Initial Problem

Start: l0
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars:
Locations: l0, l1, l2, l3, l4, l5, l6, l7
Transitions:
t₀: l0(X₀, X₁, X₂, X₃) → l2(X₀, X₁, X₂, X₃)
t₂: l1(X₀, X₁, X₂, X₃) → l3(X₀, 1, X₂, X₃) :|: 0 ≤ X₀
t₃: l1(X₀, X₁, X₂, X₃) → l4(X₀, X₁, X₂, X₃) :|: X₀ < 0
t₁: l2(X₀, X₁, X₂, X₃) → l1(X₂, X₁, X₂, X₃)
t₅: l3(X₀, X₁, X₂, X₃) → l5(X₀, X₁, X₂, X₃) :|: X₀ ≤ X₁
t₄: l3(X₀, X₁, X₂, X₃) → l6(X₀, X₁, X₂, X₃) :|: X₁ < X₀
t₈: l4(X₀, X₁, X₂, X₃) → l7(X₀, X₁, X₂, X₃)
t₇: l5(X₀, X₁, X₂, X₃) → l1(X₀-1, X₁, X₂, X₃)
t₆: l6(X₀, X₁, X₂, X₃) → l3(X₀, 2⋅X₁, X₂, X₃)

Preprocessing

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

Found invariant 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 l6

Found invariant X₀ ≤ X₂ ∧ 1+X₀ ≤ 0 for location l7

Found invariant 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 0 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 0 ≤ X₀ for location l5

Found invariant X₀ ≤ X₂ for location l1

Found invariant X₀ ≤ X₂ ∧ 1+X₀ ≤ 0 for location l4

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

Problem after Preprocessing

Start: l0
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: l0, l1, l2, l3, l4, l5, l6, l7
Transitions:
t₁₈: l0(X₀, X₁, X₂) → l2(X₀, X₁, X₂)
t₁₉: l1(X₀, X₁, X₂) → l3(X₀, 1, X₂) :|: 0 ≤ X₀ ∧ X₀ ≤ X₂
t₂₀: l1(X₀, X₁, X₂) → l4(X₀, X₁, X₂) :|: X₀ < 0 ∧ X₀ ≤ X₂
t₂₁: l2(X₀, X₁, X₂) → l1(X₂, X₁, X₂)
t₂₃: l3(X₀, X₁, X₂) → l5(X₀, X₁, X₂) :|: X₀ ≤ X₁ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 0 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 0 ≤ X₀
t₂₂: l3(X₀, X₁, X₂) → l6(X₀, X₁, X₂) :|: X₁ < X₀ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 0 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 0 ≤ X₀
t₂₄: l4(X₀, X₁, X₂) → l7(X₀, X₁, X₂) :|: X₀ ≤ X₂ ∧ 1+X₀ ≤ 0
t₂₅: l5(X₀, X₁, X₂) → l1(X₀-1, X₁, X₂) :|: 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 0 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 0 ≤ X₀
t₂₆: l6(X₀, X₁, X₂) → l3(X₀, 2⋅X₁, X₂) :|: 2 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 4 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 2 ≤ X₀

MPRF for transition t₁₉: l1(X₀, X₁, X₂) → l3(X₀, 1, X₂) :|: 0 ≤ X₀ ∧ X₀ ≤ X₂ of depth 1:

new bound:

X₂+1 {O(n)}

MPRF:

l5 [X₀ ]
l1 [X₀+1 ]
l6 [X₀ ]
l3 [X₀ ]

MPRF for transition t₂₃: l3(X₀, X₁, X₂) → l5(X₀, X₁, X₂) :|: X₀ ≤ X₁ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 0 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 0 ≤ X₀ of depth 1:

new bound:

X₂+1 {O(n)}

MPRF:

l5 [X₀ ]
l1 [X₀+1 ]
l6 [X₀+1 ]
l3 [X₀+1 ]

MPRF for transition t₂₅: l5(X₀, X₁, X₂) → l1(X₀-1, X₁, X₂) :|: 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 0 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 0 ≤ X₀ of depth 1:

new bound:

X₂+1 {O(n)}

MPRF:

l5 [X₀+1 ]
l1 [X₀+1 ]
l6 [X₀+1 ]
l3 [X₀+1 ]

