ring_theory.polynomial.hermite.basic ⟷ Mathlib.RingTheory.Polynomial.Hermite.Basic

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

@@ -3,7 +3,7 @@ Copyright (c) 2023 Luke Mantle. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Luke Mantle
 -/
-import Data.Polynomial.Derivative
+import Algebra.Polynomial.Derivative
 import Data.Nat.Parity
 import Data.Nat.Factorial.DoubleFactorial
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

@@ -194,7 +194,7 @@ theorem coeff_hermite_explicit :
   | n + 1, 0 => by
     convert coeff_hermite_succ_zero (2 * n + 1) using 1
     rw [coeff_hermite_explicit n 1, (by ring_nf : 2 * (n + 1) - 1 = 2 * n + 1),
-      Nat.doubleFactorial_add_one, Nat.choose_zero_right, Nat.choose_one_right, pow_succ]
+      Nat.doubleFactorial_add_one, Nat.choose_zero_right, Nat.choose_one_right, pow_succ']
     push_cast
     ring
   | n + 1, k + 1 =>
@@ -211,7 +211,7 @@ theorem coeff_hermite_explicit :
       -- Factor out (-1)'s.
       rw [mul_comm (↑k + _), sub_eq_add_neg]
       nth_rw 3 [neg_eq_neg_one_mul]
-      simp only [mul_assoc, ← mul_add, pow_succ]
+      simp only [mul_assoc, ← mul_add, pow_succ']
       congr 2
       -- Factor out double factorials.
       norm_cast

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

@@ -168,14 +168,14 @@ theorem hermite_monic (n : ℕ) : (hermite n).Monic :=
 theorem coeff_hermite_of_odd_add {n k : ℕ} (hnk : Odd (n + k)) : coeff (hermite n) k = 0 :=
   by
   induction' n with n ih generalizing k
-  · rw [zero_add] at hnk 
+  · rw [zero_add] at hnk
     exact coeff_hermite_of_lt hnk.pos
   · cases k
-    · rw [Nat.succ_add_eq_add_succ] at hnk 
+    · rw [Nat.succ_add_eq_add_succ] at hnk
       rw [coeff_hermite_succ_zero, ih hnk, neg_zero]
     · rw [coeff_hermite_succ_succ, ih, ih, MulZeroClass.mul_zero, sub_zero]
-      · rwa [Nat.succ_add_eq_add_succ] at hnk 
-      · rw [(by rw [Nat.succ_add, Nat.add_succ] : n.succ + k.succ = n + k + 2)] at hnk 
+      · rwa [Nat.succ_add_eq_add_succ] at hnk
+      · rw [(by rw [Nat.succ_add, Nat.add_succ] : n.succ + k.succ = n + k + 2)] at hnk
         exact (nat.odd_add.mp hnk).mpr even_two
 #align polynomial.coeff_hermite_of_odd_add Polynomial.coeff_hermite_of_odd_add
 -/
@@ -238,7 +238,7 @@ theorem coeff_hermite_of_even_add {n k : ℕ} (hnk : Even (n + k)) :
     coeff (hermite n) k = (-1) ^ ((n - k) / 2) * (n - k - 1)‼ * Nat.choose n k :=
   by
   cases' le_or_lt k n with h_le h_lt
-  · rw [Nat.even_add, ← Nat.even_sub h_le] at hnk 
+  · rw [Nat.even_add, ← Nat.even_sub h_le] at hnk
     obtain ⟨m, hm⟩ := hnk
     rw [(by linarith : n = 2 * m + k), Nat.add_sub_cancel,
       Nat.mul_div_cancel_left _ (Nat.succ_pos 1), coeff_hermite_explicit]

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

@@ -171,10 +171,10 @@ theorem coeff_hermite_of_odd_add {n k : ℕ} (hnk : Odd (n + k)) : coeff (hermit
   · rw [zero_add] at hnk 
     exact coeff_hermite_of_lt hnk.pos
   · cases k
-    · rw [Nat.succ_add_eq_succ_add] at hnk 
+    · rw [Nat.succ_add_eq_add_succ] at hnk 
       rw [coeff_hermite_succ_zero, ih hnk, neg_zero]
     · rw [coeff_hermite_succ_succ, ih, ih, MulZeroClass.mul_zero, sub_zero]
-      · rwa [Nat.succ_add_eq_succ_add] at hnk 
+      · rwa [Nat.succ_add_eq_add_succ] at hnk 
       · rw [(by rw [Nat.succ_add, Nat.add_succ] : n.succ + k.succ = n + k + 2)] at hnk 
         exact (nat.odd_add.mp hnk).mpr even_two
 #align polynomial.coeff_hermite_of_odd_add Polynomial.coeff_hermite_of_odd_add

mathlib commit https://github.com/leanprover-community/mathlib/commit/ce64cd319bb6b3e82f31c2d38e79080d377be451

@@ -3,9 +3,9 @@ Copyright (c) 2023 Luke Mantle. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Luke Mantle
 -/
-import Mathbin.Data.Polynomial.Derivative
-import Mathbin.Data.Nat.Parity
-import Mathbin.Data.Nat.Factorial.DoubleFactorial
+import Data.Polynomial.Derivative
+import Data.Nat.Parity
+import Data.Nat.Factorial.DoubleFactorial
 
 #align_import ring_theory.polynomial.hermite.basic from "leanprover-community/mathlib"@"d07a9c875ed7139abfde6a333b2be205c5bd404e"
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/8ea5598db6caeddde6cb734aa179cc2408dbd345

@@ -2,16 +2,13 @@
 Copyright (c) 2023 Luke Mantle. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Luke Mantle
-
-! This file was ported from Lean 3 source module ring_theory.polynomial.hermite.basic
-! leanprover-community/mathlib commit d07a9c875ed7139abfde6a333b2be205c5bd404e
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.Data.Polynomial.Derivative
 import Mathbin.Data.Nat.Parity
 import Mathbin.Data.Nat.Factorial.DoubleFactorial
 
+#align_import ring_theory.polynomial.hermite.basic from "leanprover-community/mathlib"@"d07a9c875ed7139abfde6a333b2be205c5bd404e"
+
 /-!
 # Hermite polynomials
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/9f55d0d4363ae59948c33864cbc52e0b12e0e8ce

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Luke Mantle
 
 ! This file was ported from Lean 3 source module ring_theory.polynomial.hermite.basic
-! leanprover-community/mathlib commit 938d3db9c278f8a52c0f964a405806f0f2b09b74
+! leanprover-community/mathlib commit d07a9c875ed7139abfde6a333b2be205c5bd404e
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -15,6 +15,9 @@ import Mathbin.Data.Nat.Factorial.DoubleFactorial
 /-!
 # Hermite polynomials
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 This file defines `polynomial.hermite n`, the nth probabilist's Hermite polynomial.
 
