KoAT2 Proof Output

Initial Problem

Preprocessing

Found invariant X₃ ≤ X₀ ∧ X₁ ≤ X₂ for location stop

Found invariant X₃ ≤ X₀ ∧ 1 ≤ X₃ ∧ 1+X₂ ≤ X₃ ∧ 1+X₁ ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ X₂ ≤ 0 ∧ X₂ ≤ X₁ ∧ X₁+X₂ ≤ 0 ∧ 1+X₂ ≤ X₀ ∧ X₁ ≤ X₂ ∧ X₁ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₀ for location lbl6

Found invariant X₃ ≤ X₀ ∧ X₀ ≤ X₃ ∧ X₂ ≤ X₁ ∧ X₁ ≤ X₂ for location start

Found invariant 1+X₃ ≤ X₀ ∧ 2 ≤ X₂+X₃ ∧ 2 ≤ X₁+X₃ ∧ 3 ≤ X₀+X₃ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 3 ≤ X₀+X₂ ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 2 ≤ X₀ for location lbl111

Found invariant X₃ ≤ X₂ ∧ X₃ ≤ X₁ ∧ X₃ ≤ X₀ ∧ 1 ≤ X₃ ∧ 2 ≤ X₂+X₃ ∧ 2 ≤ X₁+X₃ ∧ X₁ ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location cut

Found invariant 1+X₃ ≤ X₂ ∧ X₃ ≤ X₀ ∧ 1 ≤ X₃ ∧ 3 ≤ X₂+X₃ ∧ 2 ≤ X₁+X₃ ∧ 2 ≤ X₀+X₃ ∧ 2 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 3 ≤ X₀+X₂ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location lbl101

