The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^3).

(0) Obligation:

The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^3).


The TRS R consists of the following rules:

le(0, Y) → true
le(s(X), 0) → false
le(s(X), s(Y)) → le(X, Y)
minus(0, Y) → 0
minus(s(X), Y) → ifMinus(le(s(X), Y), s(X), Y)
ifMinus(true, s(X), Y) → 0
ifMinus(false, s(X), Y) → s(minus(X, Y))
quot(0, s(Y)) → 0
quot(s(X), s(Y)) → s(quot(minus(X, Y), s(Y)))

Rewrite Strategy: INNERMOST

(1) TrsToWeightedTrsProof (BOTH BOUNDS(ID, ID) transformation)

Transformed TRS to weighted TRS

(2) Obligation:

The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^3).


The TRS R consists of the following rules:

le(0, Y) → true [1]
le(s(X), 0) → false [1]
le(s(X), s(Y)) → le(X, Y) [1]
minus(0, Y) → 0 [1]
minus(s(X), Y) → ifMinus(le(s(X), Y), s(X), Y) [1]
ifMinus(true, s(X), Y) → 0 [1]
ifMinus(false, s(X), Y) → s(minus(X, Y)) [1]
quot(0, s(Y)) → 0 [1]
quot(s(X), s(Y)) → s(quot(minus(X, Y), s(Y))) [1]

Rewrite Strategy: INNERMOST

(3) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)

Infered types.

(4) Obligation:

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

le(0, Y) → true [1]
le(s(X), 0) → false [1]
le(s(X), s(Y)) → le(X, Y) [1]
minus(0, Y) → 0 [1]
minus(s(X), Y) → ifMinus(le(s(X), Y), s(X), Y) [1]
ifMinus(true, s(X), Y) → 0 [1]
ifMinus(false, s(X), Y) → s(minus(X, Y)) [1]
quot(0, s(Y)) → 0 [1]
quot(s(X), s(Y)) → s(quot(minus(X, Y), s(Y))) [1]

The TRS has the following type information:

le :: 0:s → 0:s → true:false 0 :: 0:s true :: true:false s :: 0:s → 0:s false :: true:false minus :: 0:s → 0:s → 0:s ifMinus :: true:false → 0:s → 0:s → 0:s quot :: 0:s → 0:s → 0:s

Rewrite Strategy: INNERMOST

(5) CompletionProof (UPPER BOUND(ID) transformation)

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:

quot

(c) The following functions are completely defined:

minus
le
ifMinus

Due to the following rules being added:

ifMinus(v0, v1, v2) → 0 [0]

And the following fresh constants: none

(6) 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: le(0, Y) → true [1] le(s(X), 0) → false [1] le(s(X), s(Y)) → le(X, Y) [1] minus(0, Y) → 0 [1] minus(s(X), Y) → ifMinus(le(s(X), Y), s(X), Y) [1] ifMinus(true, s(X), Y) → 0 [1] ifMinus(false, s(X), Y) → s(minus(X, Y)) [1] quot(0, s(Y)) → 0 [1] quot(s(X), s(Y)) → s(quot(minus(X, Y), s(Y))) [1] ifMinus(v0, v1, v2) → 0 [0] The TRS has the following type information: le :: 0:s → 0:s → true:false 0 :: 0:s true :: true:false s :: 0:s → 0:s false :: true:false minus :: 0:s → 0:s → 0:s ifMinus :: true:false → 0:s → 0:s → 0:s quot :: 0:s → 0:s → 0:s Rewrite Strategy: INNERMOST

(7) NarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Narrowed the inner basic terms of all right-hand sides by a single narrowing step.

(8) 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: le(0, Y) → true [1] le(s(X), 0) → false [1] le(s(X), s(Y)) → le(X, Y) [1] minus(0, Y) → 0 [1] minus(s(X), 0) → ifMinus(false, s(X), 0) [2] minus(s(X), s(Y')) → ifMinus(le(X, Y'), s(X), s(Y')) [2] ifMinus(true, s(X), Y) → 0 [1] ifMinus(false, s(X), Y) → s(minus(X, Y)) [1] quot(0, s(Y)) → 0 [1] quot(s(0), s(Y)) → s(quot(0, s(Y))) [2] quot(s(s(X')), s(Y)) → s(quot(ifMinus(le(s(X'), Y), s(X'), Y), s(Y))) [2] ifMinus(v0, v1, v2) → 0 [0] The TRS has the following type information: le :: 0:s → 0:s → true:false 0 :: 0:s true :: true:false s :: 0:s → 0:s false :: true:false minus :: 0:s → 0:s → 0:s ifMinus :: true:false → 0:s → 0:s → 0:s quot :: 0:s → 0:s → 0:s Rewrite Strategy: INNERMOST