 ## Main definitions

mathlib commit https://github.com/leanprover-community/mathlib/commit/9fb8964792b4237dac6200193a0d533f1b3f7423

@@ -47,36 +47,46 @@ open Polynomial
 
 namespace Polynomial
 
+#print Polynomial.hermite /-
 /-- the nth probabilist's Hermite polynomial -/
 noncomputable def hermite : ℕ → Polynomial ℤ
   | 0 => 1
   | n + 1 => X * hermite n - (hermite n).derivative
 #align polynomial.hermite Polynomial.hermite
+-/
 
+#print Polynomial.hermite_succ /-
 /-- The recursion `hermite (n+1) = (x - d/dx) (hermite n)` -/
 @[simp]
 theorem hermite_succ (n : ℕ) : hermite (n + 1) = X * hermite n - (hermite n).derivative := by
   rw [hermite]
 #align polynomial.hermite_succ Polynomial.hermite_succ
+-/
 
+#print Polynomial.hermite_eq_iterate /-
 theorem hermite_eq_iterate (n : ℕ) : hermite n = ((fun p => X * p - p.derivative)^[n]) 1 :=
   by
   induction' n with n ih
   · rfl
   · rw [Function.iterate_succ_apply', ← ih, hermite_succ]
 #align polynomial.hermite_eq_iterate Polynomial.hermite_eq_iterate
+-/
 
+#print Polynomial.hermite_zero /-
 @[simp]
 theorem hermite_zero : hermite 0 = C 1 :=
   rfl
 #align polynomial.hermite_zero Polynomial.hermite_zero
+-/
 
+#print Polynomial.hermite_one /-
 @[simp]
 theorem hermite_one : hermite 1 = X :=
   by
   rw [hermite_succ, hermite_zero]
   simp only [map_one, mul_one, derivative_one, sub_zero]
 #align polynomial.hermite_one Polynomial.hermite_one
+-/
 
 /-! ### Lemmas about `polynomial.coeff` -/
 
@@ -148,9 +158,11 @@ theorem leadingCoeff_hermite (n : ℕ) : (hermite n).leadingCoeff = 1 := by
 #align polynomial.leading_coeff_hermite Polynomial.leadingCoeff_hermite
 -/
 
+#print Polynomial.hermite_monic /-
 theorem hermite_monic (n : ℕ) : (hermite n).Monic :=
   leadingCoeff_hermite n
 #align polynomial.hermite_monic Polynomial.hermite_monic
+-/
 
 #print Polynomial.coeff_hermite_of_odd_add /-
 theorem coeff_hermite_of_odd_add {n k : ℕ} (hnk : Odd (n + k)) : coeff (hermite n) k = 0 :=
@@ -174,6 +186,7 @@ section CoeffExplicit
 
 open scoped Nat
 
+#print Polynomial.coeff_hermite_explicit /-
 /-- Because of `coeff_hermite_of_odd_add`, every nonzero coefficient is described as follows. -/
 theorem coeff_hermite_explicit :
     ∀ n k : ℕ, coeff (hermite (2 * n + k)) k = (-1) ^ n * (2 * n - 1)‼ * Nat.choose (2 * n + k) k
@@ -218,7 +231,9 @@ theorem coeff_hermite_explicit :
     · rw [coeff_hermite_explicit (n + 1) k]
     · rw [(by ring : 2 * (n + 1) + k = 2 * n + (k + 2)), coeff_hermite_explicit n (k + 2)]
 #align polynomial.coeff_hermite_explicit Polynomial.coeff_hermite_explicit
+-/
 
+#print Polynomial.coeff_hermite_of_even_add /-
 theorem coeff_hermite_of_even_add {n k : ℕ} (hnk : Even (n + k)) :
     coeff (hermite n) k = (-1) ^ ((n - k) / 2) * (n - k - 1)‼ * Nat.choose n k :=
   by
@@ -229,7 +244,9 @@ theorem coeff_hermite_of_even_add {n k : ℕ} (hnk : Even (n + k)) :
       Nat.mul_div_cancel_left _ (Nat.succ_pos 1), coeff_hermite_explicit]
   · simp [Nat.choose_eq_zero_of_lt h_lt, coeff_hermite_of_lt h_lt]
 #align polynomial.coeff_hermite_of_even_add Polynomial.coeff_hermite_of_even_add
+-/
 
+#print Polynomial.coeff_hermite /-
 theorem coeff_hermite (n k : ℕ) :
     coeff (hermite n) k =
       if Even (n + k) then (-1) ^ ((n - k) / 2) * (n - k - 1)‼ * Nat.choose n k else 0 :=
@@ -238,6 +255,7 @@ theorem coeff_hermite (n k : ℕ) :
   exact coeff_hermite_of_even_add h
   exact coeff_hermite_of_odd_add (nat.odd_iff_not_even.mpr h)
 #align polynomial.coeff_hermite Polynomial.coeff_hermite
+-/
 
 end CoeffExplicit
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/9fb8964792b4237dac6200193a0d533f1b3f7423

