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)
t₁: l1(X₀, X₁, X₂) → l1(X₀-1, 2⋅X₁, X₂) :|: 1 ≤ X₀
t₂: l1(X₀, X₁, X₂) → l2(X₀, X₁, X₂)
t₃: l2(X₀, X₁, X₂) → l2(X₀, 2⋅X₁, 3⋅X₂) :|: 1+X₂ ≤ X₁ ∧ 1 ≤ X₂

Preprocessing

Problem after Preprocessing

MPRF for transition t₁: l1(X₀, X₁, X₂) → l1(X₀-1, 2⋅X₁, X₂) :|: 1 ≤ X₀ of depth 1:

new bound:

X₀ {O(n)}

MPRF:

• l1: [X₀]

TWN: t₃: l2→l2

cycle: [t₃: l2→l2]
original loop: (1+X₂ ≤ X₁ ∧ 1 ≤ X₂,(X₁,X₂) -> (2⋅X₁,3⋅X₂))
transformed loop: (1+X₂ ≤ X₁ ∧ 1 ≤ X₂,(X₁,X₂) -> (2⋅X₁,3⋅X₂))
loop: (1+X₂ ≤ X₁ ∧ 1 ≤ X₂,(X₁,X₂) -> (2⋅X₁,3⋅X₂))
order: [X₂; X₁]
closed-form:

X₂: X₂⋅(3)^n
X₁: X₁⋅(2)^n

Termination: true
Formula:

0 ≤ 1 ∧ 1 ≤ 0 ∧ 1 ≤ X₂ ∧ 0 ≤ X₁ ∧ X₁ ≤ 0 ∧ 0 ≤ X₂ ∧ X₂ ≤ 0
∨ 1 ≤ 0 ∧ 1 ≤ X₂ ∧ 0 ≤ X₁ ∧ X₁ ≤ 0 ∧ 0 ≤ X₂ ∧ X₂ ≤ 0
∨ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 0 ≤ X₂ ∧ X₂ ≤ 0
∨ 1 ≤ X₂ ∧ 1+X₂ ≤ 0

Stabilization-Threshold for: 1+X₂ ≤ X₁

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

TWN - Lifting for [3: l2->l2] of log(X₁)+4 {O(log(n))}

relevant size-bounds w.r.t. t₂: l1→l2:
X₁: 2^(X₀)⋅X₁+X₁ {O(EXP)}
Runtime-bound of t₂: 1 {O(1)}
Results in: X₀+2⋅log(X₁)+4 {O(n)}

All Bounds

Timebounds

Overall timebound:2⋅X₀+2⋅log(X₁)+6 {O(n)}
t₀: 1 {O(1)}
t₁: X₀ {O(n)}
t₂: 1 {O(1)}
t₃: X₀+2⋅log(X₁)+4 {O(n)}

Costbounds

Overall costbound: 2⋅X₀+2⋅log(X₁)+6 {O(n)}
t₀: 1 {O(1)}
t₁: X₀ {O(n)}
t₂: 1 {O(1)}
t₃: X₀+2⋅log(X₁)+4 {O(n)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₁, X₀: X₀ {O(n)}
t₁, X₁: 2^(X₀)⋅X₁ {O(EXP)}
t₁, X₂: X₂ {O(n)}
t₂, X₀: 2⋅X₀ {O(n)}
t₂, X₁: 2^(X₀)⋅X₁+X₁ {O(EXP)}
t₂, X₂: 2⋅X₂ {O(n)}
t₃, X₀: 2⋅X₀ {O(n)}
t₃, X₁: 16⋅2^(2⋅log(X₁))⋅2^(X₀)⋅2^(X₀)⋅X₁+16⋅2^(2⋅log(X₁))⋅2^(X₀)⋅X₁ {O(EXP)}
t₃, X₂: 162⋅3^(2⋅log(X₁))⋅3^(X₀)⋅X₂ {O(EXP)}