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

(0) 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:

@(Cons(x, xs), ys) → Cons(x, @(xs, ys))
@(Nil, ys) → ys
game(p1, Cons(x', xs'), Cons(Capture, xs)) → game(Cons(x', p1), xs', xs)
game(p1, p2, Cons(Swap, xs)) → game(p2, p1, xs)
equal(Capture, Capture) → True
equal(Capture, Swap) → False
equal(Swap, Capture) → False
equal(Swap, Swap) → True
game(p1, p2, Nil) → @(p1, p2)
goal(p1, p2, moves) → game(p1, p2, moves)

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^1).


The TRS R consists of the following rules:

@(Cons(x, xs), ys) → Cons(x, @(xs, ys)) [1]
@(Nil, ys) → ys [1]
game(p1, Cons(x', xs'), Cons(Capture, xs)) → game(Cons(x', p1), xs', xs) [1]
game(p1, p2, Cons(Swap, xs)) → game(p2, p1, xs) [1]
equal(Capture, Capture) → True [1]
equal(Capture, Swap) → False [1]
equal(Swap, Capture) → False [1]
equal(Swap, Swap) → True [1]
game(p1, p2, Nil) → @(p1, p2) [1]
goal(p1, p2, moves) → game(p1, p2, moves) [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:

@(Cons(x, xs), ys) → Cons(x, @(xs, ys)) [1]
@(Nil, ys) → ys [1]
game(p1, Cons(x', xs'), Cons(Capture, xs)) → game(Cons(x', p1), xs', xs) [1]
game(p1, p2, Cons(Swap, xs)) → game(p2, p1, xs) [1]
equal(Capture, Capture) → True [1]
equal(Capture, Swap) → False [1]
equal(Swap, Capture) → False [1]
equal(Swap, Swap) → True [1]
game(p1, p2, Nil) → @(p1, p2) [1]
goal(p1, p2, moves) → game(p1, p2, moves) [1]

The TRS has the following type information:

@ :: Cons:Nil → Cons:Nil → Cons:Nil Cons :: Capture:Swap → Cons:Nil → Cons:Nil Nil :: Cons:Nil game :: Cons:Nil → Cons:Nil → Cons:Nil → Cons:Nil Capture :: Capture:Swap Swap :: Capture:Swap equal :: Capture:Swap → Capture:Swap → True:False True :: True:False False :: True:False goal :: Cons:Nil → Cons:Nil → Cons:Nil → Cons:Nil

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:

@
game
equal
goal

(c) The following functions are completely defined:
none

Due to the following rules being added:
none

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: @(Cons(x, xs), ys) → Cons(x, @(xs, ys)) [1] @(Nil, ys) → ys [1] game(p1, Cons(x', xs'), Cons(Capture, xs)) → game(Cons(x', p1), xs', xs) [1] game(p1, p2, Cons(Swap, xs)) → game(p2, p1, xs) [1] equal(Capture, Capture) → True [1] equal(Capture, Swap) → False [1] equal(Swap, Capture) → False [1] equal(Swap, Swap) → True [1] game(p1, p2, Nil) → @(p1, p2) [1] goal(p1, p2, moves) → game(p1, p2, moves) [1] The TRS has the following type information: @ :: Cons:Nil → Cons:Nil → Cons:Nil Cons :: Capture:Swap → Cons:Nil → Cons:Nil Nil :: Cons:Nil game :: Cons:Nil → Cons:Nil → Cons:Nil → Cons:Nil Capture :: Capture:Swap Swap :: Capture:Swap equal :: Capture:Swap → Capture:Swap → True:False True :: True:False False :: True:False goal :: Cons:Nil → Cons:Nil → Cons:Nil → Cons:Nil 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: @(Cons(x, xs), ys) → Cons(x, @(xs, ys)) [1] @(Nil, ys) → ys [1] game(p1, Cons(x', xs'), Cons(Capture, xs)) → game(Cons(x', p1), xs', xs) [1] game(p1, p2, Cons(Swap, xs)) → game(p2, p1, xs) [1] equal(Capture, Capture) → True [1] equal(Capture, Swap) → False [1] equal(Swap, Capture) → False [1] equal(Swap, Swap) → True [1] game(p1, p2, Nil) → @(p1, p2) [1] goal(p1, p2, moves) → game(p1, p2, moves) [1] The TRS has the following type information: @ :: Cons:Nil → Cons:Nil → Cons:Nil Cons :: Capture:Swap → Cons:Nil → Cons:Nil Nil :: Cons:Nil game :: Cons:Nil → Cons:Nil → Cons:Nil → Cons:Nil Capture :: Capture:Swap Swap :: Capture:Swap equal :: Capture:Swap → Capture:Swap → True:False True :: True:False False :: True:False goal :: Cons:Nil → Cons:Nil → Cons:Nil → Cons:Nil 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:

Nil => 0
Capture => 0
Swap => 1
True => 1
False => 0

(10) Obligation:

Complexity RNTS consisting of the following rules:

@(z, z') -{ 1 }→ ys :|: z' = ys, ys >= 0, z = 0
@(z, z') -{ 1 }→ 1 + x + @(xs, ys) :|: z = 1 + x + xs, xs >= 0, z' = ys, ys >= 0, x >= 0
equal(z, z') -{ 1 }→ 1 :|: z = 0, z' = 0
equal(z, z') -{ 1 }→ 1 :|: z = 1, z' = 1
equal(z, z') -{ 1 }→ 0 :|: z' = 1, z = 0
equal(z, z') -{ 1 }→ 0 :|: z = 1, z' = 0
game(z, z', z'') -{ 1 }→ game(p2, p1, xs) :|: xs >= 0, p1 >= 0, z' = p2, z = p1, z'' = 1 + 1 + xs, p2 >= 0
game(z, z', z'') -{ 1 }→ game(1 + x' + p1, xs', xs) :|: xs >= 0, p1 >= 0, z'' = 1 + 0 + xs, x' >= 0, xs' >= 0, z' = 1 + x' + xs', z = p1
game(z, z', z'') -{ 1 }→ @(p1, p2) :|: z'' = 0, p1 >= 0, z' = p2, z = p1, p2 >= 0
goal(z, z', z'') -{ 1 }→ game(p1, p2, moves) :|: p1 >= 0, z' = p2, z'' = moves, moves >= 0, z = p1, p2 >= 0

(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:

@(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
@(z, z') -{ 1 }→ 1 + x + @(xs, z') :|: z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0
equal(z, z') -{ 1 }→ 1 :|: z = 0, z' = 0
equal(z, z') -{ 1 }→ 1 :|: z = 1, z' = 1
equal(z, z') -{ 1 }→ 0 :|: z' = 1, z = 0
equal(z, z') -{ 1 }→ 0 :|: z = 1, z' = 0
game(z, z', z'') -{ 1 }→ game(z', z, z'' - 2) :|: z'' - 2 >= 0, z >= 0, z' >= 0
game(z, z', z'') -{ 1 }→ game(1 + x' + z, xs', z'' - 1) :|: z'' - 1 >= 0, z >= 0, x' >= 0, xs' >= 0, z' = 1 + x' + xs'
game(z, z', z'') -{ 1 }→ @(z, z') :|: z'' = 0, z >= 0, z' >= 0
goal(z, z', z'') -{ 1 }→ game(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0

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

Found the following analysis order by SCC decomposition:

{ equal } { @ } { game } { goal }

(14) Obligation:

Complexity RNTS consisting of the following rules:

@(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
@(z, z') -{ 1 }→ 1 + x + @(xs, z') :|: z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0
equal(z, z') -{ 1 }→ 1 :|: z = 0, z' = 0
equal(z, z') -{ 1 }→ 1 :|: z = 1, z' = 1
equal(z, z') -{ 1 }→ 0 :|: z' = 1, z = 0
equal(z, z') -{ 1 }→ 0 :|: z = 1, z' = 0
game(z, z', z'') -{ 1 }→ game(z', z, z'' - 2) :|: z'' - 2 >= 0, z >= 0, z' >= 0
game(z, z', z'') -{ 1 }→ game(1 + x' + z, xs', z'' - 1) :|: z'' - 1 >= 0, z >= 0, x' >= 0, xs' >= 0, z' = 1 + x' + xs'
game(z, z', z'') -{ 1 }→ @(z, z') :|: z'' = 0, z >= 0, z' >= 0
goal(z, z', z'') -{ 1 }→ game(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0