@@ -83,17 +83,22 @@ theorem hermite_one : hermite 1 = X :=
 
 section Coeff
 
+#print Polynomial.coeff_hermite_succ_zero /-
 theorem coeff_hermite_succ_zero (n : ℕ) : coeff (hermite (n + 1)) 0 = -coeff (hermite n) 1 := by
   simp [coeff_derivative]
 #align polynomial.coeff_hermite_succ_zero Polynomial.coeff_hermite_succ_zero
+-/
 
+#print Polynomial.coeff_hermite_succ_succ /-
 theorem coeff_hermite_succ_succ (n k : ℕ) :
     coeff (hermite (n + 1)) (k + 1) = coeff (hermite n) k - (k + 2) * coeff (hermite n) (k + 2) :=
   by
   rw [hermite_succ, coeff_sub, coeff_X_mul, coeff_derivative, mul_comm]
   norm_cast
 #align polynomial.coeff_hermite_succ_succ Polynomial.coeff_hermite_succ_succ
+-/
 
+#print Polynomial.coeff_hermite_of_lt /-
 theorem coeff_hermite_of_lt {n k : ℕ} (hnk : n < k) : coeff (hermite n) k = 0 :=
   by
   obtain ⟨k, rfl⟩ := Nat.exists_eq_add_of_lt hnk
@@ -104,7 +109,9 @@ theorem coeff_hermite_of_lt {n k : ℕ} (hnk : n < k) : coeff (hermite n) k = 0
     rw [Nat.succ_eq_add_one, coeff_hermite_succ_succ, add_right_comm, this, ih k, ih (k + 2),
       MulZeroClass.mul_zero, sub_zero]
 #align polynomial.coeff_hermite_of_lt Polynomial.coeff_hermite_of_lt
+-/
 
+#print Polynomial.coeff_hermite_self /-
 @[simp]
 theorem coeff_hermite_self (n : ℕ) : coeff (hermite n) n = 1 :=
   by
@@ -113,7 +120,9 @@ theorem coeff_hermite_self (n : ℕ) : coeff (hermite n) n = 1 :=
   · rw [coeff_hermite_succ_succ, ih, coeff_hermite_of_lt, MulZeroClass.mul_zero, sub_zero]
     simp
 #align polynomial.coeff_hermite_self Polynomial.coeff_hermite_self
+-/
 
+#print Polynomial.degree_hermite /-
 @[simp]
 theorem degree_hermite (n : ℕ) : (hermite n).degree = n :=
   by
@@ -123,21 +132,27 @@ theorem degree_hermite (n : ℕ) : (hermite n).degree = n :=
     exact coeff_hermite_of_lt
   · simp [coeff_hermite_self n]
 #align polynomial.degree_hermite Polynomial.degree_hermite
+-/
 
+#print Polynomial.natDegree_hermite /-
 @[simp]
 theorem natDegree_hermite {n : ℕ} : (hermite n).natDegree = n :=
   natDegree_eq_of_degree_eq_some (degree_hermite n)
 #align polynomial.nat_degree_hermite Polynomial.natDegree_hermite
+-/
 
+#print Polynomial.leadingCoeff_hermite /-
 @[simp]
 theorem leadingCoeff_hermite (n : ℕ) : (hermite n).leadingCoeff = 1 := by
   rw [← coeff_nat_degree, nat_degree_hermite, coeff_hermite_self]
 #align polynomial.leading_coeff_hermite Polynomial.leadingCoeff_hermite
+-/
 
 theorem hermite_monic (n : ℕ) : (hermite n).Monic :=
   leadingCoeff_hermite n
 #align polynomial.hermite_monic Polynomial.hermite_monic
 
+#print Polynomial.coeff_hermite_of_odd_add /-
 theorem coeff_hermite_of_odd_add {n k : ℕ} (hnk : Odd (n + k)) : coeff (hermite n) k = 0 :=
   by
   induction' n with n ih generalizing k
@@ -151,6 +166,7 @@ theorem coeff_hermite_of_odd_add {n k : ℕ} (hnk : Odd (n + k)) : coeff (hermit
       · rw [(by rw [Nat.succ_add, Nat.add_succ] : n.succ + k.succ = n + k + 2)] at hnk 
         exact (nat.odd_add.mp hnk).mpr even_two
 #align polynomial.coeff_hermite_of_odd_add Polynomial.coeff_hermite_of_odd_add
+-/
 
 end Coeff
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/cca40788df1b8755d5baf17ab2f27dacc2e17acb

@@ -141,14 +141,14 @@ theorem hermite_monic (n : ℕ) : (hermite n).Monic :=
 theorem coeff_hermite_of_odd_add {n k : ℕ} (hnk : Odd (n + k)) : coeff (hermite n) k = 0 :=
   by
   induction' n with n ih generalizing k
-  · rw [zero_add] at hnk
+  · rw [zero_add] at hnk 
     exact coeff_hermite_of_lt hnk.pos
   · cases k
-    · rw [Nat.succ_add_eq_succ_add] at hnk
+    · rw [Nat.succ_add_eq_succ_add] at hnk 
       rw [coeff_hermite_succ_zero, ih hnk, neg_zero]
     · rw [coeff_hermite_succ_succ, ih, ih, MulZeroClass.mul_zero, sub_zero]
-      · rwa [Nat.succ_add_eq_succ_add] at hnk
-      · rw [(by rw [Nat.succ_add, Nat.add_succ] : n.succ + k.succ = n + k + 2)] at hnk
+      · rwa [Nat.succ_add_eq_succ_add] at hnk 
+      · rw [(by rw [Nat.succ_add, Nat.add_succ] : n.succ + k.succ = n + k + 2)] at hnk 
         exact (nat.odd_add.mp hnk).mpr even_two
 #align polynomial.coeff_hermite_of_odd_add Polynomial.coeff_hermite_of_odd_add
 