(9) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID) transformation)

Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
The constant constructors are abstracted as follows:

0 => 0
true => 1
false => 0

(10) Obligation:

Complexity RNTS consisting of the following rules:

ifMinus(z, z', z'') -{ 1 }→ 0 :|: Y >= 0, z = 1, z'' = Y, z' = 1 + X, X >= 0
ifMinus(z, z', z'') -{ 0 }→ 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
ifMinus(z, z', z'') -{ 1 }→ 1 + minus(X, Y) :|: Y >= 0, z'' = Y, z' = 1 + X, X >= 0, z = 0
le(z, z') -{ 1 }→ le(X, Y) :|: z = 1 + X, Y >= 0, z' = 1 + Y, X >= 0
le(z, z') -{ 1 }→ 1 :|: z' = Y, Y >= 0, z = 0
le(z, z') -{ 1 }→ 0 :|: z = 1 + X, X >= 0, z' = 0
minus(z, z') -{ 2 }→ ifMinus(le(X, Y'), 1 + X, 1 + Y') :|: z = 1 + X, Y' >= 0, X >= 0, z' = 1 + Y'
minus(z, z') -{ 2 }→ ifMinus(0, 1 + X, 0) :|: z = 1 + X, X >= 0, z' = 0
minus(z, z') -{ 1 }→ 0 :|: z' = Y, Y >= 0, z = 0
quot(z, z') -{ 1 }→ 0 :|: Y >= 0, z' = 1 + Y, z = 0
quot(z, z') -{ 2 }→ 1 + quot(ifMinus(le(1 + X', Y), 1 + X', Y), 1 + Y) :|: Y >= 0, z' = 1 + Y, X' >= 0, z = 1 + (1 + X')
quot(z, z') -{ 2 }→ 1 + quot(0, 1 + Y) :|: Y >= 0, z = 1 + 0, z' = 1 + Y

(11) SimplificationProof (BOTH BOUNDS(ID, ID) transformation)

Simplified the RNTS by moving equalities from the constraints into the right-hand sides.

(12) Obligation:

Complexity RNTS consisting of the following rules:

ifMinus(z, z', z'') -{ 1 }→ 0 :|: z'' >= 0, z = 1, z' - 1 >= 0
ifMinus(z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
ifMinus(z, z', z'') -{ 1 }→ 1 + minus(z' - 1, z'') :|: z'' >= 0, z' - 1 >= 0, z = 0
le(z, z') -{ 1 }→ le(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
le(z, z') -{ 1 }→ 1 :|: z' >= 0, z = 0
le(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0
minus(z, z') -{ 2 }→ ifMinus(le(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
minus(z, z') -{ 2 }→ ifMinus(0, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
minus(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
quot(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0
quot(z, z') -{ 2 }→ 1 + quot(ifMinus(le(1 + (z - 2), z' - 1), 1 + (z - 2), z' - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 2 >= 0
quot(z, z') -{ 2 }→ 1 + quot(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 1 + 0

(13) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID) transformation)

Found the following analysis order by SCC decomposition:

{ le } { ifMinus, minus } { quot }

(14) Obligation:

Complexity RNTS consisting of the following rules:

ifMinus(z, z', z'') -{ 1 }→ 0 :|: z'' >= 0, z = 1, z' - 1 >= 0
ifMinus(z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
ifMinus(z, z', z'') -{ 1 }→ 1 + minus(z' - 1, z'') :|: z'' >= 0, z' - 1 >= 0, z = 0
le(z, z') -{ 1 }→ le(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
le(z, z') -{ 1 }→ 1 :|: z' >= 0, z = 0
le(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0
minus(z, z') -{ 2 }→ ifMinus(le(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
minus(z, z') -{ 2 }→ ifMinus(0, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
minus(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
quot(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0
quot(z, z') -{ 2 }→ 1 + quot(ifMinus(le(1 + (z - 2), z' - 1), 1 + (z - 2), z' - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 2 >= 0
quot(z, z') -{ 2 }→ 1 + quot(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 1 + 0

Function symbols to be analyzed: {le}, {ifMinus,minus}, {quot}

(15) IntTrsBoundProof (UPPER BOUND(ID) transformation)

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

(16) Obligation:

Complexity RNTS consisting of the following rules:

ifMinus(z, z', z'') -{ 1 }→ 0 :|: z'' >= 0, z = 1, z' - 1 >= 0
ifMinus(z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
ifMinus(z, z', z'') -{ 1 }→ 1 + minus(z' - 1, z'') :|: z'' >= 0, z' - 1 >= 0, z = 0
le(z, z') -{ 1 }→ le(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
le(z, z') -{ 1 }→ 1 :|: z' >= 0, z = 0
le(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0
minus(z, z') -{ 2 }→ ifMinus(le(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
minus(z, z') -{ 2 }→ ifMinus(0, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
minus(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
quot(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0
quot(z, z') -{ 2 }→ 1 + quot(ifMinus(le(1 + (z - 2), z' - 1), 1 + (z - 2), z' - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 2 >= 0
quot(z, z') -{ 2 }→ 1 + quot(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 1 + 0

Function symbols to be analyzed: {le}, {ifMinus,minus}, {quot}
Previous analysis results are:

le: runtime: ?, size: O(1) [1]

(17) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed RUNTIME bound using PUBS for: le
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: 1 + z'

(18) Obligation:

Complexity RNTS consisting of the following rules:

ifMinus(z, z', z'') -{ 1 }→ 0 :|: z'' >= 0, z = 1, z' - 1 >= 0
ifMinus(z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
ifMinus(z, z', z'') -{ 1 }→ 1 + minus(z' - 1, z'') :|: z'' >= 0, z' - 1 >= 0, z = 0
le(z, z') -{ 1 }→ le(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
le(z, z') -{ 1 }→ 1 :|: z' >= 0, z = 0
le(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0
minus(z, z') -{ 2 }→ ifMinus(le(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
minus(z, z') -{ 2 }→ ifMinus(0, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
minus(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
quot(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0
quot(z, z') -{ 2 }→ 1 + quot(ifMinus(le(1 + (z - 2), z' - 1), 1 + (z - 2), z' - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 2 >= 0
quot(z, z') -{ 2 }→ 1 + quot(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 1 + 0

Function symbols to be analyzed: {ifMinus,minus}, {quot}
Previous analysis results are:

le: runtime: O(n1) [1 + z'], size: O(1) [1]

(19) ResultPropagationProof (UPPER BOUND(ID) transformation)

Applied inner abstraction using the recently inferred runtime/size bounds where possible.

(20) Obligation:

Complexity RNTS consisting of the following rules:

ifMinus(z, z', z'') -{ 1 }→ 0 :|: z'' >= 0, z = 1, z' - 1 >= 0
ifMinus(z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
ifMinus(z, z', z'') -{ 1 }→ 1 + minus(z' - 1, z'') :|: z'' >= 0, z' - 1 >= 0, z = 0
le(z, z') -{ 1 + z' }→ s :|: s >= 0, s <= 1, z' - 1 >= 0, z - 1 >= 0
le(z, z') -{ 1 }→ 1 :|: z' >= 0, z = 0
le(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0
minus(z, z') -{ 2 + z' }→ ifMinus(s', 1 + (z - 1), 1 + (z' - 1)) :|: s' >= 0, s' <= 1, z' - 1 >= 0, z - 1 >= 0
minus(z, z') -{ 2 }→ ifMinus(0, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
minus(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
quot(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0
quot(z, z') -{ 2 + z' }→ 1 + quot(ifMinus(s'', 1 + (z - 2), z' - 1), 1 + (z' - 1)) :|: s'' >= 0, s'' <= 1, z' - 1 >= 0, z - 2 >= 0
quot(z, z') -{ 2 }→ 1 + quot(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 1 + 0

Function symbols to be analyzed: {ifMinus,minus}, {quot}
Previous analysis results are:

le: runtime: O(n1) [1 + z'], size: O(1) [1]

(21) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed SIZE bound using KoAT for: ifMinus
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: z'

Computed SIZE bound using CoFloCo for: minus
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: z

(22) Obligation:

Complexity RNTS consisting of the following rules:

ifMinus(z, z', z'') -{ 1 }→ 0 :|: z'' >= 0, z = 1, z' - 1 >= 0
ifMinus(z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
ifMinus(z, z', z'') -{ 1 }→ 1 + minus(z' - 1, z'') :|: z'' >= 0, z' - 1 >= 0, z = 0
le(z, z') -{ 1 + z' }→ s :|: s >= 0, s <= 1, z' - 1 >= 0, z - 1 >= 0
le(z, z') -{ 1 }→ 1 :|: z' >= 0, z = 0
le(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0
minus(z, z') -{ 2 + z' }→ ifMinus(s', 1 + (z - 1), 1 + (z' - 1)) :|: s' >= 0, s' <= 1, z' - 1 >= 0, z - 1 >= 0
minus(z, z') -{ 2 }→ ifMinus(0, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
minus(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
quot(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0
quot(z, z') -{ 2 + z' }→ 1 + quot(ifMinus(s'', 1 + (z - 2), z' - 1), 1 + (z' - 1)) :|: s'' >= 0, s'' <= 1, z' - 1 >= 0, z - 2 >= 0
quot(z, z') -{ 2 }→ 1 + quot(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 1 + 0

