WORST_CASE(?,O(n^1))
proof of input_MOfsysFWEw.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 47 ms]
(10) CdtProblem
(11) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
(12) BOUNDS(1, 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:
   F(1) -> c2
   IF(false, s(z0), s(z1)) -> c5
   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) 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)) -> 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))
We considered the (Usable) Rules:none
And the 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))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(F(x_1)) = [2] + [2]x_1
   POL(G(x_1, x_2)) = [1] + [2]x_1 + [3]x_2
   POL(c(x_1)) = [3] + x_1
   POL(c3(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(c8(x_1)) = x_1
   POL(s(x_1)) = [1] + x_1

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(10)
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:none
K 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))
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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(11) SIsEmptyProof (BOTH BOUNDS(ID, ID))
The set S is empty
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(12)
BOUNDS(1, 1)