Initial Problem

Start: l0
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: l0, l1, l2, l3, l4, l5
Transitions:
t₀: l0(X₀, X₁, X₂) → l2(X₀, X₁, X₂)
t₂: l1(X₀, X₁, X₂) → l3(X₀, X₁, X₂) :|: X₀ < X₁
t₃: l1(X₀, X₁, X₂) → l4(X₀, X₁, X₂) :|: X₁ ≤ X₀
t₁: l2(X₀, X₁, X₂) → l1(X₂, X₁, X₂)
t₄: l3(X₀, X₁, X₂) → l1(X₀+1, X₁, X₂)
t₅: l4(X₀, X₁, X₂) → l5(X₀, X₁, X₂)

Preprocessing

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

Found invariant X₂ ≤ X₀ for location l1

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

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

Problem after Preprocessing

Start: l0
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: l0, l1, l2, l3, l4, l5
Transitions:
t₀: l0(X₀, X₁, X₂) → l2(X₀, X₁, X₂)
t₂: l1(X₀, X₁, X₂) → l3(X₀, X₁, X₂) :|: X₀ < X₁ ∧ X₂ ≤ X₀
t₃: l1(X₀, X₁, X₂) → l4(X₀, X₁, X₂) :|: X₁ ≤ X₀ ∧ X₂ ≤ X₀
t₁: l2(X₀, X₁, X₂) → l1(X₂, X₁, X₂)
t₄: l3(X₀, X₁, X₂) → l1(X₀+1, X₁, X₂) :|: 1+X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ 1+X₀ ≤ X₁
t₅: l4(X₀, X₁, X₂) → l5(X₀, X₁, X₂) :|: X₂ ≤ X₀ ∧ X₁ ≤ X₀

MPRF for transition t₂: l1(X₀, X₁, X₂) → l3(X₀, X₁, X₂) :|: X₀ < X₁ ∧ X₂ ≤ X₀ of depth 1:

new bound:

X₁+X₂ {O(n)}

MPRF:

l3 [X₁-X₀-1 ]
l1 [X₁-X₀ ]

MPRF for transition t₄: l3(X₀, X₁, X₂) → l1(X₀+1, X₁, X₂) :|: 1+X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ 1+X₀ ≤ X₁ of depth 1:

new bound:

X₁+X₂ {O(n)}

MPRF:

l3 [X₁-X₀ ]
l1 [X₁-X₀ ]

All Bounds

Timebounds

Overall timebound:2⋅X₁+2⋅X₂+4 {O(n)}
t₀: 1 {O(1)}
t₂: X₁+X₂ {O(n)}
t₃: 1 {O(1)}
t₁: 1 {O(1)}
t₄: X₁+X₂ {O(n)}
t₅: 1 {O(1)}

Costbounds

Overall costbound: 2⋅X₁+2⋅X₂+4 {O(n)}
t₀: 1 {O(1)}
t₂: X₁+X₂ {O(n)}
t₃: 1 {O(1)}
t₁: 1 {O(1)}
t₄: X₁+X₂ {O(n)}
t₅: 1 {O(1)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₂, X₀: 2⋅X₂+X₁ {O(n)}
t₂, X₁: X₁ {O(n)}
t₂, X₂: X₂ {O(n)}
t₃, X₀: 3⋅X₂+X₁ {O(n)}
t₃, X₁: 2⋅X₁ {O(n)}
t₃, X₂: 2⋅X₂ {O(n)}
t₁, X₀: X₂ {O(n)}
t₁, X₁: X₁ {O(n)}
t₁, X₂: X₂ {O(n)}
t₄, X₀: 2⋅X₂+X₁ {O(n)}
t₄, X₁: X₁ {O(n)}
t₄, X₂: X₂ {O(n)}
t₅, X₀: 3⋅X₂+X₁ {O(n)}
t₅, X₁: 2⋅X₁ {O(n)}
t₅, X₂: 2⋅X₂ {O(n)}