Initial Problem

Start: l0
Program_Vars: X₀, X₁, X₂, X₃, X₄, X₅
Temp_Vars:
Locations: l0, l1, l2, l3, l4, l5
Transitions:
t₀: l0(X₀, X₁, X₂, X₃, X₄, X₅) → l2(X₀, X₁, X₂, X₃, X₄, X₅)
t₂: l1(X₀, X₁, X₂, X₃, X₄, X₅) → l3(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₀ < 0
t₃: l1(X₀, X₁, X₂, X₃, X₄, X₅) → l3(X₀, X₁, X₂, X₃, X₄, X₅) :|: 0 < X₀
t₄: l1(X₀, X₁, X₂, X₃, X₄, X₅) → l4(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₀ ≤ 0 ∧ 0 ≤ X₀
t₁: l2(X₀, X₁, X₂, X₃, X₄, X₅) → l1(1, X₄, X₂, X₃, X₄, X₅)
t₅: l3(X₀, X₁, X₂, X₃, X₄, X₅) → l1(0, X₁+1, X₂, X₃, X₄, X₅) :|: X₅ ≤ X₁
t₆: l3(X₀, X₁, X₂, X₃, X₄, X₅) → l1(X₀, X₁+1, X₂, X₃, X₄, X₅) :|: X₁ < X₅
t₇: l4(X₀, X₁, X₂, X₃, X₄, X₅) → l5(X₀, X₁, X₂, X₃, X₄, X₅)

Preprocessing

Eliminate variables {X₂,X₃} that do not contribute to the problem

Found invariant X₄ ≤ X₁ ∧ X₀ ≤ 0 ∧ 0 ≤ X₀ for location l5

Found invariant X₄ ≤ X₁ ∧ X₀ ≤ 1 ∧ 0 ≤ X₀ for location l1

Found invariant X₄ ≤ X₁ ∧ X₀ ≤ 0 ∧ 0 ≤ X₀ for location l4

Found invariant X₄ ≤ X₁ ∧ X₀ ≤ 1 ∧ 1 ≤ X₀ for location l3

Cut unsatisfiable transition t₁₇: l1→l3

Problem after Preprocessing

Start: l0
Program_Vars: X₀, X₁, X₄, X₅
Temp_Vars:
Locations: l0, l1, l2, l3, l4, l5
Transitions:
t₁₆: l0(X₀, X₁, X₄, X₅) → l2(X₀, X₁, X₄, X₅)
t₁₈: l1(X₀, X₁, X₄, X₅) → l3(X₀, X₁, X₄, X₅) :|: 0 < X₀ ∧ X₄ ≤ X₁ ∧ X₀ ≤ 1 ∧ 0 ≤ X₀
t₁₉: l1(X₀, X₁, X₄, X₅) → l4(X₀, X₁, X₄, X₅) :|: X₀ ≤ 0 ∧ 0 ≤ X₀ ∧ X₄ ≤ X₁ ∧ X₀ ≤ 1 ∧ 0 ≤ X₀
t₂₀: l2(X₀, X₁, X₄, X₅) → l1(1, X₄, X₄, X₅)
t₂₁: l3(X₀, X₁, X₄, X₅) → l1(0, X₁+1, X₄, X₅) :|: X₅ ≤ X₁ ∧ X₄ ≤ X₁ ∧ X₀ ≤ 1 ∧ 1 ≤ X₀
t₂₂: l3(X₀, X₁, X₄, X₅) → l1(X₀, X₁+1, X₄, X₅) :|: X₁ < X₅ ∧ X₄ ≤ X₁ ∧ X₀ ≤ 1 ∧ 1 ≤ X₀
t₂₃: l4(X₀, X₁, X₄, X₅) → l5(X₀, X₁, X₄, X₅) :|: X₄ ≤ X₁ ∧ X₀ ≤ 0 ∧ 0 ≤ X₀

MPRF for transition t₂₁: l3(X₀, X₁, X₄, X₅) → l1(0, X₁+1, X₄, X₅) :|: X₅ ≤ X₁ ∧ X₄ ≤ X₁ ∧ X₀ ≤ 1 ∧ 1 ≤ X₀ of depth 1:

new bound:

1 {O(1)}

MPRF:

l3 [1 ]
l1 [X₀ ]

MPRF for transition t₂₂: l3(X₀, X₁, X₄, X₅) → l1(X₀, X₁+1, X₄, X₅) :|: X₁ < X₅ ∧ X₄ ≤ X₁ ∧ X₀ ≤ 1 ∧ 1 ≤ X₀ of depth 1:

new bound:

X₄+X₅ {O(n)}

MPRF:

l3 [X₅-X₁ ]
l1 [X₅-X₁ ]

Found invariant 1 ≤ 0 for location l5

Found invariant 1 ≤ 0 for location l1

Found invariant 1 ≤ 0 for location l4

Found invariant 1 ≤ 0 for location l3

Found invariant X₄ ≤ X₁ ∧ X₀ ≤ 0 ∧ 0 ≤ X₀ for location l5

Found invariant X₄ ≤ X₁ ∧ X₀ ≤ 1 ∧ 0 ≤ X₀ for location l1

Found invariant X₄ ≤ X₁ ∧ X₀ ≤ 0 ∧ 0 ≤ X₀ for location l4

Found invariant X₄ ≤ X₁ ∧ X₀ ≤ 1 ∧ 1 ≤ X₀ for location l3

knowledge_propagation leads to new time bound X₄+X₅+1 {O(n)} for transition t₁₈: l1(X₀, X₁, X₄, X₅) → l3(X₀, X₁, X₄, X₅) :|: 0 < X₀ ∧ X₄ ≤ X₁ ∧ X₀ ≤ 1 ∧ 0 ≤ X₀

All Bounds

Timebounds

Overall timebound:2⋅X₄+2⋅X₅+6 {O(n)}
t₁₆: 1 {O(1)}
t₁₈: X₄+X₅+1 {O(n)}
t₁₉: 1 {O(1)}
t₂₀: 1 {O(1)}
t₂₁: 1 {O(1)}
t₂₂: X₄+X₅ {O(n)}
t₂₃: 1 {O(1)}

Costbounds

Overall costbound: 2⋅X₄+2⋅X₅+6 {O(n)}
t₁₆: 1 {O(1)}
t₁₈: X₄+X₅+1 {O(n)}
t₁₉: 1 {O(1)}
t₂₀: 1 {O(1)}
t₂₁: 1 {O(1)}
t₂₂: X₄+X₅ {O(n)}
t₂₃: 1 {O(1)}

Sizebounds

t₁₆, X₀: X₀ {O(n)}
t₁₆, X₁: X₁ {O(n)}
t₁₆, X₄: X₄ {O(n)}
t₁₆, X₅: X₅ {O(n)}
t₁₈, X₀: 1 {O(1)}
t₁₈, X₁: 2⋅X₄+X₅ {O(n)}
t₁₈, X₄: X₄ {O(n)}
t₁₈, X₅: X₅ {O(n)}
t₁₉, X₀: 0 {O(1)}
t₁₉, X₁: 2⋅X₄+X₅+1 {O(n)}
t₁₉, X₄: X₄ {O(n)}
t₁₉, X₅: X₅ {O(n)}
t₂₀, X₀: 1 {O(1)}
t₂₀, X₁: X₄ {O(n)}
t₂₀, X₄: X₄ {O(n)}
t₂₀, X₅: X₅ {O(n)}
t₂₁, X₀: 0 {O(1)}
t₂₁, X₁: 2⋅X₄+X₅+1 {O(n)}
t₂₁, X₄: X₄ {O(n)}
t₂₁, X₅: X₅ {O(n)}
t₂₂, X₀: 1 {O(1)}
t₂₂, X₁: 2⋅X₄+X₅ {O(n)}
t₂₂, X₄: X₄ {O(n)}
t₂₂, X₅: X₅ {O(n)}
t₂₃, X₀: 0 {O(1)}
t₂₃, X₁: 2⋅X₄+X₅+1 {O(n)}
t₂₃, X₄: X₄ {O(n)}
t₂₃, X₅: X₅ {O(n)}