Problem after Preprocessing

Start: start0
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars:
Locations: cut, lbl101, lbl111, lbl6, start, start0, stop
Transitions:
t₁₂: cut(X₀, X₁, X₂, X₃) → stop(X₀, X₁, X₂, X₃) :|: 1 ≤ X₁ ∧ X₁ ≤ X₀ ∧ X₃ ≤ X₁ ∧ X₁ ≤ X₂ ∧ X₁ ≤ X₃ ∧ 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 1 ≤ X₃ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₀+X₃ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₁+X₃ ∧ 2 ≤ X₂+X₃ ∧ X₃ ≤ X₀ ∧ X₃ ≤ X₂
t₆: lbl101(X₀, X₁, X₂, X₃) → cut(X₀, X₁, X₂, X₃) :|: 1 ≤ X₁ ∧ X₁ ≤ X₀ ∧ X₃ ≤ X₁ ∧ X₁ ≤ X₃ ∧ 2⋅X₁ ≤ X₂ ∧ 1 ≤ X₀ ∧ 1+X₁ ≤ X₂ ∧ 1+X₃ ≤ X₂ ∧ 1 ≤ X₃ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₃ ∧ 2 ≤ X₁+X₃ ∧ 2 ≤ X₂ ∧ 3 ≤ X₀+X₂ ∧ 3 ≤ X₁+X₂ ∧ 3 ≤ X₂+X₃ ∧ X₃ ≤ X₀
t₇: lbl101(X₀, X₁, X₂, X₃) → lbl101(X₀, X₁-X₃, X₂, X₃) :|: 1 ≤ X₁ ∧ 1+X₃ ≤ X₁ ∧ 1 ≤ X₃ ∧ X₃ ≤ X₀ ∧ X₁+X₃ ≤ X₂ ∧ 1 ≤ X₀ ∧ 1+X₁ ≤ X₂ ∧ 1+X₃ ≤ X₂ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₃ ∧ 2 ≤ X₁+X₃ ∧ 2 ≤ X₂ ∧ 3 ≤ X₀+X₂ ∧ 3 ≤ X₁+X₂ ∧ 3 ≤ X₂+X₃
t₈: lbl101(X₀, X₁, X₂, X₃) → lbl111(X₀, X₁, X₂, X₃-X₁) :|: 1 ≤ X₁ ∧ 1+X₁ ≤ X₃ ∧ 1 ≤ X₃ ∧ X₃ ≤ X₀ ∧ X₁+X₃ ≤ X₂ ∧ 1 ≤ X₀ ∧ 1+X₁ ≤ X₂ ∧ 1+X₃ ≤ X₂ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₃ ∧ 2 ≤ X₁+X₃ ∧ 2 ≤ X₂ ∧ 3 ≤ X₀+X₂ ∧ 3 ≤ X₁+X₂ ∧ 3 ≤ X₂+X₃
t₉: lbl111(X₀, X₁, X₂, X₃) → cut(X₀, X₁, X₂, X₃) :|: 1 ≤ X₁ ∧ 2⋅X₁ ≤ X₀ ∧ X₃ ≤ X₁ ∧ X₁ ≤ X₂ ∧ X₁ ≤ X₃ ∧ 1+X₁ ≤ X₀ ∧ 1+X₃ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₁+X₃ ∧ 2 ≤ X₂+X₃ ∧ 3 ≤ X₀+X₁ ∧ 3 ≤ X₀+X₂ ∧ 3 ≤ X₀+X₃
t₁₀: lbl111(X₀, X₁, X₂, X₃) → lbl101(X₀, X₁-X₃, X₂, X₃) :|: 1 ≤ X₁ ∧ 1+X₃ ≤ X₁ ∧ 1 ≤ X₃ ∧ X₁+X₃ ≤ X₀ ∧ X₁ ≤ X₂ ∧ 1+X₁ ≤ X₀ ∧ 1+X₃ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₁+X₃ ∧ 2 ≤ X₂+X₃ ∧ 3 ≤ X₀+X₁ ∧ 3 ≤ X₀+X₂ ∧ 3 ≤ X₀+X₃
t₁₁: lbl111(X₀, X₁, X₂, X₃) → lbl111(X₀, X₁, X₂, X₃-X₁) :|: 1 ≤ X₁ ∧ 1+X₁ ≤ X₃ ∧ 1 ≤ X₃ ∧ X₁+X₃ ≤ X₀ ∧ X₁ ≤ X₂ ∧ 1+X₁ ≤ X₀ ∧ 1+X₃ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₁+X₃ ∧ 2 ≤ X₂+X₃ ∧ 3 ≤ X₀+X₁ ∧ 3 ≤ X₀+X₂ ∧ 3 ≤ X₀+X₃
t₅: lbl6(X₀, X₁, X₂, X₃) → stop(X₀, X₁, X₂, X₃) :|: 1 ≤ X₀ ∧ X₃ ≤ X₀ ∧ X₀ ≤ X₃ ∧ X₂ ≤ X₁ ∧ X₁ ≤ X₂ ∧ X₂ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 1+X₂ ≤ X₀ ∧ 1+X₁ ≤ X₃ ∧ 1+X₂ ≤ X₃ ∧ 1 ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ X₁ ≤ 0 ∧ X₁+X₂ ≤ 0
t₂: start(X₀, X₁, X₂, X₃) → cut(X₀, X₁, X₂, X₃) :|: 1 ≤ X₀ ∧ X₁ ≤ X₀ ∧ X₂ ≤ X₀ ∧ X₃ ≤ X₀ ∧ X₀ ≤ X₁ ∧ X₀ ≤ X₂ ∧ X₀ ≤ X₃ ∧ X₂ ≤ X₁ ∧ X₁ ≤ X₂
t₃: start(X₀, X₁, X₂, X₃) → lbl101(X₀, X₁-X₃, X₂, X₃) :|: 1 ≤ X₀ ∧ 1+X₀ ≤ X₂ ∧ X₃ ≤ X₀ ∧ X₀ ≤ X₃ ∧ X₂ ≤ X₁ ∧ X₁ ≤ X₂
t₄: start(X₀, X₁, X₂, X₃) → lbl111(X₀, X₁, X₂, X₃-X₁) :|: 1+X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ X₃ ≤ X₀ ∧ X₀ ≤ X₃ ∧ X₂ ≤ X₁ ∧ X₁ ≤ X₂
t₁: start(X₀, X₁, X₂, X₃) → lbl6(X₀, X₁, X₂, X₃) :|: 1 ≤ X₀ ∧ X₃ ≤ X₀ ∧ X₀ ≤ X₃ ∧ X₂ ≤ X₁ ∧ X₁ ≤ X₂ ∧ X₂ ≤ 0
t₀: start(X₀, X₁, X₂, X₃) → stop(X₀, X₁, X₂, X₃) :|: X₃ ≤ X₀ ∧ X₀ ≤ 0 ∧ X₀ ≤ X₃ ∧ X₂ ≤ X₁ ∧ X₁ ≤ X₂
t₁₃: start0(X₀, X₁, X₂, X₃) → start(X₀, X₂, X₂, X₀)

