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+X₂, -6⋅X₁-2⋅X₂, 2⋅(X₀)²+X₃) :|: 1 ≤ X₀
t₂: l1(X₀, X₁, X₂, X₃) → l2(X₀, X₁, X₂, X₃) :|: X₀ ≤ 0
t₃: l2(X₀, X₁, X₂, X₃) → l2(X₀, X₁-1, X₂, X₃-1) :|: 1 ≤ X₁+X₃

Preprocessing

Found invariant X₀ ≤ 0 for location l2

Problem after Preprocessing

MPRF for transition t₁: l1(X₀, X₁, X₂, X₃) → l1(X₀-1, 3⋅X₁+X₂, -6⋅X₁-2⋅X₂, 2⋅Temp_Int₇₃+X₃) :|: 1 ≤ X₀ ∧ 1 ≤ Temp_Int₇₃ ∧ X₀ ≤ Temp_Int₇₃ 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, X₃-1) :|: 1 ≤ X₁+X₃ ∧ X₀ ≤ 0 of depth 1:

new bound:

5⋅X₀⋅X₀⋅X₀+20⋅X₀⋅X₀+2⋅X₃+22⋅X₀+28⋅X₁+9⋅X₂+9 {O(n^3)}

MPRF:

• l2: [X₁+X₃]

Found invariant 0 ≤ 1+X₁+X₃ ∧ X₀ ≤ 0 for location l2_v1

Found invariant X₀ ≤ 0 for location l2

All Bounds

Timebounds

Overall timebound:5⋅X₀⋅X₀⋅X₀+20⋅X₀⋅X₀+2⋅X₃+23⋅X₀+28⋅X₁+9⋅X₂+11 {O(n^3)}
t₀: 1 {O(1)}
t₁: X₀ {O(n)}
t₂: 1 {O(1)}
t₃: 5⋅X₀⋅X₀⋅X₀+20⋅X₀⋅X₀+2⋅X₃+22⋅X₀+28⋅X₁+9⋅X₂+9 {O(n^3)}

Costbounds

Overall costbound: 5⋅X₀⋅X₀⋅X₀+20⋅X₀⋅X₀+2⋅X₃+23⋅X₀+28⋅X₁+9⋅X₂+11 {O(n^3)}
t₀: 1 {O(1)}
t₁: X₀ {O(n)}
t₂: 1 {O(1)}
t₃: 5⋅X₀⋅X₀⋅X₀+20⋅X₀⋅X₀+2⋅X₃+22⋅X₀+28⋅X₁+9⋅X₂+9 {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: 27⋅X₁+9⋅X₂ {O(n)}
t₁, X₂: 22⋅X₂+66⋅X₁ {O(n)}
t₁, X₃: 5⋅X₀⋅X₀⋅X₀+20⋅X₀⋅X₀+22⋅X₀+X₃+9 {O(n^3)}
t₂, X₀: 2⋅X₀ {O(n)}
t₂, X₁: 28⋅X₁+9⋅X₂ {O(n)}
t₂, X₂: 23⋅X₂+66⋅X₁ {O(n)}
t₂, X₃: 5⋅X₀⋅X₀⋅X₀+20⋅X₀⋅X₀+2⋅X₃+22⋅X₀+9 {O(n^3)}
t₃, X₀: 2⋅X₀ {O(n)}
t₃, X₁: 5⋅X₀⋅X₀⋅X₀+20⋅X₀⋅X₀+18⋅X₂+2⋅X₃+22⋅X₀+56⋅X₁+9 {O(n^3)}
t₃, X₂: 23⋅X₂+66⋅X₁ {O(n)}
t₃, X₃: 10⋅X₀⋅X₀⋅X₀+40⋅X₀⋅X₀+28⋅X₁+4⋅X₃+44⋅X₀+9⋅X₂+18 {O(n^3)}