KoAT2 Proof Output

Initial Problem

Start: l0
Program_Vars: X₀, X₁
Temp_Vars:
Locations: l0, l1
Transitions:
t₀: l0(X₀, X₁) → l1(X₀, X₁)
t₁: l1(X₀, X₁) → l1(X₀, 1+X₁-2⋅X₀) :|: 1 ≤ X₀+X₁ ∧ X₁ ≤ X₀

Preprocessing

Problem after Preprocessing

TWN: t₁: l1→l1

cycle: [t₁: l1→l1]
original loop: (1 ≤ X₀+X₁ ∧ X₁ ≤ X₀,(X₀,X₁) -> (X₀,1+X₁-2⋅X₀))
transformed loop: (1 ≤ X₀+X₁ ∧ X₁ ≤ X₀,(X₀,X₁) -> (X₀,1+X₁-2⋅X₀))
loop: (1 ≤ X₀+X₁ ∧ X₁ ≤ X₀,(X₀,X₁) -> (X₀,1+X₁-2⋅X₀))
order: [X₀; X₁]
closed-form:

X₀: X₀
X₁: X₁ + [[n != 0]]⋅(1-2⋅X₀)⋅n^1

Termination: true
Formula:

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

Stabilization-Threshold for: X₁ ≤ X₀

alphas_abs: 1+X₀+X₁
M: 0
N: 1
Bound: 2⋅X₀+2⋅X₁+4 {O(n)}

Stabilization-Threshold for: 1 ≤ X₀+X₁

alphas_abs: X₀+X₁
M: 0
N: 1
Bound: 2⋅X₀+2⋅X₁+2 {O(n)}

TWN - Lifting for [1: l1->l1] of 4⋅X₀+4⋅X₁+8 {O(n)}

relevant size-bounds w.r.t. t₀: l0→l1:
X₀: X₀ {O(n)}
X₁: X₁ {O(n)}
Runtime-bound of t₀: 1 {O(1)}
Results in: 4⋅X₀+4⋅X₁+8 {O(n)}

All Bounds

Timebounds

Overall timebound:4⋅X₀+4⋅X₁+9 {O(n)}
t₀: 1 {O(1)}
t₁: 4⋅X₀+4⋅X₁+8 {O(n)}

Costbounds

Overall costbound: 4⋅X₀+4⋅X₁+9 {O(n)}
t₀: 1 {O(1)}
t₁: 4⋅X₀+4⋅X₁+8 {O(n)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₁, X₀: X₀ {O(n)}
t₁, X₁: 6⋅X₀ {O(n)}