KoAT2 Proof Output

Initial Problem

Start: l0
Program_Vars: X₀, X₁, X₂, X₃, X₄
Temp_Vars:
Locations: l0, l1, l2
Transitions:
t₀: l0(X₀, X₁, X₂, X₃, X₄) → l1(X₃, X₄, X₂, X₃, X₄)
t₁: l1(X₀, X₁, X₂, X₃, X₄) → l2(X₀, X₁, X₂, X₃, X₄) :|: 1 ≤ X₂
t₃: l2(X₀, X₁, X₂, X₃, X₄) → l1(X₃, X₄, X₂-1, X₃, X₄)
t₂: l2(X₀, X₁, X₂, X₃, X₄) → l2(5⋅X₀+(X₂)², 2⋅X₁, X₂, X₃, X₄) :|: 1 ≤ X₀ ∧ 1+X₀ ≤ X₁

Preprocessing

Found invariant X₄ ≤ X₁ ∧ X₁ ≤ X₄ ∧ X₃ ≤ X₀ ∧ X₀ ≤ X₃ for location l1

Found invariant X₄ ≤ X₁ ∧ X₃ ≤ X₀ ∧ 1 ≤ X₂ for location l2

Problem after Preprocessing

Start: l0
Program_Vars: X₀, X₁, X₂, X₃, X₄
Temp_Vars:
Locations: l0, l1, l2
Transitions:
t₀: l0(X₀, X₁, X₂, X₃, X₄) → l1(X₃, X₄, X₂, X₃, X₄)
t₁: l1(X₀, X₁, X₂, X₃, X₄) → l2(X₀, X₁, X₂, X₃, X₄) :|: 1 ≤ X₂ ∧ X₃ ≤ X₀ ∧ X₀ ≤ X₃ ∧ X₄ ≤ X₁ ∧ X₁ ≤ X₄
t₃: l2(X₀, X₁, X₂, X₃, X₄) → l1(X₃, X₄, X₂-1, X₃, X₄) :|: 1 ≤ X₂ ∧ X₃ ≤ X₀ ∧ X₄ ≤ X₁
t₂: l2(X₀, X₁, X₂, X₃, X₄) → l2(5⋅X₀+(X₂)², 2⋅X₁, X₂, X₃, X₄) :|: 1 ≤ X₀ ∧ 1+X₀ ≤ X₁ ∧ 1 ≤ X₂ ∧ X₃ ≤ X₀ ∧ X₄ ≤ X₁

MPRF for transition t₁: l1(X₀, X₁, X₂, X₃, X₄) → l2(X₀, X₁, X₂, X₃, X₄) :|: 1 ≤ X₂ ∧ X₃ ≤ X₀ ∧ X₀ ≤ X₃ ∧ X₄ ≤ X₁ ∧ X₁ ≤ X₄ of depth 1:

new bound:

X₂ {O(n)}

MPRF:

• l1: [X₂]
• l2: [X₂-1]

MPRF for transition t₃: l2(X₀, X₁, X₂, X₃, X₄) → l1(X₃, X₄, X₂-1, X₃, X₄) :|: 1 ≤ X₂ ∧ X₃ ≤ X₀ ∧ X₄ ≤ X₁ of depth 1:

new bound:

X₂ {O(n)}

MPRF:

• l1: [X₂]
• l2: [X₂]

