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) :|: 1 ≤ X₀
t₁: l1(X₀, X₁, X₂) → l1(5⋅X₀+(X₂)², 2⋅X₁, X₂) :|: 1+X₀ ≤ (X₁)²

Preprocessing

Found invariant 1 ≤ X₀ for location l1

Problem after Preprocessing

TWN: t₁: l1→l1

cycle: [t₁: l1→l1]
original loop: (1+X₀ ≤ (X₁)² ∧ 1 ≤ X₀,(X₀,X₁,X₂) -> (5⋅X₀+(X₂)²,2⋅X₁,X₂))
transformed loop: (1+X₀ ≤ (X₁)² ∧ 1 ≤ X₀,(X₀,X₁,X₂) -> (5⋅X₀+(X₂)²,2⋅X₁,X₂))
loop: (1+X₀ ≤ (X₁)² ∧ 1 ≤ X₀,(X₀,X₁,X₂) -> (5⋅X₀+(X₂)²,2⋅X₁,X₂))
order: [X₂; X₁; X₀]
closed-form:

X₂: X₂
X₁: X₁⋅(2)^n
X₀: X₀⋅(5)^n + [[n != 0]]⋅1/4⋅(X₂)²⋅(5)^n + [[n != 0]]⋅-1/4⋅(X₂)²

Termination: true
Formula:

(X₂)² ≤ 4 ∧ 1 ≤ X₀ ∧ 4 ≤ (X₂)² ∧ 0 ≤ 4⋅X₀+(X₂)² ∧ 4⋅X₀+(X₂)² ≤ 0 ∧ 0 ≤ (X₁)² ∧ (X₁)² ≤ 0
∨ 1 ≤ X₀ ∧ 1+4⋅X₀+(X₂)² ≤ 0
∨ 1 ≤ X₀ ∧ 1 ≤ 4⋅(X₁)² ∧ 0 ≤ 4⋅X₀+(X₂)² ∧ 4⋅X₀+(X₂)² ≤ 0
∨ 1 ≤ X₀ ∧ 5 ≤ (X₂)² ∧ 0 ≤ 4⋅X₀+(X₂)² ∧ 4⋅X₀+(X₂)² ≤ 0 ∧ 0 ≤ (X₁)² ∧ (X₁)² ≤ 0

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

alphas_abs: 4⋅(X₁)²+(X₂)²
M': 1
N: 1
Bound: 2⋅log(X₁)+2⋅log(X₂)+5 {O(log(n))}

TWN - Lifting for [1: l1->l1] of 2⋅log(X₁)+2⋅log(X₂)+7 {O(log(n))}

relevant size-bounds w.r.t. t₀: l0→l1:
X₁: X₁ {O(n)}
X₂: X₂ {O(n)}
Runtime-bound of t₀: 1 {O(1)}
Results in: 2⋅log(X₁)+2⋅log(X₂)+7 {O(log(n))}

All Bounds

Timebounds

Overall timebound:2⋅log(X₁)+2⋅log(X₂)+8 {O(log(n))}
t₀: 1 {O(1)}
t₁: 2⋅log(X₁)+2⋅log(X₂)+7 {O(log(n))}

Costbounds

Overall costbound: 2⋅log(X₁)+2⋅log(X₂)+8 {O(log(n))}
t₀: 1 {O(1)}
t₁: 2⋅log(X₁)+2⋅log(X₂)+7 {O(log(n))}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₁, X₀: 3906250⋅5^(2⋅log(X₁))⋅5^(2⋅log(X₂))⋅X₂⋅X₂+5^(2⋅log(X₁))⋅5^(2⋅log(X₂))⋅9765625⋅X₀+X₂⋅X₂ {O(EXP)}
t₁, X₁: 128⋅2^(2⋅log(X₁))⋅2^(2⋅log(X₂))⋅X₁ {O(EXP)}
t₁, X₂: X₂ {O(n)}