Function symbols to be analyzed: {equal}, {@}, {game}, {goal}

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

Computed SIZE bound using CoFloCo for: equal
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:

@(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
@(z, z') -{ 1 }→ 1 + x + @(xs, z') :|: z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0
equal(z, z') -{ 1 }→ 1 :|: z = 0, z' = 0
equal(z, z') -{ 1 }→ 1 :|: z = 1, z' = 1
equal(z, z') -{ 1 }→ 0 :|: z' = 1, z = 0
equal(z, z') -{ 1 }→ 0 :|: z = 1, z' = 0
game(z, z', z'') -{ 1 }→ game(z', z, z'' - 2) :|: z'' - 2 >= 0, z >= 0, z' >= 0
game(z, z', z'') -{ 1 }→ game(1 + x' + z, xs', z'' - 1) :|: z'' - 1 >= 0, z >= 0, x' >= 0, xs' >= 0, z' = 1 + x' + xs'
game(z, z', z'') -{ 1 }→ @(z, z') :|: z'' = 0, z >= 0, z' >= 0
goal(z, z', z'') -{ 1 }→ game(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0

Function symbols to be analyzed: {equal}, {@}, {game}, {goal}
Previous analysis results are:

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

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

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

(18) Obligation:

Complexity RNTS consisting of the following rules:

@(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
@(z, z') -{ 1 }→ 1 + x + @(xs, z') :|: z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0
equal(z, z') -{ 1 }→ 1 :|: z = 0, z' = 0
equal(z, z') -{ 1 }→ 1 :|: z = 1, z' = 1
equal(z, z') -{ 1 }→ 0 :|: z' = 1, z = 0
equal(z, z') -{ 1 }→ 0 :|: z = 1, z' = 0
game(z, z', z'') -{ 1 }→ game(z', z, z'' - 2) :|: z'' - 2 >= 0, z >= 0, z' >= 0
game(z, z', z'') -{ 1 }→ game(1 + x' + z, xs', z'' - 1) :|: z'' - 1 >= 0, z >= 0, x' >= 0, xs' >= 0, z' = 1 + x' + xs'
game(z, z', z'') -{ 1 }→ @(z, z') :|: z'' = 0, z >= 0, z' >= 0
goal(z, z', z'') -{ 1 }→ game(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0

Function symbols to be analyzed: {@}, {game}, {goal}
Previous analysis results are:

equal: runtime: O(1) [1], 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:

@(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
@(z, z') -{ 1 }→ 1 + x + @(xs, z') :|: z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0
equal(z, z') -{ 1 }→ 1 :|: z = 0, z' = 0
equal(z, z') -{ 1 }→ 1 :|: z = 1, z' = 1
equal(z, z') -{ 1 }→ 0 :|: z' = 1, z = 0
equal(z, z') -{ 1 }→ 0 :|: z = 1, z' = 0
game(z, z', z'') -{ 1 }→ game(z', z, z'' - 2) :|: z'' - 2 >= 0, z >= 0, z' >= 0
game(z, z', z'') -{ 1 }→ game(1 + x' + z, xs', z'' - 1) :|: z'' - 1 >= 0, z >= 0, x' >= 0, xs' >= 0, z' = 1 + x' + xs'
game(z, z', z'') -{ 1 }→ @(z, z') :|: z'' = 0, z >= 0, z' >= 0
goal(z, z', z'') -{ 1 }→ game(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0

Function symbols to be analyzed: {@}, {game}, {goal}
Previous analysis results are:

equal: runtime: O(1) [1], size: O(1) [1]

