Initial Problem

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

Preprocessing

Found invariant X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 0 ≤ X₀ for location l1

Problem after Preprocessing

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

knowledge_propagation leads to new time bound 1 {O(1)} for transition t₀: l2(X₀, X₁) → l1(X₀, X₁) :|: 1 ≤ X₀ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁

Found invariant X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 0 ≤ X₀ for location l1

Time-Bound by TWN-Loops:

TWN-Loops: t₁ 2⋅X₀+6 {O(n)}

TWN-Loops:

entry: t₀: l2(X₀, X₁) → l1(X₀, X₁) :|: 1 ≤ X₀ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁
results in twn-loop: twn:Inv: [X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 0 ≤ X₀] , (X₀,X₁) -> (X₀-1,X₁-1) :|: 1 ≤ X₀
order: [X₀; X₁]
closed-form:
X₀: X₀ + [[n != 0]] * -1 * n^1
X₁: X₁ + [[n != 0]] * -1 * n^1

Termination: true
Formula:

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

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

relevant size-bounds w.r.t. t₀:
X₀: X₀ {O(n)}
Runtime-bound of t₀: 1 {O(1)}
Results in: 2⋅X₀+6 {O(n)}

2⋅X₀+6 {O(n)}

Found invariant 1 ≤ 0 for location l2

Found invariant 1 ≤ 0 for location l1

Found invariant X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ for location l1

Time-Bound by TWN-Loops:

TWN-Loops: t₂ 12⋅X₀+42 {O(n)}

TWN-Loops:

entry: t₁: l1(X₀, X₁) → l1(X₀-1, X₁-1) :|: 1 ≤ X₀ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 0 ≤ X₀
results in twn-loop: twn:Inv: [X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 0 ≤ X₀] , (X₀,X₁) -> (X₀,X₁) :|: X₀ ≤ 0 ∧ 1 ≤ X₀ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁
entry: t₃: l0(X₀, X₁) → l2(X₀, X₁)
results in twn-loop: twn:Inv: [X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 0 ≤ X₀] , (X₀,X₁) -> (X₀,X₁) :|: 1 ≤ X₀ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁ ∧ X₀ ≤ 0
order: [X₀; X₁]
closed-form:
X₀: X₀
X₁: X₁

Termination: true
Formula:

X₀ < X₁ ∧ X₁ < X₀ ∧ 1 < X₀ ∧ X₀ < 0
∨ X₀ < X₁ ∧ X₁ < X₀ ∧ 1 < X₀ ∧ X₀ ≤ 0 ∧ 0 ≤ X₀
∨ X₀ < X₁ ∧ X₁ < X₀ ∧ 1 ≤ X₀ ∧ X₀ ≤ 1 ∧ X₀ < 0
∨ X₀ < X₁ ∧ X₁ < X₀ ∧ 1 ≤ X₀ ∧ X₀ ≤ 1 ∧ X₀ ≤ 0 ∧ 0 ≤ X₀
∨ X₀ < X₁ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁ ∧ 1 < X₀ ∧ X₀ < 0
∨ X₀ < X₁ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁ ∧ 1 < X₀ ∧ X₀ ≤ 0 ∧ 0 ≤ X₀
∨ X₀ < X₁ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ ∧ X₀ ≤ 1 ∧ X₀ < 0
∨ X₀ < X₁ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ ∧ X₀ ≤ 1 ∧ X₀ ≤ 0 ∧ 0 ≤ X₀
∨ X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ X₁ < X₀ ∧ 1 < X₀ ∧ X₀ < 0
∨ X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ X₁ < X₀ ∧ 1 < X₀ ∧ X₀ ≤ 0 ∧ 0 ≤ X₀
∨ X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ X₁ < X₀ ∧ 1 ≤ X₀ ∧ X₀ ≤ 1 ∧ X₀ < 0
∨ X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ X₁ < X₀ ∧ 1 ≤ X₀ ∧ X₀ ≤ 1 ∧ X₀ ≤ 0 ∧ 0 ≤ X₀
∨ X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁ ∧ 1 < X₀ ∧ X₀ < 0
∨ X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁ ∧ 1 < X₀ ∧ X₀ ≤ 0 ∧ 0 ≤ X₀
∨ X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ ∧ X₀ ≤ 1 ∧ X₀ < 0
∨ X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ ∧ X₀ ≤ 1 ∧ X₀ ≤ 0 ∧ 0 ≤ X₀

