Initial Problem

Start: l0
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars:
Locations: l0, l1, l10, l11, l12, l13, l2, l3, l4, l5, l6, l7, l8, l9
Transitions:
t₀: l0(X₀, X₁, X₂, X₃) → l2(X₀, X₁, X₂, X₃)
t₃: l1(X₀, X₁, X₂, X₃) → l4(X₀, X₁, X₂, X₃)
t₁₀: l10(X₀, X₁, X₂, X₃) → l11(X₀, X₁, X₂, X₃) :|: 0 < X₀
t₁₁: l10(X₀, X₁, X₂, X₃) → l12(X₀, X₁, X₂, X₃) :|: X₀ ≤ 0
t₁₂: l11(X₀, X₁, X₂, X₃) → l10(X₁, X₀-1, X₂, X₃)
t₁₃: l12(X₀, X₁, X₂, X₃) → l13(X₀, X₁, X₂, X₃)
t₁: l2(X₀, X₁, X₂, X₃) → l3(X₀, X₁, X₂, X₃)
t₂: l3(X₀, X₁, X₂, X₃) → l1(X₀, X₁, X₂, X₃)
t₄: l4(X₀, X₁, X₂, X₃) → l5(X₀, X₁, X₂, X₃)
t₅: l5(X₀, X₁, X₂, X₃) → l6(X₀, X₁, X₂, X₃)
t₆: l6(X₀, X₁, X₂, X₃) → l7(X₀, X₁, X₂, X₃)
t₇: l7(X₀, X₁, X₂, X₃) → l8(X₀, X₁, X₂, X₃)
t₈: l8(X₀, X₁, X₂, X₃) → l9(X₀, X₁, X₂, X₃)
t₉: l9(X₀, X₁, X₂, X₃) → l10(X₂, X₃, X₂, X₃)

Preprocessing

Found invariant 1 ≤ X₀ for location l11

Found invariant X₀ ≤ 0 for location l12

Found invariant X₀ ≤ 0 for location l13

Problem after Preprocessing

Start: l0
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars:
Locations: l0, l1, l10, l11, l12, l13, l2, l3, l4, l5, l6, l7, l8, l9
Transitions:
t₀: l0(X₀, X₁, X₂, X₃) → l2(X₀, X₁, X₂, X₃)
t₃: l1(X₀, X₁, X₂, X₃) → l4(X₀, X₁, X₂, X₃)
t₁₀: l10(X₀, X₁, X₂, X₃) → l11(X₀, X₁, X₂, X₃) :|: 0 < X₀
t₁₁: l10(X₀, X₁, X₂, X₃) → l12(X₀, X₁, X₂, X₃) :|: X₀ ≤ 0
t₁₂: l11(X₀, X₁, X₂, X₃) → l10(X₁, X₀-1, X₂, X₃) :|: 1 ≤ X₀
t₁₃: l12(X₀, X₁, X₂, X₃) → l13(X₀, X₁, X₂, X₃) :|: X₀ ≤ 0
t₁: l2(X₀, X₁, X₂, X₃) → l3(X₀, X₁, X₂, X₃)
t₂: l3(X₀, X₁, X₂, X₃) → l1(X₀, X₁, X₂, X₃)
t₄: l4(X₀, X₁, X₂, X₃) → l5(X₀, X₁, X₂, X₃)
t₅: l5(X₀, X₁, X₂, X₃) → l6(X₀, X₁, X₂, X₃)
t₆: l6(X₀, X₁, X₂, X₃) → l7(X₀, X₁, X₂, X₃)
t₇: l7(X₀, X₁, X₂, X₃) → l8(X₀, X₁, X₂, X₃)
t₈: l8(X₀, X₁, X₂, X₃) → l9(X₀, X₁, X₂, X₃)
t₉: l9(X₀, X₁, X₂, X₃) → l10(X₂, X₃, X₂, X₃)

Analysing control-flow refined program

Found invariant 1 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 0 ≤ X₁ for location n_l10___2

Found invariant X₀ ≤ X₂ ∧ X₀ ≤ 0 for location l12

Found invariant X₀ ≤ X₂ ∧ X₀ ≤ 0 for location l13

Found invariant X₃ ≤ X₁ ∧ X₁ ≤ X₃ ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₂ for location l10

