Initial Problem
Start: l0
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: l0, l1, l2
Transitions:
t₀: l0(X₀, X₁, X₂) → l1(X₀, X₁, X₂) :|: 0 < X₀
t₁: l1(X₀, X₁, X₂) → l1(X₀+X₂, X₁, X₂-1) :|: 0 < X₂
t₂: l1(X₀, X₁, X₂) → l2(X₀, X₁, X₂) :|: X₂ ≤ 0
t₃: l2(X₀, X₁, X₂) → l2(5⋅X₀+(X₂)², 2⋅X₁, X₂) :|: X₀ < (X₁)²
Preprocessing
Found invariant X₂ ≤ 0 ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₀ for location l2
Found invariant 1 ≤ X₀ for location l1
Problem after Preprocessing
Start: l0
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: l0, l1, l2
Transitions:
t₀: l0(X₀, X₁, X₂) → l1(X₀, X₁, X₂) :|: 0 < X₀
t₁: l1(X₀, X₁, X₂) → l1(X₀+X₂, X₁, X₂-1) :|: 0 < X₂ ∧ 1 ≤ X₀
t₂: l1(X₀, X₁, X₂) → l2(X₀, X₁, X₂) :|: X₂ ≤ 0 ∧ 1 ≤ X₀
t₃: l2(X₀, X₁, X₂) → l2(5⋅X₀+(X₂)², 2⋅X₁, X₂) :|: X₀ < (X₁)² ∧ X₂ ≤ 0 ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₀
MPRF for transition t₁: l1(X₀, X₁, X₂) → l1(X₀+X₂, X₁, X₂-1) :|: 0 < X₂ ∧ 1 ≤ X₀ of depth 1:
new bound:
X₂ {O(n)}
TWN: t₃: l2→l2
cycle: [t₃: l2→l2]
loop: (X₀ < (X₁)²,(X₀,X₁,X₂) -> (5⋅X₀+(X₂)²,2⋅X₁,X₂)
order: [X₂; X₀; X₁]
closed-form:
X₂: X₂
X₀: X₀ * 5^n + [[n != 0]] * 1/4⋅(X₂)² * 5^n + [[n != 0]] * -1/4⋅(X₂)²
X₁: X₁ * 2^n
Termination: true
Formula:
4⋅X₀+(X₂)² < 0
∨ 0 < 4⋅(X₁)² ∧ 4⋅X₀+(X₂)² ≤ 0 ∧ 0 ≤ 4⋅X₀+(X₂)²
∨ 0 < (X₂)² ∧ 4⋅X₀+(X₂)² ≤ 0 ∧ 0 ≤ 4⋅X₀+(X₂)² ∧ 0 ≤ 4⋅(X₁)² ∧ 4⋅(X₁)² ≤ 0
Stabilization-Threshold for: X₀ < (X₁)²
alphas_abs: 4⋅(X₁)²+(X₂)²
M: 11
N: 1
Bound: 2⋅X₂⋅X₂+8⋅X₁⋅X₁+12 {O(n^2)}
TWN - Lifting for t₃: l2→l2 of 2⋅X₂⋅X₂+8⋅X₁⋅X₁+14 {O(n^2)}
relevant size-bounds w.r.t. t₂:
X₁: 2⋅X₁ {O(n)}
X₂: 2⋅X₂ {O(n)}
Runtime-bound of t₂: 1 {O(1)}
Results in: 32⋅X₁⋅X₁+8⋅X₂⋅X₂+14 {O(n^2)}
Analysing control-flow refined program
Eliminate variables {X₁} that do not contribute to the problem
Found invariant X₁ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₀ for location l2
Found invariant X₁ ≤ 0 for location n_l2___1
Found invariant 1 ≤ X₀ for location l1
MPRF for transition t₅₂: l1(X₀, X₁) → l1(X₀+X₁, X₁-1) :|: 0 < 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:32⋅X₁⋅X₁+8⋅X₂⋅X₂+X₂+16 {O(n^2)}
t₀: 1 {O(1)}
t₁: X₂ {O(n)}
t₂: 1 {O(1)}
t₃: 32⋅X₁⋅X₁+8⋅X₂⋅X₂+14 {O(n^2)}
Costbounds
Overall costbound: 32⋅X₁⋅X₁+8⋅X₂⋅X₂+X₂+16 {O(n^2)}
t₀: 1 {O(1)}
t₁: X₂ {O(n)}
t₂: 1 {O(1)}
t₃: 32⋅X₁⋅X₁+8⋅X₂⋅X₂+14 {O(n^2)}
Sizebounds
t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₁, X₀: 2⋅X₂⋅X₂+2⋅X₂+X₀ {O(n^2)}
t₁, X₁: X₁ {O(n)}
t₁, X₂: X₂ {O(n)}
t₂, X₀: 2⋅X₂⋅X₂+2⋅X₀+2⋅X₂ {O(n^2)}
t₂, X₁: 2⋅X₁ {O(n)}
t₂, X₂: 2⋅X₂ {O(n)}
t₃, X₁: 2⋅2^(32⋅X₁⋅X₁+8⋅X₂⋅X₂+14)⋅X₁ {O(EXP)}
t₃, X₂: 2⋅X₂ {O(n)}