KoAT2 Proof Output

Initial Problem

Start: l0
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: l0, l1
Transitions:
t₀: l0(X₀, X₁, X₂) → l1(X₀, X₁, X₂)
t₁: l1(X₀, X₁, X₂) → l1(2⋅X₀, 3⋅X₁, 4⋅X₂) :|: 1+(X₂)² ≤ X₀*X₁ ∧ 1+X₂ ≤ 0
t₂: l1(X₀, X₁, X₂) → l1(2⋅X₀, 3⋅X₁, 4⋅X₂) :|: 1+(X₂)² ≤ X₀*X₁ ∧ 1 ≤ X₂

Preprocessing

Problem after Preprocessing

TWN: t₁: l1→l1

cycle: [t₁: l1→l1; t₂: l1→l1]
original loop: (1+(X₂)² ≤ X₀*X₁ ∧ 1+X₂ ≤ 0 ∨ 1+(X₂)² ≤ X₀*X₁ ∧ 1 ≤ X₂,(X₀,X₁,X₂) -> (2⋅X₀,3⋅X₁,4⋅X₂))
transformed loop: (1+(X₂)² ≤ X₀*X₁ ∧ 1+X₂ ≤ 0 ∨ 1+(X₂)² ≤ X₀*X₁ ∧ 1 ≤ X₂,(X₀,X₁,X₂) -> (2⋅X₀,3⋅X₁,4⋅X₂))
loop: (1+(X₂)² ≤ X₀*X₁ ∧ 1+X₂ ≤ 0 ∨ 1+(X₂)² ≤ X₀*X₁ ∧ 1 ≤ X₂,(X₀,X₁,X₂) -> (2⋅X₀,3⋅X₁,4⋅X₂))
order: [X₂; X₁; X₀]
closed-form:

X₂: X₂⋅(4)^n
X₁: X₁⋅(3)^n
X₀: X₀⋅(2)^n

Termination: true
Formula:

0 ≤ 1 ∧ 1 ≤ 0 ∧ 1 ≤ X₂ ∧ 0 ≤ X₀*X₁ ∧ X₀*X₁ ≤ 0 ∧ 0 ≤ (X₂)² ∧ (X₂)² ≤ 0
∨ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 1+X₂ ≤ 0 ∧ 0 ≤ X₀*X₁ ∧ X₀*X₁ ≤ 0 ∧ 0 ≤ (X₂)² ∧ (X₂)² ≤ 0
∨ 1 ≤ 0 ∧ 1 ≤ X₂ ∧ 0 ≤ X₀*X₁ ∧ X₀*X₁ ≤ 0 ∧ 0 ≤ (X₂)² ∧ (X₂)² ≤ 0
∨ 1 ≤ 0 ∧ 1+X₂ ≤ 0 ∧ 0 ≤ X₀*X₁ ∧ X₀*X₁ ≤ 0 ∧ 0 ≤ (X₂)² ∧ (X₂)² ≤ 0
∨ 1 ≤ X₀*X₁ ∧ 1 ≤ X₂ ∧ 0 ≤ (X₂)² ∧ (X₂)² ≤ 0
∨ 1 ≤ X₀*X₁ ∧ 1+X₂ ≤ 0 ∧ 0 ≤ (X₂)² ∧ (X₂)² ≤ 0
∨ 1 ≤ X₂ ∧ 1+(X₂)² ≤ 0
∨ 1+X₂ ≤ 0 ∧ 1+(X₂)² ≤ 0

Stabilization-Threshold for: 1+(X₂)² ≤ X₀*X₁

alphas_abs: X₀*X₁
M': 1
N: 1
Bound: log(X₀)+log(X₁)+2 {O(log(n))}

TWN - Lifting for [1: l1->l1; 2: l1->l1] of log(X₀)+log(X₁)+4 {O(log(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: log(X₀)+log(X₁)+4 {O(log(n))}

All Bounds

Timebounds

Overall timebound:2⋅log(X₀)+2⋅log(X₁)+9 {O(log(n))}
t₀: 1 {O(1)}
t₁: log(X₀)+log(X₁)+4 {O(log(n))}
t₂: log(X₀)+log(X₁)+4 {O(log(n))}

Costbounds

Overall costbound: 2⋅log(X₀)+2⋅log(X₁)+9 {O(log(n))}
t₀: 1 {O(1)}
t₁: log(X₀)+log(X₁)+4 {O(log(n))}
t₂: log(X₀)+log(X₁)+4 {O(log(n))}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₁, X₀: 16⋅X₀⋅X₀⋅X₁ {O(n^3)}
t₁, X₁: 81⋅X₀⋅X₁⋅X₁ {O(n^3)}
t₁, X₂: 256⋅X₀⋅X₁⋅X₂ {O(n^3)}
t₂, X₀: 16⋅X₀⋅X₀⋅X₁ {O(n^3)}
t₂, X₁: 81⋅X₀⋅X₁⋅X₁ {O(n^3)}
t₂, X₂: 256⋅X₀⋅X₁⋅X₂ {O(n^3)}