Context stringlengths 57 6.04k | file_name stringlengths 21 79 | start int64 14 1.49k | end int64 18 1.5k | theorem stringlengths 25 1.55k | proof stringlengths 5 7.36k | num_lines int64 1 150 | complexity_score float64 2.72 139,370,958,066,637,970,000,000,000,000,000,000,000,000,000,000,000,000,000B | diff_level int64 0 2 | file_diff_level float64 0 2 | theorem_same_file int64 1 32 | rank_file int64 0 2.51k |
|---|---|---|---|---|---|---|---|---|---|---|---|
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Data.Nat.Factorial.DoubleFactorial
#align_import ring_theory.polynomial.hermite.basic from "leanprover-community/mathlib"@"938d3db9c278f8a52c0f964a405806f0f2b09b74"
noncomputable section
open Polynomial
namespace Polynomial
noncomputable def hermite : ℕ → Polynomial ℤ
| 0 => 1
| n + 1 => X * hermite n - derivative (hermite n)
#align polynomial.hermite Polynomial.hermite
@[simp]
theorem hermite_succ (n : ℕ) : hermite (n + 1) = X * hermite n - derivative (hermite n) := by
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
induction' n with n ih
· rfl
· rw [Function.iterate_succ_apply', ← ih, hermite_succ]
#align polynomial.hermite_eq_iterate Polynomial.hermite_eq_iterate
@[simp]
theorem hermite_zero : hermite 0 = C 1 :=
rfl
#align polynomial.hermite_zero Polynomial.hermite_zero
-- 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]
simp only [map_one, mul_one, derivative_one, sub_zero]
#align polynomial.hermite_one Polynomial.hermite_one
section coeff
| Mathlib/RingTheory/Polynomial/Hermite/Basic.lean | 82 | 83 | theorem coeff_hermite_succ_zero (n : ℕ) : coeff (hermite (n + 1)) 0 = -coeff (hermite n) 1 := by |
simp [coeff_derivative]
| 1 | 2.718282 | 0 | 1.1 | 10 | 1,191 |
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Data.Nat.Factorial.DoubleFactorial
#align_import ring_theory.polynomial.hermite.basic from "leanprover-community/mathlib"@"938d3db9c278f8a52c0f964a405806f0f2b09b74"
noncomputable section
open Polynomial
namespace Polynomial
noncomputable def hermite : ℕ → Polynomial ℤ
| 0 => 1
| n + 1 => X * hermite n - derivative (hermite n)
#align polynomial.hermite Polynomial.hermite
@[simp]
theorem hermite_succ (n : ℕ) : hermite (n + 1) = X * hermite n - derivative (hermite n) := by
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
induction' n with n ih
· rfl
· rw [Function.iterate_succ_apply', ← ih, hermite_succ]
#align polynomial.hermite_eq_iterate Polynomial.hermite_eq_iterate
@[simp]
theorem hermite_zero : hermite 0 = C 1 :=
rfl
#align polynomial.hermite_zero Polynomial.hermite_zero
-- 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]
simp only [map_one, mul_one, derivative_one, sub_zero]
#align polynomial.hermite_one Polynomial.hermite_one
section coeff
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
| Mathlib/RingTheory/Polynomial/Hermite/Basic.lean | 86 | 89 | 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
| 2 | 7.389056 | 1 | 1.1 | 10 | 1,191 |
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Data.Nat.Factorial.DoubleFactorial
#align_import ring_theory.polynomial.hermite.basic from "leanprover-community/mathlib"@"938d3db9c278f8a52c0f964a405806f0f2b09b74"
noncomputable section
open Polynomial
namespace Polynomial
noncomputable def hermite : ℕ → Polynomial ℤ
| 0 => 1
| n + 1 => X * hermite n - derivative (hermite n)
#align polynomial.hermite Polynomial.hermite
@[simp]
theorem hermite_succ (n : ℕ) : hermite (n + 1) = X * hermite n - derivative (hermite n) := by
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
induction' n with n ih
· rfl
· rw [Function.iterate_succ_apply', ← ih, hermite_succ]
#align polynomial.hermite_eq_iterate Polynomial.hermite_eq_iterate
@[simp]
theorem hermite_zero : hermite 0 = C 1 :=
rfl
#align polynomial.hermite_zero Polynomial.hermite_zero
-- 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]
simp only [map_one, mul_one, derivative_one, sub_zero]
#align polynomial.hermite_one Polynomial.hermite_one
section coeff
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
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
| Mathlib/RingTheory/Polynomial/Hermite/Basic.lean | 92 | 99 | 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
clear hnk
induction' n with n ih generalizing k
· apply coeff_C
· have : n + k + 1 + 2 = n + (k + 2) + 1 := by ring
rw [coeff_hermite_succ_succ, add_right_comm, this, ih k, ih (k + 2),
mul_zero, sub_zero]
| 7 | 1,096.633158 | 2 | 1.1 | 10 | 1,191 |
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Data.Nat.Factorial.DoubleFactorial
#align_import ring_theory.polynomial.hermite.basic from "leanprover-community/mathlib"@"938d3db9c278f8a52c0f964a405806f0f2b09b74"
noncomputable section
open Polynomial
namespace Polynomial
noncomputable def hermite : ℕ → Polynomial ℤ
| 0 => 1
| n + 1 => X * hermite n - derivative (hermite n)
#align polynomial.hermite Polynomial.hermite
@[simp]
theorem hermite_succ (n : ℕ) : hermite (n + 1) = X * hermite n - derivative (hermite n) := by
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
induction' n with n ih
· rfl
· rw [Function.iterate_succ_apply', ← ih, hermite_succ]
#align polynomial.hermite_eq_iterate Polynomial.hermite_eq_iterate
@[simp]
theorem hermite_zero : hermite 0 = C 1 :=
rfl
#align polynomial.hermite_zero Polynomial.hermite_zero
-- 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]
simp only [map_one, mul_one, derivative_one, sub_zero]
#align polynomial.hermite_one Polynomial.hermite_one
section coeff
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
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
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
clear hnk
induction' n with n ih generalizing k
· apply coeff_C
· have : n + k + 1 + 2 = n + (k + 2) + 1 := by ring
rw [coeff_hermite_succ_succ, add_right_comm, this, ih k, ih (k + 2),
mul_zero, sub_zero]
#align polynomial.coeff_hermite_of_lt Polynomial.coeff_hermite_of_lt
@[simp]
| Mathlib/RingTheory/Polynomial/Hermite/Basic.lean | 103 | 107 | 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, mul_zero, sub_zero]
simp
| 4 | 54.59815 | 2 | 1.1 | 10 | 1,191 |
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Data.Nat.Factorial.DoubleFactorial
#align_import ring_theory.polynomial.hermite.basic from "leanprover-community/mathlib"@"938d3db9c278f8a52c0f964a405806f0f2b09b74"
noncomputable section
open Polynomial
namespace Polynomial
noncomputable def hermite : ℕ → Polynomial ℤ
| 0 => 1
| n + 1 => X * hermite n - derivative (hermite n)
#align polynomial.hermite Polynomial.hermite
@[simp]
theorem hermite_succ (n : ℕ) : hermite (n + 1) = X * hermite n - derivative (hermite n) := by
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
induction' n with n ih
· rfl
· rw [Function.iterate_succ_apply', ← ih, hermite_succ]
#align polynomial.hermite_eq_iterate Polynomial.hermite_eq_iterate
@[simp]
theorem hermite_zero : hermite 0 = C 1 :=
rfl
#align polynomial.hermite_zero Polynomial.hermite_zero
-- 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]
simp only [map_one, mul_one, derivative_one, sub_zero]
#align polynomial.hermite_one Polynomial.hermite_one
section coeff
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
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
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
clear hnk
induction' n with n ih generalizing k
· apply coeff_C
· have : n + k + 1 + 2 = n + (k + 2) + 1 := by ring
rw [coeff_hermite_succ_succ, add_right_comm, this, ih k, ih (k + 2),
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, mul_zero, sub_zero]
simp
#align polynomial.coeff_hermite_self Polynomial.coeff_hermite_self
@[simp]
| Mathlib/RingTheory/Polynomial/Hermite/Basic.lean | 111 | 116 | 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
exact coeff_hermite_of_lt hnm
· simp [coeff_hermite_self n]
| 5 | 148.413159 | 2 | 1.1 | 10 | 1,191 |
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Data.Nat.Factorial.DoubleFactorial
#align_import ring_theory.polynomial.hermite.basic from "leanprover-community/mathlib"@"938d3db9c278f8a52c0f964a405806f0f2b09b74"
noncomputable section
open Polynomial
namespace Polynomial
noncomputable def hermite : ℕ → Polynomial ℤ
| 0 => 1
| n + 1 => X * hermite n - derivative (hermite n)
#align polynomial.hermite Polynomial.hermite
@[simp]
theorem hermite_succ (n : ℕ) : hermite (n + 1) = X * hermite n - derivative (hermite n) := by
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
induction' n with n ih
· rfl
· rw [Function.iterate_succ_apply', ← ih, hermite_succ]
#align polynomial.hermite_eq_iterate Polynomial.hermite_eq_iterate
@[simp]
theorem hermite_zero : hermite 0 = C 1 :=
rfl
#align polynomial.hermite_zero Polynomial.hermite_zero
-- 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]
simp only [map_one, mul_one, derivative_one, sub_zero]
#align polynomial.hermite_one Polynomial.hermite_one
section coeff
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
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
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
clear hnk
induction' n with n ih generalizing k
· apply coeff_C
· have : n + k + 1 + 2 = n + (k + 2) + 1 := by ring
rw [coeff_hermite_succ_succ, add_right_comm, this, ih k, ih (k + 2),
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, mul_zero, sub_zero]
simp
#align polynomial.coeff_hermite_self Polynomial.coeff_hermite_self
@[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
exact coeff_hermite_of_lt hnm
· simp [coeff_hermite_self n]
#align polynomial.degree_hermite Polynomial.degree_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
@[simp]
| Mathlib/RingTheory/Polynomial/Hermite/Basic.lean | 125 | 126 | theorem leadingCoeff_hermite (n : ℕ) : (hermite n).leadingCoeff = 1 := by |
rw [← coeff_natDegree, natDegree_hermite, coeff_hermite_self]
| 1 | 2.718282 | 0 | 1.1 | 10 | 1,191 |
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Data.Nat.Factorial.DoubleFactorial
#align_import ring_theory.polynomial.hermite.basic from "leanprover-community/mathlib"@"938d3db9c278f8a52c0f964a405806f0f2b09b74"
noncomputable section
open Polynomial
namespace Polynomial
noncomputable def hermite : ℕ → Polynomial ℤ
| 0 => 1
| n + 1 => X * hermite n - derivative (hermite n)
#align polynomial.hermite Polynomial.hermite
@[simp]
theorem hermite_succ (n : ℕ) : hermite (n + 1) = X * hermite n - derivative (hermite n) := by
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
induction' n with n ih
· rfl
· rw [Function.iterate_succ_apply', ← ih, hermite_succ]
#align polynomial.hermite_eq_iterate Polynomial.hermite_eq_iterate
@[simp]
theorem hermite_zero : hermite 0 = C 1 :=
rfl
#align polynomial.hermite_zero Polynomial.hermite_zero
-- 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]
simp only [map_one, mul_one, derivative_one, sub_zero]
#align polynomial.hermite_one Polynomial.hermite_one
section coeff
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
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
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
clear hnk
induction' n with n ih generalizing k
· apply coeff_C
· have : n + k + 1 + 2 = n + (k + 2) + 1 := by ring
rw [coeff_hermite_succ_succ, add_right_comm, this, ih k, ih (k + 2),
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, mul_zero, sub_zero]
simp
#align polynomial.coeff_hermite_self Polynomial.coeff_hermite_self
@[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
exact coeff_hermite_of_lt hnm
· simp [coeff_hermite_self n]
#align polynomial.degree_hermite Polynomial.degree_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
@[simp]
theorem leadingCoeff_hermite (n : ℕ) : (hermite n).leadingCoeff = 1 := by
rw [← coeff_natDegree, natDegree_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
| Mathlib/RingTheory/Polynomial/Hermite/Basic.lean | 133 | 143 | 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 k] at hnk
exact coeff_hermite_of_lt hnk.pos
· cases' k with k
· 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_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
| 10 | 22,026.465795 | 2 | 1.1 | 10 | 1,191 |
import Mathlib.Data.Matrix.Basic
import Mathlib.LinearAlgebra.Matrix.Trace
#align_import data.matrix.basis from "leanprover-community/mathlib"@"320df450e9abeb5fc6417971e75acb6ae8bc3794"
variable {l m n : Type*}
variable {R α : Type*}
namespace Matrix
open Matrix
variable [DecidableEq l] [DecidableEq m] [DecidableEq n]
variable [Semiring α]
def stdBasisMatrix (i : m) (j : n) (a : α) : Matrix m n α := fun i' j' =>
if i = i' ∧ j = j' then a else 0
#align matrix.std_basis_matrix Matrix.stdBasisMatrix
@[simp]
| Mathlib/Data/Matrix/Basis.lean | 37 | 41 | theorem smul_stdBasisMatrix [SMulZeroClass R α] (r : R) (i : m) (j : n) (a : α) :
r • stdBasisMatrix i j a = stdBasisMatrix i j (r • a) := by |
unfold stdBasisMatrix
ext
simp [smul_ite]
| 3 | 20.085537 | 1 | 1.111111 | 9 | 1,192 |
import Mathlib.Data.Matrix.Basic
import Mathlib.LinearAlgebra.Matrix.Trace
#align_import data.matrix.basis from "leanprover-community/mathlib"@"320df450e9abeb5fc6417971e75acb6ae8bc3794"
variable {l m n : Type*}
variable {R α : Type*}
namespace Matrix
open Matrix
variable [DecidableEq l] [DecidableEq m] [DecidableEq n]
variable [Semiring α]
def stdBasisMatrix (i : m) (j : n) (a : α) : Matrix m n α := fun i' j' =>
if i = i' ∧ j = j' then a else 0
#align matrix.std_basis_matrix Matrix.stdBasisMatrix
@[simp]
theorem smul_stdBasisMatrix [SMulZeroClass R α] (r : R) (i : m) (j : n) (a : α) :
r • stdBasisMatrix i j a = stdBasisMatrix i j (r • a) := by
unfold stdBasisMatrix
ext
simp [smul_ite]
#align matrix.smul_std_basis_matrix Matrix.smul_stdBasisMatrix
@[simp]
| Mathlib/Data/Matrix/Basis.lean | 45 | 48 | theorem stdBasisMatrix_zero (i : m) (j : n) : stdBasisMatrix i j (0 : α) = 0 := by |
unfold stdBasisMatrix
ext
simp
| 3 | 20.085537 | 1 | 1.111111 | 9 | 1,192 |
import Mathlib.Data.Matrix.Basic
import Mathlib.LinearAlgebra.Matrix.Trace
#align_import data.matrix.basis from "leanprover-community/mathlib"@"320df450e9abeb5fc6417971e75acb6ae8bc3794"
variable {l m n : Type*}
variable {R α : Type*}
namespace Matrix
open Matrix
variable [DecidableEq l] [DecidableEq m] [DecidableEq n]
variable [Semiring α]
def stdBasisMatrix (i : m) (j : n) (a : α) : Matrix m n α := fun i' j' =>
if i = i' ∧ j = j' then a else 0
#align matrix.std_basis_matrix Matrix.stdBasisMatrix
@[simp]
theorem smul_stdBasisMatrix [SMulZeroClass R α] (r : R) (i : m) (j : n) (a : α) :
r • stdBasisMatrix i j a = stdBasisMatrix i j (r • a) := by
unfold stdBasisMatrix
ext
simp [smul_ite]
#align matrix.smul_std_basis_matrix Matrix.smul_stdBasisMatrix
@[simp]
theorem stdBasisMatrix_zero (i : m) (j : n) : stdBasisMatrix i j (0 : α) = 0 := by
unfold stdBasisMatrix
ext
simp
#align matrix.std_basis_matrix_zero Matrix.stdBasisMatrix_zero
| Mathlib/Data/Matrix/Basis.lean | 51 | 54 | theorem stdBasisMatrix_add (i : m) (j : n) (a b : α) :
stdBasisMatrix i j (a + b) = stdBasisMatrix i j a + stdBasisMatrix i j b := by |
unfold stdBasisMatrix; ext
split_ifs with h <;> simp [h]
| 2 | 7.389056 | 1 | 1.111111 | 9 | 1,192 |
import Mathlib.Data.Matrix.Basic
import Mathlib.LinearAlgebra.Matrix.Trace
#align_import data.matrix.basis from "leanprover-community/mathlib"@"320df450e9abeb5fc6417971e75acb6ae8bc3794"
variable {l m n : Type*}
variable {R α : Type*}
namespace Matrix
open Matrix
variable [DecidableEq l] [DecidableEq m] [DecidableEq n]
variable [Semiring α]
def stdBasisMatrix (i : m) (j : n) (a : α) : Matrix m n α := fun i' j' =>
if i = i' ∧ j = j' then a else 0
#align matrix.std_basis_matrix Matrix.stdBasisMatrix
@[simp]
theorem smul_stdBasisMatrix [SMulZeroClass R α] (r : R) (i : m) (j : n) (a : α) :
r • stdBasisMatrix i j a = stdBasisMatrix i j (r • a) := by
unfold stdBasisMatrix
ext
simp [smul_ite]
#align matrix.smul_std_basis_matrix Matrix.smul_stdBasisMatrix
@[simp]
theorem stdBasisMatrix_zero (i : m) (j : n) : stdBasisMatrix i j (0 : α) = 0 := by
unfold stdBasisMatrix
ext
simp
#align matrix.std_basis_matrix_zero Matrix.stdBasisMatrix_zero
theorem stdBasisMatrix_add (i : m) (j : n) (a b : α) :
stdBasisMatrix i j (a + b) = stdBasisMatrix i j a + stdBasisMatrix i j b := by
unfold stdBasisMatrix; ext
split_ifs with h <;> simp [h]
#align matrix.std_basis_matrix_add Matrix.stdBasisMatrix_add
| Mathlib/Data/Matrix/Basis.lean | 57 | 63 | theorem mulVec_stdBasisMatrix [Fintype m] (i : n) (j : m) (c : α) (x : m → α) :
mulVec (stdBasisMatrix i j c) x = Function.update (0 : n → α) i (c * x j) := by |
ext i'
simp [stdBasisMatrix, mulVec, dotProduct]
rcases eq_or_ne i i' with rfl|h
· simp
simp [h, h.symm]
| 5 | 148.413159 | 2 | 1.111111 | 9 | 1,192 |
import Mathlib.Data.Matrix.Basic
import Mathlib.LinearAlgebra.Matrix.Trace
#align_import data.matrix.basis from "leanprover-community/mathlib"@"320df450e9abeb5fc6417971e75acb6ae8bc3794"
variable {l m n : Type*}
variable {R α : Type*}
namespace Matrix
open Matrix
variable [DecidableEq l] [DecidableEq m] [DecidableEq n]
variable [Semiring α]
def stdBasisMatrix (i : m) (j : n) (a : α) : Matrix m n α := fun i' j' =>
if i = i' ∧ j = j' then a else 0
#align matrix.std_basis_matrix Matrix.stdBasisMatrix
@[simp]
theorem smul_stdBasisMatrix [SMulZeroClass R α] (r : R) (i : m) (j : n) (a : α) :
r • stdBasisMatrix i j a = stdBasisMatrix i j (r • a) := by
unfold stdBasisMatrix
ext
simp [smul_ite]
#align matrix.smul_std_basis_matrix Matrix.smul_stdBasisMatrix
@[simp]
theorem stdBasisMatrix_zero (i : m) (j : n) : stdBasisMatrix i j (0 : α) = 0 := by
unfold stdBasisMatrix
ext
simp
#align matrix.std_basis_matrix_zero Matrix.stdBasisMatrix_zero
theorem stdBasisMatrix_add (i : m) (j : n) (a b : α) :
stdBasisMatrix i j (a + b) = stdBasisMatrix i j a + stdBasisMatrix i j b := by
unfold stdBasisMatrix; ext
split_ifs with h <;> simp [h]
#align matrix.std_basis_matrix_add Matrix.stdBasisMatrix_add
theorem mulVec_stdBasisMatrix [Fintype m] (i : n) (j : m) (c : α) (x : m → α) :
mulVec (stdBasisMatrix i j c) x = Function.update (0 : n → α) i (c * x j) := by
ext i'
simp [stdBasisMatrix, mulVec, dotProduct]
rcases eq_or_ne i i' with rfl|h
· simp
simp [h, h.symm]
| Mathlib/Data/Matrix/Basis.lean | 65 | 79 | theorem matrix_eq_sum_std_basis [Fintype m] [Fintype n] (x : Matrix m n α) :
x = ∑ i : m, ∑ j : n, stdBasisMatrix i j (x i j) := by |
ext i j; symm
iterate 2 rw [Finset.sum_apply]
-- Porting note: was `convert`
refine (Fintype.sum_eq_single i ?_).trans ?_; swap
· -- Porting note: `simp` seems unwilling to apply `Fintype.sum_apply`
simp (config := { unfoldPartialApp := true }) only [stdBasisMatrix]
rw [Fintype.sum_apply, Fintype.sum_apply]
simp
· intro j' hj'
-- Porting note: `simp` seems unwilling to apply `Fintype.sum_apply`
simp (config := { unfoldPartialApp := true }) only [stdBasisMatrix]
rw [Fintype.sum_apply, Fintype.sum_apply]
simp [hj']
| 13 | 442,413.392009 | 2 | 1.111111 | 9 | 1,192 |
import Mathlib.Data.Matrix.Basic
import Mathlib.LinearAlgebra.Matrix.Trace
#align_import data.matrix.basis from "leanprover-community/mathlib"@"320df450e9abeb5fc6417971e75acb6ae8bc3794"
variable {l m n : Type*}
variable {R α : Type*}
namespace Matrix
open Matrix
variable [DecidableEq l] [DecidableEq m] [DecidableEq n]
variable [Semiring α]
def stdBasisMatrix (i : m) (j : n) (a : α) : Matrix m n α := fun i' j' =>
if i = i' ∧ j = j' then a else 0
#align matrix.std_basis_matrix Matrix.stdBasisMatrix
@[simp]
theorem smul_stdBasisMatrix [SMulZeroClass R α] (r : R) (i : m) (j : n) (a : α) :
r • stdBasisMatrix i j a = stdBasisMatrix i j (r • a) := by
unfold stdBasisMatrix
ext
simp [smul_ite]
#align matrix.smul_std_basis_matrix Matrix.smul_stdBasisMatrix
@[simp]
theorem stdBasisMatrix_zero (i : m) (j : n) : stdBasisMatrix i j (0 : α) = 0 := by
unfold stdBasisMatrix
ext
simp
#align matrix.std_basis_matrix_zero Matrix.stdBasisMatrix_zero
theorem stdBasisMatrix_add (i : m) (j : n) (a b : α) :
stdBasisMatrix i j (a + b) = stdBasisMatrix i j a + stdBasisMatrix i j b := by
unfold stdBasisMatrix; ext
split_ifs with h <;> simp [h]
#align matrix.std_basis_matrix_add Matrix.stdBasisMatrix_add
theorem mulVec_stdBasisMatrix [Fintype m] (i : n) (j : m) (c : α) (x : m → α) :
mulVec (stdBasisMatrix i j c) x = Function.update (0 : n → α) i (c * x j) := by
ext i'
simp [stdBasisMatrix, mulVec, dotProduct]
rcases eq_or_ne i i' with rfl|h
· simp
simp [h, h.symm]
theorem matrix_eq_sum_std_basis [Fintype m] [Fintype n] (x : Matrix m n α) :
x = ∑ i : m, ∑ j : n, stdBasisMatrix i j (x i j) := by
ext i j; symm
iterate 2 rw [Finset.sum_apply]
-- Porting note: was `convert`
refine (Fintype.sum_eq_single i ?_).trans ?_; swap
· -- Porting note: `simp` seems unwilling to apply `Fintype.sum_apply`
simp (config := { unfoldPartialApp := true }) only [stdBasisMatrix]
rw [Fintype.sum_apply, Fintype.sum_apply]
simp
· intro j' hj'
-- Porting note: `simp` seems unwilling to apply `Fintype.sum_apply`
simp (config := { unfoldPartialApp := true }) only [stdBasisMatrix]
rw [Fintype.sum_apply, Fintype.sum_apply]
simp [hj']
#align matrix.matrix_eq_sum_std_basis Matrix.matrix_eq_sum_std_basis
-- TODO: tie this up with the `Basis` machinery of linear algebra
-- this is not completely trivial because we are indexing by two types, instead of one
-- TODO: add `std_basis_vec`
| Mathlib/Data/Matrix/Basis.lean | 85 | 94 | theorem std_basis_eq_basis_mul_basis (i : m) (j : n) :
stdBasisMatrix i j (1 : α) =
vecMulVec (fun i' => ite (i = i') 1 0) fun j' => ite (j = j') 1 0 := by |
ext i' j'
-- Porting note: was `norm_num [std_basis_matrix, vec_mul_vec]` though there are no numerals
-- involved.
simp only [stdBasisMatrix, vecMulVec, mul_ite, mul_one, mul_zero, of_apply]
-- Porting note: added next line
simp_rw [@and_comm (i = i')]
exact ite_and _ _ _ _
| 7 | 1,096.633158 | 2 | 1.111111 | 9 | 1,192 |
import Mathlib.Data.Matrix.Basic
import Mathlib.LinearAlgebra.Matrix.Trace
#align_import data.matrix.basis from "leanprover-community/mathlib"@"320df450e9abeb5fc6417971e75acb6ae8bc3794"
variable {l m n : Type*}
variable {R α : Type*}
namespace Matrix
open Matrix
variable [DecidableEq l] [DecidableEq m] [DecidableEq n]
variable [Semiring α]
def stdBasisMatrix (i : m) (j : n) (a : α) : Matrix m n α := fun i' j' =>
if i = i' ∧ j = j' then a else 0
#align matrix.std_basis_matrix Matrix.stdBasisMatrix
@[simp]
theorem smul_stdBasisMatrix [SMulZeroClass R α] (r : R) (i : m) (j : n) (a : α) :
r • stdBasisMatrix i j a = stdBasisMatrix i j (r • a) := by
unfold stdBasisMatrix
ext
simp [smul_ite]
#align matrix.smul_std_basis_matrix Matrix.smul_stdBasisMatrix
@[simp]
theorem stdBasisMatrix_zero (i : m) (j : n) : stdBasisMatrix i j (0 : α) = 0 := by
unfold stdBasisMatrix
ext
simp
#align matrix.std_basis_matrix_zero Matrix.stdBasisMatrix_zero
theorem stdBasisMatrix_add (i : m) (j : n) (a b : α) :
stdBasisMatrix i j (a + b) = stdBasisMatrix i j a + stdBasisMatrix i j b := by
unfold stdBasisMatrix; ext
split_ifs with h <;> simp [h]
#align matrix.std_basis_matrix_add Matrix.stdBasisMatrix_add
theorem mulVec_stdBasisMatrix [Fintype m] (i : n) (j : m) (c : α) (x : m → α) :
mulVec (stdBasisMatrix i j c) x = Function.update (0 : n → α) i (c * x j) := by
ext i'
simp [stdBasisMatrix, mulVec, dotProduct]
rcases eq_or_ne i i' with rfl|h
· simp
simp [h, h.symm]
theorem matrix_eq_sum_std_basis [Fintype m] [Fintype n] (x : Matrix m n α) :
x = ∑ i : m, ∑ j : n, stdBasisMatrix i j (x i j) := by
ext i j; symm
iterate 2 rw [Finset.sum_apply]
-- Porting note: was `convert`
refine (Fintype.sum_eq_single i ?_).trans ?_; swap
· -- Porting note: `simp` seems unwilling to apply `Fintype.sum_apply`
simp (config := { unfoldPartialApp := true }) only [stdBasisMatrix]
rw [Fintype.sum_apply, Fintype.sum_apply]
simp
· intro j' hj'
-- Porting note: `simp` seems unwilling to apply `Fintype.sum_apply`
simp (config := { unfoldPartialApp := true }) only [stdBasisMatrix]
rw [Fintype.sum_apply, Fintype.sum_apply]
simp [hj']
#align matrix.matrix_eq_sum_std_basis Matrix.matrix_eq_sum_std_basis
-- TODO: tie this up with the `Basis` machinery of linear algebra
-- this is not completely trivial because we are indexing by two types, instead of one
-- TODO: add `std_basis_vec`
theorem std_basis_eq_basis_mul_basis (i : m) (j : n) :
stdBasisMatrix i j (1 : α) =
vecMulVec (fun i' => ite (i = i') 1 0) fun j' => ite (j = j') 1 0 := by
ext i' j'
-- Porting note: was `norm_num [std_basis_matrix, vec_mul_vec]` though there are no numerals
-- involved.
simp only [stdBasisMatrix, vecMulVec, mul_ite, mul_one, mul_zero, of_apply]
-- Porting note: added next line
simp_rw [@and_comm (i = i')]
exact ite_and _ _ _ _
#align matrix.std_basis_eq_basis_mul_basis Matrix.std_basis_eq_basis_mul_basis
-- todo: the old proof used fintypes, I don't know `Finsupp` but this feels generalizable
@[elab_as_elim]
protected theorem induction_on' [Finite m] [Finite n] {P : Matrix m n α → Prop} (M : Matrix m n α)
(h_zero : P 0) (h_add : ∀ p q, P p → P q → P (p + q))
(h_std_basis : ∀ (i : m) (j : n) (x : α), P (stdBasisMatrix i j x)) : P M := by
cases nonempty_fintype m; cases nonempty_fintype n
rw [matrix_eq_sum_std_basis M, ← Finset.sum_product']
apply Finset.sum_induction _ _ h_add h_zero
· intros
apply h_std_basis
#align matrix.induction_on' Matrix.induction_on'
@[elab_as_elim]
protected theorem induction_on [Finite m] [Finite n] [Nonempty m] [Nonempty n]
{P : Matrix m n α → Prop} (M : Matrix m n α) (h_add : ∀ p q, P p → P q → P (p + q))
(h_std_basis : ∀ i j x, P (stdBasisMatrix i j x)) : P M :=
Matrix.induction_on' M
(by
inhabit m
inhabit n
simpa using h_std_basis default default 0)
h_add h_std_basis
#align matrix.induction_on Matrix.induction_on
namespace StdBasisMatrix
section
variable (i : m) (j : n) (c : α) (i' : m) (j' : n)
@[simp]
theorem apply_same : stdBasisMatrix i j c i j = c :=
if_pos (And.intro rfl rfl)
#align matrix.std_basis_matrix.apply_same Matrix.StdBasisMatrix.apply_same
@[simp]
| Mathlib/Data/Matrix/Basis.lean | 133 | 135 | theorem apply_of_ne (h : ¬(i = i' ∧ j = j')) : stdBasisMatrix i j c i' j' = 0 := by |
simp only [stdBasisMatrix, and_imp, ite_eq_right_iff]
tauto
| 2 | 7.389056 | 1 | 1.111111 | 9 | 1,192 |
import Mathlib.Data.Matrix.Basic
import Mathlib.LinearAlgebra.Matrix.Trace
#align_import data.matrix.basis from "leanprover-community/mathlib"@"320df450e9abeb5fc6417971e75acb6ae8bc3794"
variable {l m n : Type*}
variable {R α : Type*}
namespace Matrix
open Matrix
variable [DecidableEq l] [DecidableEq m] [DecidableEq n]
variable [Semiring α]
def stdBasisMatrix (i : m) (j : n) (a : α) : Matrix m n α := fun i' j' =>
if i = i' ∧ j = j' then a else 0
#align matrix.std_basis_matrix Matrix.stdBasisMatrix
@[simp]
theorem smul_stdBasisMatrix [SMulZeroClass R α] (r : R) (i : m) (j : n) (a : α) :
r • stdBasisMatrix i j a = stdBasisMatrix i j (r • a) := by
unfold stdBasisMatrix
ext
simp [smul_ite]
#align matrix.smul_std_basis_matrix Matrix.smul_stdBasisMatrix
@[simp]
theorem stdBasisMatrix_zero (i : m) (j : n) : stdBasisMatrix i j (0 : α) = 0 := by
unfold stdBasisMatrix
ext
simp
#align matrix.std_basis_matrix_zero Matrix.stdBasisMatrix_zero
theorem stdBasisMatrix_add (i : m) (j : n) (a b : α) :
stdBasisMatrix i j (a + b) = stdBasisMatrix i j a + stdBasisMatrix i j b := by
unfold stdBasisMatrix; ext
split_ifs with h <;> simp [h]
#align matrix.std_basis_matrix_add Matrix.stdBasisMatrix_add
theorem mulVec_stdBasisMatrix [Fintype m] (i : n) (j : m) (c : α) (x : m → α) :
mulVec (stdBasisMatrix i j c) x = Function.update (0 : n → α) i (c * x j) := by
ext i'
simp [stdBasisMatrix, mulVec, dotProduct]
rcases eq_or_ne i i' with rfl|h
· simp
simp [h, h.symm]
theorem matrix_eq_sum_std_basis [Fintype m] [Fintype n] (x : Matrix m n α) :
x = ∑ i : m, ∑ j : n, stdBasisMatrix i j (x i j) := by
ext i j; symm
iterate 2 rw [Finset.sum_apply]
-- Porting note: was `convert`
refine (Fintype.sum_eq_single i ?_).trans ?_; swap
· -- Porting note: `simp` seems unwilling to apply `Fintype.sum_apply`
simp (config := { unfoldPartialApp := true }) only [stdBasisMatrix]
rw [Fintype.sum_apply, Fintype.sum_apply]
simp
· intro j' hj'
-- Porting note: `simp` seems unwilling to apply `Fintype.sum_apply`
simp (config := { unfoldPartialApp := true }) only [stdBasisMatrix]
rw [Fintype.sum_apply, Fintype.sum_apply]
simp [hj']
#align matrix.matrix_eq_sum_std_basis Matrix.matrix_eq_sum_std_basis
-- TODO: tie this up with the `Basis` machinery of linear algebra
-- this is not completely trivial because we are indexing by two types, instead of one
-- TODO: add `std_basis_vec`
theorem std_basis_eq_basis_mul_basis (i : m) (j : n) :
stdBasisMatrix i j (1 : α) =
vecMulVec (fun i' => ite (i = i') 1 0) fun j' => ite (j = j') 1 0 := by
ext i' j'
-- Porting note: was `norm_num [std_basis_matrix, vec_mul_vec]` though there are no numerals
-- involved.
simp only [stdBasisMatrix, vecMulVec, mul_ite, mul_one, mul_zero, of_apply]
-- Porting note: added next line
simp_rw [@and_comm (i = i')]
exact ite_and _ _ _ _
#align matrix.std_basis_eq_basis_mul_basis Matrix.std_basis_eq_basis_mul_basis
-- todo: the old proof used fintypes, I don't know `Finsupp` but this feels generalizable
@[elab_as_elim]
protected theorem induction_on' [Finite m] [Finite n] {P : Matrix m n α → Prop} (M : Matrix m n α)
(h_zero : P 0) (h_add : ∀ p q, P p → P q → P (p + q))
(h_std_basis : ∀ (i : m) (j : n) (x : α), P (stdBasisMatrix i j x)) : P M := by
cases nonempty_fintype m; cases nonempty_fintype n
rw [matrix_eq_sum_std_basis M, ← Finset.sum_product']
apply Finset.sum_induction _ _ h_add h_zero
· intros
apply h_std_basis
#align matrix.induction_on' Matrix.induction_on'
@[elab_as_elim]
protected theorem induction_on [Finite m] [Finite n] [Nonempty m] [Nonempty n]
{P : Matrix m n α → Prop} (M : Matrix m n α) (h_add : ∀ p q, P p → P q → P (p + q))
(h_std_basis : ∀ i j x, P (stdBasisMatrix i j x)) : P M :=
Matrix.induction_on' M
(by
inhabit m
inhabit n
simpa using h_std_basis default default 0)
h_add h_std_basis
#align matrix.induction_on Matrix.induction_on
namespace StdBasisMatrix
section
variable (i : m) (j : n) (c : α) (i' : m) (j' : n)
@[simp]
theorem apply_same : stdBasisMatrix i j c i j = c :=
if_pos (And.intro rfl rfl)
#align matrix.std_basis_matrix.apply_same Matrix.StdBasisMatrix.apply_same
@[simp]
theorem apply_of_ne (h : ¬(i = i' ∧ j = j')) : stdBasisMatrix i j c i' j' = 0 := by
simp only [stdBasisMatrix, and_imp, ite_eq_right_iff]
tauto
#align matrix.std_basis_matrix.apply_of_ne Matrix.StdBasisMatrix.apply_of_ne
@[simp]
| Mathlib/Data/Matrix/Basis.lean | 139 | 140 | theorem apply_of_row_ne {i i' : m} (hi : i ≠ i') (j j' : n) (a : α) :
stdBasisMatrix i j a i' j' = 0 := by | simp [hi]
| 1 | 2.718282 | 0 | 1.111111 | 9 | 1,192 |
import Mathlib.Data.Matrix.Basic
import Mathlib.LinearAlgebra.Matrix.Trace
#align_import data.matrix.basis from "leanprover-community/mathlib"@"320df450e9abeb5fc6417971e75acb6ae8bc3794"
variable {l m n : Type*}
variable {R α : Type*}
namespace Matrix
open Matrix
variable [DecidableEq l] [DecidableEq m] [DecidableEq n]
variable [Semiring α]
def stdBasisMatrix (i : m) (j : n) (a : α) : Matrix m n α := fun i' j' =>
if i = i' ∧ j = j' then a else 0
#align matrix.std_basis_matrix Matrix.stdBasisMatrix
@[simp]
theorem smul_stdBasisMatrix [SMulZeroClass R α] (r : R) (i : m) (j : n) (a : α) :
r • stdBasisMatrix i j a = stdBasisMatrix i j (r • a) := by
unfold stdBasisMatrix
ext
simp [smul_ite]
#align matrix.smul_std_basis_matrix Matrix.smul_stdBasisMatrix
@[simp]
theorem stdBasisMatrix_zero (i : m) (j : n) : stdBasisMatrix i j (0 : α) = 0 := by
unfold stdBasisMatrix
ext
simp
#align matrix.std_basis_matrix_zero Matrix.stdBasisMatrix_zero
theorem stdBasisMatrix_add (i : m) (j : n) (a b : α) :
stdBasisMatrix i j (a + b) = stdBasisMatrix i j a + stdBasisMatrix i j b := by
unfold stdBasisMatrix; ext
split_ifs with h <;> simp [h]
#align matrix.std_basis_matrix_add Matrix.stdBasisMatrix_add
theorem mulVec_stdBasisMatrix [Fintype m] (i : n) (j : m) (c : α) (x : m → α) :
mulVec (stdBasisMatrix i j c) x = Function.update (0 : n → α) i (c * x j) := by
ext i'
simp [stdBasisMatrix, mulVec, dotProduct]
rcases eq_or_ne i i' with rfl|h
· simp
simp [h, h.symm]
theorem matrix_eq_sum_std_basis [Fintype m] [Fintype n] (x : Matrix m n α) :
x = ∑ i : m, ∑ j : n, stdBasisMatrix i j (x i j) := by
ext i j; symm
iterate 2 rw [Finset.sum_apply]
-- Porting note: was `convert`
refine (Fintype.sum_eq_single i ?_).trans ?_; swap
· -- Porting note: `simp` seems unwilling to apply `Fintype.sum_apply`
simp (config := { unfoldPartialApp := true }) only [stdBasisMatrix]
rw [Fintype.sum_apply, Fintype.sum_apply]
simp
· intro j' hj'
-- Porting note: `simp` seems unwilling to apply `Fintype.sum_apply`
simp (config := { unfoldPartialApp := true }) only [stdBasisMatrix]
rw [Fintype.sum_apply, Fintype.sum_apply]
simp [hj']
#align matrix.matrix_eq_sum_std_basis Matrix.matrix_eq_sum_std_basis
-- TODO: tie this up with the `Basis` machinery of linear algebra
-- this is not completely trivial because we are indexing by two types, instead of one
-- TODO: add `std_basis_vec`
theorem std_basis_eq_basis_mul_basis (i : m) (j : n) :
stdBasisMatrix i j (1 : α) =
vecMulVec (fun i' => ite (i = i') 1 0) fun j' => ite (j = j') 1 0 := by
ext i' j'
-- Porting note: was `norm_num [std_basis_matrix, vec_mul_vec]` though there are no numerals
-- involved.