Function symbols to be analyzed: {ifMinus,minus}, {quot}
Previous analysis results are:

le: runtime: O(n1) [1 + z'], size: O(1) [1] ifMinus: runtime: ?, size: O(n1) [z'] minus: runtime: ?, size: O(n1) [z]

(23) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed RUNTIME bound using CoFloCo for: ifMinus
after applying outer abstraction to obtain an ITS,
resulting in: O(n²) with polynomial bound: 5 + 3·z' + z'·z'' + z''

Computed RUNTIME bound using PUBS for: minus
after applying outer abstraction to obtain an ITS,
resulting in: O(n²) with polynomial bound: 7 + 3·z + z·z' + 2·z'

(24) Obligation:

Complexity RNTS consisting of the following rules:

ifMinus(z, z', z'') -{ 1 }→ 0 :|: z'' >= 0, z = 1, z' - 1 >= 0
ifMinus(z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
ifMinus(z, z', z'') -{ 1 }→ 1 + minus(z' - 1, z'') :|: z'' >= 0, z' - 1 >= 0, z = 0
le(z, z') -{ 1 + z' }→ s :|: s >= 0, s <= 1, z' - 1 >= 0, z - 1 >= 0
le(z, z') -{ 1 }→ 1 :|: z' >= 0, z = 0
le(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0
minus(z, z') -{ 2 + z' }→ ifMinus(s', 1 + (z - 1), 1 + (z' - 1)) :|: s' >= 0, s' <= 1, z' - 1 >= 0, z - 1 >= 0
minus(z, z') -{ 2 }→ ifMinus(0, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
minus(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
quot(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0
quot(z, z') -{ 2 + z' }→ 1 + quot(ifMinus(s'', 1 + (z - 2), z' - 1), 1 + (z' - 1)) :|: s'' >= 0, s'' <= 1, z' - 1 >= 0, z - 2 >= 0
quot(z, z') -{ 2 }→ 1 + quot(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 1 + 0

Function symbols to be analyzed: {quot}
Previous analysis results are:

le: runtime: O(n1) [1 + z'], size: O(1) [1] ifMinus: runtime: O(n2) [5 + 3·z' + z'·z'' + z''], size: O(n1) [z'] minus: runtime: O(n2) [7 + 3·z + z·z' + 2·z'], size: O(n1) [z]

(25) ResultPropagationProof (UPPER BOUND(ID) transformation)

Applied inner abstraction using the recently inferred runtime/size bounds where possible.

(26) Obligation:

Complexity RNTS consisting of the following rules:

ifMinus(z, z', z'') -{ 1 }→ 0 :|: z'' >= 0, z = 1, z' - 1 >= 0
ifMinus(z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
ifMinus(z, z', z'') -{ 5 + 3·z' + z'·z'' + z'' }→ 1 + s3 :|: s3 >= 0, s3 <= 1 * (z' - 1), z'' >= 0, z' - 1 >= 0, z = 0
le(z, z') -{ 1 + z' }→ s :|: s >= 0, s <= 1, z' - 1 >= 0, z - 1 >= 0
le(z, z') -{ 1 }→ 1 :|: z' >= 0, z = 0
le(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0
minus(z, z') -{ 7 + 3·z }→ s1 :|: s1 >= 0, s1 <= 1 * (1 + (z - 1)), z - 1 >= 0, z' = 0
minus(z, z') -{ 7 + 3·z + z·z' + 2·z' }→ s2 :|: s2 >= 0, s2 <= 1 * (1 + (z - 1)), s' >= 0, s' <= 1, z' - 1 >= 0, z - 1 >= 0
minus(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
quot(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0
quot(z, z') -{ 4 + 2·z + z·z' + z' }→ 1 + quot(s4, 1 + (z' - 1)) :|: s4 >= 0, s4 <= 1 * (1 + (z - 2)), s'' >= 0, s'' <= 1, z' - 1 >= 0, z - 2 >= 0
quot(z, z') -{ 2 }→ 1 + quot(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 1 + 0