MPRF for transition t₇: lbl101(X₀, X₁, X₂, X₃) → lbl101(X₀, X₁-X₃, X₂, X₃) :|: 1 ≤ X₁ ∧ 1+X₃ ≤ X₁ ∧ 1 ≤ X₃ ∧ X₃ ≤ X₀ ∧ X₁+X₃ ≤ X₂ ∧ 1 ≤ X₀ ∧ 1+X₁ ≤ X₂ ∧ 1+X₃ ≤ X₂ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₃ ∧ 2 ≤ X₁+X₃ ∧ 2 ≤ X₂ ∧ 3 ≤ X₀+X₂ ∧ 3 ≤ X₁+X₂ ∧ 3 ≤ X₂+X₃ of depth 1:

new bound:

2⋅X₂ {O(n)}

MPRF:

• lbl101: [X₁]
• lbl111: [X₁]

MPRF for transition t₈: lbl101(X₀, X₁, X₂, X₃) → lbl111(X₀, X₁, X₂, X₃-X₁) :|: 1 ≤ X₁ ∧ 1+X₁ ≤ X₃ ∧ 1 ≤ X₃ ∧ X₃ ≤ X₀ ∧ X₁+X₃ ≤ X₂ ∧ 1 ≤ X₀ ∧ 1+X₁ ≤ X₂ ∧ 1+X₃ ≤ X₂ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₃ ∧ 2 ≤ X₁+X₃ ∧ 2 ≤ X₂ ∧ 3 ≤ X₀+X₂ ∧ 3 ≤ X₁+X₂ ∧ 3 ≤ X₂+X₃ of depth 1:

new bound:

2⋅X₂+1 {O(n)}

MPRF:

• lbl101: [X₁]
• lbl111: [X₁-1]

MPRF for transition t₁₀: lbl111(X₀, X₁, X₂, X₃) → lbl101(X₀, X₁-X₃, X₂, X₃) :|: 1 ≤ X₁ ∧ 1+X₃ ≤ X₁ ∧ 1 ≤ X₃ ∧ X₁+X₃ ≤ X₀ ∧ X₁ ≤ X₂ ∧ 1+X₁ ≤ X₀ ∧ 1+X₃ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₁+X₃ ∧ 2 ≤ X₂+X₃ ∧ 3 ≤ X₀+X₁ ∧ 3 ≤ X₀+X₂ ∧ 3 ≤ X₀+X₃ of depth 1:

new bound:

2⋅X₀+1 {O(n)}

MPRF:

• lbl101: [X₃-1]
• lbl111: [X₃]

MPRF for transition t₁₁: lbl111(X₀, X₁, X₂, X₃) → lbl111(X₀, X₁, X₂, X₃-X₁) :|: 1 ≤ X₁ ∧ 1+X₁ ≤ X₃ ∧ 1 ≤ X₃ ∧ X₁+X₃ ≤ X₀ ∧ X₁ ≤ X₂ ∧ 1+X₁ ≤ X₀ ∧ 1+X₃ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₁+X₃ ∧ 2 ≤ X₂+X₃ ∧ 3 ≤ X₀+X₁ ∧ 3 ≤ X₀+X₂ ∧ 3 ≤ X₀+X₃ of depth 1:

new bound:

2⋅X₀ {O(n)}

MPRF:

• lbl101: [X₃]
• lbl111: [X₃]

