z0), c(g(s(z0), z1)), c(z1))), IF(f(z0), c(g(s(z0), z1)), c(z1)), F(z0))
   G(z0, c(z1)) -> c8(G(z0, if(f(z0), c(g(s(z0), z1)), c(z1))), IF(f(z0), c(g(s(z0), z1)), c(z1)), G(s(z0), z1))
S tuples:
   F(s(z0)) -> c3(F(z0))
   G(z0, c(z1)) -> c6(G(z0, z1))
   G(z0, c(z1)) -> c7(G(z0, if(f(z0), c(g(s(z0), z1)), c(z1))), IF(f(z0), c(g(s(z0), z1)), c(z1)), F(z0))
   G(z0, c(z1)) -> c8(G(z0, if(f(z0), c(g(s(z0), z1)), c(z1))), IF(f(z0), c(g(s(z0), z1)), c(z1)), G(s(z0), z1))
K tuples:none
Defined Rule Symbols:   f_1, if_3, g_2

Defined Pair Symbols:   F_1, G_2

Compound Symbols:   c3_1, c6_1, c7_3, c8_3


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(5) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 4 trailing tuple parts
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(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(0) -> true
   f(1) -> false
   f(s(z0)) -> f(z0)
   if(true, s(z0), s(z1)) -> s(z0)
   if(false, s(z0), s(z1)) -> s(z1)
   g(z0, c(z1)) -> c(g(z0, z1))
   g(z0, c(z1)) -> g(z0, if(f(z0), c(g(s(z0), z1)), c(z1)))
Tuples:
   F(s(z0)) -> c3(F(z0))
   G(z0, c(z1)) -> c6(G(z0, z1))
   G(z0, c(z1)) -> c7(F(z0))
   G(z0, c(z1)) -> c8(G(s(z0), z1))
S tuples:
   F(s(z0)) -> c3(F(z0))
   G(z0, c(z1)) -> c6(G(z0, z1))
   G(z0, c(z1)) -> c7(F(z0))
   G(z0, c(z1)) -> c8(G(s(z0), z1))
K tuples:none
Defined Rule Symbols:   f_1, if_3, g_2

Defined Pair Symbols:   F_1, G_2

Compound Symbols:   c3_1, c6_1, c7_1, c8_1


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(7) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   f(0) -> true
   f(1) -> false
   f(s(z0)) -> f(z0)
   if(true, s(z0), s(z1)) -> s(z0)
   if(false, s(z0), s(z1)) -> s(z1)
   g(z0, c(z1)) -> c(g(z0, z1))
   g(z0, c(z1)) -> g(z0, if(f(z0), c(g(s(z0), z1)), c(z1)))

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(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   F(s(z0)) -> c3(F(z0))
   G(z0, c(z1)) -> c6(G(z0, z1))
   G(z0, c(z1)) -> c7(F(z0))
   G(z0, c(z1)) -> c8(G(s(z0), z1))
S tuples:
   F(s(z0)) -> c3(F(z0))
   G(z0, c(z1)) -> c6(G(z0, z1))
   G(z0, c(z1)) -> c7(F(z0))
   G(z0, c(z1)) -> c8(G(s(z0), z1))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   F_1, G_2

Compound Symbols:   c3_1, c6_1, c7_1, c8_1


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(9) CdtToCpxRelTrsProof (BOTH BOUNDS(ID, ID))
Converted S to standard rules, and D \ S as well as R to relative rules.
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(10)
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:

   F(s(z0)) -> c3(F(z0))
   G(z0, c(z1)) -> c6(G(z0, z1))
   G(z0, c(z1)) -> c7(F(z0))
   G(z0, c(z1)) -> c8(G(s(z0), z1))

S is empty.
Rewrite Strategy: INNERMOST
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(11) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
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(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:

   F(s(z0)) -> c3(F(z0))
   G(z0, c(z1)) -> c6(G(z0, z1))
   G(z0, c(z1)) -> c7(F(z0))
   G(z0, c(z1)) -> c8(G(s(z0), z1))

S is empty.
Rewrite Strategy: INNERMOST
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(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 2. 

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
c30(0) -> 0
c0(0) -> 0
c60(0) -> 0
c70(0) -> 0
c80(0) -> 0
F0(0) -> 1
G0(0, 0) -> 2
F1(0) -> 3
c31(3) -> 1
G1(0, 0) -> 4
c61(4) -> 2
F1(0) -> 5
c71(5) -> 2
s1(0) -> 7
G1(7, 0) -> 6
c81(6) -> 2
c31(3) -> 3
c31(3) -> 5
c61(4) -> 4
G1(7, 0) -> 4
c61(4) -> 6
c71(5) -> 4
F1(7) -> 5
c71(5) -> 6
c81(6) -> 4
s1(7) -> 7
c81(6) -> 6
F2(0) -> 8
c32(8) -> 5
F2(7) -> 8
c31(3) -> 8
c32(8) -> 8

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(14)
BOUNDS(1, n^1)