Research proof index

Access into a heterogeneous append on the right, by value: if m ≤ ↑j, then happend u v j is (heterogeneously) the (↑j - m)-th entry of v.

Fin.induction on m + n for i ≤ m steps is equivalent to Fin.induction on m for i steps.

Fin.induction on m + n for m + i steps is equivalent to Fin.induction on n on i steps on the result of Fin.induction on m.

The preimage of m under succSuccAbove i j hij given that m is not equal to i or j. -/ def predPredAbove (i j : Fin (n + 1 + 1)) (hij : i ≠ j) (m : Fin (n + 1 + 1)) (hm : m ≠ i ∧ m ≠ j) : Fin n := if h1 : m.1 < i.1 ∧ m.1 < j.1 then ⟨m, by grind⟩ else if h2 : m.1 - 1 < i.1 ∧ j.1 ≤ m.1 then ⟨m - 1, by grind⟩ else if h3 : i.1 - 1 ≤ m.1 ∧ m.1 < j.1 t...

The concatenation of the first m elements and the last n - m elements of a tuple is the same as the original tuple.

There is a measurable exceptional set $G' \subseteq G$ with $2\mu(G') \le \mu(G)$ such that, for every measurable $f$ bounded by $\mathbf{1}_F$, the integral over $G \setminus G'$ of the finitary oscillatory singular integral, summed only over the scales from $\sigma_1(x)$ to $\sigma_2(x)$, is at most $C(a,q)\mu(G)^{1-1/q}\mu(F)^{1/q}$.