simp only [stdBasisMatrix, vecMulVec, mul_ite, mul_one, mul_zero, of_apply]
-- Porting note: added next line
simp_rw [@and_comm (i = i')]
exact ite_and _ _ _ _
#align matrix.std_basis_eq_basis_mul_basis Matrix.std_basis_eq_basis_mul_basis
-- todo: the old proof used fintypes, I don't know `Finsupp` but this feels generalizable
@[elab_as_elim]
protected theorem induction_on' [Finite m] [Finite n] {P : Matrix m n α → Prop} (M : Matrix m n α)
(h_zero : P 0) (h_add : ∀ p q, P p → P q → P (p + q))
(h_std_basis : ∀ (i : m) (j : n) (x : α), P (stdBasisMatrix i j x)) : P M := by
cases nonempty_fintype m; cases nonempty_fintype n
rw [matrix_eq_sum_std_basis M, ← Finset.sum_product']
apply Finset.sum_induction _ _ h_add h_zero
· intros
apply h_std_basis
#align matrix.induction_on' Matrix.induction_on'
@[elab_as_elim]
protected theorem induction_on [Finite m] [Finite n] [Nonempty m] [Nonempty n]
{P : Matrix m n α → Prop} (M : Matrix m n α) (h_add : ∀ p q, P p → P q → P (p + q))
(h_std_basis : ∀ i j x, P (stdBasisMatrix i j x)) : P M :=
Matrix.induction_on' M
(by
inhabit m
inhabit n
simpa using h_std_basis default default 0)
h_add h_std_basis
#align matrix.induction_on Matrix.induction_on
namespace StdBasisMatrix
section
variable (i : m) (j : n) (c : α) (i' : m) (j' : n)
@[simp]
theorem apply_same : stdBasisMatrix i j c i j = c :=
if_pos (And.intro rfl rfl)
#align matrix.std_basis_matrix.apply_same Matrix.StdBasisMatrix.apply_same
@[simp]
theorem apply_of_ne (h : ¬(i = i' ∧ j = j')) : stdBasisMatrix i j c i' j' = 0 := by
simp only [stdBasisMatrix, and_imp, ite_eq_right_iff]
tauto
#align matrix.std_basis_matrix.apply_of_ne Matrix.StdBasisMatrix.apply_of_ne
@[simp]
theorem apply_of_row_ne {i i' : m} (hi : i ≠ i') (j j' : n) (a : α) :
stdBasisMatrix i j a i' j' = 0 := by simp [hi]
#align matrix.std_basis_matrix.apply_of_row_ne Matrix.StdBasisMatrix.apply_of_row_ne
@[simp]
| Mathlib/Data/Matrix/Basis.lean | 144 | 145 | theorem apply_of_col_ne (i i' : m) {j j' : n} (hj : j ≠ j') (a : α) :
stdBasisMatrix i j a i' j' = 0 := by | simp [hj]
| 1 | 2.718282 | 0 | 1.111111 | 9 | 1,192 |
import Mathlib.Algebra.Polynomial.Mirror
import Mathlib.Analysis.Complex.Polynomial
#align_import data.polynomial.unit_trinomial from "leanprover-community/mathlib"@"302eab4f46abb63de520828de78c04cb0f9b5836"
namespace Polynomial
open scoped Polynomial
open Finset
section Semiring
variable {R : Type*} [Semiring R] (k m n : ℕ) (u v w : R)
noncomputable def trinomial :=
C u * X ^ k + C v * X ^ m + C w * X ^ n
#align polynomial.trinomial Polynomial.trinomial
theorem trinomial_def : trinomial k m n u v w = C u * X ^ k + C v * X ^ m + C w * X ^ n :=
rfl
#align polynomial.trinomial_def Polynomial.trinomial_def
variable {k m n u v w}
| Mathlib/Algebra/Polynomial/UnitTrinomial.lean | 49 | 52 | theorem trinomial_leading_coeff' (hkm : k < m) (hmn : m < n) :
(trinomial k m n u v w).coeff n = w := by |
rw [trinomial_def, coeff_add, coeff_add, coeff_C_mul_X_pow, coeff_C_mul_X_pow, coeff_C_mul_X_pow,
if_neg (hkm.trans hmn).ne', if_neg hmn.ne', if_pos rfl, zero_add, zero_add]
| 2 | 7.389056 | 1 | 1.111111 | 9 | 1,193 |
import Mathlib.Algebra.Polynomial.Mirror
import Mathlib.Analysis.Complex.Polynomial
#align_import data.polynomial.unit_trinomial from "leanprover-community/mathlib"@"302eab4f46abb63de520828de78c04cb0f9b5836"
namespace Polynomial
open scoped Polynomial
open Finset
section Semiring
variable {R : Type*} [Semiring R] (k m n : ℕ) (u v w : R)
noncomputable def trinomial :=
C u * X ^ k + C v * X ^ m + C w * X ^ n
#align polynomial.trinomial Polynomial.trinomial
theorem trinomial_def : trinomial k m n u v w = C u * X ^ k + C v * X ^ m + C w * X ^ n :=
rfl
#align polynomial.trinomial_def Polynomial.trinomial_def
variable {k m n u v w}
theorem trinomial_leading_coeff' (hkm : k < m) (hmn : m < n) :
(trinomial k m n u v w).coeff n = w := by
rw [trinomial_def, coeff_add, coeff_add, coeff_C_mul_X_pow, coeff_C_mul_X_pow, coeff_C_mul_X_pow,
if_neg (hkm.trans hmn).ne', if_neg hmn.ne', if_pos rfl, zero_add, zero_add]
#align polynomial.trinomial_leading_coeff' Polynomial.trinomial_leading_coeff'
| Mathlib/Algebra/Polynomial/UnitTrinomial.lean | 55 | 58 | theorem trinomial_middle_coeff (hkm : k < m) (hmn : m < n) :
(trinomial k m n u v w).coeff m = v := by |
rw [trinomial_def, coeff_add, coeff_add, coeff_C_mul_X_pow, coeff_C_mul_X_pow, coeff_C_mul_X_pow,
if_neg hkm.ne', if_pos rfl, if_neg hmn.ne, zero_add, add_zero]
| 2 | 7.389056 | 1 | 1.111111 | 9 | 1,193 |
import Mathlib.Algebra.Polynomial.Mirror
import Mathlib.Analysis.Complex.Polynomial
#align_import data.polynomial.unit_trinomial from "leanprover-community/mathlib"@"302eab4f46abb63de520828de78c04cb0f9b5836"
namespace Polynomial
open scoped Polynomial
open Finset
section Semiring
variable {R : Type*} [Semiring R] (k m n : ℕ) (u v w : R)
noncomputable def trinomial :=
C u * X ^ k + C v * X ^ m + C w * X ^ n
#align polynomial.trinomial Polynomial.trinomial
theorem trinomial_def : trinomial k m n u v w = C u * X ^ k + C v * X ^ m + C w * X ^ n :=
rfl
#align polynomial.trinomial_def Polynomial.trinomial_def
variable {k m n u v w}
theorem trinomial_leading_coeff' (hkm : k < m) (hmn : m < n) :
(trinomial k m n u v w).coeff n = w := by
rw [trinomial_def, coeff_add, coeff_add, coeff_C_mul_X_pow, coeff_C_mul_X_pow, coeff_C_mul_X_pow,
if_neg (hkm.trans hmn).ne', if_neg hmn.ne', if_pos rfl, zero_add, zero_add]
#align polynomial.trinomial_leading_coeff' Polynomial.trinomial_leading_coeff'
theorem trinomial_middle_coeff (hkm : k < m) (hmn : m < n) :
(trinomial k m n u v w).coeff m = v := by
rw [trinomial_def, coeff_add, coeff_add, coeff_C_mul_X_pow, coeff_C_mul_X_pow, coeff_C_mul_X_pow,
if_neg hkm.ne', if_pos rfl, if_neg hmn.ne, zero_add, add_zero]
#align polynomial.trinomial_middle_coeff Polynomial.trinomial_middle_coeff
| Mathlib/Algebra/Polynomial/UnitTrinomial.lean | 61 | 64 | theorem trinomial_trailing_coeff' (hkm : k < m) (hmn : m < n) :
(trinomial k m n u v w).coeff k = u := by |
rw [trinomial_def, coeff_add, coeff_add, coeff_C_mul_X_pow, coeff_C_mul_X_pow, coeff_C_mul_X_pow,
if_pos rfl, if_neg hkm.ne, if_neg (hkm.trans hmn).ne, add_zero, add_zero]
| 2 | 7.389056 | 1 | 1.111111 | 9 | 1,193 |
import Mathlib.Algebra.Polynomial.Mirror
import Mathlib.Analysis.Complex.Polynomial
#align_import data.polynomial.unit_trinomial from "leanprover-community/mathlib"@"302eab4f46abb63de520828de78c04cb0f9b5836"
namespace Polynomial
open scoped Polynomial
open Finset
section Semiring
variable {R : Type*} [Semiring R] (k m n : ℕ) (u v w : R)
noncomputable def trinomial :=
C u * X ^ k + C v * X ^ m + C w * X ^ n
#align polynomial.trinomial Polynomial.trinomial
theorem trinomial_def : trinomial k m n u v w = C u * X ^ k + C v * X ^ m + C w * X ^ n :=
rfl
#align polynomial.trinomial_def Polynomial.trinomial_def
variable {k m n u v w}
theorem trinomial_leading_coeff' (hkm : k < m) (hmn : m < n) :
(trinomial k m n u v w).coeff n = w := by
rw [trinomial_def, coeff_add, coeff_add, coeff_C_mul_X_pow, coeff_C_mul_X_pow, coeff_C_mul_X_pow,
if_neg (hkm.trans hmn).ne', if_neg hmn.ne', if_pos rfl, zero_add, zero_add]
#align polynomial.trinomial_leading_coeff' Polynomial.trinomial_leading_coeff'
theorem trinomial_middle_coeff (hkm : k < m) (hmn : m < n) :
(trinomial k m n u v w).coeff m = v := by
rw [trinomial_def, coeff_add, coeff_add, coeff_C_mul_X_pow, coeff_C_mul_X_pow, coeff_C_mul_X_pow,
if_neg hkm.ne', if_pos rfl, if_neg hmn.ne, zero_add, add_zero]
#align polynomial.trinomial_middle_coeff Polynomial.trinomial_middle_coeff
theorem trinomial_trailing_coeff' (hkm : k < m) (hmn : m < n) :
(trinomial k m n u v w).coeff k = u := by
rw [trinomial_def, coeff_add, coeff_add, coeff_C_mul_X_pow, coeff_C_mul_X_pow, coeff_C_mul_X_pow,
if_pos rfl, if_neg hkm.ne, if_neg (hkm.trans hmn).ne, add_zero, add_zero]
#align polynomial.trinomial_trailing_coeff' Polynomial.trinomial_trailing_coeff'
| Mathlib/Algebra/Polynomial/UnitTrinomial.lean | 67 | 78 | theorem trinomial_natDegree (hkm : k < m) (hmn : m < n) (hw : w ≠ 0) :
(trinomial k m n u v w).natDegree = n := by |
refine
natDegree_eq_of_degree_eq_some
((Finset.sup_le fun i h => ?_).antisymm <|
le_degree_of_ne_zero <| by rwa [trinomial_leading_coeff' hkm hmn])
replace h := support_trinomial' k m n u v w h
rw [mem_insert, mem_insert, mem_singleton] at h
rcases h with (rfl | rfl | rfl)
· exact WithBot.coe_le_coe.mpr (hkm.trans hmn).le
· exact WithBot.coe_le_coe.mpr hmn.le
· exact le_rfl
| 10 | 22,026.465795 | 2 | 1.111111 | 9 | 1,193 |
import Mathlib.Algebra.Polynomial.Mirror
import Mathlib.Analysis.Complex.Polynomial
#align_import data.polynomial.unit_trinomial from "leanprover-community/mathlib"@"302eab4f46abb63de520828de78c04cb0f9b5836"
namespace Polynomial
open scoped Polynomial
open Finset
section Semiring
variable {R : Type*} [Semiring R] (k m n : ℕ) (u v w : R)
noncomputable def trinomial :=
C u * X ^ k + C v * X ^ m + C w * X ^ n
#align polynomial.trinomial Polynomial.trinomial
theorem trinomial_def : trinomial k m n u v w = C u * X ^ k + C v * X ^ m + C w * X ^ n :=
rfl
#align polynomial.trinomial_def Polynomial.trinomial_def
variable {k m n u v w}
theorem trinomial_leading_coeff' (hkm : k < m) (hmn : m < n) :
(trinomial k m n u v w).coeff n = w := by
rw [trinomial_def, coeff_add, coeff_add, coeff_C_mul_X_pow, coeff_C_mul_X_pow, coeff_C_mul_X_pow,
if_neg (hkm.trans hmn).ne', if_neg hmn.ne', if_pos rfl, zero_add, zero_add]
#align polynomial.trinomial_leading_coeff' Polynomial.trinomial_leading_coeff'
theorem trinomial_middle_coeff (hkm : k < m) (hmn : m < n) :
(trinomial k m n u v w).coeff m = v := by
rw [trinomial_def, coeff_add, coeff_add, coeff_C_mul_X_pow, coeff_C_mul_X_pow, coeff_C_mul_X_pow,
if_neg hkm.ne', if_pos rfl, if_neg hmn.ne, zero_add, add_zero]
#align polynomial.trinomial_middle_coeff Polynomial.trinomial_middle_coeff
theorem trinomial_trailing_coeff' (hkm : k < m) (hmn : m < n) :
(trinomial k m n u v w).coeff k = u := by
rw [trinomial_def, coeff_add, coeff_add, coeff_C_mul_X_pow, coeff_C_mul_X_pow, coeff_C_mul_X_pow,
if_pos rfl, if_neg hkm.ne, if_neg (hkm.trans hmn).ne, add_zero, add_zero]
#align polynomial.trinomial_trailing_coeff' Polynomial.trinomial_trailing_coeff'
theorem trinomial_natDegree (hkm : k < m) (hmn : m < n) (hw : w ≠ 0) :
(trinomial k m n u v w).natDegree = n := by
refine
natDegree_eq_of_degree_eq_some
((Finset.sup_le fun i h => ?_).antisymm <|
le_degree_of_ne_zero <| by rwa [trinomial_leading_coeff' hkm hmn])
replace h := support_trinomial' k m n u v w h
rw [mem_insert, mem_insert, mem_singleton] at h
rcases h with (rfl | rfl | rfl)
· exact WithBot.coe_le_coe.mpr (hkm.trans hmn).le
· exact WithBot.coe_le_coe.mpr hmn.le
· exact le_rfl
#align polynomial.trinomial_nat_degree Polynomial.trinomial_natDegree
| Mathlib/Algebra/Polynomial/UnitTrinomial.lean | 81 | 92 | theorem trinomial_natTrailingDegree (hkm : k < m) (hmn : m < n) (hu : u ≠ 0) :
(trinomial k m n u v w).natTrailingDegree = k := by |
refine
natTrailingDegree_eq_of_trailingDegree_eq_some
((Finset.le_inf fun i h => ?_).antisymm <|
trailingDegree_le_of_ne_zero <| by rwa [trinomial_trailing_coeff' hkm hmn]).symm
replace h := support_trinomial' k m n u v w h
rw [mem_insert, mem_insert, mem_singleton] at h
rcases h with (rfl | rfl | rfl)
· exact le_rfl
· exact WithTop.coe_le_coe.mpr hkm.le
· exact WithTop.coe_le_coe.mpr (hkm.trans hmn).le
| 10 | 22,026.465795 | 2 | 1.111111 | 9 | 1,193 |
import Mathlib.Algebra.Polynomial.Mirror
import Mathlib.Analysis.Complex.Polynomial
#align_import data.polynomial.unit_trinomial from "leanprover-community/mathlib"@"302eab4f46abb63de520828de78c04cb0f9b5836"
namespace Polynomial
open scoped Polynomial
open Finset
section Semiring
variable {R : Type*} [Semiring R] (k m n : ℕ) (u v w : R)
noncomputable def trinomial :=
C u * X ^ k + C v * X ^ m + C w * X ^ n
#align polynomial.trinomial Polynomial.trinomial
theorem trinomial_def : trinomial k m n u v w = C u * X ^ k + C v * X ^ m + C w * X ^ n :=
rfl
#align polynomial.trinomial_def Polynomial.trinomial_def
variable {k m n u v w}
theorem trinomial_leading_coeff' (hkm : k < m) (hmn : m < n) :
(trinomial k m n u v w).coeff n = w := by
rw [trinomial_def, coeff_add, coeff_add, coeff_C_mul_X_pow, coeff_C_mul_X_pow, coeff_C_mul_X_pow,
if_neg (hkm.trans hmn).ne', if_neg hmn.ne', if_pos rfl, zero_add, zero_add]
#align polynomial.trinomial_leading_coeff' Polynomial.trinomial_leading_coeff'
theorem trinomial_middle_coeff (hkm : k < m) (hmn : m < n) :
(trinomial k m n u v w).coeff m = v := by
rw [trinomial_def, coeff_add, coeff_add, coeff_C_mul_X_pow, coeff_C_mul_X_pow, coeff_C_mul_X_pow,
if_neg hkm.ne', if_pos rfl, if_neg hmn.ne, zero_add, add_zero]
#align polynomial.trinomial_middle_coeff Polynomial.trinomial_middle_coeff
theorem trinomial_trailing_coeff' (hkm : k < m) (hmn : m < n) :
(trinomial k m n u v w).coeff k = u := by
rw [trinomial_def, coeff_add, coeff_add, coeff_C_mul_X_pow, coeff_C_mul_X_pow, coeff_C_mul_X_pow,
if_pos rfl, if_neg hkm.ne, if_neg (hkm.trans hmn).ne, add_zero, add_zero]
#align polynomial.trinomial_trailing_coeff' Polynomial.trinomial_trailing_coeff'
theorem trinomial_natDegree (hkm : k < m) (hmn : m < n) (hw : w ≠ 0) :
(trinomial k m n u v w).natDegree = n := by
refine
natDegree_eq_of_degree_eq_some
((Finset.sup_le fun i h => ?_).antisymm <|
le_degree_of_ne_zero <| by rwa [trinomial_leading_coeff' hkm hmn])
replace h := support_trinomial' k m n u v w h
rw [mem_insert, mem_insert, mem_singleton] at h
rcases h with (rfl | rfl | rfl)
· exact WithBot.coe_le_coe.mpr (hkm.trans hmn).le
· exact WithBot.coe_le_coe.mpr hmn.le
· exact le_rfl
#align polynomial.trinomial_nat_degree Polynomial.trinomial_natDegree
theorem trinomial_natTrailingDegree (hkm : k < m) (hmn : m < n) (hu : u ≠ 0) :
(trinomial k m n u v w).natTrailingDegree = k := by
refine
natTrailingDegree_eq_of_trailingDegree_eq_some
((Finset.le_inf fun i h => ?_).antisymm <|
trailingDegree_le_of_ne_zero <| by rwa [trinomial_trailing_coeff' hkm hmn]).symm
replace h := support_trinomial' k m n u v w h
rw [mem_insert, mem_insert, mem_singleton] at h
rcases h with (rfl | rfl | rfl)
· exact le_rfl
· exact WithTop.coe_le_coe.mpr hkm.le
· exact WithTop.coe_le_coe.mpr (hkm.trans hmn).le
#align polynomial.trinomial_nat_trailing_degree Polynomial.trinomial_natTrailingDegree
| Mathlib/Algebra/Polynomial/UnitTrinomial.lean | 95 | 97 | theorem trinomial_leadingCoeff (hkm : k < m) (hmn : m < n) (hw : w ≠ 0) :
(trinomial k m n u v w).leadingCoeff = w := by |
rw [leadingCoeff, trinomial_natDegree hkm hmn hw, trinomial_leading_coeff' hkm hmn]
| 1 | 2.718282 | 0 | 1.111111 | 9 | 1,193 |
import Mathlib.Algebra.Polynomial.Mirror
import Mathlib.Analysis.Complex.Polynomial
#align_import data.polynomial.unit_trinomial from "leanprover-community/mathlib"@"302eab4f46abb63de520828de78c04cb0f9b5836"
namespace Polynomial
open scoped Polynomial
open Finset
section Semiring
variable {R : Type*} [Semiring R] (k m n : ℕ) (u v w : R)
noncomputable def trinomial :=
C u * X ^ k + C v * X ^ m + C w * X ^ n
#align polynomial.trinomial Polynomial.trinomial
theorem trinomial_def : trinomial k m n u v w = C u * X ^ k + C v * X ^ m + C w * X ^ n :=
rfl
#align polynomial.trinomial_def Polynomial.trinomial_def
variable {k m n u v w}
theorem trinomial_leading_coeff' (hkm : k < m) (hmn : m < n) :
(trinomial k m n u v w).coeff n = w := by
rw [trinomial_def, coeff_add, coeff_add, coeff_C_mul_X_pow, coeff_C_mul_X_pow, coeff_C_mul_X_pow,
if_neg (hkm.trans hmn).ne', if_neg hmn.ne', if_pos rfl, zero_add, zero_add]
#align polynomial.trinomial_leading_coeff' Polynomial.trinomial_leading_coeff'
theorem trinomial_middle_coeff (hkm : k < m) (hmn : m < n) :
(trinomial k m n u v w).coeff m = v := by
rw [trinomial_def, coeff_add, coeff_add, coeff_C_mul_X_pow, coeff_C_mul_X_pow, coeff_C_mul_X_pow,
if_neg hkm.ne', if_pos rfl, if_neg hmn.ne, zero_add, add_zero]
#align polynomial.trinomial_middle_coeff Polynomial.trinomial_middle_coeff
theorem trinomial_trailing_coeff' (hkm : k < m) (hmn : m < n) :
(trinomial k m n u v w).coeff k = u := by
rw [trinomial_def, coeff_add, coeff_add, coeff_C_mul_X_pow, coeff_C_mul_X_pow, coeff_C_mul_X_pow,
if_pos rfl, if_neg hkm.ne, if_neg (hkm.trans hmn).ne, add_zero, add_zero]
#align polynomial.trinomial_trailing_coeff' Polynomial.trinomial_trailing_coeff'
theorem trinomial_natDegree (hkm : k < m) (hmn : m < n) (hw : w ≠ 0) :
(trinomial k m n u v w).natDegree = n := by
refine
natDegree_eq_of_degree_eq_some
((Finset.sup_le fun i h => ?_).antisymm <|
le_degree_of_ne_zero <| by rwa [trinomial_leading_coeff' hkm hmn])
replace h := support_trinomial' k m n u v w h
rw [mem_insert, mem_insert, mem_singleton] at h
rcases h with (rfl | rfl | rfl)
· exact WithBot.coe_le_coe.mpr (hkm.trans hmn).le
· exact WithBot.coe_le_coe.mpr hmn.le
· exact le_rfl
#align polynomial.trinomial_nat_degree Polynomial.trinomial_natDegree
theorem trinomial_natTrailingDegree (hkm : k < m) (hmn : m < n) (hu : u ≠ 0) :
(trinomial k m n u v w).natTrailingDegree = k := by
refine
natTrailingDegree_eq_of_trailingDegree_eq_some
((Finset.le_inf fun i h => ?_).antisymm <|
trailingDegree_le_of_ne_zero <| by rwa [trinomial_trailing_coeff' hkm hmn]).symm
replace h := support_trinomial' k m n u v w h
rw [mem_insert, mem_insert, mem_singleton] at h
rcases h with (rfl | rfl | rfl)
· exact le_rfl
· exact WithTop.coe_le_coe.mpr hkm.le
· exact WithTop.coe_le_coe.mpr (hkm.trans hmn).le
#align polynomial.trinomial_nat_trailing_degree Polynomial.trinomial_natTrailingDegree
theorem trinomial_leadingCoeff (hkm : k < m) (hmn : m < n) (hw : w ≠ 0) :
(trinomial k m n u v w).leadingCoeff = w := by
rw [leadingCoeff, trinomial_natDegree hkm hmn hw, trinomial_leading_coeff' hkm hmn]
#align polynomial.trinomial_leading_coeff Polynomial.trinomial_leadingCoeff
| Mathlib/Algebra/Polynomial/UnitTrinomial.lean | 100 | 102 | theorem trinomial_trailingCoeff (hkm : k < m) (hmn : m < n) (hu : u ≠ 0) :
(trinomial k m n u v w).trailingCoeff = u := by |
rw [trailingCoeff, trinomial_natTrailingDegree hkm hmn hu, trinomial_trailing_coeff' hkm hmn]
| 1 | 2.718282 | 0 | 1.111111 | 9 | 1,193 |
import Mathlib.Algebra.Polynomial.Mirror
import Mathlib.Analysis.Complex.Polynomial
#align_import data.polynomial.unit_trinomial from "leanprover-community/mathlib"@"302eab4f46abb63de520828de78c04cb0f9b5836"
namespace Polynomial
open scoped Polynomial
open Finset
section Semiring
variable {R : Type*} [Semiring R] (k m n : ℕ) (u v w : R)
noncomputable def trinomial :=
C u * X ^ k + C v * X ^ m + C w * X ^ n
#align polynomial.trinomial Polynomial.trinomial
theorem trinomial_def : trinomial k m n u v w = C u * X ^ k + C v * X ^ m + C w * X ^ n :=
rfl
#align polynomial.trinomial_def Polynomial.trinomial_def
variable {k m n u v w}
theorem trinomial_leading_coeff' (hkm : k < m) (hmn : m < n) :
(trinomial k m n u v w).coeff n = w := by
rw [trinomial_def, coeff_add, coeff_add, coeff_C_mul_X_pow, coeff_C_mul_X_pow, coeff_C_mul_X_pow,
if_neg (hkm.trans hmn).ne', if_neg hmn.ne', if_pos rfl, zero_add, zero_add]
#align polynomial.trinomial_leading_coeff' Polynomial.trinomial_leading_coeff'
theorem trinomial_middle_coeff (hkm : k < m) (hmn : m < n) :
(trinomial k m n u v w).coeff m = v := by
rw [trinomial_def, coeff_add, coeff_add, coeff_C_mul_X_pow, coeff_C_mul_X_pow, coeff_C_mul_X_pow,
if_neg hkm.ne', if_pos rfl, if_neg hmn.ne, zero_add, add_zero]
#align polynomial.trinomial_middle_coeff Polynomial.trinomial_middle_coeff
theorem trinomial_trailing_coeff' (hkm : k < m) (hmn : m < n) :
(trinomial k m n u v w).coeff k = u := by
rw [trinomial_def, coeff_add, coeff_add, coeff_C_mul_X_pow, coeff_C_mul_X_pow, coeff_C_mul_X_pow,
if_pos rfl, if_neg hkm.ne, if_neg (hkm.trans hmn).ne, add_zero, add_zero]
#align polynomial.trinomial_trailing_coeff' Polynomial.trinomial_trailing_coeff'
theorem trinomial_natDegree (hkm : k < m) (hmn : m < n) (hw : w ≠ 0) :
(trinomial k m n u v w).natDegree = n := by
refine
natDegree_eq_of_degree_eq_some
((Finset.sup_le fun i h => ?_).antisymm <|
le_degree_of_ne_zero <| by rwa [trinomial_leading_coeff' hkm hmn])
replace h := support_trinomial' k m n u v w h
rw [mem_insert, mem_insert, mem_singleton] at h
rcases h with (rfl | rfl | rfl)
· exact WithBot.coe_le_coe.mpr (hkm.trans hmn).le
· exact WithBot.coe_le_coe.mpr hmn.le
· exact le_rfl
#align polynomial.trinomial_nat_degree Polynomial.trinomial_natDegree
theorem trinomial_natTrailingDegree (hkm : k < m) (hmn : m < n) (hu : u ≠ 0) :
(trinomial k m n u v w).natTrailingDegree = k := by
refine
natTrailingDegree_eq_of_trailingDegree_eq_some
((Finset.le_inf fun i h => ?_).antisymm <|
trailingDegree_le_of_ne_zero <| by rwa [trinomial_trailing_coeff' hkm hmn]).symm
replace h := support_trinomial' k m n u v w h
rw [mem_insert, mem_insert, mem_singleton] at h
rcases h with (rfl | rfl | rfl)
· exact le_rfl
· exact WithTop.coe_le_coe.mpr hkm.le
· exact WithTop.coe_le_coe.mpr (hkm.trans hmn).le
#align polynomial.trinomial_nat_trailing_degree Polynomial.trinomial_natTrailingDegree
theorem trinomial_leadingCoeff (hkm : k < m) (hmn : m < n) (hw : w ≠ 0) :
(trinomial k m n u v w).leadingCoeff = w := by
rw [leadingCoeff, trinomial_natDegree hkm hmn hw, trinomial_leading_coeff' hkm hmn]
#align polynomial.trinomial_leading_coeff Polynomial.trinomial_leadingCoeff
theorem trinomial_trailingCoeff (hkm : k < m) (hmn : m < n) (hu : u ≠ 0) :
(trinomial k m n u v w).trailingCoeff = u := by
rw [trailingCoeff, trinomial_natTrailingDegree hkm hmn hu, trinomial_trailing_coeff' hkm hmn]
#align polynomial.trinomial_trailing_coeff Polynomial.trinomial_trailingCoeff
| Mathlib/Algebra/Polynomial/UnitTrinomial.lean | 105 | 107 | theorem trinomial_monic (hkm : k < m) (hmn : m < n) : (trinomial k m n u v 1).Monic := by |
nontriviality R
exact trinomial_leadingCoeff hkm hmn one_ne_zero
| 2 | 7.389056 | 1 | 1.111111 | 9 | 1,193 |
import Mathlib.Algebra.Polynomial.Mirror
import Mathlib.Analysis.Complex.Polynomial
#align_import data.polynomial.unit_trinomial from "leanprover-community/mathlib"@"302eab4f46abb63de520828de78c04cb0f9b5836"
namespace Polynomial
open scoped Polynomial
open Finset
section Semiring
variable {R : Type*} [Semiring R] (k m n : ℕ) (u v w : R)
noncomputable def trinomial :=
C u * X ^ k + C v * X ^ m + C w * X ^ n
#align polynomial.trinomial Polynomial.trinomial
theorem trinomial_def : trinomial k m n u v w = C u * X ^ k + C v * X ^ m + C w * X ^ n :=
rfl
#align polynomial.trinomial_def Polynomial.trinomial_def
variable {k m n u v w}
theorem trinomial_leading_coeff' (hkm : k < m) (hmn : m < n) :
(trinomial k m n u v w).coeff n = w := by
rw [trinomial_def, coeff_add, coeff_add, coeff_C_mul_X_pow, coeff_C_mul_X_pow, coeff_C_mul_X_pow,
if_neg (hkm.trans hmn).ne', if_neg hmn.ne', if_pos rfl, zero_add, zero_add]
#align polynomial.trinomial_leading_coeff' Polynomial.trinomial_leading_coeff'
theorem trinomial_middle_coeff (hkm : k < m) (hmn : m < n) :
(trinomial k m n u v w).coeff m = v := by
rw [trinomial_def, coeff_add, coeff_add, coeff_C_mul_X_pow, coeff_C_mul_X_pow, coeff_C_mul_X_pow,
if_neg hkm.ne', if_pos rfl, if_neg hmn.ne, zero_add, add_zero]
#align polynomial.trinomial_middle_coeff Polynomial.trinomial_middle_coeff
theorem trinomial_trailing_coeff' (hkm : k < m) (hmn : m < n) :
(trinomial k m n u v w).coeff k = u := by
rw [trinomial_def, coeff_add, coeff_add, coeff_C_mul_X_pow, coeff_C_mul_X_pow, coeff_C_mul_X_pow,
if_pos rfl, if_neg hkm.ne, if_neg (hkm.trans hmn).ne, add_zero, add_zero]
#align polynomial.trinomial_trailing_coeff' Polynomial.trinomial_trailing_coeff'
theorem trinomial_natDegree (hkm : k < m) (hmn : m < n) (hw : w ≠ 0) :
(trinomial k m n u v w).natDegree = n := by
refine
natDegree_eq_of_degree_eq_some
((Finset.sup_le fun i h => ?_).antisymm <|
le_degree_of_ne_zero <| by rwa [trinomial_leading_coeff' hkm hmn])
replace h := support_trinomial' k m n u v w h
rw [mem_insert, mem_insert, mem_singleton] at h
rcases h with (rfl | rfl | rfl)
· exact WithBot.coe_le_coe.mpr (hkm.trans hmn).le
· exact WithBot.coe_le_coe.mpr hmn.le
· exact le_rfl
#align polynomial.trinomial_nat_degree Polynomial.trinomial_natDegree
theorem trinomial_natTrailingDegree (hkm : k < m) (hmn : m < n) (hu : u ≠ 0) :
(trinomial k m n u v w).natTrailingDegree = k := by
refine
natTrailingDegree_eq_of_trailingDegree_eq_some
((Finset.le_inf fun i h => ?_).antisymm <|
trailingDegree_le_of_ne_zero <| by rwa [trinomial_trailing_coeff' hkm hmn]).symm
replace h := support_trinomial' k m n u v w h
rw [mem_insert, mem_insert, mem_singleton] at h
rcases h with (rfl | rfl | rfl)
· exact le_rfl
· exact WithTop.coe_le_coe.mpr hkm.le
· exact WithTop.coe_le_coe.mpr (hkm.trans hmn).le
#align polynomial.trinomial_nat_trailing_degree Polynomial.trinomial_natTrailingDegree
theorem trinomial_leadingCoeff (hkm : k < m) (hmn : m < n) (hw : w ≠ 0) :
(trinomial k m n u v w).leadingCoeff = w := by
rw [leadingCoeff, trinomial_natDegree hkm hmn hw, trinomial_leading_coeff' hkm hmn]
#align polynomial.trinomial_leading_coeff Polynomial.trinomial_leadingCoeff
theorem trinomial_trailingCoeff (hkm : k < m) (hmn : m < n) (hu : u ≠ 0) :
(trinomial k m n u v w).trailingCoeff = u := by
rw [trailingCoeff, trinomial_natTrailingDegree hkm hmn hu, trinomial_trailing_coeff' hkm hmn]
#align polynomial.trinomial_trailing_coeff Polynomial.trinomial_trailingCoeff
theorem trinomial_monic (hkm : k < m) (hmn : m < n) : (trinomial k m n u v 1).Monic := by
nontriviality R
exact trinomial_leadingCoeff hkm hmn one_ne_zero
#align polynomial.trinomial_monic Polynomial.trinomial_monic
| Mathlib/Algebra/Polynomial/UnitTrinomial.lean | 110 | 117 | theorem trinomial_mirror (hkm : k < m) (hmn : m < n) (hu : u ≠ 0) (hw : w ≠ 0) :
(trinomial k m n u v w).mirror = trinomial k (n - m + k) n w v u := by |
rw [mirror, trinomial_natTrailingDegree hkm hmn hu, reverse, trinomial_natDegree hkm hmn hw,
trinomial_def, reflect_add, reflect_add, reflect_C_mul_X_pow, reflect_C_mul_X_pow,
reflect_C_mul_X_pow, revAt_le (hkm.trans hmn).le, revAt_le hmn.le, revAt_le le_rfl, add_mul,
add_mul, mul_assoc, mul_assoc, mul_assoc, ← pow_add, ← pow_add, ← pow_add,
Nat.sub_add_cancel (hkm.trans hmn).le, Nat.sub_self, zero_add, add_comm, add_comm (C u * X ^ n),
← add_assoc, ← trinomial_def]
| 6 | 403.428793 | 2 | 1.111111 | 9 | 1,193 |
import Mathlib.Data.Finsupp.Basic
import Mathlib.Data.Finsupp.Order
#align_import data.finsupp.multiset from "leanprover-community/mathlib"@"59694bd07f0a39c5beccba34bd9f413a160782bf"
