KoAT2 Proof Output

Initial Problem

Start: start0
Program_Vars: X₀, X₁, X₂, X₃, X₄, X₅
Temp_Vars:
Locations: lbl101, lbl91, start, start0, stop
Transitions:
t₁₁: lbl101(X₀, X₁, X₂, X₃, X₄, X₅) → lbl101(X₀, 1+X₁+X₅, X₂, X₃, X₄, X₅) :|: X₁ ≤ 1+X₀+X₃ ∧ 1+X₀+X₂ ≤ X₁ ∧ 0 ≤ X₀ ∧ X₅ ≤ X₀ ∧ X₀ ≤ X₅ ∧ X₁ ≤ X₃ ∧ X₃ ≤ X₄
t₁₀: lbl101(X₀, X₁, X₂, X₃, X₄, X₅) → lbl91(X₀, X₁, X₂, X₃-1-X₅, X₄, X₅) :|: X₁ ≤ 1+X₀+X₃ ∧ 1+X₀+X₂ ≤ X₁ ∧ 0 ≤ X₀ ∧ X₅ ≤ X₀ ∧ X₀ ≤ X₅ ∧ X₁ ≤ X₃ ∧ X₃ ≤ X₄
t₈: lbl101(X₀, X₁, X₂, X₃, X₄, X₅) → stop(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₁ ≤ 1+X₀+X₃ ∧ 1+X₀+X₂ ≤ X₁ ∧ 1+X₃ ≤ X₁ ∧ 0 ≤ X₀ ∧ X₅ ≤ X₀ ∧ X₀ ≤ X₅ ∧ X₃ ≤ X₄
t₉: lbl101(X₀, X₁, X₂, X₃, X₄, X₅) → stop(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₁ ≤ 1+X₀+X₃ ∧ 1+X₀ ≤ 0 ∧ 1+X₀+X₂ ≤ X₁ ∧ 0 ≤ X₀ ∧ X₅ ≤ X₀ ∧ X₀ ≤ X₅ ∧ X₁ ≤ X₃ ∧ X₃ ≤ X₄
t₇: lbl91(X₀, X₁, X₂, X₃, X₄, X₅) → lbl101(X₀, 1+X₁+X₅, X₂, X₃, X₄, X₅) :|: X₁ ≤ 1+X₀+X₃ ∧ 1+X₀+X₃ ≤ X₄ ∧ 0 ≤ X₀ ∧ X₅ ≤ X₀ ∧ X₀ ≤ X₅ ∧ X₂ ≤ X₁ ∧ X₁ ≤ X₃
t₆: lbl91(X₀, X₁, X₂, X₃, X₄, X₅) → lbl91(X₀, X₁, X₂, X₃-1-X₅, X₄, X₅) :|: X₁ ≤ 1+X₀+X₃ ∧ 1+X₀+X₃ ≤ X₄ ∧ 0 ≤ X₀ ∧ X₅ ≤ X₀ ∧ X₀ ≤ X₅ ∧ X₂ ≤ X₁ ∧ X₁ ≤ X₃
t₄: lbl91(X₀, X₁, X₂, X₃, X₄, X₅) → stop(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₁ ≤ 1+X₀+X₃ ∧ 1+X₀+X₃ ≤ X₄ ∧ 1+X₃ ≤ X₁ ∧ 0 ≤ X₀ ∧ X₅ ≤ X₀ ∧ X₀ ≤ X₅ ∧ X₂ ≤ X₁
t₅: lbl91(X₀, X₁, X₂, X₃, X₄, X₅) → stop(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₁ ≤ 1+X₀+X₃ ∧ 1+X₀ ≤ 0 ∧ 1+X₀+X₃ ≤ X₄ ∧ 0 ≤ X₀ ∧ X₅ ≤ X₀ ∧ X₀ ≤ X₅ ∧ X₂ ≤ X₁ ∧ X₁ ≤ X₃
t₃: start(X₀, X₁, X₂, X₃, X₄, X₅) → lbl101(X₀, 1+X₁+X₅, X₂, X₃, X₄, X₅) :|: 0 ≤ X₀ ∧ X₅ ≤ X₀ ∧ X₀ ≤ X₅ ∧ X₂ ≤ X₁ ∧ X₁ ≤ X₂ ∧ X₂ ≤ X₄ ∧ X₄ ≤ X₃ ∧ X₃ ≤ X₄
t₂: start(X₀, X₁, X₂, X₃, X₄, X₅) → lbl91(X₀, X₁, X₂, X₃-1-X₅, X₄, X₅) :|: 0 ≤ X₀ ∧ X₅ ≤ X₀ ∧ X₀ ≤ X₅ ∧ X₂ ≤ X₁ ∧ X₁ ≤ X₂ ∧ X₂ ≤ X₄ ∧ X₄ ≤ X₃ ∧ X₃ ≤ X₄
t₀: start(X₀, X₁, X₂, X₃, X₄, X₅) → stop(X₀, X₁, X₂, X₃, X₄, X₅) :|: 1+X₀ ≤ 0 ∧ X₅ ≤ X₀ ∧ X₀ ≤ X₅ ∧ X₂ ≤ X₁ ∧ X₁ ≤ X₂ ∧ X₄ ≤ X₃ ∧ X₃ ≤ X₄
t₁: start(X₀, X₁, X₂, X₃, X₄, X₅) → stop(X₀, X₁, X₂, X₃, X₄, X₅) :|: 1+X₄ ≤ X₂ ∧ 0 ≤ X₀ ∧ X₅ ≤ X₀ ∧ X₀ ≤ X₅ ∧ X₂ ≤ X₁ ∧ X₁ ≤ X₂ ∧ X₄ ≤ X₃ ∧ X₃ ≤ X₄
t₁₂: start0(X₀, X₁, X₂, X₃, X₄, X₅) → start(X₀, X₂, X₂, X₄, X₄, X₀)

