Initial Problem

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

Preprocessing

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

Found invariant X₀ ≤ X₁ for location l1

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

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

Problem after Preprocessing

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

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

Found invariant X₀ ≤ X₁ for location l1

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

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

Time-Bound by TWN-Loops:

TWN-Loops: t₂ 2⋅X₁+4 {O(n)}

TWN-Loops:

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

Termination: true
Formula:

1 < 0
∨ 0 < X₀ ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 0 ≤ X₀ ∧ X₀ ≤ 0

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

relevant size-bounds w.r.t. t₁:
X₀: X₁ {O(n)}
Runtime-bound of t₁: 1 {O(1)}
Results in: 2⋅X₁+4 {O(n)}

2⋅X₁+4 {O(n)}

Time-Bound by TWN-Loops:

TWN-Loops: t₄ 2⋅X₁+4 {O(n)}

relevant size-bounds w.r.t. t₁:
X₀: X₁ {O(n)}
Runtime-bound of t₁: 1 {O(1)}
Results in: 2⋅X₁+4 {O(n)}

2⋅X₁+4 {O(n)}

All Bounds

Timebounds

Overall timebound:4⋅X₁+12 {O(n)}
t₀: 1 {O(1)}
t₂: 2⋅X₁+4 {O(n)}
t₃: 1 {O(1)}
t₁: 1 {O(1)}
t₄: 2⋅X₁+4 {O(n)}
t₅: 1 {O(1)}

Costbounds

Overall costbound: 4⋅X₁+12 {O(n)}
t₀: 1 {O(1)}
t₂: 2⋅X₁+4 {O(n)}
t₃: 1 {O(1)}
t₁: 1 {O(1)}
t₄: 2⋅X₁+4 {O(n)}
t₅: 1 {O(1)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₂, X₀: X₁+1 {O(n)}
t₂, X₁: X₁ {O(n)}
t₃, X₀: 2⋅X₁+1 {O(n)}
t₃, X₁: 2⋅X₁ {O(n)}
t₁, X₀: X₁ {O(n)}
t₁, X₁: X₁ {O(n)}
t₄, X₀: X₁+1 {O(n)}
t₄, X₁: X₁ {O(n)}
t₅, X₀: 2⋅X₁+1 {O(n)}
t₅, X₁: 2⋅X₁ {O(n)}