ring_theory.witt_vector.mul_p
⟷
Mathlib.RingTheory.WittVector.MulP
The following section lists changes to this file in mathlib3 and mathlib4 that occured after the initial port. Most recent changes are shown first. Hovering over a commit will show all commits associated with the same mathlib3 commit.
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mathlib commit https://github.com/leanprover-community/mathlib/commit/ce64cd319bb6b3e82f31c2d38e79080d377be451
@@ -3,7 +3,7 @@ Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
-import Mathbin.RingTheory.WittVector.IsPoly
+import RingTheory.WittVector.IsPoly
#align_import ring_theory.witt_vector.mul_p from "leanprover-community/mathlib"@"6b31d1eebd64eab86d5bd9936bfaada6ca8b5842"
mathlib commit https://github.com/leanprover-community/mathlib/commit/8ea5598db6caeddde6cb734aa179cc2408dbd345
@@ -2,14 +2,11 @@
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-
-! This file was ported from Lean 3 source module ring_theory.witt_vector.mul_p
-! leanprover-community/mathlib commit 6b31d1eebd64eab86d5bd9936bfaada6ca8b5842
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
-/
import Mathbin.RingTheory.WittVector.IsPoly
+#align_import ring_theory.witt_vector.mul_p from "leanprover-community/mathlib"@"6b31d1eebd64eab86d5bd9936bfaada6ca8b5842"
+
/-!
## Multiplication by `n` in the ring of Witt vectors
mathlib commit https://github.com/leanprover-community/mathlib/commit/9fb8964792b4237dac6200193a0d533f1b3f7423
@@ -36,7 +36,6 @@ namespace WittVector
variable {p : ℕ} {R : Type _} [hp : Fact p.Prime] [CommRing R]
--- mathport name: expr𝕎
local notation "𝕎" => WittVector p
-- type as `\bbW`
@@ -44,19 +43,20 @@ open MvPolynomial
noncomputable section
-include hp
-
variable (p)
+#print WittVector.wittMulN /-
/-- `witt_mul_n p n` is the family of polynomials that computes
the coefficients of `x * n` in terms of the coefficients of the Witt vector `x`. -/
noncomputable def wittMulN : ℕ → ℕ → MvPolynomial ℕ ℤ
| 0 => 0
| n + 1 => fun k => bind₁ (Function.uncurry <| ![witt_mul_n n, X]) (wittAdd p k)
#align witt_vector.witt_mul_n WittVector.wittMulN
+-/
variable {p}
+#print WittVector.mulN_coeff /-
theorem mulN_coeff (n : ℕ) (x : 𝕎 R) (k : ℕ) : (x * n).coeff k = aeval x.coeff (wittMulN p n k) :=
by
induction' n with n ih generalizing k
@@ -71,6 +71,7 @@ theorem mulN_coeff (n : ℕ) (x : 𝕎 R) (k : ℕ) : (x * n).coeff k = aeval x.
· simp only [Function.uncurry, Matrix.cons_val_zero, ih]
· simp only [Function.uncurry, Matrix.cons_val_one, Matrix.head_cons, aeval_X]
#align witt_vector.mul_n_coeff WittVector.mulN_coeff
+-/
variable (p)
@@ -82,6 +83,7 @@ theorem mulN_isPoly (n : ℕ) : IsPoly p fun R _Rcr x => x * n :=
#align witt_vector.mul_n_is_poly WittVector.mulN_isPoly
-/
+#print WittVector.bind₁_wittMulN_wittPolynomial /-
@[simp]
theorem bind₁_wittMulN_wittPolynomial (n k : ℕ) :
bind₁ (wittMulN p n) (wittPolynomial p ℤ k) = n * wittPolynomial p ℤ k :=
@@ -94,6 +96,7 @@ theorem bind₁_wittMulN_wittPolynomial (n k : ℕ) :
simp only [ih, Function.uncurry, Function.comp, bind₁_X_left, AlgHom.id_apply,
Matrix.cons_val_zero, Matrix.head_cons, Matrix.cons_val_one]
#align witt_vector.bind₁_witt_mul_n_witt_polynomial WittVector.bind₁_wittMulN_wittPolynomial
+-/
end WittVector
mathlib commit https://github.com/leanprover-community/mathlib/commit/a3e83f0fa4391c8740f7d773a7a9b74e311ae2a3
@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
! This file was ported from Lean 3 source module ring_theory.witt_vector.mul_p
-! leanprover-community/mathlib commit 7abfbc92eec87190fba3ed3d5ec58e7c167e7144
+! leanprover-community/mathlib commit 6b31d1eebd64eab86d5bd9936bfaada6ca8b5842
! Please do not edit these lines, except to modify the commit id
! if you have ported upstream changes.
