Initial Problem
Start: l0
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: l0, l1
Transitions:
t₀: l0(X₀, X₁, X₂) → l1(X₀, X₁, X₂) :|: 0 < X₀
t₁: l1(X₀, X₁, X₂) → l1(5⋅X₀+(X₂)², 2⋅X₁, X₂) :|: X₀ < (X₁)²
Preprocessing
Found invariant 1 ≤ X₀ for location l1
Problem after Preprocessing
Start: l0
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: l0, l1
Transitions:
t₀: l0(X₀, X₁, X₂) → l1(X₀, X₁, X₂) :|: 0 < X₀
t₁: l1(X₀, X₁, X₂) → l1(5⋅X₀+(X₂)², 2⋅X₁, X₂) :|: X₀ < (X₁)² ∧ 1 ≤ X₀
TWN: t₁: l1→l1
cycle: [t₁: l1→l1]
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 t₁: l1→l1 of 2⋅X₂⋅X₂+8⋅X₁⋅X₁+14 {O(n^2)}
relevant size-bounds w.r.t. t₀:
X₁: X₁ {O(n)}
X₂: X₂ {O(n)}
Runtime-bound of t₀: 1 {O(1)}
Results in: 2⋅X₂⋅X₂+8⋅X₁⋅X₁+14 {O(n^2)}
Analysing control-flow refined program
Eliminate variables {X₁,X₂} that do not contribute to the problem
CFR did not improve the program. Rolling back
CFR did not improve the program. Rolling back
All Bounds
Timebounds
Overall timebound:2⋅X₂⋅X₂+8⋅X₁⋅X₁+15 {O(n^2)}
t₀: 1 {O(1)}
t₁: 2⋅X₂⋅X₂+8⋅X₁⋅X₁+14 {O(n^2)}
Costbounds
Overall costbound: 2⋅X₂⋅X₂+8⋅X₁⋅X₁+15 {O(n^2)}
t₀: 1 {O(1)}
t₁: 2⋅X₂⋅X₂+8⋅X₁⋅X₁+14 {O(n^2)}
Sizebounds
t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₁, X₁: 2^(2⋅X₂⋅X₂+8⋅X₁⋅X₁+14)⋅X₁ {O(EXP)}
t₁, X₂: X₂ {O(n)}