Initial Problem

Start: l0
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars: E
Locations: l0, l1, l2, l3
Transitions:
t₀: l0(X₀, X₁, X₂, X₃) → l1(0, X₁, X₂, X₃)
t₆: l1(X₀, X₁, X₂, X₃) → l2(X₀, X₁, X₂, X₃) :|: X₁ ≤ X₀
t₁: l1(X₀, X₁, X₂, X₃) → l3(X₀, X₁, X₀+1, X₃) :|: X₀+1 ≤ X₁
t₅: l3(X₀, X₁, X₂, X₃) → l1(X₀+1, X₁, X₂, X₃) :|: X₁ ≤ X₂
t₂: l3(X₀, X₁, X₂, X₃) → l3(X₀, X₁, X₂+1, 0) :|: X₂+1 ≤ X₁
t₃: l3(X₀, X₁, X₂, X₃) → l3(X₀, X₁-1, X₂, E) :|: X₂+1 ≤ X₁ ∧ E+1 ≤ 0
t₄: l3(X₀, X₁, X₂, X₃) → l3(X₀, X₁-1, X₂, E) :|: X₂+1 ≤ X₁ ∧ 1 ≤ E

Preprocessing

Eliminate variables {X₃} that do not contribute to the problem

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

Found invariant 0 ≤ X₀ for location l1

Found invariant X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 0 ≤ X₀ for location l3

Problem after Preprocessing

Start: l0
Program_Vars: X₀, X₁, X₂
Temp_Vars: E
Locations: l0, l1, l2, l3
Transitions:
t₁₅: l0(X₀, X₁, X₂) → l1(0, X₁, X₂)
t₁₇: l1(X₀, X₁, X₂) → l2(X₀, X₁, X₂) :|: X₁ ≤ X₀ ∧ 0 ≤ X₀
t₁₆: l1(X₀, X₁, X₂) → l3(X₀, X₁, X₀+1) :|: X₀+1 ≤ X₁ ∧ 0 ≤ X₀
t₂₁: l3(X₀, X₁, X₂) → l1(X₀+1, X₁, X₂) :|: X₁ ≤ X₂ ∧ X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 0 ≤ X₀
t₁₈: l3(X₀, X₁, X₂) → l3(X₀, X₁, X₂+1) :|: X₂+1 ≤ X₁ ∧ X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 0 ≤ X₀
t₁₉: l3(X₀, X₁, X₂) → l3(X₀, X₁-1, X₂) :|: X₂+1 ≤ X₁ ∧ E+1 ≤ 0 ∧ X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 0 ≤ X₀
t₂₀: l3(X₀, X₁, X₂) → l3(X₀, X₁-1, X₂) :|: X₂+1 ≤ X₁ ∧ 1 ≤ E ∧ X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 0 ≤ X₀

MPRF for transition t₁₆: l1(X₀, X₁, X₂) → l3(X₀, X₁, X₀+1) :|: X₀+1 ≤ X₁ ∧ 0 ≤ X₀ of depth 1:

new bound:

X₁ {O(n)}

MPRF:

l3 [X₁-X₀-1 ]
l1 [X₁-X₀ ]

MPRF for transition t₁₉: l3(X₀, X₁, X₂) → l3(X₀, X₁-1, X₂) :|: X₂+1 ≤ X₁ ∧ E+1 ≤ 0 ∧ X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 0 ≤ X₀ of depth 1:

new bound:

X₁ {O(n)}

MPRF:

l3 [X₁ ]
l1 [X₁ ]

MPRF for transition t₂₀: l3(X₀, X₁, X₂) → l3(X₀, X₁-1, X₂) :|: X₂+1 ≤ X₁ ∧ 1 ≤ E ∧ X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 0 ≤ X₀ of depth 1:

new bound:

X₁ {O(n)}

MPRF:

l3 [X₁ ]
l1 [X₁ ]

MPRF for transition t₂₁: l3(X₀, X₁, X₂) → l1(X₀+1, X₁, X₂) :|: X₁ ≤ X₂ ∧ X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 0 ≤ X₀ of depth 1:

new bound:

X₁ {O(n)}

MPRF:

l3 [X₁-X₀ ]
l1 [X₁-X₀ ]

MPRF for transition t₁₈: l3(X₀, X₁, X₂) → l3(X₀, X₁, X₂+1) :|: X₂+1 ≤ X₁ ∧ X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 0 ≤ X₀ of depth 1:

new bound:

2⋅X₁⋅X₁+X₁ {O(n^2)}

MPRF:

l1 [X₁-X₀ ]
l3 [X₁+1-X₂ ]

