Initial Problem

Start: l0
Program_Vars: X₀, X₁, X₂, X₃, X₄
Temp_Vars:
Locations: l0, l1, l2
Transitions:
t₀: l0(X₀, X₁, X₂, X₃, X₄) → l1(X₀, X₁, X₂, X₃, X₄)
t₁: l1(X₀, X₁, X₂, X₃, X₄) → l1(X₀+X₁, X₁+X₂, X₂-1, X₃, X₄) :|: 1 ≤ X₀
t₂: l1(X₀, X₁, X₂, X₃, X₄) → l2(X₀, X₁, X₂, X₃, X₄) :|: X₀ ≤ 0
t₃: l2(X₀, X₁, X₂, X₃, X₄) → l2(X₀, X₁, X₂, X₃+X₄, X₄-1) :|: 1 ≤ X₃

Preprocessing

Found invariant X₀ ≤ 0 for location l2

Problem after Preprocessing

Start: l0
Program_Vars: X₀, X₁, X₂, X₃, X₄
Temp_Vars:
Locations: l0, l1, l2
Transitions:
t₀: l0(X₀, X₁, X₂, X₃, X₄) → l1(X₀, X₁, X₂, X₃, X₄)
t₁: l1(X₀, X₁, X₂, X₃, X₄) → l1(X₀+X₁, X₁+X₂, X₂-1, X₃, X₄) :|: 1 ≤ X₀
t₂: l1(X₀, X₁, X₂, X₃, X₄) → l2(X₀, X₁, X₂, X₃, X₄) :|: X₀ ≤ 0
t₃: l2(X₀, X₁, X₂, X₃, X₄) → l2(X₀, X₁, X₂, X₃+X₄, X₄-1) :|: 1 ≤ X₃ ∧ X₀ ≤ 0

Found invariant X₀ ≤ 0 for location l2

Time-Bound by TWN-Loops:

TWN-Loops: t₁ 12⋅X₀+12⋅X₁+6⋅X₂+12 {O(n)}

TWN-Loops:

entry: t₀: l0(X₀, X₁, X₂, X₃, X₄) → l1(X₀, X₁, X₂, X₃, X₄)
results in twn-loop: twn: (X₀,X₁,X₂,X₃,X₄) -> (X₀+X₁,X₁+X₂,X₂-1,X₃,X₄) :|: 1 ≤ X₀
order: [X₂; X₁; X₀]
closed-form:
X₂: X₂ + [[n != 0]] * -1 * n^1
X₁: X₁ + [[n != 0]] * X₂ * n^1 + [[n != 0, n != 1]] * -1/2 * n^2 + [[n != 0, n != 1]] * 1/2 * n^1
X₀: X₀ + [[n != 0]] * X₁ * n^1 + [[n != 0, n != 1]] * 1/2⋅X₂ * n^2 + [[n != 0, n != 1]] * -1/2⋅X₂ * n^1 + [[n != 0, n != 1, n != 2]] * -1/6 * n^3 + [[n != 0, n != 1, n != 2]] * 1/2 * n^2 + [[n != 0, n != 1, n != 2]] * -1/3 * n^1

Termination: true
Formula:

1 < 0
∨ 0 < 3⋅X₂+3 ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ 3⋅X₂+2 < 6⋅X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 0 ≤ 3⋅X₂+3 ∧ 3⋅X₂+3 ≤ 0
∨ 6 < 6⋅X₀ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 0 ≤ 3⋅X₂+3 ∧ 3⋅X₂+3 ≤ 0 ∧ 3⋅X₂+2 ≤ 6⋅X₁ ∧ 6⋅X₁ ≤ 3⋅X₂+2
∨ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 0 ≤ 3⋅X₂+3 ∧ 3⋅X₂+3 ≤ 0 ∧ 3⋅X₂+2 ≤ 6⋅X₁ ∧ 6⋅X₁ ≤ 3⋅X₂+2 ∧ 6 ≤ 6⋅X₀ ∧ 6⋅X₀ ≤ 6