open Finset
variable {α β ι : Type*}
namespace Finsupp
def toMultiset : (α →₀ ℕ) →+ Multiset α where
toFun f := Finsupp.sum f fun a n => n • {a}
-- Porting note: times out if h is not specified
map_add' _f _g := sum_add_index' (h := fun a n => n • ({a} : Multiset α))
(fun _ ↦ zero_nsmul _) (fun _ ↦ add_nsmul _)
map_zero' := sum_zero_index
theorem toMultiset_zero : toMultiset (0 : α →₀ ℕ) = 0 :=
rfl
#align finsupp.to_multiset_zero Finsupp.toMultiset_zero
theorem toMultiset_add (m n : α →₀ ℕ) : toMultiset (m + n) = toMultiset m + toMultiset n :=
toMultiset.map_add m n
#align finsupp.to_multiset_add Finsupp.toMultiset_add
theorem toMultiset_apply (f : α →₀ ℕ) : toMultiset f = f.sum fun a n => n • {a} :=
rfl
#align finsupp.to_multiset_apply Finsupp.toMultiset_apply
@[simp]
| Mathlib/Data/Finsupp/Multiset.lean | 52 | 53 | theorem toMultiset_single (a : α) (n : ℕ) : toMultiset (single a n) = n • {a} := by |
rw [toMultiset_apply, sum_single_index]; apply zero_nsmul
| 1 | 2.718282 | 0 | 1.111111 | 9 | 1,194 |
import Mathlib.Data.Finsupp.Basic
import Mathlib.Data.Finsupp.Order
#align_import data.finsupp.multiset from "leanprover-community/mathlib"@"59694bd07f0a39c5beccba34bd9f413a160782bf"
open Finset
variable {α β ι : Type*}
namespace Finsupp
def toMultiset : (α →₀ ℕ) →+ Multiset α where
toFun f := Finsupp.sum f fun a n => n • {a}
-- Porting note: times out if h is not specified
map_add' _f _g := sum_add_index' (h := fun a n => n • ({a} : Multiset α))
(fun _ ↦ zero_nsmul _) (fun _ ↦ add_nsmul _)
map_zero' := sum_zero_index
theorem toMultiset_zero : toMultiset (0 : α →₀ ℕ) = 0 :=
rfl
#align finsupp.to_multiset_zero Finsupp.toMultiset_zero
theorem toMultiset_add (m n : α →₀ ℕ) : toMultiset (m + n) = toMultiset m + toMultiset n :=
toMultiset.map_add m n
#align finsupp.to_multiset_add Finsupp.toMultiset_add
theorem toMultiset_apply (f : α →₀ ℕ) : toMultiset f = f.sum fun a n => n • {a} :=
rfl
#align finsupp.to_multiset_apply Finsupp.toMultiset_apply
@[simp]
theorem toMultiset_single (a : α) (n : ℕ) : toMultiset (single a n) = n • {a} := by
rw [toMultiset_apply, sum_single_index]; apply zero_nsmul
#align finsupp.to_multiset_single Finsupp.toMultiset_single
theorem toMultiset_sum {f : ι → α →₀ ℕ} (s : Finset ι) :
Finsupp.toMultiset (∑ i ∈ s, f i) = ∑ i ∈ s, Finsupp.toMultiset (f i) :=
map_sum Finsupp.toMultiset _ _
#align finsupp.to_multiset_sum Finsupp.toMultiset_sum
| Mathlib/Data/Finsupp/Multiset.lean | 61 | 63 | theorem toMultiset_sum_single (s : Finset ι) (n : ℕ) :
Finsupp.toMultiset (∑ i ∈ s, single i n) = n • s.val := by |
simp_rw [toMultiset_sum, Finsupp.toMultiset_single, sum_nsmul, sum_multiset_singleton]
| 1 | 2.718282 | 0 | 1.111111 | 9 | 1,194 |
import Mathlib.Data.Finsupp.Basic
import Mathlib.Data.Finsupp.Order
#align_import data.finsupp.multiset from "leanprover-community/mathlib"@"59694bd07f0a39c5beccba34bd9f413a160782bf"
open Finset
variable {α β ι : Type*}
namespace Finsupp
def toMultiset : (α →₀ ℕ) →+ Multiset α where
toFun f := Finsupp.sum f fun a n => n • {a}
-- Porting note: times out if h is not specified
map_add' _f _g := sum_add_index' (h := fun a n => n • ({a} : Multiset α))
(fun _ ↦ zero_nsmul _) (fun _ ↦ add_nsmul _)
map_zero' := sum_zero_index
theorem toMultiset_zero : toMultiset (0 : α →₀ ℕ) = 0 :=
rfl
#align finsupp.to_multiset_zero Finsupp.toMultiset_zero
theorem toMultiset_add (m n : α →₀ ℕ) : toMultiset (m + n) = toMultiset m + toMultiset n :=
toMultiset.map_add m n
#align finsupp.to_multiset_add Finsupp.toMultiset_add
theorem toMultiset_apply (f : α →₀ ℕ) : toMultiset f = f.sum fun a n => n • {a} :=
rfl
#align finsupp.to_multiset_apply Finsupp.toMultiset_apply
@[simp]
theorem toMultiset_single (a : α) (n : ℕ) : toMultiset (single a n) = n • {a} := by
rw [toMultiset_apply, sum_single_index]; apply zero_nsmul
#align finsupp.to_multiset_single Finsupp.toMultiset_single
theorem toMultiset_sum {f : ι → α →₀ ℕ} (s : Finset ι) :
Finsupp.toMultiset (∑ i ∈ s, f i) = ∑ i ∈ s, Finsupp.toMultiset (f i) :=
map_sum Finsupp.toMultiset _ _
#align finsupp.to_multiset_sum Finsupp.toMultiset_sum
theorem toMultiset_sum_single (s : Finset ι) (n : ℕ) :
Finsupp.toMultiset (∑ i ∈ s, single i n) = n • s.val := by
simp_rw [toMultiset_sum, Finsupp.toMultiset_single, sum_nsmul, sum_multiset_singleton]
#align finsupp.to_multiset_sum_single Finsupp.toMultiset_sum_single
@[simp]
| Mathlib/Data/Finsupp/Multiset.lean | 67 | 68 | theorem card_toMultiset (f : α →₀ ℕ) : Multiset.card (toMultiset f) = f.sum fun _ => id := by |
simp [toMultiset_apply, map_finsupp_sum, Function.id_def]
| 1 | 2.718282 | 0 | 1.111111 | 9 | 1,194 |
import Mathlib.Data.Finsupp.Basic
import Mathlib.Data.Finsupp.Order
#align_import data.finsupp.multiset from "leanprover-community/mathlib"@"59694bd07f0a39c5beccba34bd9f413a160782bf"
open Finset
variable {α β ι : Type*}
namespace Finsupp
def toMultiset : (α →₀ ℕ) →+ Multiset α where
toFun f := Finsupp.sum f fun a n => n • {a}
-- Porting note: times out if h is not specified
map_add' _f _g := sum_add_index' (h := fun a n => n • ({a} : Multiset α))
(fun _ ↦ zero_nsmul _) (fun _ ↦ add_nsmul _)
map_zero' := sum_zero_index
theorem toMultiset_zero : toMultiset (0 : α →₀ ℕ) = 0 :=
rfl
#align finsupp.to_multiset_zero Finsupp.toMultiset_zero
theorem toMultiset_add (m n : α →₀ ℕ) : toMultiset (m + n) = toMultiset m + toMultiset n :=
toMultiset.map_add m n
#align finsupp.to_multiset_add Finsupp.toMultiset_add
theorem toMultiset_apply (f : α →₀ ℕ) : toMultiset f = f.sum fun a n => n • {a} :=
rfl
#align finsupp.to_multiset_apply Finsupp.toMultiset_apply
@[simp]
theorem toMultiset_single (a : α) (n : ℕ) : toMultiset (single a n) = n • {a} := by
rw [toMultiset_apply, sum_single_index]; apply zero_nsmul
#align finsupp.to_multiset_single Finsupp.toMultiset_single
theorem toMultiset_sum {f : ι → α →₀ ℕ} (s : Finset ι) :
Finsupp.toMultiset (∑ i ∈ s, f i) = ∑ i ∈ s, Finsupp.toMultiset (f i) :=
map_sum Finsupp.toMultiset _ _
#align finsupp.to_multiset_sum Finsupp.toMultiset_sum
theorem toMultiset_sum_single (s : Finset ι) (n : ℕ) :
Finsupp.toMultiset (∑ i ∈ s, single i n) = n • s.val := by
simp_rw [toMultiset_sum, Finsupp.toMultiset_single, sum_nsmul, sum_multiset_singleton]
#align finsupp.to_multiset_sum_single Finsupp.toMultiset_sum_single
@[simp]
theorem card_toMultiset (f : α →₀ ℕ) : Multiset.card (toMultiset f) = f.sum fun _ => id := by
simp [toMultiset_apply, map_finsupp_sum, Function.id_def]
#align finsupp.card_to_multiset Finsupp.card_toMultiset
| Mathlib/Data/Finsupp/Multiset.lean | 71 | 79 | theorem toMultiset_map (f : α →₀ ℕ) (g : α → β) :
f.toMultiset.map g = toMultiset (f.mapDomain g) := by |
refine f.induction ?_ ?_
· rw [toMultiset_zero, Multiset.map_zero, mapDomain_zero, toMultiset_zero]
· intro a n f _ _ ih
rw [toMultiset_add, Multiset.map_add, ih, mapDomain_add, mapDomain_single,
toMultiset_single, toMultiset_add, toMultiset_single, ← Multiset.coe_mapAddMonoidHom,
(Multiset.mapAddMonoidHom g).map_nsmul]
rfl
| 7 | 1,096.633158 | 2 | 1.111111 | 9 | 1,194 |
import Mathlib.Data.Finsupp.Basic
import Mathlib.Data.Finsupp.Order
#align_import data.finsupp.multiset from "leanprover-community/mathlib"@"59694bd07f0a39c5beccba34bd9f413a160782bf"
open Finset
variable {α β ι : Type*}
namespace Finsupp
def toMultiset : (α →₀ ℕ) →+ Multiset α where
toFun f := Finsupp.sum f fun a n => n • {a}
-- Porting note: times out if h is not specified
map_add' _f _g := sum_add_index' (h := fun a n => n • ({a} : Multiset α))
(fun _ ↦ zero_nsmul _) (fun _ ↦ add_nsmul _)
map_zero' := sum_zero_index
theorem toMultiset_zero : toMultiset (0 : α →₀ ℕ) = 0 :=
rfl
#align finsupp.to_multiset_zero Finsupp.toMultiset_zero
theorem toMultiset_add (m n : α →₀ ℕ) : toMultiset (m + n) = toMultiset m + toMultiset n :=
toMultiset.map_add m n
#align finsupp.to_multiset_add Finsupp.toMultiset_add
theorem toMultiset_apply (f : α →₀ ℕ) : toMultiset f = f.sum fun a n => n • {a} :=
rfl
#align finsupp.to_multiset_apply Finsupp.toMultiset_apply
@[simp]
theorem toMultiset_single (a : α) (n : ℕ) : toMultiset (single a n) = n • {a} := by
rw [toMultiset_apply, sum_single_index]; apply zero_nsmul
#align finsupp.to_multiset_single Finsupp.toMultiset_single
theorem toMultiset_sum {f : ι → α →₀ ℕ} (s : Finset ι) :
Finsupp.toMultiset (∑ i ∈ s, f i) = ∑ i ∈ s, Finsupp.toMultiset (f i) :=
map_sum Finsupp.toMultiset _ _
#align finsupp.to_multiset_sum Finsupp.toMultiset_sum
theorem toMultiset_sum_single (s : Finset ι) (n : ℕ) :
Finsupp.toMultiset (∑ i ∈ s, single i n) = n • s.val := by
simp_rw [toMultiset_sum, Finsupp.toMultiset_single, sum_nsmul, sum_multiset_singleton]
#align finsupp.to_multiset_sum_single Finsupp.toMultiset_sum_single
@[simp]
theorem card_toMultiset (f : α →₀ ℕ) : Multiset.card (toMultiset f) = f.sum fun _ => id := by
simp [toMultiset_apply, map_finsupp_sum, Function.id_def]
#align finsupp.card_to_multiset Finsupp.card_toMultiset
theorem toMultiset_map (f : α →₀ ℕ) (g : α → β) :
f.toMultiset.map g = toMultiset (f.mapDomain g) := by
refine f.induction ?_ ?_
· rw [toMultiset_zero, Multiset.map_zero, mapDomain_zero, toMultiset_zero]
· intro a n f _ _ ih
rw [toMultiset_add, Multiset.map_add, ih, mapDomain_add, mapDomain_single,
toMultiset_single, toMultiset_add, toMultiset_single, ← Multiset.coe_mapAddMonoidHom,
(Multiset.mapAddMonoidHom g).map_nsmul]
rfl
#align finsupp.to_multiset_map Finsupp.toMultiset_map
@[to_additive (attr := simp)]
| Mathlib/Data/Finsupp/Multiset.lean | 83 | 90 | theorem prod_toMultiset [CommMonoid α] (f : α →₀ ℕ) :
f.toMultiset.prod = f.prod fun a n => a ^ n := by |
refine f.induction ?_ ?_
· rw [toMultiset_zero, Multiset.prod_zero, Finsupp.prod_zero_index]
· intro a n f _ _ ih
rw [toMultiset_add, Multiset.prod_add, ih, toMultiset_single, Multiset.prod_nsmul,
Finsupp.prod_add_index' pow_zero pow_add, Finsupp.prod_single_index, Multiset.prod_singleton]
exact pow_zero a
| 6 | 403.428793 | 2 | 1.111111 | 9 | 1,194 |
import Mathlib.Data.Finsupp.Basic
import Mathlib.Data.Finsupp.Order
#align_import data.finsupp.multiset from "leanprover-community/mathlib"@"59694bd07f0a39c5beccba34bd9f413a160782bf"
open Finset
variable {α β ι : Type*}
namespace Finsupp
def toMultiset : (α →₀ ℕ) →+ Multiset α where
toFun f := Finsupp.sum f fun a n => n • {a}
-- Porting note: times out if h is not specified
map_add' _f _g := sum_add_index' (h := fun a n => n • ({a} : Multiset α))
(fun _ ↦ zero_nsmul _) (fun _ ↦ add_nsmul _)
map_zero' := sum_zero_index
theorem toMultiset_zero : toMultiset (0 : α →₀ ℕ) = 0 :=
rfl
#align finsupp.to_multiset_zero Finsupp.toMultiset_zero
theorem toMultiset_add (m n : α →₀ ℕ) : toMultiset (m + n) = toMultiset m + toMultiset n :=
toMultiset.map_add m n
#align finsupp.to_multiset_add Finsupp.toMultiset_add
theorem toMultiset_apply (f : α →₀ ℕ) : toMultiset f = f.sum fun a n => n • {a} :=
rfl
#align finsupp.to_multiset_apply Finsupp.toMultiset_apply
@[simp]
theorem toMultiset_single (a : α) (n : ℕ) : toMultiset (single a n) = n • {a} := by
rw [toMultiset_apply, sum_single_index]; apply zero_nsmul
#align finsupp.to_multiset_single Finsupp.toMultiset_single
theorem toMultiset_sum {f : ι → α →₀ ℕ} (s : Finset ι) :
Finsupp.toMultiset (∑ i ∈ s, f i) = ∑ i ∈ s, Finsupp.toMultiset (f i) :=
map_sum Finsupp.toMultiset _ _
#align finsupp.to_multiset_sum Finsupp.toMultiset_sum
theorem toMultiset_sum_single (s : Finset ι) (n : ℕ) :
Finsupp.toMultiset (∑ i ∈ s, single i n) = n • s.val := by
simp_rw [toMultiset_sum, Finsupp.toMultiset_single, sum_nsmul, sum_multiset_singleton]
#align finsupp.to_multiset_sum_single Finsupp.toMultiset_sum_single
@[simp]
theorem card_toMultiset (f : α →₀ ℕ) : Multiset.card (toMultiset f) = f.sum fun _ => id := by
simp [toMultiset_apply, map_finsupp_sum, Function.id_def]
#align finsupp.card_to_multiset Finsupp.card_toMultiset
theorem toMultiset_map (f : α →₀ ℕ) (g : α → β) :
f.toMultiset.map g = toMultiset (f.mapDomain g) := by
refine f.induction ?_ ?_
· rw [toMultiset_zero, Multiset.map_zero, mapDomain_zero, toMultiset_zero]
· intro a n f _ _ ih
rw [toMultiset_add, Multiset.map_add, ih, mapDomain_add, mapDomain_single,
toMultiset_single, toMultiset_add, toMultiset_single, ← Multiset.coe_mapAddMonoidHom,
(Multiset.mapAddMonoidHom g).map_nsmul]
rfl
#align finsupp.to_multiset_map Finsupp.toMultiset_map
@[to_additive (attr := simp)]
theorem prod_toMultiset [CommMonoid α] (f : α →₀ ℕ) :
f.toMultiset.prod = f.prod fun a n => a ^ n := by
refine f.induction ?_ ?_
· rw [toMultiset_zero, Multiset.prod_zero, Finsupp.prod_zero_index]
· intro a n f _ _ ih
rw [toMultiset_add, Multiset.prod_add, ih, toMultiset_single, Multiset.prod_nsmul,
Finsupp.prod_add_index' pow_zero pow_add, Finsupp.prod_single_index, Multiset.prod_singleton]
exact pow_zero a
#align finsupp.prod_to_multiset Finsupp.prod_toMultiset
@[simp]
| Mathlib/Data/Finsupp/Multiset.lean | 94 | 101 | theorem toFinset_toMultiset [DecidableEq α] (f : α →₀ ℕ) : f.toMultiset.toFinset = f.support := by |
refine f.induction ?_ ?_
· rw [toMultiset_zero, Multiset.toFinset_zero, support_zero]
· intro a n f ha hn ih
rw [toMultiset_add, Multiset.toFinset_add, ih, toMultiset_single, support_add_eq,
support_single_ne_zero _ hn, Multiset.toFinset_nsmul _ _ hn, Multiset.toFinset_singleton]
refine Disjoint.mono_left support_single_subset ?_
rwa [Finset.disjoint_singleton_left]
| 7 | 1,096.633158 | 2 | 1.111111 | 9 | 1,194 |
import Mathlib.Data.Finsupp.Basic
import Mathlib.Data.Finsupp.Order
#align_import data.finsupp.multiset from "leanprover-community/mathlib"@"59694bd07f0a39c5beccba34bd9f413a160782bf"
open Finset
variable {α β ι : Type*}
namespace Finsupp
def toMultiset : (α →₀ ℕ) →+ Multiset α where
toFun f := Finsupp.sum f fun a n => n • {a}
-- Porting note: times out if h is not specified
map_add' _f _g := sum_add_index' (h := fun a n => n • ({a} : Multiset α))
(fun _ ↦ zero_nsmul _) (fun _ ↦ add_nsmul _)
map_zero' := sum_zero_index
theorem toMultiset_zero : toMultiset (0 : α →₀ ℕ) = 0 :=
rfl
#align finsupp.to_multiset_zero Finsupp.toMultiset_zero
theorem toMultiset_add (m n : α →₀ ℕ) : toMultiset (m + n) = toMultiset m + toMultiset n :=
toMultiset.map_add m n
#align finsupp.to_multiset_add Finsupp.toMultiset_add
theorem toMultiset_apply (f : α →₀ ℕ) : toMultiset f = f.sum fun a n => n • {a} :=
rfl
#align finsupp.to_multiset_apply Finsupp.toMultiset_apply
@[simp]
theorem toMultiset_single (a : α) (n : ℕ) : toMultiset (single a n) = n • {a} := by
rw [toMultiset_apply, sum_single_index]; apply zero_nsmul
#align finsupp.to_multiset_single Finsupp.toMultiset_single
theorem toMultiset_sum {f : ι → α →₀ ℕ} (s : Finset ι) :
Finsupp.toMultiset (∑ i ∈ s, f i) = ∑ i ∈ s, Finsupp.toMultiset (f i) :=
map_sum Finsupp.toMultiset _ _
#align finsupp.to_multiset_sum Finsupp.toMultiset_sum
theorem toMultiset_sum_single (s : Finset ι) (n : ℕ) :
Finsupp.toMultiset (∑ i ∈ s, single i n) = n • s.val := by
simp_rw [toMultiset_sum, Finsupp.toMultiset_single, sum_nsmul, sum_multiset_singleton]
#align finsupp.to_multiset_sum_single Finsupp.toMultiset_sum_single
@[simp]
theorem card_toMultiset (f : α →₀ ℕ) : Multiset.card (toMultiset f) = f.sum fun _ => id := by
simp [toMultiset_apply, map_finsupp_sum, Function.id_def]
#align finsupp.card_to_multiset Finsupp.card_toMultiset
theorem toMultiset_map (f : α →₀ ℕ) (g : α → β) :
f.toMultiset.map g = toMultiset (f.mapDomain g) := by
refine f.induction ?_ ?_
· rw [toMultiset_zero, Multiset.map_zero, mapDomain_zero, toMultiset_zero]
· intro a n f _ _ ih
rw [toMultiset_add, Multiset.map_add, ih, mapDomain_add, mapDomain_single,
toMultiset_single, toMultiset_add, toMultiset_single, ← Multiset.coe_mapAddMonoidHom,
(Multiset.mapAddMonoidHom g).map_nsmul]
rfl
#align finsupp.to_multiset_map Finsupp.toMultiset_map
@[to_additive (attr := simp)]
theorem prod_toMultiset [CommMonoid α] (f : α →₀ ℕ) :
f.toMultiset.prod = f.prod fun a n => a ^ n := by
refine f.induction ?_ ?_
· rw [toMultiset_zero, Multiset.prod_zero, Finsupp.prod_zero_index]
· intro a n f _ _ ih
rw [toMultiset_add, Multiset.prod_add, ih, toMultiset_single, Multiset.prod_nsmul,
Finsupp.prod_add_index' pow_zero pow_add, Finsupp.prod_single_index, Multiset.prod_singleton]
exact pow_zero a
#align finsupp.prod_to_multiset Finsupp.prod_toMultiset
@[simp]
theorem toFinset_toMultiset [DecidableEq α] (f : α →₀ ℕ) : f.toMultiset.toFinset = f.support := by
refine f.induction ?_ ?_
· rw [toMultiset_zero, Multiset.toFinset_zero, support_zero]
· intro a n f ha hn ih
rw [toMultiset_add, Multiset.toFinset_add, ih, toMultiset_single, support_add_eq,
support_single_ne_zero _ hn, Multiset.toFinset_nsmul _ _ hn, Multiset.toFinset_singleton]
refine Disjoint.mono_left support_single_subset ?_
rwa [Finset.disjoint_singleton_left]
#align finsupp.to_finset_to_multiset Finsupp.toFinset_toMultiset
@[simp]
| Mathlib/Data/Finsupp/Multiset.lean | 105 | 114 | theorem count_toMultiset [DecidableEq α] (f : α →₀ ℕ) (a : α) : (toMultiset f).count a = f a :=
calc
(toMultiset f).count a = Finsupp.sum f (fun x n => (n • {x} : Multiset α).count a) := by |
rw [toMultiset_apply]; exact map_sum (Multiset.countAddMonoidHom a) _ f.support
_ = f.sum fun x n => n * ({x} : Multiset α).count a := by simp only [Multiset.count_nsmul]
_ = f a * ({a} : Multiset α).count a :=
sum_eq_single _
(fun a' _ H => by simp only [Multiset.count_singleton, if_false, H.symm, mul_zero])
(fun _ => zero_mul _)
_ = f a := by rw [Multiset.count_singleton_self, mul_one]
| 7 | 1,096.633158 | 2 | 1.111111 | 9 | 1,194 |
import Mathlib.Data.Finsupp.Basic
import Mathlib.Data.Finsupp.Order
#align_import data.finsupp.multiset from "leanprover-community/mathlib"@"59694bd07f0a39c5beccba34bd9f413a160782bf"
open Finset
variable {α β ι : Type*}
namespace Finsupp
def toMultiset : (α →₀ ℕ) →+ Multiset α where
toFun f := Finsupp.sum f fun a n => n • {a}
-- Porting note: times out if h is not specified
map_add' _f _g := sum_add_index' (h := fun a n => n • ({a} : Multiset α))
(fun _ ↦ zero_nsmul _) (fun _ ↦ add_nsmul _)
map_zero' := sum_zero_index
theorem toMultiset_zero : toMultiset (0 : α →₀ ℕ) = 0 :=
rfl
#align finsupp.to_multiset_zero Finsupp.toMultiset_zero
theorem toMultiset_add (m n : α →₀ ℕ) : toMultiset (m + n) = toMultiset m + toMultiset n :=
toMultiset.map_add m n
#align finsupp.to_multiset_add Finsupp.toMultiset_add
theorem toMultiset_apply (f : α →₀ ℕ) : toMultiset f = f.sum fun a n => n • {a} :=
rfl
#align finsupp.to_multiset_apply Finsupp.toMultiset_apply
@[simp]
theorem toMultiset_single (a : α) (n : ℕ) : toMultiset (single a n) = n • {a} := by
rw [toMultiset_apply, sum_single_index]; apply zero_nsmul
#align finsupp.to_multiset_single Finsupp.toMultiset_single
theorem toMultiset_sum {f : ι → α →₀ ℕ} (s : Finset ι) :
Finsupp.toMultiset (∑ i ∈ s, f i) = ∑ i ∈ s, Finsupp.toMultiset (f i) :=
map_sum Finsupp.toMultiset _ _
#align finsupp.to_multiset_sum Finsupp.toMultiset_sum
theorem toMultiset_sum_single (s : Finset ι) (n : ℕ) :
Finsupp.toMultiset (∑ i ∈ s, single i n) = n • s.val := by
simp_rw [toMultiset_sum, Finsupp.toMultiset_single, sum_nsmul, sum_multiset_singleton]
#align finsupp.to_multiset_sum_single Finsupp.toMultiset_sum_single
@[simp]
theorem card_toMultiset (f : α →₀ ℕ) : Multiset.card (toMultiset f) = f.sum fun _ => id := by
simp [toMultiset_apply, map_finsupp_sum, Function.id_def]
#align finsupp.card_to_multiset Finsupp.card_toMultiset
theorem toMultiset_map (f : α →₀ ℕ) (g : α → β) :
f.toMultiset.map g = toMultiset (f.mapDomain g) := by
refine f.induction ?_ ?_
· rw [toMultiset_zero, Multiset.map_zero, mapDomain_zero, toMultiset_zero]
· intro a n f _ _ ih
rw [toMultiset_add, Multiset.map_add, ih, mapDomain_add, mapDomain_single,
toMultiset_single, toMultiset_add, toMultiset_single, ← Multiset.coe_mapAddMonoidHom,
(Multiset.mapAddMonoidHom g).map_nsmul]
rfl
#align finsupp.to_multiset_map Finsupp.toMultiset_map
@[to_additive (attr := simp)]
theorem prod_toMultiset [CommMonoid α] (f : α →₀ ℕ) :
f.toMultiset.prod = f.prod fun a n => a ^ n := by
refine f.induction ?_ ?_
· rw [toMultiset_zero, Multiset.prod_zero, Finsupp.prod_zero_index]
· intro a n f _ _ ih
rw [toMultiset_add, Multiset.prod_add, ih, toMultiset_single, Multiset.prod_nsmul,
Finsupp.prod_add_index' pow_zero pow_add, Finsupp.prod_single_index, Multiset.prod_singleton]
exact pow_zero a
#align finsupp.prod_to_multiset Finsupp.prod_toMultiset
@[simp]
theorem toFinset_toMultiset [DecidableEq α] (f : α →₀ ℕ) : f.toMultiset.toFinset = f.support := by
refine f.induction ?_ ?_
· rw [toMultiset_zero, Multiset.toFinset_zero, support_zero]
· intro a n f ha hn ih
rw [toMultiset_add, Multiset.toFinset_add, ih, toMultiset_single, support_add_eq,
support_single_ne_zero _ hn, Multiset.toFinset_nsmul _ _ hn, Multiset.toFinset_singleton]
refine Disjoint.mono_left support_single_subset ?_
rwa [Finset.disjoint_singleton_left]
#align finsupp.to_finset_to_multiset Finsupp.toFinset_toMultiset
@[simp]
theorem count_toMultiset [DecidableEq α] (f : α →₀ ℕ) (a : α) : (toMultiset f).count a = f a :=
calc
(toMultiset f).count a = Finsupp.sum f (fun x n => (n • {x} : Multiset α).count a) := by
rw [toMultiset_apply]; exact map_sum (Multiset.countAddMonoidHom a) _ f.support
_ = f.sum fun x n => n * ({x} : Multiset α).count a := by simp only [Multiset.count_nsmul]
_ = f a * ({a} : Multiset α).count a :=
sum_eq_single _
(fun a' _ H => by simp only [Multiset.count_singleton, if_false, H.symm, mul_zero])
(fun _ => zero_mul _)
_ = f a := by rw [Multiset.count_singleton_self, mul_one]
#align finsupp.count_to_multiset Finsupp.count_toMultiset
| Mathlib/Data/Finsupp/Multiset.lean | 117 | 120 | theorem toMultiset_sup [DecidableEq α] (f g : α →₀ ℕ) :
toMultiset (f ⊔ g) = toMultiset f ∪ toMultiset g := by |
ext
simp_rw [Multiset.count_union, Finsupp.count_toMultiset, Finsupp.sup_apply, sup_eq_max]
| 2 | 7.389056 | 1 | 1.111111 | 9 | 1,194 |
import Mathlib.Data.Finsupp.Basic
import Mathlib.Data.Finsupp.Order
#align_import data.finsupp.multiset from "leanprover-community/mathlib"@"59694bd07f0a39c5beccba34bd9f413a160782bf"
open Finset
variable {α β ι : Type*}
namespace Finsupp
def toMultiset : (α →₀ ℕ) →+ Multiset α where
toFun f := Finsupp.sum f fun a n => n • {a}
-- Porting note: times out if h is not specified
map_add' _f _g := sum_add_index' (h := fun a n => n • ({a} : Multiset α))
(fun _ ↦ zero_nsmul _) (fun _ ↦ add_nsmul _)
map_zero' := sum_zero_index
theorem toMultiset_zero : toMultiset (0 : α →₀ ℕ) = 0 :=
rfl
#align finsupp.to_multiset_zero Finsupp.toMultiset_zero
theorem toMultiset_add (m n : α →₀ ℕ) : toMultiset (m + n) = toMultiset m + toMultiset n :=
toMultiset.map_add m n
#align finsupp.to_multiset_add Finsupp.toMultiset_add
theorem toMultiset_apply (f : α →₀ ℕ) : toMultiset f = f.sum fun a n => n • {a} :=
rfl
#align finsupp.to_multiset_apply Finsupp.toMultiset_apply
@[simp]
theorem toMultiset_single (a : α) (n : ℕ) : toMultiset (single a n) = n • {a} := by
rw [toMultiset_apply, sum_single_index]; apply zero_nsmul
#align finsupp.to_multiset_single Finsupp.toMultiset_single
theorem toMultiset_sum {f : ι → α →₀ ℕ} (s : Finset ι) :
Finsupp.toMultiset (∑ i ∈ s, f i) = ∑ i ∈ s, Finsupp.toMultiset (f i) :=
map_sum Finsupp.toMultiset _ _
#align finsupp.to_multiset_sum Finsupp.toMultiset_sum
theorem toMultiset_sum_single (s : Finset ι) (n : ℕ) :
Finsupp.toMultiset (∑ i ∈ s, single i n) = n • s.val := by
simp_rw [toMultiset_sum, Finsupp.toMultiset_single, sum_nsmul, sum_multiset_singleton]
#align finsupp.to_multiset_sum_single Finsupp.toMultiset_sum_single
@[simp]
theorem card_toMultiset (f : α →₀ ℕ) : Multiset.card (toMultiset f) = f.sum fun _ => id := by
simp [toMultiset_apply, map_finsupp_sum, Function.id_def]
#align finsupp.card_to_multiset Finsupp.card_toMultiset
theorem toMultiset_map (f : α →₀ ℕ) (g : α → β) :
f.toMultiset.map g = toMultiset (f.mapDomain g) := by
refine f.induction ?_ ?_
· rw [toMultiset_zero, Multiset.map_zero, mapDomain_zero, toMultiset_zero]
· intro a n f _ _ ih
rw [toMultiset_add, Multiset.map_add, ih, mapDomain_add, mapDomain_single,
toMultiset_single, toMultiset_add, toMultiset_single, ← Multiset.coe_mapAddMonoidHom,
(Multiset.mapAddMonoidHom g).map_nsmul]
rfl
#align finsupp.to_multiset_map Finsupp.toMultiset_map
@[to_additive (attr := simp)]
theorem prod_toMultiset [CommMonoid α] (f : α →₀ ℕ) :
f.toMultiset.prod = f.prod fun a n => a ^ n := by
refine f.induction ?_ ?_
· rw [toMultiset_zero, Multiset.prod_zero, Finsupp.prod_zero_index]
· intro a n f _ _ ih
rw [toMultiset_add, Multiset.prod_add, ih, toMultiset_single, Multiset.prod_nsmul,
Finsupp.prod_add_index' pow_zero pow_add, Finsupp.prod_single_index, Multiset.prod_singleton]
exact pow_zero a
#align finsupp.prod_to_multiset Finsupp.prod_toMultiset
@[simp]
theorem toFinset_toMultiset [DecidableEq α] (f : α →₀ ℕ) : f.toMultiset.toFinset = f.support := by
refine f.induction ?_ ?_
· rw [toMultiset_zero, Multiset.toFinset_zero, support_zero]
· intro a n f ha hn ih
rw [toMultiset_add, Multiset.toFinset_add, ih, toMultiset_single, support_add_eq,
support_single_ne_zero _ hn, Multiset.toFinset_nsmul _ _ hn, Multiset.toFinset_singleton]
refine Disjoint.mono_left support_single_subset ?_
rwa [Finset.disjoint_singleton_left]
#align finsupp.to_finset_to_multiset Finsupp.toFinset_toMultiset
@[simp]
theorem count_toMultiset [DecidableEq α] (f : α →₀ ℕ) (a : α) : (toMultiset f).count a = f a :=
calc
(toMultiset f).count a = Finsupp.sum f (fun x n => (n • {x} : Multiset α).count a) := by
rw [toMultiset_apply]; exact map_sum (Multiset.countAddMonoidHom a) _ f.support
_ = f.sum fun x n => n * ({x} : Multiset α).count a := by simp only [Multiset.count_nsmul]
_ = f a * ({a} : Multiset α).count a :=
sum_eq_single _
(fun a' _ H => by simp only [Multiset.count_singleton, if_false, H.symm, mul_zero])
(fun _ => zero_mul _)
_ = f a := by rw [Multiset.count_singleton_self, mul_one]
#align finsupp.count_to_multiset Finsupp.count_toMultiset
theorem toMultiset_sup [DecidableEq α] (f g : α →₀ ℕ) :
toMultiset (f ⊔ g) = toMultiset f ∪ toMultiset g := by
ext
simp_rw [Multiset.count_union, Finsupp.count_toMultiset, Finsupp.sup_apply, sup_eq_max]
| Mathlib/Data/Finsupp/Multiset.lean | 122 | 125 | theorem toMultiset_inf [DecidableEq α] (f g : α →₀ ℕ) :
toMultiset (f ⊓ g) = toMultiset f ∩ toMultiset g := by |
ext
simp_rw [Multiset.count_inter, Finsupp.count_toMultiset, Finsupp.inf_apply, inf_eq_min]
| 2 | 7.389056 | 1 | 1.111111 | 9 | 1,194 |
