Initial Problem

Start: l0
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars:
Locations: l0, l1, l2, l3, l4, l5
Transitions:
t₀: l0(X₀, X₁, X₂, X₃) → l2(X₀, X₁, X₂, X₃)
t₂: l1(X₀, X₁, X₂, X₃) → l3(X₀, X₁, X₂, X₃) :|: 0 ≤ X₀
t₃: l1(X₀, X₁, X₂, X₃) → l4(X₀, X₁, X₂, X₃) :|: X₀ < 0
t₁: l2(X₀, X₁, X₂, X₃) → l1(X₂, X₃, X₂, X₃)
t₄: l3(X₀, X₁, X₂, X₃) → l1(X₀+X₁, X₁-1, X₂, X₃)
t₅: l4(X₀, X₁, X₂, X₃) → l5(X₀, X₁, X₂, X₃)

Preprocessing

Found invariant X₁ ≤ X₃ ∧ 1+X₀ ≤ 0 for location l5

Found invariant X₁ ≤ X₃ for location l1

Found invariant X₁ ≤ X₃ ∧ 1+X₀ ≤ 0 for location l4

Found invariant X₁ ≤ X₃ ∧ 0 ≤ X₀ for location l3

Problem after Preprocessing

Start: l0
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars:
Locations: l0, l1, l2, l3, l4, l5
Transitions:
t₀: l0(X₀, X₁, X₂, X₃) → l2(X₀, X₁, X₂, X₃)
t₂: l1(X₀, X₁, X₂, X₃) → l3(X₀, X₁, X₂, X₃) :|: 0 ≤ X₀ ∧ X₁ ≤ X₃
t₃: l1(X₀, X₁, X₂, X₃) → l4(X₀, X₁, X₂, X₃) :|: X₀ < 0 ∧ X₁ ≤ X₃
t₁: l2(X₀, X₁, X₂, X₃) → l1(X₂, X₃, X₂, X₃)
t₄: l3(X₀, X₁, X₂, X₃) → l1(X₀+X₁, X₁-1, X₂, X₃) :|: X₁ ≤ X₃ ∧ 0 ≤ X₀
t₅: l4(X₀, X₁, X₂, X₃) → l5(X₀, X₁, X₂, X₃) :|: X₁ ≤ X₃ ∧ 1+X₀ ≤ 0

Found invariant X₁ ≤ X₃ ∧ 1+X₀ ≤ 0 for location l5

Found invariant X₁ ≤ X₃ for location l1

Found invariant X₁ ≤ X₃ ∧ 1+X₀ ≤ 0 for location l4

Found invariant X₁ ≤ X₃ ∧ 0 ≤ X₀ for location l3

Time-Bound by TWN-Loops:

TWN-Loops: t₂ 4⋅X₂+4⋅X₃+7 {O(n)}

TWN-Loops:

entry: t₁: l2(X₀, X₁, X₂, X₃) → l1(X₂, X₃, X₂, X₃)
results in twn-loop: twn:Inv: [X₁ ≤ X₃ ∧ X₁ ≤ X₃ ∧ 0 ≤ X₀] , (X₀,X₁,X₂,X₃) -> (X₀+X₁,X₁-1,X₂,X₃) :|: 0 ≤ X₀
order: [X₁; X₀; X₃]
closed-form:
X₁: X₁ + [[n != 0]] * -1 * n^1
X₀: X₀ + [[n != 0]] * X₁ * n^1 + [[n != 0, n != 1]] * -1/2 * n^2 + [[n != 0, n != 1]] * 1/2 * n^1
X₃: X₃

Termination: true
Formula:

1 < 0
∨ 0 < 2⋅X₁+1 ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ 0 < 2⋅X₀ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 0 ≤ 2⋅X₁+1 ∧ 2⋅X₁+1 ≤ 0
∨ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 0 ≤ 2⋅X₁+1 ∧ 2⋅X₁+1 ≤ 0 ∧ 0 ≤ 2⋅X₀ ∧ 2⋅X₀ ≤ 0

Stabilization-Threshold for: 0 ≤ X₀
alphas_abs: 1+2⋅X₀+2⋅X₁
M: 0
N: 2
Bound: 4⋅X₀+4⋅X₁+5 {O(n)}

relevant size-bounds w.r.t. t₁:
X₀: X₂ {O(n)}
X₁: X₃ {O(n)}
Runtime-bound of t₁: 1 {O(1)}
Results in: 4⋅X₂+4⋅X₃+7 {O(n)}

4⋅X₂+4⋅X₃+7 {O(n)}

Time-Bound by TWN-Loops:

TWN-Loops: t₄ 4⋅X₂+4⋅X₃+7 {O(n)}

relevant size-bounds w.r.t. t₁:
X₀: X₂ {O(n)}
X₁: X₃ {O(n)}
Runtime-bound of t₁: 1 {O(1)}
Results in: 4⋅X₂+4⋅X₃+7 {O(n)}

4⋅X₂+4⋅X₃+7 {O(n)}

All Bounds

Timebounds

Overall timebound:8⋅X₂+8⋅X₃+18 {O(n)}
t₀: 1 {O(1)}
t₂: 4⋅X₂+4⋅X₃+7 {O(n)}
t₃: 1 {O(1)}
t₁: 1 {O(1)}
t₄: 4⋅X₂+4⋅X₃+7 {O(n)}
t₅: 1 {O(1)}

Costbounds

Overall costbound: 8⋅X₂+8⋅X₃+18 {O(n)}
t₀: 1 {O(1)}
t₂: 4⋅X₂+4⋅X₃+7 {O(n)}
t₃: 1 {O(1)}
t₁: 1 {O(1)}
t₄: 4⋅X₂+4⋅X₃+7 {O(n)}
t₅: 1 {O(1)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₀, X₃: X₃ {O(n)}
t₂, X₀: 16⋅X₂⋅X₂+20⋅X₃⋅X₃+36⋅X₂⋅X₃+61⋅X₂+68⋅X₃+56 {O(n^2)}
t₂, X₁: 4⋅X₂+5⋅X₃+7 {O(n)}
t₂, X₂: X₂ {O(n)}
t₂, X₃: X₃ {O(n)}
t₃, X₀: 16⋅X₂⋅X₂+20⋅X₃⋅X₃+36⋅X₂⋅X₃+62⋅X₂+68⋅X₃+56 {O(n^2)}
t₃, X₁: 4⋅X₂+6⋅X₃+7 {O(n)}
t₃, X₂: 2⋅X₂ {O(n)}
t₃, X₃: 2⋅X₃ {O(n)}
t₁, X₀: X₂ {O(n)}
t₁, X₁: X₃ {O(n)}
t₁, X₂: X₂ {O(n)}
t₁, X₃: X₃ {O(n)}
t₄, X₀: 16⋅X₂⋅X₂+20⋅X₃⋅X₃+36⋅X₂⋅X₃+61⋅X₂+68⋅X₃+56 {O(n^2)}
t₄, X₁: 4⋅X₂+5⋅X₃+7 {O(n)}
t₄, X₂: X₂ {O(n)}
t₄, X₃: X₃ {O(n)}
t₅, X₀: 16⋅X₂⋅X₂+20⋅X₃⋅X₃+36⋅X₂⋅X₃+62⋅X₂+68⋅X₃+56 {O(n^2)}
t₅, X₁: 4⋅X₂+6⋅X₃+7 {O(n)}
t₅, X₂: 2⋅X₂ {O(n)}
t₅, X₃: 2⋅X₃ {O(n)}