Preprocessing

Cut unsatisfiable transition [t₅: lbl91→stop; t₉: lbl101→stop]

Found invariant X₅ ≤ X₀ ∧ X₀ ≤ X₅ ∧ X₃ ≤ X₄ ∧ X₂ ≤ X₁ for location stop

Found invariant X₅ ≤ X₀ ∧ X₀ ≤ X₅ ∧ X₄ ≤ X₃ ∧ X₃ ≤ X₄ ∧ X₂ ≤ X₁ ∧ X₁ ≤ X₂ for location start

Found invariant X₅ ≤ X₀ ∧ 0 ≤ X₅ ∧ 0 ≤ X₀+X₅ ∧ X₀ ≤ X₅ ∧ X₃ ≤ X₄ ∧ X₂ ≤ X₄ ∧ X₂ ≤ X₃ ∧ 1+X₂ ≤ X₁ ∧ 0 ≤ X₀ for location lbl101

Found invariant X₅ ≤ X₀ ∧ 0 ≤ X₅ ∧ 0 ≤ X₀+X₅ ∧ X₀ ≤ X₅ ∧ 1+X₃ ≤ X₄ ∧ X₂ ≤ X₄ ∧ X₁ ≤ X₄ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₀ for location lbl91

Problem after Preprocessing

Start: start0
Program_Vars: X₀, X₁, X₂, X₃, X₄, X₅
Temp_Vars:
Locations: lbl101, lbl91, start, start0, stop
Transitions:
t₁₁: lbl101(X₀, X₁, X₂, X₃, X₄, X₅) → lbl101(X₀, 1+X₁+X₅, X₂, X₃, X₄, X₅) :|: X₁ ≤ 1+X₀+X₃ ∧ 1+X₀+X₂ ≤ X₁ ∧ 0 ≤ X₀ ∧ X₅ ≤ X₀ ∧ X₀ ≤ X₅ ∧ X₁ ≤ X₃ ∧ X₃ ≤ X₄ ∧ 1+X₂ ≤ X₁ ∧ 0 ≤ X₀+X₅ ∧ X₂ ≤ X₃ ∧ X₂ ≤ X₄ ∧ 0 ≤ X₅
t₁₀: lbl101(X₀, X₁, X₂, X₃, X₄, X₅) → lbl91(X₀, X₁, X₂, X₃-1-X₅, X₄, X₅) :|: X₁ ≤ 1+X₀+X₃ ∧ 1+X₀+X₂ ≤ X₁ ∧ 0 ≤ X₀ ∧ X₅ ≤ X₀ ∧ X₀ ≤ X₅ ∧ X₁ ≤ X₃ ∧ X₃ ≤ X₄ ∧ 1+X₂ ≤ X₁ ∧ 0 ≤ X₀+X₅ ∧ X₂ ≤ X₃ ∧ X₂ ≤ X₄ ∧ 0 ≤ X₅
t₈: lbl101(X₀, X₁, X₂, X₃, X₄, X₅) → stop(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₁ ≤ 1+X₀+X₃ ∧ 1+X₀+X₂ ≤ X₁ ∧ 1+X₃ ≤ X₁ ∧ 0 ≤ X₀ ∧ X₅ ≤ X₀ ∧ X₀ ≤ X₅ ∧ X₃ ≤ X₄ ∧ 1+X₂ ≤ X₁ ∧ 0 ≤ X₀+X₅ ∧ X₂ ≤ X₃ ∧ X₂ ≤ X₄ ∧ 0 ≤ X₅
