Initial Problem
Start: l0
Program_Vars: X₀, X₁
Temp_Vars:
Locations: l0, l1
Transitions:
t₀: l0(X₀, X₁) → l1(X₀, X₀) :|: 1 ≤ X₀
t₁: l1(X₀, X₁) → l1(X₀, X₁-1) :|: 1 ≤ X₁
t₂: l1(X₀, X₁) → l1(X₀-1, X₀-1) :|: X₁ ≤ 0 ∧ 2 ≤ X₀
Preprocessing
Found invariant X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location l1
Problem after Preprocessing
Start: l0
Program_Vars: X₀, X₁
Temp_Vars:
Locations: l0, l1
Transitions:
t₀: l0(X₀, X₁) → l1(X₀, X₀) :|: 1 ≤ X₀
t₁: l1(X₀, X₁) → l1(X₀, X₁-1) :|: 1 ≤ X₁ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀
t₂: l1(X₀, X₁) → l1(X₀-1, X₀-1) :|: X₁ ≤ 0 ∧ 2 ≤ X₀ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀
MPRF for transition t₂: l1(X₀, X₁) → l1(X₀-1, X₀-1) :|: X₁ ≤ 0 ∧ 2 ≤ X₀ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ of depth 1:
new bound:
X₀ {O(n)}
TWN: t₁: l1→l1
cycle: [t₁: l1→l1]
loop: (1 ≤ X₁,(X₁) -> (X₁-1)
order: [X₁]
closed-form:
X₁: X₁ + [[n != 0]] * -1 * n^1
Termination: true
Formula:
1 < 0
∨ 1 < X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 1 ≤ X₁ ∧ X₁ ≤ 1
Stabilization-Threshold for: 1 ≤ X₁
alphas_abs: 1+X₁
M: 0
N: 1
Bound: 2⋅X₁+4 {O(n)}
loop: (1 ≤ X₁,(X₁) -> (X₁-1)
order: [X₁]
closed-form:
X₁: X₁ + [[n != 0]] * -1 * n^1
Termination: true
Formula:
1 < 0
∨ 1 < X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 1 ≤ X₁ ∧ X₁ ≤ 1
Stabilization-Threshold for: 1 ≤ X₁
alphas_abs: 1+X₁
M: 0
N: 1
Bound: 2⋅X₁+4 {O(n)}
TWN - Lifting for t₁: l1→l1 of 2⋅X₁+6 {O(n)}
relevant size-bounds w.r.t. t₂:
X₁: X₀ {O(n)}
Runtime-bound of t₂: X₀ {O(n)}
Results in: 2⋅X₀⋅X₀+6⋅X₀ {O(n^2)}
TWN - Lifting for t₁: l1→l1 of 2⋅X₁+6 {O(n)}
relevant size-bounds w.r.t. t₀:
X₁: X₀ {O(n)}
Runtime-bound of t₀: 1 {O(1)}
Results in: 2⋅X₀+6 {O(n)}
Analysing control-flow refined program
Cut unsatisfiable transition t₄₁: n_l1___2→l1
Found invariant 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ X₀ ≤ 1+X₁ ∧ 1 ≤ X₀ for location n_l1___2
Found invariant X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ for location l1
Found invariant 2+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀ for location n_l1___1
MPRF for transition t₃₉: n_l1___1(X₀, X₁) → l1(X₀-1, X₀-1) :|: 2 ≤ X₀ ∧ 1+X₁ ≤ X₀ ∧ 2 ≤ X₀ ∧ X₁ ≤ 0 ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₁ ∧ 2+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀ of depth 1:
new bound:
X₀+1 {O(n)}
MPRF for transition t₄₀: n_l1___2(X₀, X₁) → n_l1___1(X₀, X₁-1) :|: 1+X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ X₁ ≤ X₀ ∧ 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ X₀ ≤ 1+X₁ ∧ 1 ≤ X₀ of depth 1:
new bound:
X₀ {O(n)}