Stabilization-Threshold for: 1 ≤ X₀
alphas_abs: 3+6⋅X₀+6⋅X₁+3⋅X₂
M: 0
N: 3
Bound: 12⋅X₀+12⋅X₁+6⋅X₂+10 {O(n)}

relevant size-bounds w.r.t. t₀:
X₀: X₀ {O(n)}
X₁: X₁ {O(n)}
X₂: X₂ {O(n)}
Runtime-bound of t₀: 1 {O(1)}
Results in: 12⋅X₀+12⋅X₁+6⋅X₂+12 {O(n)}

12⋅X₀+12⋅X₁+6⋅X₂+12 {O(n)}

Found invariant X₀ ≤ 0 for location l2

Time-Bound by TWN-Loops:

TWN-Loops: t₃ 8⋅X₃+8⋅X₄+7 {O(n)}

TWN-Loops:

entry: t₂: l1(X₀, X₁, X₂, X₃, X₄) → l2(X₀, X₁, X₂, X₃, X₄) :|: X₀ ≤ 0
results in twn-loop: twn:Inv: [X₀ ≤ 0] , (X₀,X₁,X₂,X₃,X₄) -> (X₀,X₁,X₂,X₃+X₄,X₄-1) :|: 1 ≤ X₃
order: [X₀; X₄; X₃]
closed-form:
X₀: X₀
X₄: X₄ + [[n != 0]] * -1 * n^1
X₃: X₃ + [[n != 0]] * X₄ * n^1 + [[n != 0, n != 1]] * -1/2 * n^2 + [[n != 0, n != 1]] * 1/2 * n^1

Termination: true
Formula:

1 < 0
∨ 0 < 2⋅X₄+1 ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ 2 < 2⋅X₃ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 0 ≤ 2⋅X₄+1 ∧ 2⋅X₄+1 ≤ 0
∨ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 0 ≤ 2⋅X₄+1 ∧ 2⋅X₄+1 ≤ 0 ∧ 2 ≤ 2⋅X₃ ∧ 2⋅X₃ ≤ 2

Stabilization-Threshold for: 1 ≤ X₃
alphas_abs: 1+2⋅X₃+2⋅X₄
M: 0
N: 2
Bound: 4⋅X₃+4⋅X₄+5 {O(n)}

relevant size-bounds w.r.t. t₂:
X₃: 2⋅X₃ {O(n)}
X₄: 2⋅X₄ {O(n)}
Runtime-bound of t₂: 1 {O(1)}
Results in: 8⋅X₃+8⋅X₄+7 {O(n)}

8⋅X₃+8⋅X₄+7 {O(n)}

All Bounds

Timebounds

Overall timebound:12⋅X₀+12⋅X₁+6⋅X₂+8⋅X₃+8⋅X₄+21 {O(n)}
t₀: 1 {O(1)}
t₁: 12⋅X₀+12⋅X₁+6⋅X₂+12 {O(n)}
t₂: 1 {O(1)}
t₃: 8⋅X₃+8⋅X₄+7 {O(n)}

