Initial Problem

Start: l0
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars: E
Locations: l0, l1, l2
Transitions:
t₀: l0(X₀, X₁, X₂, X₃) → l1(0, X₁, X₂, X₃)
t₃: l1(X₀, X₁, X₂, X₃) → l1(0, 1+X₁, X₂-1, X₃) :|: 1+X₁ ≤ X₂ ∧ X₀ ≤ 0 ∧ 2+X₁ ≤ X₂
t₁: l1(X₀, X₁, X₂, X₃) → l2(X₀, X₁, X₂, E) :|: X₂ ≤ X₁
t₂: l1(X₀, X₁, X₂, X₃) → l2(1, 1+X₁, X₂, E) :|: X₀ ≤ 0 ∧ X₁+1 ≤ X₂ ∧ X₂ ≤ X₁+1

Preprocessing

Eliminate variables {E,X₃} that do not contribute to the problem

Found invariant X₂ ≤ X₁ ∧ X₀ ≤ 1 ∧ 0 ≤ X₀ for location l2

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

Problem after Preprocessing

Start: l0
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: l0, l1, l2
Transitions:
t₈: l0(X₀, X₁, X₂) → l1(0, X₁, X₂)
t₉: l1(X₀, X₁, X₂) → l1(0, 1+X₁, X₂-1) :|: 1+X₁ ≤ X₂ ∧ X₀ ≤ 0 ∧ 2+X₁ ≤ X₂ ∧ X₀ ≤ 0 ∧ 0 ≤ X₀
t₁₀: l1(X₀, X₁, X₂) → l2(X₀, X₁, X₂) :|: X₂ ≤ X₁ ∧ X₀ ≤ 0 ∧ 0 ≤ X₀
t₁₁: l1(X₀, X₁, X₂) → l2(1, 1+X₁, X₂) :|: X₀ ≤ 0 ∧ X₁+1 ≤ X₂ ∧ X₂ ≤ X₁+1 ∧ X₀ ≤ 0 ∧ 0 ≤ X₀

MPRF for transition t₉: l1(X₀, X₁, X₂) → l1(0, 1+X₁, X₂-1) :|: 1+X₁ ≤ X₂ ∧ X₀ ≤ 0 ∧ 2+X₁ ≤ X₂ ∧ X₀ ≤ 0 ∧ 0 ≤ X₀ of depth 1:

new bound:

X₁+X₂ {O(n)}

All Bounds

Timebounds

Overall timebound:X₁+X₂+3 {O(n)}
t₈: 1 {O(1)}
t₉: X₁+X₂ {O(n)}
t₁₀: 1 {O(1)}
t₁₁: 1 {O(1)}

Costbounds

Overall costbound: X₁+X₂+3 {O(n)}
t₈: 1 {O(1)}
t₉: X₁+X₂ {O(n)}
t₁₀: 1 {O(1)}
t₁₁: 1 {O(1)}

Sizebounds

t₈, X₀: 0 {O(1)}
t₈, X₁: X₁ {O(n)}
t₈, X₂: X₂ {O(n)}
t₉, X₀: 0 {O(1)}
t₉, X₁: 2⋅X₁+X₂ {O(n)}
t₉, X₂: 2⋅X₂+X₁ {O(n)}
t₁₀, X₀: 0 {O(1)}
t₁₀, X₁: 3⋅X₁+X₂ {O(n)}
t₁₀, X₂: 3⋅X₂+X₁ {O(n)}
t₁₁, X₀: 1 {O(1)}
t₁₁, X₁: 3⋅X₁+X₂+2 {O(n)}
t₁₁, X₂: 3⋅X₂+X₁ {O(n)}