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Int.GCD
import Mathlib.RingTheory.Coprime.Basic
#align_import ring_theory.coprime.lemmas from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226"
universe u v
section IsCoprime
variable {R : Type u} {I : Type v} [CommSemiring R] {x y z : R} {s : I → R} {t : Finset I}
section
| Mathlib/RingTheory/Coprime/Lemmas.lean | 33 | 40 | theorem Int.isCoprime_iff_gcd_eq_one {m n : ℤ} : IsCoprime m n ↔ Int.gcd m n = 1 := by |
constructor
· rintro ⟨a, b, h⟩
have : 1 = m * a + n * b := by rwa [mul_comm m, mul_comm n, eq_comm]
exact Nat.dvd_one.mp (Int.gcd_dvd_iff.mpr ⟨a, b, this⟩)
· rw [← Int.ofNat_inj, IsCoprime, Int.gcd_eq_gcd_ab, mul_comm m, mul_comm n, Nat.cast_one]
intro h
exact ⟨_, _, h⟩
| 7 | 1,096.633158 | 2 | 1.111111 | 18 | 1,195 |
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Int.GCD
import Mathlib.RingTheory.Coprime.Basic
#align_import ring_theory.coprime.lemmas from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226"
universe u v
section IsCoprime
variable {R : Type u} {I : Type v} [CommSemiring R] {x y z : R} {s : I → R} {t : Finset I}
section
theorem Int.isCoprime_iff_gcd_eq_one {m n : ℤ} : IsCoprime m n ↔ Int.gcd m n = 1 := by
constructor
· rintro ⟨a, b, h⟩
have : 1 = m * a + n * b := by rwa [mul_comm m, mul_comm n, eq_comm]
exact Nat.dvd_one.mp (Int.gcd_dvd_iff.mpr ⟨a, b, this⟩)
· rw [← Int.ofNat_inj, IsCoprime, Int.gcd_eq_gcd_ab, mul_comm m, mul_comm n, Nat.cast_one]
intro h
exact ⟨_, _, h⟩
| Mathlib/RingTheory/Coprime/Lemmas.lean | 42 | 43 | theorem Nat.isCoprime_iff_coprime {m n : ℕ} : IsCoprime (m : ℤ) n ↔ Nat.Coprime m n := by |
rw [Int.isCoprime_iff_gcd_eq_one, Int.gcd_natCast_natCast]
| 1 | 2.718282 | 0 | 1.111111 | 18 | 1,195 |
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Int.GCD
import Mathlib.RingTheory.Coprime.Basic
#align_import ring_theory.coprime.lemmas from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226"
universe u v
section IsCoprime
variable {R : Type u} {I : Type v} [CommSemiring R] {x y z : R} {s : I → R} {t : Finset I}
section
theorem Int.isCoprime_iff_gcd_eq_one {m n : ℤ} : IsCoprime m n ↔ Int.gcd m n = 1 := by
constructor
· rintro ⟨a, b, h⟩
have : 1 = m * a + n * b := by rwa [mul_comm m, mul_comm n, eq_comm]
exact Nat.dvd_one.mp (Int.gcd_dvd_iff.mpr ⟨a, b, this⟩)
· rw [← Int.ofNat_inj, IsCoprime, Int.gcd_eq_gcd_ab, mul_comm m, mul_comm n, Nat.cast_one]
intro h
exact ⟨_, _, h⟩
theorem Nat.isCoprime_iff_coprime {m n : ℕ} : IsCoprime (m : ℤ) n ↔ Nat.Coprime m n := by
rw [Int.isCoprime_iff_gcd_eq_one, Int.gcd_natCast_natCast]
#align nat.is_coprime_iff_coprime Nat.isCoprime_iff_coprime
alias ⟨IsCoprime.nat_coprime, Nat.Coprime.isCoprime⟩ := Nat.isCoprime_iff_coprime
#align is_coprime.nat_coprime IsCoprime.nat_coprime
#align nat.coprime.is_coprime Nat.Coprime.isCoprime
| Mathlib/RingTheory/Coprime/Lemmas.lean | 50 | 54 | theorem Nat.Coprime.cast {R : Type*} [CommRing R] {a b : ℕ} (h : Nat.Coprime a b) :
IsCoprime (a : R) (b : R) := by |
rw [← isCoprime_iff_coprime] at h
rw [← Int.cast_natCast a, ← Int.cast_natCast b]
exact IsCoprime.intCast h
| 3 | 20.085537 | 1 | 1.111111 | 18 | 1,195 |
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Int.GCD
import Mathlib.RingTheory.Coprime.Basic
#align_import ring_theory.coprime.lemmas from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226"
universe u v
section IsCoprime
variable {R : Type u} {I : Type v} [CommSemiring R] {x y z : R} {s : I → R} {t : Finset I}
section
theorem Int.isCoprime_iff_gcd_eq_one {m n : ℤ} : IsCoprime m n ↔ Int.gcd m n = 1 := by
constructor
· rintro ⟨a, b, h⟩
have : 1 = m * a + n * b := by rwa [mul_comm m, mul_comm n, eq_comm]
exact Nat.dvd_one.mp (Int.gcd_dvd_iff.mpr ⟨a, b, this⟩)
· rw [← Int.ofNat_inj, IsCoprime, Int.gcd_eq_gcd_ab, mul_comm m, mul_comm n, Nat.cast_one]
intro h
exact ⟨_, _, h⟩
theorem Nat.isCoprime_iff_coprime {m n : ℕ} : IsCoprime (m : ℤ) n ↔ Nat.Coprime m n := by
rw [Int.isCoprime_iff_gcd_eq_one, Int.gcd_natCast_natCast]
#align nat.is_coprime_iff_coprime Nat.isCoprime_iff_coprime
alias ⟨IsCoprime.nat_coprime, Nat.Coprime.isCoprime⟩ := Nat.isCoprime_iff_coprime
#align is_coprime.nat_coprime IsCoprime.nat_coprime
#align nat.coprime.is_coprime Nat.Coprime.isCoprime
theorem Nat.Coprime.cast {R : Type*} [CommRing R] {a b : ℕ} (h : Nat.Coprime a b) :
IsCoprime (a : R) (b : R) := by
rw [← isCoprime_iff_coprime] at h
rw [← Int.cast_natCast a, ← Int.cast_natCast b]
exact IsCoprime.intCast h
theorem ne_zero_or_ne_zero_of_nat_coprime {A : Type u} [CommRing A] [Nontrivial A] {a b : ℕ}
(h : Nat.Coprime a b) : (a : A) ≠ 0 ∨ (b : A) ≠ 0 :=
IsCoprime.ne_zero_or_ne_zero (R := A) <| by
simpa only [map_natCast] using IsCoprime.map (Nat.Coprime.isCoprime h) (Int.castRingHom A)
| Mathlib/RingTheory/Coprime/Lemmas.lean | 61 | 66 | theorem IsCoprime.prod_left : (∀ i ∈ t, IsCoprime (s i) x) → IsCoprime (∏ i ∈ t, s i) x := by |
classical
refine Finset.induction_on t (fun _ ↦ isCoprime_one_left) fun b t hbt ih H ↦ ?_
rw [Finset.prod_insert hbt]
rw [Finset.forall_mem_insert] at H
exact H.1.mul_left (ih H.2)
| 5 | 148.413159 | 2 | 1.111111 | 18 | 1,195 |
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Int.GCD
import Mathlib.RingTheory.Coprime.Basic
#align_import ring_theory.coprime.lemmas from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226"
universe u v
section IsCoprime
variable {R : Type u} {I : Type v} [CommSemiring R] {x y z : R} {s : I → R} {t : Finset I}
section
theorem Int.isCoprime_iff_gcd_eq_one {m n : ℤ} : IsCoprime m n ↔ Int.gcd m n = 1 := by
constructor
· rintro ⟨a, b, h⟩
have : 1 = m * a + n * b := by rwa [mul_comm m, mul_comm n, eq_comm]
exact Nat.dvd_one.mp (Int.gcd_dvd_iff.mpr ⟨a, b, this⟩)
· rw [← Int.ofNat_inj, IsCoprime, Int.gcd_eq_gcd_ab, mul_comm m, mul_comm n, Nat.cast_one]
intro h
exact ⟨_, _, h⟩
theorem Nat.isCoprime_iff_coprime {m n : ℕ} : IsCoprime (m : ℤ) n ↔ Nat.Coprime m n := by
rw [Int.isCoprime_iff_gcd_eq_one, Int.gcd_natCast_natCast]
#align nat.is_coprime_iff_coprime Nat.isCoprime_iff_coprime
alias ⟨IsCoprime.nat_coprime, Nat.Coprime.isCoprime⟩ := Nat.isCoprime_iff_coprime
#align is_coprime.nat_coprime IsCoprime.nat_coprime
#align nat.coprime.is_coprime Nat.Coprime.isCoprime
theorem Nat.Coprime.cast {R : Type*} [CommRing R] {a b : ℕ} (h : Nat.Coprime a b) :
IsCoprime (a : R) (b : R) := by
rw [← isCoprime_iff_coprime] at h
rw [← Int.cast_natCast a, ← Int.cast_natCast b]
exact IsCoprime.intCast h
theorem ne_zero_or_ne_zero_of_nat_coprime {A : Type u} [CommRing A] [Nontrivial A] {a b : ℕ}
(h : Nat.Coprime a b) : (a : A) ≠ 0 ∨ (b : A) ≠ 0 :=
IsCoprime.ne_zero_or_ne_zero (R := A) <| by
simpa only [map_natCast] using IsCoprime.map (Nat.Coprime.isCoprime h) (Int.castRingHom A)
theorem IsCoprime.prod_left : (∀ i ∈ t, IsCoprime (s i) x) → IsCoprime (∏ i ∈ t, s i) x := by
classical
refine Finset.induction_on t (fun _ ↦ isCoprime_one_left) fun b t hbt ih H ↦ ?_
rw [Finset.prod_insert hbt]
rw [Finset.forall_mem_insert] at H
exact H.1.mul_left (ih H.2)
#align is_coprime.prod_left IsCoprime.prod_left
| Mathlib/RingTheory/Coprime/Lemmas.lean | 69 | 70 | theorem IsCoprime.prod_right : (∀ i ∈ t, IsCoprime x (s i)) → IsCoprime x (∏ i ∈ t, s i) := by |
simpa only [isCoprime_comm] using IsCoprime.prod_left (R := R)
| 1 | 2.718282 | 0 | 1.111111 | 18 | 1,195 |
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Int.GCD
import Mathlib.RingTheory.Coprime.Basic
#align_import ring_theory.coprime.lemmas from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226"
universe u v
section IsCoprime
variable {R : Type u} {I : Type v} [CommSemiring R] {x y z : R} {s : I → R} {t : Finset I}
section
theorem Int.isCoprime_iff_gcd_eq_one {m n : ℤ} : IsCoprime m n ↔ Int.gcd m n = 1 := by
constructor
· rintro ⟨a, b, h⟩
have : 1 = m * a + n * b := by rwa [mul_comm m, mul_comm n, eq_comm]
exact Nat.dvd_one.mp (Int.gcd_dvd_iff.mpr ⟨a, b, this⟩)
· rw [← Int.ofNat_inj, IsCoprime, Int.gcd_eq_gcd_ab, mul_comm m, mul_comm n, Nat.cast_one]
intro h
exact ⟨_, _, h⟩
theorem Nat.isCoprime_iff_coprime {m n : ℕ} : IsCoprime (m : ℤ) n ↔ Nat.Coprime m n := by
rw [Int.isCoprime_iff_gcd_eq_one, Int.gcd_natCast_natCast]
#align nat.is_coprime_iff_coprime Nat.isCoprime_iff_coprime
alias ⟨IsCoprime.nat_coprime, Nat.Coprime.isCoprime⟩ := Nat.isCoprime_iff_coprime
#align is_coprime.nat_coprime IsCoprime.nat_coprime
#align nat.coprime.is_coprime Nat.Coprime.isCoprime
theorem Nat.Coprime.cast {R : Type*} [CommRing R] {a b : ℕ} (h : Nat.Coprime a b) :
IsCoprime (a : R) (b : R) := by
rw [← isCoprime_iff_coprime] at h
rw [← Int.cast_natCast a, ← Int.cast_natCast b]
exact IsCoprime.intCast h
theorem ne_zero_or_ne_zero_of_nat_coprime {A : Type u} [CommRing A] [Nontrivial A] {a b : ℕ}
(h : Nat.Coprime a b) : (a : A) ≠ 0 ∨ (b : A) ≠ 0 :=
IsCoprime.ne_zero_or_ne_zero (R := A) <| by
simpa only [map_natCast] using IsCoprime.map (Nat.Coprime.isCoprime h) (Int.castRingHom A)
theorem IsCoprime.prod_left : (∀ i ∈ t, IsCoprime (s i) x) → IsCoprime (∏ i ∈ t, s i) x := by
classical
refine Finset.induction_on t (fun _ ↦ isCoprime_one_left) fun b t hbt ih H ↦ ?_
rw [Finset.prod_insert hbt]
rw [Finset.forall_mem_insert] at H
exact H.1.mul_left (ih H.2)
#align is_coprime.prod_left IsCoprime.prod_left
theorem IsCoprime.prod_right : (∀ i ∈ t, IsCoprime x (s i)) → IsCoprime x (∏ i ∈ t, s i) := by
simpa only [isCoprime_comm] using IsCoprime.prod_left (R := R)
#align is_coprime.prod_right IsCoprime.prod_right
| Mathlib/RingTheory/Coprime/Lemmas.lean | 73 | 76 | theorem IsCoprime.prod_left_iff : IsCoprime (∏ i ∈ t, s i) x ↔ ∀ i ∈ t, IsCoprime (s i) x := by |
classical
refine Finset.induction_on t (iff_of_true isCoprime_one_left fun _ ↦ by simp) fun b t hbt ih ↦ ?_
rw [Finset.prod_insert hbt, IsCoprime.mul_left_iff, ih, Finset.forall_mem_insert]
| 3 | 20.085537 | 1 | 1.111111 | 18 | 1,195 |
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Int.GCD
import Mathlib.RingTheory.Coprime.Basic
#align_import ring_theory.coprime.lemmas from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226"
universe u v
section IsCoprime
variable {R : Type u} {I : Type v} [CommSemiring R] {x y z : R} {s : I → R} {t : Finset I}
section
theorem Int.isCoprime_iff_gcd_eq_one {m n : ℤ} : IsCoprime m n ↔ Int.gcd m n = 1 := by
constructor
· rintro ⟨a, b, h⟩
have : 1 = m * a + n * b := by rwa [mul_comm m, mul_comm n, eq_comm]
exact Nat.dvd_one.mp (Int.gcd_dvd_iff.mpr ⟨a, b, this⟩)
· rw [← Int.ofNat_inj, IsCoprime, Int.gcd_eq_gcd_ab, mul_comm m, mul_comm n, Nat.cast_one]
intro h
exact ⟨_, _, h⟩
theorem Nat.isCoprime_iff_coprime {m n : ℕ} : IsCoprime (m : ℤ) n ↔ Nat.Coprime m n := by
rw [Int.isCoprime_iff_gcd_eq_one, Int.gcd_natCast_natCast]
#align nat.is_coprime_iff_coprime Nat.isCoprime_iff_coprime
alias ⟨IsCoprime.nat_coprime, Nat.Coprime.isCoprime⟩ := Nat.isCoprime_iff_coprime
#align is_coprime.nat_coprime IsCoprime.nat_coprime
#align nat.coprime.is_coprime Nat.Coprime.isCoprime
theorem Nat.Coprime.cast {R : Type*} [CommRing R] {a b : ℕ} (h : Nat.Coprime a b) :
IsCoprime (a : R) (b : R) := by
rw [← isCoprime_iff_coprime] at h
rw [← Int.cast_natCast a, ← Int.cast_natCast b]
exact IsCoprime.intCast h
theorem ne_zero_or_ne_zero_of_nat_coprime {A : Type u} [CommRing A] [Nontrivial A] {a b : ℕ}
(h : Nat.Coprime a b) : (a : A) ≠ 0 ∨ (b : A) ≠ 0 :=
IsCoprime.ne_zero_or_ne_zero (R := A) <| by
simpa only [map_natCast] using IsCoprime.map (Nat.Coprime.isCoprime h) (Int.castRingHom A)
theorem IsCoprime.prod_left : (∀ i ∈ t, IsCoprime (s i) x) → IsCoprime (∏ i ∈ t, s i) x := by
classical
refine Finset.induction_on t (fun _ ↦ isCoprime_one_left) fun b t hbt ih H ↦ ?_
rw [Finset.prod_insert hbt]
rw [Finset.forall_mem_insert] at H
exact H.1.mul_left (ih H.2)
#align is_coprime.prod_left IsCoprime.prod_left
theorem IsCoprime.prod_right : (∀ i ∈ t, IsCoprime x (s i)) → IsCoprime x (∏ i ∈ t, s i) := by
simpa only [isCoprime_comm] using IsCoprime.prod_left (R := R)
#align is_coprime.prod_right IsCoprime.prod_right
theorem IsCoprime.prod_left_iff : IsCoprime (∏ i ∈ t, s i) x ↔ ∀ i ∈ t, IsCoprime (s i) x := by
classical
refine Finset.induction_on t (iff_of_true isCoprime_one_left fun _ ↦ by simp) fun b t hbt ih ↦ ?_
rw [Finset.prod_insert hbt, IsCoprime.mul_left_iff, ih, Finset.forall_mem_insert]
#align is_coprime.prod_left_iff IsCoprime.prod_left_iff
| Mathlib/RingTheory/Coprime/Lemmas.lean | 79 | 80 | theorem IsCoprime.prod_right_iff : IsCoprime x (∏ i ∈ t, s i) ↔ ∀ i ∈ t, IsCoprime x (s i) := by |
simpa only [isCoprime_comm] using IsCoprime.prod_left_iff (R := R)
| 1 | 2.718282 | 0 | 1.111111 | 18 | 1,195 |
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Int.GCD
import Mathlib.RingTheory.Coprime.Basic
#align_import ring_theory.coprime.lemmas from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226"
universe u v
section IsCoprime
variable {R : Type u} {I : Type v} [CommSemiring R] {x y z : R} {s : I → R} {t : Finset I}
section
theorem Int.isCoprime_iff_gcd_eq_one {m n : ℤ} : IsCoprime m n ↔ Int.gcd m n = 1 := by
constructor
· rintro ⟨a, b, h⟩
have : 1 = m * a + n * b := by rwa [mul_comm m, mul_comm n, eq_comm]
exact Nat.dvd_one.mp (Int.gcd_dvd_iff.mpr ⟨a, b, this⟩)
· rw [← Int.ofNat_inj, IsCoprime, Int.gcd_eq_gcd_ab, mul_comm m, mul_comm n, Nat.cast_one]
intro h
exact ⟨_, _, h⟩
theorem Nat.isCoprime_iff_coprime {m n : ℕ} : IsCoprime (m : ℤ) n ↔ Nat.Coprime m n := by
rw [Int.isCoprime_iff_gcd_eq_one, Int.gcd_natCast_natCast]
#align nat.is_coprime_iff_coprime Nat.isCoprime_iff_coprime
alias ⟨IsCoprime.nat_coprime, Nat.Coprime.isCoprime⟩ := Nat.isCoprime_iff_coprime
#align is_coprime.nat_coprime IsCoprime.nat_coprime
#align nat.coprime.is_coprime Nat.Coprime.isCoprime
theorem Nat.Coprime.cast {R : Type*} [CommRing R] {a b : ℕ} (h : Nat.Coprime a b) :
IsCoprime (a : R) (b : R) := by
rw [← isCoprime_iff_coprime] at h
rw [← Int.cast_natCast a, ← Int.cast_natCast b]
exact IsCoprime.intCast h
theorem ne_zero_or_ne_zero_of_nat_coprime {A : Type u} [CommRing A] [Nontrivial A] {a b : ℕ}
(h : Nat.Coprime a b) : (a : A) ≠ 0 ∨ (b : A) ≠ 0 :=
IsCoprime.ne_zero_or_ne_zero (R := A) <| by
simpa only [map_natCast] using IsCoprime.map (Nat.Coprime.isCoprime h) (Int.castRingHom A)
theorem IsCoprime.prod_left : (∀ i ∈ t, IsCoprime (s i) x) → IsCoprime (∏ i ∈ t, s i) x := by
classical
refine Finset.induction_on t (fun _ ↦ isCoprime_one_left) fun b t hbt ih H ↦ ?_
rw [Finset.prod_insert hbt]
rw [Finset.forall_mem_insert] at H
exact H.1.mul_left (ih H.2)
#align is_coprime.prod_left IsCoprime.prod_left
theorem IsCoprime.prod_right : (∀ i ∈ t, IsCoprime x (s i)) → IsCoprime x (∏ i ∈ t, s i) := by
simpa only [isCoprime_comm] using IsCoprime.prod_left (R := R)
#align is_coprime.prod_right IsCoprime.prod_right
theorem IsCoprime.prod_left_iff : IsCoprime (∏ i ∈ t, s i) x ↔ ∀ i ∈ t, IsCoprime (s i) x := by
classical
refine Finset.induction_on t (iff_of_true isCoprime_one_left fun _ ↦ by simp) fun b t hbt ih ↦ ?_
rw [Finset.prod_insert hbt, IsCoprime.mul_left_iff, ih, Finset.forall_mem_insert]
#align is_coprime.prod_left_iff IsCoprime.prod_left_iff
theorem IsCoprime.prod_right_iff : IsCoprime x (∏ i ∈ t, s i) ↔ ∀ i ∈ t, IsCoprime x (s i) := by
simpa only [isCoprime_comm] using IsCoprime.prod_left_iff (R := R)
#align is_coprime.prod_right_iff IsCoprime.prod_right_iff
theorem IsCoprime.of_prod_left (H1 : IsCoprime (∏ i ∈ t, s i) x) (i : I) (hit : i ∈ t) :
IsCoprime (s i) x :=
IsCoprime.prod_left_iff.1 H1 i hit
#align is_coprime.of_prod_left IsCoprime.of_prod_left
theorem IsCoprime.of_prod_right (H1 : IsCoprime x (∏ i ∈ t, s i)) (i : I) (hit : i ∈ t) :
IsCoprime x (s i) :=
IsCoprime.prod_right_iff.1 H1 i hit
#align is_coprime.of_prod_right IsCoprime.of_prod_right
-- Porting note: removed names of things due to linter, but they seem helpful
| Mathlib/RingTheory/Coprime/Lemmas.lean | 94 | 108 | theorem Finset.prod_dvd_of_coprime :
(t : Set I).Pairwise (IsCoprime on s) → (∀ i ∈ t, s i ∣ z) → (∏ x ∈ t, s x) ∣ z := by |
classical
exact Finset.induction_on t (fun _ _ ↦ one_dvd z)
(by
intro a r har ih Hs Hs1
rw [Finset.prod_insert har]
have aux1 : a ∈ (↑(insert a r) : Set I) := Finset.mem_insert_self a r
refine
(IsCoprime.prod_right fun i hir ↦
Hs aux1 (Finset.mem_insert_of_mem hir) <| by
rintro rfl
exact har hir).mul_dvd
(Hs1 a aux1) (ih (Hs.mono ?_) fun i hi ↦ Hs1 i <| Finset.mem_insert_of_mem hi)
simp only [Finset.coe_insert, Set.subset_insert])
| 13 | 442,413.392009 | 2 | 1.111111 | 18 | 1,195 |
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Int.GCD
import Mathlib.RingTheory.Coprime.Basic
#align_import ring_theory.coprime.lemmas from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226"
universe u v
section IsCoprime
variable {R : Type u} {I : Type v} [CommSemiring R] {x y z : R} {s : I → R} {t : Finset I}
section
theorem Int.isCoprime_iff_gcd_eq_one {m n : ℤ} : IsCoprime m n ↔ Int.gcd m n = 1 := by
constructor
· rintro ⟨a, b, h⟩
have : 1 = m * a + n * b := by rwa [mul_comm m, mul_comm n, eq_comm]
exact Nat.dvd_one.mp (Int.gcd_dvd_iff.mpr ⟨a, b, this⟩)
· rw [← Int.ofNat_inj, IsCoprime, Int.gcd_eq_gcd_ab, mul_comm m, mul_comm n, Nat.cast_one]
intro h
exact ⟨_, _, h⟩
theorem Nat.isCoprime_iff_coprime {m n : ℕ} : IsCoprime (m : ℤ) n ↔ Nat.Coprime m n := by
rw [Int.isCoprime_iff_gcd_eq_one, Int.gcd_natCast_natCast]
#align nat.is_coprime_iff_coprime Nat.isCoprime_iff_coprime
alias ⟨IsCoprime.nat_coprime, Nat.Coprime.isCoprime⟩ := Nat.isCoprime_iff_coprime
#align is_coprime.nat_coprime IsCoprime.nat_coprime
#align nat.coprime.is_coprime Nat.Coprime.isCoprime
theorem Nat.Coprime.cast {R : Type*} [CommRing R] {a b : ℕ} (h : Nat.Coprime a b) :
IsCoprime (a : R) (b : R) := by
rw [← isCoprime_iff_coprime] at h
rw [← Int.cast_natCast a, ← Int.cast_natCast b]
exact IsCoprime.intCast h
theorem ne_zero_or_ne_zero_of_nat_coprime {A : Type u} [CommRing A] [Nontrivial A] {a b : ℕ}
(h : Nat.Coprime a b) : (a : A) ≠ 0 ∨ (b : A) ≠ 0 :=
IsCoprime.ne_zero_or_ne_zero (R := A) <| by
simpa only [map_natCast] using IsCoprime.map (Nat.Coprime.isCoprime h) (Int.castRingHom A)
theorem IsCoprime.prod_left : (∀ i ∈ t, IsCoprime (s i) x) → IsCoprime (∏ i ∈ t, s i) x := by
classical
refine Finset.induction_on t (fun _ ↦ isCoprime_one_left) fun b t hbt ih H ↦ ?_
rw [Finset.prod_insert hbt]
rw [Finset.forall_mem_insert] at H
exact H.1.mul_left (ih H.2)
#align is_coprime.prod_left IsCoprime.prod_left
theorem IsCoprime.prod_right : (∀ i ∈ t, IsCoprime x (s i)) → IsCoprime x (∏ i ∈ t, s i) := by
simpa only [isCoprime_comm] using IsCoprime.prod_left (R := R)
#align is_coprime.prod_right IsCoprime.prod_right
theorem IsCoprime.prod_left_iff : IsCoprime (∏ i ∈ t, s i) x ↔ ∀ i ∈ t, IsCoprime (s i) x := by
classical
refine Finset.induction_on t (iff_of_true isCoprime_one_left fun _ ↦ by simp) fun b t hbt ih ↦ ?_
rw [Finset.prod_insert hbt, IsCoprime.mul_left_iff, ih, Finset.forall_mem_insert]
#align is_coprime.prod_left_iff IsCoprime.prod_left_iff
theorem IsCoprime.prod_right_iff : IsCoprime x (∏ i ∈ t, s i) ↔ ∀ i ∈ t, IsCoprime x (s i) := by
simpa only [isCoprime_comm] using IsCoprime.prod_left_iff (R := R)
#align is_coprime.prod_right_iff IsCoprime.prod_right_iff
theorem IsCoprime.of_prod_left (H1 : IsCoprime (∏ i ∈ t, s i) x) (i : I) (hit : i ∈ t) :
IsCoprime (s i) x :=
IsCoprime.prod_left_iff.1 H1 i hit
#align is_coprime.of_prod_left IsCoprime.of_prod_left
theorem IsCoprime.of_prod_right (H1 : IsCoprime x (∏ i ∈ t, s i)) (i : I) (hit : i ∈ t) :
IsCoprime x (s i) :=
IsCoprime.prod_right_iff.1 H1 i hit
#align is_coprime.of_prod_right IsCoprime.of_prod_right
-- Porting note: removed names of things due to linter, but they seem helpful
theorem Finset.prod_dvd_of_coprime :
(t : Set I).Pairwise (IsCoprime on s) → (∀ i ∈ t, s i ∣ z) → (∏ x ∈ t, s x) ∣ z := by
classical
exact Finset.induction_on t (fun _ _ ↦ one_dvd z)
(by
intro a r har ih Hs Hs1
rw [Finset.prod_insert har]
have aux1 : a ∈ (↑(insert a r) : Set I) := Finset.mem_insert_self a r
refine
(IsCoprime.prod_right fun i hir ↦
Hs aux1 (Finset.mem_insert_of_mem hir) <| by
rintro rfl
exact har hir).mul_dvd
(Hs1 a aux1) (ih (Hs.mono ?_) fun i hi ↦ Hs1 i <| Finset.mem_insert_of_mem hi)
simp only [Finset.coe_insert, Set.subset_insert])
#align finset.prod_dvd_of_coprime Finset.prod_dvd_of_coprime
theorem Fintype.prod_dvd_of_coprime [Fintype I] (Hs : Pairwise (IsCoprime on s))
(Hs1 : ∀ i, s i ∣ z) : (∏ x, s x) ∣ z :=
Finset.prod_dvd_of_coprime (Hs.set_pairwise _) fun i _ ↦ Hs1 i
#align fintype.prod_dvd_of_coprime Fintype.prod_dvd_of_coprime
end
open Finset
| Mathlib/RingTheory/Coprime/Lemmas.lean | 120 | 175 | theorem exists_sum_eq_one_iff_pairwise_coprime [DecidableEq I] (h : t.Nonempty) :
(∃ μ : I → R, (∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j) = 1) ↔
Pairwise (IsCoprime on fun i : t ↦ s i) := by |
induction h using Finset.Nonempty.cons_induction with
| singleton =>
simp [exists_apply_eq, Pairwise, Function.onFun]
| cons a t hat h ih =>
rw [pairwise_cons']
have mem : ∀ x ∈ t, a ∈ insert a t \ {x} := fun x hx ↦ by
rw [mem_sdiff, mem_singleton]
exact ⟨mem_insert_self _ _, fun ha ↦ hat (ha ▸ hx)⟩
constructor
· rintro ⟨μ, hμ⟩
rw [sum_cons, cons_eq_insert, sdiff_singleton_eq_erase, erase_insert hat] at hμ
refine ⟨ih.mp ⟨Pi.single h.choose (μ a * s h.choose) + μ * fun _ ↦ s a, ?_⟩, fun b hb ↦ ?_⟩
· rw [prod_eq_mul_prod_diff_singleton h.choose_spec, ← mul_assoc, ←
@if_pos _ _ h.choose_spec R (_ * _) 0, ← sum_pi_single', ← sum_add_distrib] at hμ
rw [← hμ, sum_congr rfl]
intro x hx
dsimp -- Porting note: terms were showing as sort of `HAdd.hadd` instead of `+`
-- this whole proof pretty much breaks and has to be rewritten from scratch
rw [add_mul]
congr 1
· by_cases hx : x = h.choose
· rw [hx, Pi.single_eq_same, Pi.single_eq_same]
· rw [Pi.single_eq_of_ne hx, Pi.single_eq_of_ne hx, zero_mul]
· rw [mul_assoc]
congr
rw [prod_eq_prod_diff_singleton_mul (mem x hx) _, mul_comm]
congr 2
rw [sdiff_sdiff_comm, sdiff_singleton_eq_erase a, erase_insert hat]
· have : IsCoprime (s b) (s a) :=
⟨μ a * ∏ i ∈ t \ {b}, s i, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j, ?_⟩
· exact ⟨this.symm, this⟩
rw [mul_assoc, ← prod_eq_prod_diff_singleton_mul hb, sum_mul, ← hμ, sum_congr rfl]
intro x hx
rw [mul_assoc]
congr
rw [prod_eq_prod_diff_singleton_mul (mem x hx) _]
congr 2
rw [sdiff_sdiff_comm, sdiff_singleton_eq_erase a, erase_insert hat]
· rintro ⟨hs, Hb⟩
obtain ⟨μ, hμ⟩ := ih.mpr hs
obtain ⟨u, v, huv⟩ := IsCoprime.prod_left fun b hb ↦ (Hb b hb).right
use fun i ↦ if i = a then u else v * μ i
have hμ' : (∑ i ∈ t, v * ((μ i * ∏ j ∈ t \ {i}, s j) * s a)) = v * s a := by
rw [← mul_sum, ← sum_mul, hμ, one_mul]
rw [sum_cons, cons_eq_insert, sdiff_singleton_eq_erase, erase_insert hat, if_pos rfl,
← huv, ← hμ', sum_congr rfl]
intro x hx
rw [mul_assoc, if_neg fun ha : x = a ↦ hat (ha.casesOn hx)]
rw [mul_assoc]
congr
rw [prod_eq_prod_diff_singleton_mul (mem x hx) _]
congr 2
rw [sdiff_sdiff_comm, sdiff_singleton_eq_erase a, erase_insert hat]
| 53 | 104,137,594,330,290,870,000,000 | 2 | 1.111111 | 18 | 1,195 |
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Int.GCD
import Mathlib.RingTheory.Coprime.Basic
#align_import ring_theory.coprime.lemmas from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226"
universe u v
section RelPrime
variable {α I} [CommMonoid α] [DecompositionMonoid α] {x y z : α} {s : I → α} {t : Finset I}
| Mathlib/RingTheory/Coprime/Lemmas.lean | 235 | 240 | theorem IsRelPrime.prod_left : (∀ i ∈ t, IsRelPrime (s i) x) → IsRelPrime (∏ i ∈ t, s i) x := by |
classical
refine Finset.induction_on t (fun _ ↦ isRelPrime_one_left) fun b t hbt ih H ↦ ?_
rw [Finset.prod_insert hbt]
rw [Finset.forall_mem_insert] at H
exact H.1.mul_left (ih H.2)
| 5 | 148.413159 | 2 | 1.111111 | 18 | 1,195 |
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Int.GCD
import Mathlib.RingTheory.Coprime.Basic
#align_import ring_theory.coprime.lemmas from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226"
universe u v
section RelPrime
variable {α I} [CommMonoid α] [DecompositionMonoid α] {x y z : α} {s : I → α} {t : Finset I}
theorem IsRelPrime.prod_left : (∀ i ∈ t, IsRelPrime (s i) x) → IsRelPrime (∏ i ∈ t, s i) x := by
classical
refine Finset.induction_on t (fun _ ↦ isRelPrime_one_left) fun b t hbt ih H ↦ ?_
rw [Finset.prod_insert hbt]
rw [Finset.forall_mem_insert] at H
exact H.1.mul_left (ih H.2)
| Mathlib/RingTheory/Coprime/Lemmas.lean | 242 | 243 | theorem IsRelPrime.prod_right : (∀ i ∈ t, IsRelPrime x (s i)) → IsRelPrime x (∏ i ∈ t, s i) := by |
simpa only [isRelPrime_comm] using IsRelPrime.prod_left (α := α)
| 1 | 2.718282 | 0 | 1.111111 | 18 | 1,195 |
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Int.GCD
import Mathlib.RingTheory.Coprime.Basic
#align_import ring_theory.coprime.lemmas from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226"
universe u v
section RelPrime
variable {α I} [CommMonoid α] [DecompositionMonoid α] {x y z : α} {s : I → α} {t : Finset I}
theorem IsRelPrime.prod_left : (∀ i ∈ t, IsRelPrime (s i) x) → IsRelPrime (∏ i ∈ t, s i) x := by
classical
refine Finset.induction_on t (fun _ ↦ isRelPrime_one_left) fun b t hbt ih H ↦ ?_
rw [Finset.prod_insert hbt]
rw [Finset.forall_mem_insert] at H
exact H.1.mul_left (ih H.2)
theorem IsRelPrime.prod_right : (∀ i ∈ t, IsRelPrime x (s i)) → IsRelPrime x (∏ i ∈ t, s i) := by
simpa only [isRelPrime_comm] using IsRelPrime.prod_left (α := α)
| Mathlib/RingTheory/Coprime/Lemmas.lean | 245 | 248 | theorem IsRelPrime.prod_left_iff : IsRelPrime (∏ i ∈ t, s i) x ↔ ∀ i ∈ t, IsRelPrime (s i) x := by |
classical
refine Finset.induction_on t (iff_of_true isRelPrime_one_left fun _ ↦ by simp) fun b t hbt ih ↦ ?_
rw [Finset.prod_insert hbt, IsRelPrime.mul_left_iff, ih, Finset.forall_mem_insert]
| 3 | 20.085537 | 1 | 1.111111 | 18 | 1,195 |
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Int.GCD
import Mathlib.RingTheory.Coprime.Basic
#align_import ring_theory.coprime.lemmas from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226"
universe u v
section RelPrime
variable {α I} [CommMonoid α] [DecompositionMonoid α] {x y z : α} {s : I → α} {t : Finset I}
theorem IsRelPrime.prod_left : (∀ i ∈ t, IsRelPrime (s i) x) → IsRelPrime (∏ i ∈ t, s i) x := by
classical
refine Finset.induction_on t (fun _ ↦ isRelPrime_one_left) fun b t hbt ih H ↦ ?_
rw [Finset.prod_insert hbt]
rw [Finset.forall_mem_insert] at H
exact H.1.mul_left (ih H.2)
theorem IsRelPrime.prod_right : (∀ i ∈ t, IsRelPrime x (s i)) → IsRelPrime x (∏ i ∈ t, s i) := by
