KoAT2 Proof Output

Initial Problem

Start: l0
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars:
Locations: l0, l1
Transitions:
t₀: l0(X₀, X₁, X₂, X₃) → l1(X₀, X₁, X₂, X₃)
t₁: l1(X₀, X₁, X₂, X₃) → l1(2, 1+X₁, -X₀-2⋅X₂, X₁) :|: 1 ≤ X₂+X₃

Preprocessing

Problem after Preprocessing

TWN: t₁: l1→l1

cycle: [t₁: l1→l1]
original loop: (1 ≤ X₂+X₃,(X₀,X₁,X₂,X₃) -> (2,1+X₁,-X₀-2⋅X₂,X₁))
transformed loop: (1 ≤ X₂+X₃,(X₀,X₁,X₂,X₃) -> (2,1+X₁,-X₀-2⋅X₂,X₁))
loop: (1 ≤ X₂+X₃,(X₀,X₁,X₂,X₃) -> (2,1+X₁,-X₀-2⋅X₂,X₁))
order: [X₁; X₀; X₃; X₂]
closed-form:

X₁: X₁ + [[n != 0]]⋅2⋅n^1
X₀: [[n == 0]]⋅X₀ + [[n != 0]]⋅2
X₃: [[n == 0]]⋅X₃ + [[n != 0]]⋅(1+X₁) + [[n != 0, n != 1]]⋅2⋅n^1 + [[n != 0, n != 1]]⋅-2
X₂: X₂⋅(4)^n + [[n != 0]]⋅(1/2⋅X₀-2/3)⋅(4)^n + [[n != 0]]⋅2/3 + [[n != 0, n != 1]]⋅1/3⋅(4)^n + [[n != 0, n != 1]]⋅-4/3

Termination: true
Formula:

3⋅X₁ ≤ 8 ∧ 3⋅X₁ ≤ 5 ∧ 3⋅X₀+6⋅X₂ ≤ 2 ∧ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 2 ≤ 3⋅X₀+6⋅X₂ ∧ 5 ≤ 3⋅X₁ ∧ 8 ≤ 3⋅X₁
∨ 3⋅X₁ ≤ 8 ∧ 3⋅X₀+6⋅X₂ ≤ 2 ∧ 0 ≤ 1 ∧ 3⋅X₀+6⋅X₂ ≤ 1 ∧ 1 ≤ 0 ∧ 2 ≤ 3⋅X₀+6⋅X₂ ∧ 8 ≤ 3⋅X₁
∨ 3⋅X₁ ≤ 8 ∧ 3⋅X₀+6⋅X₂ ≤ 2 ∧ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 2 ≤ 3⋅X₀+6⋅X₂ ∧ 2 ≤ X₁ ∧ 8 ≤ 3⋅X₁
∨ 3⋅X₁ ≤ 8 ∧ 3⋅X₀+6⋅X₂ ≤ 2 ∧ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 2 ≤ 3⋅X₀+6⋅X₂ ∧ 8 ≤ 3⋅X₁
∨ 3⋅X₁ ≤ 5 ∧ 3⋅X₀+6⋅X₂ ≤ 2 ∧ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 1 ≤ X₀+2⋅X₂ ∧ 2 ≤ 3⋅X₀+6⋅X₂ ∧ 5 ≤ 3⋅X₁
∨ 3⋅X₁ ≤ 5 ∧ 3⋅X₀+6⋅X₂ ≤ 2 ∧ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 2 ≤ 3⋅X₀+6⋅X₂ ∧ 5 ≤ 3⋅X₁
∨ 3⋅X₁ ≤ 5 ∧ 3⋅X₀+6⋅X₂ ≤ 2 ∧ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 2 ≤ 3⋅X₀+6⋅X₂ ∧ 5 ≤ 3⋅X₁ ∧ 17 ≤ 6⋅X₁
∨ 3⋅X₀+6⋅X₂ ≤ 2 ∧ 0 ≤ 1 ∧ 3⋅X₀+6⋅X₂ ≤ 1 ∧ 1 ≤ 0 ∧ 2 ≤ 3⋅X₀+6⋅X₂ ∧ 17 ≤ 6⋅X₁
∨ 3⋅X₀+6⋅X₂ ≤ 2 ∧ 0 ≤ 1 ∧ 3⋅X₀+6⋅X₂ ≤ 1 ∧ 2 ≤ 3⋅X₀+6⋅X₂
∨ 3⋅X₀+6⋅X₂ ≤ 2 ∧ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 1 ≤ X₀+2⋅X₂ ∧ 2 ≤ 3⋅X₀+6⋅X₂ ∧ 2 ≤ X₁
∨ 3⋅X₀+6⋅X₂ ≤ 2 ∧ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 2 ≤ 3⋅X₀+6⋅X₂ ∧ 2 ≤ X₁
∨ 3⋅X₀+6⋅X₂ ≤ 2 ∧ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 2 ≤ 3⋅X₀+6⋅X₂ ∧ 2 ≤ X₁ ∧ 17 ≤ 6⋅X₁
∨ 3⋅X₀+6⋅X₂ ≤ 2 ∧ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 2 ≤ 3⋅X₀+6⋅X₂ ∧ 17 ≤ 6⋅X₁
∨ 3⋅X₀+6⋅X₂ ≤ 2 ∧ 0 ≤ 1 ∧ 1 ≤ X₀+2⋅X₂ ∧ 2 ≤ 3⋅X₀+6⋅X₂
∨ 3⋅X₀+6⋅X₂ ≤ 2 ∧ 0 ≤ 1 ∧ 2 ≤ 3⋅X₀+6⋅X₂
∨ 3⋅X₀+6⋅X₂ ≤ 1 ∧ 1 ≤ X₀+2⋅X₂

Stabilization-Threshold for: 1+X₀+2⋅X₂ ≤ X₁

alphas_abs: 6+3⋅X₁
M: 0
N: 2
Bound: 6⋅X₁+15 {O(n)}

Stabilization-Threshold for: 1 ≤ X₂+X₃

alphas_abs: 12+6⋅X₁
M: 0
N: 2
Bound: 12⋅X₁+27 {O(n)}

TWN - Lifting for [1: l1->l1] of 36⋅X₁+89 {O(n)}

relevant size-bounds w.r.t. t₀: l0→l1:
X₁: X₁ {O(n)}
Runtime-bound of t₀: 1 {O(1)}
Results in: 36⋅X₁+89 {O(n)}

All Bounds

Timebounds

Overall timebound:36⋅X₁+90 {O(n)}
t₀: 1 {O(1)}
t₁: 36⋅X₁+89 {O(n)}

Costbounds

Overall costbound: 36⋅X₁+90 {O(n)}
t₀: 1 {O(1)}
t₁: 36⋅X₁+89 {O(n)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₀, X₃: X₃ {O(n)}
t₁, X₀: 2 {O(1)}
t₁, X₁: 37⋅X₁+89 {O(n)}
t₁, X₂: 111414603535684224740921180160⋅2^(36⋅X₁)+22282920707136844948184236032⋅2^(36⋅X₁)⋅X₀⋅X₁+2^(36⋅X₁)⋅44565841414273689896368472064⋅X₁+2^(36⋅X₁)⋅55707301767842112370460590080⋅X₀+2^(36⋅X₁)⋅618970019642690137449562112⋅X₂ {O(EXP)}
t₁, X₃: 38⋅X₁+89 {O(n)}