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(X₃, X₁, X₂, X₃)
t₄: l3(X₀, X₁, X₂, X₃) → l1(X₀-1, X₁, X₂, X₃)
t₅: l4(X₀, X₁, X₂, X₃) → l5(X₀, X₁, X₂, X₃)

Preprocessing

Eliminate variables {X₁,X₂} that do not contribute to the problem

Found invariant X₀ ≤ X₃ ∧ X₀ ≤ 0 for location l5

Found invariant X₀ ≤ X₃ for location l1

Found invariant X₀ ≤ X₃ ∧ X₀ ≤ 0 for location l4

Found invariant 1 ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ 1 ≤ X₀ for location l3

Problem after Preprocessing

Start: l0
Program_Vars: X₀, X₃
Temp_Vars:
Locations: l0, l1, l2, l3, l4, l5
Transitions:
t₁₂: l0(X₀, X₃) → l2(X₀, X₃)
t₁₃: l1(X₀, X₃) → l3(X₀, X₃) :|: 0 < X₀ ∧ X₀ ≤ X₃
t₁₄: l1(X₀, X₃) → l4(X₀, X₃) :|: X₀ ≤ 0 ∧ X₀ ≤ X₃
t₁₅: l2(X₀, X₃) → l1(X₃, X₃)
t₁₆: l3(X₀, X₃) → l1(X₀-1, X₃) :|: 1 ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ 1 ≤ X₀
t₁₇: l4(X₀, X₃) → l5(X₀, X₃) :|: X₀ ≤ X₃ ∧ X₀ ≤ 0

Found invariant X₀ ≤ X₃ ∧ X₀ ≤ 0 for location l5

Found invariant X₀ ≤ X₃ for location l1

Found invariant X₀ ≤ X₃ ∧ X₀ ≤ 0 for location l4

Found invariant 1 ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ 1 ≤ X₀ for location l3

Time-Bound by TWN-Loops:

TWN-Loops: t₁₃ 2⋅X₃+4 {O(n)}

TWN-Loops:

entry: t₁₅: l2(X₀, X₃) → l1(X₃, X₃)
results in twn-loop: twn:Inv: [X₀ ≤ X₃ ∧ 1 ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ 1 ≤ X₀] , (X₀,X₃) -> (X₀-1,X₃) :|: 0 < X₀
order: [X₀; X₃]
closed-form:
X₀: X₀ + [[n != 0]] * -1 * n^1
X₃: X₃

Termination: true
Formula:

1 < 0
∨ 0 < X₀ ∧ 1 ≤ 0 ∧ 0 ≤ 1

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

relevant size-bounds w.r.t. t₁₅:
X₀: X₃ {O(n)}
Runtime-bound of t₁₅: 1 {O(1)}
Results in: 2⋅X₃+4 {O(n)}

2⋅X₃+4 {O(n)}

Time-Bound by TWN-Loops:

TWN-Loops: t₁₆ 2⋅X₃+4 {O(n)}

relevant size-bounds w.r.t. t₁₅:
X₀: X₃ {O(n)}
Runtime-bound of t₁₅: 1 {O(1)}
Results in: 2⋅X₃+4 {O(n)}

2⋅X₃+4 {O(n)}

All Bounds

Timebounds

Overall timebound:4⋅X₃+12 {O(n)}
t₁₂: 1 {O(1)}
t₁₃: 2⋅X₃+4 {O(n)}
t₁₄: 1 {O(1)}
t₁₅: 1 {O(1)}
t₁₆: 2⋅X₃+4 {O(n)}
t₁₇: 1 {O(1)}

Costbounds

Overall costbound: 4⋅X₃+12 {O(n)}
t₁₂: 1 {O(1)}
t₁₃: 2⋅X₃+4 {O(n)}
t₁₄: 1 {O(1)}
t₁₅: 1 {O(1)}
t₁₆: 2⋅X₃+4 {O(n)}
t₁₇: 1 {O(1)}

Sizebounds

t₁₂, X₀: X₀ {O(n)}
t₁₂, X₃: X₃ {O(n)}
t₁₃, X₀: X₃ {O(n)}
t₁₃, X₃: X₃ {O(n)}
t₁₄, X₀: 2⋅X₃ {O(n)}
t₁₄, X₃: 2⋅X₃ {O(n)}
t₁₅, X₀: X₃ {O(n)}
t₁₅, X₃: X₃ {O(n)}
t₁₆, X₀: X₃ {O(n)}
t₁₆, X₃: X₃ {O(n)}
t₁₇, X₀: 2⋅X₃ {O(n)}
t₁₇, X₃: 2⋅X₃ {O(n)}