Found invariant X₃ ≤ X₁ ∧ X₁ ≤ X₃ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₀ for location n_l11___3

Found invariant 1 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location n_l11___1

MPRF for transition t₁₀₆: n_l10___2(X₀, X₁, X₂, X₃) → n_l11___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_l11___1 [X₀+X₁-1 ]
n_l10___2 [X₀+X₁ ]

MPRF for transition t₁₀₈: n_l11___1(X₀, X₁, X₂, X₃) → n_l10___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_l11___1 [X₀+X₁ ]
n_l10___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, l10, l12, l13, l2, l3, l4, l5, l6, l7, l8, l9, n_l10___2, n_l11___1, n_l11___3
Transitions:
t₀: l0(X₀, X₁, X₂, X₃) → l2(X₀, X₁, X₂, X₃)
t₃: l1(X₀, X₁, X₂, X₃) → l4(X₀, X₁, X₂, X₃)
t₁₁: l10(X₀, X₁, X₂, X₃) → l12(X₀, X₁, X₂, X₃) :|: X₀ ≤ 0 ∧ X₃ ≤ X₁ ∧ X₁ ≤ X₃ ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₂
t₁₀₇: l10(X₀, X₁, X₂, X₃) → n_l11___3(X₀, X₁, X₂, X₃) :|: X₁ ≤ X₃ ∧ X₃ ≤ X₁ ∧ X₀ ≤ X₂ ∧ X₂ ≤ X₀ ∧ 0 < X₀ ∧ X₃ ≤ X₁ ∧ X₁ ≤ X₃ ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₂
t₁₃: l12(X₀, X₁, X₂, X₃) → l13(X₀, X₁, X₂, X₃) :|: X₀ ≤ 0 ∧ X₀ ≤ X₂ ∧ X₀ ≤ 0
t₁: l2(X₀, X₁, X₂, X₃) → l3(X₀, X₁, X₂, X₃)
t₂: l3(X₀, X₁, X₂, X₃) → l1(X₀, X₁, X₂, X₃)
t₄: l4(X₀, X₁, X₂, X₃) → l5(X₀, X₁, X₂, X₃)
t₅: l5(X₀, X₁, X₂, X₃) → l6(X₀, X₁, X₂, X₃)
t₆: l6(X₀, X₁, X₂, X₃) → l7(X₀, X₁, X₂, X₃)
t₇: l7(X₀, X₁, X₂, X₃) → l8(X₀, X₁, X₂, X₃)
t₈: l8(X₀, X₁, X₂, X₃) → l9(X₀, X₁, X₂, X₃)
t₉: l9(X₀, X₁, X₂, X₃) → l10(X₂, X₃, X₂, X₃)
t₁₁₃: n_l10___2(X₀, X₁, X₂, X₃) → l12(X₀, X₁, X₂, X₃) :|: X₀ ≤ 0 ∧ 1 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 0 ≤ X₁
t₁₀₆: n_l10___2(X₀, X₁, X₂, X₃) → n_l11___1(X₀, X₁, X₂, X₃) :|: 0 ≤ X₁ ∧ 0 < X₀ ∧ 1 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 0 ≤ X₁
t₁₀₈: n_l11___1(X₀, X₁, X₂, X₃) → n_l10___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
t₁₀₉: n_l11___3(X₀, X₁, X₂, X₃) → n_l10___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+15 {O(n)}
t₀: 1 {O(1)}
t₃: 1 {O(1)}
t₁₁: 1 {O(1)}
t₁₀₇: 1 {O(1)}
t₁₃: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 1 {O(1)}
t₄: 1 {O(1)}
t₅: 1 {O(1)}
t₆: 1 {O(1)}
t₇: 1 {O(1)}
t₈: 1 {O(1)}
t₉: 1 {O(1)}
t₁₀₆: X₂+X₃ {O(n)}
t₁₁₃: 1 {O(1)}
t₁₀₈: X₂+X₃ {O(n)}
t₁₀₉: 1 {O(1)}

