KoAT2 Proof Output

Initial Problem

Start: l0
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars:
Locations: l0, l1
Transitions:
t₀: l0(X₀, X₁, X₂, X₃) → l1(X₀, X₁, X₂, X₃)
t₁: l1(X₀, X₁, X₂, X₃) → l1(2⋅X₀+2⋅(X₀)², 2⋅X₁-(X₀)², 3⋅X₂, 4⋅X₃) :|: 1+X₃ ≤ X₀+2⋅X₁ ∧ 1 ≤ X₃
t₂: l1(X₀, X₁, X₂, X₃) → l1(2⋅X₀+2⋅(X₀)², 2⋅X₁-(X₀)², 3⋅X₂, 4⋅X₃) :|: 1+X₃ ≤ X₂ ∧ 1 ≤ X₃

Preprocessing

Problem after Preprocessing

TWN: t₁: l1→l1

cycle: [t₁: l1→l1; t₂: l1→l1]
original loop: (1+Temp_Int₇₂₀+X₃ ≤ 0 ∧ 1 ≤ X₃ ∨ 1+X₃ ≤ X₂ ∧ 1 ≤ X₃,(Temp_Int₇₂₀,X₂,X₃) -> (2⋅Temp_Int₇₂₀,3⋅X₂,4⋅X₃))
transformed loop: (1+Temp_Int₇₂₀+X₃ ≤ 0 ∧ 1 ≤ X₃ ∨ 1+X₃ ≤ X₂ ∧ 1 ≤ X₃,(Temp_Int₇₂₀,X₂,X₃) -> (2⋅Temp_Int₇₂₀,3⋅X₂,4⋅X₃))
loop: (1+Temp_Int₇₂₀+X₃ ≤ 0 ∧ 1 ≤ X₃ ∨ 1+X₃ ≤ X₂ ∧ 1 ≤ X₃,(Temp_Int₇₂₀,X₂,X₃) -> (2⋅Temp_Int₇₂₀,3⋅X₂,4⋅X₃))
order: [X₃; X₂; Temp_Int₇₂₀]
closed-form:

X₃: X₃⋅(4)^n
X₂: X₂⋅(3)^n
Temp_Int₇₂₀: Temp_Int₇₂₀⋅(2)^n

Termination: true
Formula:

0 ≤ 1 ∧ 1 ≤ 0 ∧ 1 ≤ X₃ ∧ 0 ≤ Temp_Int₇₂₀ ∧ Temp_Int₇₂₀ ≤ 0 ∧ 0 ≤ X₃ ∧ X₃ ≤ 0
∨ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 1 ≤ X₃ ∧ 0 ≤ X₂ ∧ X₂ ≤ 0 ∧ 0 ≤ X₃ ∧ X₃ ≤ 0
∨ 1 ≤ 0 ∧ 1 ≤ X₃ ∧ 0 ≤ Temp_Int₇₂₀ ∧ Temp_Int₇₂₀ ≤ 0 ∧ 0 ≤ X₃ ∧ X₃ ≤ 0
∨ 1 ≤ 0 ∧ 1 ≤ X₃ ∧ 0 ≤ X₂ ∧ X₂ ≤ 0 ∧ 0 ≤ X₃ ∧ X₃ ≤ 0
∨ 1+Temp_Int₇₂₀ ≤ 0 ∧ 1 ≤ X₃ ∧ 0 ≤ X₃ ∧ X₃ ≤ 0
∨ 1 ≤ X₂ ∧ 1 ≤ X₃ ∧ 0 ≤ X₃ ∧ X₃ ≤ 0
∨ 1 ≤ X₃ ∧ 1+X₃ ≤ 0

Stabilization-Threshold for: 1+X₃ ≤ X₂

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

Stabilization-Threshold for: 1+Temp_Int₇₂₀+X₃ ≤ 0

alphas_abs: Temp_Int₇₂₀
M': 1
N: 1
Bound: log(Temp_Int₇₂₀)+2 {O(log(n))}

TWN - Lifting for [1: l1->l1; 2: l1->l1] of log(X₀)+log(X₁)+log(X₂)+7 {O(log(n))}

relevant size-bounds w.r.t. t₀: l0→l1:
X₀: X₀ {O(n)}
X₁: X₁ {O(n)}
X₂: X₂ {O(n)}
Runtime-bound of t₀: 1 {O(1)}
Results in: log(X₀)+log(X₁)+log(X₂)+7 {O(log(n))}

All Bounds

Timebounds

Overall timebound:2⋅log(X₀)+2⋅log(X₁)+2⋅log(X₂)+15 {O(log(n))}
t₀: 1 {O(1)}
t₁: log(X₀)+log(X₁)+log(X₂)+7 {O(log(n))}
t₂: log(X₀)+log(X₁)+log(X₂)+7 {O(log(n))}

Costbounds

Overall costbound: 2⋅log(X₀)+2⋅log(X₁)+2⋅log(X₂)+15 {O(log(n))}
t₀: 1 {O(1)}
t₁: log(X₀)+log(X₁)+log(X₂)+7 {O(log(n))}
t₂: log(X₀)+log(X₁)+log(X₂)+7 {O(log(n))}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₀, X₃: X₃ {O(n)}
t₁, X₂: 4782969⋅X₀⋅X₀⋅X₁⋅X₁⋅X₂⋅X₂⋅X₂ {O(n^7)}
t₁, X₃: 268435456⋅X₀⋅X₀⋅X₁⋅X₁⋅X₂⋅X₂⋅X₃ {O(n^7)}
t₂, X₂: 4782969⋅X₀⋅X₀⋅X₁⋅X₁⋅X₂⋅X₂⋅X₂ {O(n^7)}
t₂, X₃: 268435456⋅X₀⋅X₀⋅X₁⋅X₁⋅X₂⋅X₂⋅X₃ {O(n^7)}