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, X₁+1, -2⋅X₂-X₀, X₁) :|: 0 < X₂+X₃

Preprocessing

Problem after Preprocessing

Time-Bound by TWN-Loops:

TWN-Loops: t₁ 36⋅X₁+89 {O(n)}

TWN-Loops:

entry: t₀: l0(X₀, X₁, X₂, X₃) → l1(X₀, X₁, X₂, X₃)

results in twn-loop: twn: (X₀,X₁,X₂,X₃) -> (2,X₁+1,-2⋅X₂-X₀,X₁) :|: 0 < X₂+X₃

order: [X₀; X₁; X₂; X₃]
closed-form:

X₀: [[n == 0]] * X₀ + [[n != 0]] * 2
X₁: X₁ + [[n != 0]] * 2 * n^1
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
X₃: [[n == 0]] * X₃ + [[n != 0]] * 1+X₁ + [[n != 0, n != 1]] * 2 * n^1 + [[n != 0, n != 1]] * -2

Termination: true
Formula:

6⋅X₂+3⋅X₀ < 2 ∧ 2 < 6⋅X₂+3⋅X₀
∨ 6⋅X₂+3⋅X₀ < 2 ∧ 0 < 12 ∧ 2 ≤ 6⋅X₂+3⋅X₀ ∧ 6⋅X₂+3⋅X₀ ≤ 2
∨ 6⋅X₂+3⋅X₀ < 2 ∧ 10 < 6⋅X₁ ∧ 2 ≤ 6⋅X₂+3⋅X₀ ∧ 6⋅X₂+3⋅X₀ ≤ 2 ∧ 0 ≤ 12 ∧ 12 ≤ 0
∨ 0 < 6 ∧ 6⋅X₂+3⋅X₀ ≤ 2 ∧ 2 ≤ 6⋅X₂+3⋅X₀ ∧ 2 < 6⋅X₂+3⋅X₀
∨ 0 < 6 ∧ 0 < 12 ∧ 2 ≤ 6⋅X₂+3⋅X₀ ∧ 6⋅X₂+3⋅X₀ ≤ 2
∨ 0 < 6 ∧ 10 < 6⋅X₁ ∧ 2 ≤ 6⋅X₂+3⋅X₀ ∧ 6⋅X₂+3⋅X₀ ≤ 2 ∧ 0 ≤ 12 ∧ 12 ≤ 0
∨ 2 < 3⋅X₁ ∧ 6⋅X₂+3⋅X₀ ≤ 2 ∧ 2 ≤ 6⋅X₂+3⋅X₀ ∧ 0 ≤ 6 ∧ 6 ≤ 0 ∧ 2 < 6⋅X₂+3⋅X₀
∨ 2 < 3⋅X₁ ∧ 0 ≤ 6 ∧ 6 ≤ 0 ∧ 0 < 12 ∧ 2 ≤ 6⋅X₂+3⋅X₀ ∧ 6⋅X₂+3⋅X₀ ≤ 2
∨ 2 < 3⋅X₁ ∧ 0 ≤ 6 ∧ 6 ≤ 0 ∧ 10 < 6⋅X₁ ∧ 2 ≤ 6⋅X₂+3⋅X₀ ∧ 6⋅X₂+3⋅X₀ ≤ 2 ∧ 0 ≤ 12 ∧ 12 ≤ 0

Stabilization-Threshold for: 2⋅X₂+X₀ < X₁

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

Stabilization-Threshold for: 0 < X₂+X₃

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

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

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: 180⋅2^(36⋅X₁+89)+2^(36⋅X₁+89)⋅36⋅X₀⋅X₁+2^(36⋅X₁+89)⋅72⋅X₁+2^(36⋅X₁+89)⋅90⋅X₀+2^(36⋅X₁+89)⋅X₂ {O(EXP)}
t₁, X₃: 38⋅X₁+89 {O(n)}