Initial Problem
Start: l0
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars:
Locations: l0, l1, l2, l3
Transitions:
t₀: l0(X₀, X₁, X₂, X₃) → l1(X₀, X₁, X₂, X₃)
t₁: l1(X₀, X₁, X₂, X₃) → l1(X₀, X₁, X₂+X₃, X₃-1) :|: 1 ≤ X₃
t₂: l1(X₀, X₁, X₂, X₃) → l2(X₀, X₁, X₂, X₃) :|: X₃ < 1
t₃: l2(X₀, X₁, X₂, X₃) → l3(X₂, X₂, X₂, X₃) :|: 1 ≤ X₂
t₅: l3(X₀, X₁, X₂, X₃) → l2(X₀, X₁, X₂-1, X₃) :|: X₀ ≤ 0
t₄: l3(X₀, X₁, X₂, X₃) → l3(X₀+X₁, X₁-1, X₂, X₃) :|: 1 ≤ X₀
Preprocessing
Found invariant X₃ ≤ 0 for location l2
Found invariant X₃ ≤ 0 ∧ 1+X₃ ≤ X₂ ∧ 1 ≤ X₂ ∧ X₁ ≤ X₂ ∧ X₁ ≤ X₀ for location l3
Problem after Preprocessing
Start: l0
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars:
Locations: l0, l1, l2, l3
Transitions:
t₀: l0(X₀, X₁, X₂, X₃) → l1(X₀, X₁, X₂, X₃)
t₁: l1(X₀, X₁, X₂, X₃) → l1(X₀, X₁, X₂+X₃, X₃-1) :|: 1 ≤ X₃
t₂: l1(X₀, X₁, X₂, X₃) → l2(X₀, X₁, X₂, X₃) :|: X₃ < 1
t₃: l2(X₀, X₁, X₂, X₃) → l3(X₂, X₂, X₂, X₃) :|: 1 ≤ X₂ ∧ X₃ ≤ 0
t₅: l3(X₀, X₁, X₂, X₃) → l2(X₀, X₁, X₂-1, X₃) :|: X₀ ≤ 0 ∧ X₃ ≤ 0 ∧ 1+X₃ ≤ X₂ ∧ 1 ≤ X₂ ∧ X₁ ≤ X₂ ∧ X₁ ≤ X₀
t₄: l3(X₀, X₁, X₂, X₃) → l3(X₀+X₁, X₁-1, X₂, X₃) :|: 1 ≤ X₀ ∧ X₃ ≤ 0 ∧ 1+X₃ ≤ X₂ ∧ 1 ≤ X₂ ∧ X₁ ≤ X₂ ∧ X₁ ≤ X₀
MPRF for transition t₁: l1(X₀, X₁, X₂, X₃) → l1(X₀, X₁, X₂+X₃, X₃-1) :|: 1 ≤ X₃ of depth 1:
new bound:
X₃ {O(n)}
MPRF:
l1 [X₃ ]
MPRF for transition t₃: l2(X₀, X₁, X₂, X₃) → l3(X₂, X₂, X₂, X₃) :|: 1 ≤ X₂ ∧ X₃ ≤ 0 of depth 1:
new bound:
2⋅X₃⋅X₃+2⋅X₂+2⋅X₃ {O(n^2)}
MPRF:
l3 [X₂-1 ]
l2 [X₂ ]
