Initial Problem

Start: l0
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: l0, l1, l2, l3, l4, l5
Transitions:
t₀: l0(X₀, X₁, X₂) → l3(X₀, X₁, X₂)
t₅: l1(X₀, X₁, X₂) → l2(X₀, X₁, X₂) :|: X₀ ≤ X₁ ∧ X₁ ≤ X₀
t₃: l1(X₀, X₁, X₂) → l4(X₀, X₁, X₂) :|: X₀ < X₁
t₄: l1(X₀, X₁, X₂) → l4(X₀, X₁, X₂) :|: X₁ < X₀
t₈: l2(X₀, X₁, X₂) → l5(X₀, X₁, X₂)
t₁: l3(X₀, X₁, X₂) → l1(X₂, X₁, X₂) :|: 0 < X₁
t₂: l3(X₀, X₁, X₂) → l2(X₀, X₁, X₂) :|: X₁ ≤ 0
t₆: l4(X₀, X₁, X₂) → l1(0, X₁, X₂) :|: X₁ < X₀
t₇: l4(X₀, X₁, X₂) → l1(X₀+1, X₁, X₂) :|: X₀ ≤ X₁

Preprocessing

Found invariant 1 ≤ X₁ for location l1

Found invariant 1 ≤ X₁ for location l4

Problem after Preprocessing

Start: l0
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: l0, l1, l2, l3, l4, l5
Transitions:
t₀: l0(X₀, X₁, X₂) → l3(X₀, X₁, X₂)
t₅: l1(X₀, X₁, X₂) → l2(X₀, X₁, X₂) :|: X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁
t₃: l1(X₀, X₁, X₂) → l4(X₀, X₁, X₂) :|: X₀ < X₁ ∧ 1 ≤ X₁
t₄: l1(X₀, X₁, X₂) → l4(X₀, X₁, X₂) :|: X₁ < X₀ ∧ 1 ≤ X₁
t₈: l2(X₀, X₁, X₂) → l5(X₀, X₁, X₂)
t₁: l3(X₀, X₁, X₂) → l1(X₂, X₁, X₂) :|: 0 < X₁
t₂: l3(X₀, X₁, X₂) → l2(X₀, X₁, X₂) :|: X₁ ≤ 0
t₆: l4(X₀, X₁, X₂) → l1(0, X₁, X₂) :|: X₁ < X₀ ∧ 1 ≤ X₁
t₇: l4(X₀, X₁, X₂) → l1(X₀+1, X₁, X₂) :|: X₀ ≤ X₁ ∧ 1 ≤ X₁

Analysing control-flow refined program

Cut unsatisfiable transition t₂₂₉: n_l1___2→l2

Cut unsatisfiable transition t₂₁₁: n_l1___4→n_l4___3

Cut unreachable locations [n_l4___3] from the program graph

Found invariant 1 ≤ X₁ ∧ 1+X₀ ≤ X₁ for location n_l4___6

Found invariant 1 ≤ X₁ ∧ X₀ ≤ X₁ for location n_l1___4

Found invariant 2 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ 2+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ X₀ ≤ 0 ∧ 0 ≤ X₀ for location n_l1___2

Found invariant X₂ ≤ X₀ ∧ 2 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 4 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 2 ≤ X₀ for location n_l4___5

Found invariant 2 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ 2+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ X₀ ≤ 0 ∧ 0 ≤ X₀ for location n_l4___1

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

MPRF for transition t₂₁₂: n_l1___4(X₀, X₁, X₂) → n_l4___6(X₀, X₁, X₂) :|: 1 ≤ X₁ ∧ 1 ≤ X₁ ∧ X₀ ≤ 1+X₁ ∧ 1 ≤ X₁ ∧ X₀ < X₁ ∧ 1 ≤ X₁ ∧ X₀ ≤ X₁ of depth 1:

new bound:

2⋅X₁+X₂+2 {O(n)}

MPRF:

n_l4___6 [X₁-X₀ ]
n_l1___4 [X₁+1-X₀ ]

MPRF for transition t₂₁₈: n_l4___6(X₀, X₁, X₂) → n_l1___4(X₀+1, X₁, X₂) :|: X₀ < X₁ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₁ ∧ 1+X₀ ≤ X₁ of depth 1:

new bound:

2⋅X₁+X₂+3 {O(n)}

MPRF:

n_l4___6 [X₁+1-X₀ ]
n_l1___4 [X₁+1-X₀ ]

CFR: Improvement to new bound with the following program:

new bound:

2⋅X₂+4⋅X₁+5 {O(n)}

cfr-program:

Start: l0
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: l0, l1, l2, l3, l5, n_l1___2, n_l1___4, n_l4___1, n_l4___5, n_l4___6
Transitions:
t₀: l0(X₀, X₁, X₂) → l3(X₀, X₁, X₂)
t₅: l1(X₀, X₁, X₂) → l2(X₀, X₁, X₂) :|: X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁
t₂₁₃: l1(X₀, X₁, X₂) → n_l4___5(X₀, X₁, X₂) :|: X₁ < X₀ ∧ 1 ≤ X₁ ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁
t₂₁₄: l1(X₀, X₁, X₂) → n_l4___6(X₀, X₁, X₂) :|: 1 ≤ X₁ ∧ X₀ < X₁ ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁
t₈: l2(X₀, X₁, X₂) → l5(X₀, X₁, X₂)
t₁: l3(X₀, X₁, X₂) → l1(X₂, X₁, X₂) :|: 0 < X₁
t₂: l3(X₀, X₁, X₂) → l2(X₀, X₁, X₂) :|: X₁ ≤ 0
t₂₁₀: n_l1___2(X₀, X₁, X₂) → n_l4___1(X₀, X₁, X₂) :|: X₀ < X₁ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁ ∧ 1+X₀ ≤ X₁ ∧ X₀ ≤ 0 ∧ 0 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁ ∧ X₀ ≤ 1+X₁ ∧ 1 ≤ X₁ ∧ X₀ < X₁ ∧ 2 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ 2+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ X₀ ≤ 0 ∧ 0 ≤ X₀
t₂₃₀: n_l1___4(X₀, X₁, X₂) → l2(X₀, X₁, X₂) :|: X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁ ∧ X₀ ≤ X₁
t₂₁₂: n_l1___4(X₀, X₁, X₂) → n_l4___6(X₀, X₁, X₂) :|: 1 ≤ X₁ ∧ 1 ≤ X₁ ∧ X₀ ≤ 1+X₁ ∧ 1 ≤ X₁ ∧ X₀ < X₁ ∧ 1 ≤ X₁ ∧ X₀ ≤ X₁
t₂₁₅: n_l4___1(X₀, X₁, X₂) → n_l1___4(X₀+1, X₁, X₂) :|: 1 ≤ X₁ ∧ X₀ ≤ 0 ∧ 0 ≤ X₀ ∧ 1 ≤ X₁ ∧ X₀ ≤ X₁ ∧ 2 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ 2+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ X₀ ≤ 0 ∧ 0 ≤ X₀
t₂₁₇: n_l4___5(X₀, X₁, X₂) → n_l1___2(0, X₁, X₂) :|: X₁ < X₀ ∧ 1 ≤ X₁ ∧ X₁ < X₀ ∧ 1 ≤ X₁ ∧ X₂ ≤ X₀ ∧ 2 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 4 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 2 ≤ X₀
t₂₁₈: n_l4___6(X₀, X₁, X₂) → n_l1___4(X₀+1, X₁, X₂) :|: X₀ < X₁ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₁ ∧ 1+X₀ ≤ X₁

All Bounds

Timebounds

Overall timebound:2⋅X₂+4⋅X₁+16 {O(n)}
t₀: 1 {O(1)}
t₅: 1 {O(1)}
t₂₁₃: 1 {O(1)}
t₂₁₄: 1 {O(1)}
t₈: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 1 {O(1)}
t₂₁₀: 1 {O(1)}
t₂₁₂: 2⋅X₁+X₂+2 {O(n)}
t₂₃₀: 1 {O(1)}
t₂₁₅: 1 {O(1)}
t₂₁₇: 1 {O(1)}
t₂₁₈: 2⋅X₁+X₂+3 {O(n)}

Costbounds

Overall costbound: 2⋅X₂+4⋅X₁+16 {O(n)}
t₀: 1 {O(1)}
t₅: 1 {O(1)}
t₂₁₃: 1 {O(1)}
t₂₁₄: 1 {O(1)}
t₈: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 1 {O(1)}
t₂₁₀: 1 {O(1)}
t₂₁₂: 2⋅X₁+X₂+2 {O(n)}
t₂₃₀: 1 {O(1)}
t₂₁₅: 1 {O(1)}
t₂₁₇: 1 {O(1)}
t₂₁₈: 2⋅X₁+X₂+3 {O(n)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₅, X₀: X₂ {O(n)}
t₅, X₁: X₁ {O(n)}
t₅, X₂: X₂ {O(n)}
t₂₁₃, X₀: X₂ {O(n)}
t₂₁₃, X₁: X₁ {O(n)}
t₂₁₃, X₂: X₂ {O(n)}
t₂₁₄, X₀: X₂ {O(n)}
t₂₁₄, X₁: X₁ {O(n)}
t₂₁₄, X₂: X₂ {O(n)}
t₈, X₀: 2⋅X₁+3⋅X₂+X₀+5 {O(n)}
t₈, X₁: 5⋅X₁ {O(n)}
t₈, X₂: 5⋅X₂ {O(n)}
t₁, X₀: X₂ {O(n)}
t₁, X₁: X₁ {O(n)}
t₁, X₂: X₂ {O(n)}
t₂, X₀: X₀ {O(n)}
t₂, X₁: X₁ {O(n)}
t₂, X₂: X₂ {O(n)}
t₂₁₀, X₀: 0 {O(1)}
t₂₁₀, X₁: X₁ {O(n)}
t₂₁₀, X₂: X₂ {O(n)}
t₂₁₂, X₀: 2⋅X₁+2⋅X₂+4 {O(n)}
t₂₁₂, X₁: 2⋅X₁ {O(n)}
t₂₁₂, X₂: 2⋅X₂ {O(n)}
t₂₃₀, X₀: 2⋅X₁+2⋅X₂+5 {O(n)}
t₂₃₀, X₁: 3⋅X₁ {O(n)}
t₂₃₀, X₂: 3⋅X₂ {O(n)}
t₂₁₅, X₀: 1 {O(1)}
t₂₁₅, X₁: X₁ {O(n)}
t₂₁₅, X₂: X₂ {O(n)}
t₂₁₇, X₀: 0 {O(1)}
t₂₁₇, X₁: X₁ {O(n)}
t₂₁₇, X₂: X₂ {O(n)}
t₂₁₈, X₀: 2⋅X₁+2⋅X₂+4 {O(n)}
t₂₁₈, X₁: 2⋅X₁ {O(n)}
t₂₁₈, X₂: 2⋅X₂ {O(n)}