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

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

(22) Obligation:

Complexity RNTS consisting of the following rules:

@(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
@(z, z') -{ 1 }→ 1 + x + @(xs, z') :|: z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0
equal(z, z') -{ 1 }→ 1 :|: z = 0, z' = 0
equal(z, z') -{ 1 }→ 1 :|: z = 1, z' = 1
equal(z, z') -{ 1 }→ 0 :|: z' = 1, z = 0
equal(z, z') -{ 1 }→ 0 :|: z = 1, z' = 0
game(z, z', z'') -{ 1 }→ game(z', z, z'' - 2) :|: z'' - 2 >= 0, z >= 0, z' >= 0
game(z, z', z'') -{ 1 }→ game(1 + x' + z, xs', z'' - 1) :|: z'' - 1 >= 0, z >= 0, x' >= 0, xs' >= 0, z' = 1 + x' + xs'
game(z, z', z'') -{ 1 }→ @(z, z') :|: z'' = 0, z >= 0, z' >= 0
goal(z, z', z'') -{ 1 }→ game(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0

Function symbols to be analyzed: {@}, {game}, {goal}
Previous analysis results are:

equal: runtime: O(1) [1], size: O(1) [1] @: runtime: ?, size: O(n1) [z + z']

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

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

(24) Obligation:

Complexity RNTS consisting of the following rules:

@(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
@(z, z') -{ 1 }→ 1 + x + @(xs, z') :|: z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0
equal(z, z') -{ 1 }→ 1 :|: z = 0, z' = 0
equal(z, z') -{ 1 }→ 1 :|: z = 1, z' = 1
equal(z, z') -{ 1 }→ 0 :|: z' = 1, z = 0
equal(z, z') -{ 1 }→ 0 :|: z = 1, z' = 0
game(z, z', z'') -{ 1 }→ game(z', z, z'' - 2) :|: z'' - 2 >= 0, z >= 0, z' >= 0
game(z, z', z'') -{ 1 }→ game(1 + x' + z, xs', z'' - 1) :|: z'' - 1 >= 0, z >= 0, x' >= 0, xs' >= 0, z' = 1 + x' + xs'
game(z, z', z'') -{ 1 }→ @(z, z') :|: z'' = 0, z >= 0, z' >= 0
goal(z, z', z'') -{ 1 }→ game(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0

Function symbols to be analyzed: {game}, {goal}
Previous analysis results are:

equal: runtime: O(1) [1], size: O(1) [1] @: runtime: O(n1) [1 + z], size: O(n1) [z + 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:

@(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
@(z, z') -{ 2 + xs }→ 1 + x + s :|: s >= 0, s <= 1 * xs + 1 * z', z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0
equal(z, z') -{ 1 }→ 1 :|: z = 0, z' = 0
equal(z, z') -{ 1 }→ 1 :|: z = 1, z' = 1
equal(z, z') -{ 1 }→ 0 :|: z' = 1, z = 0
equal(z, z') -{ 1 }→ 0 :|: z = 1, z' = 0
game(z, z', z'') -{ 2 + z }→ s' :|: s' >= 0, s' <= 1 * z + 1 * z', z'' = 0, z >= 0, z' >= 0
game(z, z', z'') -{ 1 }→ game(z', z, z'' - 2) :|: z'' - 2 >= 0, z >= 0, z' >= 0
game(z, z', z'') -{ 1 }→ game(1 + x' + z, xs', z'' - 1) :|: z'' - 1 >= 0, z >= 0, x' >= 0, xs' >= 0, z' = 1 + x' + xs'
goal(z, z', z'') -{ 1 }→ game(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0

Function symbols to be analyzed: {game}, {goal}
Previous analysis results are:

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

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

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

(28) Obligation:

Complexity RNTS consisting of the following rules:

@(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
@(z, z') -{ 2 + xs }→ 1 + x + s :|: s >= 0, s <= 1 * xs + 1 * z', z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0
equal(z, z') -{ 1 }→ 1 :|: z = 0, z' = 0
equal(z, z') -{ 1 }→ 1 :|: z = 1, z' = 1
equal(z, z') -{ 1 }→ 0 :|: z' = 1, z = 0
equal(z, z') -{ 1 }→ 0 :|: z = 1, z' = 0
game(z, z', z'') -{ 2 + z }→ s' :|: s' >= 0, s' <= 1 * z + 1 * z', z'' = 0, z >= 0, z' >= 0
game(z, z', z'') -{ 1 }→ game(z', z, z'' - 2) :|: z'' - 2 >= 0, z >= 0, z' >= 0
game(z, z', z'') -{ 1 }→ game(1 + x' + z, xs', z'' - 1) :|: z'' - 1 >= 0, z >= 0, x' >= 0, xs' >= 0, z' = 1 + x' + xs'
goal(z, z', z'') -{ 1 }→ game(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0

Function symbols to be analyzed: {game}, {goal}
Previous analysis results are:

equal: runtime: O(1) [1], size: O(1) [1] @: runtime: O(n1) [1 + z], size: O(n1) [z + z'] game: runtime: ?, size: O(n1) [z + z']

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

Computed RUNTIME bound using PUBS for: game
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: 2 + z + z' + z''

(30) Obligation:

Complexity RNTS consisting of the following rules:

@(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
@(z, z') -{ 2 + xs }→ 1 + x + s :|: s >= 0, s <= 1 * xs + 1 * z', z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0
equal(z, z') -{ 1 }→ 1 :|: z = 0, z' = 0
equal(z, z') -{ 1 }→ 1 :|: z = 1, z' = 1
equal(z, z') -{ 1 }→ 0 :|: z' = 1, z = 0
equal(z, z') -{ 1 }→ 0 :|: z = 1, z' = 0
game(z, z', z'') -{ 2 + z }→ s' :|: s' >= 0, s' <= 1 * z + 1 * z', z'' = 0, z >= 0, z' >= 0
game(z, z', z'') -{ 1 }→ game(z', z, z'' - 2) :|: z'' - 2 >= 0, z >= 0, z' >= 0
game(z, z', z'') -{ 1 }→ game(1 + x' + z, xs', z'' - 1) :|: z'' - 1 >= 0, z >= 0, x' >= 0, xs' >= 0, z' = 1 + x' + xs'
goal(z, z', z'') -{ 1 }→ game(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0

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

equal: runtime: O(1) [1], size: O(1) [1] @: runtime: O(n1) [1 + z], size: O(n1) [z + z'] game: runtime: O(n1) [2 + z + z' + z''], size: O(n1) [z + z']

(31) ResultPropagationProof (UPPER BOUND(ID) transformation)

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

(32) Obligation:

Complexity RNTS consisting of the following rules:

@(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
@(z, z') -{ 2 + xs }→ 1 + x + s :|: s >= 0, s <= 1 * xs + 1 * z', z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0
equal(z, z') -{ 1 }→ 1 :|: z = 0, z' = 0
equal(z, z') -{ 1 }→ 1 :|: z = 1, z' = 1
equal(z, z') -{ 1 }→ 0 :|: z' = 1, z = 0
equal(z, z') -{ 1 }→ 0 :|: z = 1, z' = 0
game(z, z', z'') -{ 2 + z }→ s' :|: s' >= 0, s' <= 1 * z + 1 * z', z'' = 0, z >= 0, z' >= 0
game(z, z', z'') -{ 3 + x' + xs' + z + z'' }→ s'' :|: s'' >= 0, s'' <= 1 * (1 + x' + z) + 1 * xs', z'' - 1 >= 0, z >= 0, x' >= 0, xs' >= 0, z' = 1 + x' + xs'
game(z, z', z'') -{ 1 + z + z' + z'' }→ s1 :|: s1 >= 0, s1 <= 1 * z' + 1 * z, z'' - 2 >= 0, z >= 0, z' >= 0
goal(z, z', z'') -{ 3 + z + z' + z'' }→ s2 :|: s2 >= 0, s2 <= 1 * z + 1 * z', z >= 0, z'' >= 0, z' >= 0

