Initial Problem
Start: l0
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: l0, l1, l2, l3, l4, l5, l6, l7
Transitions:
t₀: l0(X₀, X₁, X₂) → l6(X₀, X₁, X₂)
t₄: l1(X₀, X₁, X₂) → l2(X₀, X₁, X₂) :|: 1 ≤ X₂
t₅: l1(X₀, X₁, X₂) → l3(X₀, X₁, X₂) :|: X₂ ≤ 0
t₆: l2(X₀, X₁, X₂) → l1(X₀, X₁, X₂-1)
t₇: l3(X₀, X₁, X₂) → l4(X₀-1, X₁, X₂)
t₂: l4(X₀, X₁, X₂) → l1(X₀, X₁, X₁) :|: 1 ≤ X₀
t₃: l4(X₀, X₁, X₂) → l5(X₀, X₁, X₂) :|: X₀ ≤ 0
t₈: l5(X₀, X₁, X₂) → l7(X₀, X₁, X₂)
t₁: l6(X₀, X₁, X₂) → l4(X₁, X₁, X₂)
Preprocessing
Found invariant X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ for location l2
Found invariant X₀ ≤ X₁ ∧ X₀ ≤ 0 for location l7
Found invariant X₀ ≤ X₁ ∧ X₀ ≤ 0 for location l5
Found invariant X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ for location l1
Found invariant X₀ ≤ X₁ for location l4
Found invariant X₂ ≤ 0 ∧ 1+X₂ ≤ X₁ ∧ 1+X₂ ≤ X₀ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ 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:
t₀: l0(X₀, X₁, X₂) → l6(X₀, X₁, X₂)
t₄: l1(X₀, X₁, X₂) → l2(X₀, X₁, X₂) :|: 1 ≤ X₂ ∧ X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀
t₅: l1(X₀, X₁, X₂) → l3(X₀, X₁, X₂) :|: X₂ ≤ 0 ∧ X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀
t₆: l2(X₀, X₁, X₂) → l1(X₀, X₁, X₂-1) :|: X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀
t₇: l3(X₀, X₁, X₂) → l4(X₀-1, X₁, X₂) :|: X₂ ≤ 0 ∧ 1+X₂ ≤ X₁ ∧ 1+X₂ ≤ X₀ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀
t₂: l4(X₀, X₁, X₂) → l1(X₀, X₁, X₁) :|: 1 ≤ X₀ ∧ X₀ ≤ X₁
t₃: l4(X₀, X₁, X₂) → l5(X₀, X₁, X₂) :|: X₀ ≤ 0 ∧ X₀ ≤ X₁
t₈: l5(X₀, X₁, X₂) → l7(X₀, X₁, X₂) :|: X₀ ≤ X₁ ∧ X₀ ≤ 0
t₁: l6(X₀, X₁, X₂) → l4(X₁, X₁, X₂)
MPRF for transition t₂: l4(X₀, X₁, X₂) → l1(X₀, X₁, X₁) :|: 1 ≤ X₀ ∧ X₀ ≤ X₁ of depth 1:
new bound:
X₁ {O(n)}
MPRF for transition t₅: l1(X₀, X₁, X₂) → l3(X₀, X₁, X₂) :|: X₂ ≤ 0 ∧ X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ of depth 1:
new bound:
X₁ {O(n)}
MPRF for transition t₇: l3(X₀, X₁, X₂) → l4(X₀-1, X₁, X₂) :|: X₂ ≤ 0 ∧ 1+X₂ ≤ X₁ ∧ 1+X₂ ≤ X₀ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ of depth 1:
new bound:
X₁ {O(n)}
TWN: t₄: l1→l2
cycle: [t₄: l1→l2; t₆: l2→l1]
loop: (1 ≤ X₂,(X₂) -> (X₂-1)
order: [X₂]
closed-form:
X₂: X₂ + [[n != 0]] * -1 * n^1
Termination: true
Formula:
1 < 0
∨ 1 < X₂ ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 1 ≤ X₂ ∧ X₂ ≤ 1
Stabilization-Threshold for: 1 ≤ X₂
alphas_abs: 1+X₂
M: 0
N: 1
Bound: 2⋅X₂+4 {O(n)}
TWN - Lifting for t₄: l1→l2 of 2⋅X₂+6 {O(n)}
relevant size-bounds w.r.t. t₂:
X₂: 2⋅X₁ {O(n)}
Runtime-bound of t₂: X₁ {O(n)}
Results in: 4⋅X₁⋅X₁+6⋅X₁ {O(n^2)}
TWN: t₆: l2→l1
TWN - Lifting for t₆: l2→l1 of 2⋅X₂+6 {O(n)}
relevant size-bounds w.r.t. t₂:
X₂: 2⋅X₁ {O(n)}
Runtime-bound of t₂: X₁ {O(n)}
