KoAT2 Proof Output

Initial Problem

Start: l0
Program_Vars: X₀, X₁, X₂, X₃, X₄
Temp_Vars:
Locations: l0, l1, l2, l3
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+X₃, X₃+X₄, X₄-1) :|: 1 ≤ X₄
t₂: l1(X₀, X₁, X₂, X₃, X₄) → l2(X₀, X₁, X₂, X₃, X₄) :|: X₄ ≤ 0
t₃: l2(X₀, X₁, X₂, X₃, X₄) → l2(X₀, X₁, X₂-1, X₃, X₄) :|: 1 ≤ X₂
t₄: l2(X₀, X₁, X₂, X₃, X₄) → l3(X₀, X₁, X₂, X₃, X₄) :|: X₂ ≤ 0
t₅: l3(X₀, X₁, X₂, X₃, X₄) → l3(X₀+X₁, X₁-1, X₂, X₃, X₄) :|: 1 ≤ X₀

Preprocessing

Found invariant X₄ ≤ 0 for location l2

Found invariant X₄ ≤ 0 ∧ X₂+X₄ ≤ 0 ∧ X₂ ≤ 0 for location l3

Problem after Preprocessing

Start: l0
Program_Vars: X₀, X₁, X₂, X₃, X₄
Temp_Vars:
Locations: l0, l1, l2, l3
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+X₃, X₃+X₄, X₄-1) :|: 1 ≤ X₄
t₂: l1(X₀, X₁, X₂, X₃, X₄) → l2(X₀, X₁, X₂, X₃, X₄) :|: X₄ ≤ 0
t₃: l2(X₀, X₁, X₂, X₃, X₄) → l2(X₀, X₁, X₂-1, X₃, X₄) :|: 1 ≤ X₂ ∧ X₄ ≤ 0
t₄: l2(X₀, X₁, X₂, X₃, X₄) → l3(X₀, X₁, X₂, X₃, X₄) :|: X₂ ≤ 0 ∧ X₄ ≤ 0
t₅: l3(X₀, X₁, X₂, X₃, X₄) → l3(X₀+X₁, X₁-1, X₂, X₃, X₄) :|: 1 ≤ X₀ ∧ X₂ ≤ 0 ∧ X₂+X₄ ≤ 0 ∧ X₄ ≤ 0

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

new bound:

X₄ {O(n)}

MPRF:

• l1: [X₄]

MPRF for transition t₃: l2(X₀, X₁, X₂, X₃, X₄) → l2(X₀, X₁, X₂-1, X₃, X₄) :|: 1 ≤ X₂ ∧ X₄ ≤ 0 of depth 1:

new bound:

2⋅X₄⋅X₄⋅X₄+2⋅X₃⋅X₄+4⋅X₄⋅X₄+2⋅X₂+2⋅X₃+2⋅X₄ {O(n^3)}

MPRF:

• l2: [X₂]

Found invariant X₄ ≤ 0 for location l2

Found invariant X₄ ≤ 0 ∧ X₄ ≤ X₂ ∧ 0 ≤ X₂ for location l2_v1

Found invariant X₄ ≤ 0 ∧ X₂+X₄ ≤ 0 ∧ X₂ ≤ 0 for location l3

MPRF for transition t₅: l3(X₀, X₁, X₂, X₃, X₄) → l3(X₀+X₁, X₁-1, X₂, X₃, X₄) :|: 1 ≤ X₀ ∧ X₂ ≤ 0 ∧ X₂+X₄ ≤ 0 ∧ X₄ ≤ 0 of depth 2:

new bound:

32⋅X₀+32⋅X₁+9 {O(n)}

MPRF:

• l3: [1+X₁; X₀]