simpa only [isRelPrime_comm] using IsRelPrime.prod_left (α := α)
theorem IsRelPrime.prod_left_iff : IsRelPrime (∏ i ∈ t, s i) x ↔ ∀ i ∈ t, IsRelPrime (s i) x := by
classical
refine Finset.induction_on t (iff_of_true isRelPrime_one_left fun _ ↦ by simp) fun b t hbt ih ↦ ?_
rw [Finset.prod_insert hbt, IsRelPrime.mul_left_iff, ih, Finset.forall_mem_insert]
| Mathlib/RingTheory/Coprime/Lemmas.lean | 250 | 251 | theorem IsRelPrime.prod_right_iff : IsRelPrime x (∏ i ∈ t, s i) ↔ ∀ i ∈ t, IsRelPrime x (s i) := by |
simpa only [isRelPrime_comm] using IsRelPrime.prod_left_iff (α := α)
| 1 | 2.718282 | 0 | 1.111111 | 18 | 1,195 |
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Int.GCD
import Mathlib.RingTheory.Coprime.Basic
#align_import ring_theory.coprime.lemmas from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226"
universe u v
section RelPrime
variable {α I} [CommMonoid α] [DecompositionMonoid α] {x y z : α} {s : I → α} {t : Finset I}
theorem IsRelPrime.prod_left : (∀ i ∈ t, IsRelPrime (s i) x) → IsRelPrime (∏ i ∈ t, s i) x := by
classical
refine Finset.induction_on t (fun _ ↦ isRelPrime_one_left) fun b t hbt ih H ↦ ?_
rw [Finset.prod_insert hbt]
rw [Finset.forall_mem_insert] at H
exact H.1.mul_left (ih H.2)
theorem IsRelPrime.prod_right : (∀ i ∈ t, IsRelPrime x (s i)) → IsRelPrime x (∏ i ∈ t, s i) := by
simpa only [isRelPrime_comm] using IsRelPrime.prod_left (α := α)
theorem IsRelPrime.prod_left_iff : IsRelPrime (∏ i ∈ t, s i) x ↔ ∀ i ∈ t, IsRelPrime (s i) x := by
classical
refine Finset.induction_on t (iff_of_true isRelPrime_one_left fun _ ↦ by simp) fun b t hbt ih ↦ ?_
rw [Finset.prod_insert hbt, IsRelPrime.mul_left_iff, ih, Finset.forall_mem_insert]
theorem IsRelPrime.prod_right_iff : IsRelPrime x (∏ i ∈ t, s i) ↔ ∀ i ∈ t, IsRelPrime x (s i) := by
simpa only [isRelPrime_comm] using IsRelPrime.prod_left_iff (α := α)
theorem IsRelPrime.of_prod_left (H1 : IsRelPrime (∏ i ∈ t, s i) x) (i : I) (hit : i ∈ t) :
IsRelPrime (s i) x :=
IsRelPrime.prod_left_iff.1 H1 i hit
theorem IsRelPrime.of_prod_right (H1 : IsRelPrime x (∏ i ∈ t, s i)) (i : I) (hit : i ∈ t) :
IsRelPrime x (s i) :=
IsRelPrime.prod_right_iff.1 H1 i hit
| Mathlib/RingTheory/Coprime/Lemmas.lean | 261 | 275 | theorem Finset.prod_dvd_of_isRelPrime :
(t : Set I).Pairwise (IsRelPrime on s) → (∀ i ∈ t, s i ∣ z) → (∏ x ∈ t, s x) ∣ z := by |
classical
exact Finset.induction_on t (fun _ _ ↦ one_dvd z)
(by
intro a r har ih Hs Hs1
rw [Finset.prod_insert har]
have aux1 : a ∈ (↑(insert a r) : Set I) := Finset.mem_insert_self a r
refine
(IsRelPrime.prod_right fun i hir ↦
Hs aux1 (Finset.mem_insert_of_mem hir) <| by
rintro rfl
exact har hir).mul_dvd
(Hs1 a aux1) (ih (Hs.mono ?_) fun i hi ↦ Hs1 i <| Finset.mem_insert_of_mem hi)
simp only [Finset.coe_insert, Set.subset_insert])
| 13 | 442,413.392009 | 2 | 1.111111 | 18 | 1,195 |
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Int.GCD
import Mathlib.RingTheory.Coprime.Basic
#align_import ring_theory.coprime.lemmas from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226"
universe u v
section RelPrime
variable {α I} [CommMonoid α] [DecompositionMonoid α] {x y z : α} {s : I → α} {t : Finset I}
theorem IsRelPrime.prod_left : (∀ i ∈ t, IsRelPrime (s i) x) → IsRelPrime (∏ i ∈ t, s i) x := by
classical
refine Finset.induction_on t (fun _ ↦ isRelPrime_one_left) fun b t hbt ih H ↦ ?_
rw [Finset.prod_insert hbt]
rw [Finset.forall_mem_insert] at H
exact H.1.mul_left (ih H.2)
theorem IsRelPrime.prod_right : (∀ i ∈ t, IsRelPrime x (s i)) → IsRelPrime x (∏ i ∈ t, s i) := by
simpa only [isRelPrime_comm] using IsRelPrime.prod_left (α := α)
theorem IsRelPrime.prod_left_iff : IsRelPrime (∏ i ∈ t, s i) x ↔ ∀ i ∈ t, IsRelPrime (s i) x := by
classical
refine Finset.induction_on t (iff_of_true isRelPrime_one_left fun _ ↦ by simp) fun b t hbt ih ↦ ?_
rw [Finset.prod_insert hbt, IsRelPrime.mul_left_iff, ih, Finset.forall_mem_insert]
theorem IsRelPrime.prod_right_iff : IsRelPrime x (∏ i ∈ t, s i) ↔ ∀ i ∈ t, IsRelPrime x (s i) := by
simpa only [isRelPrime_comm] using IsRelPrime.prod_left_iff (α := α)
theorem IsRelPrime.of_prod_left (H1 : IsRelPrime (∏ i ∈ t, s i) x) (i : I) (hit : i ∈ t) :
IsRelPrime (s i) x :=
IsRelPrime.prod_left_iff.1 H1 i hit
theorem IsRelPrime.of_prod_right (H1 : IsRelPrime x (∏ i ∈ t, s i)) (i : I) (hit : i ∈ t) :
IsRelPrime x (s i) :=
IsRelPrime.prod_right_iff.1 H1 i hit
theorem Finset.prod_dvd_of_isRelPrime :
(t : Set I).Pairwise (IsRelPrime on s) → (∀ i ∈ t, s i ∣ z) → (∏ x ∈ t, s x) ∣ z := by
classical
exact Finset.induction_on t (fun _ _ ↦ one_dvd z)
(by
intro a r har ih Hs Hs1
rw [Finset.prod_insert har]
have aux1 : a ∈ (↑(insert a r) : Set I) := Finset.mem_insert_self a r
refine
(IsRelPrime.prod_right fun i hir ↦
Hs aux1 (Finset.mem_insert_of_mem hir) <| by
rintro rfl
exact har hir).mul_dvd
(Hs1 a aux1) (ih (Hs.mono ?_) fun i hi ↦ Hs1 i <| Finset.mem_insert_of_mem hi)
simp only [Finset.coe_insert, Set.subset_insert])
theorem Fintype.prod_dvd_of_isRelPrime [Fintype I] (Hs : Pairwise (IsRelPrime on s))
(Hs1 : ∀ i, s i ∣ z) : (∏ x, s x) ∣ z :=
Finset.prod_dvd_of_isRelPrime (Hs.set_pairwise _) fun i _ ↦ Hs1 i
| Mathlib/RingTheory/Coprime/Lemmas.lean | 281 | 289 | theorem pairwise_isRelPrime_iff_isRelPrime_prod [DecidableEq I] :
Pairwise (IsRelPrime on fun i : t ↦ s i) ↔ ∀ i ∈ t, IsRelPrime (s i) (∏ j ∈ t \ {i}, s j) := by |
refine ⟨fun hp i hi ↦ IsRelPrime.prod_right_iff.mpr fun j hj ↦ ?_, fun hp ↦ ?_⟩
· rw [Finset.mem_sdiff, Finset.mem_singleton] at hj
obtain ⟨hj, ji⟩ := hj
exact @hp ⟨i, hi⟩ ⟨j, hj⟩ fun h ↦ ji (congrArg Subtype.val h).symm
· rintro ⟨i, hi⟩ ⟨j, hj⟩ h
apply IsRelPrime.prod_right_iff.mp (hp i hi)
exact Finset.mem_sdiff.mpr ⟨hj, fun f ↦ h <| Subtype.ext (Finset.mem_singleton.mp f).symm⟩
| 7 | 1,096.633158 | 2 | 1.111111 | 18 | 1,195 |
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Int.GCD
import Mathlib.RingTheory.Coprime.Basic
#align_import ring_theory.coprime.lemmas from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226"
universe u v
section RelPrime
variable {α I} [CommMonoid α] [DecompositionMonoid α] {x y z : α} {s : I → α} {t : Finset I}
theorem IsRelPrime.prod_left : (∀ i ∈ t, IsRelPrime (s i) x) → IsRelPrime (∏ i ∈ t, s i) x := by
classical
refine Finset.induction_on t (fun _ ↦ isRelPrime_one_left) fun b t hbt ih H ↦ ?_
rw [Finset.prod_insert hbt]
rw [Finset.forall_mem_insert] at H
exact H.1.mul_left (ih H.2)
theorem IsRelPrime.prod_right : (∀ i ∈ t, IsRelPrime x (s i)) → IsRelPrime x (∏ i ∈ t, s i) := by
simpa only [isRelPrime_comm] using IsRelPrime.prod_left (α := α)
theorem IsRelPrime.prod_left_iff : IsRelPrime (∏ i ∈ t, s i) x ↔ ∀ i ∈ t, IsRelPrime (s i) x := by
classical
refine Finset.induction_on t (iff_of_true isRelPrime_one_left fun _ ↦ by simp) fun b t hbt ih ↦ ?_
rw [Finset.prod_insert hbt, IsRelPrime.mul_left_iff, ih, Finset.forall_mem_insert]
theorem IsRelPrime.prod_right_iff : IsRelPrime x (∏ i ∈ t, s i) ↔ ∀ i ∈ t, IsRelPrime x (s i) := by
simpa only [isRelPrime_comm] using IsRelPrime.prod_left_iff (α := α)
theorem IsRelPrime.of_prod_left (H1 : IsRelPrime (∏ i ∈ t, s i) x) (i : I) (hit : i ∈ t) :
IsRelPrime (s i) x :=
IsRelPrime.prod_left_iff.1 H1 i hit
theorem IsRelPrime.of_prod_right (H1 : IsRelPrime x (∏ i ∈ t, s i)) (i : I) (hit : i ∈ t) :
IsRelPrime x (s i) :=
IsRelPrime.prod_right_iff.1 H1 i hit
theorem Finset.prod_dvd_of_isRelPrime :
(t : Set I).Pairwise (IsRelPrime on s) → (∀ i ∈ t, s i ∣ z) → (∏ x ∈ t, s x) ∣ z := by
classical
exact Finset.induction_on t (fun _ _ ↦ one_dvd z)
(by
intro a r har ih Hs Hs1
rw [Finset.prod_insert har]
have aux1 : a ∈ (↑(insert a r) : Set I) := Finset.mem_insert_self a r
refine
(IsRelPrime.prod_right fun i hir ↦
Hs aux1 (Finset.mem_insert_of_mem hir) <| by
rintro rfl
exact har hir).mul_dvd
(Hs1 a aux1) (ih (Hs.mono ?_) fun i hi ↦ Hs1 i <| Finset.mem_insert_of_mem hi)
simp only [Finset.coe_insert, Set.subset_insert])
theorem Fintype.prod_dvd_of_isRelPrime [Fintype I] (Hs : Pairwise (IsRelPrime on s))
(Hs1 : ∀ i, s i ∣ z) : (∏ x, s x) ∣ z :=
Finset.prod_dvd_of_isRelPrime (Hs.set_pairwise _) fun i _ ↦ Hs1 i
theorem pairwise_isRelPrime_iff_isRelPrime_prod [DecidableEq I] :
Pairwise (IsRelPrime on fun i : t ↦ s i) ↔ ∀ i ∈ t, IsRelPrime (s i) (∏ j ∈ t \ {i}, s j) := by
refine ⟨fun hp i hi ↦ IsRelPrime.prod_right_iff.mpr fun j hj ↦ ?_, fun hp ↦ ?_⟩
· rw [Finset.mem_sdiff, Finset.mem_singleton] at hj
obtain ⟨hj, ji⟩ := hj
exact @hp ⟨i, hi⟩ ⟨j, hj⟩ fun h ↦ ji (congrArg Subtype.val h).symm
· rintro ⟨i, hi⟩ ⟨j, hj⟩ h
apply IsRelPrime.prod_right_iff.mp (hp i hi)
exact Finset.mem_sdiff.mpr ⟨hj, fun f ↦ h <| Subtype.ext (Finset.mem_singleton.mp f).symm⟩
namespace IsRelPrime
variable {m n : ℕ}
| Mathlib/RingTheory/Coprime/Lemmas.lean | 295 | 297 | theorem pow_left (H : IsRelPrime x y) : IsRelPrime (x ^ m) y := by |
rw [← Finset.card_range m, ← Finset.prod_const]
exact IsRelPrime.prod_left fun _ _ ↦ H
| 2 | 7.389056 | 1 | 1.111111 | 18 | 1,195 |
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Int.GCD
import Mathlib.RingTheory.Coprime.Basic
#align_import ring_theory.coprime.lemmas from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226"
universe u v
section RelPrime
variable {α I} [CommMonoid α] [DecompositionMonoid α] {x y z : α} {s : I → α} {t : Finset I}
theorem IsRelPrime.prod_left : (∀ i ∈ t, IsRelPrime (s i) x) → IsRelPrime (∏ i ∈ t, s i) x := by
classical
refine Finset.induction_on t (fun _ ↦ isRelPrime_one_left) fun b t hbt ih H ↦ ?_
rw [Finset.prod_insert hbt]
rw [Finset.forall_mem_insert] at H
exact H.1.mul_left (ih H.2)
theorem IsRelPrime.prod_right : (∀ i ∈ t, IsRelPrime x (s i)) → IsRelPrime x (∏ i ∈ t, s i) := by
simpa only [isRelPrime_comm] using IsRelPrime.prod_left (α := α)
theorem IsRelPrime.prod_left_iff : IsRelPrime (∏ i ∈ t, s i) x ↔ ∀ i ∈ t, IsRelPrime (s i) x := by
classical
refine Finset.induction_on t (iff_of_true isRelPrime_one_left fun _ ↦ by simp) fun b t hbt ih ↦ ?_
rw [Finset.prod_insert hbt, IsRelPrime.mul_left_iff, ih, Finset.forall_mem_insert]
theorem IsRelPrime.prod_right_iff : IsRelPrime x (∏ i ∈ t, s i) ↔ ∀ i ∈ t, IsRelPrime x (s i) := by
simpa only [isRelPrime_comm] using IsRelPrime.prod_left_iff (α := α)
theorem IsRelPrime.of_prod_left (H1 : IsRelPrime (∏ i ∈ t, s i) x) (i : I) (hit : i ∈ t) :
IsRelPrime (s i) x :=
IsRelPrime.prod_left_iff.1 H1 i hit
theorem IsRelPrime.of_prod_right (H1 : IsRelPrime x (∏ i ∈ t, s i)) (i : I) (hit : i ∈ t) :
IsRelPrime x (s i) :=
IsRelPrime.prod_right_iff.1 H1 i hit
theorem Finset.prod_dvd_of_isRelPrime :
(t : Set I).Pairwise (IsRelPrime on s) → (∀ i ∈ t, s i ∣ z) → (∏ x ∈ t, s x) ∣ z := by
classical
exact Finset.induction_on t (fun _ _ ↦ one_dvd z)
(by
intro a r har ih Hs Hs1
rw [Finset.prod_insert har]
have aux1 : a ∈ (↑(insert a r) : Set I) := Finset.mem_insert_self a r
refine
(IsRelPrime.prod_right fun i hir ↦
Hs aux1 (Finset.mem_insert_of_mem hir) <| by
rintro rfl
exact har hir).mul_dvd
(Hs1 a aux1) (ih (Hs.mono ?_) fun i hi ↦ Hs1 i <| Finset.mem_insert_of_mem hi)
simp only [Finset.coe_insert, Set.subset_insert])
theorem Fintype.prod_dvd_of_isRelPrime [Fintype I] (Hs : Pairwise (IsRelPrime on s))
(Hs1 : ∀ i, s i ∣ z) : (∏ x, s x) ∣ z :=
Finset.prod_dvd_of_isRelPrime (Hs.set_pairwise _) fun i _ ↦ Hs1 i
theorem pairwise_isRelPrime_iff_isRelPrime_prod [DecidableEq I] :
Pairwise (IsRelPrime on fun i : t ↦ s i) ↔ ∀ i ∈ t, IsRelPrime (s i) (∏ j ∈ t \ {i}, s j) := by
refine ⟨fun hp i hi ↦ IsRelPrime.prod_right_iff.mpr fun j hj ↦ ?_, fun hp ↦ ?_⟩
· rw [Finset.mem_sdiff, Finset.mem_singleton] at hj
obtain ⟨hj, ji⟩ := hj
exact @hp ⟨i, hi⟩ ⟨j, hj⟩ fun h ↦ ji (congrArg Subtype.val h).symm
· rintro ⟨i, hi⟩ ⟨j, hj⟩ h
apply IsRelPrime.prod_right_iff.mp (hp i hi)
exact Finset.mem_sdiff.mpr ⟨hj, fun f ↦ h <| Subtype.ext (Finset.mem_singleton.mp f).symm⟩
namespace IsRelPrime
variable {m n : ℕ}
theorem pow_left (H : IsRelPrime x y) : IsRelPrime (x ^ m) y := by
rw [← Finset.card_range m, ← Finset.prod_const]
exact IsRelPrime.prod_left fun _ _ ↦ H
| Mathlib/RingTheory/Coprime/Lemmas.lean | 299 | 301 | theorem pow_right (H : IsRelPrime x y) : IsRelPrime x (y ^ n) := by |
rw [← Finset.card_range n, ← Finset.prod_const]
exact IsRelPrime.prod_right fun _ _ ↦ H
| 2 | 7.389056 | 1 | 1.111111 | 18 | 1,195 |
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Int.GCD
import Mathlib.RingTheory.Coprime.Basic
#align_import ring_theory.coprime.lemmas from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226"
universe u v
section RelPrime
variable {α I} [CommMonoid α] [DecompositionMonoid α] {x y z : α} {s : I → α} {t : Finset I}
theorem IsRelPrime.prod_left : (∀ i ∈ t, IsRelPrime (s i) x) → IsRelPrime (∏ i ∈ t, s i) x := by
classical
refine Finset.induction_on t (fun _ ↦ isRelPrime_one_left) fun b t hbt ih H ↦ ?_
rw [Finset.prod_insert hbt]
rw [Finset.forall_mem_insert] at H
exact H.1.mul_left (ih H.2)
theorem IsRelPrime.prod_right : (∀ i ∈ t, IsRelPrime x (s i)) → IsRelPrime x (∏ i ∈ t, s i) := by
simpa only [isRelPrime_comm] using IsRelPrime.prod_left (α := α)
theorem IsRelPrime.prod_left_iff : IsRelPrime (∏ i ∈ t, s i) x ↔ ∀ i ∈ t, IsRelPrime (s i) x := by
classical
refine Finset.induction_on t (iff_of_true isRelPrime_one_left fun _ ↦ by simp) fun b t hbt ih ↦ ?_
rw [Finset.prod_insert hbt, IsRelPrime.mul_left_iff, ih, Finset.forall_mem_insert]
theorem IsRelPrime.prod_right_iff : IsRelPrime x (∏ i ∈ t, s i) ↔ ∀ i ∈ t, IsRelPrime x (s i) := by
simpa only [isRelPrime_comm] using IsRelPrime.prod_left_iff (α := α)
theorem IsRelPrime.of_prod_left (H1 : IsRelPrime (∏ i ∈ t, s i) x) (i : I) (hit : i ∈ t) :
IsRelPrime (s i) x :=
IsRelPrime.prod_left_iff.1 H1 i hit
theorem IsRelPrime.of_prod_right (H1 : IsRelPrime x (∏ i ∈ t, s i)) (i : I) (hit : i ∈ t) :
IsRelPrime x (s i) :=
IsRelPrime.prod_right_iff.1 H1 i hit
theorem Finset.prod_dvd_of_isRelPrime :
(t : Set I).Pairwise (IsRelPrime on s) → (∀ i ∈ t, s i ∣ z) → (∏ x ∈ t, s x) ∣ z := by
classical
exact Finset.induction_on t (fun _ _ ↦ one_dvd z)
(by
intro a r har ih Hs Hs1
rw [Finset.prod_insert har]
have aux1 : a ∈ (↑(insert a r) : Set I) := Finset.mem_insert_self a r
refine
(IsRelPrime.prod_right fun i hir ↦
Hs aux1 (Finset.mem_insert_of_mem hir) <| by
rintro rfl
exact har hir).mul_dvd
(Hs1 a aux1) (ih (Hs.mono ?_) fun i hi ↦ Hs1 i <| Finset.mem_insert_of_mem hi)
simp only [Finset.coe_insert, Set.subset_insert])
theorem Fintype.prod_dvd_of_isRelPrime [Fintype I] (Hs : Pairwise (IsRelPrime on s))
(Hs1 : ∀ i, s i ∣ z) : (∏ x, s x) ∣ z :=
Finset.prod_dvd_of_isRelPrime (Hs.set_pairwise _) fun i _ ↦ Hs1 i
theorem pairwise_isRelPrime_iff_isRelPrime_prod [DecidableEq I] :
Pairwise (IsRelPrime on fun i : t ↦ s i) ↔ ∀ i ∈ t, IsRelPrime (s i) (∏ j ∈ t \ {i}, s j) := by
refine ⟨fun hp i hi ↦ IsRelPrime.prod_right_iff.mpr fun j hj ↦ ?_, fun hp ↦ ?_⟩
· rw [Finset.mem_sdiff, Finset.mem_singleton] at hj
obtain ⟨hj, ji⟩ := hj
exact @hp ⟨i, hi⟩ ⟨j, hj⟩ fun h ↦ ji (congrArg Subtype.val h).symm
· rintro ⟨i, hi⟩ ⟨j, hj⟩ h
apply IsRelPrime.prod_right_iff.mp (hp i hi)
exact Finset.mem_sdiff.mpr ⟨hj, fun f ↦ h <| Subtype.ext (Finset.mem_singleton.mp f).symm⟩
namespace IsRelPrime
variable {m n : ℕ}
theorem pow_left (H : IsRelPrime x y) : IsRelPrime (x ^ m) y := by
rw [← Finset.card_range m, ← Finset.prod_const]
exact IsRelPrime.prod_left fun _ _ ↦ H
theorem pow_right (H : IsRelPrime x y) : IsRelPrime x (y ^ n) := by
rw [← Finset.card_range n, ← Finset.prod_const]
exact IsRelPrime.prod_right fun _ _ ↦ H
theorem pow (H : IsRelPrime x y) : IsRelPrime (x ^ m) (y ^ n) :=
H.pow_left.pow_right
| Mathlib/RingTheory/Coprime/Lemmas.lean | 306 | 309 | theorem pow_left_iff (hm : 0 < m) : IsRelPrime (x ^ m) y ↔ IsRelPrime x y := by |
refine ⟨fun h ↦ ?_, IsRelPrime.pow_left⟩
rw [← Finset.card_range m, ← Finset.prod_const] at h
exact h.of_prod_left 0 (Finset.mem_range.mpr hm)
| 3 | 20.085537 | 1 | 1.111111 | 18 | 1,195 |
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.FreeModule.StrongRankCondition
import Mathlib.LinearAlgebra.Projection
import Mathlib.LinearAlgebra.SesquilinearForm
import Mathlib.RingTheory.TensorProduct.Basic
import Mathlib.RingTheory.Ideal.LocalRing
#align_import linear_algebra.dual from "leanprover-community/mathlib"@"b1c017582e9f18d8494e5c18602a8cb4a6f843ac"
noncomputable section
namespace Module
-- Porting note: max u v universe issues so name and specific below
universe uR uA uM uM' uM''
variable (R : Type uR) (A : Type uA) (M : Type uM)
variable [CommSemiring R] [AddCommMonoid M] [Module R M]
abbrev Dual :=
M →ₗ[R] R
#align module.dual Module.Dual
def dualPairing (R M) [CommSemiring R] [AddCommMonoid M] [Module R M] :
Module.Dual R M →ₗ[R] M →ₗ[R] R :=
LinearMap.id
#align module.dual_pairing Module.dualPairing
@[simp]
theorem dualPairing_apply (v x) : dualPairing R M v x = v x :=
rfl
#align module.dual_pairing_apply Module.dualPairing_apply
namespace Dual
instance : Inhabited (Dual R M) := ⟨0⟩
def eval : M →ₗ[R] Dual R (Dual R M) :=
LinearMap.flip LinearMap.id
#align module.dual.eval Module.Dual.eval
@[simp]
theorem eval_apply (v : M) (a : Dual R M) : eval R M v a = a v :=
rfl
#align module.dual.eval_apply Module.Dual.eval_apply
variable {R M} {M' : Type uM'}
variable [AddCommMonoid M'] [Module R M']
def transpose : (M →ₗ[R] M') →ₗ[R] Dual R M' →ₗ[R] Dual R M :=
(LinearMap.llcomp R M M' R).flip
#align module.dual.transpose Module.Dual.transpose
-- Porting note: with reducible def need to specify some parameters to transpose explicitly
theorem transpose_apply (u : M →ₗ[R] M') (l : Dual R M') : transpose (R := R) u l = l.comp u :=
rfl
#align module.dual.transpose_apply Module.Dual.transpose_apply
variable {M'' : Type uM''} [AddCommMonoid M''] [Module R M'']
-- Porting note: with reducible def need to specify some parameters to transpose explicitly
theorem transpose_comp (u : M' →ₗ[R] M'') (v : M →ₗ[R] M') :
transpose (R := R) (u.comp v) = (transpose (R := R) v).comp (transpose (R := R) u) :=
rfl
#align module.dual.transpose_comp Module.Dual.transpose_comp
end Dual
section Prod
variable (M' : Type uM') [AddCommMonoid M'] [Module R M']
@[simps!]
def dualProdDualEquivDual : (Module.Dual R M × Module.Dual R M') ≃ₗ[R] Module.Dual R (M × M') :=
LinearMap.coprodEquiv R
#align module.dual_prod_dual_equiv_dual Module.dualProdDualEquivDual
@[simp]
theorem dualProdDualEquivDual_apply (φ : Module.Dual R M) (ψ : Module.Dual R M') :
dualProdDualEquivDual R M M' (φ, ψ) = φ.coprod ψ :=
rfl
#align module.dual_prod_dual_equiv_dual_apply Module.dualProdDualEquivDual_apply
end Prod
end Module
section DualMap
open Module
universe u v v'
variable {R : Type u} [CommSemiring R] {M₁ : Type v} {M₂ : Type v'}
variable [AddCommMonoid M₁] [Module R M₁] [AddCommMonoid M₂] [Module R M₂]
def LinearMap.dualMap (f : M₁ →ₗ[R] M₂) : Dual R M₂ →ₗ[R] Dual R M₁ :=
-- Porting note: with reducible def need to specify some parameters to transpose explicitly
Module.Dual.transpose (R := R) f
#align linear_map.dual_map LinearMap.dualMap
lemma LinearMap.dualMap_eq_lcomp (f : M₁ →ₗ[R] M₂) : f.dualMap = f.lcomp R := rfl
-- Porting note: with reducible def need to specify some parameters to transpose explicitly
theorem LinearMap.dualMap_def (f : M₁ →ₗ[R] M₂) : f.dualMap = Module.Dual.transpose (R := R) f :=
rfl
#align linear_map.dual_map_def LinearMap.dualMap_def
theorem LinearMap.dualMap_apply' (f : M₁ →ₗ[R] M₂) (g : Dual R M₂) : f.dualMap g = g.comp f :=
rfl
#align linear_map.dual_map_apply' LinearMap.dualMap_apply'
@[simp]
theorem LinearMap.dualMap_apply (f : M₁ →ₗ[R] M₂) (g : Dual R M₂) (x : M₁) :
f.dualMap g x = g (f x) :=
rfl
#align linear_map.dual_map_apply LinearMap.dualMap_apply
@[simp]
| Mathlib/LinearAlgebra/Dual.lean | 215 | 217 | theorem LinearMap.dualMap_id : (LinearMap.id : M₁ →ₗ[R] M₁).dualMap = LinearMap.id := by |
ext
rfl
| 2 | 7.389056 | 1 | 1.111111 | 9 | 1,196 |
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.FreeModule.StrongRankCondition
import Mathlib.LinearAlgebra.Projection
import Mathlib.LinearAlgebra.SesquilinearForm
import Mathlib.RingTheory.TensorProduct.Basic
import Mathlib.RingTheory.Ideal.LocalRing
#align_import linear_algebra.dual from "leanprover-community/mathlib"@"b1c017582e9f18d8494e5c18602a8cb4a6f843ac"
noncomputable section
namespace Module
-- Porting note: max u v universe issues so name and specific below
universe uR uA uM uM' uM''
variable (R : Type uR) (A : Type uA) (M : Type uM)
variable [CommSemiring R] [AddCommMonoid M] [Module R M]
abbrev Dual :=
M →ₗ[R] R
#align module.dual Module.Dual
def dualPairing (R M) [CommSemiring R] [AddCommMonoid M] [Module R M] :
Module.Dual R M →ₗ[R] M →ₗ[R] R :=
LinearMap.id
#align module.dual_pairing Module.dualPairing
@[simp]
theorem dualPairing_apply (v x) : dualPairing R M v x = v x :=
rfl
#align module.dual_pairing_apply Module.dualPairing_apply
namespace Dual
instance : Inhabited (Dual R M) := ⟨0⟩
def eval : M →ₗ[R] Dual R (Dual R M) :=
LinearMap.flip LinearMap.id
#align module.dual.eval Module.Dual.eval
@[simp]
theorem eval_apply (v : M) (a : Dual R M) : eval R M v a = a v :=
rfl
#align module.dual.eval_apply Module.Dual.eval_apply
variable {R M} {M' : Type uM'}
variable [AddCommMonoid M'] [Module R M']
def transpose : (M →ₗ[R] M') →ₗ[R] Dual R M' →ₗ[R] Dual R M :=
(LinearMap.llcomp R M M' R).flip
#align module.dual.transpose Module.Dual.transpose
-- Porting note: with reducible def need to specify some parameters to transpose explicitly
theorem transpose_apply (u : M →ₗ[R] M') (l : Dual R M') : transpose (R := R) u l = l.comp u :=
rfl
#align module.dual.transpose_apply Module.Dual.transpose_apply
variable {M'' : Type uM''} [AddCommMonoid M''] [Module R M'']
-- Porting note: with reducible def need to specify some parameters to transpose explicitly
theorem transpose_comp (u : M' →ₗ[R] M'') (v : M →ₗ[R] M') :
transpose (R := R) (u.comp v) = (transpose (R := R) v).comp (transpose (R := R) u) :=
rfl
#align module.dual.transpose_comp Module.Dual.transpose_comp
end Dual
section Prod
variable (M' : Type uM') [AddCommMonoid M'] [Module R M']
@[simps!]
def dualProdDualEquivDual : (Module.Dual R M × Module.Dual R M') ≃ₗ[R] Module.Dual R (M × M') :=
LinearMap.coprodEquiv R
#align module.dual_prod_dual_equiv_dual Module.dualProdDualEquivDual
@[simp]
theorem dualProdDualEquivDual_apply (φ : Module.Dual R M) (ψ : Module.Dual R M') :
dualProdDualEquivDual R M M' (φ, ψ) = φ.coprod ψ :=
rfl
#align module.dual_prod_dual_equiv_dual_apply Module.dualProdDualEquivDual_apply
end Prod
end Module
section DualMap
open Module
universe u v v'
variable {R : Type u} [CommSemiring R] {M₁ : Type v} {M₂ : Type v'}
variable [AddCommMonoid M₁] [Module R M₁] [AddCommMonoid M₂] [Module R M₂]
def LinearMap.dualMap (f : M₁ →ₗ[R] M₂) : Dual R M₂ →ₗ[R] Dual R M₁ :=
-- Porting note: with reducible def need to specify some parameters to transpose explicitly
Module.Dual.transpose (R := R) f
#align linear_map.dual_map LinearMap.dualMap
lemma LinearMap.dualMap_eq_lcomp (f : M₁ →ₗ[R] M₂) : f.dualMap = f.lcomp R := rfl
-- Porting note: with reducible def need to specify some parameters to transpose explicitly
theorem LinearMap.dualMap_def (f : M₁ →ₗ[R] M₂) : f.dualMap = Module.Dual.transpose (R := R) f :=
rfl
#align linear_map.dual_map_def LinearMap.dualMap_def
theorem LinearMap.dualMap_apply' (f : M₁ →ₗ[R] M₂) (g : Dual R M₂) : f.dualMap g = g.comp f :=
rfl
#align linear_map.dual_map_apply' LinearMap.dualMap_apply'
@[simp]
theorem LinearMap.dualMap_apply (f : M₁ →ₗ[R] M₂) (g : Dual R M₂) (x : M₁) :
f.dualMap g x = g (f x) :=
rfl
#align linear_map.dual_map_apply LinearMap.dualMap_apply
@[simp]
theorem LinearMap.dualMap_id : (LinearMap.id : M₁ →ₗ[R] M₁).dualMap = LinearMap.id := by
ext
rfl
#align linear_map.dual_map_id LinearMap.dualMap_id
theorem LinearMap.dualMap_comp_dualMap {M₃ : Type*} [AddCommGroup M₃] [Module R M₃]
(f : M₁ →ₗ[R] M₂) (g : M₂ →ₗ[R] M₃) : f.dualMap.comp g.dualMap = (g.comp f).dualMap :=
rfl
#align linear_map.dual_map_comp_dual_map LinearMap.dualMap_comp_dualMap
| Mathlib/LinearAlgebra/Dual.lean | 226 | 231 | theorem LinearMap.dualMap_injective_of_surjective {f : M₁ →ₗ[R] M₂} (hf : Function.Surjective f) :
Function.Injective f.dualMap := by |
intro φ ψ h
ext x
obtain ⟨y, rfl⟩ := hf x
exact congr_arg (fun g : Module.Dual R M₁ => g y) h
| 4 | 54.59815 | 2 | 1.111111 | 9 | 1,196 |
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.FreeModule.StrongRankCondition
import Mathlib.LinearAlgebra.Projection
import Mathlib.LinearAlgebra.SesquilinearForm
import Mathlib.RingTheory.TensorProduct.Basic
import Mathlib.RingTheory.Ideal.LocalRing
#align_import linear_algebra.dual from "leanprover-community/mathlib"@"b1c017582e9f18d8494e5c18602a8cb4a6f843ac"
noncomputable section
namespace Module
-- Porting note: max u v universe issues so name and specific below
universe uR uA uM uM' uM''
variable (R : Type uR) (A : Type uA) (M : Type uM)
variable [CommSemiring R] [AddCommMonoid M] [Module R M]
abbrev Dual :=
M →ₗ[R] R
#align module.dual Module.Dual
def dualPairing (R M) [CommSemiring R] [AddCommMonoid M] [Module R M] :
Module.Dual R M →ₗ[R] M →ₗ[R] R :=
LinearMap.id
#align module.dual_pairing Module.dualPairing
@[simp]
theorem dualPairing_apply (v x) : dualPairing R M v x = v x :=
rfl
#align module.dual_pairing_apply Module.dualPairing_apply
namespace Dual
instance : Inhabited (Dual R M) := ⟨0⟩
def eval : M →ₗ[R] Dual R (Dual R M) :=
LinearMap.flip LinearMap.id
#align module.dual.eval Module.Dual.eval
@[simp]
theorem eval_apply (v : M) (a : Dual R M) : eval R M v a = a v :=
rfl
#align module.dual.eval_apply Module.Dual.eval_apply
variable {R M} {M' : Type uM'}
variable [AddCommMonoid M'] [Module R M']
def transpose : (M →ₗ[R] M') →ₗ[R] Dual R M' →ₗ[R] Dual R M :=
(LinearMap.llcomp R M M' R).flip
#align module.dual.transpose Module.Dual.transpose
-- Porting note: with reducible def need to specify some parameters to transpose explicitly
theorem transpose_apply (u : M →ₗ[R] M') (l : Dual R M') : transpose (R := R) u l = l.comp u :=
rfl
#align module.dual.transpose_apply Module.Dual.transpose_apply
variable {M'' : Type uM''} [AddCommMonoid M''] [Module R M'']
-- Porting note: with reducible def need to specify some parameters to transpose explicitly
theorem transpose_comp (u : M' →ₗ[R] M'') (v : M →ₗ[R] M') :
transpose (R := R) (u.comp v) = (transpose (R := R) v).comp (transpose (R := R) u) :=
rfl
#align module.dual.transpose_comp Module.Dual.transpose_comp
end Dual
section Prod
variable (M' : Type uM') [AddCommMonoid M'] [Module R M']
@[simps!]
