Initial Problem

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

Preprocessing

Found invariant X₁ ≤ X₀ ∧ X₀ ≤ X₁ for location l5

Found invariant X₁ ≤ X₀ ∧ X₀ ≤ 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) → l2(X₀, X₁, X₂)
t₂: l1(X₀, X₁, X₂) → l3(X₀, X₁, X₂) :|: X₀ < X₁
t₃: l1(X₀, X₁, X₂) → l3(X₀, X₁, X₂) :|: X₁ < X₀
t₄: l1(X₀, X₁, X₂) → l4(X₀, X₁, X₂) :|: X₀ ≤ X₁ ∧ X₁ ≤ X₀
t₁: l2(X₀, X₁, X₂) → l1(X₂, X₁, X₂)
t₅: l3(X₀, X₁, X₂) → l1(X₀-1, X₁, X₂) :|: X₁ < X₀
t₆: l3(X₀, X₁, X₂) → l1(X₀+1, X₁, X₂) :|: X₀ ≤ X₁
t₇: l4(X₀, X₁, X₂) → l5(X₀, X₁, X₂) :|: X₁ ≤ X₀ ∧ X₀ ≤ X₁

MPRF for transition t₂: l1(X₀, X₁, X₂) → l3(X₀, X₁, X₂) :|: X₀ < X₁ of depth 1:

new bound:

X₁+X₂ {O(n)}

MPRF:

l3 [X₁-X₀-1 ]
l1 [X₁-X₀ ]

MPRF for transition t₆: l3(X₀, X₁, X₂) → l1(X₀+1, X₁, X₂) :|: X₀ ≤ X₁ of depth 1:

new bound:

X₁+X₂+1 {O(n)}

MPRF:

l3 [X₁+1-X₀ ]
l1 [X₁+1-X₀ ]

MPRF for transition t₃: l1(X₀, X₁, X₂) → l3(X₀, X₁, X₂) :|: X₁ < X₀ of depth 1:

new bound:

2⋅X₁⋅X₁+2⋅X₂⋅X₂+4⋅X₁⋅X₂+6⋅X₁+6⋅X₂+5 {O(n^2)}

MPRF:

l3 [X₀+1-X₁ ]
l1 [X₀+2-X₁ ]

MPRF for transition t₅: l3(X₀, X₁, X₂) → l1(X₀-1, X₁, X₂) :|: X₁ < X₀ of depth 1:

new bound:

2⋅X₁⋅X₁+2⋅X₂⋅X₂+4⋅X₁⋅X₂+5⋅X₁+5⋅X₂+3 {O(n^2)}

MPRF:

l3 [X₀+1-X₁ ]
l1 [X₀+1-X₁ ]

Analysing control-flow refined program

Cut unsatisfiable transition t₇₅: n_l1___2→n_l3___4

Cut unsatisfiable transition t₇₆: n_l1___5→n_l3___3

Cut unreachable locations [n_l3___3] from the program graph

Found invariant 2+X₂ ≤ X₁ ∧ 1+X₂ ≤ X₀ ∧ 1+X₀ ≤ X₁ for location n_l3___4

Found invariant X₂ ≤ X₀ ∧ 1+X₁ ≤ X₂ ∧ X₀ ≤ X₂ ∧ 1+X₁ ≤ X₀ for location n_l3___6

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

Found invariant X₁ ≤ X₀ ∧ X₀ ≤ X₁ for location l5

Found invariant 1+X₂ ≤ X₁ ∧ 1+X₂ ≤ X₀ ∧ X₀ ≤ X₁ for location n_l1___5

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

Found invariant X₁ ≤ X₀ ∧ X₀ ≤ X₁ for location l4

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

Found invariant 1+X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₂ ∧ 1+X₀ ≤ X₁ for location n_l3___7

MPRF for transition t₇₄: n_l1___2(X₀, X₁, X₂) → n_l3___1(X₀, X₁, X₂) :|: X₁ < X₀ ∧ 1+X₁ ≤ X₂ ∧ 1+X₀ ≤ X₂ ∧ X₁ ≤ X₀ of depth 1:

new bound:

X₁+X₂+2 {O(n)}

MPRF:

n_l3___1 [X₀-X₁ ]
n_l1___2 [X₀+1-X₁ ]

