Initial Problem

Start: l0
Program_Vars: X₀, X₁
Temp_Vars:
Locations: l0, l1, l2, l3, l4, l5, l6, l7
Transitions:
t₀: l0(X₀, X₁) → l6(X₀, X₁)
t₉: l1(X₀, X₁) → l3(X₀-X₁, X₁)
t₁₀: l2(X₀, X₁) → l3(X₀, X₁-X₀)
t₄: l3(X₀, X₁) → l4(X₀, X₁) :|: X₁+1 ≤ X₀
t₅: l3(X₀, X₁) → l4(X₀, X₁) :|: X₀+1 ≤ X₁
t₆: l3(X₀, X₁) → l5(X₀, X₁) :|: X₁ ≤ X₀ ∧ X₀ ≤ X₁
t₇: l4(X₀, X₁) → l1(X₀, X₁) :|: X₁+1 ≤ X₀
t₈: l4(X₀, X₁) → l2(X₀, X₁) :|: X₀ ≤ X₁
t₁₁: l5(X₀, X₁) → l7(X₀, X₁)
t₃: l6(X₀, X₁) → l3(X₁, X₀) :|: 1 ≤ X₀ ∧ 1 ≤ X₁
t₁: l6(X₀, X₁) → l5(X₀, X₁) :|: X₀ ≤ 0
t₂: l6(X₀, X₁) → l5(X₀, X₁) :|: X₁ ≤ 0

Preprocessing

Found invariant 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ X₀ ≤ X₁ for location l2

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

Found invariant 1 ≤ X₀+X₁ for location l4

Found invariant 1 ≤ X₀+X₁ for location l3

Problem after Preprocessing

Start: l0
Program_Vars: X₀, X₁
Temp_Vars:
Locations: l0, l1, l2, l3, l4, l5, l6, l7
Transitions:
t₀: l0(X₀, X₁) → l6(X₀, X₁)
t₉: l1(X₀, X₁) → l3(X₀-X₁, X₁) :|: 1+X₁ ≤ X₀ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀
t₁₀: l2(X₀, X₁) → l3(X₀, X₁-X₀) :|: 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ X₀ ≤ X₁
t₄: l3(X₀, X₁) → l4(X₀, X₁) :|: X₁+1 ≤ X₀ ∧ 1 ≤ X₀+X₁
t₅: l3(X₀, X₁) → l4(X₀, X₁) :|: X₀+1 ≤ X₁ ∧ 1 ≤ X₀+X₁
t₆: l3(X₀, X₁) → l5(X₀, X₁) :|: X₁ ≤ X₀ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀+X₁
t₇: l4(X₀, X₁) → l1(X₀, X₁) :|: X₁+1 ≤ X₀ ∧ 1 ≤ X₀+X₁
t₈: l4(X₀, X₁) → l2(X₀, X₁) :|: X₀ ≤ X₁ ∧ 1 ≤ X₀+X₁
t₁₁: l5(X₀, X₁) → l7(X₀, X₁)
t₃: l6(X₀, X₁) → l3(X₁, X₀) :|: 1 ≤ X₀ ∧ 1 ≤ X₁
t₁: l6(X₀, X₁) → l5(X₀, X₁) :|: X₀ ≤ 0
t₂: l6(X₀, X₁) → l5(X₀, X₁) :|: X₁ ≤ 0

Analysing control-flow refined program

Found invariant 2 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 1 ≤ X₀ for location n_l4___3

Found invariant 1+X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 2 ≤ X₀ for location n_l4___4

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

Found invariant 2 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 1 ≤ X₀ for location n_l2___1

Found invariant 2 ≤ X₀+X₁ for location l3

MPRF for transition t₃₉₄: l3(X₀, X₁) → n_l4___3(X₀, X₁) :|: 1 ≤ X₀+X₁ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+2⋅X₁ ∧ 1 ≤ 2⋅X₀+X₁ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 2 ≤ X₀+X₁ of depth 1:

new bound:

X₀+1 {O(n)}

MPRF:

l3 [X₁+1 ]
n_l4___3 [X₁ ]
n_l2___1 [X₁+1-X₀ ]
n_l4___4 [X₁+1 ]
n_l1___2 [X₁+1 ]

MPRF for transition t₃₉₅: l3(X₀, X₁) → n_l4___4(X₀, X₁) :|: 1 ≤ X₀+X₁ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+2⋅X₁ ∧ 1 ≤ 2⋅X₀+X₁ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₁ ≤ X₀ ∧ 2 ≤ X₀+X₁ of depth 1:

new bound:

X₁+1 {O(n)}

MPRF:

l3 [X₀+1 ]
n_l4___3 [X₀+1 ]
n_l2___1 [X₀+1 ]
n_l4___4 [X₀+1-X₁ ]
n_l1___2 [X₀+1-X₁ ]

MPRF for transition t₃₉₂: n_l1___2(X₀, X₁) → l3(X₀-X₁, X₁) :|: 1+X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₁ ≤ X₀ ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 2 ≤ X₀ of depth 1:

new bound:

X₁+1 {O(n)}

MPRF:

l3 [X₀-1 ]
n_l4___3 [X₀-1 ]
n_l2___1 [X₀-1 ]
n_l4___4 [X₀-1 ]
n_l1___2 [X₀-1 ]

MPRF for transition t₃₉₃: n_l2___1(X₀, X₁) → l3(X₀, X₁-X₀) :|: 1+X₀ ≤ X₁ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₁ ∧ X₀ ≤ X₁ ∧ 2 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 1 ≤ X₀ of depth 1:

new bound:

X₀ {O(n)}

MPRF:

l3 [X₁ ]
n_l4___3 [X₁ ]
n_l2___1 [X₁+1-X₀ ]
n_l4___4 [X₁ ]
n_l1___2 [X₁ ]

MPRF for transition t₃₉₆: n_l4___3(X₀, X₁) → n_l2___1(X₀, X₁) :|: 1+X₀ ≤ X₁ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 2 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 1 ≤ X₀ of depth 1:

new bound:

X₀ {O(n)}

MPRF:

l3 [X₁ ]
n_l4___3 [X₁ ]
n_l2___1 [X₁-X₀ ]
n_l4___4 [X₁ ]
n_l1___2 [X₁ ]

MPRF for transition t₃₉₇: n_l4___4(X₀, X₁) → n_l1___2(X₀, X₁) :|: 1+X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₁ ≤ X₀ ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 2 ≤ X₀ of depth 1:

new bound:

X₀+X₁+2 {O(n)}

MPRF:

l3 [X₀+X₁-2 ]
n_l4___3 [X₀+X₁-2 ]
n_l2___1 [X₀+X₁-2 ]
n_l4___4 [X₀-1 ]
n_l1___2 [X₀-2 ]

CFR: Improvement to new bound with the following program:

new bound:

3⋅X₁+4⋅X₀+5 {O(n)}

cfr-program:

Start: l0
Program_Vars: X₀, X₁
Temp_Vars:
Locations: l0, l3, l5, l6, l7, n_l1___2, n_l2___1, n_l4___3, n_l4___4
Transitions:
t₀: l0(X₀, X₁) → l6(X₀, X₁)
t₆: l3(X₀, X₁) → l5(X₀, X₁) :|: X₁ ≤ X₀ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₁
t₃₉₄: l3(X₀, X₁) → n_l4___3(X₀, X₁) :|: 1 ≤ X₀+X₁ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+2⋅X₁ ∧ 1 ≤ 2⋅X₀+X₁ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 2 ≤ X₀+X₁
t₃₉₅: l3(X₀, X₁) → n_l4___4(X₀, X₁) :|: 1 ≤ X₀+X₁ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+2⋅X₁ ∧ 1 ≤ 2⋅X₀+X₁ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₁ ≤ X₀ ∧ 2 ≤ X₀+X₁
t₁₁: l5(X₀, X₁) → l7(X₀, X₁)
t₃: l6(X₀, X₁) → l3(X₁, X₀) :|: 1 ≤ X₀ ∧ 1 ≤ X₁
t₁: l6(X₀, X₁) → l5(X₀, X₁) :|: X₀ ≤ 0
t₂: l6(X₀, X₁) → l5(X₀, X₁) :|: X₁ ≤ 0
t₃₉₂: n_l1___2(X₀, X₁) → l3(X₀-X₁, X₁) :|: 1+X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₁ ≤ X₀ ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 2 ≤ X₀
t₃₉₃: n_l2___1(X₀, X₁) → l3(X₀, X₁-X₀) :|: 1+X₀ ≤ X₁ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₁ ∧ X₀ ≤ X₁ ∧ 2 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 1 ≤ X₀
t₃₉₆: n_l4___3(X₀, X₁) → n_l2___1(X₀, X₁) :|: 1+X₀ ≤ X₁ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 2 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 1 ≤ X₀
t₃₉₇: n_l4___4(X₀, X₁) → n_l1___2(X₀, X₁) :|: 1+X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₁ ≤ X₀ ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 2 ≤ X₀

All Bounds

Timebounds

Overall timebound:3⋅X₁+4⋅X₀+11 {O(n)}
t₀: 1 {O(1)}
t₆: 1 {O(1)}
t₃₉₄: X₀+1 {O(n)}
t₃₉₅: X₁+1 {O(n)}
t₁₁: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 1 {O(1)}
t₃: 1 {O(1)}
t₃₉₂: X₁+1 {O(n)}
t₃₉₃: X₀ {O(n)}
t₃₉₆: X₀ {O(n)}
t₃₉₇: X₀+X₁+2 {O(n)}

Costbounds

Overall costbound: 3⋅X₁+4⋅X₀+11 {O(n)}
t₀: 1 {O(1)}
t₆: 1 {O(1)}
t₃₉₄: X₀+1 {O(n)}
t₃₉₅: X₁+1 {O(n)}
t₁₁: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 1 {O(1)}
t₃: 1 {O(1)}
t₃₉₂: X₁+1 {O(n)}
t₃₉₃: X₀ {O(n)}
t₃₉₆: X₀ {O(n)}
t₃₉₇: X₀+X₁+2 {O(n)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₆, X₀: 5⋅X₁ {O(n)}
t₆, X₁: 5⋅X₀ {O(n)}
t₃₉₄, X₀: 2⋅X₁ {O(n)}
t₃₉₄, X₁: 2⋅X₀ {O(n)}
t₃₉₅, X₀: 2⋅X₁ {O(n)}
t₃₉₅, X₁: 2⋅X₀ {O(n)}
t₁₁, X₀: 2⋅X₀+5⋅X₁ {O(n)}
t₁₁, X₁: 2⋅X₁+5⋅X₀ {O(n)}
t₁, X₀: X₀ {O(n)}
t₁, X₁: X₁ {O(n)}
t₂, X₀: X₀ {O(n)}
t₂, X₁: X₁ {O(n)}
t₃, X₀: X₁ {O(n)}
t₃, X₁: X₀ {O(n)}
t₃₉₂, X₀: 2⋅X₁ {O(n)}
t₃₉₂, X₁: 2⋅X₀ {O(n)}
t₃₉₃, X₀: 2⋅X₁ {O(n)}
t₃₉₃, X₁: 2⋅X₀ {O(n)}
t₃₉₆, X₀: 2⋅X₁ {O(n)}
t₃₉₆, X₁: 2⋅X₀ {O(n)}
t₃₉₇, X₀: 2⋅X₁ {O(n)}
t₃₉₇, X₁: 2⋅X₀ {O(n)}