MPRF for transition t₂₂: l3(X₀, X₁, X₂) → l6(X₀, X₁, X₂) :|: X₁ < X₀ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 0 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 0 ≤ X₀ of depth 1:

new bound:

X₂⋅X₂+2⋅X₂ {O(n^2)}

MPRF:

l1 [X₂ ]
l5 [-X₁ ]
l6 [X₂-X₁-1 ]
l3 [X₂-X₁ ]

MPRF for transition t₂₆: l6(X₀, X₁, X₂) → l3(X₀, 2⋅X₁, X₂) :|: 2 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 4 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 2 ≤ X₀ of depth 1:

new bound:

X₂⋅X₂+3⋅X₂+1 {O(n^2)}

MPRF:

l1 [X₀ ]
l5 [X₀-X₁ ]
l6 [X₀-X₁ ]
l3 [X₀-X₁ ]

Analysing control-flow refined program

Found invariant 2 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 4 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ X₁ ≤ 1 ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 2 ≤ X₀ for location n_l6___3

Found invariant 3 ≤ X₂ ∧ 5 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 6 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1+X₁ ≤ X₀ ∧ 2 ≤ X₁ ∧ 5 ≤ X₀+X₁ ∧ 3 ≤ X₀ for location n_l6___1

Found invariant X₀ ≤ X₂ ∧ 1+X₀ ≤ 0 for location l7

Found invariant 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 0 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 0 ≤ X₀ for location l5

Found invariant 2 ≤ X₂ ∧ 4 ≤ X₁+X₂ ∧ 4 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 2 ≤ X₁ ∧ 4 ≤ X₀+X₁ ∧ 2 ≤ X₀ for location n_l3___2

Found invariant X₀ ≤ X₂ for location l1

Found invariant X₀ ≤ X₂ ∧ 1+X₀ ≤ 0 for location l4

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

knowledge_propagation leads to new time bound X₂+1 {O(n)} for transition t₆₁: l3(X₀, X₁, X₂) → n_l6___3(X₀, X₁, X₂) :|: X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ X₁ ≤ 1 ∧ 1 ≤ X₁ ∧ 0 ≤ X₀ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ X₁ < X₀ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ X₁ ≤ 1+X₂ ∧ 0 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ X₁ ≤ 1 ∧ X₁ ≤ 1+X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 0 ≤ X₀

knowledge_propagation leads to new time bound X₂+1 {O(n)} for transition t₆₃: n_l6___3(X₀, X₁, X₂) → n_l3___2(X₀, 2⋅X₁, X₂) :|: X₀ ≤ X₂ ∧ 1 < X₀ ∧ X₁ ≤ 1 ∧ 1 ≤ X₁ ∧ 1 ≤ X₁ ∧ 1+X₁ ≤ X₀ ∧ X₀ ≤ X₂ ∧ 2 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 4 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ X₁ ≤ 1 ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 2 ≤ X₀

MPRF for transition t₆₀: n_l3___2(X₀, X₁, X₂) → n_l6___1(X₀, X₁, X₂) :|: X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₁ ∧ 2+X₁ ≤ 2⋅X₀ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ X₁ < X₀ ∧ X₀ ≤ X₂ ∧ 2 ≤ X₂ ∧ 4 ≤ X₁+X₂ ∧ 4 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 2 ≤ X₁ ∧ 4 ≤ X₀+X₁ ∧ 2 ≤ X₀ of depth 1:

new bound:

X₂⋅X₂+2⋅X₂ {O(n^2)}

MPRF:

l1 [X₂ ]
l3 [X₂ ]
l5 [X₂-X₁ ]
n_l6___1 [X₂-X₁-1 ]
n_l6___3 [X₂ ]
n_l3___2 [X₂-X₁ ]

MPRF for transition t₆₇: n_l3___2(X₀, X₁, X₂) → l5(X₀, X₁, X₂) :|: X₀ ≤ X₁ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 0 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 0 ≤ X₀ ∧ 2 ≤ X₂ ∧ 4 ≤ X₁+X₂ ∧ 4 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 2 ≤ X₁ ∧ 4 ≤ X₀+X₁ ∧ 2 ≤ X₀ of depth 1:

new bound:

