Initial Problem

Start: l0
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: l0, l1, l2, l3, l4, l5
Transitions:
t₀: l0(X₀, X₁, X₂) → l2(X₀, X₁, X₂)
t₂: l1(X₀, X₁, X₂) → l3(X₀, X₁, X₂) :|: 0 < X₁ ∧ X₀ < X₁
t₃: l1(X₀, X₁, X₂) → l4(X₀, X₁, X₂) :|: X₁ ≤ 0
t₄: l1(X₀, X₁, X₂) → l4(X₀, X₁, X₂) :|: X₁ ≤ X₀
t₁: l2(X₀, X₁, X₂) → l1(X₂, X₁, X₂)
t₅: l3(X₀, X₁, X₂) → l1(X₀+X₁, X₁, X₂)
t₆: l4(X₀, X₁, X₂) → l5(X₀, X₁, X₂)

Preprocessing

Found invariant X₂ ≤ X₀ for location l5

Found invariant X₂ ≤ X₀ for location l1

Found invariant X₂ ≤ X₀ for location l4

Found invariant 1+X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₁ ∧ 1+X₀ ≤ X₁ for location l3

Problem after Preprocessing

Start: l0
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: l0, l1, l2, l3, l4, l5
Transitions:
t₀: l0(X₀, X₁, X₂) → l2(X₀, X₁, X₂)
t₂: l1(X₀, X₁, X₂) → l3(X₀, X₁, X₂) :|: 0 < X₁ ∧ X₀ < X₁ ∧ X₂ ≤ X₀
t₃: l1(X₀, X₁, X₂) → l4(X₀, X₁, X₂) :|: X₁ ≤ 0 ∧ X₂ ≤ X₀
t₄: l1(X₀, X₁, X₂) → l4(X₀, X₁, X₂) :|: X₁ ≤ X₀ ∧ X₂ ≤ X₀
t₁: l2(X₀, X₁, X₂) → l1(X₂, X₁, X₂)
t₅: l3(X₀, X₁, X₂) → l1(X₀+X₁, X₁, X₂) :|: 1+X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₁ ∧ 1+X₀ ≤ X₁
t₆: l4(X₀, X₁, X₂) → l5(X₀, X₁, X₂) :|: X₂ ≤ X₀

Found invariant X₂ ≤ X₀ for location l5

Found invariant X₂ ≤ X₀ for location l1

Found invariant X₂ ≤ X₀ for location l4

Found invariant 1+X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₁ ∧ 1+X₀ ≤ X₁ for location l3

Time-Bound by TWN-Loops:

TWN-Loops: t₂ 2⋅X₁+2⋅X₂+5 {O(n)}

TWN-Loops:

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

Termination: true
Formula:

0 < X₁ ∧ X₁ < 0
∨ X₁ < 0 ∧ 0 < X₁ ∧ 1+X₀ < X₁ ∧ X₁ ≤ 0 ∧ 0 ≤ X₁
∨ X₁ < 0 ∧ 0 < X₁ ∧ X₁ ≤ 0 ∧ 0 ≤ X₁ ∧ 1+X₀ ≤ X₁ ∧ X₁ ≤ 1+X₀
∨ 0 < X₁ ∧ X₂ < X₀ ∧ 0 ≤ X₁ ∧ X₁ ≤ 0 ∧ X₁ < 0
∨ X₁ < 0 ∧ 0 < X₁ ∧ X₂ < X₀ ∧ 1+X₀ < X₁ ∧ X₁ ≤ 0 ∧ 0 ≤ X₁
∨ X₁ < 0 ∧ 0 < X₁ ∧ X₂ < X₀ ∧ X₁ ≤ 0 ∧ 0 ≤ X₁ ∧ 1+X₀ ≤ X₁ ∧ X₁ ≤ 1+X₀
∨ 0 < X₁ ∧ 0 ≤ X₁ ∧ X₁ ≤ 0 ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₂ ∧ X₁ < 0
∨ X₁ < 0 ∧ 0 < X₁ ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₂ ∧ 1+X₀ < X₁ ∧ X₁ ≤ 0 ∧ 0 ≤ X₁
∨ X₁ < 0 ∧ 0 < X₁ ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₂ ∧ X₁ ≤ 0 ∧ 0 ≤ X₁ ∧ 1+X₀ ≤ X₁ ∧ X₁ ≤ 1+X₀
∨ X₀ < X₁ ∧ X₁ ≤ 0 ∧ 0 ≤ X₁ ∧ 0 < X₁ ∧ X₁ < 0
∨ X₀ < X₁ ∧ 0 < X₁ ∧ 1+X₀ < X₁ ∧ X₁ ≤ 0 ∧ 0 ≤ X₁
∨ X₀ < X₁ ∧ 0 < X₁ ∧ X₁ ≤ 0 ∧ 0 ≤ X₁ ∧ 1+X₀ ≤ X₁ ∧ X₁ ≤ 1+X₀
∨ X₀ < X₁ ∧ 0 < X₁ ∧ X₂ < X₀ ∧ 0 ≤ X₁ ∧ X₁ ≤ 0 ∧ X₁ < 0
∨ X₀ < X₁ ∧ 0 < X₁ ∧ X₂ < X₀ ∧ 1+X₀ < X₁ ∧ X₁ ≤ 0 ∧ 0 ≤ X₁
∨ X₀ < X₁ ∧ 0 < X₁ ∧ X₂ < X₀ ∧ X₁ ≤ 0 ∧ 0 ≤ X₁ ∧ 1+X₀ ≤ X₁ ∧ X₁ ≤ 1+X₀
∨ X₀ < X₁ ∧ 0 < X₁ ∧ 0 ≤ X₁ ∧ X₁ ≤ 0 ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₂ ∧ X₁ < 0
∨ X₀ < X₁ ∧ 0 < X₁ ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₂ ∧ 1+X₀ < X₁ ∧ X₁ ≤ 0 ∧ 0 ≤ X₁
∨ X₀ < X₁ ∧ 0 < X₁ ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₂ ∧ X₁ ≤ 0 ∧ 0 ≤ X₁ ∧ 1+X₀ ≤ X₁ ∧ X₁ ≤ 1+X₀

