KoAT2 Proof Output

Initial Problem

Start: l0
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: l0, l1, l2
Transitions:
t₀: l0(X₀, X₁, X₂) → l1(X₀, X₁, X₂) :|: 1 ≤ X₀
t₁: l1(X₀, X₁, X₂) → l1(X₀+X₂, X₁, X₂-1) :|: 1 ≤ X₂
t₂: l1(X₀, X₁, X₂) → l2(X₀, X₁, X₂) :|: X₂ ≤ 0
t₃: l2(X₀, X₁, X₂) → l2(5⋅X₀+(X₂)², 2⋅X₁, X₂) :|: 1+X₀ ≤ (X₁)²

Preprocessing

Found invariant 1 ≤ X₀ for location l1

Found invariant X₂ ≤ 0 ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₀ for location l2

Problem after Preprocessing

Start: l0
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: l0, l1, l2
Transitions:
t₀: l0(X₀, X₁, X₂) → l1(X₀, X₁, X₂) :|: 1 ≤ X₀
t₁: l1(X₀, X₁, X₂) → l1(X₀+X₂, X₁, X₂-1) :|: 1 ≤ X₂ ∧ 1 ≤ X₀
t₂: l1(X₀, X₁, X₂) → l2(X₀, X₁, X₂) :|: X₂ ≤ 0 ∧ 1 ≤ X₀
t₃: l2(X₀, X₁, X₂) → l2(5⋅X₀+(X₂)², 2⋅X₁, X₂) :|: 1+X₀ ≤ (X₁)² ∧ 1 ≤ X₀ ∧ 1+X₂ ≤ X₀ ∧ X₂ ≤ 0

MPRF for transition t₁: l1(X₀, X₁, X₂) → l1(X₀+X₂, X₁, X₂-1) :|: 1 ≤ X₂ ∧ 1 ≤ X₀ of depth 1:

new bound:

X₂ {O(n)}

MPRF:

• l1: [X₂]

TWN: t₃: l2→l2

cycle: [t₃: l2→l2]
original loop: (1+X₀ ≤ (X₁)² ∧ 1 ≤ X₀ ∧ 1+X₂ ≤ X₀ ∧ X₂ ≤ 0,(X₀,X₁,X₂) -> (5⋅X₀+(X₂)²,2⋅X₁,X₂))
transformed loop: (1+X₀ ≤ (X₁)² ∧ 1 ≤ X₀ ∧ 1+X₂ ≤ X₀ ∧ X₂ ≤ 0,(X₀,X₁,X₂) -> (5⋅X₀+(X₂)²,2⋅X₁,X₂))
loop: (1+X₀ ≤ (X₁)² ∧ 1 ≤ X₀ ∧ 1+X₂ ≤ X₀ ∧ X₂ ≤ 0,(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 ∧ 1+X₂ ≤ X₀ ∧ 4 ≤ (X₂)² ∧ 0 ≤ 4⋅X₀+(X₂)² ∧ 4⋅X₀+(X₂)² ≤ 0 ∧ 0 ≤ (X₁)² ∧ (X₁)² ≤ 0 ∧ X₂ ≤ 0
∨ 1 ≤ X₀ ∧ 1+X₂ ≤ X₀ ∧ 1+4⋅X₀+(X₂)² ≤ 0 ∧ X₂ ≤ 0
∨ 1 ≤ X₀ ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ 4⋅(X₁)² ∧ 0 ≤ 4⋅X₀+(X₂)² ∧ 4⋅X₀+(X₂)² ≤ 0 ∧ X₂ ≤ 0
∨ 1 ≤ X₀ ∧ 1+X₂ ≤ X₀ ∧ 5 ≤ (X₂)² ∧ 0 ≤ 4⋅X₀+(X₂)² ∧ 4⋅X₀+(X₂)² ≤ 0 ∧ 0 ≤ (X₁)² ∧ (X₁)² ≤ 0 ∧ X₂ ≤ 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 [3: l2->l2] of 2⋅log(X₁)+2⋅log(X₂)+7 {O(log(n))}

relevant size-bounds w.r.t. t₂: l1→l2:
X₁: 2⋅X₁ {O(n)}
X₂: 2⋅X₂ {O(n)}
Runtime-bound of t₂: 1 {O(1)}
Results in: 2⋅log(X₁)+2⋅log(X₂)+11 {O(log(n))}

All Bounds

Timebounds

Overall timebound:X₂+2⋅log(X₁)+2⋅log(X₂)+13 {O(n)}
t₀: 1 {O(1)}
t₁: X₂ {O(n)}
t₂: 1 {O(1)}
t₃: 2⋅log(X₁)+2⋅log(X₂)+11 {O(log(n))}

Costbounds

Overall costbound: X₂+2⋅log(X₁)+2⋅log(X₂)+13 {O(n)}
t₀: 1 {O(1)}
t₁: X₂ {O(n)}
t₂: 1 {O(1)}
t₃: 2⋅log(X₁)+2⋅log(X₂)+11 {O(log(n))}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₁, X₀: 2⋅X₂⋅X₂+2⋅X₂+X₀ {O(n^2)}
t₁, X₁: X₁ {O(n)}
t₁, X₂: X₂ {O(n)}
t₂, X₀: 2⋅X₂⋅X₂+2⋅X₀+2⋅X₂ {O(n^2)}
t₂, X₁: 2⋅X₁ {O(n)}
t₂, X₂: 2⋅X₂ {O(n)}
t₃, X₀: 5^(2⋅log(X₁))⋅5^(2⋅log(X₂))⋅6103515625000⋅X₂⋅X₂+5^(2⋅log(X₁))⋅5^(2⋅log(X₂))⋅7629394531250⋅X₀+5^(2⋅log(X₁))⋅5^(2⋅log(X₂))⋅7629394531250⋅X₂+5^(2⋅log(X₁))⋅5^(2⋅log(X₂))⋅7629394531250⋅X₂⋅X₂+4⋅X₂⋅X₂ {O(EXP)}
t₃, X₁: 2^(2⋅log(X₁))⋅2^(2⋅log(X₂))⋅4096⋅X₁ {O(EXP)}
t₃, X₂: 2⋅X₂ {O(n)}