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) → l3(X₀, X₁, X₂, X₃)
t₄: l1(X₀, X₁, X₂, X₃) → l2(X₀, X₁, X₂, X₃) :|: X₀ ≤ 0
t₃: l1(X₀, X₁, X₂, X₃) → l4(X₀, X₁, X₂, X₃) :|: 0 < X₀
t₇: l2(X₀, X₁, X₂, X₃) → l5(X₀, X₁, X₂, X₃)
t₁: l3(X₀, X₁, X₂, X₃) → l1(X₂, X₁, X₂, X₃) :|: 0 < X₃
t₂: l3(X₀, X₁, X₂, X₃) → l2(X₀, X₁, X₂, X₃) :|: X₃ ≤ 0
t₅: l4(X₀, X₁, X₂, X₃) → l1(X₃, X₁, X₂, X₃) :|: X₃ < X₀
t₆: l4(X₀, X₁, X₂, X₃) → l1(X₀-1, X₁, X₂, X₃) :|: X₀ ≤ X₃

Preprocessing

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

Found invariant 1 ≤ X₃ ∧ X₀ ≤ X₂ for location l1

Found invariant 1 ≤ X₃ ∧ 2 ≤ X₂+X₃ ∧ 2 ≤ X₀+X₃ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₀ for location l4

Problem after Preprocessing

Start: l0
Program_Vars: X₀, X₂, X₃
Temp_Vars:
Locations: l0, l1, l2, l3, l4, l5
Transitions:
t₁₆: l0(X₀, X₂, X₃) → l3(X₀, X₂, X₃)
t₁₈: l1(X₀, X₂, X₃) → l2(X₀, X₂, X₃) :|: X₀ ≤ 0 ∧ 1 ≤ X₃ ∧ X₀ ≤ X₂
t₁₇: l1(X₀, X₂, X₃) → l4(X₀, X₂, X₃) :|: 0 < X₀ ∧ 1 ≤ X₃ ∧ X₀ ≤ X₂
t₁₉: l2(X₀, X₂, X₃) → l5(X₀, X₂, X₃)
t₂₀: l3(X₀, X₂, X₃) → l1(X₂, X₂, X₃) :|: 0 < X₃
t₂₁: l3(X₀, X₂, X₃) → l2(X₀, X₂, X₃) :|: X₃ ≤ 0
t₂₂: l4(X₀, X₂, X₃) → l1(X₃, X₂, X₃) :|: X₃ < X₀ ∧ 1 ≤ X₃ ∧ 2 ≤ X₂+X₃ ∧ 2 ≤ X₀+X₃ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₀
t₂₃: l4(X₀, X₂, X₃) → l1(X₀-1, X₂, X₃) :|: X₀ ≤ X₃ ∧ 1 ≤ X₃ ∧ 2 ≤ X₂+X₃ ∧ 2 ≤ X₀+X₃ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₀

MPRF for transition t₁₇: l1(X₀, X₂, X₃) → l4(X₀, X₂, X₃) :|: 0 < X₀ ∧ 1 ≤ X₃ ∧ X₀ ≤ X₂ of depth 1:

new bound:

X₂ {O(n)}

MPRF:

l4 [X₀-1 ]
l1 [X₀ ]

MPRF for transition t₂₂: l4(X₀, X₂, X₃) → l1(X₃, X₂, X₃) :|: X₃ < X₀ ∧ 1 ≤ X₃ ∧ 2 ≤ X₂+X₃ ∧ 2 ≤ X₀+X₃ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₀ of depth 1:

new bound:

X₂ {O(n)}

MPRF:

l4 [X₀ ]
l1 [X₀ ]

MPRF for transition t₂₃: l4(X₀, X₂, X₃) → l1(X₀-1, X₂, X₃) :|: X₀ ≤ X₃ ∧ 1 ≤ X₃ ∧ 2 ≤ X₂+X₃ ∧ 2 ≤ X₀+X₃ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₀ of depth 1:

new bound:

X₂ {O(n)}

MPRF:

l4 [X₀ ]
l1 [X₀ ]

All Bounds

Timebounds

Overall timebound:3⋅X₂+5 {O(n)}
t₁₆: 1 {O(1)}
t₁₇: X₂ {O(n)}
t₁₈: 1 {O(1)}
t₁₉: 1 {O(1)}
t₂₀: 1 {O(1)}
t₂₁: 1 {O(1)}
t₂₂: X₂ {O(n)}
t₂₃: X₂ {O(n)}

Costbounds

Overall costbound: 3⋅X₂+5 {O(n)}
t₁₆: 1 {O(1)}
t₁₇: X₂ {O(n)}
t₁₈: 1 {O(1)}
t₁₉: 1 {O(1)}
t₂₀: 1 {O(1)}
t₂₁: 1 {O(1)}
t₂₂: X₂ {O(n)}
t₂₃: X₂ {O(n)}

Sizebounds

t₁₆, X₀: X₀ {O(n)}
t₁₆, X₂: X₂ {O(n)}
t₁₆, X₃: X₃ {O(n)}
t₁₇, X₀: X₂ {O(n)}
t₁₇, X₂: X₂ {O(n)}
t₁₇, X₃: X₃ {O(n)}
t₁₈, X₀: 2⋅X₂ {O(n)}
t₁₈, X₂: 2⋅X₂ {O(n)}
t₁₈, X₃: 2⋅X₃ {O(n)}
t₁₉, X₀: 2⋅X₂+X₀ {O(n)}
t₁₉, X₂: 3⋅X₂ {O(n)}
t₁₉, X₃: 3⋅X₃ {O(n)}
t₂₀, X₀: X₂ {O(n)}
t₂₀, X₂: X₂ {O(n)}
t₂₀, X₃: X₃ {O(n)}
t₂₁, X₀: X₀ {O(n)}
t₂₁, X₂: X₂ {O(n)}
t₂₁, X₃: X₃ {O(n)}
t₂₂, X₀: X₂ {O(n)}
t₂₂, X₂: X₂ {O(n)}
t₂₂, X₃: X₃ {O(n)}
t₂₃, X₀: X₂ {O(n)}
t₂₃, X₂: X₂ {O(n)}
t₂₃, X₃: X₃ {O(n)}