MPRF for transition t₅: l3(X₀, X₁, X₂, X₃) → l2(X₀, X₁, X₂-1, X₃) :|: X₀ ≤ 0 ∧ X₃ ≤ 0 ∧ 1+X₃ ≤ X₂ ∧ 1 ≤ X₂ ∧ X₁ ≤ X₂ ∧ X₁ ≤ X₀ of depth 1:
new bound:
2⋅X₃⋅X₃+2⋅X₂+2⋅X₃ {O(n^2)}
MPRF:
l3 [X₂ ]
l2 [X₂ ]
MPRF for transition t₄: l3(X₀, X₁, X₂, X₃) → l3(X₀+X₁, X₁-1, X₂, X₃) :|: 1 ≤ X₀ ∧ X₃ ≤ 0 ∧ 1+X₃ ≤ X₂ ∧ 1 ≤ X₂ ∧ X₁ ≤ X₂ ∧ X₁ ≤ X₀ of depth 2:
new bound:
32⋅X₃⋅X₃⋅X₃⋅X₃+64⋅X₂⋅X₃⋅X₃+64⋅X₃⋅X₃⋅X₃+32⋅X₂⋅X₂+64⋅X₂⋅X₃+98⋅X₃⋅X₃+66⋅X₂+66⋅X₃+25 {O(n^4)}
MPRF:
l2 [X₂+3 ; 0 ]
l3 [X₁+3 ; X₀+X₂-2⋅X₁ ]
Analysing control-flow refined program
Found invariant X₃ ≤ 0 for location l2
Found invariant X₃ ≤ 0 ∧ 1+X₃ ≤ X₂ ∧ 1+X₃ ≤ X₁ ∧ 1+X₃ ≤ X₀ ∧ X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ for location n_l3___4
Found invariant X₃ ≤ 0 ∧ 1+X₃ ≤ X₂ ∧ X₃ ≤ X₁ ∧ 2+X₃ ≤ X₀ ∧ X₂ ≤ 1+X₁ ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 3 ≤ X₀+X₂ ∧ 2+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀ for location n_l3___3
Found invariant X₃ ≤ 0 ∧ X₃ ≤ X₂ ∧ 2+X₁+X₃ ≤ 0 ∧ X₀+X₃ ≤ 0 ∧ 0 ≤ X₂ ∧ 2+X₁ ≤ X₂ ∧ X₀ ≤ X₂ ∧ 2+X₁ ≤ 0 ∧ 2+X₁ ≤ X₀ ∧ 2+X₀+X₁ ≤ 0 ∧ X₀ ≤ 0 for location n_l2___1
Found invariant X₃ ≤ 0 ∧ 1+X₃ ≤ X₂ ∧ 1 ≤ X₂ ∧ 2+X₁ ≤ X₂ ∧ 2+X₁ ≤ X₀ for location n_l3___2
MPRF for transition t₅₅: n_l2___1(X₀, X₁, X₂, X₃) → n_l3___4(X₂, X₂, X₂, X₃) :|: X₃ ≤ 0 ∧ X₃ ≤ 0 ∧ 0 ≤ X₂ ∧ X₃ ≤ 0 ∧ X₁ ≤ X₀ ∧ X₀ ≤ 0 ∧ 1 ≤ X₂ ∧ X₃ ≤ 0 ∧ X₃ ≤ 0 ∧ X₃ ≤ X₂ ∧ 2+X₁+X₃ ≤ 0 ∧ X₀+X₃ ≤ 0 ∧ 0 ≤ X₂ ∧ 2+X₁ ≤ X₂ ∧ X₀ ≤ X₂ ∧ 2+X₁ ≤ 0 ∧ 2+X₁ ≤ X₀ ∧ 2+X₀+X₁ ≤ 0 ∧ X₀ ≤ 0 of depth 1:
new bound:
2⋅X₃⋅X₃+2⋅X₂+2⋅X₃ {O(n^2)}
MPRF:
n_l2___1 [X₂+1 ]
n_l3___2 [X₂ ]