All Bounds

Timebounds

Overall timebound:4⋅X₀+4⋅X₂+12 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 1 {O(1)}
t₃: 1 {O(1)}
t₄: 1 {O(1)}
t₅: 1 {O(1)}
t₆: 1 {O(1)}
t₇: 2⋅X₂ {O(n)}
t₈: 2⋅X₂+1 {O(n)}
t₉: 1 {O(1)}
t₁₀: 2⋅X₀+1 {O(n)}
t₁₁: 2⋅X₀ {O(n)}
t₁₂: 1 {O(1)}
t₁₃: 1 {O(1)}

Costbounds

Overall costbound: 4⋅X₀+4⋅X₂+12 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 1 {O(1)}
t₃: 1 {O(1)}
t₄: 1 {O(1)}
t₅: 1 {O(1)}
t₆: 1 {O(1)}
t₇: 2⋅X₂ {O(n)}
t₈: 2⋅X₂+1 {O(n)}
t₉: 1 {O(1)}
t₁₀: 2⋅X₀+1 {O(n)}
t₁₁: 2⋅X₀ {O(n)}
t₁₂: 1 {O(1)}
t₁₃: 1 {O(1)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₂ {O(n)}
t₀, X₂: X₂ {O(n)}
t₀, X₃: X₀ {O(n)}
t₁, X₀: X₀ {O(n)}
t₁, X₁: X₂ {O(n)}
t₁, X₂: X₂ {O(n)}
t₁, X₃: X₀ {O(n)}
t₂, X₀: X₀ {O(n)}
t₂, X₁: X₂ {O(n)}
t₂, X₂: X₂ {O(n)}
t₂, X₃: X₀ {O(n)}
t₃, X₀: X₀ {O(n)}
t₃, X₁: X₂ {O(n)}
t₃, X₂: X₂ {O(n)}
t₃, X₃: X₀ {O(n)}
t₄, X₀: X₀ {O(n)}
t₄, X₁: X₂ {O(n)}
t₄, X₂: X₂ {O(n)}
t₄, X₃: X₀ {O(n)}
t₅, X₀: X₀ {O(n)}
t₅, X₁: X₂ {O(n)}
t₅, X₂: X₂ {O(n)}
t₅, X₃: X₀ {O(n)}
t₆, X₀: 5⋅X₀ {O(n)}
t₆, X₁: 5⋅X₂ {O(n)}
t₆, X₂: 5⋅X₂ {O(n)}
t₆, X₃: 5⋅X₀ {O(n)}
t₇, X₀: 2⋅X₀ {O(n)}
t₇, X₁: 2⋅X₂ {O(n)}
t₇, X₂: 2⋅X₂ {O(n)}
t₇, X₃: 2⋅X₀ {O(n)}
t₈, X₀: 2⋅X₀ {O(n)}
t₈, X₁: 2⋅X₂ {O(n)}
t₈, X₂: 2⋅X₂ {O(n)}
t₈, X₃: 2⋅X₀ {O(n)}
t₉, X₀: 5⋅X₀ {O(n)}
t₉, X₁: 5⋅X₂ {O(n)}
t₉, X₂: 5⋅X₂ {O(n)}
t₉, X₃: 5⋅X₀ {O(n)}
t₁₀, X₀: 2⋅X₀ {O(n)}
t₁₀, X₁: 2⋅X₂ {O(n)}
t₁₀, X₂: 2⋅X₂ {O(n)}
t₁₀, X₃: 2⋅X₀ {O(n)}
t₁₁, X₀: 2⋅X₀ {O(n)}
t₁₁, X₁: 2⋅X₂ {O(n)}
t₁₁, X₂: 2⋅X₂ {O(n)}
t₁₁, X₃: 2⋅X₀ {O(n)}
t₁₂, X₀: 11⋅X₀ {O(n)}
t₁₂, X₁: 11⋅X₂ {O(n)}
t₁₂, X₂: 11⋅X₂ {O(n)}
t₁₂, X₃: 11⋅X₀ {O(n)}
t₁₃, X₀: X₀ {O(n)}
t₁₃, X₁: X₂ {O(n)}
t₁₃, X₂: X₂ {O(n)}
t₁₃, X₃: X₀ {O(n)}