Function symbols to be analyzed: {quot}
Previous analysis results are:

le: runtime: O(n1) [1 + z'], size: O(1) [1] ifMinus: runtime: O(n2) [5 + 3·z' + z'·z'' + z''], size: O(n1) [z'] minus: runtime: O(n2) [7 + 3·z + z·z' + 2·z'], size: O(n1) [z]

(27) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed SIZE bound using KoAT for: quot
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: z

(28) Obligation:

Complexity RNTS consisting of the following rules:

ifMinus(z, z', z'') -{ 1 }→ 0 :|: z'' >= 0, z = 1, z' - 1 >= 0
ifMinus(z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
ifMinus(z, z', z'') -{ 5 + 3·z' + z'·z'' + z'' }→ 1 + s3 :|: s3 >= 0, s3 <= 1 * (z' - 1), z'' >= 0, z' - 1 >= 0, z = 0
le(z, z') -{ 1 + z' }→ s :|: s >= 0, s <= 1, z' - 1 >= 0, z - 1 >= 0
le(z, z') -{ 1 }→ 1 :|: z' >= 0, z = 0
le(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0
minus(z, z') -{ 7 + 3·z }→ s1 :|: s1 >= 0, s1 <= 1 * (1 + (z - 1)), z - 1 >= 0, z' = 0
minus(z, z') -{ 7 + 3·z + z·z' + 2·z' }→ s2 :|: s2 >= 0, s2 <= 1 * (1 + (z - 1)), s' >= 0, s' <= 1, z' - 1 >= 0, z - 1 >= 0
minus(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
quot(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0
quot(z, z') -{ 4 + 2·z + z·z' + z' }→ 1 + quot(s4, 1 + (z' - 1)) :|: s4 >= 0, s4 <= 1 * (1 + (z - 2)), s'' >= 0, s'' <= 1, z' - 1 >= 0, z - 2 >= 0
quot(z, z') -{ 2 }→ 1 + quot(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 1 + 0

Function symbols to be analyzed: {quot}
Previous analysis results are:

le: runtime: O(n1) [1 + z'], size: O(1) [1] ifMinus: runtime: O(n2) [5 + 3·z' + z'·z'' + z''], size: O(n1) [z'] minus: runtime: O(n2) [7 + 3·z + z·z' + 2·z'], size: O(n1) [z] quot: runtime: ?, size: O(n1) [z]

(29) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed RUNTIME bound using PUBS for: quot
after applying outer abstraction to obtain an ITS,
resulting in: O(n³) with polynomial bound: 1 + 4·z + z·z' + 2·z² + z²·z'

(30) Obligation:

Complexity RNTS consisting of the following rules:

ifMinus(z, z', z'') -{ 1 }→ 0 :|: z'' >= 0, z = 1, z' - 1 >= 0
ifMinus(z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
ifMinus(z, z', z'') -{ 5 + 3·z' + z'·z'' + z'' }→ 1 + s3 :|: s3 >= 0, s3 <= 1 * (z' - 1), z'' >= 0, z' - 1 >= 0, z = 0
le(z, z') -{ 1 + z' }→ s :|: s >= 0, s <= 1, z' - 1 >= 0, z - 1 >= 0
le(z, z') -{ 1 }→ 1 :|: z' >= 0, z = 0
le(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0
minus(z, z') -{ 7 + 3·z }→ s1 :|: s1 >= 0, s1 <= 1 * (1 + (z - 1)), z - 1 >= 0, z' = 0
minus(z, z') -{ 7 + 3·z + z·z' + 2·z' }→ s2 :|: s2 >= 0, s2 <= 1 * (1 + (z - 1)), s' >= 0, s' <= 1, z' - 1 >= 0, z - 1 >= 0
minus(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
quot(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0
quot(z, z') -{ 4 + 2·z + z·z' + z' }→ 1 + quot(s4, 1 + (z' - 1)) :|: s4 >= 0, s4 <= 1 * (1 + (z - 2)), s'' >= 0, s'' <= 1, z' - 1 >= 0, z - 2 >= 0
quot(z, z') -{ 2 }→ 1 + quot(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 1 + 0

Function symbols to be analyzed:
Previous analysis results are:

le: runtime: O(n1) [1 + z'], size: O(1) [1] ifMinus: runtime: O(n2) [5 + 3·z' + z'·z'' + z''], size: O(n1) [z'] minus: runtime: O(n2) [7 + 3·z + z·z' + 2·z'], size: O(n1) [z] quot: runtime: O(n3) [1 + 4·z + z·z' + 2·z2 + z2·z'], size: O(n1) [z]

(31) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(32) BOUNDS(1, n^3)