n_l3___4 [X₂ ]
n_l3___3 [X₂ ]
MPRF for transition t₅₇: n_l3___2(X₀, X₁, X₂, X₃) → n_l2___1(X₀, X₁, X₂-1, X₃) :|: X₁ ≤ X₀ ∧ X₃ ≤ 0 ∧ 1 ≤ X₂ ∧ X₁ ≤ X₂ ∧ X₃ ≤ 0 ∧ 1 ≤ X₂ ∧ 2+X₁ ≤ X₀ ∧ 2+2⋅X₁ ≤ X₀ ∧ 1+X₁ ≤ X₂ ∧ 1 ≤ X₂ ∧ X₃ ≤ 0 ∧ X₁ ≤ X₀ ∧ X₀ ≤ 0 ∧ X₃ ≤ 0 ∧ 1+X₃ ≤ X₂ ∧ 1 ≤ X₂ ∧ 2+X₁ ≤ X₂ ∧ 2+X₁ ≤ X₀ of depth 1:
new bound:
2⋅X₃⋅X₃+2⋅X₂+2⋅X₃ {O(n^2)}
MPRF:
n_l2___1 [X₂ ]
n_l3___2 [X₂ ]
n_l3___4 [X₂ ]
n_l3___3 [X₂ ]
MPRF for transition t₅₉: n_l3___3(X₀, X₁, X₂, X₃) → n_l3___2(X₀+X₁, X₁-1, X₂, X₃) :|: X₁ ≤ X₀ ∧ X₃ ≤ 0 ∧ 1 ≤ X₂ ∧ X₁ ≤ X₂ ∧ 1 ≤ X₀ ∧ X₃ ≤ 0 ∧ 1 ≤ X₂ ∧ 1 ≤ X₀ ∧ X₁ ≤ X₀ ∧ X₁ ≤ X₂ ∧ X₃ ≤ 0 ∧ 1 ≤ X₂ ∧ 2+X₁ ≤ X₀ ∧ 2+2⋅X₁ ≤ X₀ ∧ 1+X₁ ≤ X₂ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₂ ∧ 1 ≤ X₀ ∧ X₃ ≤ 0 ∧ X₁ ≤ X₂ ∧ X₃ ≤ 0 ∧ 1+X₃ ≤ X₂ ∧ X₃ ≤ X₁ ∧ 2+X₃ ≤ X₀ ∧ X₂ ≤ 1+X₁ ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 3 ≤ X₀+X₂ ∧ 2+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀ of depth 1:
new bound:
2⋅X₃⋅X₃+2⋅X₂+2⋅X₃ {O(n^2)}
MPRF:
n_l2___1 [X₂ ]
n_l3___2 [X₂-1 ]
n_l3___4 [X₁ ]
n_l3___3 [X₁+1 ]
MPRF for transition t₆₀: n_l3___4(X₀, X₁, X₂, X₃) → n_l3___3(X₀+X₁, X₁-1, X₂, X₃) :|: X₁ ≤ X₀ ∧ X₃ ≤ 0 ∧ 1 ≤ X₂ ∧ X₁ ≤ X₂ ∧ 1 ≤ X₀ ∧ X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₂ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₀ ∧ X₃ ≤ 0 ∧ X₃ ≤ 0 ∧ 1 ≤ X₂ ∧ 1 ≤ X₀ ∧ X₁ ≤ X₀ ∧ X₁ ≤ X₂ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₂ ∧ 1 ≤ X₀ ∧ X₃ ≤ 0 ∧ X₁ ≤ X₂ ∧ X₃ ≤ 0 ∧ 1+X₃ ≤ X₂ ∧ 1+X₃ ≤ X₁ ∧ 1+X₃ ≤ X₀ ∧ X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ of depth 1:
new bound:
2⋅X₃⋅X₃+2⋅X₂+2⋅X₃ {O(n^2)}
MPRF:
n_l2___1 [X₂ ]
n_l3___2 [X₂-1 ]
n_l3___4 [X₀ ]
n_l3___3 [X₂-1 ]
MPRF for transition t₅₈: n_l3___2(X₀, X₁, X₂, X₃) → n_l3___2(X₀+X₁, X₁-1, X₂, X₃) :|: X₁ ≤ X₀ ∧ X₃ ≤ 0 ∧ 1 ≤ X₂ ∧ X₁ ≤ X₂ ∧ X₃ ≤ 0 ∧ 1 ≤ X₂ ∧ 2+X₁ ≤ X₀ ∧ 2+2⋅X₁ ≤ X₀ ∧ 1+X₁ ≤ X₂ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₂ ∧ 1 ≤ X₀ ∧ X₃ ≤ 0 ∧ X₁ ≤ X₂ ∧ X₃ ≤ 0 ∧ 1+X₃ ≤ X₂ ∧ 1 ≤ X₂ ∧ 2+X₁ ≤ X₂ ∧ 2+X₁ ≤ X₀ of depth 2:
new bound:
256⋅X₃⋅X₃⋅X₃⋅X₃+512⋅X₂⋅X₃⋅X₃+512⋅X₃⋅X₃⋅X₃+256⋅X₂⋅X₂+386⋅X₃⋅X₃+512⋅X₂⋅X₃+130⋅X₂+130⋅X₃+1 {O(n^4)}
MPRF:
n_l2___1 [2⋅X₂ ; 6⋅X₂ ]
n_l3___2 [X₁+3 ; X₀+2⋅X₂-2⋅X₁ ]
n_l3___4 [X₁+X₂ ; 4⋅X₀+2⋅X₂ ]
n_l3___3 [X₁+2 ; X₀+2⋅X₂+2-X₁ ]
CFR did not improve the program. Rolling back
All Bounds
Timebounds
Overall timebound:32⋅X₃⋅X₃⋅X₃⋅X₃+64⋅X₂⋅X₃⋅X₃+64⋅X₃⋅X₃⋅X₃+102⋅X₃⋅X₃+32⋅X₂⋅X₂+64⋅X₂⋅X₃+70⋅X₂+71⋅X₃+27 {O(n^4)}
t₀: 1 {O(1)}
t₁: X₃ {O(n)}
t₂: 1 {O(1)}
t₃: 2⋅X₃⋅X₃+2⋅X₂+2⋅X₃ {O(n^2)}
t₄: 32⋅X₃⋅X₃⋅X₃⋅X₃+64⋅X₂⋅X₃⋅X₃+64⋅X₃⋅X₃⋅X₃+32⋅X₂⋅X₂+64⋅X₂⋅X₃+98⋅X₃⋅X₃+66⋅X₂+66⋅X₃+25 {O(n^4)}
t₅: 2⋅X₃⋅X₃+2⋅X₂+2⋅X₃ {O(n^2)}
Costbounds
Overall costbound: 32⋅X₃⋅X₃⋅X₃⋅X₃+64⋅X₂⋅X₃⋅X₃+64⋅X₃⋅X₃⋅X₃+102⋅X₃⋅X₃+32⋅X₂⋅X₂+64⋅X₂⋅X₃+70⋅X₂+71⋅X₃+27 {O(n^4)}
t₀: 1 {O(1)}
t₁: X₃ {O(n)}
t₂: 1 {O(1)}
t₃: 2⋅X₃⋅X₃+2⋅X₂+2⋅X₃ {O(n^2)}
t₄: 32⋅X₃⋅X₃⋅X₃⋅X₃+64⋅X₂⋅X₃⋅X₃+64⋅X₃⋅X₃⋅X₃+32⋅X₂⋅X₂+64⋅X₂⋅X₃+98⋅X₃⋅X₃+66⋅X₂+66⋅X₃+25 {O(n^4)}
t₅: 2⋅X₃⋅X₃+2⋅X₂+2⋅X₃ {O(n^2)}
Sizebounds
t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₀, X₃: X₃ {O(n)}
t₁, X₀: X₀ {O(n)}
t₁, X₁: X₁ {O(n)}