Costbounds

Overall costbound: 12⋅X₀+12⋅X₁+6⋅X₂+8⋅X₃+8⋅X₄+21 {O(n)}
t₀: 1 {O(1)}
t₁: 12⋅X₀+12⋅X₁+6⋅X₂+12 {O(n)}
t₂: 1 {O(1)}
t₃: 8⋅X₃+8⋅X₄+7 {O(n)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₀, X₃: X₃ {O(n)}
t₀, X₄: X₄ {O(n)}
t₁, X₀: 1584⋅X₀⋅X₂⋅X₂+1584⋅X₁⋅X₂⋅X₂+1728⋅X₀⋅X₀⋅X₀+1728⋅X₁⋅X₁⋅X₁+288⋅X₂⋅X₂⋅X₂+2880⋅X₀⋅X₀⋅X₂+2880⋅X₁⋅X₁⋅X₂+5184⋅X₀⋅X₀⋅X₁+5184⋅X₀⋅X₁⋅X₁+5760⋅X₀⋅X₁⋅X₂+10968⋅X₀⋅X₁+1680⋅X₂⋅X₂+5472⋅X₀⋅X₀+5496⋅X₁⋅X₁+6096⋅X₀⋅X₂+6108⋅X₁⋅X₂+3224⋅X₂+5773⋅X₀+5798⋅X₁+2028 {O(n^3)}
t₁, X₁: 144⋅X₀⋅X₀+144⋅X₁⋅X₁+168⋅X₀⋅X₂+168⋅X₁⋅X₂+288⋅X₀⋅X₁+48⋅X₂⋅X₂+176⋅X₂+300⋅X₀+301⋅X₁+156 {O(n^2)}
t₁, X₂: 12⋅X₀+12⋅X₁+7⋅X₂+12 {O(n)}
t₁, X₃: X₃ {O(n)}
t₁, X₄: X₄ {O(n)}
t₂, X₀: 1584⋅X₀⋅X₂⋅X₂+1584⋅X₁⋅X₂⋅X₂+1728⋅X₀⋅X₀⋅X₀+1728⋅X₁⋅X₁⋅X₁+288⋅X₂⋅X₂⋅X₂+2880⋅X₀⋅X₀⋅X₂+2880⋅X₁⋅X₁⋅X₂+5184⋅X₀⋅X₀⋅X₁+5184⋅X₀⋅X₁⋅X₁+5760⋅X₀⋅X₁⋅X₂+10968⋅X₀⋅X₁+1680⋅X₂⋅X₂+5472⋅X₀⋅X₀+5496⋅X₁⋅X₁+6096⋅X₀⋅X₂+6108⋅X₁⋅X₂+3224⋅X₂+5774⋅X₀+5798⋅X₁+2028 {O(n^3)}
t₂, X₁: 144⋅X₀⋅X₀+144⋅X₁⋅X₁+168⋅X₀⋅X₂+168⋅X₁⋅X₂+288⋅X₀⋅X₁+48⋅X₂⋅X₂+176⋅X₂+300⋅X₀+302⋅X₁+156 {O(n^2)}
t₂, X₂: 12⋅X₀+12⋅X₁+8⋅X₂+12 {O(n)}
t₂, X₃: 2⋅X₃ {O(n)}
t₂, X₄: 2⋅X₄ {O(n)}
t₃, X₀: 1584⋅X₀⋅X₂⋅X₂+1584⋅X₁⋅X₂⋅X₂+1728⋅X₀⋅X₀⋅X₀+1728⋅X₁⋅X₁⋅X₁+288⋅X₂⋅X₂⋅X₂+2880⋅X₀⋅X₀⋅X₂+2880⋅X₁⋅X₁⋅X₂+5184⋅X₀⋅X₀⋅X₁+5184⋅X₀⋅X₁⋅X₁+5760⋅X₀⋅X₁⋅X₂+10968⋅X₀⋅X₁+1680⋅X₂⋅X₂+5472⋅X₀⋅X₀+5496⋅X₁⋅X₁+6096⋅X₀⋅X₂+6108⋅X₁⋅X₂+3224⋅X₂+5774⋅X₀+5798⋅X₁+2028 {O(n^3)}
t₃, X₁: 144⋅X₀⋅X₀+144⋅X₁⋅X₁+168⋅X₀⋅X₂+168⋅X₁⋅X₂+288⋅X₀⋅X₁+48⋅X₂⋅X₂+176⋅X₂+300⋅X₀+302⋅X₁+156 {O(n^2)}
t₃, X₂: 12⋅X₀+12⋅X₁+8⋅X₂+12 {O(n)}
t₃, X₃: 160⋅X₃⋅X₄+64⋅X₃⋅X₃+96⋅X₄⋅X₄+122⋅X₃+152⋅X₄+56 {O(n^2)}
t₃, X₄: 10⋅X₄+8⋅X₃+7 {O(n)}