TWN: t₂: l2→l2

cycle: [t₂: l2→l2]
original loop: (1 ≤ X₀ ∧ 1+X₀ ≤ X₁ ∧ 1 ≤ X₂ ∧ X₃ ≤ X₀ ∧ X₄ ≤ X₁,(X₀,X₁,X₂,X₃,X₄) -> (5⋅X₀+(X₂)²,2⋅X₁,X₂,X₃,X₄))
transformed loop: (1 ≤ X₀ ∧ 1+X₀ ≤ X₁ ∧ 1 ≤ X₂ ∧ X₃ ≤ X₀ ∧ X₄ ≤ X₁,(X₀,X₁,X₂,X₃,X₄) -> (5⋅X₀+(X₂)²,2⋅X₁,X₂,X₃,X₄))
loop: (1 ≤ X₀ ∧ 1+X₀ ≤ X₁ ∧ 1 ≤ X₂ ∧ X₃ ≤ X₀ ∧ X₄ ≤ X₁,(X₀,X₁,X₂,X₃,X₄) -> (5⋅X₀+(X₂)²,2⋅X₁,X₂,X₃,X₄))
order: [X₂; X₄; X₃; X₁; X₀]
closed-form:

X₂: X₂
X₄: X₄
X₃: X₃
X₁: X₁⋅(2)^n
X₀: X₀⋅(5)^n + [[n != 0]]⋅1/4⋅(X₂)²⋅(5)^n + [[n != 0]]⋅-1/4⋅(X₂)²

Termination: true
Formula:

(X₂)² ≤ 4 ∧ 1 ≤ 4⋅X₀+(X₂)² ∧ 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 4 ≤ (X₂)² ∧ 0 ≤ 4⋅X₀+(X₂)² ∧ 4⋅X₀+(X₂)² ≤ 0 ∧ 0 ≤ X₁ ∧ X₁ ≤ 0
∨ (X₂)² ≤ 4 ∧ 1 ≤ 4⋅X₀+(X₂)² ∧ 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 1+X₄ ≤ 0 ∧ 4 ≤ (X₂)² ∧ 0 ≤ 4⋅X₀+(X₂)² ∧ 4⋅X₀+(X₂)² ≤ 0 ∧ 0 ≤ X₁ ∧ X₁ ≤ 0
∨ (X₂)² ≤ 4 ∧ 1 ≤ 4⋅X₀+(X₂)² ∧ 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 4 ≤ (X₂)² ∧ 0 ≤ 4⋅X₀+(X₂)² ∧ 4⋅X₀+(X₂)² ≤ 0 ∧ 0 ≤ X₁ ∧ X₁ ≤ 0 ∧ 0 ≤ X₄ ∧ X₄ ≤ 0
∨ (X₂)² ≤ 4 ∧ 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 1+(X₂)²+4⋅X₃ ≤ 0 ∧ 4 ≤ (X₂)² ∧ 0 ≤ 4⋅X₀+(X₂)² ∧ 4⋅X₀+(X₂)² ≤ 0 ∧ 0 ≤ X₁ ∧ X₁ ≤ 0
∨ (X₂)² ≤ 4 ∧ 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 4 ≤ (X₂)² ∧ 0 ≤ 4⋅X₀+(X₂)² ∧ 4⋅X₀+(X₂)² ≤ 0 ∧ 0 ≤ X₁ ∧ X₁ ≤ 0 ∧ 0 ≤ (X₂)²+4⋅X₃ ∧ (X₂)²+4⋅X₃ ≤ 0
∨ (X₂)² ≤ 4 ∧ 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 1+(X₂)²+4⋅X₃ ≤ 0 ∧ 1+X₄ ≤ 0 ∧ 4 ≤ (X₂)² ∧ 0 ≤ 4⋅X₀+(X₂)² ∧ 4⋅X₀+(X₂)² ≤ 0 ∧ 0 ≤ X₁ ∧ X₁ ≤ 0
∨ (X₂)² ≤ 4 ∧ 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 1+(X₂)²+4⋅X₃ ≤ 0 ∧ 4 ≤ (X₂)² ∧ 0 ≤ 4⋅X₀+(X₂)² ∧ 4⋅X₀+(X₂)² ≤ 0 ∧ 0 ≤ X₁ ∧ X₁ ≤ 0 ∧ 0 ≤ X₄ ∧ X₄ ≤ 0
∨ (X₂)² ≤ 4 ∧ 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 1+X₄ ≤ 0 ∧ 4 ≤ (X₂)² ∧ 0 ≤ 4⋅X₀+(X₂)² ∧ 4⋅X₀+(X₂)² ≤ 0 ∧ 0 ≤ X₁ ∧ X₁ ≤ 0 ∧ 0 ≤ (X₂)²+4⋅X₃ ∧ (X₂)²+4⋅X₃ ≤ 0
∨ (X₂)² ≤ 4 ∧ 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 4 ≤ (X₂)² ∧ 0 ≤ 4⋅X₀+(X₂)² ∧ 4⋅X₀+(X₂)² ≤ 0 ∧ 0 ≤ X₁ ∧ X₁ ≤ 0 ∧ 0 ≤ (X₂)²+4⋅X₃ ∧ (X₂)²+4⋅X₃ ≤ 0 ∧ 0 ≤ X₄ ∧ X₄ ≤ 0
∨ 1 ≤ 4⋅X₀+(X₂)² ∧ 1 ≤ X₀ ∧ 1+4⋅X₀+(X₂)² ≤ 0 ∧ 1 ≤ X₁ ∧ 1 ≤ X₂
∨ 1 ≤ 4⋅X₀+(X₂)² ∧ 1 ≤ X₀ ∧ 1+4⋅X₀+(X₂)² ≤ 0 ∧ 1 ≤ X₂ ∧ 1+X₄ ≤ 0 ∧ 0 ≤ X₁ ∧ X₁ ≤ 0
