Initial Problem

Start: l0
Program_Vars: X₀, X₁
Temp_Vars:
Locations: l0, l1, l2, l3
Transitions:
t₆: l0(X₀, X₁) → l3(X₀, X₁)
t₂: l1(X₀, X₁) → l1(X₀-1, X₁) :|: 1 ≤ X₀
t₃: l1(X₀, X₁) → l3(X₀, X₁) :|: X₀ ≤ 0
t₄: l2(X₀, X₁) → l2(X₀, X₁-1) :|: 1 ≤ X₁
t₅: l2(X₀, X₁) → l3(X₀, X₁) :|: X₁ ≤ 0
t₀: l3(X₀, X₁) → l1(X₀, X₁) :|: 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ X₁+1 ≤ X₀
t₁: l3(X₀, X₁) → l2(X₀, X₁) :|: 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ X₀ ≤ X₁

Preprocessing

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

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

Problem after Preprocessing

Start: l0
Program_Vars: X₀, X₁
Temp_Vars:
Locations: l0, l1, l2, l3
Transitions:
t₆: l0(X₀, X₁) → l3(X₀, X₁)
t₂: l1(X₀, X₁) → l1(X₀-1, X₁) :|: 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 0 ≤ X₀
t₃: l1(X₀, X₁) → l3(X₀, X₁) :|: X₀ ≤ 0 ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 0 ≤ X₀
t₄: l2(X₀, X₁) → l2(X₀, X₁-1) :|: 1 ≤ X₁ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀
t₅: l2(X₀, X₁) → l3(X₀, X₁) :|: X₁ ≤ 0 ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀
t₀: l3(X₀, X₁) → l1(X₀, X₁) :|: 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ X₁+1 ≤ X₀
t₁: l3(X₀, X₁) → l2(X₀, X₁) :|: 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ X₀ ≤ X₁

knowledge_propagation leads to new time bound 1 {O(1)} for transition t₀: l3(X₀, X₁) → l1(X₀, X₁) :|: 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ X₁+1 ≤ X₀

knowledge_propagation leads to new time bound 1 {O(1)} for transition t₁: l3(X₀, X₁) → l2(X₀, X₁) :|: 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ X₀ ≤ X₁

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

new bound:

X₀+X₁ {O(n)}

MPRF:

l1 [X₀+X₁ ]
l3 [X₀+X₁ ]
l2 [X₀ ]

MPRF for transition t₃: l1(X₀, X₁) → l3(X₀, X₁) :|: X₀ ≤ 0 ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 0 ≤ X₀ of depth 1:

new bound:

X₀+1 {O(n)}

MPRF:

l1 [1 ]
l3 [X₀-1 ]
l2 [X₀-1 ]

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

new bound:

2⋅X₁ {O(n)}

MPRF:

l1 [2⋅X₁ ]
l3 [2⋅X₁ ]
l2 [X₁+1 ]

MPRF for transition t₅: l2(X₀, X₁) → l3(X₀, X₁) :|: X₁ ≤ 0 ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ of depth 1:

new bound:

X₁ {O(n)}

MPRF:

l1 [X₁ ]
l3 [X₁ ]
l2 [1 ]

All Bounds

Timebounds

Overall timebound:2⋅X₀+4⋅X₁+4 {O(n)}
t₆: 1 {O(1)}
t₂: X₀+X₁ {O(n)}
t₃: X₀+1 {O(n)}
t₄: 2⋅X₁ {O(n)}
t₅: X₁ {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}

Costbounds

Overall costbound: 2⋅X₀+4⋅X₁+4 {O(n)}
t₆: 1 {O(1)}
t₂: X₀+X₁ {O(n)}
t₃: X₀+1 {O(n)}
t₄: 2⋅X₁ {O(n)}
t₅: X₁ {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}

Sizebounds

t₆, X₀: X₀ {O(n)}
t₆, X₁: X₁ {O(n)}
t₂, X₀: X₀ {O(n)}
t₂, X₁: X₁ {O(n)}
t₃, X₀: 0 {O(1)}
t₃, X₁: X₁ {O(n)}
t₄, X₀: X₀ {O(n)}
t₄, X₁: X₁ {O(n)}
t₅, X₀: X₀ {O(n)}
t₅, X₁: 0 {O(1)}
t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₁, X₀: X₀ {O(n)}
t₁, X₁: X₁ {O(n)}