@@ -207,7 +207,7 @@ theorem coeff_hermite_of_even_add {n k : ℕ} (hnk : Even (n + k)) :
     coeff (hermite n) k = (-1) ^ ((n - k) / 2) * (n - k - 1)‼ * Nat.choose n k :=
   by
   cases' le_or_lt k n with h_le h_lt
-  · rw [Nat.even_add, ← Nat.even_sub h_le] at hnk
+  · rw [Nat.even_add, ← Nat.even_sub h_le] at hnk 
     obtain ⟨m, hm⟩ := hnk
     rw [(by linarith : n = 2 * m + k), Nat.add_sub_cancel,
       Nat.mul_div_cancel_left _ (Nat.succ_pos 1), coeff_hermite_explicit]

mathlib commit https://github.com/leanprover-community/mathlib/commit/cca40788df1b8755d5baf17ab2f27dacc2e17acb

@@ -4,12 +4,13 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Luke Mantle
 
 ! This file was ported from Lean 3 source module ring_theory.polynomial.hermite.basic
-! leanprover-community/mathlib commit 066ecdb4834c7a4693e0f0e5154935a6f3d3f90c
+! leanprover-community/mathlib commit 938d3db9c278f8a52c0f964a405806f0f2b09b74
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
 import Mathbin.Data.Polynomial.Derivative
 import Mathbin.Data.Nat.Parity
+import Mathbin.Data.Nat.Factorial.DoubleFactorial
 
 /-!
 # Hermite polynomials
@@ -24,9 +25,14 @@ This file defines `polynomial.hermite n`, the nth probabilist's Hermite polynomi
 ## Results
 
 * `polynomial.hermite_succ`: the recursion `hermite (n+1) = (x - d/dx) (hermite n)`
+* `polynomial.coeff_hermite_explicit`: a closed formula for (nonvanishing) coefficients in terms
+  of binomial coefficients and double factorials.
 * `polynomial.coeff_hermite_of_odd_add`: for `n`,`k` where `n+k` is odd, `(hermite n).coeff k` is
   zero.
+* `polynomial.coeff_hermite_of_even_add`: a closed formula for `(hermite n).coeff k` when `n+k` is
+  even, equivalent to `polynomial.coeff_hermite_explicit`.
 * `polynomial.monic_hermite`: for all `n`, `hermite n` is monic.
+* `polynomial.degree_hermite`: for all `n`, `hermite n` has degree `n`.
 
 ## References
 
@@ -148,5 +154,76 @@ theorem coeff_hermite_of_odd_add {n k : ℕ} (hnk : Odd (n + k)) : coeff (hermit
 
 end Coeff
 
+section CoeffExplicit
+
+open scoped Nat
+
+/-- Because of `coeff_hermite_of_odd_add`, every nonzero coefficient is described as follows. -/
+theorem coeff_hermite_explicit :
+    ∀ n k : ℕ, coeff (hermite (2 * n + k)) k = (-1) ^ n * (2 * n - 1)‼ * Nat.choose (2 * n + k) k
+  | 0, _ => by simp
+  | n + 1, 0 => by
+    convert coeff_hermite_succ_zero (2 * n + 1) using 1
+    rw [coeff_hermite_explicit n 1, (by ring_nf : 2 * (n + 1) - 1 = 2 * n + 1),
+      Nat.doubleFactorial_add_one, Nat.choose_zero_right, Nat.choose_one_right, pow_succ]
+    push_cast
+    ring
+  | n + 1, k + 1 =>
+    by
+    let hermite_explicit : ℕ → ℕ → ℤ := fun n k =>
+      (-1) ^ n * (2 * n - 1)‼ * Nat.choose (2 * n + k) k
+    have hermite_explicit_recur :
+      ∀ n k : ℕ,
+        hermite_explicit (n + 1) (k + 1) =
+          hermite_explicit (n + 1) k - (k + 2) * hermite_explicit n (k + 2) :=
+      by
+      intro n k
+      simp only [hermite_explicit]
+      -- Factor out (-1)'s.
+      rw [mul_comm (↑k + _), sub_eq_add_neg]
+      nth_rw 3 [neg_eq_neg_one_mul]
+      simp only [mul_assoc, ← mul_add, pow_succ]
+      congr 2
+      -- Factor out double factorials.
+      norm_cast
+      rw [(by ring_nf : 2 * (n + 1) - 1 = 2 * n + 1), Nat.doubleFactorial_add_one,
+        mul_comm (2 * n + 1)]
+      simp only [mul_assoc, ← mul_add]
+      congr 1
+      -- Match up binomial coefficients using `nat.choose_succ_right_eq`.
+      rw [(by ring : 2 * (n + 1) + (k + 1) = 2 * n + 1 + (k + 1) + 1),
+        (by ring : 2 * (n + 1) + k = 2 * n + 1 + (k + 1)),
+        (by ring : 2 * n + (k + 2) = 2 * n + 1 + (k + 1))]
+      rw [Nat.choose, Nat.choose_succ_right_eq (2 * n + 1 + (k + 1)) (k + 1), Nat.add_sub_cancel]
+      ring
+    change _ = hermite_explicit _ _
+    rw [← add_assoc, coeff_hermite_succ_succ, hermite_explicit_recur]
+    congr
+    · rw [coeff_hermite_explicit (n + 1) k]
+    · rw [(by ring : 2 * (n + 1) + k = 2 * n + (k + 2)), coeff_hermite_explicit n (k + 2)]
+#align polynomial.coeff_hermite_explicit Polynomial.coeff_hermite_explicit
+
+theorem coeff_hermite_of_even_add {n k : ℕ} (hnk : Even (n + k)) :
+    coeff (hermite n) k = (-1) ^ ((n - k) / 2) * (n - k - 1)‼ * Nat.choose n k :=
+  by
+  cases' le_or_lt k n with h_le h_lt
+  · rw [Nat.even_add, ← Nat.even_sub h_le] at hnk
+    obtain ⟨m, hm⟩ := hnk
+    rw [(by linarith : n = 2 * m + k), Nat.add_sub_cancel,
+      Nat.mul_div_cancel_left _ (Nat.succ_pos 1), coeff_hermite_explicit]
+  · simp [Nat.choose_eq_zero_of_lt h_lt, coeff_hermite_of_lt h_lt]
+#align polynomial.coeff_hermite_of_even_add Polynomial.coeff_hermite_of_even_add
+
+theorem coeff_hermite (n k : ℕ) :
+    coeff (hermite n) k =
+      if Even (n + k) then (-1) ^ ((n - k) / 2) * (n - k - 1)‼ * Nat.choose n k else 0 :=
+  by
+  split_ifs with h
+  exact coeff_hermite_of_even_add h
+  exact coeff_hermite_of_odd_add (nat.odd_iff_not_even.mpr h)
+#align polynomial.coeff_hermite Polynomial.coeff_hermite
+
+end CoeffExplicit
+
 end Polynomial
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/403190b5419b3f03f1a2893ad9352ca7f7d8bff6

