KoAT2 Proof Output

Initial Problem

Start: l0
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars:
Locations: l0, l1, l2
Transitions:
t₀: l0(X₀, X₁, X₂, X₃) → l1(X₀, X₁, X₂, X₃)
t₁: l1(X₀, X₁, X₂, X₃) → l1(X₀-1, 3⋅X₁+2⋅X₂, -5⋅X₁-3⋅X₂, (X₀)²+X₃) :|: 1 ≤ X₀
t₂: l1(X₀, X₁, X₂, X₃) → l2(X₀, X₁, X₂, X₃)
t₃: l2(X₀, X₁, X₂, X₃) → l2(X₀, X₁-1, X₂-1, X₃-1) :|: 1 ≤ X₁+X₂+X₃

Preprocessing

Problem after Preprocessing

MPRF for transition t₁: l1(X₀, X₁, X₂, X₃) → l1(X₀-1, 3⋅X₁+2⋅X₂, -5⋅X₁-3⋅X₂, Temp_Int₅₂+X₃) :|: 1 ≤ X₀ ∧ 1 ≤ Temp_Int₅₂ ∧ X₀ ≤ Temp_Int₅₂ of depth 1:

new bound:

X₀ {O(n)}

MPRF:

• l1: [X₀]

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

new bound:

3⋅X₀⋅X₀⋅X₀+12⋅X₀⋅X₀+14⋅X₀+2⋅X₃+39⋅X₂+61⋅X₁+6 {O(n^3)}

MPRF:

• l2: [X₁+X₂+X₃]

All Bounds

Timebounds

Overall timebound:3⋅X₀⋅X₀⋅X₀+12⋅X₀⋅X₀+15⋅X₀+2⋅X₃+39⋅X₂+61⋅X₁+8 {O(n^3)}
t₀: 1 {O(1)}
t₁: X₀ {O(n)}
t₂: 1 {O(1)}
t₃: 3⋅X₀⋅X₀⋅X₀+12⋅X₀⋅X₀+14⋅X₀+2⋅X₃+39⋅X₂+61⋅X₁+6 {O(n^3)}

Costbounds

Overall costbound: 3⋅X₀⋅X₀⋅X₀+12⋅X₀⋅X₀+15⋅X₀+2⋅X₃+39⋅X₂+61⋅X₁+8 {O(n^3)}
t₀: 1 {O(1)}
t₁: X₀ {O(n)}
t₂: 1 {O(1)}
t₃: 3⋅X₀⋅X₀⋅X₀+12⋅X₀⋅X₀+14⋅X₀+2⋅X₃+39⋅X₂+61⋅X₁+6 {O(n^3)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₀, X₃: X₃ {O(n)}
t₁, X₀: X₀ {O(n)}
t₁, X₁: 14⋅X₂+22⋅X₁ {O(n)}
t₁, X₂: 24⋅X₂+38⋅X₁ {O(n)}
t₁, X₃: 3⋅X₀⋅X₀⋅X₀+12⋅X₀⋅X₀+14⋅X₀+X₃+6 {O(n^3)}
t₂, X₀: 2⋅X₀ {O(n)}
t₂, X₁: 14⋅X₂+23⋅X₁ {O(n)}
t₂, X₂: 25⋅X₂+38⋅X₁ {O(n)}
t₂, X₃: 3⋅X₀⋅X₀⋅X₀+12⋅X₀⋅X₀+14⋅X₀+2⋅X₃+6 {O(n^3)}
t₃, X₀: 2⋅X₀ {O(n)}
t₃, X₁: 3⋅X₀⋅X₀⋅X₀+12⋅X₀⋅X₀+14⋅X₀+2⋅X₃+53⋅X₂+84⋅X₁+6 {O(n^3)}
t₃, X₂: 3⋅X₀⋅X₀⋅X₀+12⋅X₀⋅X₀+14⋅X₀+2⋅X₃+64⋅X₂+99⋅X₁+6 {O(n^3)}
t₃, X₃: 6⋅X₀⋅X₀⋅X₀+24⋅X₀⋅X₀+28⋅X₀+39⋅X₂+4⋅X₃+61⋅X₁+12 {O(n^3)}