t₇: lbl91(X₀, X₁, X₂, X₃, X₄, X₅) → lbl101(X₀, 1+X₁+X₅, X₂, X₃, X₄, X₅) :|: X₁ ≤ 1+X₀+X₃ ∧ 1+X₀+X₃ ≤ X₄ ∧ 0 ≤ X₀ ∧ X₅ ≤ X₀ ∧ X₀ ≤ X₅ ∧ X₂ ≤ X₁ ∧ X₁ ≤ X₃ ∧ 1+X₃ ≤ X₄ ∧ 0 ≤ X₀+X₅ ∧ X₁ ≤ X₄ ∧ X₂ ≤ X₄ ∧ 0 ≤ X₅
t₆: lbl91(X₀, X₁, X₂, X₃, X₄, X₅) → lbl91(X₀, X₁, X₂, X₃-1-X₅, X₄, X₅) :|: X₁ ≤ 1+X₀+X₃ ∧ 1+X₀+X₃ ≤ X₄ ∧ 0 ≤ X₀ ∧ X₅ ≤ X₀ ∧ X₀ ≤ X₅ ∧ X₂ ≤ X₁ ∧ X₁ ≤ X₃ ∧ 1+X₃ ≤ X₄ ∧ 0 ≤ X₀+X₅ ∧ X₁ ≤ X₄ ∧ X₂ ≤ X₄ ∧ 0 ≤ X₅
t₄: lbl91(X₀, X₁, X₂, X₃, X₄, X₅) → stop(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₁ ≤ 1+X₀+X₃ ∧ 1+X₀+X₃ ≤ X₄ ∧ 1+X₃ ≤ X₁ ∧ 0 ≤ X₀ ∧ X₅ ≤ X₀ ∧ X₀ ≤ X₅ ∧ X₂ ≤ X₁ ∧ 1+X₃ ≤ X₄ ∧ 0 ≤ X₀+X₅ ∧ X₁ ≤ X₄ ∧ X₂ ≤ X₄ ∧ 0 ≤ X₅
t₃: start(X₀, X₁, X₂, X₃, X₄, X₅) → lbl101(X₀, 1+X₁+X₅, X₂, X₃, X₄, X₅) :|: 0 ≤ X₀ ∧ X₅ ≤ X₀ ∧ X₀ ≤ X₅ ∧ X₂ ≤ X₁ ∧ X₁ ≤ X₂ ∧ X₂ ≤ X₄ ∧ X₄ ≤ X₃ ∧ X₃ ≤ X₄
t₂: start(X₀, X₁, X₂, X₃, X₄, X₅) → lbl91(X₀, X₁, X₂, X₃-1-X₅, X₄, X₅) :|: 0 ≤ X₀ ∧ X₅ ≤ X₀ ∧ X₀ ≤ X₅ ∧ X₂ ≤ X₁ ∧ X₁ ≤ X₂ ∧ X₂ ≤ X₄ ∧ X₄ ≤ X₃ ∧ X₃ ≤ X₄
t₀: start(X₀, X₁, X₂, X₃, X₄, X₅) → stop(X₀, X₁, X₂, X₃, X₄, X₅) :|: 1+X₀ ≤ 0 ∧ X₅ ≤ X₀ ∧ X₀ ≤ X₅ ∧ X₂ ≤ X₁ ∧ X₁ ≤ X₂ ∧ X₄ ≤ X₃ ∧ X₃ ≤ X₄
t₁: start(X₀, X₁, X₂, X₃, X₄, X₅) → stop(X₀, X₁, X₂, X₃, X₄, X₅) :|: 1+X₄ ≤ X₂ ∧ 0 ≤ X₀ ∧ X₅ ≤ X₀ ∧ X₀ ≤ X₅ ∧ X₂ ≤ X₁ ∧ X₁ ≤ X₂ ∧ X₄ ≤ X₃ ∧ X₃ ≤ X₄
t₁₂: start0(X₀, X₁, X₂, X₃, X₄, X₅) → start(X₀, X₂, X₂, X₄, X₄, X₀)

