WORST_CASE(?,O(n^1))
proof of input_rjiIYPb3ZA.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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) 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), 273 ms]
(26) CpxRNTS
(27) IntTrsBoundProof [UPPER BOUND(ID), 37 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:

   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
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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:
   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


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(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, s(z1), z2) -> c3
   F(s(z0), s(z1), z2, z3) -> c5(F(s(z0), minus(z1, z0), z2, z3))
   PERFECTP(0) -> c
   F(0, z0, 0, z1) -> c2

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(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


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(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))

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(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


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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:

   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 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:

   F(s(z0), 0, z1, z2) -> c4(F(z0, z2, minus(z1, s(z0)), z2)) [1]
   F(s(z0), s(z1), z2, z3) -> c6(F(z0, z3, z2, z3)) [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:

   F(s(z0), 0, z1, z2) -> c4(F(z0, z2, minus(z1, s(z0)), z2)) [1]
   F(s(z0), s(z1), z2, z3) -> c6(F(z0, z3, z2, z3)) [1]

The TRS has the following type information:
F :: s:0 -> s:0 -> minus -> s:0 -> c4:c6
s :: s:0 -> s:0
0 :: s:0
c4 :: c4:c6 -> c4:c6
minus :: minus -> s:0 -> minus
c6 :: c4:c6 -> c4:c6

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:

   F_4

(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:

   F(s(z0), 0, z1, z2) -> c4(F(z0, z2, minus(z1, s(z0)), z2)) [1]
   F(s(z0), s(z1), z2, z3) -> c6(F(z0, z3, z2, z3)) [1]

The TRS has the following type information:
F :: s:0 -> s:0 -> minus -> s:0 -> c4:c6
s :: s:0 -> s:0
0 :: s:0
c4 :: c4:c6 -> c4:c6
minus :: minus -> s:0 -> minus
c6 :: c4:c6 -> c4:c6
const :: c4:c6
const1 :: minus

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:

   F(s(z0), 0, z1, z2) -> c4(F(z0, z2, minus(z1, s(z0)), z2)) [1]
   F(s(z0), s(z1), z2, z3) -> c6(F(z0, z3, z2, z3)) [1]

The TRS has the following type information:
F :: s:0 -> s:0 -> minus -> s:0 -> c4:c6
s :: s:0 -> s:0
0 :: s:0
c4 :: c4:c6 -> c4:c6
minus :: minus -> s:0 -> minus
c6 :: c4:c6 -> c4:c6
const :: c4:c6
const1 :: minus

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:

   F(z, z', z'', z4) -{ 1 }-> 1 + F(z0, z2, 1 + z1 + (1 + z0), z2) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z2 >= 0, z' = 0, z'' = z1, z4 = z2
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z0, z3, z2, z3) :|: z'' = z2, z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1, z4 = z3, z2 >= 0, z3 >= 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:

   F(z, z', z'', z4) -{ 1 }-> 1 + F(z - 1, z4, z'', z4) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z - 1, z4, 1 + z'' + (1 + (z - 1)), z4) :|: z'' >= 0, z - 1 >= 0, z4 >= 0, z' = 0


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

   { F }

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

   F(z, z', z'', z4) -{ 1 }-> 1 + F(z - 1, z4, z'', z4) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z - 1, z4, 1 + z'' + (1 + (z - 1)), z4) :|: z'' >= 0, z - 1 >= 0, z4 >= 0, z' = 0

Function symbols to be analyzed: {F}

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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:

   F(z, z', z'', z4) -{ 1 }-> 1 + F(z - 1, z4, z'', z4) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z - 1, z4, 1 + z'' + (1 + (z - 1)), z4) :|: z'' >= 0, z - 1 >= 0, z4 >= 0, z' = 0

Function symbols to be analyzed: {F}

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(25) IntTrsBoundProof (UPPER BOUND(ID))

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

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

   F(z, z', z'', z4) -{ 1 }-> 1 + F(z - 1, z4, z'', z4) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z - 1, z4, 1 + z'' + (1 + (z - 1)), z4) :|: z'' >= 0, z - 1 >= 0, z4 >= 0, z' = 0

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

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(27) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: F
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: z

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

   F(z, z', z'', z4) -{ 1 }-> 1 + F(z - 1, z4, z'', z4) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0, z4 >= 0
   F(z, z', z'', z4) -{ 1 }-> 1 + F(z - 1, z4, 1 + z'' + (1 + (z - 1)), z4) :|: z'' >= 0, z - 1 >= 0, z4 >= 0, z' = 0

Function symbols to be analyzed: 
Previous analysis results are:
  F: runtime: O(n^1) [z], size: O(1) [0]

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