X₂+1 {O(n)}

MPRF:

l3 [X₀+1 ]
l1 [X₀+1 ]
l5 [X₀ ]
n_l6___1 [X₀+1 ]
n_l6___3 [X₀+X₁ ]
n_l3___2 [X₀+1 ]

MPRF for transition t₆₂: n_l6___1(X₀, X₁, X₂) → n_l3___2(X₀, 2⋅X₁, X₂) :|: X₀ ≤ X₂ ∧ X₁ < X₀ ∧ 2 ≤ X₁ ∧ 1 ≤ X₁ ∧ 1+X₁ ≤ X₀ ∧ X₀ ≤ X₂ ∧ 3 ≤ X₂ ∧ 5 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 6 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1+X₁ ≤ X₀ ∧ 2 ≤ X₁ ∧ 5 ≤ X₀+X₁ ∧ 3 ≤ X₀ of depth 1:

new bound:

X₂⋅X₂+2⋅X₂ {O(n^2)}

MPRF:

l1 [X₂ ]
l3 [X₂ ]
l5 [-X₁ ]
n_l6___1 [X₀-X₁ ]
n_l6___3 [X₀ ]
n_l3___2 [X₀-X₁ ]

CFR did not improve the program. Rolling back

All Bounds

Timebounds

Overall timebound:2⋅X₂⋅X₂+8⋅X₂+8 {O(n^2)}
t₁₈: 1 {O(1)}
t₁₉: X₂+1 {O(n)}
t₂₀: 1 {O(1)}
t₂₁: 1 {O(1)}
t₂₂: X₂⋅X₂+2⋅X₂ {O(n^2)}
t₂₃: X₂+1 {O(n)}
t₂₄: 1 {O(1)}
t₂₅: X₂+1 {O(n)}
t₂₆: X₂⋅X₂+3⋅X₂+1 {O(n^2)}

Costbounds

Overall costbound: 2⋅X₂⋅X₂+8⋅X₂+8 {O(n^2)}
t₁₈: 1 {O(1)}
t₁₉: X₂+1 {O(n)}
t₂₀: 1 {O(1)}
t₂₁: 1 {O(1)}
t₂₂: X₂⋅X₂+2⋅X₂ {O(n^2)}
t₂₃: X₂+1 {O(n)}
t₂₄: 1 {O(1)}
t₂₅: X₂+1 {O(n)}
t₂₆: X₂⋅X₂+3⋅X₂+1 {O(n^2)}

Sizebounds

t₁₈, X₀: X₀ {O(n)}
t₁₈, X₁: X₁ {O(n)}
t₁₈, X₂: X₂ {O(n)}
t₁₉, X₀: X₂+1 {O(n)}
t₁₉, X₁: 1 {O(1)}
t₁₉, X₂: X₂ {O(n)}
t₂₀, X₀: 2⋅X₂+1 {O(n)}
t₂₀, X₁: 2^(X₂⋅X₂+3⋅X₂+1)+X₁+1 {O(EXP)}
t₂₀, X₂: 2⋅X₂ {O(n)}
t₂₁, X₀: X₂ {O(n)}
t₂₁, X₁: X₁ {O(n)}
t₂₁, X₂: X₂ {O(n)}
t₂₂, X₀: X₂+1 {O(n)}
t₂₂, X₁: 2^(X₂⋅X₂+3⋅X₂+1) {O(EXP)}
t₂₂, X₂: X₂ {O(n)}
t₂₃, X₀: X₂+1 {O(n)}
t₂₃, X₁: 2^(X₂⋅X₂+3⋅X₂+1)+1 {O(EXP)}
t₂₃, X₂: X₂ {O(n)}
t₂₄, X₀: 2⋅X₂+1 {O(n)}
t₂₄, X₁: 2^(X₂⋅X₂+3⋅X₂+1)+X₁+1 {O(EXP)}
t₂₄, X₂: 2⋅X₂ {O(n)}
t₂₅, X₀: X₂+1 {O(n)}
t₂₅, X₁: 2^(X₂⋅X₂+3⋅X₂+1)+1 {O(EXP)}
t₂₅, X₂: X₂ {O(n)}
t₂₆, X₀: X₂+1 {O(n)}
t₂₆, X₁: 2^(X₂⋅X₂+3⋅X₂+1) {O(EXP)}
t₂₆, X₂: X₂ {O(n)}