-/
@@ -13,6 +13,9 @@ import Mathbin.RingTheory.WittVector.IsPoly
/-!
## Multiplication by `n` in the ring of Witt vectors
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
In this file we show that multiplication by `n` in the ring of Witt vectors
is a polynomial function. We then use this fact to show that the composition of Frobenius
and Verschiebung is equal to multiplication by `p`.
mathlib commit https://github.com/leanprover-community/mathlib/commit/31c24aa72e7b3e5ed97a8412470e904f82b81004
@@ -54,7 +54,7 @@ noncomputable def wittMulN : ℕ → ℕ → MvPolynomial ℕ ℤ
variable {p}
-theorem mul_n_coeff (n : ℕ) (x : 𝕎 R) (k : ℕ) : (x * n).coeff k = aeval x.coeff (wittMulN p n k) :=
+theorem mulN_coeff (n : ℕ) (x : 𝕎 R) (k : ℕ) : (x * n).coeff k = aeval x.coeff (wittMulN p n k) :=
by
induction' n with n ih generalizing k
·
@@ -67,15 +67,17 @@ theorem mul_n_coeff (n : ℕ) (x : 𝕎 R) (k : ℕ) : (x * n).coeff k = aeval x
fin_cases b
· simp only [Function.uncurry, Matrix.cons_val_zero, ih]
· simp only [Function.uncurry, Matrix.cons_val_one, Matrix.head_cons, aeval_X]
-#align witt_vector.mul_n_coeff WittVector.mul_n_coeff
+#align witt_vector.mul_n_coeff WittVector.mulN_coeff
variable (p)
+#print WittVector.mulN_isPoly /-
/-- Multiplication by `n` is a polynomial function. -/
@[is_poly]
-theorem mulNIsPoly (n : ℕ) : IsPoly p fun R _Rcr x => x * n :=
+theorem mulN_isPoly (n : ℕ) : IsPoly p fun R _Rcr x => x * n :=
⟨⟨wittMulN p n, fun R _Rcr x => by funext k; exact mul_n_coeff n x k⟩⟩
-#align witt_vector.mul_n_is_poly WittVector.mulNIsPoly
+#align witt_vector.mul_n_is_poly WittVector.mulN_isPoly
+-/
@[simp]
theorem bind₁_wittMulN_wittPolynomial (n k : ℕ) :
mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb
@@ -74,9 +74,7 @@ variable (p)
/-- Multiplication by `n` is a polynomial function. -/
@[is_poly]
theorem mulNIsPoly (n : ℕ) : IsPoly p fun R _Rcr x => x * n :=
- ⟨⟨wittMulN p n, fun R _Rcr x => by
- funext k
- exact mul_n_coeff n x k⟩⟩
+ ⟨⟨wittMulN p n, fun R _Rcr x => by funext k; exact mul_n_coeff n x k⟩⟩
#align witt_vector.mul_n_is_poly WittVector.mulNIsPoly
@[simp]
mathlib commit https://github.com/leanprover-community/mathlib/commit/1a313d8bba1bad05faba71a4a4e9742ab5bd9efd
@@ -49,7 +49,7 @@ variable (p)
the coefficients of `x * n` in terms of the coefficients of the Witt vector `x`. -/
noncomputable def wittMulN : ℕ → ℕ → MvPolynomial ℕ ℤ
| 0 => 0
- | n + 1 => fun k => bind₁ (Function.uncurry <| ![witt_mul_n n, x]) (wittAdd p k)
+ | n + 1 => fun k => bind₁ (Function.uncurry <| ![witt_mul_n n, X]) (wittAdd p k)
#align witt_vector.witt_mul_n WittVector.wittMulN
variable {p}
mathlib commit https://github.com/leanprover-community/mathlib/commit/3180fab693e2cee3bff62675571264cb8778b212
@@ -58,8 +58,8 @@ theorem mul_n_coeff (n : ℕ) (x : 𝕎 R) (k : ℕ) : (x * n).coeff k = aeval x
by
induction' n with n ih generalizing k
·
- simp only [Nat.zero_eq, Nat.cast_zero, mul_zero, zero_coeff, witt_mul_n, AlgHom.map_zero,
- Pi.zero_apply]
