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:
t₁₄: l0(X₀, X₁, X₂) → l1(0, X₁, X₂)
t₁₅: 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
t₁₇: 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 for transition t₁₇: 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 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 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 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)}
Chain transitions t₁₇: l3→l1 and t₁₆: l1→l3 to t₅₇: l3→l3
Chain transitions t₁₄: l0→l1 and t₁₆: l1→l3 to t₅₈: l0→l3
Chain transitions t₁₄: l0→l1 and t₁₅: l1→l2 to t₅₉: l0→l2
Chain transitions t₁₇: l3→l1 and t₁₅: l1→l2 to t₆₀: l3→l2
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 ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 0 ≤ X₀ for location l3
knowledge_propagation leads to new time bound 2⋅X₁⋅X₁+3⋅X₁ {O(n^2)} for transition t₅₇: l3(X₀, X₁, X₂) -{2}> l3(1+X₀, X₁, 2+X₀) :|: X₁ ≤ X₂ ∧ 2+X₀ ≤ X₁ ∧ X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 0 ≤ X₀ ∧ 0 ≤ 1+X₀ ∧ X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 0 ≤ X₀
CFR did not improve the program. Rolling back
CFR did not improve the program. Rolling back
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 t₁₁₉: n_l3___1(X₀, X₁, X₂) → n_l3___1(X₀, X₁, X₂+1) :|: 2+X₀ ≤ X₂ ∧ 1+X₂ ≤ X₁ ∧ 1+X₀ ≤ X₂ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₀ ∧ 1+X₀ ≤ X₂ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₀ ∧ 1+X₀ ≤ X₂ ∧ 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:
2⋅X₁⋅X₁+2⋅X₁ {O(n^2)}
MPRF for transition t₁₂₀: n_l3___1(X₀, X₁, X₂) → n_l3___1(X₀, X₁-1, X₂) :|: 2+X₀ ≤ X₂ ∧ 1+X₂ ≤ X₁ ∧ 1+X₀ ≤ X₂ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₀ ∧ 1+X₀ ≤ X₂ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₀ ∧ 1+X₀ ≤ X₂ ∧ 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 for transition t₁₂₁: n_l3___1(X₀, X₁, X₂) → n_l3___1(X₀, X₁-1, X₂) :|: 2+X₀ ≤ X₂ ∧ 1+X₂ ≤ X₁ ∧ 1+X₀ ≤ X₂ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₀ ∧ 1+X₀ ≤ X₂ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₀ ∧ 1+X₀ ≤ X₂ ∧ 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 for transition t₁₂₂: l3(X₀, X₁, X₂) → n_l3___1(X₀, X₁, X₂+1) :|: X₂ ≤ 1+X₀ ∧ 1+X₂ ≤ X₁ ∧ 1+X₀ ≤ X₂ ∧ 0 ≤ X₀ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₂ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₀ ∧ 1+X₀ ≤ X₂ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₀ ∧ 1+X₀ ≤ X₂ ∧ 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 for transition t₁₂₃: l3(X₀, X₁, X₂) → l3(X₀, X₁-1, X₂) :|: X₂ ≤ 1+X₀ ∧ 1+X₂ ≤ X₁ ∧ 1+X₀ ≤ X₂ ∧ 0 ≤ X₀ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₂ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₀ ∧ 1+X₀ ≤ X₂ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₀ ∧ 1+X₀ ≤ X₂ ∧ 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 for transition t₁₂₄: l3(X₀, X₁, X₂) → l3(X₀, X₁-1, X₂) :|: X₂ ≤ 1+X₀ ∧ 1+X₂ ≤ X₁ ∧ 1+X₀ ≤ X₂ ∧ 0 ≤ X₀ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₂ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₀ ∧ 1+X₀ ≤ X₂ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₀ ∧ 1+X₀ ≤ X₂ ∧ 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 for transition t₁₃₇: 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)}
CFR did not improve the program. Rolling back
CFR did not improve the program. Rolling back
All Bounds
Timebounds
Overall timebound:2⋅X₁⋅X₁+5⋅X₁+2 {O(n^2)}
t₁₄: 1 {O(1)}
t₁₅: 1 {O(1)}
t₁₆: X₁ {O(n)}
t₁₇: X₁ {O(n)}
t₁₈: 2⋅X₁⋅X₁+X₁ {O(n^2)}
t₁₉: X₁ {O(n)}
t₂₀: X₁ {O(n)}
Costbounds
Overall costbound: 2⋅X₁⋅X₁+5⋅X₁+2 {O(n^2)}
t₁₄: 1 {O(1)}
t₁₅: 1 {O(1)}
t₁₆: X₁ {O(n)}
t₁₇: X₁ {O(n)}
t₁₈: 2⋅X₁⋅X₁+X₁ {O(n^2)}
t₁₉: X₁ {O(n)}
t₂₀: X₁ {O(n)}
Sizebounds
t₁₄, X₀: 0 {O(1)}
t₁₄, X₁: X₁ {O(n)}
t₁₄, X₂: X₂ {O(n)}
t₁₅, X₀: X₁ {O(n)}
t₁₅, X₁: 2⋅X₁ {O(n)}
t₁₅, X₂: 6⋅X₁⋅X₁+7⋅X₁+X₂+8 {O(n^2)}
t₁₆, X₀: X₁ {O(n)}
t₁₆, X₁: X₁ {O(n)}
t₁₆, X₂: X₁+2 {O(n)}
t₁₇, X₀: X₁ {O(n)}
t₁₇, X₁: X₁ {O(n)}
t₁₇, X₂: 6⋅X₁⋅X₁+7⋅X₁+8 {O(n^2)}
t₁₈, X₀: X₁ {O(n)}
t₁₈, X₁: X₁ {O(n)}
t₁₈, X₂: 2⋅X₁⋅X₁+2⋅X₁+2 {O(n^2)}
t₁₉, X₀: X₁ {O(n)}
t₁₉, X₁: X₁ {O(n)}
t₁₉, X₂: 2⋅X₁⋅X₁+2⋅X₁+2 {O(n^2)}
t₂₀, X₀: X₁ {O(n)}
t₂₀, X₁: X₁ {O(n)}
t₂₀, X₂: 2⋅X₁⋅X₁+2⋅X₁+2 {O(n^2)}