t₁, X₂: 2⋅X₃⋅X₃+2⋅X₃+X₂ {O(n^2)}
t₁, X₃: X₃ {O(n)}
t₂, X₀: 2⋅X₀ {O(n)}
t₂, X₁: 2⋅X₁ {O(n)}
t₂, X₂: 2⋅X₃⋅X₃+2⋅X₂+2⋅X₃ {O(n^2)}
t₂, X₃: 2⋅X₃ {O(n)}
t₃, X₀: 4⋅X₃⋅X₃+4⋅X₂+4⋅X₃ {O(n^2)}
t₃, X₁: 4⋅X₃⋅X₃+4⋅X₂+4⋅X₃ {O(n^2)}
t₃, X₂: 2⋅X₃⋅X₃+2⋅X₂+2⋅X₃ {O(n^2)}
t₃, X₃: 2⋅X₃ {O(n)}
t₄, X₀: 1024⋅X₃⋅X₃⋅X₃⋅X₃⋅X₃⋅X₃⋅X₃⋅X₃+4096⋅X₂⋅X₃⋅X₃⋅X₃⋅X₃⋅X₃⋅X₃+4096⋅X₃⋅X₃⋅X₃⋅X₃⋅X₃⋅X₃⋅X₃+10624⋅X₃⋅X₃⋅X₃⋅X₃⋅X₃⋅X₃+12288⋅X₂⋅X₃⋅X₃⋅X₃⋅X₃⋅X₃+6144⋅X₂⋅X₂⋅X₃⋅X₃⋅X₃⋅X₃+12288⋅X₂⋅X₂⋅X₃⋅X₃⋅X₃+17536⋅X₃⋅X₃⋅X₃⋅X₃⋅X₃+25728⋅X₂⋅X₃⋅X₃⋅X₃⋅X₃+4096⋅X₂⋅X₂⋅X₂⋅X₃⋅X₃+1024⋅X₂⋅X₂⋅X₂⋅X₂+19584⋅X₂⋅X₂⋅X₃⋅X₃+20980⋅X₃⋅X₃⋅X₃⋅X₃+30976⋅X₂⋅X₃⋅X₃⋅X₃+4096⋅X₂⋅X₂⋅X₂⋅X₃+13440⋅X₂⋅X₂⋅X₃+17512⋅X₃⋅X₃⋅X₃+26472⋅X₂⋅X₃⋅X₃+4480⋅X₂⋅X₂⋅X₂+10094⋅X₃⋅X₃+13032⋅X₂⋅X₃+6516⋅X₂⋅X₂+3578⋅X₂+3578⋅X₃+650 {O(n^8)}
t₄, X₁: 32⋅X₃⋅X₃⋅X₃⋅X₃+64⋅X₂⋅X₃⋅X₃+64⋅X₃⋅X₃⋅X₃+102⋅X₃⋅X₃+32⋅X₂⋅X₂+64⋅X₂⋅X₃+70⋅X₂+70⋅X₃+25 {O(n^4)}
t₄, X₂: 2⋅X₃⋅X₃+2⋅X₂+2⋅X₃ {O(n^2)}
t₄, X₃: 2⋅X₃ {O(n)}
t₅, X₀: 1024⋅X₃⋅X₃⋅X₃⋅X₃⋅X₃⋅X₃⋅X₃⋅X₃+4096⋅X₂⋅X₃⋅X₃⋅X₃⋅X₃⋅X₃⋅X₃+4096⋅X₃⋅X₃⋅X₃⋅X₃⋅X₃⋅X₃⋅X₃+10624⋅X₃⋅X₃⋅X₃⋅X₃⋅X₃⋅X₃+12288⋅X₂⋅X₃⋅X₃⋅X₃⋅X₃⋅X₃+6144⋅X₂⋅X₂⋅X₃⋅X₃⋅X₃⋅X₃+12288⋅X₂⋅X₂⋅X₃⋅X₃⋅X₃+17536⋅X₃⋅X₃⋅X₃⋅X₃⋅X₃+25728⋅X₂⋅X₃⋅X₃⋅X₃⋅X₃+4096⋅X₂⋅X₂⋅X₂⋅X₃⋅X₃+1024⋅X₂⋅X₂⋅X₂⋅X₂+19584⋅X₂⋅X₂⋅X₃⋅X₃+20980⋅X₃⋅X₃⋅X₃⋅X₃+30976⋅X₂⋅X₃⋅X₃⋅X₃+4096⋅X₂⋅X₂⋅X₂⋅X₃+13440⋅X₂⋅X₂⋅X₃+17512⋅X₃⋅X₃⋅X₃+26472⋅X₂⋅X₃⋅X₃+4480⋅X₂⋅X₂⋅X₂+10094⋅X₃⋅X₃+13032⋅X₂⋅X₃+6516⋅X₂⋅X₂+3578⋅X₂+3578⋅X₃+650 {O(n^8)}
t₅, X₁: 32⋅X₃⋅X₃⋅X₃⋅X₃+64⋅X₂⋅X₃⋅X₃+64⋅X₃⋅X₃⋅X₃+102⋅X₃⋅X₃+32⋅X₂⋅X₂+64⋅X₂⋅X₃+70⋅X₂+70⋅X₃+25 {O(n^4)}
t₅, X₂: 2⋅X₃⋅X₃+2⋅X₂+2⋅X₃ {O(n^2)}
t₅, X₃: 2⋅X₃ {O(n)}