KoAT2 Proof Output

Initial Problem

Start: l0
Program_Vars: X₀, X₁
Temp_Vars:
Locations: l0, l1
Transitions:
t₀: l0(X₀, X₁) → l1(X₀, X₁) :|: 1 ≤ X₀
t₁: l1(X₀, X₁) → l1(3⋅X₀, 2⋅X₁) :|: 1+X₀ ≤ X₁

Preprocessing

Found invariant 1 ≤ X₀ for location l1

Problem after Preprocessing

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

TWN: t₁: l1→l1

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

X₁: X₁⋅(2)^n
X₀: X₀⋅(3)^n

Termination: true
Formula:

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

Stabilization-Threshold for: 1+X₀ ≤ X₁

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

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

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

All Bounds

Timebounds

Overall timebound:log(X₁)+5 {O(log(n))}
t₀: 1 {O(1)}
t₁: log(X₁)+4 {O(log(n))}

Costbounds

Overall costbound: log(X₁)+5 {O(log(n))}
t₀: 1 {O(1)}
t₁: log(X₁)+4 {O(log(n))}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₁, X₀: 81⋅X₀⋅X₁ {O(n^2)}
t₁, X₁: 16⋅X₁⋅X₁ {O(n^2)}