Analysing control-flow refined program

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

Found invariant 0 ≤ X₀ for location l1

Found invariant X₂ ≤ X₁ ∧ 2 ≤ X₂ ∧ 4 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ 2+X₀ ≤ X₂ ∧ 2 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 2+X₀ ≤ X₁ ∧ 0 ≤ X₀ for location n_l3___1

Found invariant X₂ ≤ X₁ ∧ X₂ ≤ 1+X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 0 ≤ X₀ for location l3

MPRF for transition t₈₃: l3(X₀, X₁, X₂) → n_l3___1(X₀, X₁, X₂+1) :|: 1+X₀ ≤ X₂ ∧ 0 ≤ X₀ ∧ 1+X₀ ≤ X₂ ∧ X₂ ≤ 1+X₀ ∧ 1 ≤ X₂ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₀ ∧ 1+X₀ ≤ X₂ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₀ ∧ 1+X₀ ≤ X₂ ∧ X₂ ≤ X₁ ∧ 1+X₂ ≤ X₁ ∧ 0 ≤ X₀ ∧ 1+X₀ ≤ X₂ ∧ X₂ ≤ X₁ ∧ X₂ ≤ 1+X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 0 ≤ X₀ of depth 1:

new bound:

X₁ {O(n)}

MPRF:

l3 [X₁-X₀ ]
n_l3___1 [X₁-X₀-1 ]
l1 [X₁-X₀ ]

MPRF for transition t₈₄: l3(X₀, X₁, X₂) → l3(X₀, X₁-1, X₂) :|: 1+X₀ ≤ X₂ ∧ 0 ≤ X₀ ∧ 1+X₀ ≤ X₂ ∧ X₂ ≤ 1+X₀ ∧ 1 ≤ X₂ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₀ ∧ 1+X₀ ≤ X₂ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₀ ∧ 1+X₀ ≤ X₂ ∧ X₂ ≤ X₁ ∧ 1+X₂ ≤ X₁ ∧ 1+X₀ ≤ X₂ ∧ 0 ≤ X₀ ∧ X₂ ≤ X₁ ∧ X₂ ≤ 1+X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 0 ≤ X₀ of depth 1:

new bound:

X₁ {O(n)}

MPRF:

l3 [X₁+1-X₂ ]
n_l3___1 [X₁-X₀ ]
l1 [X₁-X₀ ]

MPRF for transition t₈₅: l3(X₀, X₁, X₂) → l3(X₀, X₁-1, X₂) :|: 1+X₀ ≤ X₂ ∧ 0 ≤ X₀ ∧ 1+X₀ ≤ X₂ ∧ X₂ ≤ 1+X₀ ∧ 1 ≤ X₂ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₀ ∧ 1+X₀ ≤ X₂ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₀ ∧ 1+X₀ ≤ X₂ ∧ X₂ ≤ X₁ ∧ 1+X₂ ≤ X₁ ∧ 1+X₀ ≤ X₂ ∧ 0 ≤ X₀ ∧ X₂ ≤ X₁ ∧ X₂ ≤ 1+X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 0 ≤ X₀ of depth 1:

new bound:

X₁ {O(n)}

MPRF:

l3 [X₁+1-X₂ ]
n_l3___1 [X₁-X₀ ]
l1 [X₁-X₀ ]

MPRF for transition t₈₀: n_l3___1(X₀, X₁, X₂) → n_l3___1(X₀, X₁, X₂+1) :|: 1+X₀ ≤ X₂ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀ ∧ 2+X₀ ≤ X₂ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₀ ∧ 1+X₀ ≤ X₂ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₀ ∧ 1+X₀ ≤ X₂ ∧ X₂ ≤ X₁ ∧ 1+X₂ ≤ X₁ ∧ 0 ≤ X₀ ∧ 1+X₀ ≤ X₂ ∧ X₂ ≤ X₁ ∧ 2 ≤ X₂ ∧ 4 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ 2+X₀ ≤ X₂ ∧ 2 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 2+X₀ ≤ X₁ ∧ 0 ≤ X₀ of depth 1:

new bound:

3⋅X₁⋅X₁+2⋅X₁ {O(n^2)}

MPRF:

l3 [X₀+X₁-X₂ ]
n_l3___1 [X₁+1-X₂ ]
l1 [0 ]