∨ 1 ≤ 4⋅X₀+(X₂)² ∧ 1 ≤ X₀ ∧ 1+4⋅X₀+(X₂)² ≤ 0 ∧ 1 ≤ X₂ ∧ 0 ≤ X₁ ∧ X₁ ≤ 0 ∧ 0 ≤ X₄ ∧ X₄ ≤ 0
∨ 1 ≤ 4⋅X₀+(X₂)² ∧ 1 ≤ X₀ ∧ 1 ≤ 4⋅X₁ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 0 ≤ 4⋅X₀+(X₂)² ∧ 4⋅X₀+(X₂)² ≤ 0
∨ 1 ≤ 4⋅X₀+(X₂)² ∧ 1 ≤ X₀ ∧ 1 ≤ 4⋅X₁ ∧ 1 ≤ X₂ ∧ 1+X₄ ≤ 0 ∧ 0 ≤ 4⋅X₀+(X₂)² ∧ 4⋅X₀+(X₂)² ≤ 0 ∧ 0 ≤ X₁ ∧ X₁ ≤ 0
∨ 1 ≤ 4⋅X₀+(X₂)² ∧ 1 ≤ X₀ ∧ 1 ≤ 4⋅X₁ ∧ 1 ≤ X₂ ∧ 0 ≤ 4⋅X₀+(X₂)² ∧ 4⋅X₀+(X₂)² ≤ 0 ∧ 0 ≤ X₁ ∧ X₁ ≤ 0 ∧ 0 ≤ X₄ ∧ X₄ ≤ 0
∨ 1 ≤ 4⋅X₀+(X₂)² ∧ 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 5 ≤ (X₂)² ∧ 0 ≤ 4⋅X₀+(X₂)² ∧ 4⋅X₀+(X₂)² ≤ 0 ∧ 0 ≤ X₁ ∧ X₁ ≤ 0
∨ 1 ≤ 4⋅X₀+(X₂)² ∧ 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 1+X₄ ≤ 0 ∧ 5 ≤ (X₂)² ∧ 0 ≤ 4⋅X₀+(X₂)² ∧ 4⋅X₀+(X₂)² ≤ 0 ∧ 0 ≤ X₁ ∧ X₁ ≤ 0
∨ 1 ≤ 4⋅X₀+(X₂)² ∧ 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 5 ≤ (X₂)² ∧ 0 ≤ 4⋅X₀+(X₂)² ∧ 4⋅X₀+(X₂)² ≤ 0 ∧ 0 ≤ X₁ ∧ X₁ ≤ 0 ∧ 0 ≤ X₄ ∧ X₄ ≤ 0
∨ 1 ≤ X₀ ∧ 1+4⋅X₀+(X₂)² ≤ 0 ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 1+(X₂)²+4⋅X₃ ≤ 0 ∧ 0 ≤ 4⋅X₀+(X₂)² ∧ 4⋅X₀+(X₂)² ≤ 0
∨ 1 ≤ X₀ ∧ 1+4⋅X₀+(X₂)² ≤ 0 ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 0 ≤ 4⋅X₀+(X₂)² ∧ 4⋅X₀+(X₂)² ≤ 0 ∧ 0 ≤ (X₂)²+4⋅X₃ ∧ (X₂)²+4⋅X₃ ≤ 0
∨ 1 ≤ X₀ ∧ 1+4⋅X₀+(X₂)² ≤ 0 ∧ 1 ≤ X₂ ∧ 1+(X₂)²+4⋅X₃ ≤ 0 ∧ 1+X₄ ≤ 0 ∧ 0 ≤ 4⋅X₀+(X₂)² ∧ 4⋅X₀+(X₂)² ≤ 0 ∧ 0 ≤ X₁ ∧ X₁ ≤ 0
∨ 1 ≤ X₀ ∧ 1+4⋅X₀+(X₂)² ≤ 0 ∧ 1 ≤ X₂ ∧ 1+(X₂)²+4⋅X₃ ≤ 0 ∧ 0 ≤ 4⋅X₀+(X₂)² ∧ 4⋅X₀+(X₂)² ≤ 0 ∧ 0 ≤ X₁ ∧ X₁ ≤ 0 ∧ 0 ≤ X₄ ∧ X₄ ≤ 0
∨ 1 ≤ X₀ ∧ 1+4⋅X₀+(X₂)² ≤ 0 ∧ 1 ≤ X₂ ∧ 1+X₄ ≤ 0 ∧ 0 ≤ 4⋅X₀+(X₂)² ∧ 4⋅X₀+(X₂)² ≤ 0 ∧ 0 ≤ X₁ ∧ X₁ ≤ 0 ∧ 0 ≤ (X₂)²+4⋅X₃ ∧ (X₂)²+4⋅X₃ ≤ 0
∨ 1 ≤ X₀ ∧ 1+4⋅X₀+(X₂)² ≤ 0 ∧ 1 ≤ X₂ ∧ 0 ≤ 4⋅X₀+(X₂)² ∧ 4⋅X₀+(X₂)² ≤ 0 ∧ 0 ≤ X₁ ∧ X₁ ≤ 0 ∧ 0 ≤ (X₂)²+4⋅X₃ ∧ (X₂)²+4⋅X₃ ≤ 0 ∧ 0 ≤ X₄ ∧ X₄ ≤ 0