MPRF for transition t₈₀: n_l3___1(X₀, X₁, X₂) → n_l1___2(X₀-1, X₁, X₂) :|: X₁ < X₀ ∧ X₁ < X₀ ∧ 2+X₁ ≤ X₂ ∧ 1+X₀ ≤ X₂ ∧ 1+X₁ ≤ X₀ of depth 1:

new bound:

X₁+X₂+1 {O(n)}

MPRF:

n_l3___1 [X₀-X₁ ]
n_l1___2 [X₀-X₁ ]

MPRF for transition t₇₇: n_l1___5(X₀, X₁, X₂) → n_l3___4(X₀, X₁, X₂) :|: X₀ ≤ 1+X₁ ∧ X₀ < X₁ ∧ 1+X₂ ≤ X₁ ∧ 1+X₂ ≤ X₀ ∧ X₀ ≤ X₁ of depth 1:

new bound:

X₁+X₂+2 {O(n)}

MPRF:

n_l3___4 [X₁-X₀ ]
n_l1___5 [X₁+1-X₀ ]

MPRF for transition t₈₂: n_l3___4(X₀, X₁, X₂) → n_l1___5(X₀+1, X₁, X₂) :|: X₀ < X₁ ∧ X₀ ≤ X₁ ∧ 2+X₂ ≤ X₁ ∧ 1+X₂ ≤ X₀ ∧ 1+X₀ ≤ X₁ of depth 1:

new bound:

X₁+X₂+2 {O(n)}

MPRF:

n_l3___4 [X₁+1-X₀ ]
n_l1___5 [X₁+1-X₀ ]

CFR: Improvement to new bound with the following program:

new bound:

4⋅X₁+4⋅X₂+7 {O(n)}

cfr-program:

Start: l0
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: l0, l1, l2, l4, l5, n_l1___2, n_l1___5, n_l3___1, n_l3___4, n_l3___6, n_l3___7
Transitions:
t₀: l0(X₀, X₁, X₂) → l2(X₀, X₁, X₂)
t₄: l1(X₀, X₁, X₂) → l4(X₀, X₁, X₂) :|: X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₂
t₇₈: l1(X₀, X₁, X₂) → n_l3___6(X₀, X₁, X₂) :|: X₀ ≤ X₂ ∧ X₂ ≤ X₀ ∧ X₁ < X₀ ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₂
t₇₉: l1(X₀, X₁, X₂) → n_l3___7(X₀, X₁, X₂) :|: X₀ ≤ X₂ ∧ X₂ ≤ X₀ ∧ X₀ < X₁ ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₂
t₁: l2(X₀, X₁, X₂) → l1(X₂, X₁, X₂)
t₇: l4(X₀, X₁, X₂) → l5(X₀, X₁, X₂) :|: X₁ ≤ X₀ ∧ X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁
t₉₅: n_l1___2(X₀, X₁, X₂) → l4(X₀, X₁, X₂) :|: X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ 1+X₁ ≤ X₂ ∧ 1+X₀ ≤ X₂ ∧ X₁ ≤ X₀
t₇₄: n_l1___2(X₀, X₁, X₂) → n_l3___1(X₀, X₁, X₂) :|: X₁ < X₀ ∧ 1+X₁ ≤ X₂ ∧ 1+X₀ ≤ X₂ ∧ X₁ ≤ X₀
t₉₆: n_l1___5(X₀, X₁, X₂) → l4(X₀, X₁, X₂) :|: X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ 1+X₂ ≤ X₁ ∧ 1+X₂ ≤ X₀ ∧ X₀ ≤ X₁
t₇₇: n_l1___5(X₀, X₁, X₂) → n_l3___4(X₀, X₁, X₂) :|: X₀ ≤ 1+X₁ ∧ X₀ < X₁ ∧ 1+X₂ ≤ X₁ ∧ 1+X₂ ≤ X₀ ∧ X₀ ≤ X₁
t₈₀: n_l3___1(X₀, X₁, X₂) → n_l1___2(X₀-1, X₁, X₂) :|: X₁ < X₀ ∧ X₁ < X₀ ∧ 2+X₁ ≤ X₂ ∧ 1+X₀ ≤ X₂ ∧ 1+X₁ ≤ X₀
t₈₂: n_l3___4(X₀, X₁, X₂) → n_l1___5(X₀+1, X₁, X₂) :|: X₀ < X₁ ∧ X₀ ≤ X₁ ∧ 2+X₂ ≤ X₁ ∧ 1+X₂ ≤ X₀ ∧ 1+X₀ ≤ X₁
t₈₃: n_l3___6(X₀, X₁, X₂) → n_l1___2(X₀-1, X₁, X₂) :|: X₁ < X₀ ∧ X₀ ≤ X₂ ∧ X₂ ≤ X₀ ∧ X₁ < X₀ ∧ X₂ ≤ X₀ ∧ 1+X₁ ≤ X₂ ∧ X₀ ≤ X₂ ∧ 1+X₁ ≤ X₀
t₈₄: n_l3___7(X₀, X₁, X₂) → n_l1___5(X₀+1, X₁, X₂) :|: X₀ < X₁ ∧ X₀ ≤ X₂ ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₁ ∧ 1+X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₂ ∧ 1+X₀ ≤ X₁

