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(-X₀, -X₁, X₂, 4⋅X₃, 9⋅X₄) :|: 1+(X₃)² ≤ (X₂)⁵+X₄ ∧ 1+X₃ ≤ 0
t₂: l1(X₀, X₁, X₂, X₃, X₄) → l1(-X₀, -X₁, X₂, 4⋅X₃, 9⋅X₄) :|: 1+(X₃)² ≤ (X₂)⁵+X₄ ∧ 1 ≤ X₃

Preprocessing

Eliminate variables [X₀; X₁] that do not contribute to the problem

Problem after Preprocessing

Start: l0
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: l0, l1
Transitions:
t₉: l0(X₀, X₁, X₂) → l1(X₀, X₁, X₂)
t₁₀: l1(X₀, X₁, X₂) → l1(X₀, 4⋅X₁, 9⋅X₂) :|: 1+(X₁)² ≤ (X₀)⁵+X₂ ∧ 1+X₁ ≤ 0
t₁₁: l1(X₀, X₁, X₂) → l1(X₀, 4⋅X₁, 9⋅X₂) :|: 1+(X₁)² ≤ (X₀)⁵+X₂ ∧ 1 ≤ X₁

TWN: t₁₀: l1→l1

cycle: [t₁₀: l1→l1; t₁₁: l1→l1]
original loop: (1+(X₁)² ≤ (X₀)⁵+X₂ ∧ 1+X₁ ≤ 0 ∨ 1+(X₁)² ≤ (X₀)⁵+X₂ ∧ 1 ≤ X₁,(X₀,X₁,X₂) -> (X₀,4⋅X₁,9⋅X₂))
transformed loop: (1+(X₁)² ≤ (X₀)⁵+X₂ ∧ 1+X₁ ≤ 0 ∨ 1+(X₁)² ≤ (X₀)⁵+X₂ ∧ 1 ≤ X₁,(X₀,X₁,X₂) -> (X₀,4⋅X₁,9⋅X₂))
loop: (1+(X₁)² ≤ (X₀)⁵+X₂ ∧ 1+X₁ ≤ 0 ∨ 1+(X₁)² ≤ (X₀)⁵+X₂ ∧ 1 ≤ X₁,(X₀,X₁,X₂) -> (X₀,4⋅X₁,9⋅X₂))
order: [X₂; X₁; X₀]
closed-form:

X₂: X₂⋅(9)^n
X₁: X₁⋅(4)^n
X₀: X₀

Termination: true
Formula:

(X₀)⁵ ≤ 1 ∧ 1 ≤ (X₀)⁵ ∧ 1 ≤ X₁ ∧ 0 ≤ (X₁)² ∧ (X₁)² ≤ 0 ∧ 0 ≤ X₂ ∧ X₂ ≤ 0
∨ (X₀)⁵ ≤ 1 ∧ 1 ≤ (X₀)⁵ ∧ 1+X₁ ≤ 0 ∧ 0 ≤ (X₁)² ∧ (X₁)² ≤ 0 ∧ 0 ≤ X₂ ∧ X₂ ≤ 0
∨ 1 ≤ X₁ ∧ 1+(X₁)² ≤ 0
∨ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 0 ≤ (X₁)² ∧ (X₁)² ≤ 0
∨ 1 ≤ X₁ ∧ 2 ≤ (X₀)⁵ ∧ 0 ≤ (X₁)² ∧ (X₁)² ≤ 0 ∧ 0 ≤ X₂ ∧ X₂ ≤ 0
∨ 1+X₁ ≤ 0 ∧ 1+(X₁)² ≤ 0
∨ 1+X₁ ≤ 0 ∧ 1 ≤ X₂ ∧ 0 ≤ (X₁)² ∧ (X₁)² ≤ 0
∨ 1+X₁ ≤ 0 ∧ 2 ≤ (X₀)⁵ ∧ 0 ≤ (X₁)² ∧ (X₁)² ≤ 0 ∧ 0 ≤ X₂ ∧ X₂ ≤ 0

Stabilization-Threshold for: 1+(X₁)² ≤ (X₀)⁵+X₂

alphas_abs: (X₀)⁵+X₂
M': 1
N: 1
Bound: 5⋅log(X₀)+log(X₂)+3 {O(log(n))}

TWN - Lifting for [10: l1->l1; 11: l1->l1] of 5⋅log(X₀)+log(X₂)+5 {O(log(n))}

relevant size-bounds w.r.t. t₉: l0→l1:
X₀: X₀ {O(n)}
X₂: X₂ {O(n)}
Runtime-bound of t₉: 1 {O(1)}
Results in: 5⋅log(X₀)+log(X₂)+5 {O(log(n))}

All Bounds

Timebounds

Overall timebound:10⋅log(X₀)+2⋅log(X₂)+11 {O(log(n))}
t₉: 1 {O(1)}
t₁₀: 5⋅log(X₀)+log(X₂)+5 {O(log(n))}
t₁₁: 5⋅log(X₀)+log(X₂)+5 {O(log(n))}

Costbounds

Overall costbound: 10⋅log(X₀)+2⋅log(X₂)+11 {O(log(n))}
t₉: 1 {O(1)}
t₁₀: 5⋅log(X₀)+log(X₂)+5 {O(log(n))}
t₁₁: 5⋅log(X₀)+log(X₂)+5 {O(log(n))}

Sizebounds

t₉, X₀: X₀ {O(n)}
t₉, X₁: X₁ {O(n)}
t₉, X₂: X₂ {O(n)}
t₁₀, X₀: X₀ {O(n)}
t₁₀, X₁: 1024⋅4^(5⋅log(X₀))⋅X₁⋅X₂ {O(EXP)}
t₁₀, X₂: 59049⋅9^(5⋅log(X₀))⋅X₂⋅X₂ {O(EXP)}
t₁₁, X₀: X₀ {O(n)}
t₁₁, X₁: 1024⋅4^(5⋅log(X₀))⋅X₁⋅X₂ {O(EXP)}
t₁₁, X₂: 59049⋅9^(5⋅log(X₀))⋅X₂⋅X₂ {O(EXP)}