∨ 1 ≤ X₀ ∧ 1 ≤ 4⋅X₁ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 1+(X₂)²+4⋅X₃ ≤ 0 ∧ 0 ≤ 4⋅X₀+(X₂)² ∧ 4⋅X₀+(X₂)² ≤ 0
∨ 1 ≤ X₀ ∧ 1 ≤ 4⋅X₁ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 0 ≤ 4⋅X₀+(X₂)² ∧ 4⋅X₀+(X₂)² ≤ 0 ∧ 0 ≤ (X₂)²+4⋅X₃ ∧ (X₂)²+4⋅X₃ ≤ 0
∨ 1 ≤ X₀ ∧ 1 ≤ 4⋅X₁ ∧ 1 ≤ X₂ ∧ 1+(X₂)²+4⋅X₃ ≤ 0 ∧ 1+X₄ ≤ 0 ∧ 0 ≤ 4⋅X₀+(X₂)² ∧ 4⋅X₀+(X₂)² ≤ 0 ∧ 0 ≤ X₁ ∧ X₁ ≤ 0
∨ 1 ≤ X₀ ∧ 1 ≤ 4⋅X₁ ∧ 1 ≤ X₂ ∧ 1+(X₂)²+4⋅X₃ ≤ 0 ∧ 0 ≤ 4⋅X₀+(X₂)² ∧ 4⋅X₀+(X₂)² ≤ 0 ∧ 0 ≤ X₁ ∧ X₁ ≤ 0 ∧ 0 ≤ X₄ ∧ X₄ ≤ 0
∨ 1 ≤ X₀ ∧ 1 ≤ 4⋅X₁ ∧ 1 ≤ X₂ ∧ 1+X₄ ≤ 0 ∧ 0 ≤ 4⋅X₀+(X₂)² ∧ 4⋅X₀+(X₂)² ≤ 0 ∧ 0 ≤ X₁ ∧ X₁ ≤ 0 ∧ 0 ≤ (X₂)²+4⋅X₃ ∧ (X₂)²+4⋅X₃ ≤ 0
∨ 1 ≤ X₀ ∧ 1 ≤ 4⋅X₁ ∧ 1 ≤ X₂ ∧ 0 ≤ 4⋅X₀+(X₂)² ∧ 4⋅X₀+(X₂)² ≤ 0 ∧ 0 ≤ X₁ ∧ X₁ ≤ 0 ∧ 0 ≤ (X₂)²+4⋅X₃ ∧ (X₂)²+4⋅X₃ ≤ 0 ∧ 0 ≤ X₄ ∧ X₄ ≤ 0
∨ 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 1+(X₂)²+4⋅X₃ ≤ 0 ∧ 5 ≤ (X₂)² ∧ 0 ≤ 4⋅X₀+(X₂)² ∧ 4⋅X₀+(X₂)² ≤ 0 ∧ 0 ≤ X₁ ∧ X₁ ≤ 0
∨ 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 5 ≤ (X₂)² ∧ 0 ≤ 4⋅X₀+(X₂)² ∧ 4⋅X₀+(X₂)² ≤ 0 ∧ 0 ≤ X₁ ∧ X₁ ≤ 0 ∧ 0 ≤ (X₂)²+4⋅X₃ ∧ (X₂)²+4⋅X₃ ≤ 0
∨ 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 1+(X₂)²+4⋅X₃ ≤ 0 ∧ 1+X₄ ≤ 0 ∧ 5 ≤ (X₂)² ∧ 0 ≤ 4⋅X₀+(X₂)² ∧ 4⋅X₀+(X₂)² ≤ 0 ∧ 0 ≤ X₁ ∧ X₁ ≤ 0
∨ 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 1+(X₂)²+4⋅X₃ ≤ 0 ∧ 5 ≤ (X₂)² ∧ 0 ≤ 4⋅X₀+(X₂)² ∧ 4⋅X₀+(X₂)² ≤ 0 ∧ 0 ≤ X₁ ∧ X₁ ≤ 0 ∧ 0 ≤ X₄ ∧ X₄ ≤ 0
∨ 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 1+X₄ ≤ 0 ∧ 5 ≤ (X₂)² ∧ 0 ≤ 4⋅X₀+(X₂)² ∧ 4⋅X₀+(X₂)² ≤ 0 ∧ 0 ≤ X₁ ∧ X₁ ≤ 0 ∧ 0 ≤ (X₂)²+4⋅X₃ ∧ (X₂)²+4⋅X₃ ≤ 0
∨ 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 5 ≤ (X₂)² ∧ 0 ≤ 4⋅X₀+(X₂)² ∧ 4⋅X₀+(X₂)² ≤ 0 ∧ 0 ≤ X₁ ∧ X₁ ≤ 0 ∧ 0 ≤ (X₂)²+4⋅X₃ ∧ (X₂)²+4⋅X₃ ≤ 0 ∧ 0 ≤ X₄ ∧ X₄ ≤ 0