Results in: 4⋅X₁⋅X₁+6⋅X₁ {O(n^2)}
Chain transitions t₂: l4→l1 and t₅: l1→l3 to t₅₄: l4→l3
Chain transitions t₆: l2→l1 and t₅: l1→l3 to t₅₅: l2→l3
Chain transitions t₆: l2→l1 and t₄: l1→l2 to t₅₆: l2→l2
Chain transitions t₂: l4→l1 and t₄: l1→l2 to t₅₇: l4→l2
Chain transitions t₅₄: l4→l3 and t₇: l3→l4 to t₅₈: l4→l4
Chain transitions t₅₅: l2→l3 and t₇: l3→l4 to t₅₉: l2→l4
Analysing control-flow refined program
Cut unsatisfiable transition t₅₄: l4→l3
Cut unsatisfiable transition t₅₈: l4→l4
Found invariant X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ for location l2
Found invariant X₀ ≤ X₁ ∧ X₀ ≤ 0 for location l7
Found invariant X₀ ≤ X₁ ∧ X₀ ≤ 0 for location l5
Found invariant X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ for location l1
Found invariant X₀ ≤ X₁ for location l4
Found invariant X₂ ≤ 0 ∧ 1+X₂ ≤ X₁ ∧ 1+X₂ ≤ X₀ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ for location l3
MPRF for transition t₅₇: l4(X₀, X₁, X₂) -{2}> l2(X₀, X₁, X₁) :|: 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ X₀ ≤ X₁ ∧ 0 ≤ 0 ∧ 0 ≤ X₁ ∧ 1 ≤ 2⋅X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ ∧ X₀ ≤ X₁ of depth 1:
new bound:
X₁ {O(n)}
MPRF for transition t₅₉: l2(X₀, X₁, X₂) -{3}> l4(X₀-1, X₁, X₂-1) :|: X₂ ≤ 1 ∧ X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ ∧ X₂ ≤ X₁+1 ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ ∧ X₂ ≤ 1 ∧ X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ ∧ X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ of depth 1:
new bound:
X₁ {O(n)}
TWN: t₅₆: l2→l2
cycle: [t₅₆: l2→l2]
loop: (2 ≤ X₂,(X₂) -> (X₂-1)
order: [X₂]
closed-form:
X₂: X₂ + [[n != 0]] * -1 * n^1
Termination: true
Formula:
1 < 0
∨ 2 < X₂ ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 2 ≤ X₂ ∧ X₂ ≤ 2
Stabilization-Threshold for: 2 ≤ X₂
alphas_abs: 2+X₂
M: 0
N: 1
Bound: 2⋅X₂+6 {O(n)}
TWN - Lifting for t₅₆: l2→l2 of 2⋅X₂+8 {O(n)}
relevant size-bounds w.r.t. t₅₇:
X₂: 2⋅X₁ {O(n)}
Runtime-bound of t₅₇: X₁ {O(n)}
Results in: 4⋅X₁⋅X₁+8⋅X₁ {O(n^2)}
CFR did not improve the program. Rolling back
CFR did not improve the program. Rolling back
Analysing control-flow refined program
Cut unsatisfiable transition t₅: l1→l3
Found invariant 1+X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ for location n_l1___2
Found invariant 1+X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ for location n_l2___1
Found invariant X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ for location n_l2___3
Found invariant X₀ ≤ X₁ ∧ X₀ ≤ 0 for location l7
Found invariant X₀ ≤ X₁ ∧ X₀ ≤ 0 for location l5
Found invariant X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ for location l1
Found invariant X₀ ≤ X₁ for location l4
Found invariant X₂ ≤ 0 ∧ 1+X₂ ≤ X₁ ∧ 1+X₂ ≤ X₀ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ for location l3
