Initial Problem

Start: l0
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars:
Locations: l0, l1, l2, l3, l4, l5
Transitions:
t₀: l0(X₀, X₁, X₂, X₃) → l2(X₀, X₁, X₂, X₃)
t₂: l1(X₀, X₁, X₂, X₃) → l3(X₀, X₁, X₂, X₃) :|: X₀ < X₂
t₃: l1(X₀, X₁, X₂, X₃) → l4(X₀, X₁, X₂, X₃) :|: X₂ ≤ X₀
t₁: l2(X₀, X₁, X₂, X₃) → l1(X₃, 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)

Preprocessing

Eliminate variables {X₁} that do not contribute to the problem

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:
t₁₁: l0(X₀, X₁, X₂) → l2(X₀, X₁, X₂)
t₁₂: l1(X₀, X₁, X₂) → l3(X₀, X₁, X₂) :|: X₀ < X₁ ∧ X₂ ≤ X₀
t₁₃: l1(X₀, X₁, X₂) → l4(X₀, X₁, X₂) :|: X₁ ≤ X₀ ∧ X₂ ≤ X₀
t₁₄: l2(X₀, X₁, X₂) → l1(X₂, X₁, X₂)
t₁₅: l3(X₀, X₁, X₂) → l1(X₀+1, X₁, X₂) :|: 1+X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ 1+X₀ ≤ X₁
t₁₆: l4(X₀, X₁, X₂) → l5(X₀, X₁, X₂) :|: X₂ ≤ X₀ ∧ X₁ ≤ X₀

MPRF for transition t₁₂: l1(X₀, X₁, X₂) → l3(X₀, X₁, X₂) :|: X₀ < X₁ ∧ X₂ ≤ X₀ of depth 1:

new bound:

X₁+X₂ {O(n)}

MPRF for transition t₁₅: 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)}

All Bounds

Timebounds

Overall timebound:2⋅X₁+2⋅X₂+4 {O(n)}
t₁₁: 1 {O(1)}
t₁₂: X₁+X₂ {O(n)}
t₁₃: 1 {O(1)}
t₁₄: 1 {O(1)}
t₁₅: X₁+X₂ {O(n)}
t₁₆: 1 {O(1)}

Costbounds

Overall costbound: 2⋅X₁+2⋅X₂+4 {O(n)}
t₁₁: 1 {O(1)}
t₁₂: X₁+X₂ {O(n)}
t₁₃: 1 {O(1)}
t₁₄: 1 {O(1)}
t₁₅: X₁+X₂ {O(n)}
t₁₆: 1 {O(1)}

Sizebounds

t₁₁, X₀: X₀ {O(n)}
t₁₁, X₁: X₁ {O(n)}
t₁₁, X₂: X₂ {O(n)}
t₁₂, X₀: 2⋅X₂+X₁ {O(n)}
t₁₂, X₁: X₁ {O(n)}
t₁₂, X₂: X₂ {O(n)}
t₁₃, X₀: 3⋅X₂+X₁ {O(n)}
t₁₃, X₁: 2⋅X₁ {O(n)}
t₁₃, X₂: 2⋅X₂ {O(n)}
t₁₄, X₀: X₂ {O(n)}
t₁₄, X₁: X₁ {O(n)}
t₁₄, X₂: X₂ {O(n)}
t₁₅, X₀: 2⋅X₂+X₁ {O(n)}
t₁₅, X₁: X₁ {O(n)}
t₁₅, X₂: X₂ {O(n)}
t₁₆, X₀: 3⋅X₂+X₁ {O(n)}
t₁₆, X₁: 2⋅X₁ {O(n)}
t₁₆, X₂: 2⋅X₂ {O(n)}