WORST_CASE(?,O(n^1))
proof of input_cbBhDSCqZ6.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) CdtProblem
(7) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
(8) CpxTRS
(9) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
(10) CpxWeightedTrs
(11) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
(12) CpxTypedWeightedTrs
(13) CompletionProof [UPPER BOUND(ID), 0 ms]
(14) CpxTypedWeightedCompleteTrs
(15) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
(16) CpxTypedWeightedCompleteTrs
(17) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
(18) CpxRNTS
(19) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
(20) CpxRNTS
(21) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
(22) CpxRNTS
(23) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(24) CpxRNTS
(25) IntTrsBoundProof [UPPER BOUND(ID), 474 ms]
(26) CpxRNTS
(27) IntTrsBoundProof [UPPER BOUND(ID), 230 ms]
(28) CpxRNTS
(29) FinalProof [FINISHED, 0 ms]
(30) 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:

   div(0, y) -> 0
   div(x, y) -> quot(x, y, y)
   quot(0, s(y), z) -> 0
   quot(s(x), s(y), z) -> quot(x, y, z)
   quot(x, 0, s(z)) -> s(div(x, s(z)))

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:
   div(0, z0) -> 0
   div(z0, z1) -> quot(z0, z1, z1)
   quot(0, s(z0), z1) -> 0
   quot(s(z0), s(z1), z2) -> quot(z0, z1, z2)
   quot(z0, 0, s(z1)) -> s(div(z0, s(z1)))
Tuples:
   DIV(0, z0) -> c
   DIV(z0, z1) -> c1(QUOT(z0, z1, z1))
   QUOT(0, s(z0), z1) -> c2
   QUOT(s(z0), s(z1), z2) -> c3(QUOT(z0, z1, z2))
   QUOT(z0, 0, s(z1)) -> c4(DIV(z0, s(z1)))
S tuples:
   DIV(0, z0) -> c
   DIV(z0, z1) -> c1(QUOT(z0, z1, z1))
   QUOT(0, s(z0), z1) -> c2
   QUOT(s(z0), s(z1), z2) -> c3(QUOT(z0, z1, z2))
   QUOT(z0, 0, s(z1)) -> c4(DIV(z0, s(z1)))
K tuples:none
Defined Rule Symbols:   div_2, quot_3

Defined Pair Symbols:   DIV_2, QUOT_3

Compound Symbols:   c, c1_1, c2, c3_1, c4_1


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(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing nodes:
   DIV(0, z0) -> c
   QUOT(0, s(z0), z1) -> c2

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(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   div(0, z0) -> 0
   div(z0, z1) -> quot(z0, z1, z1)
   quot(0, s(z0), z1) -> 0
   quot(s(z0), s(z1), z2) -> quot(z0, z1, z2)
   quot(z0, 0, s(z1)) -> s(div(z0, s(z1)))
Tuples:
   DIV(z0, z1) -> c1(QUOT(z0, z1, z1))
   QUOT(s(z0), s(z1), z2) -> c3(QUOT(z0, z1, z2))
   QUOT(z0, 0, s(z1)) -> c4(DIV(z0, s(z1)))
S tuples:
   DIV(z0, z1) -> c1(QUOT(z0, z1, z1))
   QUOT(s(z0), s(z1), z2) -> c3(QUOT(z0, z1, z2))
   QUOT(z0, 0, s(z1)) -> c4(DIV(z0, s(z1)))
K tuples:none
Defined Rule Symbols:   div_2, quot_3

Defined Pair Symbols:   DIV_2, QUOT_3

Compound Symbols:   c1_1, c3_1, c4_1


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(5) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   div(0, z0) -> 0
   div(z0, z1) -> quot(z0, z1, z1)
   quot(0, s(z0), z1) -> 0
   quot(s(z0), s(z1), z2) -> quot(z0, z1, z2)
   quot(z0, 0, s(z1)) -> s(div(z0, s(z1)))