MPRF for transition t₆: lbl91(X₀, X₁, X₂, X₃, X₄, X₅) → lbl91(X₀, X₁, X₂, X₃-1-X₅, X₄, X₅) :|: X₁ ≤ 1+X₀+X₃ ∧ 1+X₀+X₃ ≤ X₄ ∧ 0 ≤ X₀ ∧ X₅ ≤ X₀ ∧ X₀ ≤ X₅ ∧ X₂ ≤ X₁ ∧ X₁ ≤ X₃ ∧ 1+X₃ ≤ X₄ ∧ 0 ≤ X₀+X₅ ∧ X₁ ≤ X₄ ∧ X₂ ≤ X₄ ∧ 0 ≤ X₅ of depth 1:

new bound:

2⋅X₂+2⋅X₄+X₀+2 {O(n)}

MPRF:

• lbl101: [X₃-X₂]
• lbl91: [1+X₃-X₂]

MPRF for transition t₇: lbl91(X₀, X₁, X₂, X₃, X₄, X₅) → lbl101(X₀, 1+X₁+X₅, X₂, X₃, X₄, X₅) :|: X₁ ≤ 1+X₀+X₃ ∧ 1+X₀+X₃ ≤ X₄ ∧ 0 ≤ X₀ ∧ X₅ ≤ X₀ ∧ X₀ ≤ X₅ ∧ X₂ ≤ X₁ ∧ X₁ ≤ X₃ ∧ 1+X₃ ≤ X₄ ∧ 0 ≤ X₀+X₅ ∧ X₁ ≤ X₄ ∧ X₂ ≤ X₄ ∧ 0 ≤ X₅ of depth 1:

new bound:

2⋅X₂+2⋅X₄+X₀+1 {O(n)}

MPRF:

• lbl101: [X₄-X₁]
• lbl91: [X₄-X₁]

MPRF for transition t₁₀: lbl101(X₀, X₁, X₂, X₃, X₄, X₅) → lbl91(X₀, X₁, X₂, X₃-1-X₅, X₄, X₅) :|: X₁ ≤ 1+X₀+X₃ ∧ 1+X₀+X₂ ≤ X₁ ∧ 0 ≤ X₀ ∧ X₅ ≤ X₀ ∧ X₀ ≤ X₅ ∧ X₁ ≤ X₃ ∧ X₃ ≤ X₄ ∧ 1+X₂ ≤ X₁ ∧ 0 ≤ X₀+X₅ ∧ X₂ ≤ X₃ ∧ X₂ ≤ X₄ ∧ 0 ≤ X₅ of depth 1:

new bound:

2⋅X₂+2⋅X₄+X₀+2 {O(n)}

MPRF:

• lbl101: [1+X₄-X₁]
• lbl91: [X₄-X₁]

MPRF for transition t₁₁: lbl101(X₀, X₁, X₂, X₃, X₄, X₅) → lbl101(X₀, 1+X₁+X₅, X₂, X₃, X₄, X₅) :|: X₁ ≤ 1+X₀+X₃ ∧ 1+X₀+X₂ ≤ X₁ ∧ 0 ≤ X₀ ∧ X₅ ≤ X₀ ∧ X₀ ≤ X₅ ∧ X₁ ≤ X₃ ∧ X₃ ≤ X₄ ∧ 1+X₂ ≤ X₁ ∧ 0 ≤ X₀+X₅ ∧ X₂ ≤ X₃ ∧ X₂ ≤ X₄ ∧ 0 ≤ X₅ of depth 1:

new bound:

2⋅X₂+2⋅X₄+X₀+2 {O(n)}

MPRF:

• lbl101: [1+X₄-X₁]
• lbl91: [X₄-X₁]

All Bounds

Timebounds

Overall timebound:4⋅X₀+8⋅X₂+8⋅X₄+14 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 1 {O(1)}
t₃: 1 {O(1)}
t₄: 1 {O(1)}
t₆: 2⋅X₂+2⋅X₄+X₀+2 {O(n)}
t₇: 2⋅X₂+2⋅X₄+X₀+1 {O(n)}
t₈: 1 {O(1)}
t₁₀: 2⋅X₂+2⋅X₄+X₀+2 {O(n)}
t₁₁: 2⋅X₂+2⋅X₄+X₀+2 {O(n)}
t₁₂: 1 {O(1)}

