Initial Problem
Start: l0
Program_Vars: X₀, X₁
Temp_Vars:
Locations: l0, l1, l2
Transitions:
t₃: l0(X₀, X₁) → l2(X₀, X₁)
t₁: l1(X₀, X₁) → l1(X₀, X₁+1) :|: 1+X₁ ≤ X₀
t₂: l1(X₀, X₁) → l2(X₀-1, X₁) :|: X₀ ≤ X₁
t₀: l2(X₀, X₁) → l1(X₀, 0) :|: 0 ≤ X₀
Preprocessing
Found invariant X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location l1
Problem after Preprocessing
Start: l0
Program_Vars: X₀, X₁
Temp_Vars:
Locations: l0, l1, l2
Transitions:
t₃: l0(X₀, X₁) → l2(X₀, X₁)
t₁: l1(X₀, X₁) → l1(X₀, X₁+1) :|: 1+X₁ ≤ X₀ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀
t₂: l1(X₀, X₁) → l2(X₀-1, X₁) :|: X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀
t₀: l2(X₀, X₁) → l1(X₀, 0) :|: 0 ≤ X₀
MPRF for transition t₀: l2(X₀, X₁) → l1(X₀, 0) :|: 0 ≤ X₀ of depth 1:
new bound:
X₀+1 {O(n)}
MPRF for transition t₂: l1(X₀, X₁) → l2(X₀-1, X₁) :|: X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀ of depth 1:
new bound:
X₀+1 {O(n)}
TWN: t₁: l1→l1
cycle: [t₁: l1→l1]
loop: (1+X₁ ≤ X₀,(X₀,X₁) -> (X₀,X₁+1)
order: [X₀; X₁]
closed-form:
X₀: X₀
X₁: X₁ + [[n != 0]] * n^1
Termination: true
Formula:
1 < 0
∨ 1+X₁ < X₀ ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 1+X₁ ≤ X₀ ∧ X₀ ≤ 1+X₁
Stabilization-Threshold for: 1+X₁ ≤ X₀
alphas_abs: 1+X₁+X₀
M: 0
N: 1
Bound: 2⋅X₀+2⋅X₁+4 {O(n)}
TWN - Lifting for t₁: l1→l1 of 2⋅X₀+2⋅X₁+6 {O(n)}
relevant size-bounds w.r.t. t₀:
X₀: X₀+1 {O(n)}
X₁: 0 {O(1)}
Runtime-bound of t₀: X₀+1 {O(n)}
Results in: 2⋅X₀⋅X₀+10⋅X₀+8 {O(n^2)}
Chain transitions t₂: l1→l2 and t₀: l2→l1 to t₂₅: l1→l1
Chain transitions t₃: l0→l2 and t₀: l2→l1 to t₂₆: l0→l1
Analysing control-flow refined program
Found invariant X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location l1
knowledge_propagation leads to new time bound 2⋅X₀⋅X₀+10⋅X₀+8 {O(n^2)} for transition t₂₅: l1(X₀, X₁) -{2}> l1(X₀-1, 0) :|: X₀ ≤ X₁ ∧ 1 ≤ X₀ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ 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 ≤ 0 ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location l1
Found invariant X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location n_l1___1
knowledge_propagation leads to new time bound X₀+1 {O(n)} for transition t₄₉: l1(X₀, X₁) → n_l1___1(X₀, X₁+1) :|: X₁ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀ ∧ 0 ≤ X₁ ∧ X₁ ≤ 0 ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀
MPRF for transition t₄₈: n_l1___1(X₀, X₁) → n_l1___1(X₀, X₁+1) :|: 1 ≤ X₁ ∧ 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ of depth 1:
new bound:
2⋅X₀⋅X₀+5⋅X₀+2 {O(n^2)}
MPRF for transition t₅₂: n_l1___1(X₀, X₁) → l2(X₀-1, X₁) :|: X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ of depth 1:
new bound:
X₀ {O(n)}
CFR did not improve the program. Rolling back
CFR did not improve the program. Rolling back
All Bounds
Timebounds
Overall timebound:2⋅X₀⋅X₀+12⋅X₀+11 {O(n^2)}
t₀: X₀+1 {O(n)}
t₁: 2⋅X₀⋅X₀+10⋅X₀+8 {O(n^2)}
t₂: X₀+1 {O(n)}
t₃: 1 {O(1)}
Costbounds
Overall costbound: 2⋅X₀⋅X₀+12⋅X₀+11 {O(n^2)}
t₀: X₀+1 {O(n)}
t₁: 2⋅X₀⋅X₀+10⋅X₀+8 {O(n^2)}
t₂: X₀+1 {O(n)}
t₃: 1 {O(1)}
Sizebounds
t₀, X₀: X₀+1 {O(n)}
t₀, X₁: 0 {O(1)}
t₁, X₀: X₀+1 {O(n)}
t₁, X₁: 2⋅X₀⋅X₀+10⋅X₀+8 {O(n^2)}
t₂, X₀: X₀+1 {O(n)}
t₂, X₁: 2⋅X₀⋅X₀+10⋅X₀+8 {O(n^2)}
t₃, X₀: X₀ {O(n)}
t₃, X₁: X₁ {O(n)}