def dualProdDualEquivDual : (Module.Dual R M × Module.Dual R M') ≃ₗ[R] Module.Dual R (M × M') :=
LinearMap.coprodEquiv R
#align module.dual_prod_dual_equiv_dual Module.dualProdDualEquivDual
@[simp]
theorem dualProdDualEquivDual_apply (φ : Module.Dual R M) (ψ : Module.Dual R M') :
dualProdDualEquivDual R M M' (φ, ψ) = φ.coprod ψ :=
rfl
#align module.dual_prod_dual_equiv_dual_apply Module.dualProdDualEquivDual_apply
end Prod
end Module
namespace Basis
universe u v w
open Module Module.Dual Submodule LinearMap Cardinal Function
universe uR uM uK uV uι
variable {R : Type uR} {M : Type uM} {K : Type uK} {V : Type uV} {ι : Type uι}
section CommSemiring
variable [CommSemiring R] [AddCommMonoid M] [Module R M] [DecidableEq ι]
variable (b : Basis ι R M)
def toDual : M →ₗ[R] Module.Dual R M :=
b.constr ℕ fun v => b.constr ℕ fun w => if w = v then (1 : R) else 0
#align basis.to_dual Basis.toDual
| Mathlib/LinearAlgebra/Dual.lean | 303 | 305 | theorem toDual_apply (i j : ι) : b.toDual (b i) (b j) = if i = j then 1 else 0 := by |
erw [constr_basis b, constr_basis b]
simp only [eq_comm]
| 2 | 7.389056 | 1 | 1.111111 | 9 | 1,196 |
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.FreeModule.StrongRankCondition
import Mathlib.LinearAlgebra.Projection
import Mathlib.LinearAlgebra.SesquilinearForm
import Mathlib.RingTheory.TensorProduct.Basic
import Mathlib.RingTheory.Ideal.LocalRing
#align_import linear_algebra.dual from "leanprover-community/mathlib"@"b1c017582e9f18d8494e5c18602a8cb4a6f843ac"
noncomputable section
namespace Module
-- Porting note: max u v universe issues so name and specific below
universe uR uA uM uM' uM''
variable (R : Type uR) (A : Type uA) (M : Type uM)
variable [CommSemiring R] [AddCommMonoid M] [Module R M]
abbrev Dual :=
M →ₗ[R] R
#align module.dual Module.Dual
def dualPairing (R M) [CommSemiring R] [AddCommMonoid M] [Module R M] :
Module.Dual R M →ₗ[R] M →ₗ[R] R :=
LinearMap.id
#align module.dual_pairing Module.dualPairing
@[simp]
theorem dualPairing_apply (v x) : dualPairing R M v x = v x :=
rfl
#align module.dual_pairing_apply Module.dualPairing_apply
namespace Dual
instance : Inhabited (Dual R M) := ⟨0⟩
def eval : M →ₗ[R] Dual R (Dual R M) :=
LinearMap.flip LinearMap.id
#align module.dual.eval Module.Dual.eval
@[simp]
theorem eval_apply (v : M) (a : Dual R M) : eval R M v a = a v :=
rfl
#align module.dual.eval_apply Module.Dual.eval_apply
variable {R M} {M' : Type uM'}
variable [AddCommMonoid M'] [Module R M']
def transpose : (M →ₗ[R] M') →ₗ[R] Dual R M' →ₗ[R] Dual R M :=
(LinearMap.llcomp R M M' R).flip
#align module.dual.transpose Module.Dual.transpose
-- Porting note: with reducible def need to specify some parameters to transpose explicitly
theorem transpose_apply (u : M →ₗ[R] M') (l : Dual R M') : transpose (R := R) u l = l.comp u :=
rfl
#align module.dual.transpose_apply Module.Dual.transpose_apply
variable {M'' : Type uM''} [AddCommMonoid M''] [Module R M'']
-- Porting note: with reducible def need to specify some parameters to transpose explicitly
theorem transpose_comp (u : M' →ₗ[R] M'') (v : M →ₗ[R] M') :
transpose (R := R) (u.comp v) = (transpose (R := R) v).comp (transpose (R := R) u) :=
rfl
#align module.dual.transpose_comp Module.Dual.transpose_comp
end Dual
section Prod
variable (M' : Type uM') [AddCommMonoid M'] [Module R M']
@[simps!]
def dualProdDualEquivDual : (Module.Dual R M × Module.Dual R M') ≃ₗ[R] Module.Dual R (M × M') :=
LinearMap.coprodEquiv R
#align module.dual_prod_dual_equiv_dual Module.dualProdDualEquivDual
@[simp]
theorem dualProdDualEquivDual_apply (φ : Module.Dual R M) (ψ : Module.Dual R M') :
dualProdDualEquivDual R M M' (φ, ψ) = φ.coprod ψ :=
rfl
#align module.dual_prod_dual_equiv_dual_apply Module.dualProdDualEquivDual_apply
end Prod
end Module
namespace Basis
universe u v w
open Module Module.Dual Submodule LinearMap Cardinal Function
universe uR uM uK uV uι
variable {R : Type uR} {M : Type uM} {K : Type uK} {V : Type uV} {ι : Type uι}
section CommSemiring
variable [CommSemiring R] [AddCommMonoid M] [Module R M] [DecidableEq ι]
variable (b : Basis ι R M)
def toDual : M →ₗ[R] Module.Dual R M :=
b.constr ℕ fun v => b.constr ℕ fun w => if w = v then (1 : R) else 0
#align basis.to_dual Basis.toDual
theorem toDual_apply (i j : ι) : b.toDual (b i) (b j) = if i = j then 1 else 0 := by
erw [constr_basis b, constr_basis b]
simp only [eq_comm]
#align basis.to_dual_apply Basis.toDual_apply
@[simp]
| Mathlib/LinearAlgebra/Dual.lean | 309 | 316 | theorem toDual_total_left (f : ι →₀ R) (i : ι) :
b.toDual (Finsupp.total ι M R b f) (b i) = f i := by |
rw [Finsupp.total_apply, Finsupp.sum, _root_.map_sum, LinearMap.sum_apply]
simp_rw [LinearMap.map_smul, LinearMap.smul_apply, toDual_apply, smul_eq_mul, mul_boole,
Finset.sum_ite_eq']
split_ifs with h
· rfl
· rw [Finsupp.not_mem_support_iff.mp h]
| 6 | 403.428793 | 2 | 1.111111 | 9 | 1,196 |
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.FreeModule.StrongRankCondition
import Mathlib.LinearAlgebra.Projection
import Mathlib.LinearAlgebra.SesquilinearForm
import Mathlib.RingTheory.TensorProduct.Basic
import Mathlib.RingTheory.Ideal.LocalRing
#align_import linear_algebra.dual from "leanprover-community/mathlib"@"b1c017582e9f18d8494e5c18602a8cb4a6f843ac"
noncomputable section
namespace Module
-- Porting note: max u v universe issues so name and specific below
universe uR uA uM uM' uM''
variable (R : Type uR) (A : Type uA) (M : Type uM)
variable [CommSemiring R] [AddCommMonoid M] [Module R M]
abbrev Dual :=
M →ₗ[R] R
#align module.dual Module.Dual
def dualPairing (R M) [CommSemiring R] [AddCommMonoid M] [Module R M] :
Module.Dual R M →ₗ[R] M →ₗ[R] R :=
LinearMap.id
#align module.dual_pairing Module.dualPairing
@[simp]
theorem dualPairing_apply (v x) : dualPairing R M v x = v x :=
rfl
#align module.dual_pairing_apply Module.dualPairing_apply
namespace Dual
instance : Inhabited (Dual R M) := ⟨0⟩
def eval : M →ₗ[R] Dual R (Dual R M) :=
LinearMap.flip LinearMap.id
#align module.dual.eval Module.Dual.eval
@[simp]
theorem eval_apply (v : M) (a : Dual R M) : eval R M v a = a v :=
rfl
#align module.dual.eval_apply Module.Dual.eval_apply
variable {R M} {M' : Type uM'}
variable [AddCommMonoid M'] [Module R M']
def transpose : (M →ₗ[R] M') →ₗ[R] Dual R M' →ₗ[R] Dual R M :=
(LinearMap.llcomp R M M' R).flip
#align module.dual.transpose Module.Dual.transpose
-- Porting note: with reducible def need to specify some parameters to transpose explicitly
theorem transpose_apply (u : M →ₗ[R] M') (l : Dual R M') : transpose (R := R) u l = l.comp u :=
rfl
#align module.dual.transpose_apply Module.Dual.transpose_apply
variable {M'' : Type uM''} [AddCommMonoid M''] [Module R M'']
-- Porting note: with reducible def need to specify some parameters to transpose explicitly
theorem transpose_comp (u : M' →ₗ[R] M'') (v : M →ₗ[R] M') :
transpose (R := R) (u.comp v) = (transpose (R := R) v).comp (transpose (R := R) u) :=
rfl
#align module.dual.transpose_comp Module.Dual.transpose_comp
end Dual
section Prod
variable (M' : Type uM') [AddCommMonoid M'] [Module R M']
@[simps!]
def dualProdDualEquivDual : (Module.Dual R M × Module.Dual R M') ≃ₗ[R] Module.Dual R (M × M') :=
LinearMap.coprodEquiv R
#align module.dual_prod_dual_equiv_dual Module.dualProdDualEquivDual
@[simp]
theorem dualProdDualEquivDual_apply (φ : Module.Dual R M) (ψ : Module.Dual R M') :
dualProdDualEquivDual R M M' (φ, ψ) = φ.coprod ψ :=
rfl
#align module.dual_prod_dual_equiv_dual_apply Module.dualProdDualEquivDual_apply
end Prod
end Module
namespace Basis
universe u v w
open Module Module.Dual Submodule LinearMap Cardinal Function
universe uR uM uK uV uι
variable {R : Type uR} {M : Type uM} {K : Type uK} {V : Type uV} {ι : Type uι}
section CommSemiring
variable [CommSemiring R] [AddCommMonoid M] [Module R M] [DecidableEq ι]
variable (b : Basis ι R M)
def toDual : M →ₗ[R] Module.Dual R M :=
b.constr ℕ fun v => b.constr ℕ fun w => if w = v then (1 : R) else 0
#align basis.to_dual Basis.toDual
theorem toDual_apply (i j : ι) : b.toDual (b i) (b j) = if i = j then 1 else 0 := by
erw [constr_basis b, constr_basis b]
simp only [eq_comm]
#align basis.to_dual_apply Basis.toDual_apply
@[simp]
theorem toDual_total_left (f : ι →₀ R) (i : ι) :
b.toDual (Finsupp.total ι M R b f) (b i) = f i := by
rw [Finsupp.total_apply, Finsupp.sum, _root_.map_sum, LinearMap.sum_apply]
simp_rw [LinearMap.map_smul, LinearMap.smul_apply, toDual_apply, smul_eq_mul, mul_boole,
Finset.sum_ite_eq']
split_ifs with h
· rfl
· rw [Finsupp.not_mem_support_iff.mp h]
#align basis.to_dual_total_left Basis.toDual_total_left
@[simp]
| Mathlib/LinearAlgebra/Dual.lean | 320 | 326 | theorem toDual_total_right (f : ι →₀ R) (i : ι) :
b.toDual (b i) (Finsupp.total ι M R b f) = f i := by |
rw [Finsupp.total_apply, Finsupp.sum, _root_.map_sum]
simp_rw [LinearMap.map_smul, toDual_apply, smul_eq_mul, mul_boole, Finset.sum_ite_eq]
split_ifs with h
· rfl
· rw [Finsupp.not_mem_support_iff.mp h]
| 5 | 148.413159 | 2 | 1.111111 | 9 | 1,196 |
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.FreeModule.StrongRankCondition
import Mathlib.LinearAlgebra.Projection
import Mathlib.LinearAlgebra.SesquilinearForm
import Mathlib.RingTheory.TensorProduct.Basic
import Mathlib.RingTheory.Ideal.LocalRing
#align_import linear_algebra.dual from "leanprover-community/mathlib"@"b1c017582e9f18d8494e5c18602a8cb4a6f843ac"
noncomputable section
namespace Module
-- Porting note: max u v universe issues so name and specific below
universe uR uA uM uM' uM''
variable (R : Type uR) (A : Type uA) (M : Type uM)
variable [CommSemiring R] [AddCommMonoid M] [Module R M]
abbrev Dual :=
M →ₗ[R] R
#align module.dual Module.Dual
def dualPairing (R M) [CommSemiring R] [AddCommMonoid M] [Module R M] :
Module.Dual R M →ₗ[R] M →ₗ[R] R :=
LinearMap.id
#align module.dual_pairing Module.dualPairing
@[simp]
theorem dualPairing_apply (v x) : dualPairing R M v x = v x :=
rfl
#align module.dual_pairing_apply Module.dualPairing_apply
namespace Dual
instance : Inhabited (Dual R M) := ⟨0⟩
def eval : M →ₗ[R] Dual R (Dual R M) :=
LinearMap.flip LinearMap.id
#align module.dual.eval Module.Dual.eval
@[simp]
theorem eval_apply (v : M) (a : Dual R M) : eval R M v a = a v :=
rfl
#align module.dual.eval_apply Module.Dual.eval_apply
variable {R M} {M' : Type uM'}
variable [AddCommMonoid M'] [Module R M']
def transpose : (M →ₗ[R] M') →ₗ[R] Dual R M' →ₗ[R] Dual R M :=
(LinearMap.llcomp R M M' R).flip
#align module.dual.transpose Module.Dual.transpose
-- Porting note: with reducible def need to specify some parameters to transpose explicitly
theorem transpose_apply (u : M →ₗ[R] M') (l : Dual R M') : transpose (R := R) u l = l.comp u :=
rfl
#align module.dual.transpose_apply Module.Dual.transpose_apply
variable {M'' : Type uM''} [AddCommMonoid M''] [Module R M'']
-- Porting note: with reducible def need to specify some parameters to transpose explicitly
theorem transpose_comp (u : M' →ₗ[R] M'') (v : M →ₗ[R] M') :
transpose (R := R) (u.comp v) = (transpose (R := R) v).comp (transpose (R := R) u) :=
rfl
#align module.dual.transpose_comp Module.Dual.transpose_comp
end Dual
section Prod
variable (M' : Type uM') [AddCommMonoid M'] [Module R M']
@[simps!]
def dualProdDualEquivDual : (Module.Dual R M × Module.Dual R M') ≃ₗ[R] Module.Dual R (M × M') :=
LinearMap.coprodEquiv R
#align module.dual_prod_dual_equiv_dual Module.dualProdDualEquivDual
@[simp]
theorem dualProdDualEquivDual_apply (φ : Module.Dual R M) (ψ : Module.Dual R M') :
dualProdDualEquivDual R M M' (φ, ψ) = φ.coprod ψ :=
rfl
#align module.dual_prod_dual_equiv_dual_apply Module.dualProdDualEquivDual_apply
end Prod
end Module
namespace Basis
universe u v w
open Module Module.Dual Submodule LinearMap Cardinal Function
universe uR uM uK uV uι
variable {R : Type uR} {M : Type uM} {K : Type uK} {V : Type uV} {ι : Type uι}
section CommSemiring
variable [CommSemiring R] [AddCommMonoid M] [Module R M] [DecidableEq ι]
variable (b : Basis ι R M)
def toDual : M →ₗ[R] Module.Dual R M :=
b.constr ℕ fun v => b.constr ℕ fun w => if w = v then (1 : R) else 0
#align basis.to_dual Basis.toDual
theorem toDual_apply (i j : ι) : b.toDual (b i) (b j) = if i = j then 1 else 0 := by
erw [constr_basis b, constr_basis b]
simp only [eq_comm]
#align basis.to_dual_apply Basis.toDual_apply
@[simp]
theorem toDual_total_left (f : ι →₀ R) (i : ι) :
b.toDual (Finsupp.total ι M R b f) (b i) = f i := by
rw [Finsupp.total_apply, Finsupp.sum, _root_.map_sum, LinearMap.sum_apply]
simp_rw [LinearMap.map_smul, LinearMap.smul_apply, toDual_apply, smul_eq_mul, mul_boole,
Finset.sum_ite_eq']
split_ifs with h
· rfl
· rw [Finsupp.not_mem_support_iff.mp h]
#align basis.to_dual_total_left Basis.toDual_total_left
@[simp]
theorem toDual_total_right (f : ι →₀ R) (i : ι) :
b.toDual (b i) (Finsupp.total ι M R b f) = f i := by
rw [Finsupp.total_apply, Finsupp.sum, _root_.map_sum]
simp_rw [LinearMap.map_smul, toDual_apply, smul_eq_mul, mul_boole, Finset.sum_ite_eq]
split_ifs with h
· rfl
· rw [Finsupp.not_mem_support_iff.mp h]
#align basis.to_dual_total_right Basis.toDual_total_right
| Mathlib/LinearAlgebra/Dual.lean | 329 | 330 | theorem toDual_apply_left (m : M) (i : ι) : b.toDual m (b i) = b.repr m i := by |
rw [← b.toDual_total_left, b.total_repr]
| 1 | 2.718282 | 0 | 1.111111 | 9 | 1,196 |
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.FreeModule.StrongRankCondition
import Mathlib.LinearAlgebra.Projection
import Mathlib.LinearAlgebra.SesquilinearForm
import Mathlib.RingTheory.TensorProduct.Basic
import Mathlib.RingTheory.Ideal.LocalRing
#align_import linear_algebra.dual from "leanprover-community/mathlib"@"b1c017582e9f18d8494e5c18602a8cb4a6f843ac"
noncomputable section
namespace Module
-- Porting note: max u v universe issues so name and specific below
universe uR uA uM uM' uM''
variable (R : Type uR) (A : Type uA) (M : Type uM)
variable [CommSemiring R] [AddCommMonoid M] [Module R M]
abbrev Dual :=
M →ₗ[R] R
#align module.dual Module.Dual
def dualPairing (R M) [CommSemiring R] [AddCommMonoid M] [Module R M] :
Module.Dual R M →ₗ[R] M →ₗ[R] R :=
LinearMap.id
#align module.dual_pairing Module.dualPairing
@[simp]
theorem dualPairing_apply (v x) : dualPairing R M v x = v x :=
rfl
#align module.dual_pairing_apply Module.dualPairing_apply
namespace Dual
instance : Inhabited (Dual R M) := ⟨0⟩
def eval : M →ₗ[R] Dual R (Dual R M) :=
LinearMap.flip LinearMap.id
#align module.dual.eval Module.Dual.eval
@[simp]
theorem eval_apply (v : M) (a : Dual R M) : eval R M v a = a v :=
rfl
#align module.dual.eval_apply Module.Dual.eval_apply
variable {R M} {M' : Type uM'}
variable [AddCommMonoid M'] [Module R M']
def transpose : (M →ₗ[R] M') →ₗ[R] Dual R M' →ₗ[R] Dual R M :=
(LinearMap.llcomp R M M' R).flip
#align module.dual.transpose Module.Dual.transpose
-- Porting note: with reducible def need to specify some parameters to transpose explicitly
theorem transpose_apply (u : M →ₗ[R] M') (l : Dual R M') : transpose (R := R) u l = l.comp u :=
rfl
#align module.dual.transpose_apply Module.Dual.transpose_apply
variable {M'' : Type uM''} [AddCommMonoid M''] [Module R M'']
-- Porting note: with reducible def need to specify some parameters to transpose explicitly
theorem transpose_comp (u : M' →ₗ[R] M'') (v : M →ₗ[R] M') :
transpose (R := R) (u.comp v) = (transpose (R := R) v).comp (transpose (R := R) u) :=
rfl
#align module.dual.transpose_comp Module.Dual.transpose_comp
end Dual
section Prod
variable (M' : Type uM') [AddCommMonoid M'] [Module R M']
@[simps!]
def dualProdDualEquivDual : (Module.Dual R M × Module.Dual R M') ≃ₗ[R] Module.Dual R (M × M') :=
LinearMap.coprodEquiv R
#align module.dual_prod_dual_equiv_dual Module.dualProdDualEquivDual
@[simp]
theorem dualProdDualEquivDual_apply (φ : Module.Dual R M) (ψ : Module.Dual R M') :
dualProdDualEquivDual R M M' (φ, ψ) = φ.coprod ψ :=
rfl
#align module.dual_prod_dual_equiv_dual_apply Module.dualProdDualEquivDual_apply
end Prod
end Module
namespace Basis
universe u v w
open Module Module.Dual Submodule LinearMap Cardinal Function
universe uR uM uK uV uι
variable {R : Type uR} {M : Type uM} {K : Type uK} {V : Type uV} {ι : Type uι}
section CommSemiring
variable [CommSemiring R] [AddCommMonoid M] [Module R M] [DecidableEq ι]
variable (b : Basis ι R M)
def toDual : M →ₗ[R] Module.Dual R M :=
b.constr ℕ fun v => b.constr ℕ fun w => if w = v then (1 : R) else 0
#align basis.to_dual Basis.toDual
theorem toDual_apply (i j : ι) : b.toDual (b i) (b j) = if i = j then 1 else 0 := by
erw [constr_basis b, constr_basis b]
simp only [eq_comm]
#align basis.to_dual_apply Basis.toDual_apply
@[simp]
theorem toDual_total_left (f : ι →₀ R) (i : ι) :
b.toDual (Finsupp.total ι M R b f) (b i) = f i := by
rw [Finsupp.total_apply, Finsupp.sum, _root_.map_sum, LinearMap.sum_apply]
simp_rw [LinearMap.map_smul, LinearMap.smul_apply, toDual_apply, smul_eq_mul, mul_boole,
Finset.sum_ite_eq']
split_ifs with h
· rfl
· rw [Finsupp.not_mem_support_iff.mp h]
#align basis.to_dual_total_left Basis.toDual_total_left
@[simp]
theorem toDual_total_right (f : ι →₀ R) (i : ι) :
b.toDual (b i) (Finsupp.total ι M R b f) = f i := by
rw [Finsupp.total_apply, Finsupp.sum, _root_.map_sum]
simp_rw [LinearMap.map_smul, toDual_apply, smul_eq_mul, mul_boole, Finset.sum_ite_eq]
split_ifs with h
· rfl
· rw [Finsupp.not_mem_support_iff.mp h]
#align basis.to_dual_total_right Basis.toDual_total_right
theorem toDual_apply_left (m : M) (i : ι) : b.toDual m (b i) = b.repr m i := by
rw [← b.toDual_total_left, b.total_repr]
#align basis.to_dual_apply_left Basis.toDual_apply_left
| Mathlib/LinearAlgebra/Dual.lean | 333 | 334 | theorem toDual_apply_right (i : ι) (m : M) : b.toDual (b i) m = b.repr m i := by |
rw [← b.toDual_total_right, b.total_repr]
| 1 | 2.718282 | 0 | 1.111111 | 9 | 1,196 |
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.FreeModule.StrongRankCondition
import Mathlib.LinearAlgebra.Projection
import Mathlib.LinearAlgebra.SesquilinearForm
import Mathlib.RingTheory.TensorProduct.Basic
import Mathlib.RingTheory.Ideal.LocalRing
#align_import linear_algebra.dual from "leanprover-community/mathlib"@"b1c017582e9f18d8494e5c18602a8cb4a6f843ac"
noncomputable section
namespace Module
-- Porting note: max u v universe issues so name and specific below
universe uR uA uM uM' uM''
variable (R : Type uR) (A : Type uA) (M : Type uM)
variable [CommSemiring R] [AddCommMonoid M] [Module R M]
abbrev Dual :=
M →ₗ[R] R
#align module.dual Module.Dual
def dualPairing (R M) [CommSemiring R] [AddCommMonoid M] [Module R M] :
Module.Dual R M →ₗ[R] M →ₗ[R] R :=
LinearMap.id
#align module.dual_pairing Module.dualPairing
@[simp]
theorem dualPairing_apply (v x) : dualPairing R M v x = v x :=
rfl
#align module.dual_pairing_apply Module.dualPairing_apply
namespace Dual
instance : Inhabited (Dual R M) := ⟨0⟩
def eval : M →ₗ[R] Dual R (Dual R M) :=
LinearMap.flip LinearMap.id
#align module.dual.eval Module.Dual.eval
@[simp]
theorem eval_apply (v : M) (a : Dual R M) : eval R M v a = a v :=
rfl
#align module.dual.eval_apply Module.Dual.eval_apply
variable {R M} {M' : Type uM'}
variable [AddCommMonoid M'] [Module R M']
def transpose : (M →ₗ[R] M') →ₗ[R] Dual R M' →ₗ[R] Dual R M :=
(LinearMap.llcomp R M M' R).flip
#align module.dual.transpose Module.Dual.transpose
-- Porting note: with reducible def need to specify some parameters to transpose explicitly
theorem transpose_apply (u : M →ₗ[R] M') (l : Dual R M') : transpose (R := R) u l = l.comp u :=
rfl
#align module.dual.transpose_apply Module.Dual.transpose_apply
variable {M'' : Type uM''} [AddCommMonoid M''] [Module R M'']
-- Porting note: with reducible def need to specify some parameters to transpose explicitly
theorem transpose_comp (u : M' →ₗ[R] M'') (v : M →ₗ[R] M') :
transpose (R := R) (u.comp v) = (transpose (R := R) v).comp (transpose (R := R) u) :=
rfl
#align module.dual.transpose_comp Module.Dual.transpose_comp
end Dual
section Prod
variable (M' : Type uM') [AddCommMonoid M'] [Module R M']
@[simps!]
def dualProdDualEquivDual : (Module.Dual R M × Module.Dual R M') ≃ₗ[R] Module.Dual R (M × M') :=
LinearMap.coprodEquiv R
#align module.dual_prod_dual_equiv_dual Module.dualProdDualEquivDual
@[simp]
theorem dualProdDualEquivDual_apply (φ : Module.Dual R M) (ψ : Module.Dual R M') :
dualProdDualEquivDual R M M' (φ, ψ) = φ.coprod ψ :=
rfl
#align module.dual_prod_dual_equiv_dual_apply Module.dualProdDualEquivDual_apply
end Prod
end Module
namespace Basis
universe u v w
open Module Module.Dual Submodule LinearMap Cardinal Function
universe uR uM uK uV uι
variable {R : Type uR} {M : Type uM} {K : Type uK} {V : Type uV} {ι : Type uι}
section CommSemiring
variable [CommSemiring R] [AddCommMonoid M] [Module R M] [DecidableEq ι]
variable (b : Basis ι R M)
def toDual : M →ₗ[R] Module.Dual R M :=
b.constr ℕ fun v => b.constr ℕ fun w => if w = v then (1 : R) else 0
#align basis.to_dual Basis.toDual
theorem toDual_apply (i j : ι) : b.toDual (b i) (b j) = if i = j then 1 else 0 := by
erw [constr_basis b, constr_basis b]
simp only [eq_comm]
#align basis.to_dual_apply Basis.toDual_apply
@[simp]
theorem toDual_total_left (f : ι →₀ R) (i : ι) :
b.toDual (Finsupp.total ι M R b f) (b i) = f i := by
rw [Finsupp.total_apply, Finsupp.sum, _root_.map_sum, LinearMap.sum_apply]
simp_rw [LinearMap.map_smul, LinearMap.smul_apply, toDual_apply, smul_eq_mul, mul_boole,
Finset.sum_ite_eq']
split_ifs with h
· rfl
· rw [Finsupp.not_mem_support_iff.mp h]
#align basis.to_dual_total_left Basis.toDual_total_left
@[simp]
theorem toDual_total_right (f : ι →₀ R) (i : ι) :
b.toDual (b i) (Finsupp.total ι M R b f) = f i := by
rw [Finsupp.total_apply, Finsupp.sum, _root_.map_sum]
simp_rw [LinearMap.map_smul, toDual_apply, smul_eq_mul, mul_boole, Finset.sum_ite_eq]
split_ifs with h
· rfl
· rw [Finsupp.not_mem_support_iff.mp h]
#align basis.to_dual_total_right Basis.toDual_total_right
theorem toDual_apply_left (m : M) (i : ι) : b.toDual m (b i) = b.repr m i := by
rw [← b.toDual_total_left, b.total_repr]
#align basis.to_dual_apply_left Basis.toDual_apply_left
theorem toDual_apply_right (i : ι) (m : M) : b.toDual (b i) m = b.repr m i := by
rw [← b.toDual_total_right, b.total_repr]
#align basis.to_dual_apply_right Basis.toDual_apply_right
| Mathlib/LinearAlgebra/Dual.lean | 337 | 339 | theorem coe_toDual_self (i : ι) : b.toDual (b i) = b.coord i := by |
ext
apply toDual_apply_right
| 2 | 7.389056 | 1 | 1.111111 | 9 | 1,196 |
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.FreeModule.StrongRankCondition
import Mathlib.LinearAlgebra.Projection
import Mathlib.LinearAlgebra.SesquilinearForm
import Mathlib.RingTheory.TensorProduct.Basic
import Mathlib.RingTheory.Ideal.LocalRing
#align_import linear_algebra.dual from "leanprover-community/mathlib"@"b1c017582e9f18d8494e5c18602a8cb4a6f843ac"
noncomputable section
namespace Module
-- Porting note: max u v universe issues so name and specific below
universe uR uA uM uM' uM''
variable (R : Type uR) (A : Type uA) (M : Type uM)
variable [CommSemiring R] [AddCommMonoid M] [Module R M]
abbrev Dual :=
M →ₗ[R] R
#align module.dual Module.Dual
def dualPairing (R M) [CommSemiring R] [AddCommMonoid M] [Module R M] :
Module.Dual R M →ₗ[R] M →ₗ[R] R :=
LinearMap.id
#align module.dual_pairing Module.dualPairing
@[simp]
theorem dualPairing_apply (v x) : dualPairing R M v x = v x :=
rfl
#align module.dual_pairing_apply Module.dualPairing_apply
namespace Dual
instance : Inhabited (Dual R M) := ⟨0⟩
def eval : M →ₗ[R] Dual R (Dual R M) :=
LinearMap.flip LinearMap.id
#align module.dual.eval Module.Dual.eval
@[simp]
theorem eval_apply (v : M) (a : Dual R M) : eval R M v a = a v :=
rfl
#align module.dual.eval_apply Module.Dual.eval_apply
variable {R M} {M' : Type uM'}
variable [AddCommMonoid M'] [Module R M']
def transpose : (M →ₗ[R] M') →ₗ[R] Dual R M' →ₗ[R] Dual R M :=
(LinearMap.llcomp R M M' R).flip
#align module.dual.transpose Module.Dual.transpose
-- Porting note: with reducible def need to specify some parameters to transpose explicitly
theorem transpose_apply (u : M →ₗ[R] M') (l : Dual R M') : transpose (R := R) u l = l.comp u :=
rfl
#align module.dual.transpose_apply Module.Dual.transpose_apply
variable {M'' : Type uM''} [AddCommMonoid M''] [Module R M'']
-- Porting note: with reducible def need to specify some parameters to transpose explicitly
theorem transpose_comp (u : M' →ₗ[R] M'') (v : M →ₗ[R] M') :
transpose (R := R) (u.comp v) = (transpose (R := R) v).comp (transpose (R := R) u) :=
rfl
#align module.dual.transpose_comp Module.Dual.transpose_comp
end Dual
section Prod
variable (M' : Type uM') [AddCommMonoid M'] [Module R M']
@[simps!]
