Initial Problem

Start: l0
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: l0, l1, l2
Transitions:
t₀: l0(X₀, X₁, X₂) → l1(X₀, X₁, X₂) :|: 0 < X₀
t₁: l1(X₀, X₁, X₂) → l1(X₀+X₂, X₁, X₂-1) :|: 0 < X₂
t₂: l1(X₀, X₁, X₂) → l2(X₀, X₁, X₂) :|: X₂ ≤ 0
t₃: l2(X₀, X₁, X₂) → l2(5⋅X₀+(X₂)², 2⋅X₁, X₂) :|: X₀ < (X₁)²

Preprocessing

Found invariant X₂ ≤ 0 ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₀ for location l2

Found invariant 1 ≤ X₀ for location l1

Problem after Preprocessing

Start: l0
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: l0, l1, l2
Transitions:
t₀: l0(X₀, X₁, X₂) → l1(X₀, X₁, X₂) :|: 0 < X₀
t₁: l1(X₀, X₁, X₂) → l1(X₀+X₂, X₁, X₂-1) :|: 0 < X₂ ∧ 1 ≤ X₀
t₂: l1(X₀, X₁, X₂) → l2(X₀, X₁, X₂) :|: X₂ ≤ 0 ∧ 1 ≤ X₀
t₃: l2(X₀, X₁, X₂) → l2(5⋅X₀+(X₂)², 2⋅X₁, X₂) :|: X₀ < (X₁)² ∧ X₂ ≤ 0 ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₀

Found invariant X₂ ≤ 0 ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₀ for location l2

Found invariant 1 ≤ X₀ for location l1

Time-Bound by TWN-Loops:

TWN-Loops: t₁ 2⋅X₂+4 {O(n)}

TWN-Loops:

entry: t₀: l0(X₀, X₁, X₂) → l1(X₀, X₁, X₂) :|: 0 < X₀
results in twn-loop: twn:Inv: [1 ≤ X₀] , (X₀,X₁,X₂) -> (X₀+X₂,X₁,X₂-1) :|: 0 < X₂
order: [X₂; X₀]
closed-form:
X₂: X₂ + [[n != 0]] * -1 * n^1
X₀: X₀ + [[n != 0]] * X₂ * n^1 + [[n != 0, n != 1]] * -1/2 * n^2 + [[n != 0, n != 1]] * 1/2 * n^1

Termination: true
Formula:

1 < 0
∨ 1 < 0 ∧ 0 < 2⋅X₂+1 ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ 1 < 0 ∧ 2 < 2⋅X₀ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 0 ≤ 2⋅X₂+1 ∧ 2⋅X₂+1 ≤ 0
∨ 1 < 0 ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 0 ≤ 2⋅X₂+1 ∧ 2⋅X₂+1 ≤ 0 ∧ 2 ≤ 2⋅X₀ ∧ 2⋅X₀ ≤ 2
∨ 0 < X₂ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 1 < 0
∨ 0 < X₂ ∧ 0 < 2⋅X₂+1 ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ 0 < X₂ ∧ 2 < 2⋅X₀ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 0 ≤ 2⋅X₂+1 ∧ 2⋅X₂+1 ≤ 0
∨ 0 < X₂ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 0 ≤ 2⋅X₂+1 ∧ 2⋅X₂+1 ≤ 0 ∧ 2 ≤ 2⋅X₀ ∧ 2⋅X₀ ≤ 2

Stabilization-Threshold for: 0 < X₂
alphas_abs: X₂
M: 0
N: 1
Bound: 2⋅X₂+2 {O(n)}

relevant size-bounds w.r.t. t₀:
X₂: X₂ {O(n)}
Runtime-bound of t₀: 1 {O(1)}
Results in: 2⋅X₂+4 {O(n)}

2⋅X₂+4 {O(n)}

Found invariant X₂ ≤ 0 ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₀ for location l2

Found invariant 1 ≤ X₀ for location l1

Time-Bound by TWN-Loops:

TWN-Loops: t₃ 32⋅X₁⋅X₁+8⋅X₂⋅X₂+14 {O(n^2)}

TWN-Loops:

entry: t₂: l1(X₀, X₁, X₂) → l2(X₀, X₁, X₂) :|: X₂ ≤ 0 ∧ 1 ≤ X₀
results in twn-loop: twn:Inv: [X₂ ≤ 0 ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₀] , (X₀,X₁,X₂) -> (5⋅X₀+(X₂)²,2⋅X₁,X₂) :|: X₀ < (X₁)²
order: [X₂; X₀; X₁]
closed-form:
X₂: X₂
X₀: X₀ * 5^n + [[n != 0]] * 1/4⋅(X₂)² * 5^n + [[n != 0]] * -1/4⋅(X₂)²
X₁: X₁ * 2^n

Termination: true
Formula:

4⋅X₀+(X₂)² < 0
∨ 0 < 4⋅(X₁)² ∧ 4⋅X₀+(X₂)² ≤ 0 ∧ 0 ≤ 4⋅X₀+(X₂)²
∨ 0 < (X₂)² ∧ 4⋅X₀+(X₂)² ≤ 0 ∧ 0 ≤ 4⋅X₀+(X₂)² ∧ 0 ≤ 4⋅(X₁)² ∧ 4⋅(X₁)² ≤ 0