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

equal: runtime: O(1) [1], size: O(1) [1] @: runtime: O(n1) [1 + z], size: O(n1) [z + z'] game: runtime: O(n1) [2 + z + z' + z''], size: O(n1) [z + z']

(33) IntTrsBoundProof (UPPER BOUND(ID) transformation)

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

(34) Obligation:

Complexity RNTS consisting of the following rules:

@(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
@(z, z') -{ 2 + xs }→ 1 + x + s :|: s >= 0, s <= 1 * xs + 1 * z', z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0
equal(z, z') -{ 1 }→ 1 :|: z = 0, z' = 0
equal(z, z') -{ 1 }→ 1 :|: z = 1, z' = 1
equal(z, z') -{ 1 }→ 0 :|: z' = 1, z = 0
equal(z, z') -{ 1 }→ 0 :|: z = 1, z' = 0
game(z, z', z'') -{ 2 + z }→ s' :|: s' >= 0, s' <= 1 * z + 1 * z', z'' = 0, z >= 0, z' >= 0
game(z, z', z'') -{ 3 + x' + xs' + z + z'' }→ s'' :|: s'' >= 0, s'' <= 1 * (1 + x' + z) + 1 * xs', z'' - 1 >= 0, z >= 0, x' >= 0, xs' >= 0, z' = 1 + x' + xs'
game(z, z', z'') -{ 1 + z + z' + z'' }→ s1 :|: s1 >= 0, s1 <= 1 * z' + 1 * z, z'' - 2 >= 0, z >= 0, z' >= 0
goal(z, z', z'') -{ 3 + z + z' + z'' }→ s2 :|: s2 >= 0, s2 <= 1 * z + 1 * z', z >= 0, z'' >= 0, z' >= 0

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

equal: runtime: O(1) [1], size: O(1) [1] @: runtime: O(n1) [1 + z], size: O(n1) [z + z'] game: runtime: O(n1) [2 + z + z' + z''], size: O(n1) [z + z'] goal: runtime: ?, size: O(n1) [z + z']

(35) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed RUNTIME bound using CoFloCo for: goal
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: 3 + z + z' + z''

(36) Obligation:

Complexity RNTS consisting of the following rules:

@(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
@(z, z') -{ 2 + xs }→ 1 + x + s :|: s >= 0, s <= 1 * xs + 1 * z', z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0
equal(z, z') -{ 1 }→ 1 :|: z = 0, z' = 0
equal(z, z') -{ 1 }→ 1 :|: z = 1, z' = 1
equal(z, z') -{ 1 }→ 0 :|: z' = 1, z = 0
equal(z, z') -{ 1 }→ 0 :|: z = 1, z' = 0
game(z, z', z'') -{ 2 + z }→ s' :|: s' >= 0, s' <= 1 * z + 1 * z', z'' = 0, z >= 0, z' >= 0
game(z, z', z'') -{ 3 + x' + xs' + z + z'' }→ s'' :|: s'' >= 0, s'' <= 1 * (1 + x' + z) + 1 * xs', z'' - 1 >= 0, z >= 0, x' >= 0, xs' >= 0, z' = 1 + x' + xs'
game(z, z', z'') -{ 1 + z + z' + z'' }→ s1 :|: s1 >= 0, s1 <= 1 * z' + 1 * z, z'' - 2 >= 0, z >= 0, z' >= 0
goal(z, z', z'') -{ 3 + z + z' + z'' }→ s2 :|: s2 >= 0, s2 <= 1 * z + 1 * z', z >= 0, z'' >= 0, z' >= 0

Function symbols to be analyzed:
Previous analysis results are:

equal: runtime: O(1) [1], size: O(1) [1] @: runtime: O(n1) [1 + z], size: O(n1) [z + z'] game: runtime: O(n1) [2 + z + z' + z''], size: O(n1) [z + z'] goal: runtime: O(n1) [3 + z + z' + z''], size: O(n1) [z + z']

(37) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(38) BOUNDS(1, n^1)