KoAT2 Proof Output

Initial Problem

Start: start0
Program_Vars: X₀, X₁, X₂, X₃, X₄, X₅
Temp_Vars:
Locations: cut, lbl71, start, start0, stop
Transitions:
t₆: cut(X₀, X₁, X₂, X₃, X₄, X₅) → cut(X₀, X₁, 0, X₃, 1+X₄, X₅) :|: X₀ ≤ 1 ∧ X₀ ≤ 1+X₂ ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₄ ∧ 2+X₄ ≤ X₀ ∧ 3+X₄ ≤ X₀ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁
t₅: cut(X₀, X₁, X₂, X₃, X₄, X₅) → lbl71(X₀, X₁, 1, X₃, X₄, X₅) :|: X₀ ≤ 1+X₂ ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₄ ∧ 2 ≤ X₀ ∧ 2+X₄ ≤ X₀ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁
t₇: cut(X₀, X₁, X₂, X₃, X₄, X₅) → stop(X₀, X₁, 0, X₃, 1+X₄, X₅) :|: X₀ ≤ 2+X₄ ∧ X₀ ≤ 1 ∧ X₀ ≤ 1+X₂ ∧ 1+X₂ ≤ X₀ ∧ 2+X₄ ≤ X₀ ∧ 3 ≤ X₀ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁
t₃: lbl71(X₀, X₁, X₂, X₃, X₄, X₅) → cut(X₀, X₁, X₂, X₃, 1+X₄, X₅) :|: X₀ ≤ 1+X₂ ∧ 1+X₂ ≤ X₀ ∧ 2 ≤ X₀ ∧ 2+X₄ ≤ X₀ ∧ 3+X₄ ≤ X₀ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁ ∧ 0 ≤ X₄
t₂: lbl71(X₀, X₁, X₂, X₃, X₄, X₅) → lbl71(X₀, X₁, 1+X₂, X₃, X₄, X₅) :|: 1+X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2+X₂ ≤ X₀ ∧ 2+X₄ ≤ X₀ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁ ∧ 0 ≤ X₄
t₄: lbl71(X₀, X₁, X₂, X₃, X₄, X₅) → stop(X₀, X₁, X₂, X₃, 1+X₄, X₅) :|: X₀ ≤ 2+X₄ ∧ X₀ ≤ 1+X₂ ∧ 1+X₂ ≤ X₀ ∧ 2 ≤ X₀ ∧ 2+X₄ ≤ X₀ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁
t₀: start(X₀, X₁, X₂, X₃, X₄, X₅) → lbl71(X₀, X₁, 1, X₃, 0, X₅) :|: 2 ≤ X₀ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁ ∧ X₃ ≤ X₂ ∧ X₂ ≤ X₃ ∧ X₅ ≤ X₄ ∧ X₄ ≤ X₅
t₁: start(X₀, X₁, X₂, X₃, X₄, X₅) → stop(X₀, X₁, 0, X₃, 1, X₅) :|: X₀ ≤ 1 ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁ ∧ X₃ ≤ X₂ ∧ X₂ ≤ X₃ ∧ X₅ ≤ X₄ ∧ X₄ ≤ X₅
t₈: start0(X₀, X₁, X₂, X₃, X₄, X₅) → start(X₀, X₀, X₃, X₃, X₅, X₅)

Preprocessing

Cut unsatisfiable transition [t₆: cut→cut; t₇: cut→stop]

Found invariant X₄ ≤ 1+X₂ ∧ 1 ≤ X₄ ∧ 1 ≤ X₂+X₄ ∧ X₂ ≤ X₄ ∧ X₁ ≤ 1+X₄ ∧ X₀ ≤ 1+X₄ ∧ 0 ≤ X₂ ∧ X₁ ≤ 1+X₂ ∧ X₀ ≤ 1+X₂ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁ for location stop