Stabilization-Threshold for: X₀ < (X₁)²
alphas_abs: 4⋅(X₁)²+(X₂)²
M: 11
N: 1
Bound: 2⋅X₂⋅X₂+8⋅X₁⋅X₁+12 {O(n^2)}

relevant size-bounds w.r.t. t₂:
X₁: 2⋅X₁ {O(n)}
X₂: 2⋅X₂ {O(n)}
Runtime-bound of t₂: 1 {O(1)}
Results in: 32⋅X₁⋅X₁+8⋅X₂⋅X₂+14 {O(n^2)}

32⋅X₁⋅X₁+8⋅X₂⋅X₂+14 {O(n^2)}

Analysing control-flow refined program

Eliminate variables {X₁} that do not contribute to the problem

Found invariant X₂ ≤ 0 ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₀ for location l2

Found invariant X₂ ≤ 0 for location n_l2___1

Found invariant 1 ≤ X₀ for location l1

Found invariant X₂ ≤ 0 ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₀ for location l2

Found invariant X₂ ≤ 0 for location n_l2___1

Found invariant 1 ≤ X₀ for location l1

Time-Bound by TWN-Loops:

TWN-Loops: t₅₉ 2⋅X₂+4 {O(n)}

TWN-Loops:

entry: t₅₈: l0(X₀, X₂) → l1(X₀, X₂) :|: 0 < X₀
results in twn-loop: twn:Inv: [1 ≤ X₀ ∧ 1 ≤ X₀] , (X₀,X₂) -> (X₀+X₂,X₂-1) :|: 0 < X₂
order: [X₂; X₀]
closed-form:
X₂: X₂ + [[n != 0]] * -1 * n^1
X₀: X₀ + [[n != 0]] * X₂ * n^1 + [[n != 0, n != 1]] * -1/2 * n^2 + [[n != 0, n != 1]] * 1/2 * n^1

Termination: true
Formula:

1 < 0
∨ 1 < 0 ∧ 0 < 2⋅X₂+1 ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ 1 < 0 ∧ 2 < 2⋅X₀ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 0 ≤ 2⋅X₂+1 ∧ 2⋅X₂+1 ≤ 0
∨ 1 < 0 ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 0 ≤ 2⋅X₂+1 ∧ 2⋅X₂+1 ≤ 0 ∧ 2 ≤ 2⋅X₀ ∧ 2⋅X₀ ≤ 2
∨ 0 < X₂ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 1 < 0
∨ 0 < X₂ ∧ 0 < 2⋅X₂+1 ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ 0 < X₂ ∧ 2 < 2⋅X₀ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 0 ≤ 2⋅X₂+1 ∧ 2⋅X₂+1 ≤ 0
∨ 0 < X₂ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 0 ≤ 2⋅X₂+1 ∧ 2⋅X₂+1 ≤ 0 ∧ 2 ≤ 2⋅X₀ ∧ 2⋅X₀ ≤ 2

Stabilization-Threshold for: 0 < X₂
alphas_abs: X₂
M: 0
N: 1
Bound: 2⋅X₂+2 {O(n)}

relevant size-bounds w.r.t. t₅₈:
X₂: X₂ {O(n)}
Runtime-bound of t₅₈: 1 {O(1)}
Results in: 2⋅X₂+4 {O(n)}

2⋅X₂+4 {O(n)}

CFR did not improve the program. Rolling back

All Bounds

Timebounds

Overall timebound:32⋅X₁⋅X₁+8⋅X₂⋅X₂+2⋅X₂+20 {O(n^2)}
t₀: 1 {O(1)}
t₁: 2⋅X₂+4 {O(n)}
t₂: 1 {O(1)}
t₃: 32⋅X₁⋅X₁+8⋅X₂⋅X₂+14 {O(n^2)}

Costbounds

Overall costbound: 32⋅X₁⋅X₁+8⋅X₂⋅X₂+2⋅X₂+20 {O(n^2)}
t₀: 1 {O(1)}
t₁: 2⋅X₂+4 {O(n)}
t₂: 1 {O(1)}
t₃: 32⋅X₁⋅X₁+8⋅X₂⋅X₂+14 {O(n^2)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₁, X₀: 4⋅X₂⋅X₂+10⋅X₂+X₀ {O(n^2)}
t₁, X₁: X₁ {O(n)}
t₁, X₂: X₂ {O(n)}
t₂, X₀: 4⋅X₂⋅X₂+10⋅X₂+2⋅X₀ {O(n^2)}
t₂, X₁: 2⋅X₁ {O(n)}
t₂, X₂: 2⋅X₂ {O(n)}
t₃, X₁: 2⋅2^(32⋅X₁⋅X₁+8⋅X₂⋅X₂+14)⋅X₁ {O(EXP)}
t₃, X₂: 2⋅X₂ {O(n)}