Initial Problem

Start: l0
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars:
Locations: l0, l1, l2
Transitions:
t₀: l0(X₀, X₁, X₂, X₃) → l1(X₁, X₁, X₃, X₃)
t₁: l1(X₀, X₁, X₂, X₃) → l1(X₀-1, X₁, X₂-1, X₃) :|: X₀+1 ≤ 0
t₂: l1(X₀, X₁, X₂, X₃) → l1(X₀-1, X₁, X₂-1, X₃) :|: 1 ≤ X₀
t₃: l1(X₀, X₁, X₂, X₃) → l2(0, X₁, X₂, X₃) :|: X₁+1 ≤ X₃ ∧ X₀ ≤ 0 ∧ 0 ≤ X₀
t₄: l1(X₀, X₁, X₂, X₃) → l2(0, X₁, X₂, X₃) :|: 1+X₃ ≤ X₁ ∧ X₀ ≤ 0 ∧ 0 ≤ X₀
t₅: l1(X₀, X₁, X₂, X₃) → l2(0, X₁, X₂, X₁) :|: X₀ ≤ 0 ∧ 0 ≤ X₀ ∧ X₁ ≤ X₃ ∧ X₃ ≤ X₁

Preprocessing

Eliminate variables {X₂} that do not contribute to the problem

Found invariant 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ X₀ ≤ 0 ∧ 0 ≤ X₀ for location l2

Found invariant X₀ ≤ X₁ for location l1

Problem after Preprocessing

Start: l0
Program_Vars: X₀, X₁, X₃
Temp_Vars:
Locations: l0, l1, l2
Transitions:
t₁₆: l0(X₀, X₁, X₃) → l1(X₁, X₁, X₃)
t₁₇: l1(X₀, X₁, X₃) → l1(X₀-1, X₁, X₃) :|: X₀+1 ≤ 0 ∧ X₀ ≤ X₁
t₁₈: l1(X₀, X₁, X₃) → l1(X₀-1, X₁, X₃) :|: 1 ≤ X₀ ∧ X₀ ≤ X₁
t₁₉: l1(X₀, X₁, X₃) → l2(0, X₁, X₃) :|: X₁+1 ≤ X₃ ∧ X₀ ≤ 0 ∧ 0 ≤ X₀ ∧ X₀ ≤ X₁
t₂₀: l1(X₀, X₁, X₃) → l2(0, X₁, X₃) :|: 1+X₃ ≤ X₁ ∧ X₀ ≤ 0 ∧ 0 ≤ X₀ ∧ X₀ ≤ X₁
t₂₁: l1(X₀, X₁, X₃) → l2(0, X₁, X₁) :|: X₀ ≤ 0 ∧ 0 ≤ X₀ ∧ X₁ ≤ X₃ ∧ X₃ ≤ X₁ ∧ X₀ ≤ X₁

MPRF for transition t₁₈: l1(X₀, X₁, X₃) → l1(X₀-1, X₁, X₃) :|: 1 ≤ X₀ ∧ X₀ ≤ X₁ of depth 1:

new bound:

X₁ {O(n)}

MPRF:

l1 [X₀ ]

Found invariant 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ X₀ ≤ 0 ∧ 0 ≤ X₀ for location l2

Found invariant X₀ ≤ X₁ for location l1

Analysing control-flow refined program

Cut unsatisfiable transition t₈₉: n_l1___2→l2

Cut unsatisfiable transition t₉₁: n_l1___2→l2

Cut unsatisfiable transition t₉₃: n_l1___2→l2

Found invariant 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ X₀ ≤ 0 ∧ 0 ≤ X₀ for location l2

Found invariant 1+X₁ ≤ 0 ∧ 3+X₀+X₁ ≤ 0 ∧ 1+X₀ ≤ X₁ ∧ 2+X₀ ≤ 0 for location n_l1___2

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

Found invariant 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 0 ≤ X₀ for location n_l1___1

MPRF for transition t₇₄: n_l1___1(X₀, X₁, X₃) → n_l1___1(X₀-1, X₁, X₃) :|: X₀ ≤ X₁ ∧ 0 ≤ X₀ ∧ 1+X₀ ≤ X₁ ∧ 1 ≤ X₀ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 0 ≤ X₀ of depth 1:

new bound:

X₁+1 {O(n)}

MPRF:

n_l1___1 [X₀+1 ]

Found invariant 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ X₀ ≤ 0 ∧ 0 ≤ X₀ for location l2

Found invariant 1+X₁ ≤ 0 ∧ 3+X₀+X₁ ≤ 0 ∧ 1+X₀ ≤ X₁ ∧ 2+X₀ ≤ 0 for location n_l1___2

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

Found invariant 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 0 ≤ X₀ for location n_l1___1

CFR did not improve the program. Rolling back

All Bounds

Timebounds

Overall timebound:inf {Infinity}
t₁₆: 1 {O(1)}
t₁₇: inf {Infinity}
t₁₈: X₁ {O(n)}
t₁₉: 1 {O(1)}
t₂₀: 1 {O(1)}
t₂₁: 1 {O(1)}

Costbounds

Overall costbound: inf {Infinity}
t₁₆: 1 {O(1)}
t₁₇: inf {Infinity}
t₁₈: X₁ {O(n)}
t₁₉: 1 {O(1)}
t₂₀: 1 {O(1)}
t₂₁: 1 {O(1)}

Sizebounds

t₁₆, X₀: X₁ {O(n)}
t₁₆, X₁: X₁ {O(n)}
t₁₆, X₃: X₃ {O(n)}
t₁₇, X₁: X₁ {O(n)}
t₁₇, X₃: X₃ {O(n)}
t₁₈, X₀: X₁ {O(n)}
t₁₈, X₁: X₁ {O(n)}
t₁₈, X₃: X₃ {O(n)}
t₁₉, X₀: 0 {O(1)}
t₁₉, X₁: 2⋅X₁ {O(n)}
t₁₉, X₃: 2⋅X₃ {O(n)}
t₂₀, X₀: 0 {O(1)}
t₂₀, X₁: 2⋅X₁ {O(n)}
t₂₀, X₃: 2⋅X₃ {O(n)}
t₂₁, X₀: 0 {O(1)}
t₂₁, X₁: 2⋅X₁ {O(n)}
t₂₁, X₃: 2⋅X₁ {O(n)}