KoAT2 Proof Output

Initial Problem

Start: start0
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars:
Locations: lbl81, lbl91, start, start0, stop
Transitions:
t₆: lbl81(X₀, X₁, X₂, X₃) → lbl81(X₀, X₁, X₂, 1+X₃) :|: 1+X₀ ≤ X₃ ∧ 2+X₃ ≤ X₂ ∧ 3+X₃ ≤ X₂ ∧ 0 ≤ X₀ ∧ X₂ ≤ X₁ ∧ X₁ ≤ X₂
t₇: lbl81(X₀, X₁, X₂, X₃) → lbl91(X₀, 1+X₁, X₂, X₃) :|: 1+X₀ ≤ X₃ ∧ 2+X₃ ≤ X₂ ∧ 3+X₂ ≤ X₃ ∧ 0 ≤ X₀ ∧ X₂ ≤ X₁ ∧ X₁ ≤ X₂
t₅: lbl81(X₀, X₁, X₂, X₃) → stop(X₀, X₁, X₂, X₃) :|: X₂ ≤ 2+X₃ ∧ 2+X₃ ≤ X₂ ∧ 3+X₀ ≤ X₂ ∧ 0 ≤ X₀ ∧ X₂ ≤ X₁ ∧ X₁ ≤ X₂
t₉: lbl91(X₀, X₁, X₂, X₃) → lbl81(X₀, X₁, X₂, 1+X₃) :|: 1+X₂ ≤ X₁ ∧ 2+X₁ ≤ X₀ ∧ 3+X₀ ≤ X₁ ∧ X₃ ≤ X₀ ∧ X₀ ≤ X₃ ∧ 0 ≤ X₂
t₁₀: lbl91(X₀, X₁, X₂, X₃) → lbl91(X₀, 1+X₁, X₂, X₃) :|: 1+X₂ ≤ X₁ ∧ 2+X₁ ≤ X₀ ∧ 3+X₁ ≤ X₀ ∧ X₃ ≤ X₀ ∧ X₀ ≤ X₃ ∧ 0 ≤ X₂
t₈: lbl91(X₀, X₁, X₂, X₃) → stop(X₀, X₁, X₂, X₃) :|: X₀ ≤ 2+X₁ ∧ 2+X₁ ≤ X₀ ∧ 3+X₂ ≤ X₀ ∧ X₃ ≤ X₀ ∧ X₀ ≤ X₃ ∧ 0 ≤ X₂
t₃: start(X₀, X₁, X₂, X₃) → lbl81(X₀, X₁, X₂, 1+X₃) :|: 3+X₀ ≤ X₂ ∧ 0 ≤ X₀ ∧ X₃ ≤ X₀ ∧ X₀ ≤ X₃ ∧ X₂ ≤ X₁ ∧ X₁ ≤ X₂
t₄: start(X₀, X₁, X₂, X₃) → lbl91(X₀, 1+X₁, X₂, X₃) :|: 3+X₂ ≤ X₀ ∧ X₃ ≤ X₀ ∧ X₀ ≤ X₃ ∧ X₂ ≤ X₁ ∧ X₁ ≤ X₂ ∧ 0 ≤ X₂
t₀: start(X₀, X₁, X₂, X₃) → stop(X₀, X₁, X₂, X₃) :|: 1+X₀ ≤ 0 ∧ X₃ ≤ X₀ ∧ X₀ ≤ X₃ ∧ X₂ ≤ X₁ ∧ X₁ ≤ X₂
t₁: start(X₀, X₁, X₂, X₃) → stop(X₀, X₁, X₂, X₃) :|: 1+X₂ ≤ 0 ∧ X₃ ≤ X₀ ∧ X₀ ≤ X₃ ∧ X₂ ≤ X₁ ∧ X₁ ≤ X₂
t₂: start(X₀, X₁, X₂, X₃) → stop(X₀, X₁, X₂, X₃) :|: X₂ ≤ 2+X₀ ∧ X₀ ≤ 2+X₂ ∧ 0 ≤ X₀ ∧ X₃ ≤ X₀ ∧ X₀ ≤ X₃ ∧ X₂ ≤ X₁ ∧ X₁ ≤ X₂ ∧ 0 ≤ X₂
t₁₁: start0(X₀, X₁, X₂, X₃) → start(X₀, X₂, X₂, X₀)

Preprocessing

Cut unsatisfiable transition [t₇: lbl81→lbl91; t₉: lbl91→lbl81]

Found invariant 2+X₃ ≤ X₂ ∧ 2+X₃ ≤ X₁ ∧ 1 ≤ X₃ ∧ 4 ≤ X₂+X₃ ∧ 4 ≤ X₁+X₃ ∧ 1 ≤ X₀+X₃ ∧ 1+X₀ ≤ X₃ ∧ X₂ ≤ X₁ ∧ 3 ≤ X₂ ∧ 6 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 3 ≤ X₀+X₂ ∧ 3+X₀ ≤ X₂ ∧ 3 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 3+X₀ ≤ X₁ ∧ 0 ≤ X₀ for location lbl81