def dualProdDualEquivDual : (Module.Dual R M × Module.Dual R M') ≃ₗ[R] Module.Dual R (M × M') :=
LinearMap.coprodEquiv R
#align module.dual_prod_dual_equiv_dual Module.dualProdDualEquivDual
@[simp]
theorem dualProdDualEquivDual_apply (φ : Module.Dual R M) (ψ : Module.Dual R M') :
dualProdDualEquivDual R M M' (φ, ψ) = φ.coprod ψ :=
rfl
#align module.dual_prod_dual_equiv_dual_apply Module.dualProdDualEquivDual_apply
end Prod
end Module
namespace Basis
universe u v w
open Module Module.Dual Submodule LinearMap Cardinal Function
universe uR uM uK uV uι
variable {R : Type uR} {M : Type uM} {K : Type uK} {V : Type uV} {ι : Type uι}
section
variable [CommSemiring R] [AddCommMonoid M] [Module R M] [Fintype ι]
variable (b : Basis ι R M)
@[simp]
| Mathlib/LinearAlgebra/Dual.lean | 388 | 392 | theorem sum_dual_apply_smul_coord (f : Module.Dual R M) :
(∑ x, f (b x) • b.coord x) = f := by |
ext m
simp_rw [LinearMap.sum_apply, LinearMap.smul_apply, smul_eq_mul, mul_comm (f _), ← smul_eq_mul, ←
f.map_smul, ← _root_.map_sum, Basis.coord_apply, Basis.sum_repr]
| 3 | 20.085537 | 1 | 1.111111 | 9 | 1,196 |
import Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho
import Mathlib.LinearAlgebra.Orientation
#align_import analysis.inner_product_space.orientation from "leanprover-community/mathlib"@"bd65478311e4dfd41f48bf38c7e3b02fb75d0163"
noncomputable section
variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
open FiniteDimensional
open scoped RealInnerProductSpace
namespace OrthonormalBasis
variable {ι : Type*} [Fintype ι] [DecidableEq ι] [ne : Nonempty ι] (e f : OrthonormalBasis ι ℝ E)
(x : Orientation ℝ E ι)
| Mathlib/Analysis/InnerProductSpace/Orientation.lean | 54 | 60 | theorem det_to_matrix_orthonormalBasis_of_same_orientation
(h : e.toBasis.orientation = f.toBasis.orientation) : e.toBasis.det f = 1 := by |
apply (e.det_to_matrix_orthonormalBasis_real f).resolve_right
have : 0 < e.toBasis.det f := by
rw [e.toBasis.orientation_eq_iff_det_pos] at h
simpa using h
linarith
| 5 | 148.413159 | 2 | 1.111111 | 9 | 1,197 |
import Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho
import Mathlib.LinearAlgebra.Orientation
#align_import analysis.inner_product_space.orientation from "leanprover-community/mathlib"@"bd65478311e4dfd41f48bf38c7e3b02fb75d0163"
noncomputable section
variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
open FiniteDimensional
open scoped RealInnerProductSpace
namespace OrthonormalBasis
variable {ι : Type*} [Fintype ι] [DecidableEq ι] [ne : Nonempty ι] (e f : OrthonormalBasis ι ℝ E)
(x : Orientation ℝ E ι)
theorem det_to_matrix_orthonormalBasis_of_same_orientation
(h : e.toBasis.orientation = f.toBasis.orientation) : e.toBasis.det f = 1 := by
apply (e.det_to_matrix_orthonormalBasis_real f).resolve_right
have : 0 < e.toBasis.det f := by
rw [e.toBasis.orientation_eq_iff_det_pos] at h
simpa using h
linarith
#align orthonormal_basis.det_to_matrix_orthonormal_basis_of_same_orientation OrthonormalBasis.det_to_matrix_orthonormalBasis_of_same_orientation
| Mathlib/Analysis/InnerProductSpace/Orientation.lean | 65 | 69 | theorem det_to_matrix_orthonormalBasis_of_opposite_orientation
(h : e.toBasis.orientation ≠ f.toBasis.orientation) : e.toBasis.det f = -1 := by |
contrapose! h
simp [e.toBasis.orientation_eq_iff_det_pos,
(e.det_to_matrix_orthonormalBasis_real f).resolve_right h]
| 3 | 20.085537 | 1 | 1.111111 | 9 | 1,197 |
import Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho
import Mathlib.LinearAlgebra.Orientation
#align_import analysis.inner_product_space.orientation from "leanprover-community/mathlib"@"bd65478311e4dfd41f48bf38c7e3b02fb75d0163"
noncomputable section
variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
open FiniteDimensional
open scoped RealInnerProductSpace
namespace OrthonormalBasis
variable {ι : Type*} [Fintype ι] [DecidableEq ι] [ne : Nonempty ι] (e f : OrthonormalBasis ι ℝ E)
(x : Orientation ℝ E ι)
theorem det_to_matrix_orthonormalBasis_of_same_orientation
(h : e.toBasis.orientation = f.toBasis.orientation) : e.toBasis.det f = 1 := by
apply (e.det_to_matrix_orthonormalBasis_real f).resolve_right
have : 0 < e.toBasis.det f := by
rw [e.toBasis.orientation_eq_iff_det_pos] at h
simpa using h
linarith
#align orthonormal_basis.det_to_matrix_orthonormal_basis_of_same_orientation OrthonormalBasis.det_to_matrix_orthonormalBasis_of_same_orientation
theorem det_to_matrix_orthonormalBasis_of_opposite_orientation
(h : e.toBasis.orientation ≠ f.toBasis.orientation) : e.toBasis.det f = -1 := by
contrapose! h
simp [e.toBasis.orientation_eq_iff_det_pos,
(e.det_to_matrix_orthonormalBasis_real f).resolve_right h]
#align orthonormal_basis.det_to_matrix_orthonormal_basis_of_opposite_orientation OrthonormalBasis.det_to_matrix_orthonormalBasis_of_opposite_orientation
variable {e f}
| Mathlib/Analysis/InnerProductSpace/Orientation.lean | 76 | 84 | theorem same_orientation_iff_det_eq_det :
e.toBasis.det = f.toBasis.det ↔ e.toBasis.orientation = f.toBasis.orientation := by |
constructor
· intro h
dsimp [Basis.orientation]
congr
· intro h
rw [e.toBasis.det.eq_smul_basis_det f.toBasis]
simp [e.det_to_matrix_orthonormalBasis_of_same_orientation f h]
| 7 | 1,096.633158 | 2 | 1.111111 | 9 | 1,197 |
import Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho
import Mathlib.LinearAlgebra.Orientation
#align_import analysis.inner_product_space.orientation from "leanprover-community/mathlib"@"bd65478311e4dfd41f48bf38c7e3b02fb75d0163"
noncomputable section
variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
open FiniteDimensional
open scoped RealInnerProductSpace
namespace OrthonormalBasis
variable {ι : Type*} [Fintype ι] [DecidableEq ι] [ne : Nonempty ι] (e f : OrthonormalBasis ι ℝ E)
(x : Orientation ℝ E ι)
theorem det_to_matrix_orthonormalBasis_of_same_orientation
(h : e.toBasis.orientation = f.toBasis.orientation) : e.toBasis.det f = 1 := by
apply (e.det_to_matrix_orthonormalBasis_real f).resolve_right
have : 0 < e.toBasis.det f := by
rw [e.toBasis.orientation_eq_iff_det_pos] at h
simpa using h
linarith
#align orthonormal_basis.det_to_matrix_orthonormal_basis_of_same_orientation OrthonormalBasis.det_to_matrix_orthonormalBasis_of_same_orientation
theorem det_to_matrix_orthonormalBasis_of_opposite_orientation
(h : e.toBasis.orientation ≠ f.toBasis.orientation) : e.toBasis.det f = -1 := by
contrapose! h
simp [e.toBasis.orientation_eq_iff_det_pos,
(e.det_to_matrix_orthonormalBasis_real f).resolve_right h]
#align orthonormal_basis.det_to_matrix_orthonormal_basis_of_opposite_orientation OrthonormalBasis.det_to_matrix_orthonormalBasis_of_opposite_orientation
variable {e f}
theorem same_orientation_iff_det_eq_det :
e.toBasis.det = f.toBasis.det ↔ e.toBasis.orientation = f.toBasis.orientation := by
constructor
· intro h
dsimp [Basis.orientation]
congr
· intro h
rw [e.toBasis.det.eq_smul_basis_det f.toBasis]
simp [e.det_to_matrix_orthonormalBasis_of_same_orientation f h]
#align orthonormal_basis.same_orientation_iff_det_eq_det OrthonormalBasis.same_orientation_iff_det_eq_det
variable (e f)
| Mathlib/Analysis/InnerProductSpace/Orientation.lean | 91 | 96 | theorem det_eq_neg_det_of_opposite_orientation (h : e.toBasis.orientation ≠ f.toBasis.orientation) :
e.toBasis.det = -f.toBasis.det := by |
rw [e.toBasis.det.eq_smul_basis_det f.toBasis]
-- Porting note: added `neg_one_smul` with explicit type
simp [e.det_to_matrix_orthonormalBasis_of_opposite_orientation f h,
neg_one_smul ℝ (M := E [⋀^ι]→ₗ[ℝ] ℝ)]
| 4 | 54.59815 | 2 | 1.111111 | 9 | 1,197 |
import Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho
import Mathlib.LinearAlgebra.Orientation
#align_import analysis.inner_product_space.orientation from "leanprover-community/mathlib"@"bd65478311e4dfd41f48bf38c7e3b02fb75d0163"
noncomputable section
variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
open FiniteDimensional
open scoped RealInnerProductSpace
namespace OrthonormalBasis
variable {ι : Type*} [Fintype ι] [DecidableEq ι] [ne : Nonempty ι] (e f : OrthonormalBasis ι ℝ E)
(x : Orientation ℝ E ι)
theorem det_to_matrix_orthonormalBasis_of_same_orientation
(h : e.toBasis.orientation = f.toBasis.orientation) : e.toBasis.det f = 1 := by
apply (e.det_to_matrix_orthonormalBasis_real f).resolve_right
have : 0 < e.toBasis.det f := by
rw [e.toBasis.orientation_eq_iff_det_pos] at h
simpa using h
linarith
#align orthonormal_basis.det_to_matrix_orthonormal_basis_of_same_orientation OrthonormalBasis.det_to_matrix_orthonormalBasis_of_same_orientation
theorem det_to_matrix_orthonormalBasis_of_opposite_orientation
(h : e.toBasis.orientation ≠ f.toBasis.orientation) : e.toBasis.det f = -1 := by
contrapose! h
simp [e.toBasis.orientation_eq_iff_det_pos,
(e.det_to_matrix_orthonormalBasis_real f).resolve_right h]
#align orthonormal_basis.det_to_matrix_orthonormal_basis_of_opposite_orientation OrthonormalBasis.det_to_matrix_orthonormalBasis_of_opposite_orientation
variable {e f}
theorem same_orientation_iff_det_eq_det :
e.toBasis.det = f.toBasis.det ↔ e.toBasis.orientation = f.toBasis.orientation := by
constructor
· intro h
dsimp [Basis.orientation]
congr
· intro h
rw [e.toBasis.det.eq_smul_basis_det f.toBasis]
simp [e.det_to_matrix_orthonormalBasis_of_same_orientation f h]
#align orthonormal_basis.same_orientation_iff_det_eq_det OrthonormalBasis.same_orientation_iff_det_eq_det
variable (e f)
theorem det_eq_neg_det_of_opposite_orientation (h : e.toBasis.orientation ≠ f.toBasis.orientation) :
e.toBasis.det = -f.toBasis.det := by
rw [e.toBasis.det.eq_smul_basis_det f.toBasis]
-- Porting note: added `neg_one_smul` with explicit type
simp [e.det_to_matrix_orthonormalBasis_of_opposite_orientation f h,
neg_one_smul ℝ (M := E [⋀^ι]→ₗ[ℝ] ℝ)]
#align orthonormal_basis.det_eq_neg_det_of_opposite_orientation OrthonormalBasis.det_eq_neg_det_of_opposite_orientation
section AdjustToOrientation
| Mathlib/Analysis/InnerProductSpace/Orientation.lean | 103 | 105 | theorem orthonormal_adjustToOrientation : Orthonormal ℝ (e.toBasis.adjustToOrientation x) := by |
apply e.orthonormal.orthonormal_of_forall_eq_or_eq_neg
simpa using e.toBasis.adjustToOrientation_apply_eq_or_eq_neg x
| 2 | 7.389056 | 1 | 1.111111 | 9 | 1,197 |
import Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho
import Mathlib.LinearAlgebra.Orientation
#align_import analysis.inner_product_space.orientation from "leanprover-community/mathlib"@"bd65478311e4dfd41f48bf38c7e3b02fb75d0163"
noncomputable section
variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
open FiniteDimensional
open scoped RealInnerProductSpace
namespace OrthonormalBasis
variable {ι : Type*} [Fintype ι] [DecidableEq ι] [ne : Nonempty ι] (e f : OrthonormalBasis ι ℝ E)
(x : Orientation ℝ E ι)
theorem det_to_matrix_orthonormalBasis_of_same_orientation
(h : e.toBasis.orientation = f.toBasis.orientation) : e.toBasis.det f = 1 := by
apply (e.det_to_matrix_orthonormalBasis_real f).resolve_right
have : 0 < e.toBasis.det f := by
rw [e.toBasis.orientation_eq_iff_det_pos] at h
simpa using h
linarith
#align orthonormal_basis.det_to_matrix_orthonormal_basis_of_same_orientation OrthonormalBasis.det_to_matrix_orthonormalBasis_of_same_orientation
theorem det_to_matrix_orthonormalBasis_of_opposite_orientation
(h : e.toBasis.orientation ≠ f.toBasis.orientation) : e.toBasis.det f = -1 := by
contrapose! h
simp [e.toBasis.orientation_eq_iff_det_pos,
(e.det_to_matrix_orthonormalBasis_real f).resolve_right h]
#align orthonormal_basis.det_to_matrix_orthonormal_basis_of_opposite_orientation OrthonormalBasis.det_to_matrix_orthonormalBasis_of_opposite_orientation
variable {e f}
theorem same_orientation_iff_det_eq_det :
e.toBasis.det = f.toBasis.det ↔ e.toBasis.orientation = f.toBasis.orientation := by
constructor
· intro h
dsimp [Basis.orientation]
congr
· intro h
rw [e.toBasis.det.eq_smul_basis_det f.toBasis]
simp [e.det_to_matrix_orthonormalBasis_of_same_orientation f h]
#align orthonormal_basis.same_orientation_iff_det_eq_det OrthonormalBasis.same_orientation_iff_det_eq_det
variable (e f)
theorem det_eq_neg_det_of_opposite_orientation (h : e.toBasis.orientation ≠ f.toBasis.orientation) :
e.toBasis.det = -f.toBasis.det := by
rw [e.toBasis.det.eq_smul_basis_det f.toBasis]
-- Porting note: added `neg_one_smul` with explicit type
simp [e.det_to_matrix_orthonormalBasis_of_opposite_orientation f h,
neg_one_smul ℝ (M := E [⋀^ι]→ₗ[ℝ] ℝ)]
#align orthonormal_basis.det_eq_neg_det_of_opposite_orientation OrthonormalBasis.det_eq_neg_det_of_opposite_orientation
section AdjustToOrientation
theorem orthonormal_adjustToOrientation : Orthonormal ℝ (e.toBasis.adjustToOrientation x) := by
apply e.orthonormal.orthonormal_of_forall_eq_or_eq_neg
simpa using e.toBasis.adjustToOrientation_apply_eq_or_eq_neg x
#align orthonormal_basis.orthonormal_adjust_to_orientation OrthonormalBasis.orthonormal_adjustToOrientation
def adjustToOrientation : OrthonormalBasis ι ℝ E :=
(e.toBasis.adjustToOrientation x).toOrthonormalBasis (e.orthonormal_adjustToOrientation x)
#align orthonormal_basis.adjust_to_orientation OrthonormalBasis.adjustToOrientation
theorem toBasis_adjustToOrientation :
(e.adjustToOrientation x).toBasis = e.toBasis.adjustToOrientation x :=
(e.toBasis.adjustToOrientation x).toBasis_toOrthonormalBasis _
#align orthonormal_basis.to_basis_adjust_to_orientation OrthonormalBasis.toBasis_adjustToOrientation
@[simp]
| Mathlib/Analysis/InnerProductSpace/Orientation.lean | 122 | 124 | theorem orientation_adjustToOrientation : (e.adjustToOrientation x).toBasis.orientation = x := by |
rw [e.toBasis_adjustToOrientation]
exact e.toBasis.orientation_adjustToOrientation x
| 2 | 7.389056 | 1 | 1.111111 | 9 | 1,197 |
import Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho
import Mathlib.LinearAlgebra.Orientation
#align_import analysis.inner_product_space.orientation from "leanprover-community/mathlib"@"bd65478311e4dfd41f48bf38c7e3b02fb75d0163"
noncomputable section
variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
open FiniteDimensional
open scoped RealInnerProductSpace
namespace OrthonormalBasis
variable {ι : Type*} [Fintype ι] [DecidableEq ι] [ne : Nonempty ι] (e f : OrthonormalBasis ι ℝ E)
(x : Orientation ℝ E ι)
theorem det_to_matrix_orthonormalBasis_of_same_orientation
(h : e.toBasis.orientation = f.toBasis.orientation) : e.toBasis.det f = 1 := by
apply (e.det_to_matrix_orthonormalBasis_real f).resolve_right
have : 0 < e.toBasis.det f := by
rw [e.toBasis.orientation_eq_iff_det_pos] at h
simpa using h
linarith
#align orthonormal_basis.det_to_matrix_orthonormal_basis_of_same_orientation OrthonormalBasis.det_to_matrix_orthonormalBasis_of_same_orientation
theorem det_to_matrix_orthonormalBasis_of_opposite_orientation
(h : e.toBasis.orientation ≠ f.toBasis.orientation) : e.toBasis.det f = -1 := by
contrapose! h
simp [e.toBasis.orientation_eq_iff_det_pos,
(e.det_to_matrix_orthonormalBasis_real f).resolve_right h]
#align orthonormal_basis.det_to_matrix_orthonormal_basis_of_opposite_orientation OrthonormalBasis.det_to_matrix_orthonormalBasis_of_opposite_orientation
variable {e f}
theorem same_orientation_iff_det_eq_det :
e.toBasis.det = f.toBasis.det ↔ e.toBasis.orientation = f.toBasis.orientation := by
constructor
· intro h
dsimp [Basis.orientation]
congr
· intro h
rw [e.toBasis.det.eq_smul_basis_det f.toBasis]
simp [e.det_to_matrix_orthonormalBasis_of_same_orientation f h]
#align orthonormal_basis.same_orientation_iff_det_eq_det OrthonormalBasis.same_orientation_iff_det_eq_det
variable (e f)
theorem det_eq_neg_det_of_opposite_orientation (h : e.toBasis.orientation ≠ f.toBasis.orientation) :
e.toBasis.det = -f.toBasis.det := by
rw [e.toBasis.det.eq_smul_basis_det f.toBasis]
-- Porting note: added `neg_one_smul` with explicit type
simp [e.det_to_matrix_orthonormalBasis_of_opposite_orientation f h,
neg_one_smul ℝ (M := E [⋀^ι]→ₗ[ℝ] ℝ)]
#align orthonormal_basis.det_eq_neg_det_of_opposite_orientation OrthonormalBasis.det_eq_neg_det_of_opposite_orientation
section AdjustToOrientation
theorem orthonormal_adjustToOrientation : Orthonormal ℝ (e.toBasis.adjustToOrientation x) := by
apply e.orthonormal.orthonormal_of_forall_eq_or_eq_neg
simpa using e.toBasis.adjustToOrientation_apply_eq_or_eq_neg x
#align orthonormal_basis.orthonormal_adjust_to_orientation OrthonormalBasis.orthonormal_adjustToOrientation
def adjustToOrientation : OrthonormalBasis ι ℝ E :=
(e.toBasis.adjustToOrientation x).toOrthonormalBasis (e.orthonormal_adjustToOrientation x)
#align orthonormal_basis.adjust_to_orientation OrthonormalBasis.adjustToOrientation
theorem toBasis_adjustToOrientation :
(e.adjustToOrientation x).toBasis = e.toBasis.adjustToOrientation x :=
(e.toBasis.adjustToOrientation x).toBasis_toOrthonormalBasis _
#align orthonormal_basis.to_basis_adjust_to_orientation OrthonormalBasis.toBasis_adjustToOrientation
@[simp]
theorem orientation_adjustToOrientation : (e.adjustToOrientation x).toBasis.orientation = x := by
rw [e.toBasis_adjustToOrientation]
exact e.toBasis.orientation_adjustToOrientation x
#align orthonormal_basis.orientation_adjust_to_orientation OrthonormalBasis.orientation_adjustToOrientation
| Mathlib/Analysis/InnerProductSpace/Orientation.lean | 129 | 132 | theorem adjustToOrientation_apply_eq_or_eq_neg (i : ι) :
e.adjustToOrientation x i = e i ∨ e.adjustToOrientation x i = -e i := by |
simpa [← e.toBasis_adjustToOrientation] using
e.toBasis.adjustToOrientation_apply_eq_or_eq_neg x i
| 2 | 7.389056 | 1 | 1.111111 | 9 | 1,197 |
import Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho
import Mathlib.LinearAlgebra.Orientation
#align_import analysis.inner_product_space.orientation from "leanprover-community/mathlib"@"bd65478311e4dfd41f48bf38c7e3b02fb75d0163"
noncomputable section
variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
open FiniteDimensional
open scoped RealInnerProductSpace
namespace OrthonormalBasis
variable {ι : Type*} [Fintype ι] [DecidableEq ι] [ne : Nonempty ι] (e f : OrthonormalBasis ι ℝ E)
(x : Orientation ℝ E ι)
theorem det_to_matrix_orthonormalBasis_of_same_orientation
(h : e.toBasis.orientation = f.toBasis.orientation) : e.toBasis.det f = 1 := by
apply (e.det_to_matrix_orthonormalBasis_real f).resolve_right
have : 0 < e.toBasis.det f := by
rw [e.toBasis.orientation_eq_iff_det_pos] at h
simpa using h
linarith
#align orthonormal_basis.det_to_matrix_orthonormal_basis_of_same_orientation OrthonormalBasis.det_to_matrix_orthonormalBasis_of_same_orientation
theorem det_to_matrix_orthonormalBasis_of_opposite_orientation
(h : e.toBasis.orientation ≠ f.toBasis.orientation) : e.toBasis.det f = -1 := by
contrapose! h
simp [e.toBasis.orientation_eq_iff_det_pos,
(e.det_to_matrix_orthonormalBasis_real f).resolve_right h]
#align orthonormal_basis.det_to_matrix_orthonormal_basis_of_opposite_orientation OrthonormalBasis.det_to_matrix_orthonormalBasis_of_opposite_orientation
variable {e f}
theorem same_orientation_iff_det_eq_det :
e.toBasis.det = f.toBasis.det ↔ e.toBasis.orientation = f.toBasis.orientation := by
constructor
· intro h
dsimp [Basis.orientation]
congr
· intro h
rw [e.toBasis.det.eq_smul_basis_det f.toBasis]
simp [e.det_to_matrix_orthonormalBasis_of_same_orientation f h]
#align orthonormal_basis.same_orientation_iff_det_eq_det OrthonormalBasis.same_orientation_iff_det_eq_det
variable (e f)
theorem det_eq_neg_det_of_opposite_orientation (h : e.toBasis.orientation ≠ f.toBasis.orientation) :
e.toBasis.det = -f.toBasis.det := by
rw [e.toBasis.det.eq_smul_basis_det f.toBasis]
-- Porting note: added `neg_one_smul` with explicit type
simp [e.det_to_matrix_orthonormalBasis_of_opposite_orientation f h,
neg_one_smul ℝ (M := E [⋀^ι]→ₗ[ℝ] ℝ)]
#align orthonormal_basis.det_eq_neg_det_of_opposite_orientation OrthonormalBasis.det_eq_neg_det_of_opposite_orientation
section AdjustToOrientation
theorem orthonormal_adjustToOrientation : Orthonormal ℝ (e.toBasis.adjustToOrientation x) := by
apply e.orthonormal.orthonormal_of_forall_eq_or_eq_neg
simpa using e.toBasis.adjustToOrientation_apply_eq_or_eq_neg x
#align orthonormal_basis.orthonormal_adjust_to_orientation OrthonormalBasis.orthonormal_adjustToOrientation
def adjustToOrientation : OrthonormalBasis ι ℝ E :=
(e.toBasis.adjustToOrientation x).toOrthonormalBasis (e.orthonormal_adjustToOrientation x)
#align orthonormal_basis.adjust_to_orientation OrthonormalBasis.adjustToOrientation
theorem toBasis_adjustToOrientation :
(e.adjustToOrientation x).toBasis = e.toBasis.adjustToOrientation x :=
(e.toBasis.adjustToOrientation x).toBasis_toOrthonormalBasis _
#align orthonormal_basis.to_basis_adjust_to_orientation OrthonormalBasis.toBasis_adjustToOrientation
@[simp]
theorem orientation_adjustToOrientation : (e.adjustToOrientation x).toBasis.orientation = x := by
rw [e.toBasis_adjustToOrientation]
exact e.toBasis.orientation_adjustToOrientation x
#align orthonormal_basis.orientation_adjust_to_orientation OrthonormalBasis.orientation_adjustToOrientation
theorem adjustToOrientation_apply_eq_or_eq_neg (i : ι) :
e.adjustToOrientation x i = e i ∨ e.adjustToOrientation x i = -e i := by
simpa [← e.toBasis_adjustToOrientation] using
e.toBasis.adjustToOrientation_apply_eq_or_eq_neg x i
#align orthonormal_basis.adjust_to_orientation_apply_eq_or_eq_neg OrthonormalBasis.adjustToOrientation_apply_eq_or_eq_neg
| Mathlib/Analysis/InnerProductSpace/Orientation.lean | 135 | 138 | theorem det_adjustToOrientation :
(e.adjustToOrientation x).toBasis.det = e.toBasis.det ∨
(e.adjustToOrientation x).toBasis.det = -e.toBasis.det := by |
simpa using e.toBasis.det_adjustToOrientation x
| 1 | 2.718282 | 0 | 1.111111 | 9 | 1,197 |
import Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho
import Mathlib.LinearAlgebra.Orientation
#align_import analysis.inner_product_space.orientation from "leanprover-community/mathlib"@"bd65478311e4dfd41f48bf38c7e3b02fb75d0163"
noncomputable section
variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
open FiniteDimensional
open scoped RealInnerProductSpace
namespace OrthonormalBasis
variable {ι : Type*} [Fintype ι] [DecidableEq ι] [ne : Nonempty ι] (e f : OrthonormalBasis ι ℝ E)
(x : Orientation ℝ E ι)
theorem det_to_matrix_orthonormalBasis_of_same_orientation
(h : e.toBasis.orientation = f.toBasis.orientation) : e.toBasis.det f = 1 := by
apply (e.det_to_matrix_orthonormalBasis_real f).resolve_right
have : 0 < e.toBasis.det f := by
rw [e.toBasis.orientation_eq_iff_det_pos] at h
simpa using h
linarith
#align orthonormal_basis.det_to_matrix_orthonormal_basis_of_same_orientation OrthonormalBasis.det_to_matrix_orthonormalBasis_of_same_orientation
theorem det_to_matrix_orthonormalBasis_of_opposite_orientation
(h : e.toBasis.orientation ≠ f.toBasis.orientation) : e.toBasis.det f = -1 := by
contrapose! h
simp [e.toBasis.orientation_eq_iff_det_pos,
(e.det_to_matrix_orthonormalBasis_real f).resolve_right h]
#align orthonormal_basis.det_to_matrix_orthonormal_basis_of_opposite_orientation OrthonormalBasis.det_to_matrix_orthonormalBasis_of_opposite_orientation
variable {e f}
theorem same_orientation_iff_det_eq_det :
e.toBasis.det = f.toBasis.det ↔ e.toBasis.orientation = f.toBasis.orientation := by
constructor
· intro h
dsimp [Basis.orientation]
congr
· intro h
rw [e.toBasis.det.eq_smul_basis_det f.toBasis]
simp [e.det_to_matrix_orthonormalBasis_of_same_orientation f h]
#align orthonormal_basis.same_orientation_iff_det_eq_det OrthonormalBasis.same_orientation_iff_det_eq_det
variable (e f)
theorem det_eq_neg_det_of_opposite_orientation (h : e.toBasis.orientation ≠ f.toBasis.orientation) :
e.toBasis.det = -f.toBasis.det := by
rw [e.toBasis.det.eq_smul_basis_det f.toBasis]
-- Porting note: added `neg_one_smul` with explicit type
simp [e.det_to_matrix_orthonormalBasis_of_opposite_orientation f h,
neg_one_smul ℝ (M := E [⋀^ι]→ₗ[ℝ] ℝ)]
#align orthonormal_basis.det_eq_neg_det_of_opposite_orientation OrthonormalBasis.det_eq_neg_det_of_opposite_orientation
section AdjustToOrientation
theorem orthonormal_adjustToOrientation : Orthonormal ℝ (e.toBasis.adjustToOrientation x) := by
apply e.orthonormal.orthonormal_of_forall_eq_or_eq_neg
simpa using e.toBasis.adjustToOrientation_apply_eq_or_eq_neg x
#align orthonormal_basis.orthonormal_adjust_to_orientation OrthonormalBasis.orthonormal_adjustToOrientation
def adjustToOrientation : OrthonormalBasis ι ℝ E :=
(e.toBasis.adjustToOrientation x).toOrthonormalBasis (e.orthonormal_adjustToOrientation x)
#align orthonormal_basis.adjust_to_orientation OrthonormalBasis.adjustToOrientation
theorem toBasis_adjustToOrientation :
(e.adjustToOrientation x).toBasis = e.toBasis.adjustToOrientation x :=
(e.toBasis.adjustToOrientation x).toBasis_toOrthonormalBasis _
#align orthonormal_basis.to_basis_adjust_to_orientation OrthonormalBasis.toBasis_adjustToOrientation
@[simp]
theorem orientation_adjustToOrientation : (e.adjustToOrientation x).toBasis.orientation = x := by
rw [e.toBasis_adjustToOrientation]
exact e.toBasis.orientation_adjustToOrientation x
#align orthonormal_basis.orientation_adjust_to_orientation OrthonormalBasis.orientation_adjustToOrientation
theorem adjustToOrientation_apply_eq_or_eq_neg (i : ι) :
e.adjustToOrientation x i = e i ∨ e.adjustToOrientation x i = -e i := by
simpa [← e.toBasis_adjustToOrientation] using
e.toBasis.adjustToOrientation_apply_eq_or_eq_neg x i
#align orthonormal_basis.adjust_to_orientation_apply_eq_or_eq_neg OrthonormalBasis.adjustToOrientation_apply_eq_or_eq_neg
theorem det_adjustToOrientation :
(e.adjustToOrientation x).toBasis.det = e.toBasis.det ∨
(e.adjustToOrientation x).toBasis.det = -e.toBasis.det := by
simpa using e.toBasis.det_adjustToOrientation x
#align orthonormal_basis.det_adjust_to_orientation OrthonormalBasis.det_adjustToOrientation
| Mathlib/Analysis/InnerProductSpace/Orientation.lean | 141 | 143 | theorem abs_det_adjustToOrientation (v : ι → E) :
|(e.adjustToOrientation x).toBasis.det v| = |e.toBasis.det v| := by |
simp [toBasis_adjustToOrientation]
| 1 | 2.718282 | 0 | 1.111111 | 9 | 1,197 |
import Mathlib.GroupTheory.Coxeter.Length
import Mathlib.Data.ZMod.Parity
namespace CoxeterSystem
open List Matrix Function
variable {B : Type*}
variable {W : Type*} [Group W]
variable {M : CoxeterMatrix B} (cs : CoxeterSystem M W)
local prefix:100 "s" => cs.simple
local prefix:100 "π" => cs.wordProd
local prefix:100 "ℓ" => cs.length
def IsReflection (t : W) : Prop := ∃ w i, t = w * s i * w⁻¹
| Mathlib/GroupTheory/Coxeter/Inversion.lean | 61 | 61 | theorem isReflection_simple (i : B) : cs.IsReflection (s i) := by | use 1, i; simp
| 1 | 2.718282 | 0 | 1.111111 | 9 | 1,198 |
import Mathlib.GroupTheory.Coxeter.Length
import Mathlib.Data.ZMod.Parity
namespace CoxeterSystem
open List Matrix Function
variable {B : Type*}
variable {W : Type*} [Group W]
variable {M : CoxeterMatrix B} (cs : CoxeterSystem M W)
local prefix:100 "s" => cs.simple
local prefix:100 "π" => cs.wordProd
local prefix:100 "ℓ" => cs.length
def IsReflection (t : W) : Prop := ∃ w i, t = w * s i * w⁻¹
theorem isReflection_simple (i : B) : cs.IsReflection (s i) := by use 1, i; simp
namespace IsReflection
variable {cs}
variable {t : W} (ht : cs.IsReflection t)
| Mathlib/GroupTheory/Coxeter/Inversion.lean | 68 | 70 | theorem pow_two : t ^ 2 = 1 := by |
rcases ht with ⟨w, i, rfl⟩
simp
| 2 | 7.389056 | 1 | 1.111111 | 9 | 1,198 |
import Mathlib.GroupTheory.Coxeter.Length
import Mathlib.Data.ZMod.Parity
namespace CoxeterSystem
open List Matrix Function
variable {B : Type*}
variable {W : Type*} [Group W]
variable {M : CoxeterMatrix B} (cs : CoxeterSystem M W)
local prefix:100 "s" => cs.simple
local prefix:100 "π" => cs.wordProd
local prefix:100 "ℓ" => cs.length
def IsReflection (t : W) : Prop := ∃ w i, t = w * s i * w⁻¹
theorem isReflection_simple (i : B) : cs.IsReflection (s i) := by use 1, i; simp
namespace IsReflection
variable {cs}
variable {t : W} (ht : cs.IsReflection t)
theorem pow_two : t ^ 2 = 1 := by
rcases ht with ⟨w, i, rfl⟩
simp
| Mathlib/GroupTheory/Coxeter/Inversion.lean | 72 | 74 | theorem mul_self : t * t = 1 := by |
rcases ht with ⟨w, i, rfl⟩
simp
| 2 | 7.389056 | 1 | 1.111111 | 9 | 1,198 |
import Mathlib.GroupTheory.Coxeter.Length
import Mathlib.Data.ZMod.Parity
namespace CoxeterSystem
open List Matrix Function
variable {B : Type*}
variable {W : Type*} [Group W]
variable {M : CoxeterMatrix B} (cs : CoxeterSystem M W)
local prefix:100 "s" => cs.simple
local prefix:100 "π" => cs.wordProd
local prefix:100 "ℓ" => cs.length
def IsReflection (t : W) : Prop := ∃ w i, t = w * s i * w⁻¹
theorem isReflection_simple (i : B) : cs.IsReflection (s i) := by use 1, i; simp
namespace IsReflection
variable {cs}
variable {t : W} (ht : cs.IsReflection t)
theorem pow_two : t ^ 2 = 1 := by
rcases ht with ⟨w, i, rfl⟩
simp
theorem mul_self : t * t = 1 := by
rcases ht with ⟨w, i, rfl⟩
simp
| Mathlib/GroupTheory/Coxeter/Inversion.lean | 76 | 78 | theorem inv : t⁻¹ = t := by |
rcases ht with ⟨w, i, rfl⟩
simp [mul_assoc]
| 2 | 7.389056 | 1 | 1.111111 | 9 | 1,198 |
import Mathlib.GroupTheory.Coxeter.Length
import Mathlib.Data.ZMod.Parity
namespace CoxeterSystem
open List Matrix Function
variable {B : Type*}
variable {W : Type*} [Group W]
variable {M : CoxeterMatrix B} (cs : CoxeterSystem M W)
local prefix:100 "s" => cs.simple
local prefix:100 "π" => cs.wordProd
local prefix:100 "ℓ" => cs.length
def IsReflection (t : W) : Prop := ∃ w i, t = w * s i * w⁻¹
theorem isReflection_simple (i : B) : cs.IsReflection (s i) := by use 1, i; simp
namespace IsReflection
variable {cs}
variable {t : W} (ht : cs.IsReflection t)
theorem pow_two : t ^ 2 = 1 := by
rcases ht with ⟨w, i, rfl⟩
simp
theorem mul_self : t * t = 1 := by
rcases ht with ⟨w, i, rfl⟩
simp
theorem inv : t⁻¹ = t := by
rcases ht with ⟨w, i, rfl⟩
simp [mul_assoc]
| Mathlib/GroupTheory/Coxeter/Inversion.lean | 80 | 80 | theorem isReflection_inv : cs.IsReflection t⁻¹ := by | rwa [ht.inv]
| 1 | 2.718282 | 0 | 1.111111 | 9 | 1,198 |
import Mathlib.GroupTheory.Coxeter.Length
import Mathlib.Data.ZMod.Parity
namespace CoxeterSystem
open List Matrix Function
variable {B : Type*}
variable {W : Type*} [Group W]
variable {M : CoxeterMatrix B} (cs : CoxeterSystem M W)
local prefix:100 "s" => cs.simple
local prefix:100 "π" => cs.wordProd
local prefix:100 "ℓ" => cs.length
def IsReflection (t : W) : Prop := ∃ w i, t = w * s i * w⁻¹
theorem isReflection_simple (i : B) : cs.IsReflection (s i) := by use 1, i; simp
namespace IsReflection
variable {cs}
variable {t : W} (ht : cs.IsReflection t)
theorem pow_two : t ^ 2 = 1 := by
rcases ht with ⟨w, i, rfl⟩
simp
theorem mul_self : t * t = 1 := by
rcases ht with ⟨w, i, rfl⟩
simp
theorem inv : t⁻¹ = t := by
rcases ht with ⟨w, i, rfl⟩
simp [mul_assoc]
theorem isReflection_inv : cs.IsReflection t⁻¹ := by rwa [ht.inv]
| Mathlib/GroupTheory/Coxeter/Inversion.lean | 82 | 86 | theorem odd_length : Odd (ℓ t) := by |
suffices cs.lengthParity t = Multiplicative.ofAdd 1 by
simpa [lengthParity_eq_ofAdd_length, ZMod.eq_one_iff_odd]
rcases ht with ⟨w, i, rfl⟩
simp [lengthParity_simple]
| 4 | 54.59815 | 2 | 1.111111 | 9 | 1,198 |
import Mathlib.GroupTheory.Coxeter.Length
import Mathlib.Data.ZMod.Parity
namespace CoxeterSystem
open List Matrix Function
variable {B : Type*}
variable {W : Type*} [Group W]
variable {M : CoxeterMatrix B} (cs : CoxeterSystem M W)
local prefix:100 "s" => cs.simple
local prefix:100 "π" => cs.wordProd
local prefix:100 "ℓ" => cs.length
def IsReflection (t : W) : Prop := ∃ w i, t = w * s i * w⁻¹
theorem isReflection_simple (i : B) : cs.IsReflection (s i) := by use 1, i; simp
namespace IsReflection
variable {cs}
variable {t : W} (ht : cs.IsReflection t)
theorem pow_two : t ^ 2 = 1 := by
rcases ht with ⟨w, i, rfl⟩
simp
theorem mul_self : t * t = 1 := by
rcases ht with ⟨w, i, rfl⟩
simp
theorem inv : t⁻¹ = t := by
rcases ht with ⟨w, i, rfl⟩
simp [mul_assoc]
theorem isReflection_inv : cs.IsReflection t⁻¹ := by rwa [ht.inv]
theorem odd_length : Odd (ℓ t) := by
suffices cs.lengthParity t = Multiplicative.ofAdd 1 by
simpa [lengthParity_eq_ofAdd_length, ZMod.eq_one_iff_odd]
rcases ht with ⟨w, i, rfl⟩