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(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   DIV(z0, z1) -> c1(QUOT(z0, z1, z1))
   QUOT(s(z0), s(z1), z2) -> c3(QUOT(z0, z1, z2))
   QUOT(z0, 0, s(z1)) -> c4(DIV(z0, s(z1)))
S tuples:
   DIV(z0, z1) -> c1(QUOT(z0, z1, z1))
   QUOT(s(z0), s(z1), z2) -> c3(QUOT(z0, z1, z2))
   QUOT(z0, 0, s(z1)) -> c4(DIV(z0, s(z1)))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   DIV_2, QUOT_3

Compound Symbols:   c1_1, c3_1, c4_1


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(7) CdtToCpxRelTrsProof (BOTH BOUNDS(ID, ID))
Converted S to standard rules, and D \ S as well as R to relative rules.
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(8)
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:

   DIV(z0, z1) -> c1(QUOT(z0, z1, z1))
   QUOT(s(z0), s(z1), z2) -> c3(QUOT(z0, z1, z2))
   QUOT(z0, 0, s(z1)) -> c4(DIV(z0, s(z1)))

S is empty.
Rewrite Strategy: INNERMOST
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(9) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
Transformed relative TRS to weighted TRS
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(10)
Obligation:
The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   DIV(z0, z1) -> c1(QUOT(z0, z1, z1)) [1]
   QUOT(s(z0), s(z1), z2) -> c3(QUOT(z0, z1, z2)) [1]
   QUOT(z0, 0, s(z1)) -> c4(DIV(z0, s(z1))) [1]

Rewrite Strategy: INNERMOST
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(11) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
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(12)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   DIV(z0, z1) -> c1(QUOT(z0, z1, z1)) [1]
   QUOT(s(z0), s(z1), z2) -> c3(QUOT(z0, z1, z2)) [1]
   QUOT(z0, 0, s(z1)) -> c4(DIV(z0, s(z1))) [1]

The TRS has the following type information:
DIV :: s:0 -> s:0 -> c1
c1 :: c3:c4 -> c1
QUOT :: s:0 -> s:0 -> s:0 -> c3:c4
s :: s:0 -> s:0
c3 :: c3:c4 -> c3:c4
0 :: s:0
c4 :: c1 -> c3:c4

Rewrite Strategy: INNERMOST
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(13) CompletionProof (UPPER BOUND(ID))
The transformation into a RNTS is sound, since:

(a) The obligation is a constructor system where every type has a constant constructor,

(b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols:

   DIV_2
   QUOT_3

(c) The following functions are completely defined:
none

Due to the following rules being added:
none

And the following fresh constants:    const, const1

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(14)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   DIV(z0, z1) -> c1(QUOT(z0, z1, z1)) [1]
   QUOT(s(z0), s(z1), z2) -> c3(QUOT(z0, z1, z2)) [1]
   QUOT(z0, 0, s(z1)) -> c4(DIV(z0, s(z1))) [1]

The TRS has the following type information:
DIV :: s:0 -> s:0 -> c1
c1 :: c3:c4 -> c1
QUOT :: s:0 -> s:0 -> s:0 -> c3:c4
s :: s:0 -> s:0
c3 :: c3:c4 -> c3:c4
0 :: s:0
c4 :: c1 -> c3:c4
const :: c1
const1 :: c3:c4

Rewrite Strategy: INNERMOST
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(15) NarrowingProof (BOTH BOUNDS(ID, ID))
Narrowed the inner basic terms of all right-hand sides by a single narrowing step.
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(16)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   DIV(z0, z1) -> c1(QUOT(z0, z1, z1)) [1]
   QUOT(s(z0), s(z1), z2) -> c3(QUOT(z0, z1, z2)) [1]
   QUOT(z0, 0, s(z1)) -> c4(DIV(z0, s(z1))) [1]

