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) :|: 1 < X₀ ∧ X₀ < X₃
t₃: l1(X₀, X₁, X₂, X₃) → l4(X₀, X₁, X₂, X₃) :|: X₀ ≤ 1
t₄: l1(X₀, X₁, X₂, X₃) → l4(X₀, X₁, X₂, X₃) :|: X₃ ≤ X₀
t₁: l2(X₀, X₁, X₂, X₃) → l1(X₂, X₁, X₂, X₃)
t₅: l3(X₀, X₁, X₂, X₃) → l1((X₀)², X₁, X₂, X₃)
t₆: l4(X₀, X₁, X₂, X₃) → l5(X₀, X₁, X₂, X₃)

Preprocessing

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

Found invariant 3 ≤ X₂ ∧ 5 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ 2 ≤ X₀ for location l3

Problem after Preprocessing

Start: l0
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: l0, l1, l2, l3, l4, l5
Transitions:
t₁₃: l0(X₀, X₁, X₂) → l2(X₀, X₁, X₂)
t₁₄: l1(X₀, X₁, X₂) → l3(X₀, X₁, X₂) :|: 1 < X₀ ∧ X₀ < X₂
t₁₅: l1(X₀, X₁, X₂) → l4(X₀, X₁, X₂) :|: X₀ ≤ 1
t₁₆: l1(X₀, X₁, X₂) → l4(X₀, X₁, X₂) :|: X₂ ≤ X₀
t₁₇: l2(X₀, X₁, X₂) → l1(X₁, X₁, X₂)
t₁₈: l3(X₀, X₁, X₂) → l1((X₀)², X₁, X₂) :|: 3 ≤ X₂ ∧ 5 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ 2 ≤ X₀
t₁₉: l4(X₀, X₁, X₂) → l5(X₀, X₁, X₂)

MPRF for transition t₁₄: l1(X₀, X₁, X₂) → l3(X₀, X₁, X₂) :|: 1 < X₀ ∧ X₀ < X₂ of depth 1:

new bound:

X₁+X₂ {O(n)}

MPRF for transition t₁₈: l3(X₀, X₁, X₂) → l1(Temp_Int₉₅, X₁, X₂) :|: X₀ < Temp_Int₉₅ ∧ 4 ≤ Temp_Int₉₅ ∧ 3 ≤ X₂ ∧ 5 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ 2 ≤ X₀ of depth 1:

new bound:

X₁+X₂ {O(n)}

All Bounds

Timebounds

Overall timebound:2⋅X₁+2⋅X₂+5 {O(n)}
t₁₃: 1 {O(1)}
t₁₄: X₁+X₂ {O(n)}
t₁₅: 1 {O(1)}
t₁₆: 1 {O(1)}
t₁₇: 1 {O(1)}
t₁₈: X₁+X₂ {O(n)}
t₁₉: 1 {O(1)}

Costbounds

Overall costbound: 2⋅X₁+2⋅X₂+5 {O(n)}
t₁₃: 1 {O(1)}
t₁₄: X₁+X₂ {O(n)}
t₁₅: 1 {O(1)}
t₁₆: 1 {O(1)}
t₁₇: 1 {O(1)}
t₁₈: X₁+X₂ {O(n)}
t₁₉: 1 {O(1)}

Sizebounds

t₁₃, X₀: X₀ {O(n)}
t₁₃, X₁: X₁ {O(n)}
t₁₃, X₂: X₂ {O(n)}
t₁₄, X₁: X₁ {O(n)}
t₁₄, X₂: X₂ {O(n)}
t₁₅, X₁: 2⋅X₁ {O(n)}
t₁₅, X₂: 2⋅X₂ {O(n)}
t₁₆, X₁: 2⋅X₁ {O(n)}
t₁₆, X₂: 2⋅X₂ {O(n)}
t₁₇, X₀: X₁ {O(n)}
t₁₇, X₁: X₁ {O(n)}
t₁₇, X₂: X₂ {O(n)}
t₁₈, X₁: X₁ {O(n)}
t₁₈, X₂: X₂ {O(n)}
t₁₉, X₁: 4⋅X₁ {O(n)}
t₁₉, X₂: 4⋅X₂ {O(n)}