KoAT2 Proof Output

Initial Problem

Start: l0
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: l0, l1
Transitions:
t₀: l0(X₀, X₁, X₂) → l1(X₀, X₁, X₂) :|: 1 ≤ X₂
t₁: l1(X₀, X₁, X₂) → l1(4⋅X₀, 9⋅X₁-8⋅(X₂)³, X₂) :|: 1+X₀ ≤ 0 ∧ 1+(X₀)²+(X₂)⁵ ≤ X₁
t₂: l1(X₀, X₁, X₂) → l1(4⋅X₀, 9⋅X₁-8⋅(X₂)³, X₂) :|: 1 ≤ X₀ ∧ 1+(X₀)²+(X₂)⁵ ≤ X₁

Preprocessing

Found invariant 1 ≤ X₂ for location l1

Problem after Preprocessing

Start: l0
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: l0, l1
Transitions:
t₀: l0(X₀, X₁, X₂) → l1(X₀, X₁, X₂) :|: 1 ≤ X₂
t₁: l1(X₀, X₁, X₂) → l1(4⋅X₀, 9⋅X₁-8⋅(X₂)³, X₂) :|: 1+X₀ ≤ 0 ∧ 1+(X₀)²+(X₂)⁵ ≤ X₁ ∧ 1 ≤ X₂
t₂: l1(X₀, X₁, X₂) → l1(4⋅X₀, 9⋅X₁-8⋅(X₂)³, X₂) :|: 1 ≤ X₀ ∧ 1+(X₀)²+(X₂)⁵ ≤ X₁ ∧ 1 ≤ X₂

TWN: t₁: l1→l1

cycle: [t₁: l1→l1; t₂: l1→l1]
original loop: (1+X₀ ≤ 0 ∧ 1+(X₀)²+(X₂)⁵ ≤ X₁ ∧ 1 ≤ X₂ ∨ 1 ≤ X₀ ∧ 1+(X₀)²+(X₂)⁵ ≤ X₁ ∧ 1 ≤ X₂,(X₀,X₁,X₂) -> (4⋅X₀,9⋅X₁-8⋅(X₂)³,X₂))
transformed loop: (1+X₀ ≤ 0 ∧ 1+(X₀)²+(X₂)⁵ ≤ X₁ ∧ 1 ≤ X₂ ∨ 1 ≤ X₀ ∧ 1+(X₀)²+(X₂)⁵ ≤ X₁ ∧ 1 ≤ X₂,(X₀,X₁,X₂) -> (4⋅X₀,9⋅X₁-8⋅(X₂)³,X₂))
loop: (1+X₀ ≤ 0 ∧ 1+(X₀)²+(X₂)⁵ ≤ X₁ ∧ 1 ≤ X₂ ∨ 1 ≤ X₀ ∧ 1+(X₀)²+(X₂)⁵ ≤ X₁ ∧ 1 ≤ X₂,(X₀,X₁,X₂) -> (4⋅X₀,9⋅X₁-8⋅(X₂)³,X₂))
order: [X₂; X₁; X₀]
closed-form:

X₂: X₂
X₁: X₁⋅(9)^n + [[n != 0]]⋅-(X₂)³⋅(9)^n + [[n != 0]]⋅(X₂)³
X₀: X₀⋅(4)^n

Termination: true
Formula:

(X₂)³ ≤ 1+(X₂)⁵ ∧ 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 1+(X₂)⁵ ≤ (X₂)³ ∧ 0 ≤ (X₀)² ∧ (X₀)² ≤ 0 ∧ (X₂)³ ≤ X₁ ∧ X₁ ≤ (X₂)³
∨ (X₂)³ ≤ 1+(X₂)⁵ ∧ 1+X₀ ≤ 0 ∧ 1 ≤ X₂ ∧ 1+(X₂)⁵ ≤ (X₂)³ ∧ 0 ≤ (X₀)² ∧ (X₀)² ≤ 0 ∧ (X₂)³ ≤ X₁ ∧ X₁ ≤ (X₂)³
∨ 1 ≤ X₀ ∧ 1+(X₀)² ≤ 0 ∧ 1 ≤ X₂
∨ 1 ≤ X₀ ∧ 1+(X₂)³ ≤ X₁ ∧ 1 ≤ X₂ ∧ 0 ≤ (X₀)² ∧ (X₀)² ≤ 0
∨ 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 2+(X₂)⁵ ≤ (X₂)³ ∧ 0 ≤ (X₀)² ∧ (X₀)² ≤ 0 ∧ (X₂)³ ≤ X₁ ∧ X₁ ≤ (X₂)³
∨ 1+X₀ ≤ 0 ∧ 1+(X₀)² ≤ 0 ∧ 1 ≤ X₂
∨ 1+X₀ ≤ 0 ∧ 1+(X₂)³ ≤ X₁ ∧ 1 ≤ X₂ ∧ 0 ≤ (X₀)² ∧ (X₀)² ≤ 0
∨ 1+X₀ ≤ 0 ∧ 1 ≤ X₂ ∧ 2+(X₂)⁵ ≤ (X₂)³ ∧ 0 ≤ (X₀)² ∧ (X₀)² ≤ 0 ∧ (X₂)³ ≤ X₁ ∧ X₁ ≤ (X₂)³

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

alphas_abs: X₁+(X₂)³+(X₂)⁵
M': 1
N: 1
Bound: 8⋅log(X₂)+log(X₁)+4 {O(log(n))}

TWN - Lifting for [1: l1->l1; 2: l1->l1] of 8⋅log(X₂)+log(X₁)+6 {O(log(n))}

relevant size-bounds w.r.t. t₀: l0→l1:
X₁: X₁ {O(n)}
X₂: X₂ {O(n)}
Runtime-bound of t₀: 1 {O(1)}
Results in: 8⋅log(X₂)+log(X₁)+6 {O(log(n))}

All Bounds

Timebounds

Overall timebound:16⋅log(X₂)+2⋅log(X₁)+13 {O(log(n))}
t₀: 1 {O(1)}
t₁: 8⋅log(X₂)+log(X₁)+6 {O(log(n))}
t₂: 8⋅log(X₂)+log(X₁)+6 {O(log(n))}

Costbounds

Overall costbound: 16⋅log(X₂)+2⋅log(X₁)+13 {O(log(n))}
t₀: 1 {O(1)}
t₁: 8⋅log(X₂)+log(X₁)+6 {O(log(n))}
t₂: 8⋅log(X₂)+log(X₁)+6 {O(log(n))}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₁, X₀: 4096⋅4^(8⋅log(X₂))⋅X₀⋅X₁ {O(EXP)}
t₁, X₁: 58149737003040059690390169⋅9^(16⋅log(X₂))⋅9^(2⋅log(X₁))⋅9^(6⋅log(X₂))⋅X₁+58149737003040059690390169⋅9^(16⋅log(X₂))⋅9^(2⋅log(X₁))⋅9^(6⋅log(X₂))⋅X₂⋅X₂⋅X₂+X₂⋅X₂⋅X₂ {O(EXP)}
t₁, X₂: X₂ {O(n)}
t₂, X₀: 4096⋅4^(8⋅log(X₂))⋅X₀⋅X₁ {O(EXP)}
t₂, X₁: 58149737003040059690390169⋅9^(16⋅log(X₂))⋅9^(2⋅log(X₁))⋅9^(6⋅log(X₂))⋅X₁+58149737003040059690390169⋅9^(16⋅log(X₂))⋅9^(2⋅log(X₁))⋅9^(6⋅log(X₂))⋅X₂⋅X₂⋅X₂+X₂⋅X₂⋅X₂ {O(EXP)}
t₂, X₂: X₂ {O(n)}