Research proof index

Given a list i :: l the left-most minimal position a of i :: l wrt r as an element of Fin (insertionSortDropMinPos r i l).length.succ. -/ def insertionSortMinPosFin {α : Type} (r : α → α → Prop) [DecidableRel r] (i : α) (l : List α) : Fin (insertionSortDropMinPos r i l).length.succ := ⟨insertionSortMinPos r i l, insertionSortMin_lt_length_succ r...

Closed form for the iterated derivatives of the resolvent: the n-th derivative is (-1)ⁿ · n! · (resolvent z)ⁿ⁺¹.

The single-term coordinate chain rule for a holomorphic outer g, anti-holomorphic version, pointwise at u: ∂̄_I (g ∘ f) = deriv g (f u) · ∂̄_I f. As in dWirtingerCoord_comp_holomorphic_apply, the ∂g/∂f̄ channel vanishes and ∂g/∂f collapses to deriv g (f u).

∂̄_I z̄^J = δ_IJ. The conjugate of dWirtingerCoord_coordProj, read off through dWirtingerAntiDir_star_comp rather than recomputed. Used to assemble the coordinate-difference rule dWirtingerAntiCoord_coordDiff (§C).

For an anti-holomorphic function the anti-holomorphic Wirtinger derivative equals the complex Fréchet derivative of g at star u along the slot-I real coordinate direction, ∂̄_I (g ∘ star) = fderiv ℂ g (star u) (Pi.single I 1). Dual of dWirtingerCoord_eq_complex_fderiv_apply.

The two-term Wirtinger chain rule for dWirtingerAntiDir, the anti-holomorphic dual of dWirtingerDir_comp: ∂̄_v(g∘f) = (∂g/∂f)·∂̄_v f + (∂g/∂f̄)·∂̄_v f̄. Same outer ∂g/∂f, ∂g/∂f̄ coefficients, now each multiplying its anti-holomorphic inner derivative ∂̄_v f, ∂̄_v f̄; same proof as dWirtingerDir_comp.