KoAT2 Proof Output

Initial Problem

Start: l0
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars: T
Locations: l0, l1, l2
Transitions:
t₀: l0(X₀, X₁, X₂, X₃) → l1(X₀, X₁, X₂, X₃)
t₁: l1(X₀, X₁, X₂, X₃) → l1(2⋅X₀, 3⋅X₁, X₂, X₃) :|: 1+X₁ ≤ X₀ ∧ 1 ≤ X₁
t₂: l1(X₀, X₁, X₂, X₃) → l2(X₀, X₁, X₂, X₃)
t₃: l2(X₀, X₁, X₂, X₃) → l2(X₀, X₁, X₂+X₃, X₃-T) :|: 1 ≤ T ∧ 1 ≤ X₂

Preprocessing

Problem after Preprocessing

TWN: t₁: l1→l1

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

X₁: X₁⋅(3)^n
X₀: X₀⋅(2)^n

Termination: true
Formula:

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

Stabilization-Threshold for: 1+X₁ ≤ X₀

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

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

relevant size-bounds w.r.t. t₀: l0→l1:
X₀: X₀ {O(n)}
Runtime-bound of t₀: 1 {O(1)}
Results in: log(X₀)+4 {O(log(n))}

MPRF for transition t₃: l2(X₀, X₁, X₂, X₃) → l2(X₀, X₁, X₂+X₃, X₃-T) :|: 1 ≤ T ∧ 1 ≤ X₂ of depth 2:

new bound:

16⋅X₂+16⋅X₃+9 {O(n)}

MPRF:

• l2: [1+X₃; X₂]

All Bounds

Timebounds

Overall timebound:16⋅X₂+16⋅X₃+log(X₀)+15 {O(n)}
t₀: 1 {O(1)}
t₁: log(X₀)+4 {O(log(n))}
t₂: 1 {O(1)}
t₃: 16⋅X₂+16⋅X₃+9 {O(n)}

Costbounds

Overall costbound: 16⋅X₂+16⋅X₃+log(X₀)+15 {O(n)}
t₀: 1 {O(1)}
t₁: log(X₀)+4 {O(log(n))}
t₂: 1 {O(1)}
t₃: 16⋅X₂+16⋅X₃+9 {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: 16⋅X₀⋅X₀ {O(n^2)}
t₁, X₁: 81⋅X₀⋅X₁ {O(n^2)}
t₁, X₂: X₂ {O(n)}
t₁, X₃: X₃ {O(n)}
t₂, X₀: 16⋅X₀⋅X₀+X₀ {O(n^2)}
t₂, X₁: 81⋅X₀⋅X₁+X₁ {O(n^2)}
t₂, X₂: 2⋅X₂ {O(n)}
t₂, X₃: 2⋅X₃ {O(n)}
t₃, X₀: 16⋅X₀⋅X₀+X₀ {O(n^2)}
t₃, X₁: 81⋅X₀⋅X₁+X₁ {O(n^2)}