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 < X₄
t₃: l1(X₀, X₁, X₂, X₃, X₄, X₅) → l4(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₄ ≤ X₀
t₁: l2(X₀, X₁, X₂, X₃, X₄, X₅) → l1(X₃, X₅, X₂, X₃, X₄, X₅)
t₄: l3(X₀, X₁, X₂, X₃, X₄, X₅) → l1(X₀+1, X₁, X₂, X₃, X₄, X₅) :|: X₀ < X₁ ∧ X₀ < X₁
t₅: l3(X₀, X₁, X₂, X₃, X₄, X₅) → l1(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₀ < X₁ ∧ X₁ ≤ X₀
t₆: l3(X₀, X₁, X₂, X₃, X₄, X₅) → l1(X₀+1, X₁+1, X₂, X₃, X₄, X₅) :|: X₁ ≤ X₀ ∧ X₀ < X₁
t₇: l3(X₀, X₁, X₂, X₃, X₄, X₅) → l1(X₀, X₁+1, X₂, X₃, X₄, X₅) :|: X₁ ≤ X₀ ∧ X₁ ≤ X₀
t₈: l4(X₀, X₁, X₂, X₃, X₄, X₅) → l5(X₀, X₁, X₂, X₃, X₄, X₅)

Preprocessing

Cut unsatisfiable transition t₅: l3→l1

Cut unsatisfiable transition t₆: l3→l1

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

Found invariant X₄ ≤ X₁ ∧ X₃ ≤ X₀ ∧ X₂ ≤ X₀ for location l5

Found invariant X₄ ≤ X₁ ∧ X₂ ≤ X₀ for location l1

Found invariant X₄ ≤ X₁ ∧ X₃ ≤ X₀ ∧ X₂ ≤ X₀ for location l4

Found invariant X₄ ≤ X₁ ∧ 1+X₂ ≤ X₃ ∧ 1+X₀ ≤ X₃ ∧ X₂ ≤ X₀ for location l3

Problem after Preprocessing

Start: l0
Program_Vars: X₀, X₁, X₂, X₃, X₄
Temp_Vars:
Locations: l0, l1, l2, l3, l4, l5
Transitions:
t₁₇: l0(X₀, X₁, X₂, X₃, X₄) → l2(X₀, X₁, X₂, X₃, X₄)
t₁₈: l1(X₀, X₁, X₂, X₃, X₄) → l3(X₀, X₁, X₂, X₃, X₄) :|: X₀ < X₃ ∧ X₄ ≤ X₁ ∧ X₂ ≤ X₀
t₁₉: l1(X₀, X₁, X₂, X₃, X₄) → l4(X₀, X₁, X₂, X₃, X₄) :|: X₃ ≤ X₀ ∧ X₄ ≤ X₁ ∧ X₂ ≤ X₀
t₂₀: l2(X₀, X₁, X₂, X₃, X₄) → l1(X₂, X₄, X₂, X₃, X₄)
t₂₁: l3(X₀, X₁, X₂, X₃, X₄) → l1(X₀+1, X₁, X₂, X₃, X₄) :|: X₀ < X₁ ∧ X₀ < X₁ ∧ X₄ ≤ X₁ ∧ 1+X₂ ≤ X₃ ∧ 1+X₀ ≤ X₃ ∧ X₂ ≤ X₀
t₂₂: l3(X₀, X₁, X₂, X₃, X₄) → l1(X₀, X₁+1, X₂, X₃, X₄) :|: X₁ ≤ X₀ ∧ X₁ ≤ X₀ ∧ X₄ ≤ X₁ ∧ 1+X₂ ≤ X₃ ∧ 1+X₀ ≤ X₃ ∧ X₂ ≤ X₀
t₂₃: l4(X₀, X₁, X₂, X₃, X₄) → l5(X₀, X₁, X₂, X₃, X₄) :|: X₄ ≤ X₁ ∧ X₃ ≤ X₀ ∧ X₂ ≤ X₀

MPRF for transition t₂₁: l3(X₀, X₁, X₂, X₃, X₄) → l1(X₀+1, X₁, X₂, X₃, X₄) :|: X₀ < X₁ ∧ X₀ < X₁ ∧ X₄ ≤ X₁ ∧ 1+X₂ ≤ X₃ ∧ 1+X₀ ≤ X₃ ∧ X₂ ≤ X₀ of depth 1:

new bound:

X₂+X₃ {O(n)}

MPRF for transition t₂₂: l3(X₀, X₁, X₂, X₃, X₄) → l1(X₀, X₁+1, X₂, X₃, X₄) :|: X₁ ≤ X₀ ∧ X₁ ≤ X₀ ∧ X₄ ≤ X₁ ∧ 1+X₂ ≤ X₃ ∧ 1+X₀ ≤ X₃ ∧ X₂ ≤ X₀ of depth 1:

new bound:

X₃+X₄ {O(n)}

knowledge_propagation leads to new time bound 2⋅X₃+X₂+X₄+1 {O(n)} for transition t₁₈: l1(X₀, X₁, X₂, X₃, X₄) → l3(X₀, X₁, X₂, X₃, X₄) :|: X₀ < X₃ ∧ X₄ ≤ X₁ ∧ X₂ ≤ X₀

All Bounds

Timebounds

Overall timebound:2⋅X₂+2⋅X₄+4⋅X₃+5 {O(n)}
t₁₇: 1 {O(1)}
t₁₈: 2⋅X₃+X₂+X₄+1 {O(n)}
t₁₉: 1 {O(1)}
t₂₀: 1 {O(1)}
t₂₁: X₂+X₃ {O(n)}
t₂₂: X₃+X₄ {O(n)}
t₂₃: 1 {O(1)}

Costbounds

Overall costbound: 2⋅X₂+2⋅X₄+4⋅X₃+5 {O(n)}
t₁₇: 1 {O(1)}
t₁₈: 2⋅X₃+X₂+X₄+1 {O(n)}
t₁₉: 1 {O(1)}
t₂₀: 1 {O(1)}
t₂₁: X₂+X₃ {O(n)}
t₂₂: X₃+X₄ {O(n)}
t₂₃: 1 {O(1)}

Sizebounds

t₁₇, X₀: X₀ {O(n)}
t₁₇, X₁: X₁ {O(n)}
t₁₇, X₂: X₂ {O(n)}
t₁₇, X₃: X₃ {O(n)}
t₁₇, X₄: X₄ {O(n)}
t₁₈, X₀: 2⋅X₂+X₃ {O(n)}
t₁₈, X₁: 2⋅X₄+X₃ {O(n)}
t₁₈, X₂: X₂ {O(n)}
t₁₈, X₃: X₃ {O(n)}
t₁₈, X₄: X₄ {O(n)}
t₁₉, X₀: 3⋅X₂+X₃ {O(n)}
t₁₉, X₁: 3⋅X₄+X₃ {O(n)}
t₁₉, X₂: 2⋅X₂ {O(n)}
t₁₉, X₃: 2⋅X₃ {O(n)}
t₁₉, X₄: 2⋅X₄ {O(n)}
t₂₀, X₀: X₂ {O(n)}
t₂₀, X₁: X₄ {O(n)}
t₂₀, X₂: X₂ {O(n)}
t₂₀, X₃: X₃ {O(n)}
t₂₀, X₄: X₄ {O(n)}
t₂₁, X₀: 2⋅X₂+X₃ {O(n)}
t₂₁, X₁: 2⋅X₄+X₃ {O(n)}
t₂₁, X₂: X₂ {O(n)}
t₂₁, X₃: X₃ {O(n)}
t₂₁, X₄: X₄ {O(n)}
t₂₂, X₀: 2⋅X₂+X₃ {O(n)}
t₂₂, X₁: 2⋅X₄+X₃ {O(n)}
t₂₂, X₂: X₂ {O(n)}
t₂₂, X₃: X₃ {O(n)}
t₂₂, X₄: X₄ {O(n)}
t₂₃, X₀: 3⋅X₂+X₃ {O(n)}
t₂₃, X₁: 3⋅X₄+X₃ {O(n)}
t₂₃, X₂: 2⋅X₂ {O(n)}
t₂₃, X₃: 2⋅X₃ {O(n)}
t₂₃, X₄: 2⋅X₄ {O(n)}