All Bounds

Timebounds

Overall timebound:4⋅X₁+4⋅X₂+17 {O(n)}
t₀: 1 {O(1)}
t₄: 1 {O(1)}
t₇₈: 1 {O(1)}
t₇₉: 1 {O(1)}
t₁: 1 {O(1)}
t₇: 1 {O(1)}
t₇₄: X₁+X₂+2 {O(n)}
t₉₅: 1 {O(1)}
t₇₇: X₁+X₂+2 {O(n)}
t₉₆: 1 {O(1)}
t₈₀: X₁+X₂+1 {O(n)}
t₈₂: X₁+X₂+2 {O(n)}
t₈₃: 1 {O(1)}
t₈₄: 1 {O(1)}

Costbounds

Overall costbound: 4⋅X₁+4⋅X₂+17 {O(n)}
t₀: 1 {O(1)}
t₄: 1 {O(1)}
t₇₈: 1 {O(1)}
t₇₉: 1 {O(1)}
t₁: 1 {O(1)}
t₇: 1 {O(1)}
t₇₄: X₁+X₂+2 {O(n)}
t₉₅: 1 {O(1)}
t₇₇: X₁+X₂+2 {O(n)}
t₉₆: 1 {O(1)}
t₈₀: X₁+X₂+1 {O(n)}
t₈₂: X₁+X₂+2 {O(n)}
t₈₃: 1 {O(1)}
t₈₄: 1 {O(1)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₄, X₀: X₂ {O(n)}
t₄, X₁: X₁ {O(n)}
t₄, X₂: X₂ {O(n)}
t₇₈, X₀: X₂ {O(n)}
t₇₈, X₁: X₁ {O(n)}
t₇₈, X₂: X₂ {O(n)}
t₇₉, X₀: X₂ {O(n)}
t₇₉, X₁: X₁ {O(n)}
t₇₉, X₂: X₂ {O(n)}
t₁, X₀: X₂ {O(n)}
t₁, X₁: X₁ {O(n)}
t₁, X₂: X₂ {O(n)}
t₇, X₀: 2⋅X₁+7⋅X₂+7 {O(n)}
t₇, X₁: 5⋅X₁ {O(n)}
t₇, X₂: 5⋅X₂ {O(n)}
t₇₄, X₀: 2⋅X₂+X₁+2 {O(n)}
t₇₄, X₁: X₁ {O(n)}
t₇₄, X₂: X₂ {O(n)}
t₉₅, X₀: 3⋅X₂+X₁+3 {O(n)}
t₉₅, X₁: 2⋅X₁ {O(n)}
t₉₅, X₂: 2⋅X₂ {O(n)}
t₇₇, X₀: 2⋅X₂+X₁+3 {O(n)}
t₇₇, X₁: X₁ {O(n)}
t₇₇, X₂: X₂ {O(n)}
t₉₆, X₀: 3⋅X₂+X₁+4 {O(n)}
t₉₆, X₁: 2⋅X₁ {O(n)}
t₉₆, X₂: 2⋅X₂ {O(n)}
t₈₀, X₀: 2⋅X₂+X₁+2 {O(n)}
t₈₀, X₁: X₁ {O(n)}
t₈₀, X₂: X₂ {O(n)}
t₈₂, X₀: 2⋅X₂+X₁+3 {O(n)}
t₈₂, X₁: X₁ {O(n)}
t₈₂, X₂: X₂ {O(n)}
t₈₃, X₀: X₂+1 {O(n)}
t₈₃, X₁: X₁ {O(n)}
t₈₃, X₂: X₂ {O(n)}
t₈₄, X₀: X₂+1 {O(n)}
t₈₄, X₁: X₁ {O(n)}
t₈₄, X₂: X₂ {O(n)}