MPRF for transition t₄₂: l1(X₀, X₁) → n_l1___2(X₀, X₁-1) :|: 1 ≤ X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₁ ∧ X₁ ≤ X₀ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₀ ∧ X₁ ≤ X₀ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₀ ∧ X₁ ≤ X₀ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ of depth 1:
new bound:
X₀ {O(n)}
TWN: t₃₈: n_l1___1→n_l1___1
cycle: [t₃₈: n_l1___1→n_l1___1]
loop: (2 ≤ X₀ ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₁,(X₀,X₁) -> (X₀,X₁-1)
order: [X₀; X₁]
closed-form:
X₀: X₀
X₁: X₁ + [[n != 0]] * -1 * n^1
Termination: true
Formula:
1 < 0 ∧ 0 < 1 ∧ 2 < X₀
∨ 1 < 0 ∧ 0 < 1 ∧ 2 ≤ X₀ ∧ X₀ ≤ 2
∨ 1 < 0 ∧ 1+X₁ < X₀ ∧ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 2 < X₀
∨ 1 < 0 ∧ 1+X₁ < X₀ ∧ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 2 ≤ X₀ ∧ X₀ ≤ 2
∨ 1 < 0 ∧ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ X₀ ≤ 1+X₁ ∧ 2 < X₀
∨ 1 < 0 ∧ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ X₀ ≤ 1+X₁ ∧ 2 ≤ X₀ ∧ X₀ ≤ 2
∨ 1 < X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 0 < 1 ∧ 2 < X₀
∨ 1 < X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 0 < 1 ∧ 2 ≤ X₀ ∧ X₀ ≤ 2
∨ 1 < X₁ ∧ 1+X₁ < X₀ ∧ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 2 < X₀
∨ 1 < X₁ ∧ 1+X₁ < X₀ ∧ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 2 ≤ X₀ ∧ X₀ ≤ 2
∨ 1 < X₁ ∧ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ X₀ ≤ 1+X₁ ∧ 2 < X₀
∨ 1 < X₁ ∧ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ X₀ ≤ 1+X₁ ∧ 2 ≤ X₀ ∧ X₀ ≤ 2
∨ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 1 ≤ X₁ ∧ X₁ ≤ 1 ∧ 0 < 1 ∧ 2 < X₀
∨ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 1 ≤ X₁ ∧ X₁ ≤ 1 ∧ 0 < 1 ∧ 2 ≤ X₀ ∧ X₀ ≤ 2
∨ 1 ≤ X₁ ∧ X₁ ≤ 1 ∧ 1+X₁ < X₀ ∧ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 2 < X₀
∨ 1 ≤ X₁ ∧ X₁ ≤ 1 ∧ 1+X₁ < X₀ ∧ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 2 ≤ X₀ ∧ X₀ ≤ 2
∨ 1 ≤ X₁ ∧ X₁ ≤ 1 ∧ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ X₀ ≤ 1+X₁ ∧ 2 < X₀
∨ 1 ≤ X₁ ∧ X₁ ≤ 1 ∧ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ X₀ ≤ 1+X₁ ∧ 2 ≤ X₀ ∧ X₀ ≤ 2
Stabilization-Threshold for: 1 ≤ X₁
alphas_abs: 1+X₁
M: 0
N: 1
Bound: 2⋅X₁+4 {O(n)}
Stabilization-Threshold for: 1+X₁ ≤ X₀
alphas_abs: 1+X₀+X₁
M: 0
N: 1
Bound: 2⋅X₀+2⋅X₁+4 {O(n)}
TWN - Lifting for t₃₈: n_l1___1→n_l1___1 of 2⋅X₀+4⋅X₁+11 {O(n)}
relevant size-bounds w.r.t. t₄₀:
X₀: X₀ {O(n)}
X₁: 3⋅X₀ {O(n)}
Runtime-bound of t₄₀: X₀ {O(n)}
Results in: 14⋅X₀⋅X₀+11⋅X₀ {O(n^2)}
CFR did not improve the program. Rolling back
CFR did not improve the program. Rolling back
All Bounds
Timebounds
Overall timebound:2⋅X₀⋅X₀+9⋅X₀+7 {O(n^2)}
t₀: 1 {O(1)}
t₁: 2⋅X₀⋅X₀+8⋅X₀+6 {O(n^2)}
t₂: X₀ {O(n)}
Costbounds
Overall costbound: 2⋅X₀⋅X₀+9⋅X₀+7 {O(n^2)}
t₀: 1 {O(1)}
t₁: 2⋅X₀⋅X₀+8⋅X₀+6 {O(n^2)}
t₂: X₀ {O(n)}
Sizebounds
t₀, X₀: X₀ {O(n)}
t₀, X₁: X₀ {O(n)}
t₁, X₀: X₀ {O(n)}
t₁, X₁: 2⋅X₀ {O(n)}
t₂, X₀: X₀ {O(n)}
t₂, X₁: X₀ {O(n)}