Initial Problem

Start: l0
Program_Vars: X₀, X₁
Temp_Vars:
Locations: l0, l1, l2
Transitions:
t₃: l0(X₀, X₁) → l2(X₀, X₁)
t₁: l1(X₀, X₁) → l1(X₀-1, X₁-1) :|: 1 ≤ X₀
t₂: l1(X₀, X₁) → l2(X₀, X₁) :|: X₀ ≤ 0
t₀: l2(X₀, X₁) → l1(X₀, X₁) :|: 1 ≤ X₀ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁

Preprocessing

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

Problem after Preprocessing

Start: l0
Program_Vars: X₀, X₁
Temp_Vars:
Locations: l0, l1, l2
Transitions:
t₃: l0(X₀, X₁) → l2(X₀, X₁)
t₁: l1(X₀, X₁) → l1(X₀-1, X₁-1) :|: 1 ≤ X₀ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 0 ≤ X₀
t₂: l1(X₀, X₁) → l2(X₀, X₁) :|: X₀ ≤ 0 ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 0 ≤ X₀
t₀: l2(X₀, X₁) → l1(X₀, X₁) :|: 1 ≤ X₀ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁

knowledge_propagation leads to new time bound 1 {O(1)} for transition t₀: l2(X₀, X₁) → l1(X₀, X₁) :|: 1 ≤ X₀ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁

MPRF for transition t₁: l1(X₀, X₁) → l1(X₀-1, X₁-1) :|: 1 ≤ X₀ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 0 ≤ X₀ of depth 1:

new bound:

X₀+X₁ {O(n)}

MPRF:

l2 [X₀+X₁ ]
l1 [X₁+1 ]

MPRF for transition t₂: l1(X₀, X₁) → l2(X₀, X₁) :|: X₀ ≤ 0 ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 0 ≤ X₀ of depth 1:

new bound:

X₁ {O(n)}

MPRF:

l2 [X₁ ]
l1 [1 ]

All Bounds

Timebounds

Overall timebound:2⋅X₁+X₀+2 {O(n)}
t₃: 1 {O(1)}
t₁: X₀+X₁ {O(n)}
t₂: X₁ {O(n)}
t₀: 1 {O(1)}

Costbounds

Overall costbound: 2⋅X₁+X₀+2 {O(n)}
t₃: 1 {O(1)}
t₁: X₀+X₁ {O(n)}
t₂: X₁ {O(n)}
t₀: 1 {O(1)}

Sizebounds

t₃, X₀: X₀ {O(n)}
t₃, X₁: X₁ {O(n)}
t₁, X₀: X₀ {O(n)}
t₁, X₁: X₁ {O(n)}
t₂, X₀: 0 {O(1)}
t₂, X₁: 0 {O(1)}
t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}