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

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

Found invariant X₃ ≤ X₀ ∧ 3 ≤ X₃ ∧ 3 ≤ X₂+X₃ ∧ 3+X₂ ≤ X₃ ∧ 4 ≤ X₁+X₃ ∧ 2+X₁ ≤ X₃ ∧ 6 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ 1+X₂ ≤ X₁ ∧ 3+X₂ ≤ X₀ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 3 ≤ X₀+X₂ ∧ 2+X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 4 ≤ X₀+X₁ ∧ 3 ≤ X₀ for location lbl91

Problem after Preprocessing

Start: start0
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars:
Locations: lbl81, lbl91, start, start0, stop
Transitions:
t₆: lbl81(X₀, X₁, X₂, X₃) → lbl81(X₀, X₁, X₂, 1+X₃) :|: 1+X₀ ≤ X₃ ∧ 2+X₃ ≤ X₂ ∧ 3+X₃ ≤ X₂ ∧ 0 ≤ X₀ ∧ X₂ ≤ X₁ ∧ X₁ ≤ X₂ ∧ 1 ≤ X₀+X₃ ∧ 1 ≤ X₃ ∧ 2+X₃ ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 3 ≤ X₀+X₂ ∧ 3+X₀ ≤ X₁ ∧ 3+X₀ ≤ X₂ ∧ 3 ≤ X₁ ∧ 3 ≤ X₂ ∧ 4 ≤ X₁+X₃ ∧ 4 ≤ X₂+X₃ ∧ 6 ≤ X₁+X₂
t₅: lbl81(X₀, X₁, X₂, X₃) → stop(X₀, X₁, X₂, X₃) :|: X₂ ≤ 2+X₃ ∧ 2+X₃ ≤ X₂ ∧ 3+X₀ ≤ X₂ ∧ 0 ≤ X₀ ∧ X₂ ≤ X₁ ∧ X₁ ≤ X₂ ∧ 1 ≤ X₀+X₃ ∧ 1+X₀ ≤ X₃ ∧ 1 ≤ X₃ ∧ 2+X₃ ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 3 ≤ X₀+X₂ ∧ 3+X₀ ≤ X₁ ∧ 3 ≤ X₁ ∧ 3 ≤ X₂ ∧ 4 ≤ X₁+X₃ ∧ 4 ≤ X₂+X₃ ∧ 6 ≤ X₁+X₂
t₁₀: lbl91(X₀, X₁, X₂, X₃) → lbl91(X₀, 1+X₁, X₂, X₃) :|: 1+X₂ ≤ X₁ ∧ 2+X₁ ≤ X₀ ∧ 3+X₁ ≤ X₀ ∧ X₃ ≤ X₀ ∧ X₀ ≤ X₃ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁+X₂ ∧ 2+X₁ ≤ X₃ ∧ 3 ≤ X₀ ∧ 3 ≤ X₀+X₂ ∧ 3+X₂ ≤ X₀ ∧ 3 ≤ X₂+X₃ ∧ 3+X₂ ≤ X₃ ∧ 3 ≤ X₃ ∧ 4 ≤ X₀+X₁ ∧ 4 ≤ X₁+X₃ ∧ 6 ≤ X₀+X₃
t₈: lbl91(X₀, X₁, X₂, X₃) → stop(X₀, X₁, X₂, X₃) :|: X₀ ≤ 2+X₁ ∧ 2+X₁ ≤ X₀ ∧ 3+X₂ ≤ X₀ ∧ X₃ ≤ X₀ ∧ X₀ ≤ X₃ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁+X₂ ∧ 1+X₂ ≤ X₁ ∧ 2+X₁ ≤ X₃ ∧ 3 ≤ X₀ ∧ 3 ≤ X₀+X₂ ∧ 3 ≤ X₂+X₃ ∧ 3+X₂ ≤ X₃ ∧ 3 ≤ X₃ ∧ 4 ≤ X₀+X₁ ∧ 4 ≤ X₁+X₃ ∧ 6 ≤ X₀+X₃
t₃: start(X₀, X₁, X₂, X₃) → lbl81(X₀, X₁, X₂, 1+X₃) :|: 3+X₀ ≤ X₂ ∧ 0 ≤ X₀ ∧ X₃ ≤ X₀ ∧ X₀ ≤ X₃ ∧ X₂ ≤ X₁ ∧ X₁ ≤ X₂
t₄: start(X₀, X₁, X₂, X₃) → lbl91(X₀, 1+X₁, X₂, X₃) :|: 3+X₂ ≤ X₀ ∧ X₃ ≤ X₀ ∧ X₀ ≤ X₃ ∧ X₂ ≤ X₁ ∧ X₁ ≤ X₂ ∧ 0 ≤ X₂
t₀: start(X₀, X₁, X₂, X₃) → stop(X₀, X₁, X₂, X₃) :|: 1+X₀ ≤ 0 ∧ X₃ ≤ X₀ ∧ X₀ ≤ X₃ ∧ X₂ ≤ X₁ ∧ X₁ ≤ X₂
t₁: start(X₀, X₁, X₂, X₃) → stop(X₀, X₁, X₂, X₃) :|: 1+X₂ ≤ 0 ∧ X₃ ≤ X₀ ∧ X₀ ≤ X₃ ∧ X₂ ≤ X₁ ∧ X₁ ≤ X₂
t₂: start(X₀, X₁, X₂, X₃) → stop(X₀, X₁, X₂, X₃) :|: X₂ ≤ 2+X₀ ∧ X₀ ≤ 2+X₂ ∧ 0 ≤ X₀ ∧ X₃ ≤ X₀ ∧ X₀ ≤ X₃ ∧ X₂ ≤ X₁ ∧ X₁ ≤ X₂ ∧ 0 ≤ X₂
t₁₁: start0(X₀, X₁, X₂, X₃) → start(X₀, X₂, X₂, X₀)