Costbounds

Overall costbound: 2⋅X₂+2⋅X₃+15 {O(n)}
t₀: 1 {O(1)}
t₃: 1 {O(1)}
t₁₁: 1 {O(1)}
t₁₀₇: 1 {O(1)}
t₁₃: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 1 {O(1)}
t₄: 1 {O(1)}
t₅: 1 {O(1)}
t₆: 1 {O(1)}
t₇: 1 {O(1)}
t₈: 1 {O(1)}
t₉: 1 {O(1)}
t₁₀₆: X₂+X₃ {O(n)}
t₁₁₃: 1 {O(1)}
t₁₀₈: X₂+X₃ {O(n)}
t₁₀₉: 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: X₀ {O(n)}
t₃, X₁: X₁ {O(n)}
t₃, X₂: X₂ {O(n)}
t₃, X₃: X₃ {O(n)}
t₁₁, X₀: X₂ {O(n)}
t₁₁, X₁: X₃ {O(n)}
t₁₁, X₂: X₂ {O(n)}
t₁₁, X₃: X₃ {O(n)}
t₁₀₇, X₀: X₂ {O(n)}
t₁₀₇, X₁: X₃ {O(n)}
t₁₀₇, X₂: X₂ {O(n)}
t₁₀₇, X₃: X₃ {O(n)}
t₁₃, X₀: 2⋅X₂+X₃ {O(n)}
t₁₃, X₁: 2⋅X₃+X₂ {O(n)}
t₁₃, X₂: 3⋅X₂ {O(n)}
t₁₃, X₃: 3⋅X₃ {O(n)}
t₁, X₀: X₀ {O(n)}
t₁, X₁: X₁ {O(n)}
t₁, X₂: X₂ {O(n)}
t₁, X₃: X₃ {O(n)}
t₂, X₀: X₀ {O(n)}
t₂, X₁: X₁ {O(n)}
t₂, X₂: X₂ {O(n)}
t₂, X₃: X₃ {O(n)}
t₄, X₀: X₀ {O(n)}
t₄, X₁: X₁ {O(n)}
t₄, X₂: X₂ {O(n)}
t₄, X₃: X₃ {O(n)}
t₅, X₀: X₀ {O(n)}
t₅, X₁: X₁ {O(n)}
t₅, X₂: X₂ {O(n)}
t₅, X₃: X₃ {O(n)}
t₆, X₀: X₀ {O(n)}
t₆, X₁: X₁ {O(n)}
t₆, X₂: X₂ {O(n)}
t₆, X₃: X₃ {O(n)}
t₇, X₀: X₀ {O(n)}
t₇, X₁: X₁ {O(n)}
t₇, X₂: X₂ {O(n)}
t₇, X₃: X₃ {O(n)}
t₈, X₀: X₀ {O(n)}
t₈, X₁: X₁ {O(n)}
t₈, X₂: X₂ {O(n)}
t₈, X₃: X₃ {O(n)}
t₉, X₀: X₂ {O(n)}
t₉, X₁: X₃ {O(n)}
t₉, X₂: X₂ {O(n)}
t₉, X₃: X₃ {O(n)}
t₁₀₆, X₀: X₂+X₃ {O(n)}
t₁₀₆, X₁: X₂+X₃ {O(n)}
t₁₀₆, X₂: X₂ {O(n)}
t₁₀₆, X₃: X₃ {O(n)}
t₁₁₃, X₀: 2⋅X₃+X₂ {O(n)}
t₁₁₃, X₁: 2⋅X₂+X₃ {O(n)}
t₁₁₃, X₂: 2⋅X₂ {O(n)}
t₁₁₃, X₃: 2⋅X₃ {O(n)}
t₁₀₈, X₀: X₂+X₃ {O(n)}
t₁₀₈, X₁: X₂+X₃ {O(n)}
t₁₀₈, X₂: X₂ {O(n)}
t₁₀₈, X₃: X₃ {O(n)}
t₁₀₉, X₀: X₃ {O(n)}
t₁₀₉, X₁: X₂ {O(n)}
t₁₀₉, X₂: X₂ {O(n)}
t₁₀₉, X₃: X₃ {O(n)}