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

Found invariant 1+X₄ ≤ X₂ ∧ 2+X₄ ≤ X₁ ∧ 2+X₄ ≤ X₀ ∧ 1 ≤ X₄ ∧ 3 ≤ X₂+X₄ ∧ 4 ≤ X₁+X₄ ∧ 4 ≤ X₀+X₄ ∧ 1+X₂ ≤ X₁ ∧ 1+X₂ ≤ X₀ ∧ 2 ≤ X₂ ∧ 5 ≤ X₁+X₂ ∧ X₁ ≤ 1+X₂ ∧ 5 ≤ X₀+X₂ ∧ X₀ ≤ 1+X₂ ∧ X₁ ≤ X₀ ∧ 3 ≤ X₁ ∧ 6 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 3 ≤ X₀ for location cut

Found invariant 2+X₄ ≤ X₁ ∧ 2+X₄ ≤ X₀ ∧ 0 ≤ X₄ ∧ 1 ≤ X₂+X₄ ∧ 2 ≤ X₁+X₄ ∧ 2 ≤ X₀+X₄ ∧ 1+X₂ ≤ X₁ ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 3 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 2 ≤ X₁ ∧ 4 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 2 ≤ X₀ for location lbl71

Problem after Preprocessing

Start: start0
Program_Vars: X₀, X₁, X₂, X₃, X₄, X₅
Temp_Vars:
Locations: cut, lbl71, start, start0, stop
Transitions:
t₅: cut(X₀, X₁, X₂, X₃, X₄, X₅) → lbl71(X₀, X₁, 1, X₃, X₄, X₅) :|: X₀ ≤ 1+X₂ ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₄ ∧ 2 ≤ X₀ ∧ 2+X₄ ≤ X₀ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁ ∧ X₁ ≤ 1+X₂ ∧ 1+X₂ ≤ X₁ ∧ 1+X₄ ≤ X₂ ∧ 2+X₄ ≤ X₁ ∧ 2 ≤ X₂ ∧ 3 ≤ X₀ ∧ 3 ≤ X₁ ∧ 3 ≤ X₂+X₄ ∧ 4 ≤ X₀+X₄ ∧ 4 ≤ X₁+X₄ ∧ 5 ≤ X₀+X₂ ∧ 5 ≤ X₁+X₂ ∧ 6 ≤ X₀+X₁
t₃: lbl71(X₀, X₁, X₂, X₃, X₄, X₅) → cut(X₀, X₁, X₂, X₃, 1+X₄, X₅) :|: X₀ ≤ 1+X₂ ∧ 1+X₂ ≤ X₀ ∧ 2 ≤ X₀ ∧ 2+X₄ ≤ X₀ ∧ 3+X₄ ≤ X₀ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁ ∧ 0 ≤ X₄ ∧ 1+X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 1 ≤ X₂+X₄ ∧ 2 ≤ X₀+X₄ ∧ 2 ≤ X₁ ∧ 2 ≤ X₁+X₄ ∧ 2+X₄ ≤ X₁ ∧ 3 ≤ X₀+X₂ ∧ 3 ≤ X₁+X₂ ∧ 4 ≤ X₀+X₁
t₂: lbl71(X₀, X₁, X₂, X₃, X₄, X₅) → lbl71(X₀, X₁, 1+X₂, X₃, X₄, X₅) :|: 1+X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2+X₂ ≤ X₀ ∧ 2+X₄ ≤ X₀ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁ ∧ 0 ≤ X₄ ∧ 1+X₂ ≤ X₁ ∧ 1 ≤ X₂+X₄ ∧ 2 ≤ X₀ ∧ 2 ≤ X₀+X₄ ∧ 2 ≤ X₁ ∧ 2 ≤ X₁+X₄ ∧ 2+X₄ ≤ X₁ ∧ 3 ≤ X₀+X₂ ∧ 3 ≤ X₁+X₂ ∧ 4 ≤ X₀+X₁
t₄: lbl71(X₀, X₁, X₂, X₃, X₄, X₅) → stop(X₀, X₁, X₂, X₃, 1+X₄, X₅) :|: X₀ ≤ 2+X₄ ∧ X₀ ≤ 1+X₂ ∧ 1+X₂ ≤ X₀ ∧ 2 ≤ X₀ ∧ 2+X₄ ≤ X₀ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁ ∧ 1+X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 1 ≤ X₂+X₄ ∧ 2 ≤ X₀+X₄ ∧ 2 ≤ X₁ ∧ 2 ≤ X₁+X₄ ∧ 2+X₄ ≤ X₁ ∧ 3 ≤ X₀+X₂ ∧ 3 ≤ X₁+X₂ ∧ 4 ≤ X₀+X₁ ∧ 0 ≤ X₄
t₀: start(X₀, X₁, X₂, X₃, X₄, X₅) → lbl71(X₀, X₁, 1, X₃, 0, X₅) :|: 2 ≤ X₀ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁ ∧ X₃ ≤ X₂ ∧ X₂ ≤ X₃ ∧ X₅ ≤ X₄ ∧ X₄ ≤ X₅
t₁: start(X₀, X₁, X₂, X₃, X₄, X₅) → stop(X₀, X₁, 0, X₃, 1, X₅) :|: X₀ ≤ 1 ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁ ∧ X₃ ≤ X₂ ∧ X₂ ≤ X₃ ∧ X₅ ≤ X₄ ∧ X₄ ≤ X₅
t₈: start0(X₀, X₁, X₂, X₃, X₄, X₅) → start(X₀, X₀, X₃, X₃, X₅, X₅)

