Initial Problem

Start: l0
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars:
Locations: l0, l1, l2, l3, l4, l5, l6, l7
Transitions:
t₀: l0(X₀, X₁, X₂, X₃) → l2(X₀, X₁, X₂, X₃)
t₂: l1(X₀, X₁, X₂, X₃) → l3(X₀, X₁, X₂, X₃) :|: X₀ < 100
t₃: l1(X₀, X₁, X₂, X₃) → l4(X₀, 5, X₂, X₃) :|: 100 ≤ X₀
t₁: l2(X₀, X₁, X₂, X₃) → l1(0, X₁, X₂, X₃)
t₄: l3(X₀, X₁, X₂, X₃) → l1(X₀+1, X₁, X₂, X₃)
t₅: l4(X₀, X₁, X₂, X₃) → l5(X₀, X₁, X₂, X₃) :|: X₁ < 21
t₆: l4(X₀, X₁, X₂, X₃) → l6(X₀, X₁, X₂, X₃) :|: 21 ≤ X₁
t₇: l5(X₀, X₁, X₂, X₃) → l4(X₀, X₁+3, X₂, X₃)
t₈: l6(X₀, X₁, X₂, X₃) → l7(X₀, X₁, X₂, X₃)

Preprocessing

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

Found invariant 21 ≤ X₁ ∧ 121 ≤ X₀+X₁ ∧ 100 ≤ X₀ for location l6

Found invariant 21 ≤ X₁ ∧ 121 ≤ X₀+X₁ ∧ 100 ≤ X₀ for location l7

Found invariant X₁ ≤ 20 ∧ 80+X₁ ≤ X₀ ∧ 5 ≤ X₁ ∧ 105 ≤ X₀+X₁ ∧ 100 ≤ X₀ for location l5

Found invariant 0 ≤ X₀ for location l1

Found invariant 5 ≤ X₁ ∧ 105 ≤ X₀+X₁ ∧ 100 ≤ X₀ for location l4

Found invariant X₀ ≤ 99 ∧ 0 ≤ X₀ for location l3

Problem after Preprocessing

Start: l0
Program_Vars: X₀, X₁
Temp_Vars:
Locations: l0, l1, l2, l3, l4, l5, l6, l7
Transitions:
t₁₈: l0(X₀, X₁) → l2(X₀, X₁)
t₁₉: l1(X₀, X₁) → l3(X₀, X₁) :|: X₀ < 100 ∧ 0 ≤ X₀
t₂₀: l1(X₀, X₁) → l4(X₀, 5) :|: 100 ≤ X₀ ∧ 0 ≤ X₀
t₂₁: l2(X₀, X₁) → l1(0, X₁)
t₂₂: l3(X₀, X₁) → l1(X₀+1, X₁) :|: X₀ ≤ 99 ∧ 0 ≤ X₀
t₂₃: l4(X₀, X₁) → l5(X₀, X₁) :|: X₁ < 21 ∧ 5 ≤ X₁ ∧ 105 ≤ X₀+X₁ ∧ 100 ≤ X₀
t₂₄: l4(X₀, X₁) → l6(X₀, X₁) :|: 21 ≤ X₁ ∧ 5 ≤ X₁ ∧ 105 ≤ X₀+X₁ ∧ 100 ≤ X₀
t₂₅: l5(X₀, X₁) → l4(X₀, X₁+3) :|: X₁ ≤ 20 ∧ 80+X₁ ≤ X₀ ∧ 5 ≤ X₁ ∧ 105 ≤ X₀+X₁ ∧ 100 ≤ X₀
t₂₆: l6(X₀, X₁) → l7(X₀, X₁) :|: 21 ≤ X₁ ∧ 121 ≤ X₀+X₁ ∧ 100 ≤ X₀

MPRF for transition t₁₉: l1(X₀, X₁) → l3(X₀, X₁) :|: X₀ < 100 ∧ 0 ≤ X₀ of depth 1:

new bound:

100 {O(1)}

MPRF:

l3 [99-X₀ ]
l1 [100-X₀ ]

MPRF for transition t₂₂: l3(X₀, X₁) → l1(X₀+1, X₁) :|: X₀ ≤ 99 ∧ 0 ≤ X₀ of depth 1:

new bound:

100 {O(1)}

MPRF:

l3 [100-X₀ ]
l1 [100-X₀ ]

MPRF for transition t₂₃: l4(X₀, X₁) → l5(X₀, X₁) :|: X₁ < 21 ∧ 5 ≤ X₁ ∧ 105 ≤ X₀+X₁ ∧ 100 ≤ X₀ of depth 1:

new bound:

26 {O(1)}

MPRF:

l5 [18-X₁ ]
l4 [21-X₁ ]

MPRF for transition t₂₅: l5(X₀, X₁) → l4(X₀, X₁+3) :|: X₁ ≤ 20 ∧ 80+X₁ ≤ X₀ ∧ 5 ≤ X₁ ∧ 105 ≤ X₀+X₁ ∧ 100 ≤ X₀ of depth 1:

new bound:

26 {O(1)}

MPRF:

l5 [21-X₁ ]
l4 [21-X₁ ]

All Bounds

Timebounds

Overall timebound:257 {O(1)}
t₁₈: 1 {O(1)}
t₁₉: 100 {O(1)}
t₂₀: 1 {O(1)}
t₂₁: 1 {O(1)}
t₂₂: 100 {O(1)}
t₂₃: 26 {O(1)}
t₂₄: 1 {O(1)}
t₂₅: 26 {O(1)}
t₂₆: 1 {O(1)}

Costbounds

Overall costbound: 257 {O(1)}
t₁₈: 1 {O(1)}
t₁₉: 100 {O(1)}
t₂₀: 1 {O(1)}
t₂₁: 1 {O(1)}
t₂₂: 100 {O(1)}
t₂₃: 26 {O(1)}
t₂₄: 1 {O(1)}
t₂₅: 26 {O(1)}
t₂₆: 1 {O(1)}

Sizebounds

t₁₈, X₀: X₀ {O(n)}
t₁₈, X₁: X₁ {O(n)}
t₁₉, X₀: 99 {O(1)}
t₁₉, X₁: X₁ {O(n)}
t₂₀, X₀: 100 {O(1)}
t₂₀, X₁: 5 {O(1)}
t₂₁, X₀: 0 {O(1)}
t₂₁, X₁: X₁ {O(n)}
t₂₂, X₀: 100 {O(1)}
t₂₂, X₁: X₁ {O(n)}
t₂₃, X₀: 100 {O(1)}
t₂₃, X₁: 20 {O(1)}
t₂₄, X₀: 100 {O(1)}
t₂₄, X₁: 23 {O(1)}
t₂₅, X₀: 100 {O(1)}
t₂₅, X₁: 23 {O(1)}
t₂₆, X₀: 100 {O(1)}
t₂₆, X₁: 23 {O(1)}