Costbounds

Overall costbound: 4⋅X₀+8⋅X₂+8⋅X₄+14 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 1 {O(1)}
t₃: 1 {O(1)}
t₄: 1 {O(1)}
t₆: 2⋅X₂+2⋅X₄+X₀+2 {O(n)}
t₇: 2⋅X₂+2⋅X₄+X₀+1 {O(n)}
t₈: 1 {O(1)}
t₁₀: 2⋅X₂+2⋅X₄+X₀+2 {O(n)}
t₁₁: 2⋅X₂+2⋅X₄+X₀+2 {O(n)}
t₁₂: 1 {O(1)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₂ {O(n)}
t₀, X₂: X₂ {O(n)}
t₀, X₃: X₄ {O(n)}
t₀, X₄: X₄ {O(n)}
t₀, X₅: X₀ {O(n)}
t₁, X₀: X₀ {O(n)}
t₁, X₁: X₂ {O(n)}
t₁, X₂: X₂ {O(n)}
t₁, X₃: X₄ {O(n)}
t₁, X₄: X₄ {O(n)}
t₁, X₅: X₀ {O(n)}
t₂, X₀: X₀ {O(n)}
t₂, X₁: X₂ {O(n)}
t₂, X₂: X₂ {O(n)}
t₂, X₃: X₀+X₄+1 {O(n)}
t₂, X₄: X₄ {O(n)}
t₂, X₅: X₀ {O(n)}
t₃, X₀: X₀ {O(n)}
t₃, X₁: X₀+X₂+1 {O(n)}
t₃, X₂: X₂ {O(n)}
t₃, X₃: X₄ {O(n)}
t₃, X₄: X₄ {O(n)}
t₃, X₅: X₀ {O(n)}
t₄, X₀: 5⋅X₀ {O(n)}
t₄, X₁: 20⋅X₀⋅X₀+40⋅X₀⋅X₂+40⋅X₀⋅X₄+13⋅X₂+56⋅X₀+8⋅X₄+8 {O(n^2)}
t₄, X₂: 5⋅X₂ {O(n)}
t₄, X₃: 20⋅X₀⋅X₀+40⋅X₀⋅X₂+40⋅X₀⋅X₄+13⋅X₄+67⋅X₀+8⋅X₂+11 {O(n^2)}
t₄, X₄: 5⋅X₄ {O(n)}
t₄, X₅: 5⋅X₀ {O(n)}
t₆, X₀: 2⋅X₀ {O(n)}
t₆, X₁: 10⋅X₀⋅X₀+20⋅X₀⋅X₂+20⋅X₀⋅X₄+28⋅X₀+4⋅X₄+6⋅X₂+4 {O(n^2)}
t₆, X₂: 2⋅X₂ {O(n)}
t₆, X₃: 10⋅X₀⋅X₀+20⋅X₀⋅X₂+20⋅X₀⋅X₄+33⋅X₀+4⋅X₂+6⋅X₄+5 {O(n^2)}
t₆, X₄: 2⋅X₄ {O(n)}
t₆, X₅: 2⋅X₀ {O(n)}
t₇, X₀: 2⋅X₀ {O(n)}
t₇, X₁: 10⋅X₀⋅X₀+20⋅X₀⋅X₂+20⋅X₀⋅X₄+28⋅X₀+4⋅X₄+6⋅X₂+4 {O(n^2)}
t₇, X₂: 2⋅X₂ {O(n)}
t₇, X₃: 10⋅X₀⋅X₀+20⋅X₀⋅X₂+20⋅X₀⋅X₄+33⋅X₀+4⋅X₂+6⋅X₄+5 {O(n^2)}
t₇, X₄: 2⋅X₄ {O(n)}
t₇, X₅: 2⋅X₀ {O(n)}
t₈, X₀: 5⋅X₀ {O(n)}
t₈, X₁: 20⋅X₀⋅X₀+40⋅X₀⋅X₂+40⋅X₀⋅X₄+13⋅X₂+57⋅X₀+8⋅X₄+9 {O(n^2)}
t₈, X₂: 5⋅X₂ {O(n)}
t₈, X₃: 20⋅X₀⋅X₀+40⋅X₀⋅X₂+40⋅X₀⋅X₄+13⋅X₄+66⋅X₀+8⋅X₂+10 {O(n^2)}
t₈, X₄: 5⋅X₄ {O(n)}
t₈, X₅: 5⋅X₀ {O(n)}
t₁₀, X₀: 2⋅X₀ {O(n)}
t₁₀, X₁: 10⋅X₀⋅X₀+20⋅X₀⋅X₂+20⋅X₀⋅X₄+28⋅X₀+4⋅X₄+6⋅X₂+4 {O(n^2)}
t₁₀, X₂: 2⋅X₂ {O(n)}
t₁₀, X₃: 10⋅X₀⋅X₀+20⋅X₀⋅X₂+20⋅X₀⋅X₄+33⋅X₀+4⋅X₂+6⋅X₄+5 {O(n^2)}
t₁₀, X₄: 2⋅X₄ {O(n)}
t₁₀, X₅: 2⋅X₀ {O(n)}
t₁₁, X₀: 2⋅X₀ {O(n)}
t₁₁, X₁: 10⋅X₀⋅X₀+20⋅X₀⋅X₂+20⋅X₀⋅X₄+28⋅X₀+4⋅X₄+6⋅X₂+4 {O(n^2)}
t₁₁, X₂: 2⋅X₂ {O(n)}
t₁₁, X₃: 10⋅X₀⋅X₀+20⋅X₀⋅X₂+20⋅X₀⋅X₄+33⋅X₀+4⋅X₂+6⋅X₄+5 {O(n^2)}
t₁₁, X₄: 2⋅X₄ {O(n)}
t₁₁, X₅: 2⋅X₀ {O(n)}
t₁₂, X₀: X₀ {O(n)}
t₁₂, X₁: X₂ {O(n)}
t₁₂, X₂: X₂ {O(n)}
t₁₂, X₃: X₄ {O(n)}
t₁₂, X₄: X₄ {O(n)}
t₁₂, X₅: X₀ {O(n)}