feat: the inverse of an analytic partial homeo is also analytic (#17170)

We already have the results for the inverse of formal multilinear series, so this PR is mostly just extending our API. However, this showed that our API is suboptimal (notably with respect to the choice of constant coefficients in inverses of formal multilinear series), so we also tweak the definitions of `FormalMultilinearSeries.id` and `FormalMultilinearSeries.leftInv/rightInv` to make them more usable.



Co-authored-by: Johan Commelin <[email protected]>

Mathlib/Analysis/Analytic/Basic.lean

@@ -822,6 +822,13 @@ theorem HasFPowerSeriesOnBall.tendsto_partialSum
     Tendsto (fun n => p.partialSum n y) atTop (𝓝 (f (x + y))) :=
   (hf.hasSum hy).tendsto_sum_nat
 
+theorem HasFPowerSeriesAt.tendsto_partialSum
+    (hf : HasFPowerSeriesAt f p x) :
+    ∀ᶠ y in 𝓝 0, Tendsto (fun n => p.partialSum n y) atTop (𝓝 (f (x + y))) := by
+  rcases hf with ⟨r, hr⟩
+  filter_upwards [EMetric.ball_mem_nhds (0 : E) hr.r_pos] with y hy
+  exact hr.tendsto_partialSum hy
+
 open Finset in
 /-- If a function admits a power series expansion within a ball, then the partial sums
 `p.partialSum n z` converge to `f (x + y)` as `n → ∞` and `z → y`. Note that `x + z` doesn't need

Mathlib/Analysis/Analytic/Composition.lean

@@ -331,41 +331,46 @@ section
 variable (𝕜 E)
 
 /-- The identity formal multilinear series, with all coefficients equal to `0` except for `n = 1`
-where it is (the continuous multilinear version of) the identity. -/
-def id : FormalMultilinearSeries 𝕜 E E
-  | 0 => 0
+where it is (the continuous multilinear version of) the identity. We allow an arbitrary
+constant coefficient `x`. -/
+def id (x : E) : FormalMultilinearSeries 𝕜 E E
+  | 0 => ContinuousMultilinearMap.uncurry0 𝕜 _ x
   | 1 => (continuousMultilinearCurryFin1 𝕜 E E).symm (ContinuousLinearMap.id 𝕜 E)
   | _ => 0
 
+@[simp] theorem id_apply_zero (x : E) (v : Fin 0 → E) :
+    (FormalMultilinearSeries.id 𝕜 E x) 0 v = x := rfl
+
 /-- The first coefficient of `id 𝕜 E` is the identity. -/
 @[simp]
-theorem id_apply_one (v : Fin 1 → E) : (FormalMultilinearSeries.id 𝕜 E) 1 v = v 0 :=
+theorem id_apply_one (x : E) (v : Fin 1 → E) : (FormalMultilinearSeries.id 𝕜 E x) 1 v = v 0 :=
   rfl
 
 /-- The `n`th coefficient of `id 𝕜 E` is the identity when `n = 1`. We state this in a dependent
 way, as it will often appear in this form. -/
-theorem id_apply_one' {n : ℕ} (h : n = 1) (v : Fin n → E) :
-    (id 𝕜 E) n v = v ⟨0, h.symm ▸ zero_lt_one⟩ := by
+theorem id_apply_one' (x : E) {n : ℕ} (h : n = 1) (v : Fin n → E) :
+    (id 𝕜 E x) n v = v ⟨0, h.symm ▸ zero_lt_one⟩ := by
   subst n
   apply id_apply_one
 
 /-- For `n ≠ 1`, the `n`-th coefficient of `id 𝕜 E` is zero, by definition. -/
 @[simp]
-theorem id_apply_ne_one {n : ℕ} (h : n ≠ 1) : (FormalMultilinearSeries.id 𝕜 E) n = 0 := by
+theorem id_apply_of_one_lt (x : E) {n : ℕ} (h : 1 < n) :
+    (FormalMultilinearSeries.id 𝕜 E x) n = 0 := by
   cases' n with n
-  · rfl
+  · contradiction
   · cases n
     · contradiction
     · rfl
 