relevant size-bounds w.r.t. t₁:
Runtime-bound of t₁: 2⋅X₀+6 {O(n)}
Results in: 12⋅X₀+36 {O(n)}

order: [X₀; X₁]
closed-form:
X₀: X₀
X₁: X₁

Termination: true
Formula:

X₀ < 0 ∧ X₀ < X₁ ∧ X₁ < X₀ ∧ 1 < X₀
∨ X₀ < 0 ∧ X₀ < X₁ ∧ X₁ < X₀ ∧ 1 ≤ X₀ ∧ X₀ ≤ 1
∨ X₀ < 0 ∧ X₀ < X₁ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁ ∧ 1 < X₀
∨ X₀ < 0 ∧ X₀ < X₁ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ ∧ X₀ ≤ 1
∨ X₀ < 0 ∧ X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ X₁ < X₀ ∧ 1 < X₀
∨ X₀ < 0 ∧ X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ X₁ < X₀ ∧ 1 ≤ X₀ ∧ X₀ ≤ 1
∨ X₀ < 0 ∧ X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁ ∧ 1 < X₀
∨ X₀ < 0 ∧ X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ ∧ X₀ ≤ 1
∨ X₀ ≤ 0 ∧ 0 ≤ X₀ ∧ X₀ < X₁ ∧ X₁ < X₀ ∧ 1 < X₀
∨ X₀ ≤ 0 ∧ 0 ≤ X₀ ∧ X₀ < X₁ ∧ X₁ < X₀ ∧ 1 ≤ X₀ ∧ X₀ ≤ 1
∨ X₀ ≤ 0 ∧ 0 ≤ X₀ ∧ X₀ < X₁ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁ ∧ 1 < X₀
∨ X₀ ≤ 0 ∧ 0 ≤ X₀ ∧ X₀ < X₁ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ ∧ X₀ ≤ 1
∨ X₀ ≤ 0 ∧ 0 ≤ X₀ ∧ X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ X₁ < X₀ ∧ 1 < X₀
∨ X₀ ≤ 0 ∧ 0 ≤ X₀ ∧ X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ X₁ < X₀ ∧ 1 ≤ X₀ ∧ X₀ ≤ 1
∨ X₀ ≤ 0 ∧ 0 ≤ X₀ ∧ X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁ ∧ 1 < X₀
∨ X₀ ≤ 0 ∧ 0 ≤ X₀ ∧ X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ ∧ X₀ ≤ 1

relevant size-bounds w.r.t. t₃:
Runtime-bound of t₃: 1 {O(1)}
Results in: 6 {O(1)}

12⋅X₀+42 {O(n)}

All Bounds

Timebounds

Overall timebound:14⋅X₀+50 {O(n)}
t₃: 1 {O(1)}
t₁: 2⋅X₀+6 {O(n)}
t₂: 12⋅X₀+42 {O(n)}
t₀: 1 {O(1)}

Costbounds

Overall costbound: 14⋅X₀+50 {O(n)}
t₃: 1 {O(1)}
t₁: 2⋅X₀+6 {O(n)}
t₂: 12⋅X₀+42 {O(n)}
t₀: 1 {O(1)}

Sizebounds

t₃, X₀: X₀ {O(n)}
t₃, X₁: X₁ {O(n)}
t₁, X₀: X₀ {O(n)}
t₁, X₁: X₁ {O(n)}
t₂, X₀: 0 {O(1)}
t₂, X₁: 0 {O(1)}
t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}