@@ -3,8 +3,8 @@ Copyright (c) 2023 Luke Mantle. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Luke Mantle
 
-! This file was ported from Lean 3 source module ring_theory.polynomial.hermite
-! leanprover-community/mathlib commit b17e272059b7ba6d9ba850e274b08d2a2cde3ccf
+! This file was ported from Lean 3 source module ring_theory.polynomial.hermite.basic
+! leanprover-community/mathlib commit 066ecdb4834c7a4693e0f0e5154935a6f3d3f90c
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -23,6 +23,7 @@ This file defines `polynomial.hermite n`, the nth probabilist's Hermite polynomi
 
 ## Results
 
+* `polynomial.hermite_succ`: the recursion `hermite (n+1) = (x - d/dx) (hermite n)`
 * `polynomial.coeff_hermite_of_odd_add`: for `n`,`k` where `n+k` is odd, `(hermite n).coeff k` is
   zero.
 * `polynomial.monic_hermite`: for all `n`, `hermite n` is monic.
@@ -46,6 +47,7 @@ noncomputable def hermite : ℕ → Polynomial ℤ
   | n + 1 => X * hermite n - (hermite n).derivative
 #align polynomial.hermite Polynomial.hermite
 
+/-- The recursion `hermite (n+1) = (x - d/dx) (hermite n)` -/
 @[simp]
 theorem hermite_succ (n : ℕ) : hermite (n + 1) = X * hermite n - (hermite n).derivative := by
   rw [hermite]

A PR analogous to #12338 and #12361: reformatting proofs following the multiple goals linter of #12339.

@@ -110,8 +110,8 @@ theorem coeff_hermite_self (n : ℕ) : coeff (hermite n) n = 1 := by
 @[simp]
 theorem degree_hermite (n : ℕ) : (hermite n).degree = n := by
   rw [degree_eq_of_le_of_coeff_ne_zero]
-  simp_rw [degree_le_iff_coeff_zero, Nat.cast_lt]
-  · rintro m hnm
+  · simp_rw [degree_le_iff_coeff_zero, Nat.cast_lt]
+    rintro m hnm
     exact coeff_hermite_of_lt hnm
   · simp [coeff_hermite_self n]
 #align polynomial.degree_hermite Polynomial.degree_hermite
@@ -210,8 +210,8 @@ theorem coeff_hermite (n k : ℕ) :
     coeff (hermite n) k =
       if Even (n + k) then (-1 : ℤ) ^ ((n - k) / 2) * (n - k - 1)‼ * Nat.choose n k else 0 := by
   split_ifs with h
-  exact coeff_hermite_of_even_add h
-  exact coeff_hermite_of_odd_add (Nat.odd_iff_not_even.mpr h)
+  · exact coeff_hermite_of_even_add h
+  · exact coeff_hermite_of_odd_add (Nat.odd_iff_not_even.mpr h)
 #align polynomial.coeff_hermite Polynomial.coeff_hermite
 
 end CoeffExplicit

Polynomial and MvPolynomial are algebraic objects, hence should be under Algebra (or at least not under Data)

@@ -3,7 +3,7 @@ Copyright (c) 2023 Luke Mantle. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Luke Mantle
 -/
-import Mathlib.Data.Polynomial.Derivative
+import Mathlib.Algebra.Polynomial.Derivative
 import Mathlib.Data.Nat.Parity
 import Mathlib.Data.Nat.Factorial.DoubleFactorial
 

We change the following field in the definition of an additive commutative monoid:

 nsmul_succ : ∀ (n : ℕ) (x : G),
-  AddMonoid.nsmul (n + 1) x = x + AddMonoid.nsmul n x
+  AddMonoid.nsmul (n + 1) x = AddMonoid.nsmul n x + x

where the latter is more natural

We adjust the definitions of ^ in monoids, groups, etc. Originally there was a warning comment about why this natural order was preferred

use x * npowRec n x and not npowRec n x * x in the definition to make sure that definitional unfolding of npowRec is blocked, to avoid deep recursion issues.

but it seems to no longer apply.

Remarks on the PR :

• pow_succ and pow_succ' have switched their meanings.
• Most of the time, the proofs were adjusted by priming/unpriming one lemma, or exchanging left and right; a few proofs were more complicated to adjust.
• In particular, [Mathlib/NumberTheory/RamificationInertia.lean] used Ideal.IsPrime.mul_mem_pow which is defined in [Mathlib/RingTheory/DedekindDomain/Ideal.lean]. Changing the order of operation forced me to add the symmetric lemma Ideal.IsPrime.mem_pow_mul.
• the docstring for Cauchy condensation test in [Mathlib/Analysis/PSeries.lean] was mathematically incorrect, I added the mention that the function is antitone.

@@ -173,7 +173,7 @@ theorem coeff_hermite_explicit :
       -- Factor out (-1)'s.
       rw [mul_comm (↑k + _ : ℤ), sub_eq_add_neg]
       nth_rw 3 [neg_eq_neg_one_mul]
-      simp only [mul_assoc, ← mul_add, pow_succ]
+      simp only [mul_assoc, ← mul_add, pow_succ']
       congr 2
       -- Factor out double factorials.
       norm_cast

The termination checker has been getting more capable, and many of the termination_by or decreasing_by clauses in Mathlib are no longer needed.