 end
 
 @[simp]
-theorem comp_id (p : FormalMultilinearSeries 𝕜 E F) : p.comp (id 𝕜 E) = p := by
+theorem comp_id (p : FormalMultilinearSeries 𝕜 E F) (x : E) : p.comp (id 𝕜 E x) = p := by
   ext1 n
   dsimp [FormalMultilinearSeries.comp]
   rw [Finset.sum_eq_single (Composition.ones n)]
-  · show compAlongComposition p (id 𝕜 E) (Composition.ones n) = p n
+  · show compAlongComposition p (id 𝕜 E x) (Composition.ones n) = p n
     ext v
     rw [compAlongComposition_apply]
     apply p.congr (Composition.ones_length n)
@@ -375,50 +380,60 @@ theorem comp_id (p : FormalMultilinearSeries 𝕜 E F) : p.comp (id 𝕜 E) = p
     rw [Fin.ext_iff, Fin.coe_castLE, Fin.val_mk]
   · show
     ∀ b : Composition n,
-      b ∈ Finset.univ → b ≠ Composition.ones n → compAlongComposition p (id 𝕜 E) b = 0
+      b ∈ Finset.univ → b ≠ Composition.ones n → compAlongComposition p (id 𝕜 E x) b = 0
     intro b _ hb
     obtain ⟨k, hk, lt_k⟩ : ∃ (k : ℕ), k ∈ Composition.blocks b ∧ 1 < k :=
       Composition.ne_ones_iff.1 hb
     obtain ⟨i, hi⟩ : ∃ (i : Fin b.blocks.length), b.blocks[i] = k :=
       List.get_of_mem hk
-
     let j : Fin b.length := ⟨i.val, b.blocks_length ▸ i.prop⟩
     have A : 1 < b.blocksFun j := by convert lt_k
     ext v
     rw [compAlongComposition_apply, ContinuousMultilinearMap.zero_apply]
     apply ContinuousMultilinearMap.map_coord_zero _ j
     dsimp [applyComposition]
-    rw [id_apply_ne_one _ _ (ne_of_gt A)]
+    rw [id_apply_of_one_lt _ _ _ A]
     rfl
   · simp
 
 @[simp]
-theorem id_comp (p : FormalMultilinearSeries 𝕜 E F) (h : p 0 = 0) : (id 𝕜 F).comp p = p := by
+theorem id_comp (p : FormalMultilinearSeries 𝕜 E F) (v0 : Fin 0 → E) :
+    (id 𝕜 F (p 0 v0)).comp p = p := by
   ext1 n
   by_cases hn : n = 0
-  · rw [hn, h]
+  · rw [hn]
     ext v
-    rw [comp_coeff_zero', id_apply_ne_one _ _ zero_ne_one]
-    rfl
+    simp only [comp_coeff_zero', id_apply_zero]
+    congr
+    exact ofFn_inj.mp rfl
   · dsimp [FormalMultilinearSeries.comp]
     have n_pos : 0 < n := bot_lt_iff_ne_bot.mpr hn
     rw [Finset.sum_eq_single (Composition.single n n_pos)]
-    · show compAlongComposition (id 𝕜 F) p (Composition.single n n_pos) = p n
+    · show compAlongComposition (id 𝕜 F (p 0 v0)) p (Composition.single n n_pos) = p n
       ext v
-      rw [compAlongComposition_apply, id_apply_one' _ _ (Composition.single_length n_pos)]
+      rw [compAlongComposition_apply, id_apply_one' _ _ _ (Composition.single_length n_pos)]
       dsimp [applyComposition]
       refine p.congr rfl fun i him hin => congr_arg v <| ?_
       ext; simp
     · show
-      ∀ b : Composition n,
-        b ∈ Finset.univ → b ≠ Composition.single n n_pos → compAlongComposition (id 𝕜 F) p b = 0
+      ∀ b : Composition n, b ∈ Finset.univ → b ≠ Composition.single n n_pos →
+        compAlongComposition (id 𝕜 F (p 0 v0)) p b = 0
       intro b _ hb
-      have A : b.length ≠ 1 := by simpa [Composition.eq_single_iff_length] using hb
+      have A : 1 < b.length := by
+        have : b.length ≠ 1 := by simpa [Composition.eq_single_iff_length] using hb
+        have : 0 < b.length := Composition.length_pos_of_pos b n_pos
+        omega
       ext v
-      rw [compAlongComposition_apply, id_apply_ne_one _ _ A]
+      rw [compAlongComposition_apply, id_apply_of_one_lt _ _ _ A]
       rfl
     · simp
 