MPRF for transition t₃: lbl71(X₀, X₁, X₂, X₃, X₄, X₅) → cut(X₀, X₁, X₂, X₃, 1+X₄, X₅) :|: X₀ ≤ 1+X₂ ∧ 1+X₂ ≤ X₀ ∧ 2 ≤ X₀ ∧ 2+X₄ ≤ X₀ ∧ 3+X₄ ≤ X₀ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁ ∧ 0 ≤ X₄ ∧ 1+X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 1 ≤ X₂+X₄ ∧ 2 ≤ X₀+X₄ ∧ 2 ≤ X₁ ∧ 2 ≤ X₁+X₄ ∧ 2+X₄ ≤ X₁ ∧ 3 ≤ X₀+X₂ ∧ 3 ≤ X₁+X₂ ∧ 4 ≤ X₀+X₁ of depth 1:

new bound:

X₀+1 {O(n)}

MPRF:

• cut: [X₀-1-X₄]
• lbl71: [X₀-1-X₄]

MPRF for transition t₅: cut(X₀, X₁, X₂, X₃, X₄, X₅) → lbl71(X₀, X₁, 1, X₃, X₄, X₅) :|: X₀ ≤ 1+X₂ ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₄ ∧ 2 ≤ X₀ ∧ 2+X₄ ≤ X₀ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁ ∧ X₁ ≤ 1+X₂ ∧ 1+X₂ ≤ X₁ ∧ 1+X₄ ≤ X₂ ∧ 2+X₄ ≤ X₁ ∧ 2 ≤ X₂ ∧ 3 ≤ X₀ ∧ 3 ≤ X₁ ∧ 3 ≤ X₂+X₄ ∧ 4 ≤ X₀+X₄ ∧ 4 ≤ X₁+X₄ ∧ 5 ≤ X₀+X₂ ∧ 5 ≤ X₁+X₂ ∧ 6 ≤ X₀+X₁ of depth 1:

new bound:

X₀+2 {O(n)}

MPRF:

• cut: [X₀-1-X₄]
• lbl71: [X₀-2-X₄]

MPRF for transition t₂: lbl71(X₀, X₁, X₂, X₃, X₄, X₅) → lbl71(X₀, X₁, 1+X₂, X₃, X₄, X₅) :|: 1+X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2+X₂ ≤ X₀ ∧ 2+X₄ ≤ X₀ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁ ∧ 0 ≤ X₄ ∧ 1+X₂ ≤ X₁ ∧ 1 ≤ X₂+X₄ ∧ 2 ≤ X₀ ∧ 2 ≤ X₀+X₄ ∧ 2 ≤ X₁ ∧ 2 ≤ X₁+X₄ ∧ 2+X₄ ≤ X₁ ∧ 3 ≤ X₀+X₂ ∧ 3 ≤ X₁+X₂ ∧ 4 ≤ X₀+X₁ of depth 1:

new bound:

X₀⋅X₀+4⋅X₀+3 {O(n^2)}

MPRF:

• cut: [0]
• lbl71: [X₀-X₂]

Cut unsatisfiable transition [t₃: lbl71→cut; t₄₈: lbl71→cut]

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

Found invariant 2+X₄ ≤ X₁ ∧ 2+X₄ ≤ X₀ ∧ 0 ≤ X₄ ∧ 1 ≤ X₂+X₄ ∧ X₂ ≤ 1+X₄ ∧ 2 ≤ X₁+X₄ ∧ 2 ≤ X₀+X₄ ∧ X₂ ≤ 1 ∧ 1+X₂ ≤ X₁ ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 3 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 2 ≤ X₁ ∧ 4 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 2 ≤ X₀ for location lbl71

Found invariant 2+X₄ ≤ X₁ ∧ 2+X₄ ≤ X₀ ∧ 0 ≤ X₄ ∧ 2 ≤ X₂+X₄ ∧ 3 ≤ X₁+X₄ ∧ 3 ≤ X₀+X₄ ∧ 1+X₂ ≤ X₁ ∧ 1+X₂ ≤ X₀ ∧ 2 ≤ X₂ ∧ 5 ≤ X₁+X₂ ∧ 5 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 3 ≤ X₁ ∧ 6 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 3 ≤ X₀ for location lbl71_v1

All Bounds

Timebounds

Overall timebound:X₀⋅X₀+6⋅X₀+10 {O(n^2)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₀⋅X₀+4⋅X₀+3 {O(n^2)}
t₃: X₀+1 {O(n)}
t₄: 1 {O(1)}
t₅: X₀+2 {O(n)}
t₈: 1 {O(1)}

Costbounds

Overall costbound: X₀⋅X₀+6⋅X₀+10 {O(n^2)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₀⋅X₀+4⋅X₀+3 {O(n^2)}
t₃: X₀+1 {O(n)}
t₄: 1 {O(1)}
t₅: X₀+2 {O(n)}
t₈: 1 {O(1)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₀ {O(n)}
t₀, X₂: 1 {O(1)}
t₀, X₃: X₃ {O(n)}
t₀, X₄: 0 {O(1)}
t₀, X₅: X₅ {O(n)}
t₁, X₀: X₀ {O(n)}
t₁, X₁: X₀ {O(n)}
t₁, X₂: 0 {O(1)}
t₁, X₃: X₃ {O(n)}
t₁, X₄: 1 {O(1)}
t₁, X₅: X₅ {O(n)}
t₂, X₀: X₀ {O(n)}
t₂, X₁: X₀ {O(n)}
t₂, X₂: X₀⋅X₀+4⋅X₀+5 {O(n^2)}
t₂, X₃: X₃ {O(n)}
t₂, X₄: X₀+1 {O(n)}
t₂, X₅: X₅ {O(n)}
t₃, X₀: X₀ {O(n)}
t₃, X₁: X₀ {O(n)}
t₃, X₂: X₀⋅X₀+4⋅X₀+5 {O(n^2)}
t₃, X₃: X₃ {O(n)}
t₃, X₄: X₀+1 {O(n)}
t₃, X₅: X₅ {O(n)}
t₄, X₀: 2⋅X₀ {O(n)}
t₄, X₁: 2⋅X₀ {O(n)}
t₄, X₂: X₀⋅X₀+4⋅X₀+6 {O(n^2)}
t₄, X₃: 2⋅X₃ {O(n)}
t₄, X₄: X₀+3 {O(n)}
t₄, X₅: 2⋅X₅ {O(n)}
t₅, X₀: X₀ {O(n)}
t₅, X₁: X₀ {O(n)}
t₅, X₂: 1 {O(1)}
t₅, X₃: X₃ {O(n)}
t₅, X₄: X₀+1 {O(n)}
t₅, X₅: X₅ {O(n)}
t₈, X₀: X₀ {O(n)}
t₈, X₁: X₀ {O(n)}
t₈, X₂: X₃ {O(n)}
t₈, X₃: X₃ {O(n)}
t₈, X₄: X₅ {O(n)}
t₈, X₅: X₅ {O(n)}