(Note that termination_by? will show the automatically derived termination expression, so no information is being lost by removing these.)

Co-authored-by: Scott Morrison <scott.morrison@gmail.com>

@@ -193,8 +193,6 @@ theorem coeff_hermite_explicit :
     congr
     · rw [coeff_hermite_explicit (n + 1) k]
     · rw [(by ring : 2 * (n + 1) + k = 2 * n + (k + 2)), coeff_hermite_explicit n (k + 2)]
--- Porting note: Lean 3 worked this out automatically
-termination_by n k => (n, k)
 #align polynomial.coeff_hermite_explicit Polynomial.coeff_hermite_explicit
 
 theorem coeff_hermite_of_even_add {n k : ℕ} (hnk : Even (n + k)) :

Homogenises porting notes via capitalisation and addition of whitespace.

It makes the following changes:

• converts "--porting note" into "-- Porting note";
• converts "porting note" into "Porting note".

@@ -155,7 +155,7 @@ theorem coeff_hermite_explicit :
   | 0, _ => by simp
   | n + 1, 0 => by
     convert coeff_hermite_succ_zero (2 * n + 1) using 1
-    -- porting note: ring_nf did not solve the goal on line 165
+    -- Porting note: ring_nf did not solve the goal on line 165
     rw [coeff_hermite_explicit n 1, (by rw [Nat.left_distrib, mul_one, Nat.add_one_sub_one] :
       2 * (n + 1) - 1 = 2 * n + 1), Nat.doubleFactorial_add_one, Nat.choose_zero_right,
       Nat.choose_one_right, pow_succ]
@@ -177,7 +177,7 @@ theorem coeff_hermite_explicit :
       congr 2
       -- Factor out double factorials.
       norm_cast
-      -- porting note: ring_nf did not solve the goal on line 186
+      -- Porting note: ring_nf did not solve the goal on line 186
       rw [(by rw [Nat.left_distrib, mul_one, Nat.add_one_sub_one] : 2 * (n + 1) - 1 = 2 * n + 1),
         Nat.doubleFactorial_add_one, mul_comm (2 * n + 1)]
       simp only [mul_assoc, ← mul_add]
@@ -193,7 +193,7 @@ theorem coeff_hermite_explicit :
     congr
     · rw [coeff_hermite_explicit (n + 1) k]
     · rw [(by ring : 2 * (n + 1) + k = 2 * n + (k + 2)), coeff_hermite_explicit n (k + 2)]
--- porting note: Lean 3 worked this out automatically
+-- Porting note: Lean 3 worked this out automatically
 termination_by n k => (n, k)
 #align polynomial.coeff_hermite_explicit Polynomial.coeff_hermite_explicit
 
@@ -202,7 +202,7 @@ theorem coeff_hermite_of_even_add {n k : ℕ} (hnk : Even (n + k)) :
   rcases le_or_lt k n with h_le | h_lt
   · rw [Nat.even_add, ← Nat.even_sub h_le] at hnk
     obtain ⟨m, hm⟩ := hnk
-    -- porting note: linarith failed to find a contradiction by itself
+    -- Porting note: linarith failed to find a contradiction by itself
     rw [(by omega : n = 2 * m + k),
       Nat.add_sub_cancel, Nat.mul_div_cancel_left _ (Nat.succ_pos 1), coeff_hermite_explicit]
   · simp [Nat.choose_eq_zero_of_lt h_lt, coeff_hermite_of_lt h_lt]

I ran tryAtEachStep on all files under Mathlib to find all locations where omega succeeds. For each that was a linarith without an only, I tried replacing it with omega, and I verified that elaboration time got smaller. (In almost all cases, there was a noticeable speedup.) I also replaced some slow aesops along the way.

@@ -203,7 +203,7 @@ theorem coeff_hermite_of_even_add {n k : ℕ} (hnk : Even (n + k)) :
   · rw [Nat.even_add, ← Nat.even_sub h_le] at hnk
     obtain ⟨m, hm⟩ := hnk
     -- porting note: linarith failed to find a contradiction by itself
-    rw [(by linarith [by rwa [Nat.sub_eq_iff_eq_add h_le] at hm] : n = 2 * m + k),
+    rw [(by omega : n = 2 * m + k),
       Nat.add_sub_cancel, Nat.mul_div_cancel_left _ (Nat.succ_pos 1), coeff_hermite_explicit]
   · simp [Nat.choose_eq_zero_of_lt h_lt, coeff_hermite_of_lt h_lt]
 #align polynomial.coeff_hermite_of_even_add Polynomial.coeff_hermite_of_even_add

Co-authored-by: Patrick Massot <patrickmassot@free.fr> Co-authored-by: Scott Morrison <scott.morrison@gmail.com>

@@ -169,7 +169,7 @@ theorem coeff_hermite_explicit :
         hermite_explicit (n + 1) (k + 1) =
           hermite_explicit (n + 1) k - (k + 2) * hermite_explicit n (k + 2) := by
       intro n k
-      simp only
+      simp only [hermite_explicit]
       -- Factor out (-1)'s.
       rw [mul_comm (↑k + _ : ℤ), sub_eq_add_neg]
       nth_rw 3 [neg_eq_neg_one_mul]

Classify by adding issue number (#10618) to porting notes claiming anything semantically equivalent to simp can prove this or simp can simplify this.