+/-- Variant of `id_comp` in which the zero coefficient is given by an equality hypothesis instead
+of a definitional equality. Useful for rewriting or simplifying out in some situations. -/
+theorem id_comp' (p : FormalMultilinearSeries 𝕜 E F) (x : F) (v0 : Fin 0 → E) (h : x = p 0 v0) :
+    (id 𝕜 F x).comp p = p := by
+  simp [h]
+
 /-! ### Summability properties of the composition of formal power series -/
 
 
@@ -634,11 +649,12 @@ theorem compChangeOfVariables_sum {α : Type*} [AddCommMonoid α] (m M N : ℕ)
 
 /-- The auxiliary set corresponding to the composition of partial sums asymptotically contains
 all possible compositions. -/
-theorem compPartialSumTarget_tendsto_atTop :
-    Tendsto (fun N => compPartialSumTarget 0 N N) atTop atTop := by
+theorem compPartialSumTarget_tendsto_prod_atTop :
+    Tendsto (fun (p : ℕ × ℕ) => compPartialSumTarget 0 p.1 p.2) atTop atTop := by
   apply Monotone.tendsto_atTop_finset
   · intro m n hmn a ha
-    have : ∀ i, i < m → i < n := fun i hi => lt_of_lt_of_le hi hmn
+    have : ∀ i, i < m.1 → i < n.1 := fun i hi => lt_of_lt_of_le hi hmn.1
+    have : ∀ i, i < m.2 → i < n.2 := fun i hi => lt_of_lt_of_le hi hmn.2
     aesop
   · rintro ⟨n, c⟩
     simp only [mem_compPartialSumTarget_iff]
@@ -650,29 +666,35 @@ theorem compPartialSumTarget_tendsto_atTop :
     apply hn
     simp only [Finset.mem_image_of_mem, Finset.mem_coe, Finset.mem_univ]
 
+/-- The auxiliary set corresponding to the composition of partial sums asymptotically contains
+all possible compositions. -/
+theorem compPartialSumTarget_tendsto_atTop :
+    Tendsto (fun N => compPartialSumTarget 0 N N) atTop atTop := by
+  apply Tendsto.comp compPartialSumTarget_tendsto_prod_atTop tendsto_atTop_diagonal
+
 /-- Composing the partial sums of two multilinear series coincides with the sum over all
 compositions in `compPartialSumTarget 0 N N`. This is precisely the motivation for the
 definition of `compPartialSumTarget`. -/
 theorem comp_partialSum (q : FormalMultilinearSeries 𝕜 F G) (p : FormalMultilinearSeries 𝕜 E F)
-    (N : ℕ) (z : E) :
-    q.partialSum N (∑ i ∈ Finset.Ico 1 N, p i fun _j => z) =
-      ∑ i ∈ compPartialSumTarget 0 N N, q.compAlongComposition p i.2 fun _j => z := by
+    (M N : ℕ) (z : E) :
+    q.partialSum M (∑ i ∈ Finset.Ico 1 N, p i fun _j => z) =
+      ∑ i ∈ compPartialSumTarget 0 M N, q.compAlongComposition p i.2 fun _j => z := by
   -- we expand the composition, using the multilinearity of `q` to expand along each coordinate.
   suffices H :
-    (∑ n ∈ Finset.range N,
+    (∑ n ∈ Finset.range M,
         ∑ r ∈ Fintype.piFinset fun i : Fin n => Finset.Ico 1 N,
           q n fun i : Fin n => p (r i) fun _j => z) =
-      ∑ i ∈ compPartialSumTarget 0 N N, q.compAlongComposition p i.2 fun _j => z by
+      ∑ i ∈ compPartialSumTarget 0 M N, q.compAlongComposition p i.2 fun _j => z by
     simpa only [FormalMultilinearSeries.partialSum, ContinuousMultilinearMap.map_sum_finset] using H
   -- rewrite the first sum as a big sum over a sigma type, in the finset
   -- `compPartialSumTarget 0 N N`
   rw [Finset.range_eq_Ico, Finset.sum_sigma']
   -- use `compChangeOfVariables_sum`, saying that this change of variables respects sums
-  apply compChangeOfVariables_sum 0 N N
+  apply compChangeOfVariables_sum 0 M N
   rintro ⟨k, blocks_fun⟩ H
-  apply congr _ (compChangeOfVariables_length 0 N N H).symm
+  apply congr _ (compChangeOfVariables_length 0 M N H).symm
   intros
-  rw [← compChangeOfVariables_blocksFun 0 N N H]
+  rw [← compChangeOfVariables_blocksFun 0 M N H]
   rfl
 
 end FormalMultilinearSeries