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

Preprocessing

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

Found invariant X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1+X₀ ≤ X₁ ∧ X₀ ≤ 0 for location l5

Found invariant X₀ ≤ X₂ ∧ 1 ≤ X₁ for location l1

Found invariant X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1+X₀ ≤ X₁ ∧ X₀ ≤ 0 for location l4

Found invariant 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location l3

Problem after Preprocessing

Start: l0
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: l0, l1, l2, l3, l4, l5
Transitions:
t₁₂: l0(X₀, X₁, X₂) → l2(X₀, X₁, X₂)
t₁₃: l1(X₀, X₁, X₂) → l3(X₀, X₁, X₂) :|: 0 < X₀ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁
t₁₄: l1(X₀, X₁, X₂) → l4(X₀, X₁, X₂) :|: X₀ ≤ 0 ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁
t₁₅: l2(X₀, X₁, X₂) → l1(X₂, 1, X₂)
t₁₆: l3(X₀, X₁, X₂) → l1(X₀-X₁, X₁+1, X₂) :|: 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀
t₁₇: l4(X₀, X₁, X₂) → l5(X₀, X₁, X₂) :|: X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1+X₀ ≤ X₁ ∧ X₀ ≤ 0

MPRF for transition t₁₃: l1(X₀, X₁, X₂) → l3(X₀, X₁, X₂) :|: 0 < X₀ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ of depth 1:

new bound:

X₂ {O(n)}

MPRF:

l3 [X₀-1 ]
l1 [X₀ ]

MPRF for transition t₁₆: l3(X₀, X₁, X₂) → l1(X₀-X₁, X₁+1, X₂) :|: 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ of depth 1:

new bound:

X₂ {O(n)}

MPRF:

l3 [X₀ ]
l1 [X₀ ]

All Bounds

Timebounds

Overall timebound:2⋅X₂+4 {O(n)}
t₁₂: 1 {O(1)}
t₁₃: X₂ {O(n)}
t₁₄: 1 {O(1)}
t₁₅: 1 {O(1)}
t₁₆: X₂ {O(n)}
t₁₇: 1 {O(1)}

Costbounds

Overall costbound: 2⋅X₂+4 {O(n)}
t₁₂: 1 {O(1)}
t₁₃: X₂ {O(n)}
t₁₄: 1 {O(1)}
t₁₅: 1 {O(1)}
t₁₆: X₂ {O(n)}
t₁₇: 1 {O(1)}

Sizebounds

t₁₂, X₀: X₀ {O(n)}
t₁₂, X₁: X₁ {O(n)}
t₁₂, X₂: X₂ {O(n)}
t₁₃, X₀: 2⋅X₂+1 {O(n)}
t₁₃, X₁: X₂+1 {O(n)}
t₁₃, X₂: X₂ {O(n)}
t₁₄, X₀: 3⋅X₂+1 {O(n)}
t₁₄, X₁: X₂+2 {O(n)}
t₁₄, X₂: 2⋅X₂ {O(n)}
t₁₅, X₀: X₂ {O(n)}
t₁₅, X₁: 1 {O(1)}
t₁₅, X₂: X₂ {O(n)}
t₁₆, X₀: 2⋅X₂+1 {O(n)}
t₁₆, X₁: X₂+1 {O(n)}
t₁₆, X₂: X₂ {O(n)}
t₁₇, X₀: 3⋅X₂+1 {O(n)}
t₁₇, X₁: X₂+2 {O(n)}
t₁₇, X₂: 2⋅X₂ {O(n)}