@@ -67,7 +67,7 @@ theorem hermite_zero : hermite 0 = C 1 :=
   rfl
 #align polynomial.hermite_zero Polynomial.hermite_zero
 
--- Porting note: There was initially @[simp] on this line but it was removed
+-- Porting note (#10618): There was initially @[simp] on this line but it was removed
 -- because simp can prove this theorem
 theorem hermite_one : hermite 1 = X := by
   rw [hermite_succ, hermite_zero]

Co-authored-by: Scott Morrison <scott.morrison@gmail.com> Co-authored-by: Eric Wieser <wieser.eric@gmail.com> Co-authored-by: Joachim Breitner <mail@joachim-breitner.de>

@@ -186,20 +186,15 @@ theorem coeff_hermite_explicit :
       rw [(by ring : 2 * (n + 1) + (k + 1) = 2 * n + 1 + (k + 1) + 1),
         (by ring : 2 * (n + 1) + k = 2 * n + 1 + (k + 1)),
         (by ring : 2 * n + (k + 2) = 2 * n + 1 + (k + 1))]
-      rw [Nat.choose, Nat.choose_succ_right_eq (2 * n + 1 + (k + 1)) (k + 1), Nat.add_sub_cancel,
-        Int.negSucc_eq]
-      -- porting note: ring could not solve the goal so the lines 195, 198-200 were added.
-      ring_nf
-      simp only [sub_eq_add_neg, ← neg_mul, ← right_distrib _ _ ((-(1 : ℤ)) ^ n), ← neg_add]
-      norm_cast
-      simp only [← add_assoc, add_comm]
+      rw [Nat.choose, Nat.choose_succ_right_eq (2 * n + 1 + (k + 1)) (k + 1), Nat.add_sub_cancel]
+      ring
     change _ = hermite_explicit _ _
     rw [← add_assoc, coeff_hermite_succ_succ, hermite_explicit_recur]
     congr
     · rw [coeff_hermite_explicit (n + 1) k]
     · rw [(by ring : 2 * (n + 1) + k = 2 * n + (k + 2)), coeff_hermite_explicit n (k + 2)]
 -- porting note: Lean 3 worked this out automatically
-termination_by _ n k => (n, k)
+termination_by n k => (n, k)
 #align polynomial.coeff_hermite_explicit Polynomial.coeff_hermite_explicit
 
 theorem coeff_hermite_of_even_add {n k : ℕ} (hnk : Even (n + k)) :

I literally went through and regex'd some uses of cases', replacing them with rcases; this is meant to be a low effort PR as I hope that tools can do this in the future.

rcases is an easier replacement than cases, though with better tools we could in future do a second pass converting simple rcases added here (and existing ones) to cases.

@@ -204,7 +204,7 @@ termination_by _ n k => (n, k)
 
 theorem coeff_hermite_of_even_add {n k : ℕ} (hnk : Even (n + k)) :
     coeff (hermite n) k = (-1) ^ ((n - k) / 2) * (n - k - 1)‼ * Nat.choose n k := by
-  cases' le_or_lt k n with h_le h_lt
+  rcases le_or_lt k n with h_le | h_lt
   · rw [Nat.even_add, ← Nat.even_sub h_le] at hnk
     obtain ⟨m, hm⟩ := hnk
     -- porting note: linarith failed to find a contradiction by itself

@@ -135,10 +135,10 @@ theorem coeff_hermite_of_odd_add {n k : ℕ} (hnk : Odd (n + k)) : coeff (hermit
   · rw [Nat.zero_eq, zero_add k] at hnk
     exact coeff_hermite_of_lt hnk.pos
   · cases' k with k
-    · rw [Nat.succ_add_eq_succ_add] at hnk
+    · rw [Nat.succ_add_eq_add_succ] at hnk
       rw [coeff_hermite_succ_zero, ih hnk, neg_zero]
     · rw [coeff_hermite_succ_succ, ih, ih, mul_zero, sub_zero]
-      · rwa [Nat.succ_add_eq_succ_add] at hnk
+      · rwa [Nat.succ_add_eq_add_succ] at hnk
       · rw [(by rw [Nat.succ_add, Nat.add_succ] : n.succ + k.succ = n + k + 2)] at hnk
         exact (Nat.odd_add.mp hnk).mpr even_two
 #align polynomial.coeff_hermite_of_odd_add Polynomial.coeff_hermite_of_odd_add

@@ -156,7 +156,7 @@ theorem coeff_hermite_explicit :
   | n + 1, 0 => by
     convert coeff_hermite_succ_zero (2 * n + 1) using 1
     -- porting note: ring_nf did not solve the goal on line 165
-    rw [coeff_hermite_explicit n 1, (by rw [Nat.left_distrib, mul_one, Nat.succ_sub_one] :
+    rw [coeff_hermite_explicit n 1, (by rw [Nat.left_distrib, mul_one, Nat.add_one_sub_one] :
       2 * (n + 1) - 1 = 2 * n + 1), Nat.doubleFactorial_add_one, Nat.choose_zero_right,
       Nat.choose_one_right, pow_succ]
     push_cast
@@ -178,7 +178,7 @@ theorem coeff_hermite_explicit :
       -- Factor out double factorials.
       norm_cast
       -- porting note: ring_nf did not solve the goal on line 186
-      rw [(by rw [Nat.left_distrib, mul_one, Nat.succ_sub_one] : 2 * (n + 1) - 1 = 2 * n + 1),
+      rw [(by rw [Nat.left_distrib, mul_one, Nat.add_one_sub_one] : 2 * (n + 1) - 1 = 2 * n + 1),
         Nat.doubleFactorial_add_one, mul_comm (2 * n + 1)]
       simp only [mul_assoc, ← mul_add]
       congr 1

Search&replace MulZeroClass.mul_zero -> mul_zero, MulZeroClass.zero_mul -> zero_mul.

These were introduced by Mathport, as the full name of mul_zero is actually MulZeroClass.mul_zero (it's exported with the short name).