MPRF for transition t₈₁: n_l3___1(X₀, X₁, X₂) → n_l3___1(X₀, X₁-1, X₂) :|: 1+X₀ ≤ X₂ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀ ∧ 2+X₀ ≤ X₂ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₀ ∧ 1+X₀ ≤ X₂ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₀ ∧ 1+X₀ ≤ X₂ ∧ X₂ ≤ X₁ ∧ 1+X₂ ≤ X₁ ∧ 1+X₀ ≤ X₂ ∧ 0 ≤ X₀ ∧ X₂ ≤ X₁ ∧ 2 ≤ X₂ ∧ 4 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ 2+X₀ ≤ X₂ ∧ 2 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 2+X₀ ≤ X₁ ∧ 0 ≤ X₀ of depth 1:

new bound:

X₁ {O(n)}

MPRF:

l3 [X₁ ]
n_l3___1 [X₁ ]
l1 [X₁ ]

MPRF for transition t₈₂: n_l3___1(X₀, X₁, X₂) → n_l3___1(X₀, X₁-1, X₂) :|: 1+X₀ ≤ X₂ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀ ∧ 2+X₀ ≤ X₂ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₀ ∧ 1+X₀ ≤ X₂ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₀ ∧ 1+X₀ ≤ X₂ ∧ X₂ ≤ X₁ ∧ 1+X₂ ≤ X₁ ∧ 1+X₀ ≤ X₂ ∧ 0 ≤ X₀ ∧ X₂ ≤ X₁ ∧ 2 ≤ X₂ ∧ 4 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ 2+X₀ ≤ X₂ ∧ 2 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 2+X₀ ≤ X₁ ∧ 0 ≤ X₀ of depth 1:

new bound:

X₁ {O(n)}

MPRF:

l3 [X₁ ]
n_l3___1 [X₁ ]
l1 [X₁ ]

MPRF for transition t₉₈: n_l3___1(X₀, X₁, X₂) → l1(X₀+1, X₁, X₂) :|: X₁ ≤ X₂ ∧ X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 0 ≤ X₀ ∧ X₂ ≤ X₁ ∧ 2 ≤ X₂ ∧ 4 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ 2+X₀ ≤ X₂ ∧ 2 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 2+X₀ ≤ X₁ ∧ 0 ≤ X₀ of depth 1:

new bound:

X₁+1 {O(n)}

MPRF:

l3 [X₁-X₀-1 ]
n_l3___1 [X₁-X₀-1 ]
l1 [X₁-X₀-1 ]

CFR did not improve the program. Rolling back

All Bounds

Timebounds

Overall timebound:2⋅X₁⋅X₁+5⋅X₁+2 {O(n^2)}
t₁₅: 1 {O(1)}
t₁₆: X₁ {O(n)}
t₁₇: 1 {O(1)}
t₁₈: 2⋅X₁⋅X₁+X₁ {O(n^2)}
t₁₉: X₁ {O(n)}
t₂₀: X₁ {O(n)}
t₂₁: X₁ {O(n)}

Costbounds

Overall costbound: 2⋅X₁⋅X₁+5⋅X₁+2 {O(n^2)}
t₁₅: 1 {O(1)}
t₁₆: X₁ {O(n)}
t₁₇: 1 {O(1)}
t₁₈: 2⋅X₁⋅X₁+X₁ {O(n^2)}
t₁₉: X₁ {O(n)}
t₂₀: X₁ {O(n)}
t₂₁: X₁ {O(n)}

Sizebounds

t₁₅, X₀: 0 {O(1)}
t₁₅, X₁: X₁ {O(n)}
t₁₅, X₂: X₂ {O(n)}
t₁₆, X₀: X₁ {O(n)}
t₁₆, X₁: X₁ {O(n)}
t₁₆, X₂: X₁+2 {O(n)}
t₁₇, X₀: X₁ {O(n)}
t₁₇, X₁: 2⋅X₁ {O(n)}
t₁₇, X₂: 6⋅X₁⋅X₁+13⋅X₁+X₂+20 {O(n^2)}
t₁₈, X₀: X₁ {O(n)}
t₁₈, X₁: X₁ {O(n)}
t₁₈, X₂: 2⋅X₁⋅X₁+4⋅X₁+6 {O(n^2)}
t₁₉, X₀: X₁ {O(n)}
t₁₉, X₁: X₁ {O(n)}
t₁₉, X₂: 2⋅X₁⋅X₁+4⋅X₁+6 {O(n^2)}
t₂₀, X₀: X₁ {O(n)}
t₂₀, X₁: X₁ {O(n)}
t₂₀, X₂: 2⋅X₁⋅X₁+4⋅X₁+6 {O(n^2)}
t₂₁, X₀: X₁ {O(n)}
t₂₁, X₁: X₁ {O(n)}
t₂₁, X₂: 6⋅X₁⋅X₁+13⋅X₁+20 {O(n^2)}