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, X₃, X₂, X₃)
t₄: l3(X₀, X₁, X₂, X₃) → l1(X₁, X₀-1, X₂, X₃)
t₅: l4(X₀, X₁, X₂, X₃) → l5(X₀, X₁, X₂, X₃)

Preprocessing

Found invariant X₀ ≤ 0 for location l5

Found invariant X₀ ≤ 0 for location l4

Found invariant 1 ≤ X₀ for location l3

Problem after Preprocessing

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

Analysing control-flow refined program

Found invariant X₃ ≤ X₁ ∧ X₁ ≤ X₃ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₀ for location n_l3___3

Found invariant 1 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 0 ≤ X₁ for location n_l1___2

Found invariant X₀ ≤ X₂ ∧ X₀ ≤ 0 for location l5

Found invariant X₃ ≤ X₁ ∧ X₁ ≤ X₃ ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₂ for location l1

Found invariant X₀ ≤ X₂ ∧ X₀ ≤ 0 for location l4

Found invariant 1 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location n_l3___1

MPRF for transition t₆₆: n_l1___2(X₀, X₁, X₂, X₃) → n_l3___1(X₀, X₁, X₂, X₃) :|: 0 ≤ X₁ ∧ 0 < X₀ ∧ 1 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 0 ≤ X₁ of depth 1:

new bound:

X₂+X₃ {O(n)}

MPRF:

n_l3___1 [X₀+X₁-1 ]
n_l1___2 [X₀+X₁ ]

MPRF for transition t₆₈: n_l3___1(X₀, X₁, X₂, X₃) → n_l1___2(X₁, X₀-1, X₂, X₃) :|: 0 ≤ X₁ ∧ 0 < X₀ ∧ 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ of depth 1:

new bound:

X₂+X₃ {O(n)}

MPRF:

n_l3___1 [X₀+X₁ ]
n_l1___2 [X₀+X₁ ]

CFR: Improvement to new bound with the following program:

new bound:

2⋅X₂+2⋅X₃ {O(n)}

cfr-program:

Start: l0
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars:
Locations: l0, l1, l2, l4, l5, n_l1___2, n_l3___1, n_l3___3
Transitions:
t₀: l0(X₀, X₁, X₂, X₃) → l2(X₀, X₁, X₂, X₃)
t₃: l1(X₀, X₁, X₂, X₃) → l4(X₀, X₁, X₂, X₃) :|: X₀ ≤ 0 ∧ X₃ ≤ X₁ ∧ X₁ ≤ X₃ ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₂
t₆₇: l1(X₀, X₁, X₂, X₃) → n_l3___3(X₀, X₁, X₂, X₃) :|: X₁ ≤ X₃ ∧ X₃ ≤ X₁ ∧ X₀ ≤ X₂ ∧ X₂ ≤ X₀ ∧ 0 < X₀ ∧ X₃ ≤ X₁ ∧ X₁ ≤ X₃ ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₂
t₁: l2(X₀, X₁, X₂, X₃) → l1(X₂, X₃, X₂, X₃)
t₅: l4(X₀, X₁, X₂, X₃) → l5(X₀, X₁, X₂, X₃) :|: X₀ ≤ 0 ∧ X₀ ≤ X₂ ∧ X₀ ≤ 0
t₇₃: n_l1___2(X₀, X₁, X₂, X₃) → l4(X₀, X₁, X₂, X₃) :|: X₀ ≤ 0 ∧ 1 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 0 ≤ X₁
t₆₆: n_l1___2(X₀, X₁, X₂, X₃) → n_l3___1(X₀, X₁, X₂, X₃) :|: 0 ≤ X₁ ∧ 0 < X₀ ∧ 1 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 0 ≤ X₁
t₆₈: n_l3___1(X₀, X₁, X₂, X₃) → n_l1___2(X₁, X₀-1, X₂, X₃) :|: 0 ≤ X₁ ∧ 0 < X₀ ∧ 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀
t₆₉: n_l3___3(X₀, X₁, X₂, X₃) → n_l1___2(X₁, X₀-1, X₂, X₃) :|: 0 < X₀ ∧ X₀ ≤ X₂ ∧ X₂ ≤ X₀ ∧ X₁ ≤ X₃ ∧ X₃ ≤ X₁ ∧ 1 ≤ X₀ ∧ X₃ ≤ X₁ ∧ X₁ ≤ X₃ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₀

All Bounds

Timebounds

Overall timebound:2⋅X₂+2⋅X₃+7 {O(n)}
t₀: 1 {O(1)}
t₃: 1 {O(1)}
t₆₇: 1 {O(1)}
t₁: 1 {O(1)}
t₅: 1 {O(1)}
t₆₆: X₂+X₃ {O(n)}
t₇₃: 1 {O(1)}
t₆₈: X₂+X₃ {O(n)}
t₆₉: 1 {O(1)}

Costbounds

Overall costbound: 2⋅X₂+2⋅X₃+7 {O(n)}
t₀: 1 {O(1)}
t₃: 1 {O(1)}
t₆₇: 1 {O(1)}
t₁: 1 {O(1)}
t₅: 1 {O(1)}
t₆₆: X₂+X₃ {O(n)}
t₇₃: 1 {O(1)}
t₆₈: X₂+X₃ {O(n)}
t₆₉: 1 {O(1)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₀, X₃: X₃ {O(n)}
t₃, X₀: X₂ {O(n)}
t₃, X₁: X₃ {O(n)}
t₃, X₂: X₂ {O(n)}
t₃, X₃: X₃ {O(n)}
t₆₇, X₀: X₂ {O(n)}
t₆₇, X₁: X₃ {O(n)}
t₆₇, X₂: X₂ {O(n)}
t₆₇, X₃: X₃ {O(n)}
t₁, X₀: X₂ {O(n)}
t₁, X₁: X₃ {O(n)}
t₁, X₂: X₂ {O(n)}
t₁, X₃: X₃ {O(n)}
t₅, X₀: 2⋅X₂+X₃ {O(n)}
t₅, X₁: 2⋅X₃+X₂ {O(n)}
t₅, X₂: 3⋅X₂ {O(n)}
t₅, X₃: 3⋅X₃ {O(n)}
t₆₆, X₀: X₂+X₃ {O(n)}
t₆₆, X₁: X₂+X₃ {O(n)}
t₆₆, X₂: X₂ {O(n)}
t₆₆, X₃: X₃ {O(n)}
t₇₃, X₀: 2⋅X₃+X₂ {O(n)}
t₇₃, X₁: 2⋅X₂+X₃ {O(n)}
t₇₃, X₂: 2⋅X₂ {O(n)}
t₇₃, X₃: 2⋅X₃ {O(n)}
t₆₈, X₀: X₂+X₃ {O(n)}
t₆₈, X₁: X₂+X₃ {O(n)}
t₆₈, X₂: X₂ {O(n)}
t₆₈, X₃: X₃ {O(n)}
t₆₉, X₀: X₃ {O(n)}
t₆₉, X₁: X₂ {O(n)}
t₆₉, X₂: X₂ {O(n)}
t₆₉, X₃: X₃ {O(n)}