@@ -96,14 +96,14 @@ theorem coeff_hermite_of_lt {n k : ℕ} (hnk : n < k) : coeff (hermite n) k = 0
   · apply coeff_C
   · have : n + k + 1 + 2 = n + (k + 2) + 1 := by ring
     rw [Nat.succ_eq_add_one, coeff_hermite_succ_succ, add_right_comm, this, ih k, ih (k + 2),
-      MulZeroClass.mul_zero, sub_zero]
+      mul_zero, sub_zero]
 #align polynomial.coeff_hermite_of_lt Polynomial.coeff_hermite_of_lt
 
 @[simp]
 theorem coeff_hermite_self (n : ℕ) : coeff (hermite n) n = 1 := by
   induction' n with n ih
   · apply coeff_C
-  · rw [coeff_hermite_succ_succ, ih, coeff_hermite_of_lt, MulZeroClass.mul_zero, sub_zero]
+  · rw [coeff_hermite_succ_succ, ih, coeff_hermite_of_lt, mul_zero, sub_zero]
     simp
 #align polynomial.coeff_hermite_self Polynomial.coeff_hermite_self
 
@@ -137,7 +137,7 @@ theorem coeff_hermite_of_odd_add {n k : ℕ} (hnk : Odd (n + k)) : coeff (hermit
   · cases' k with k
     · rw [Nat.succ_add_eq_succ_add] at hnk
       rw [coeff_hermite_succ_zero, ih hnk, neg_zero]
-    · rw [coeff_hermite_succ_succ, ih, ih, MulZeroClass.mul_zero, sub_zero]
+    · rw [coeff_hermite_succ_succ, ih, ih, mul_zero, sub_zero]
       · rwa [Nat.succ_add_eq_succ_add] at hnk
       · rw [(by rw [Nat.succ_add, Nat.add_succ] : n.succ + k.succ = n + k + 2)] at hnk
         exact (Nat.odd_add.mp hnk).mpr even_two

• depends on: #5966

Co-authored-by: Eric Wieser <wieser.eric@gmail.com> Co-authored-by: Scott Morrison <scott.morrison@gmail.com>

@@ -2,16 +2,13 @@
 Copyright (c) 2023 Luke Mantle. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Luke Mantle
-
-! This file was ported from Lean 3 source module ring_theory.polynomial.hermite.basic
-! leanprover-community/mathlib commit 938d3db9c278f8a52c0f964a405806f0f2b09b74
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.Data.Polynomial.Derivative
 import Mathlib.Data.Nat.Parity
 import Mathlib.Data.Nat.Factorial.DoubleFactorial
 
+#align_import ring_theory.polynomial.hermite.basic from "leanprover-community/mathlib"@"938d3db9c278f8a52c0f964a405806f0f2b09b74"
+
 /-!
 # Hermite polynomials
 

@@ -59,7 +59,7 @@ theorem hermite_succ (n : ℕ) : hermite (n + 1) = X * hermite n - derivative (h
   rw [hermite]
 #align polynomial.hermite_succ Polynomial.hermite_succ
 
-theorem hermite_eq_iterate (n : ℕ) : hermite n = ((fun p => X * p - derivative p)^[n]) 1 := by
+theorem hermite_eq_iterate (n : ℕ) : hermite n = (fun p => X * p - derivative p)^[n] 1 := by
   induction' n with n ih
   · rfl
   · rw [Function.iterate_succ_apply', ← ih, hermite_succ]

https://github.com/leanprover-community/mathlib4/actions/runs/5349279420/jobs/9700275999 shows this working successfully (i.e. the action failing as there were ring's that should be ring_nf's

We then clean up the noisy lines in mathlib

@@ -80,7 +80,7 @@ theorem hermite_one : hermite 1 = X := by
 /-! ### Lemmas about `Polynomial.coeff` -/
 
 
-section Coeff
+section coeff
 
 theorem coeff_hermite_succ_zero (n : ℕ) : coeff (hermite (n + 1)) 0 = -coeff (hermite n) 1 := by
   simp [coeff_derivative]
@@ -146,7 +146,7 @@ theorem coeff_hermite_of_odd_add {n k : ℕ} (hnk : Odd (n + k)) : coeff (hermit
         exact (Nat.odd_add.mp hnk).mpr even_two
 #align polynomial.coeff_hermite_of_odd_add Polynomial.coeff_hermite_of_odd_add
 
-end Coeff
+end coeff
 
 section CoeffExplicit
 
@@ -190,9 +190,9 @@ theorem coeff_hermite_explicit :
         (by ring : 2 * (n + 1) + k = 2 * n + 1 + (k + 1)),
         (by ring : 2 * n + (k + 2) = 2 * n + 1 + (k + 1))]
       rw [Nat.choose, Nat.choose_succ_right_eq (2 * n + 1 + (k + 1)) (k + 1), Nat.add_sub_cancel,
-      Int.negSucc_eq]
+        Int.negSucc_eq]
       -- porting note: ring could not solve the goal so the lines 195, 198-200 were added.
-      ring
+      ring_nf
       simp only [sub_eq_add_neg, ← neg_mul, ← right_distrib _ _ ((-(1 : ℤ)) ^ n), ← neg_add]
       norm_cast
       simp only [← add_assoc, add_comm]

Co-authored-by: Scott Morrison <scott.morrison@anu.edu.au>

@@ -113,11 +113,9 @@ theorem coeff_hermite_self (n : ℕ) : coeff (hermite n) n = 1 := by
 @[simp]
 theorem degree_hermite (n : ℕ) : (hermite n).degree = n := by
   rw [degree_eq_of_le_of_coeff_ne_zero]
-  simp_rw [degree_le_iff_coeff_zero]
-   -- porting note: mathlib3 also had `simp_rw [WithBot.coe_lt_coe]` but it's not firing
-   -- so we add it manually later
+  simp_rw [degree_le_iff_coeff_zero, Nat.cast_lt]
   · rintro m hnm
-    exact coeff_hermite_of_lt (WithBot.coe_lt_coe.1 hnm)
+    exact coeff_hermite_of_lt hnm
   · simp [coeff_hermite_self n]
 #align polynomial.degree_hermite Polynomial.degree_hermite
 

Co-authored-by: Emilie Uthaiwat <emiliepathum@gmail.com> Co-authored-by: EmilieUthaiwat <102412311+EmilieUthaiwat@users.noreply.github.com>