KoAT2 Proof Output

Initial Problem

Start: l0
Program_Vars: X₀, X₁, X₂, X₃, X₄
Temp_Vars:
Locations: l0, l1
Transitions:
t₀: l0(X₀, X₁, X₂, X₃, X₄) → l1(X₀, X₁, X₂, X₃, X₄)
t₁: l1(X₀, X₁, X₂, X₃, X₄) → l1(2⋅X₀+2⋅(X₀)², 2⋅X₁-(X₀)², (X₂)³-3⋅X₃, (X₂)³-3⋅X₂, 4⋅X₄) :|: 1+X₄ ≤ X₀+2⋅X₁ ∧ 1 ≤ X₄
t₂: l1(X₀, X₁, X₂, X₃, X₄) → l1(2⋅X₀+2⋅(X₀)², 2⋅X₁-(X₀)², (X₂)³-3⋅X₃, (X₂)³-3⋅X₂, 4⋅X₄) :|: 1+X₃+X₄ ≤ X₂ ∧ 1 ≤ X₄

Preprocessing

Problem after Preprocessing

TWN: t₁: l1→l1

cycle: [t₁: l1→l1; t₂: l1→l1]
original loop: (1+Temp_Int₇₉₈+X₄ ≤ 0 ∧ 1 ≤ X₄ ∨ 1+Temp_Int₇₉₉+X₄ ≤ 0 ∧ 1 ≤ X₄,(Temp_Int₇₉₈,Temp_Int₇₉₉,X₄) -> (2⋅Temp_Int₇₉₈,3⋅Temp_Int₇₉₉,4⋅X₄))
transformed loop: (1+Temp_Int₇₉₈+X₄ ≤ 0 ∧ 1 ≤ X₄ ∨ 1+Temp_Int₇₉₉+X₄ ≤ 0 ∧ 1 ≤ X₄,(Temp_Int₇₉₈,Temp_Int₇₉₉,X₄) -> (2⋅Temp_Int₇₉₈,3⋅Temp_Int₇₉₉,4⋅X₄))
loop: (1+Temp_Int₇₉₈+X₄ ≤ 0 ∧ 1 ≤ X₄ ∨ 1+Temp_Int₇₉₉+X₄ ≤ 0 ∧ 1 ≤ X₄,(Temp_Int₇₉₈,Temp_Int₇₉₉,X₄) -> (2⋅Temp_Int₇₉₈,3⋅Temp_Int₇₉₉,4⋅X₄))
order: [X₄; Temp_Int₇₉₉; Temp_Int₇₉₈]
closed-form:

X₄: X₄⋅(4)^n
Temp_Int₇₉₉: Temp_Int₇₉₉⋅(3)^n
Temp_Int₇₉₈: Temp_Int₇₉₈⋅(2)^n

Termination: true
Formula:

0 ≤ 1 ∧ 1 ≤ 0 ∧ 1 ≤ X₄ ∧ 0 ≤ Temp_Int₇₉₈ ∧ Temp_Int₇₉₈ ≤ 0 ∧ 0 ≤ X₄ ∧ X₄ ≤ 0
∨ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 1 ≤ X₄ ∧ 0 ≤ Temp_Int₇₉₉ ∧ Temp_Int₇₉₉ ≤ 0 ∧ 0 ≤ X₄ ∧ X₄ ≤ 0
∨ 1 ≤ 0 ∧ 1 ≤ X₄ ∧ 0 ≤ Temp_Int₇₉₈ ∧ Temp_Int₇₉₈ ≤ 0 ∧ 0 ≤ X₄ ∧ X₄ ≤ 0
∨ 1 ≤ 0 ∧ 1 ≤ X₄ ∧ 0 ≤ Temp_Int₇₉₉ ∧ Temp_Int₇₉₉ ≤ 0 ∧ 0 ≤ X₄ ∧ X₄ ≤ 0
∨ 1+Temp_Int₇₉₈ ≤ 0 ∧ 1 ≤ X₄ ∧ 0 ≤ X₄ ∧ X₄ ≤ 0
∨ 1+Temp_Int₇₉₉ ≤ 0 ∧ 1 ≤ X₄ ∧ 0 ≤ X₄ ∧ X₄ ≤ 0
∨ 1 ≤ X₄ ∧ 1+X₄ ≤ 0

Stabilization-Threshold for: 1+Temp_Int₇₉₉+X₄ ≤ 0

alphas_abs: Temp_Int₇₉₉
M': 1
N: 1
Bound: log(Temp_Int₇₉₉)+2 {O(log(n))}

Stabilization-Threshold for: 1+Temp_Int₇₉₈+X₄ ≤ 0

alphas_abs: Temp_Int₇₉₈
M': 1
N: 1
Bound: log(Temp_Int₇₉₈)+2 {O(log(n))}

TWN - Lifting for [1: l1->l1; 2: l1->l1] of log(X₀)+log(X₁)+log(X₂)+log(X₃)+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)}
X₃: X₃ {O(n)}
Runtime-bound of t₀: 1 {O(1)}
Results in: log(X₀)+log(X₁)+log(X₂)+log(X₃)+7 {O(log(n))}

All Bounds

Timebounds

Overall timebound:2⋅log(X₀)+2⋅log(X₁)+2⋅log(X₂)+2⋅log(X₃)+15 {O(log(n))}
t₀: 1 {O(1)}
t₁: log(X₀)+log(X₁)+log(X₂)+log(X₃)+7 {O(log(n))}
t₂: log(X₀)+log(X₁)+log(X₂)+log(X₃)+7 {O(log(n))}

Costbounds

Overall costbound: 2⋅log(X₀)+2⋅log(X₁)+2⋅log(X₂)+2⋅log(X₃)+15 {O(log(n))}
t₀: 1 {O(1)}
t₁: log(X₀)+log(X₁)+log(X₂)+log(X₃)+7 {O(log(n))}
t₂: log(X₀)+log(X₁)+log(X₂)+log(X₃)+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)}
t₀, X₄: X₄ {O(n)}
t₁, X₄: 268435456⋅X₀⋅X₀⋅X₁⋅X₁⋅X₂⋅X₂⋅X₃⋅X₃⋅X₄ {O(n^9)}
t₂, X₄: 268435456⋅X₀⋅X₀⋅X₁⋅X₁⋅X₂⋅X₂⋅X₃⋅X₃⋅X₄ {O(n^9)}