knowledge_propagation leads to new time bound X₁ {O(n)} for transition t₁₁₅: l1(X₀, X₁, X₂) → n_l2___3(X₀, X₁, X₂) :|: 1 ≤ X₂ ∧ X₁ ≤ X₂ ∧ 1 ≤ X₂ ∧ 1 ≤ X₂ ∧ X₂ ≤ X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ ∧ X₂ ≤ X₁ ∧ 1 ≤ X₀ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ ∧ X₀ ≤ X₁ ∧ X₂ ≤ X₁ ∧ 1 ≤ X₀ ∧ X₂ ≤ X₁ ∧ X₀ ≤ X₁ ∧ X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀
knowledge_propagation leads to new time bound X₁ {O(n)} for transition t₁₁₇: n_l2___3(X₀, X₁, X₂) → n_l1___2(X₀, X₁, X₂-1) :|: X₁ ≤ X₂ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ ∧ X₂ ≤ X₁ ∧ 1 ≤ X₀ ∧ X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ X₀ ≤ X₁ ∧ X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀
MPRF for transition t₁₁₄: n_l1___2(X₀, X₁, X₂) → n_l2___1(X₀, X₁, X₂) :|: 1+X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ X₂ ≤ X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀ ∧ 0 ≤ X₂ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ ∧ X₂ ≤ X₁ ∧ X₀ ≤ X₁ ∧ 1+X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ of depth 1:
new bound:
2⋅X₁⋅X₁+X₁ {O(n^2)}
MPRF for transition t₁₁₆: n_l2___1(X₀, X₁, X₂) → n_l1___2(X₀, X₁, X₂-1) :|: 1+X₂ ≤ X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 1 ≤ X₀ ∧ X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ X₀ ≤ X₁ ∧ 1+X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ of depth 1:
new bound:
2⋅X₁⋅X₁ {O(n^2)}
MPRF for transition t₁₂₁: n_l1___2(X₀, X₁, X₂) → l3(X₀, X₁, X₂) :|: X₂ ≤ 0 ∧ X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ ∧ 1+X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ of depth 1:
new bound:
X₁ {O(n)}
CFR did not improve the program. Rolling back
CFR did not improve the program. Rolling back
All Bounds
Timebounds
Overall timebound:8⋅X₁⋅X₁+15⋅X₁+4 {O(n^2)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₁ {O(n)}
t₃: 1 {O(1)}
t₄: 4⋅X₁⋅X₁+6⋅X₁ {O(n^2)}
t₅: X₁ {O(n)}
t₆: 4⋅X₁⋅X₁+6⋅X₁ {O(n^2)}
t₇: X₁ {O(n)}
t₈: 1 {O(1)}
Costbounds
Overall costbound: 8⋅X₁⋅X₁+15⋅X₁+4 {O(n^2)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₁ {O(n)}
t₃: 1 {O(1)}
t₄: 4⋅X₁⋅X₁+6⋅X₁ {O(n^2)}
t₅: X₁ {O(n)}
t₆: 4⋅X₁⋅X₁+6⋅X₁ {O(n^2)}
t₇: X₁ {O(n)}
t₈: 1 {O(1)}
Sizebounds
t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₁, X₀: X₁ {O(n)}
t₁, X₁: X₁ {O(n)}
t₁, X₂: X₂ {O(n)}
t₂, X₀: X₁ {O(n)}
t₂, X₁: X₁ {O(n)}
t₂, X₂: 2⋅X₁ {O(n)}
t₃, X₀: 2⋅X₁ {O(n)}
t₃, X₁: 2⋅X₁ {O(n)}
t₃, X₂: X₂ {O(n)}
t₄, X₀: X₁ {O(n)}
t₄, X₁: X₁ {O(n)}
t₄, X₂: 2⋅X₁ {O(n)}
t₅, X₀: X₁ {O(n)}
t₅, X₁: X₁ {O(n)}
t₅, X₂: 0 {O(1)}
t₆, X₀: X₁ {O(n)}
t₆, X₁: X₁ {O(n)}
t₆, X₂: 2⋅X₁ {O(n)}
t₇, X₀: X₁ {O(n)}
t₇, X₁: X₁ {O(n)}
t₇, X₂: 0 {O(1)}
t₈, X₀: 2⋅X₁ {O(n)}
t₈, X₁: 2⋅X₁ {O(n)}
t₈, X₂: X₂ {O(n)}