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

relevant size-bounds w.r.t. t₁:
X₀: X₂ {O(n)}
X₁: X₁ {O(n)}
Runtime-bound of t₁: 1 {O(1)}
Results in: 2⋅X₁+2⋅X₂+5 {O(n)}

2⋅X₁+2⋅X₂+5 {O(n)}

Time-Bound by TWN-Loops:

TWN-Loops: t₅ 2⋅X₁+2⋅X₂+5 {O(n)}

relevant size-bounds w.r.t. t₁:
X₀: X₂ {O(n)}
X₁: X₁ {O(n)}
Runtime-bound of t₁: 1 {O(1)}
Results in: 2⋅X₁+2⋅X₂+5 {O(n)}

2⋅X₁+2⋅X₂+5 {O(n)}

All Bounds

Timebounds

Overall timebound:4⋅X₁+4⋅X₂+15 {O(n)}
t₀: 1 {O(1)}
t₂: 2⋅X₁+2⋅X₂+5 {O(n)}
t₃: 1 {O(1)}
t₄: 1 {O(1)}
t₁: 1 {O(1)}
t₅: 2⋅X₁+2⋅X₂+5 {O(n)}
t₆: 1 {O(1)}

Costbounds

Overall costbound: 4⋅X₁+4⋅X₂+15 {O(n)}
t₀: 1 {O(1)}
t₂: 2⋅X₁+2⋅X₂+5 {O(n)}
t₃: 1 {O(1)}
t₄: 1 {O(1)}
t₁: 1 {O(1)}
t₅: 2⋅X₁+2⋅X₂+5 {O(n)}
t₆: 1 {O(1)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₂, X₀: 2⋅X₁⋅X₁+2⋅X₁⋅X₂+6⋅X₁+X₂ {O(n^2)}
t₂, X₁: X₁ {O(n)}
t₂, X₂: X₂ {O(n)}
t₃, X₀: X₂ {O(n)}
t₃, X₁: X₁ {O(n)}
t₃, X₂: X₂ {O(n)}
t₄, X₀: 2⋅X₁⋅X₁+2⋅X₁⋅X₂+2⋅X₂+6⋅X₁ {O(n^2)}
t₄, X₁: 2⋅X₁ {O(n)}
t₄, X₂: 2⋅X₂ {O(n)}
t₁, X₀: X₂ {O(n)}
t₁, X₁: X₁ {O(n)}
t₁, X₂: X₂ {O(n)}
t₅, X₀: 2⋅X₁⋅X₁+2⋅X₁⋅X₂+6⋅X₁+X₂ {O(n^2)}
t₅, X₁: X₁ {O(n)}
t₅, X₂: X₂ {O(n)}
t₆, X₀: 2⋅X₁⋅X₁+2⋅X₁⋅X₂+3⋅X₂+6⋅X₁ {O(n^2)}
t₆, X₁: 3⋅X₁ {O(n)}
t₆, X₂: 3⋅X₂ {O(n)}