MPRF for transition t₁₀: lbl91(X₀, X₁, X₂, X₃) → lbl91(X₀, 1+X₁, X₂, X₃) :|: 1+X₂ ≤ X₁ ∧ 2+X₁ ≤ X₀ ∧ 3+X₁ ≤ X₀ ∧ X₃ ≤ X₀ ∧ X₀ ≤ X₃ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁+X₂ ∧ 2+X₁ ≤ X₃ ∧ 3 ≤ X₀ ∧ 3 ≤ X₀+X₂ ∧ 3+X₂ ≤ X₀ ∧ 3 ≤ X₂+X₃ ∧ 3+X₂ ≤ X₃ ∧ 3 ≤ X₃ ∧ 4 ≤ X₀+X₁ ∧ 4 ≤ X₁+X₃ ∧ 6 ≤ X₀+X₃ of depth 1:

new bound:

X₀+X₂+1 {O(n)}

MPRF:

• lbl91: [X₀-X₁]

MPRF for transition t₆: lbl81(X₀, X₁, X₂, X₃) → lbl81(X₀, X₁, X₂, 1+X₃) :|: 1+X₀ ≤ X₃ ∧ 2+X₃ ≤ X₂ ∧ 3+X₃ ≤ X₂ ∧ 0 ≤ X₀ ∧ X₂ ≤ X₁ ∧ X₁ ≤ X₂ ∧ 1 ≤ X₀+X₃ ∧ 1 ≤ X₃ ∧ 2+X₃ ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 3 ≤ X₀+X₂ ∧ 3+X₀ ≤ X₁ ∧ 3+X₀ ≤ X₂ ∧ 3 ≤ X₁ ∧ 3 ≤ X₂ ∧ 4 ≤ X₁+X₃ ∧ 4 ≤ X₂+X₃ ∧ 6 ≤ X₁+X₂ of depth 1:

new bound:

X₀+X₂+1 {O(n)}

MPRF:

• lbl81: [X₂-X₃]

All Bounds

Timebounds

Overall timebound:2⋅X₀+2⋅X₂+10 {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: X₀+X₂+1 {O(n)}
t₈: 1 {O(1)}
t₁₀: X₀+X₂+1 {O(n)}
t₁₁: 1 {O(1)}

Costbounds

Overall costbound: 2⋅X₀+2⋅X₂+10 {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: X₀+X₂+1 {O(n)}
t₈: 1 {O(1)}
t₁₀: X₀+X₂+1 {O(n)}
t₁₁: 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+1 {O(n)}
t₄, X₀: X₀ {O(n)}
t₄, X₁: X₂+1 {O(n)}
t₄, X₂: X₂ {O(n)}
t₄, X₃: X₀ {O(n)}
t₅, X₀: 2⋅X₀ {O(n)}
t₅, X₁: 2⋅X₂ {O(n)}
t₅, X₂: 2⋅X₂ {O(n)}
t₅, X₃: 3⋅X₀+X₂+3 {O(n)}
t₆, X₀: X₀ {O(n)}
t₆, X₁: X₂ {O(n)}
t₆, X₂: X₂ {O(n)}
t₆, X₃: 2⋅X₀+X₂+2 {O(n)}
t₈, X₀: 2⋅X₀ {O(n)}
t₈, X₁: 3⋅X₂+X₀+3 {O(n)}
t₈, X₂: 2⋅X₂ {O(n)}
t₈, X₃: 2⋅X₀ {O(n)}
t₁₀, X₀: X₀ {O(n)}
t₁₀, X₁: 2⋅X₂+X₀+2 {O(n)}
t₁₀, X₂: X₂ {O(n)}
t₁₀, X₃: X₀ {O(n)}
t₁₁, X₀: X₀ {O(n)}
t₁₁, X₁: X₂ {O(n)}
t₁₁, X₂: X₂ {O(n)}
t₁₁, X₃: X₀ {O(n)}