+ simp only [Nat.zero_eq, Nat.cast_zero, MulZeroClass.mul_zero, zero_coeff, witt_mul_n,
+ AlgHom.map_zero, Pi.zero_apply]
· rw [witt_mul_n, Nat.succ_eq_add_one, Nat.cast_add, Nat.cast_one, mul_add, mul_one, aeval_bind₁,
add_coeff]
apply eval₂_hom_congr (RingHom.ext_int _ _) _ rfl
@@ -84,7 +84,7 @@ theorem bind₁_wittMulN_wittPolynomial (n k : ℕ) :
bind₁ (wittMulN p n) (wittPolynomial p ℤ k) = n * wittPolynomial p ℤ k :=
by
induction' n with n ih
- · simp only [witt_mul_n, Nat.cast_zero, zero_mul, bind₁_zero_witt_polynomial]
+ · simp only [witt_mul_n, Nat.cast_zero, MulZeroClass.zero_mul, bind₁_zero_witt_polynomial]
· rw [witt_mul_n, ← bind₁_bind₁, witt_add, wittStructureInt_prop]
simp only [AlgHom.map_add, Nat.cast_succ, bind₁_X_right]
rw [add_mul, one_mul, bind₁_rename, bind₁_rename]
mathlib commit https://github.com/leanprover-community/mathlib/commit/bd9851ca476957ea4549eb19b40e7b5ade9428cc
MulZeroClass.
in mul_zero
/zero_mul
(#6682)
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).
@@ -50,7 +50,7 @@ variable {p}
theorem mulN_coeff (n : ℕ) (x : 𝕎 R) (k : ℕ) :
(x * n).coeff k = aeval x.coeff (wittMulN p n k) := by
induction' n with n ih generalizing k
- · simp only [Nat.zero_eq, Nat.cast_zero, MulZeroClass.mul_zero, zero_coeff, wittMulN,
+ · simp only [Nat.zero_eq, Nat.cast_zero, mul_zero, zero_coeff, wittMulN,
AlgHom.map_zero, Pi.zero_apply]
· rw [wittMulN, Nat.succ_eq_add_one, Nat.cast_add, Nat.cast_one, mul_add, mul_one, aeval_bind₁,
add_coeff]
@@ -73,7 +73,7 @@ theorem mulN_isPoly (n : ℕ) : IsPoly p fun R _Rcr x => x * n :=
theorem bind₁_wittMulN_wittPolynomial (n k : ℕ) :
bind₁ (wittMulN p n) (wittPolynomial p ℤ k) = n * wittPolynomial p ℤ k := by
induction' n with n ih
- · simp [wittMulN, Nat.cast_zero, MulZeroClass.zero_mul, bind₁_zero_wittPolynomial]
+ · simp [wittMulN, Nat.cast_zero, zero_mul, bind₁_zero_wittPolynomial]
· rw [wittMulN, ← bind₁_bind₁, wittAdd, wittStructureInt_prop]
simp only [AlgHom.map_add, Nat.cast_succ, bind₁_X_right]
rw [add_mul, one_mul, bind₁_rename, bind₁_rename]
Type _
and Sort _
(#6499)
We remove all possible occurences of Type _
and Sort _
in favor of Type*
and Sort*
.
This has nice performance benefits.
@@ -28,7 +28,7 @@ and Verschiebung is equal to multiplication by `p`.
namespace WittVector
-variable {p : ℕ} {R : Type _} [hp : Fact p.Prime] [CommRing R]
+variable {p : ℕ} {R : Type*} [hp : Fact p.Prime] [CommRing R]
local notation "𝕎" => WittVector p -- type as `\bbW`
@@ -2,14 +2,11 @@
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-
-! This file was ported from Lean 3 source module ring_theory.witt_vector.mul_p
-! leanprover-community/mathlib commit 7abfbc92eec87190fba3ed3d5ec58e7c167e7144
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
-/
import Mathlib.RingTheory.WittVector.IsPoly
+#align_import ring_theory.witt_vector.mul_p from "leanprover-community/mathlib"@"7abfbc92eec87190fba3ed3d5ec58e7c167e7144"
+
/-!
## Multiplication by `n` in the ring of Witt vectors
The unported dependencies are