Initial Problem

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

Preprocessing

Found invariant 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location l2

Found invariant 0 ≤ X₁ ∧ X₀ ≤ X₁ for location l6

Found invariant 0 ≤ X₁ ∧ X₀ ≤ X₁ for location l7

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

Found invariant 0 ≤ X₁ for location l4

Found invariant 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ 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:
t₀: l0(X₀, X₁) → l5(X₀, X₁)
t₇: l1(X₀, X₁) → l4(X₀, X₁+1) :|: 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀
t₄: l2(X₀, X₁) → l3(X₀, X₁) :|: 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀
t₆: l3(X₀, X₁) → l1(X₀, X₁) :|: 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀
t₂: l4(X₀, X₁) → l2(X₀, X₁) :|: X₁ < X₀ ∧ 0 ≤ X₁
t₃: l4(X₀, X₁) → l6(X₀, X₁) :|: X₀ ≤ X₁ ∧ 0 ≤ X₁
t₁: l5(X₀, X₁) → l4(X₀, 0)
t₈: l6(X₀, X₁) → l7(X₀, X₁) :|: 0 ≤ X₁ ∧ X₀ ≤ X₁

MPRF for transition t₇: l1(X₀, X₁) → l4(X₀, X₁+1) :|: 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ of depth 1:

new bound:

X₀ {O(n)}

MPRF:

l3 [X₀-X₁ ]
l1 [X₀-X₁ ]
l4 [X₀-X₁ ]
l2 [X₀-X₁ ]

MPRF for transition t₄: l2(X₀, X₁) → l3(X₀, X₁) :|: 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ of depth 1:

new bound:

X₀ {O(n)}

MPRF:

l3 [X₀-X₁-1 ]
l1 [X₀-X₁-1 ]
l4 [X₀-X₁ ]
l2 [X₀-X₁ ]

MPRF for transition t₆: l3(X₀, X₁) → l1(X₀, X₁) :|: 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ of depth 1:

new bound:

X₀ {O(n)}

MPRF:

l3 [X₀-X₁ ]
l1 [X₀-X₁-1 ]
l4 [X₀-X₁ ]
l2 [X₀-X₁ ]

MPRF for transition t₂: l4(X₀, X₁) → l2(X₀, X₁) :|: X₁ < X₀ ∧ 0 ≤ X₁ of depth 1:

new bound:

X₀ {O(n)}

MPRF:

l3 [X₀-X₁-1 ]
l1 [X₀-X₁-1 ]
l4 [X₀-X₁ ]
l2 [X₀-X₁-1 ]

All Bounds

Timebounds

Overall timebound:4⋅X₀+4 {O(n)}
t₀: 1 {O(1)}
t₇: X₀ {O(n)}
t₄: X₀ {O(n)}
t₆: X₀ {O(n)}
t₂: X₀ {O(n)}
t₃: 1 {O(1)}
t₁: 1 {O(1)}
t₈: 1 {O(1)}

Costbounds

Overall costbound: 4⋅X₀+4 {O(n)}
t₀: 1 {O(1)}
t₇: X₀ {O(n)}
t₄: X₀ {O(n)}
t₆: X₀ {O(n)}
t₂: X₀ {O(n)}
t₃: 1 {O(1)}
t₁: 1 {O(1)}
t₈: 1 {O(1)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₇, X₀: X₀ {O(n)}
t₇, X₁: X₀ {O(n)}
t₄, X₀: X₀ {O(n)}
t₄, X₁: X₀ {O(n)}
t₆, X₀: X₀ {O(n)}
t₆, X₁: X₀ {O(n)}
t₂, X₀: X₀ {O(n)}
t₂, X₁: X₀ {O(n)}
t₃, X₀: 2⋅X₀ {O(n)}
t₃, X₁: X₀ {O(n)}
t₁, X₀: X₀ {O(n)}
t₁, X₁: 0 {O(1)}
t₈, X₀: 2⋅X₀ {O(n)}
t₈, X₁: X₀ {O(n)}