Initial Problem

Start: l0
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars: E
Locations: l0, l1, l2
Transitions:
t₀: l0(X₀, X₁, X₂, X₃) → l1(0, X₁, X₂, X₃)
t₃: l1(X₀, X₁, X₂, X₃) → l1(0, 1+X₁, X₂-1, X₃) :|: 2+X₁ ≤ X₂ ∧ 1+X₁ ≤ X₂ ∧ X₀ ≤ 0 ∧ 0 ≤ X₀
t₁: l1(X₀, X₁, X₂, X₃) → l2(X₀, X₁, X₂, E) :|: X₂ ≤ X₁
t₂: l1(X₀, X₁, X₂, X₃) → l2(1, 1+X₁, X₂, E) :|: X₁+1 ≤ X₂ ∧ X₂ ≤ X₁+1 ∧ X₀ ≤ 0 ∧ 0 ≤ X₀

Preprocessing

Eliminate variables {E,X₃} that do not contribute to the problem

Found invariant X₂ ≤ X₁ ∧ X₀ ≤ 1 ∧ 0 ≤ X₀ for location l2

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

Problem after Preprocessing

Start: l0
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: l0, l1, l2
Transitions:
t₉: l0(X₀, X₁, X₂) → l1(0, X₁, X₂)
t₁₂: l1(X₀, X₁, X₂) → l1(0, 1+X₁, X₂-1) :|: 2+X₁ ≤ X₂ ∧ 1+X₁ ≤ X₂ ∧ X₀ ≤ 0 ∧ 0 ≤ X₀ ∧ X₀ ≤ 0 ∧ 0 ≤ X₀
t₁₀: l1(X₀, X₁, X₂) → l2(X₀, X₁, X₂) :|: X₂ ≤ X₁ ∧ X₀ ≤ 0 ∧ 0 ≤ X₀
t₁₁: l1(X₀, X₁, X₂) → l2(1, 1+X₁, X₂) :|: X₁+1 ≤ X₂ ∧ X₂ ≤ X₁+1 ∧ X₀ ≤ 0 ∧ 0 ≤ X₀ ∧ X₀ ≤ 0 ∧ 0 ≤ X₀

Found invariant X₂ ≤ X₁ ∧ X₀ ≤ 1 ∧ 0 ≤ X₀ for location l2

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

Time-Bound by TWN-Loops:

TWN-Loops: t₁₂ 4⋅X₁+4⋅X₂+12 {O(n)}

TWN-Loops:

entry: t₉: l0(X₀, X₁, X₂) → l1(0, X₁, X₂)
results in twn-loop: twn:Inv: [X₀ ≤ 0 ∧ 0 ≤ X₀] , (X₀,X₁,X₂) -> (0,1+X₁,X₂-1) :|: 2+X₁ ≤ X₂ ∧ 1+X₁ ≤ X₂ ∧ X₀ ≤ 0 ∧ 0 ≤ X₀
order: [X₀; X₁; X₂]
closed-form:
X₀: [[n == 0]] * X₀
X₁: X₁ + [[n != 0]] * n^1
X₂: X₂ + [[n != 0]] * -1 * n^1

Termination: true
Formula:

2 < 0
∨ 2 < 0 ∧ 2+X₁ < X₂ ∧ 2 ≤ 0 ∧ 0 ≤ 2
∨ 2 < 0 ∧ 2 ≤ 0 ∧ 0 ≤ 2 ∧ 2+X₁ ≤ X₂ ∧ X₂ ≤ 2+X₁
∨ 1+X₁ < X₂ ∧ 2 ≤ 0 ∧ 0 ≤ 2 ∧ 2 < 0
∨ 1+X₁ < X₂ ∧ 2+X₁ < X₂ ∧ 2 ≤ 0 ∧ 0 ≤ 2
∨ 1+X₁ < X₂ ∧ 2 ≤ 0 ∧ 0 ≤ 2 ∧ 2+X₁ ≤ X₂ ∧ X₂ ≤ 2+X₁
∨ 2 ≤ 0 ∧ 0 ≤ 2 ∧ 1+X₁ ≤ X₂ ∧ X₂ ≤ 1+X₁ ∧ 2 < 0
∨ 1+X₁ ≤ X₂ ∧ X₂ ≤ 1+X₁ ∧ 2+X₁ < X₂ ∧ 2 ≤ 0 ∧ 0 ≤ 2
∨ 1+X₁ ≤ X₂ ∧ X₂ ≤ 1+X₁ ∧ 2 ≤ 0 ∧ 0 ≤ 2 ∧ 2+X₁ ≤ X₂ ∧ X₂ ≤ 2+X₁

Stabilization-Threshold for: 1+X₁ ≤ X₂
alphas_abs: 1+X₁+X₂
M: 0
N: 1
Bound: 2⋅X₁+2⋅X₂+4 {O(n)}
Stabilization-Threshold for: 2+X₁ ≤ X₂
alphas_abs: 2+X₁+X₂
M: 0
N: 1
Bound: 2⋅X₁+2⋅X₂+6 {O(n)}

relevant size-bounds w.r.t. t₉:
X₁: X₁ {O(n)}
X₂: X₂ {O(n)}
Runtime-bound of t₉: 1 {O(1)}
Results in: 4⋅X₁+4⋅X₂+12 {O(n)}

4⋅X₁+4⋅X₂+12 {O(n)}

All Bounds

Timebounds

Overall timebound:4⋅X₁+4⋅X₂+15 {O(n)}
t₉: 1 {O(1)}
t₁₀: 1 {O(1)}
t₁₁: 1 {O(1)}
t₁₂: 4⋅X₁+4⋅X₂+12 {O(n)}

Costbounds

Overall costbound: 4⋅X₁+4⋅X₂+15 {O(n)}
t₉: 1 {O(1)}
t₁₀: 1 {O(1)}
t₁₁: 1 {O(1)}
t₁₂: 4⋅X₁+4⋅X₂+12 {O(n)}

Sizebounds

t₉, X₀: 0 {O(1)}
t₉, X₁: X₁ {O(n)}
t₉, X₂: X₂ {O(n)}
t₁₀, X₀: 0 {O(1)}
t₁₀, X₁: 4⋅X₂+6⋅X₁+12 {O(n)}
t₁₀, X₂: 4⋅X₁+6⋅X₂+12 {O(n)}
t₁₁, X₀: 1 {O(1)}
t₁₁, X₁: 4⋅X₂+6⋅X₁+14 {O(n)}
t₁₁, X₂: 4⋅X₁+6⋅X₂+12 {O(n)}
t₁₂, X₀: 0 {O(1)}
t₁₂, X₁: 4⋅X₂+5⋅X₁+12 {O(n)}
t₁₂, X₂: 4⋅X₁+5⋅X₂+12 {O(n)}