The TRS has the following type information:
DIV :: s:0 -> s:0 -> c1
c1 :: c3:c4 -> c1
QUOT :: s:0 -> s:0 -> s:0 -> c3:c4
s :: s:0 -> s:0
c3 :: c3:c4 -> c3:c4
0 :: s:0
c4 :: c1 -> c3:c4
const :: c1
const1 :: c3:c4

Rewrite Strategy: INNERMOST
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(17) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
The constant constructors are abstracted as follows:

   0 => 0
   const => 0
   const1 => 0

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(18)
Obligation:
Complexity RNTS consisting of the following rules:

   DIV(z, z') -{ 1 }-> 1 + QUOT(z0, z1, z1) :|: z = z0, z1 >= 0, z' = z1, z0 >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + QUOT(z0, z1, z2) :|: z'' = z2, z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1, z2 >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + DIV(z0, 1 + z1) :|: z = z0, z1 >= 0, z0 >= 0, z'' = 1 + z1, z' = 0


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(19) SimplificationProof (BOTH BOUNDS(ID, ID))
Simplified the RNTS by moving equalities from the constraints into the right-hand sides.
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(20)
Obligation:
Complexity RNTS consisting of the following rules:

   DIV(z, z') -{ 1 }-> 1 + QUOT(z, z', z') :|: z' >= 0, z >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + QUOT(z - 1, z' - 1, z'') :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + DIV(z, 1 + (z'' - 1)) :|: z'' - 1 >= 0, z >= 0, z' = 0


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(21) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID))
Found the following analysis order by SCC decomposition:

   { QUOT, DIV }

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(22)
Obligation:
Complexity RNTS consisting of the following rules:

   DIV(z, z') -{ 1 }-> 1 + QUOT(z, z', z') :|: z' >= 0, z >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + QUOT(z - 1, z' - 1, z'') :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + DIV(z, 1 + (z'' - 1)) :|: z'' - 1 >= 0, z >= 0, z' = 0

Function symbols to be analyzed: {QUOT,DIV}

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(23) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
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(24)
Obligation:
Complexity RNTS consisting of the following rules:

   DIV(z, z') -{ 1 }-> 1 + QUOT(z, z', z') :|: z' >= 0, z >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + QUOT(z - 1, z' - 1, z'') :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + DIV(z, 1 + (z'' - 1)) :|: z'' - 1 >= 0, z >= 0, z' = 0

Function symbols to be analyzed: {QUOT,DIV}

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(25) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: QUOT
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

Computed SIZE bound using CoFloCo for: DIV
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 1

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(26)
Obligation:
Complexity RNTS consisting of the following rules:

   DIV(z, z') -{ 1 }-> 1 + QUOT(z, z', z') :|: z' >= 0, z >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + QUOT(z - 1, z' - 1, z'') :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + DIV(z, 1 + (z'' - 1)) :|: z'' - 1 >= 0, z >= 0, z' = 0

Function symbols to be analyzed: {QUOT,DIV}
Previous analysis results are:
  QUOT: runtime: ?, size: O(1) [0]
  DIV: runtime: ?, size: O(1) [1]

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(27) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: QUOT
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 2 + 3*z + 2*z'

Computed RUNTIME bound using CoFloCo for: DIV
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 3 + 3*z + 2*z'

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(28)
Obligation:
Complexity RNTS consisting of the following rules:

   DIV(z, z') -{ 1 }-> 1 + QUOT(z, z', z') :|: z' >= 0, z >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + QUOT(z - 1, z' - 1, z'') :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0
   QUOT(z, z', z'') -{ 1 }-> 1 + DIV(z, 1 + (z'' - 1)) :|: z'' - 1 >= 0, z >= 0, z' = 0

Function symbols to be analyzed: 
Previous analysis results are:
  QUOT: runtime: O(n^1) [2 + 3*z + 2*z'], size: O(1) [0]
  DIV: runtime: O(n^1) [3 + 3*z + 2*z'], size: O(1) [1]

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(29) FinalProof (FINISHED)
Computed overall runtime complexity
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(30)
BOUNDS(1, n^1)