Stabilization-Threshold for: X₄ ≤ X₁

alphas_abs: 1+X₄
M': 1
N: 1
Bound: log(X₄)+3 {O(log(n))}

Stabilization-Threshold for: X₃ ≤ X₀

alphas_abs: 4+(X₂)²+4⋅X₃
M': 1
N: 1
Bound: 2⋅log(X₂)+log(X₃)+8 {O(log(n))}

Stabilization-Threshold for: 1+X₀ ≤ X₁

alphas_abs: 4⋅X₁+(X₂)²
M': 1
N: 1
Bound: 2⋅log(X₂)+log(X₁)+5 {O(log(n))}

TWN - Lifting for [2: l2->l2] of 4⋅log(X₂)+log(X₁)+log(X₃)+log(X₄)+18 {O(log(n))}

relevant size-bounds w.r.t. t₁: l1→l2:
X₁: 3⋅X₄ {O(n)}
X₂: X₂ {O(n)}
X₃: X₃ {O(n)}
X₄: X₄ {O(n)}
Runtime-bound of t₁: X₂ {O(n)}
Results in: 2⋅X₂⋅log(X₄)+4⋅X₂⋅log(X₂)+X₂⋅log(X₃)+20⋅X₂ {O(log(n)*n)}

All Bounds

Timebounds

Overall timebound:2⋅X₂⋅log(X₄)+4⋅X₂⋅log(X₂)+X₂⋅log(X₃)+22⋅X₂+1 {O(log(n)*n)}
t₀: 1 {O(1)}
t₁: X₂ {O(n)}
t₂: 2⋅X₂⋅log(X₄)+4⋅X₂⋅log(X₂)+X₂⋅log(X₃)+20⋅X₂ {O(log(n)*n)}
t₃: X₂ {O(n)}

Costbounds

Overall costbound: 2⋅X₂⋅log(X₄)+4⋅X₂⋅log(X₂)+X₂⋅log(X₃)+22⋅X₂+1 {O(log(n)*n)}
t₀: 1 {O(1)}
t₁: X₂ {O(n)}
t₂: 2⋅X₂⋅log(X₄)+4⋅X₂⋅log(X₂)+X₂⋅log(X₃)+20⋅X₂ {O(log(n)*n)}
t₃: X₂ {O(n)}

Sizebounds

t₀, X₀: X₃ {O(n)}
t₀, X₁: X₄ {O(n)}
t₀, X₂: X₂ {O(n)}
t₀, X₃: X₃ {O(n)}
t₀, X₄: X₄ {O(n)}
t₁, X₀: 3⋅X₃ {O(n)}
t₁, X₁: 3⋅X₄ {O(n)}
t₁, X₂: X₂ {O(n)}
t₁, X₃: X₃ {O(n)}
t₁, X₄: X₄ {O(n)}
t₂, X₀: 15⋅5^(2⋅X₂⋅log(X₄))⋅5^(20⋅X₂)⋅5^(4⋅X₂⋅log(X₂))⋅5^(X₂⋅log(X₃))⋅X₃+2⋅5^(2⋅X₂⋅log(X₄))⋅5^(20⋅X₂)⋅5^(4⋅X₂⋅log(X₂))⋅5^(X₂⋅log(X₃))⋅X₂⋅X₂+X₂⋅X₂ {O(EXP)}
t₂, X₁: 2^(2⋅X₂⋅log(X₄))⋅2^(20⋅X₂)⋅2^(4⋅X₂⋅log(X₂))⋅2^(X₂⋅log(X₃))⋅3⋅X₄ {O(EXP)}
t₂, X₂: X₂ {O(n)}
t₂, X₃: X₃ {O(n)}
t₂, X₄: X₄ {O(n)}
t₃, X₀: 2⋅X₃ {O(n)}
t₃, X₁: 2⋅X₄ {O(n)}
t₃, X₂: X₂ {O(n)}
t₃, X₃: X₃ {O(n)}
t₃, X₄: X₄ {O(n)}