All Bounds

Timebounds

Overall timebound:2⋅X₄⋅X₄⋅X₄+2⋅X₃⋅X₄+4⋅X₄⋅X₄+2⋅X₂+2⋅X₃+3⋅X₄+32⋅X₀+32⋅X₁+12 {O(n^3)}
t₀: 1 {O(1)}
t₁: X₄ {O(n)}
t₂: 1 {O(1)}
t₃: 2⋅X₄⋅X₄⋅X₄+2⋅X₃⋅X₄+4⋅X₄⋅X₄+2⋅X₂+2⋅X₃+2⋅X₄ {O(n^3)}
t₄: 1 {O(1)}
t₅: 32⋅X₀+32⋅X₁+9 {O(n)}

Costbounds

Overall costbound: 2⋅X₄⋅X₄⋅X₄+2⋅X₃⋅X₄+4⋅X₄⋅X₄+2⋅X₂+2⋅X₃+3⋅X₄+32⋅X₀+32⋅X₁+12 {O(n^3)}
t₀: 1 {O(1)}
t₁: X₄ {O(n)}
t₂: 1 {O(1)}
t₃: 2⋅X₄⋅X₄⋅X₄+2⋅X₃⋅X₄+4⋅X₄⋅X₄+2⋅X₂+2⋅X₃+2⋅X₄ {O(n^3)}
t₄: 1 {O(1)}
t₅: 32⋅X₀+32⋅X₁+9 {O(n)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₀, X₃: X₃ {O(n)}
t₀, X₄: X₄ {O(n)}
t₁, X₀: X₀ {O(n)}
t₁, X₁: X₁ {O(n)}
t₁, X₂: 2⋅X₄⋅X₄⋅X₄+2⋅X₃⋅X₄+4⋅X₄⋅X₄+2⋅X₃+2⋅X₄+X₂ {O(n^3)}
t₁, X₃: 2⋅X₄⋅X₄+2⋅X₄+X₃ {O(n^2)}
t₁, X₄: X₄ {O(n)}
t₂, X₀: 2⋅X₀ {O(n)}
t₂, X₁: 2⋅X₁ {O(n)}
t₂, X₂: 2⋅X₄⋅X₄⋅X₄+2⋅X₃⋅X₄+4⋅X₄⋅X₄+2⋅X₂+2⋅X₃+2⋅X₄ {O(n^3)}
t₂, X₃: 2⋅X₄⋅X₄+2⋅X₃+2⋅X₄ {O(n^2)}
t₂, X₄: 2⋅X₄ {O(n)}
t₃, X₀: 2⋅X₀ {O(n)}
t₃, X₁: 2⋅X₁ {O(n)}
t₃, X₂: 2⋅X₄⋅X₄⋅X₄+2⋅X₃⋅X₄+4⋅X₄⋅X₄+2⋅X₂+2⋅X₃+2⋅X₄ {O(n^3)}
t₃, X₃: 2⋅X₄⋅X₄+2⋅X₃+2⋅X₄ {O(n^2)}
t₃, X₄: 2⋅X₄ {O(n)}
t₄, X₀: 4⋅X₀ {O(n)}
t₄, X₁: 4⋅X₁ {O(n)}
t₄, X₂: 4⋅X₄⋅X₄⋅X₄+4⋅X₃⋅X₄+8⋅X₄⋅X₄+4⋅X₂+4⋅X₃+4⋅X₄ {O(n^3)}
t₄, X₃: 4⋅X₄⋅X₄+4⋅X₃+4⋅X₄ {O(n^2)}
t₄, X₄: 4⋅X₄ {O(n)}
t₅, X₀: 1024⋅X₀⋅X₀+1280⋅X₁⋅X₁+2304⋅X₀⋅X₁+612⋅X₀+688⋅X₁+90 {O(n^2)}
t₅, X₁: 32⋅X₀+36⋅X₁+9 {O(n)}
t₅, X₂: 4⋅X₄⋅X₄⋅X₄+4⋅X₃⋅X₄+8⋅X₄⋅X₄+4⋅X₂+4⋅X₃+4⋅X₄ {O(n^3)}
t₅, X₃: 4⋅X₄⋅X₄+4⋅X₃+4⋅X₄ {O(n^2)}
t₅, X₄: 4⋅X₄ {O(n)}