simp [lengthParity_simple]
| Mathlib/GroupTheory/Coxeter/Inversion.lean | 88 | 93 | theorem length_mul_left_ne (w : W) : ℓ (w * t) ≠ ℓ w := by |
suffices cs.lengthParity (w * t) ≠ cs.lengthParity w by
contrapose! this
simp only [lengthParity_eq_ofAdd_length, this]
rcases ht with ⟨w, i, rfl⟩
simp [lengthParity_simple]
| 5 | 148.413159 | 2 | 1.111111 | 9 | 1,198 |
import Mathlib.GroupTheory.Coxeter.Length
import Mathlib.Data.ZMod.Parity
namespace CoxeterSystem
open List Matrix Function
variable {B : Type*}
variable {W : Type*} [Group W]
variable {M : CoxeterMatrix B} (cs : CoxeterSystem M W)
local prefix:100 "s" => cs.simple
local prefix:100 "π" => cs.wordProd
local prefix:100 "ℓ" => cs.length
def IsReflection (t : W) : Prop := ∃ w i, t = w * s i * w⁻¹
theorem isReflection_simple (i : B) : cs.IsReflection (s i) := by use 1, i; simp
namespace IsReflection
variable {cs}
variable {t : W} (ht : cs.IsReflection t)
theorem pow_two : t ^ 2 = 1 := by
rcases ht with ⟨w, i, rfl⟩
simp
theorem mul_self : t * t = 1 := by
rcases ht with ⟨w, i, rfl⟩
simp
theorem inv : t⁻¹ = t := by
rcases ht with ⟨w, i, rfl⟩
simp [mul_assoc]
theorem isReflection_inv : cs.IsReflection t⁻¹ := by rwa [ht.inv]
theorem odd_length : Odd (ℓ t) := by
suffices cs.lengthParity t = Multiplicative.ofAdd 1 by
simpa [lengthParity_eq_ofAdd_length, ZMod.eq_one_iff_odd]
rcases ht with ⟨w, i, rfl⟩
simp [lengthParity_simple]
theorem length_mul_left_ne (w : W) : ℓ (w * t) ≠ ℓ w := by
suffices cs.lengthParity (w * t) ≠ cs.lengthParity w by
contrapose! this
simp only [lengthParity_eq_ofAdd_length, this]
rcases ht with ⟨w, i, rfl⟩
simp [lengthParity_simple]
| Mathlib/GroupTheory/Coxeter/Inversion.lean | 95 | 100 | theorem length_mul_right_ne (w : W) : ℓ (t * w) ≠ ℓ w := by |
suffices cs.lengthParity (t * w) ≠ cs.lengthParity w by
contrapose! this
simp only [lengthParity_eq_ofAdd_length, this]
rcases ht with ⟨w, i, rfl⟩
simp [lengthParity_simple]
| 5 | 148.413159 | 2 | 1.111111 | 9 | 1,198 |
import Mathlib.GroupTheory.Coxeter.Length
import Mathlib.Data.ZMod.Parity
namespace CoxeterSystem
open List Matrix Function
variable {B : Type*}
variable {W : Type*} [Group W]
variable {M : CoxeterMatrix B} (cs : CoxeterSystem M W)
local prefix:100 "s" => cs.simple
local prefix:100 "π" => cs.wordProd
local prefix:100 "ℓ" => cs.length
def IsReflection (t : W) : Prop := ∃ w i, t = w * s i * w⁻¹
theorem isReflection_simple (i : B) : cs.IsReflection (s i) := by use 1, i; simp
namespace IsReflection
variable {cs}
variable {t : W} (ht : cs.IsReflection t)
theorem pow_two : t ^ 2 = 1 := by
rcases ht with ⟨w, i, rfl⟩
simp
theorem mul_self : t * t = 1 := by
rcases ht with ⟨w, i, rfl⟩
simp
theorem inv : t⁻¹ = t := by
rcases ht with ⟨w, i, rfl⟩
simp [mul_assoc]
theorem isReflection_inv : cs.IsReflection t⁻¹ := by rwa [ht.inv]
theorem odd_length : Odd (ℓ t) := by
suffices cs.lengthParity t = Multiplicative.ofAdd 1 by
simpa [lengthParity_eq_ofAdd_length, ZMod.eq_one_iff_odd]
rcases ht with ⟨w, i, rfl⟩
simp [lengthParity_simple]
theorem length_mul_left_ne (w : W) : ℓ (w * t) ≠ ℓ w := by
suffices cs.lengthParity (w * t) ≠ cs.lengthParity w by
contrapose! this
simp only [lengthParity_eq_ofAdd_length, this]
rcases ht with ⟨w, i, rfl⟩
simp [lengthParity_simple]
theorem length_mul_right_ne (w : W) : ℓ (t * w) ≠ ℓ w := by
suffices cs.lengthParity (t * w) ≠ cs.lengthParity w by
contrapose! this
simp only [lengthParity_eq_ofAdd_length, this]
rcases ht with ⟨w, i, rfl⟩
simp [lengthParity_simple]
| Mathlib/GroupTheory/Coxeter/Inversion.lean | 102 | 105 | theorem conj (w : W) : cs.IsReflection (w * t * w⁻¹) := by |
obtain ⟨u, i, rfl⟩ := ht
use w * u, i
group
| 3 | 20.085537 | 1 | 1.111111 | 9 | 1,198 |
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Combinatorics.SimpleGraph.Maps
open Finset
namespace SimpleGraph
variable {V : Type*} [DecidableEq V] (G : SimpleGraph V) (s t : V)
namespace Iso
variable {G} {W : Type*} {G' : SimpleGraph W} (f : G ≃g G')
| Mathlib/Combinatorics/SimpleGraph/Operations.lean | 35 | 39 | theorem card_edgeFinset_eq [Fintype G.edgeSet] [Fintype G'.edgeSet] :
G.edgeFinset.card = G'.edgeFinset.card := by |
apply Finset.card_eq_of_equiv
simp only [Set.mem_toFinset]
exact f.mapEdgeSet
| 3 | 20.085537 | 1 | 1.111111 | 9 | 1,199 |
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Combinatorics.SimpleGraph.Maps
open Finset
namespace SimpleGraph
variable {V : Type*} [DecidableEq V] (G : SimpleGraph V) (s t : V)
section ReplaceVertex
def replaceVertex : SimpleGraph V where
Adj v w := if v = t then if w = t then False else G.Adj s w
else if w = t then G.Adj v s else G.Adj v w
symm v w := by dsimp only; split_ifs <;> simp [adj_comm]
lemma not_adj_replaceVertex_same : ¬(G.replaceVertex s t).Adj s t := by simp [replaceVertex]
@[simp] lemma replaceVertex_self : G.replaceVertex s s = G := by
ext; unfold replaceVertex; aesop (add simp or_iff_not_imp_left)
variable {t}
lemma adj_replaceVertex_iff_of_ne_left {w : V} (hw : w ≠ t) :
(G.replaceVertex s t).Adj s w ↔ G.Adj s w := by simp [replaceVertex, hw]
lemma adj_replaceVertex_iff_of_ne_right {w : V} (hw : w ≠ t) :
(G.replaceVertex s t).Adj t w ↔ G.Adj s w := by simp [replaceVertex, hw]
lemma adj_replaceVertex_iff_of_ne {v w : V} (hv : v ≠ t) (hw : w ≠ t) :
(G.replaceVertex s t).Adj v w ↔ G.Adj v w := by simp [replaceVertex, hv, hw]
variable {s}
| Mathlib/Combinatorics/SimpleGraph/Operations.lean | 76 | 80 | theorem edgeSet_replaceVertex_of_not_adj (hn : ¬G.Adj s t) : (G.replaceVertex s t).edgeSet =
G.edgeSet \ G.incidenceSet t ∪ (s(·, t)) '' (G.neighborSet s) := by |
ext e; refine e.inductionOn ?_
simp only [replaceVertex, mem_edgeSet, Set.mem_union, Set.mem_diff, mk'_mem_incidenceSet_iff]
intros; split_ifs; exacts [by simp_all, by aesop, by rw [adj_comm]; aesop, by aesop]
| 3 | 20.085537 | 1 | 1.111111 | 9 | 1,199 |
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Combinatorics.SimpleGraph.Maps
open Finset
namespace SimpleGraph
variable {V : Type*} [DecidableEq V] (G : SimpleGraph V) (s t : V)
section ReplaceVertex
def replaceVertex : SimpleGraph V where
Adj v w := if v = t then if w = t then False else G.Adj s w
else if w = t then G.Adj v s else G.Adj v w
symm v w := by dsimp only; split_ifs <;> simp [adj_comm]
lemma not_adj_replaceVertex_same : ¬(G.replaceVertex s t).Adj s t := by simp [replaceVertex]
@[simp] lemma replaceVertex_self : G.replaceVertex s s = G := by
ext; unfold replaceVertex; aesop (add simp or_iff_not_imp_left)
variable {t}
lemma adj_replaceVertex_iff_of_ne_left {w : V} (hw : w ≠ t) :
(G.replaceVertex s t).Adj s w ↔ G.Adj s w := by simp [replaceVertex, hw]
lemma adj_replaceVertex_iff_of_ne_right {w : V} (hw : w ≠ t) :
(G.replaceVertex s t).Adj t w ↔ G.Adj s w := by simp [replaceVertex, hw]
lemma adj_replaceVertex_iff_of_ne {v w : V} (hv : v ≠ t) (hw : w ≠ t) :
(G.replaceVertex s t).Adj v w ↔ G.Adj v w := by simp [replaceVertex, hv, hw]
variable {s}
theorem edgeSet_replaceVertex_of_not_adj (hn : ¬G.Adj s t) : (G.replaceVertex s t).edgeSet =
G.edgeSet \ G.incidenceSet t ∪ (s(·, t)) '' (G.neighborSet s) := by
ext e; refine e.inductionOn ?_
simp only [replaceVertex, mem_edgeSet, Set.mem_union, Set.mem_diff, mk'_mem_incidenceSet_iff]
intros; split_ifs; exacts [by simp_all, by aesop, by rw [adj_comm]; aesop, by aesop]
| Mathlib/Combinatorics/SimpleGraph/Operations.lean | 82 | 86 | theorem edgeSet_replaceVertex_of_adj (ha : G.Adj s t) : (G.replaceVertex s t).edgeSet =
(G.edgeSet \ G.incidenceSet t ∪ (s(·, t)) '' (G.neighborSet s)) \ {s(t, t)} := by |
ext e; refine e.inductionOn ?_
simp only [replaceVertex, mem_edgeSet, Set.mem_union, Set.mem_diff, mk'_mem_incidenceSet_iff]
intros; split_ifs; exacts [by simp_all, by aesop, by rw [adj_comm]; aesop, by aesop]
| 3 | 20.085537 | 1 | 1.111111 | 9 | 1,199 |
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Combinatorics.SimpleGraph.Maps
open Finset
namespace SimpleGraph
variable {V : Type*} [DecidableEq V] (G : SimpleGraph V) (s t : V)
section ReplaceVertex
def replaceVertex : SimpleGraph V where
Adj v w := if v = t then if w = t then False else G.Adj s w
else if w = t then G.Adj v s else G.Adj v w
symm v w := by dsimp only; split_ifs <;> simp [adj_comm]
lemma not_adj_replaceVertex_same : ¬(G.replaceVertex s t).Adj s t := by simp [replaceVertex]
@[simp] lemma replaceVertex_self : G.replaceVertex s s = G := by
ext; unfold replaceVertex; aesop (add simp or_iff_not_imp_left)
variable {t}
lemma adj_replaceVertex_iff_of_ne_left {w : V} (hw : w ≠ t) :
(G.replaceVertex s t).Adj s w ↔ G.Adj s w := by simp [replaceVertex, hw]
lemma adj_replaceVertex_iff_of_ne_right {w : V} (hw : w ≠ t) :
(G.replaceVertex s t).Adj t w ↔ G.Adj s w := by simp [replaceVertex, hw]
lemma adj_replaceVertex_iff_of_ne {v w : V} (hv : v ≠ t) (hw : w ≠ t) :
(G.replaceVertex s t).Adj v w ↔ G.Adj v w := by simp [replaceVertex, hv, hw]
variable {s}
theorem edgeSet_replaceVertex_of_not_adj (hn : ¬G.Adj s t) : (G.replaceVertex s t).edgeSet =
G.edgeSet \ G.incidenceSet t ∪ (s(·, t)) '' (G.neighborSet s) := by
ext e; refine e.inductionOn ?_
simp only [replaceVertex, mem_edgeSet, Set.mem_union, Set.mem_diff, mk'_mem_incidenceSet_iff]
intros; split_ifs; exacts [by simp_all, by aesop, by rw [adj_comm]; aesop, by aesop]
theorem edgeSet_replaceVertex_of_adj (ha : G.Adj s t) : (G.replaceVertex s t).edgeSet =
(G.edgeSet \ G.incidenceSet t ∪ (s(·, t)) '' (G.neighborSet s)) \ {s(t, t)} := by
ext e; refine e.inductionOn ?_
simp only [replaceVertex, mem_edgeSet, Set.mem_union, Set.mem_diff, mk'_mem_incidenceSet_iff]
intros; split_ifs; exacts [by simp_all, by aesop, by rw [adj_comm]; aesop, by aesop]
variable [Fintype V] [DecidableRel G.Adj]
instance : DecidableRel (G.replaceVertex s t).Adj := by unfold replaceVertex; infer_instance
| Mathlib/Combinatorics/SimpleGraph/Operations.lean | 92 | 96 | theorem edgeFinset_replaceVertex_of_not_adj (hn : ¬G.Adj s t) : (G.replaceVertex s t).edgeFinset =
G.edgeFinset \ G.incidenceFinset t ∪ (G.neighborFinset s).image (s(·, t)) := by |
simp only [incidenceFinset, neighborFinset, ← Set.toFinset_diff, ← Set.toFinset_image,
← Set.toFinset_union]
exact Set.toFinset_congr (G.edgeSet_replaceVertex_of_not_adj hn)
| 3 | 20.085537 | 1 | 1.111111 | 9 | 1,199 |
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Combinatorics.SimpleGraph.Maps
open Finset
namespace SimpleGraph
variable {V : Type*} [DecidableEq V] (G : SimpleGraph V) (s t : V)
section ReplaceVertex
def replaceVertex : SimpleGraph V where
Adj v w := if v = t then if w = t then False else G.Adj s w
else if w = t then G.Adj v s else G.Adj v w
symm v w := by dsimp only; split_ifs <;> simp [adj_comm]
lemma not_adj_replaceVertex_same : ¬(G.replaceVertex s t).Adj s t := by simp [replaceVertex]
@[simp] lemma replaceVertex_self : G.replaceVertex s s = G := by
ext; unfold replaceVertex; aesop (add simp or_iff_not_imp_left)
variable {t}
lemma adj_replaceVertex_iff_of_ne_left {w : V} (hw : w ≠ t) :
(G.replaceVertex s t).Adj s w ↔ G.Adj s w := by simp [replaceVertex, hw]
lemma adj_replaceVertex_iff_of_ne_right {w : V} (hw : w ≠ t) :
(G.replaceVertex s t).Adj t w ↔ G.Adj s w := by simp [replaceVertex, hw]
lemma adj_replaceVertex_iff_of_ne {v w : V} (hv : v ≠ t) (hw : w ≠ t) :
(G.replaceVertex s t).Adj v w ↔ G.Adj v w := by simp [replaceVertex, hv, hw]
variable {s}
theorem edgeSet_replaceVertex_of_not_adj (hn : ¬G.Adj s t) : (G.replaceVertex s t).edgeSet =
G.edgeSet \ G.incidenceSet t ∪ (s(·, t)) '' (G.neighborSet s) := by
ext e; refine e.inductionOn ?_
simp only [replaceVertex, mem_edgeSet, Set.mem_union, Set.mem_diff, mk'_mem_incidenceSet_iff]
intros; split_ifs; exacts [by simp_all, by aesop, by rw [adj_comm]; aesop, by aesop]
theorem edgeSet_replaceVertex_of_adj (ha : G.Adj s t) : (G.replaceVertex s t).edgeSet =
(G.edgeSet \ G.incidenceSet t ∪ (s(·, t)) '' (G.neighborSet s)) \ {s(t, t)} := by
ext e; refine e.inductionOn ?_
simp only [replaceVertex, mem_edgeSet, Set.mem_union, Set.mem_diff, mk'_mem_incidenceSet_iff]
intros; split_ifs; exacts [by simp_all, by aesop, by rw [adj_comm]; aesop, by aesop]
variable [Fintype V] [DecidableRel G.Adj]
instance : DecidableRel (G.replaceVertex s t).Adj := by unfold replaceVertex; infer_instance
theorem edgeFinset_replaceVertex_of_not_adj (hn : ¬G.Adj s t) : (G.replaceVertex s t).edgeFinset =
G.edgeFinset \ G.incidenceFinset t ∪ (G.neighborFinset s).image (s(·, t)) := by
simp only [incidenceFinset, neighborFinset, ← Set.toFinset_diff, ← Set.toFinset_image,
← Set.toFinset_union]
exact Set.toFinset_congr (G.edgeSet_replaceVertex_of_not_adj hn)
| Mathlib/Combinatorics/SimpleGraph/Operations.lean | 98 | 102 | theorem edgeFinset_replaceVertex_of_adj (ha : G.Adj s t) : (G.replaceVertex s t).edgeFinset =
(G.edgeFinset \ G.incidenceFinset t ∪ (G.neighborFinset s).image (s(·, t))) \ {s(t, t)} := by |
simp only [incidenceFinset, neighborFinset, ← Set.toFinset_diff, ← Set.toFinset_image,
← Set.toFinset_union, ← Set.toFinset_singleton]
exact Set.toFinset_congr (G.edgeSet_replaceVertex_of_adj ha)
| 3 | 20.085537 | 1 | 1.111111 | 9 | 1,199 |
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Combinatorics.SimpleGraph.Maps
open Finset
namespace SimpleGraph
variable {V : Type*} [DecidableEq V] (G : SimpleGraph V) (s t : V)
section ReplaceVertex
def replaceVertex : SimpleGraph V where
Adj v w := if v = t then if w = t then False else G.Adj s w
else if w = t then G.Adj v s else G.Adj v w
symm v w := by dsimp only; split_ifs <;> simp [adj_comm]
lemma not_adj_replaceVertex_same : ¬(G.replaceVertex s t).Adj s t := by simp [replaceVertex]
@[simp] lemma replaceVertex_self : G.replaceVertex s s = G := by
ext; unfold replaceVertex; aesop (add simp or_iff_not_imp_left)
variable {t}
lemma adj_replaceVertex_iff_of_ne_left {w : V} (hw : w ≠ t) :
(G.replaceVertex s t).Adj s w ↔ G.Adj s w := by simp [replaceVertex, hw]
lemma adj_replaceVertex_iff_of_ne_right {w : V} (hw : w ≠ t) :
(G.replaceVertex s t).Adj t w ↔ G.Adj s w := by simp [replaceVertex, hw]
lemma adj_replaceVertex_iff_of_ne {v w : V} (hv : v ≠ t) (hw : w ≠ t) :
(G.replaceVertex s t).Adj v w ↔ G.Adj v w := by simp [replaceVertex, hv, hw]
variable {s}
theorem edgeSet_replaceVertex_of_not_adj (hn : ¬G.Adj s t) : (G.replaceVertex s t).edgeSet =
G.edgeSet \ G.incidenceSet t ∪ (s(·, t)) '' (G.neighborSet s) := by
ext e; refine e.inductionOn ?_
simp only [replaceVertex, mem_edgeSet, Set.mem_union, Set.mem_diff, mk'_mem_incidenceSet_iff]
intros; split_ifs; exacts [by simp_all, by aesop, by rw [adj_comm]; aesop, by aesop]
theorem edgeSet_replaceVertex_of_adj (ha : G.Adj s t) : (G.replaceVertex s t).edgeSet =
(G.edgeSet \ G.incidenceSet t ∪ (s(·, t)) '' (G.neighborSet s)) \ {s(t, t)} := by
ext e; refine e.inductionOn ?_
simp only [replaceVertex, mem_edgeSet, Set.mem_union, Set.mem_diff, mk'_mem_incidenceSet_iff]
intros; split_ifs; exacts [by simp_all, by aesop, by rw [adj_comm]; aesop, by aesop]
variable [Fintype V] [DecidableRel G.Adj]
instance : DecidableRel (G.replaceVertex s t).Adj := by unfold replaceVertex; infer_instance
theorem edgeFinset_replaceVertex_of_not_adj (hn : ¬G.Adj s t) : (G.replaceVertex s t).edgeFinset =
G.edgeFinset \ G.incidenceFinset t ∪ (G.neighborFinset s).image (s(·, t)) := by
simp only [incidenceFinset, neighborFinset, ← Set.toFinset_diff, ← Set.toFinset_image,
← Set.toFinset_union]
exact Set.toFinset_congr (G.edgeSet_replaceVertex_of_not_adj hn)
theorem edgeFinset_replaceVertex_of_adj (ha : G.Adj s t) : (G.replaceVertex s t).edgeFinset =
(G.edgeFinset \ G.incidenceFinset t ∪ (G.neighborFinset s).image (s(·, t))) \ {s(t, t)} := by
simp only [incidenceFinset, neighborFinset, ← Set.toFinset_diff, ← Set.toFinset_image,
← Set.toFinset_union, ← Set.toFinset_singleton]
exact Set.toFinset_congr (G.edgeSet_replaceVertex_of_adj ha)
lemma disjoint_sdiff_neighborFinset_image :
Disjoint (G.edgeFinset \ G.incidenceFinset t) ((G.neighborFinset s).image (s(·, t))) := by
rw [disjoint_iff_ne]
intro e he
have : t ∉ e := by
rw [mem_sdiff, mem_incidenceFinset] at he
obtain ⟨_, h⟩ := he
contrapose! h
simp_all [incidenceSet]
aesop
| Mathlib/Combinatorics/SimpleGraph/Operations.lean | 115 | 124 | theorem card_edgeFinset_replaceVertex_of_not_adj (hn : ¬G.Adj s t) :
(G.replaceVertex s t).edgeFinset.card = G.edgeFinset.card + G.degree s - G.degree t := by |
have inc : G.incidenceFinset t ⊆ G.edgeFinset := by simp [incidenceFinset, incidenceSet_subset]
rw [G.edgeFinset_replaceVertex_of_not_adj hn,
card_union_of_disjoint G.disjoint_sdiff_neighborFinset_image, card_sdiff inc,
← Nat.sub_add_comm <| card_le_card inc, card_incidenceFinset_eq_degree]
congr 2
rw [card_image_of_injective, card_neighborFinset_eq_degree]
unfold Function.Injective
aesop
| 8 | 2,980.957987 | 2 | 1.111111 | 9 | 1,199 |
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Combinatorics.SimpleGraph.Maps
open Finset
namespace SimpleGraph
variable {V : Type*} [DecidableEq V] (G : SimpleGraph V) (s t : V)
section ReplaceVertex
def replaceVertex : SimpleGraph V where
Adj v w := if v = t then if w = t then False else G.Adj s w
else if w = t then G.Adj v s else G.Adj v w
symm v w := by dsimp only; split_ifs <;> simp [adj_comm]
lemma not_adj_replaceVertex_same : ¬(G.replaceVertex s t).Adj s t := by simp [replaceVertex]
@[simp] lemma replaceVertex_self : G.replaceVertex s s = G := by
ext; unfold replaceVertex; aesop (add simp or_iff_not_imp_left)
variable {t}
lemma adj_replaceVertex_iff_of_ne_left {w : V} (hw : w ≠ t) :
(G.replaceVertex s t).Adj s w ↔ G.Adj s w := by simp [replaceVertex, hw]
lemma adj_replaceVertex_iff_of_ne_right {w : V} (hw : w ≠ t) :
(G.replaceVertex s t).Adj t w ↔ G.Adj s w := by simp [replaceVertex, hw]
lemma adj_replaceVertex_iff_of_ne {v w : V} (hv : v ≠ t) (hw : w ≠ t) :
(G.replaceVertex s t).Adj v w ↔ G.Adj v w := by simp [replaceVertex, hv, hw]
variable {s}
theorem edgeSet_replaceVertex_of_not_adj (hn : ¬G.Adj s t) : (G.replaceVertex s t).edgeSet =
G.edgeSet \ G.incidenceSet t ∪ (s(·, t)) '' (G.neighborSet s) := by
ext e; refine e.inductionOn ?_
simp only [replaceVertex, mem_edgeSet, Set.mem_union, Set.mem_diff, mk'_mem_incidenceSet_iff]
intros; split_ifs; exacts [by simp_all, by aesop, by rw [adj_comm]; aesop, by aesop]
theorem edgeSet_replaceVertex_of_adj (ha : G.Adj s t) : (G.replaceVertex s t).edgeSet =
(G.edgeSet \ G.incidenceSet t ∪ (s(·, t)) '' (G.neighborSet s)) \ {s(t, t)} := by
ext e; refine e.inductionOn ?_
simp only [replaceVertex, mem_edgeSet, Set.mem_union, Set.mem_diff, mk'_mem_incidenceSet_iff]
intros; split_ifs; exacts [by simp_all, by aesop, by rw [adj_comm]; aesop, by aesop]
variable [Fintype V] [DecidableRel G.Adj]
instance : DecidableRel (G.replaceVertex s t).Adj := by unfold replaceVertex; infer_instance
theorem edgeFinset_replaceVertex_of_not_adj (hn : ¬G.Adj s t) : (G.replaceVertex s t).edgeFinset =
G.edgeFinset \ G.incidenceFinset t ∪ (G.neighborFinset s).image (s(·, t)) := by
simp only [incidenceFinset, neighborFinset, ← Set.toFinset_diff, ← Set.toFinset_image,
← Set.toFinset_union]
exact Set.toFinset_congr (G.edgeSet_replaceVertex_of_not_adj hn)
theorem edgeFinset_replaceVertex_of_adj (ha : G.Adj s t) : (G.replaceVertex s t).edgeFinset =
(G.edgeFinset \ G.incidenceFinset t ∪ (G.neighborFinset s).image (s(·, t))) \ {s(t, t)} := by
simp only [incidenceFinset, neighborFinset, ← Set.toFinset_diff, ← Set.toFinset_image,
← Set.toFinset_union, ← Set.toFinset_singleton]
exact Set.toFinset_congr (G.edgeSet_replaceVertex_of_adj ha)
lemma disjoint_sdiff_neighborFinset_image :
Disjoint (G.edgeFinset \ G.incidenceFinset t) ((G.neighborFinset s).image (s(·, t))) := by
rw [disjoint_iff_ne]
intro e he
have : t ∉ e := by
rw [mem_sdiff, mem_incidenceFinset] at he
obtain ⟨_, h⟩ := he
contrapose! h
simp_all [incidenceSet]
aesop
theorem card_edgeFinset_replaceVertex_of_not_adj (hn : ¬G.Adj s t) :
(G.replaceVertex s t).edgeFinset.card = G.edgeFinset.card + G.degree s - G.degree t := by
have inc : G.incidenceFinset t ⊆ G.edgeFinset := by simp [incidenceFinset, incidenceSet_subset]
rw [G.edgeFinset_replaceVertex_of_not_adj hn,
card_union_of_disjoint G.disjoint_sdiff_neighborFinset_image, card_sdiff inc,
← Nat.sub_add_comm <| card_le_card inc, card_incidenceFinset_eq_degree]
congr 2
rw [card_image_of_injective, card_neighborFinset_eq_degree]
unfold Function.Injective
aesop
| Mathlib/Combinatorics/SimpleGraph/Operations.lean | 126 | 135 | theorem card_edgeFinset_replaceVertex_of_adj (ha : G.Adj s t) :
(G.replaceVertex s t).edgeFinset.card = G.edgeFinset.card + G.degree s - G.degree t - 1 := by |
have inc : G.incidenceFinset t ⊆ G.edgeFinset := by simp [incidenceFinset, incidenceSet_subset]
rw [G.edgeFinset_replaceVertex_of_adj ha, card_sdiff (by simp [ha]),
card_union_of_disjoint G.disjoint_sdiff_neighborFinset_image, card_sdiff inc,
← Nat.sub_add_comm <| card_le_card inc, card_incidenceFinset_eq_degree]
congr 2
rw [card_image_of_injective, card_neighborFinset_eq_degree]
unfold Function.Injective
aesop
| 8 | 2,980.957987 | 2 | 1.111111 | 9 | 1,199 |
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Combinatorics.SimpleGraph.Maps
open Finset
namespace SimpleGraph
variable {V : Type*} [DecidableEq V] (G : SimpleGraph V) (s t : V)
section AddEdge
def edge : SimpleGraph V := fromEdgeSet {s(s, t)}
lemma edge_adj (v w : V) : (edge s t).Adj v w ↔ (v = s ∧ w = t ∨ v = t ∧ w = s) ∧ v ≠ w := by
rw [edge, fromEdgeSet_adj, Set.mem_singleton_iff, Sym2.eq_iff]
instance : DecidableRel (edge s t).Adj := fun _ _ ↦ by
rw [edge_adj]; infer_instance
lemma edge_self_eq_bot : edge s s = ⊥ := by
ext; rw [edge_adj]; aesop
@[simp]
lemma sup_edge_self : G ⊔ edge s s = G := by
rw [edge_self_eq_bot, sup_of_le_left bot_le]
variable {s t}
lemma edge_edgeSet_of_ne (h : s ≠ t) : (edge s t).edgeSet = {s(s, t)} := by
rwa [edge, edgeSet_fromEdgeSet, sdiff_eq_left, Set.disjoint_singleton_left, Set.mem_setOf_eq,
Sym2.isDiag_iff_proj_eq]
lemma sup_edge_of_adj (h : G.Adj s t) : G ⊔ edge s t = G := by
rwa [sup_eq_left, ← edgeSet_subset_edgeSet, edge_edgeSet_of_ne h.ne, Set.singleton_subset_iff,
mem_edgeSet]
variable [Fintype V] [DecidableRel G.Adj]
instance : Fintype (edge s t).edgeSet := by rw [edge]; infer_instance
| Mathlib/Combinatorics/SimpleGraph/Operations.lean | 171 | 175 | theorem edgeFinset_sup_edge [Fintype (edgeSet (G ⊔ edge s t))] (hn : ¬G.Adj s t) (h : s ≠ t) :
(G ⊔ edge s t).edgeFinset = G.edgeFinset.cons s(s, t) (by simp_all) := by |
letI := Classical.decEq V
rw [edgeFinset_sup, cons_eq_insert, insert_eq, union_comm]
simp_rw [edgeFinset, edge_edgeSet_of_ne h]; rfl
| 3 | 20.085537 | 1 | 1.111111 | 9 | 1,199 |
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Combinatorics.SimpleGraph.Maps
open Finset
namespace SimpleGraph
variable {V : Type*} [DecidableEq V] (G : SimpleGraph V) (s t : V)
section AddEdge
def edge : SimpleGraph V := fromEdgeSet {s(s, t)}
lemma edge_adj (v w : V) : (edge s t).Adj v w ↔ (v = s ∧ w = t ∨ v = t ∧ w = s) ∧ v ≠ w := by
rw [edge, fromEdgeSet_adj, Set.mem_singleton_iff, Sym2.eq_iff]
instance : DecidableRel (edge s t).Adj := fun _ _ ↦ by
rw [edge_adj]; infer_instance
lemma edge_self_eq_bot : edge s s = ⊥ := by
ext; rw [edge_adj]; aesop
@[simp]
lemma sup_edge_self : G ⊔ edge s s = G := by
rw [edge_self_eq_bot, sup_of_le_left bot_le]
variable {s t}
lemma edge_edgeSet_of_ne (h : s ≠ t) : (edge s t).edgeSet = {s(s, t)} := by
rwa [edge, edgeSet_fromEdgeSet, sdiff_eq_left, Set.disjoint_singleton_left, Set.mem_setOf_eq,
Sym2.isDiag_iff_proj_eq]
lemma sup_edge_of_adj (h : G.Adj s t) : G ⊔ edge s t = G := by
rwa [sup_eq_left, ← edgeSet_subset_edgeSet, edge_edgeSet_of_ne h.ne, Set.singleton_subset_iff,
mem_edgeSet]
variable [Fintype V] [DecidableRel G.Adj]
instance : Fintype (edge s t).edgeSet := by rw [edge]; infer_instance
theorem edgeFinset_sup_edge [Fintype (edgeSet (G ⊔ edge s t))] (hn : ¬G.Adj s t) (h : s ≠ t) :
(G ⊔ edge s t).edgeFinset = G.edgeFinset.cons s(s, t) (by simp_all) := by
letI := Classical.decEq V
rw [edgeFinset_sup, cons_eq_insert, insert_eq, union_comm]
simp_rw [edgeFinset, edge_edgeSet_of_ne h]; rfl
| Mathlib/Combinatorics/SimpleGraph/Operations.lean | 177 | 179 | theorem card_edgeFinset_sup_edge [Fintype (edgeSet (G ⊔ edge s t))] (hn : ¬G.Adj s t) (h : s ≠ t) :
(G ⊔ edge s t).edgeFinset.card = G.edgeFinset.card + 1 := by |
rw [G.edgeFinset_sup_edge hn h, card_cons]
| 1 | 2.718282 | 0 | 1.111111 | 9 | 1,199 |
import Mathlib.RingTheory.PowerSeries.Trunc
import Mathlib.RingTheory.PowerSeries.Inverse
import Mathlib.RingTheory.Derivation.Basic
namespace PowerSeries
open Polynomial Derivation Nat
section CommutativeSemiring
variable {R} [CommSemiring R]
noncomputable def derivativeFun (f : R⟦X⟧) : R⟦X⟧ := mk fun n ↦ coeff R (n + 1) f * (n + 1)
| Mathlib/RingTheory/PowerSeries/Derivative.lean | 41 | 43 | theorem coeff_derivativeFun (f : R⟦X⟧) (n : ℕ) :
coeff R n f.derivativeFun = coeff R (n + 1) f * (n + 1) := by |
rw [derivativeFun, coeff_mk]
| 1 | 2.718282 | 0 | 1.111111 | 9 | 1,200 |
import Mathlib.RingTheory.PowerSeries.Trunc
import Mathlib.RingTheory.PowerSeries.Inverse
import Mathlib.RingTheory.Derivation.Basic
namespace PowerSeries
open Polynomial Derivation Nat
section CommutativeSemiring
variable {R} [CommSemiring R]
noncomputable def derivativeFun (f : R⟦X⟧) : R⟦X⟧ := mk fun n ↦ coeff R (n + 1) f * (n + 1)
theorem coeff_derivativeFun (f : R⟦X⟧) (n : ℕ) :
coeff R n f.derivativeFun = coeff R (n + 1) f * (n + 1) := by
rw [derivativeFun, coeff_mk]
| Mathlib/RingTheory/PowerSeries/Derivative.lean | 45 | 47 | theorem derivativeFun_coe (f : R[X]) : (f : R⟦X⟧).derivativeFun = derivative f := by |
ext
rw [coeff_derivativeFun, coeff_coe, coeff_coe, coeff_derivative]
| 2 | 7.389056 | 1 | 1.111111 | 9 | 1,200 |
import Mathlib.RingTheory.PowerSeries.Trunc
import Mathlib.RingTheory.PowerSeries.Inverse
import Mathlib.RingTheory.Derivation.Basic
namespace PowerSeries
open Polynomial Derivation Nat
section CommutativeSemiring
variable {R} [CommSemiring R]
noncomputable def derivativeFun (f : R⟦X⟧) : R⟦X⟧ := mk fun n ↦ coeff R (n + 1) f * (n + 1)
theorem coeff_derivativeFun (f : R⟦X⟧) (n : ℕ) :
coeff R n f.derivativeFun = coeff R (n + 1) f * (n + 1) := by
rw [derivativeFun, coeff_mk]
theorem derivativeFun_coe (f : R[X]) : (f : R⟦X⟧).derivativeFun = derivative f := by
ext
rw [coeff_derivativeFun, coeff_coe, coeff_coe, coeff_derivative]
| Mathlib/RingTheory/PowerSeries/Derivative.lean | 49 | 53 | theorem derivativeFun_add (f g : R⟦X⟧) :
derivativeFun (f + g) = derivativeFun f + derivativeFun g := by |
ext
rw [coeff_derivativeFun, map_add, map_add, coeff_derivativeFun,
coeff_derivativeFun, add_mul]
| 3 | 20.085537 | 1 | 1.111111 | 9 | 1,200 |
import Mathlib.RingTheory.PowerSeries.Trunc
import Mathlib.RingTheory.PowerSeries.Inverse
import Mathlib.RingTheory.Derivation.Basic
namespace PowerSeries
open Polynomial Derivation Nat
section CommutativeSemiring
variable {R} [CommSemiring R]
noncomputable def derivativeFun (f : R⟦X⟧) : R⟦X⟧ := mk fun n ↦ coeff R (n + 1) f * (n + 1)
theorem coeff_derivativeFun (f : R⟦X⟧) (n : ℕ) :
coeff R n f.derivativeFun = coeff R (n + 1) f * (n + 1) := by
rw [derivativeFun, coeff_mk]
theorem derivativeFun_coe (f : R[X]) : (f : R⟦X⟧).derivativeFun = derivative f := by
ext
rw [coeff_derivativeFun, coeff_coe, coeff_coe, coeff_derivative]
theorem derivativeFun_add (f g : R⟦X⟧) :
derivativeFun (f + g) = derivativeFun f + derivativeFun g := by
ext
rw [coeff_derivativeFun, map_add, map_add, coeff_derivativeFun,
coeff_derivativeFun, add_mul]
| Mathlib/RingTheory/PowerSeries/Derivative.lean | 55 | 58 | theorem derivativeFun_C (r : R) : derivativeFun (C R r) = 0 := by |
ext n
-- Note that `map_zero` didn't get picked up, apparently due to a missing `FunLike.coe`
rw [coeff_derivativeFun, coeff_succ_C, zero_mul, (coeff R n).map_zero]
| 3 | 20.085537 | 1 | 1.111111 | 9 | 1,200 |
import Mathlib.RingTheory.PowerSeries.Trunc
import Mathlib.RingTheory.PowerSeries.Inverse
import Mathlib.RingTheory.Derivation.Basic
namespace PowerSeries
open Polynomial Derivation Nat
section CommutativeSemiring
variable {R} [CommSemiring R]
noncomputable def derivativeFun (f : R⟦X⟧) : R⟦X⟧ := mk fun n ↦ coeff R (n + 1) f * (n + 1)
theorem coeff_derivativeFun (f : R⟦X⟧) (n : ℕ) :
coeff R n f.derivativeFun = coeff R (n + 1) f * (n + 1) := by
rw [derivativeFun, coeff_mk]
theorem derivativeFun_coe (f : R[X]) : (f : R⟦X⟧).derivativeFun = derivative f := by
ext
rw [coeff_derivativeFun, coeff_coe, coeff_coe, coeff_derivative]
theorem derivativeFun_add (f g : R⟦X⟧) :
derivativeFun (f + g) = derivativeFun f + derivativeFun g := by
ext
rw [coeff_derivativeFun, map_add, map_add, coeff_derivativeFun,
coeff_derivativeFun, add_mul]
theorem derivativeFun_C (r : R) : derivativeFun (C R r) = 0 := by
ext n
-- Note that `map_zero` didn't get picked up, apparently due to a missing `FunLike.coe`
rw [coeff_derivativeFun, coeff_succ_C, zero_mul, (coeff R n).map_zero]
| Mathlib/RingTheory/PowerSeries/Derivative.lean | 60 | 68 | theorem trunc_derivativeFun (f : R⟦X⟧) (n : ℕ) :
trunc n f.derivativeFun = derivative (trunc (n + 1) f) := by |
ext d
rw [coeff_trunc]
split_ifs with h
· have : d + 1 < n + 1 := succ_lt_succ_iff.2 h
rw [coeff_derivativeFun, coeff_derivative, coeff_trunc, if_pos this]
· have : ¬d + 1 < n + 1 := by rwa [succ_lt_succ_iff]
rw [coeff_derivative, coeff_trunc, if_neg this, zero_mul]
| 7 | 1,096.633158 | 2 | 1.111111 | 9 | 1,200 |
import Mathlib.RingTheory.PowerSeries.Trunc
import Mathlib.RingTheory.PowerSeries.Inverse
import Mathlib.RingTheory.Derivation.Basic
namespace PowerSeries
open Polynomial Derivation Nat
section CommutativeSemiring
variable {R} [CommSemiring R]
noncomputable def derivativeFun (f : R⟦X⟧) : R⟦X⟧ := mk fun n ↦ coeff R (n + 1) f * (n + 1)
theorem coeff_derivativeFun (f : R⟦X⟧) (n : ℕ) :
coeff R n f.derivativeFun = coeff R (n + 1) f * (n + 1) := by
rw [derivativeFun, coeff_mk]
theorem derivativeFun_coe (f : R[X]) : (f : R⟦X⟧).derivativeFun = derivative f := by
ext
rw [coeff_derivativeFun, coeff_coe, coeff_coe, coeff_derivative]
theorem derivativeFun_add (f g : R⟦X⟧) :
derivativeFun (f + g) = derivativeFun f + derivativeFun g := by
ext
rw [coeff_derivativeFun, map_add, map_add, coeff_derivativeFun,
coeff_derivativeFun, add_mul]
theorem derivativeFun_C (r : R) : derivativeFun (C R r) = 0 := by
ext n
-- Note that `map_zero` didn't get picked up, apparently due to a missing `FunLike.coe`
rw [coeff_derivativeFun, coeff_succ_C, zero_mul, (coeff R n).map_zero]
theorem trunc_derivativeFun (f : R⟦X⟧) (n : ℕ) :
trunc n f.derivativeFun = derivative (trunc (n + 1) f) := by
ext d
rw [coeff_trunc]
split_ifs with h
· have : d + 1 < n + 1 := succ_lt_succ_iff.2 h
rw [coeff_derivativeFun, coeff_derivative, coeff_trunc, if_pos this]
· have : ¬d + 1 < n + 1 := by rwa [succ_lt_succ_iff]
rw [coeff_derivative, coeff_trunc, if_neg this, zero_mul]
--A special case of `derivativeFun_mul`, used in its proof.
private theorem derivativeFun_coe_mul_coe (f g : R[X]) : derivativeFun (f * g : R⟦X⟧) =
f * derivative g + g * derivative f := by
rw [← coe_mul, derivativeFun_coe, derivative_mul,
add_comm, mul_comm _ g, ← coe_mul, ← coe_mul, Polynomial.coe_add]
| Mathlib/RingTheory/PowerSeries/Derivative.lean | 77 | 85 | theorem derivativeFun_mul (f g : R⟦X⟧) :
derivativeFun (f * g) = f • g.derivativeFun + g • f.derivativeFun := by |
ext n
have h₁ : n < n + 1 := lt_succ_self n
have h₂ : n < n + 1 + 1 := Nat.lt_add_right _ h₁
rw [coeff_derivativeFun, map_add, coeff_mul_eq_coeff_trunc_mul_trunc _ _ (lt_succ_self _),
smul_eq_mul, smul_eq_mul, coeff_mul_eq_coeff_trunc_mul_trunc₂ g f.derivativeFun h₂ h₁,
coeff_mul_eq_coeff_trunc_mul_trunc₂ f g.derivativeFun h₂ h₁, trunc_derivativeFun,
trunc_derivativeFun, ← map_add, ← derivativeFun_coe_mul_coe, coeff_derivativeFun]
| 7 | 1,096.633158 | 2 | 1.111111 | 9 | 1,200 |
import Mathlib.RingTheory.PowerSeries.Trunc
import Mathlib.RingTheory.PowerSeries.Inverse
import Mathlib.RingTheory.Derivation.Basic
namespace PowerSeries
open Polynomial Derivation Nat
section CommutativeSemiring
variable {R} [CommSemiring R]
noncomputable def derivativeFun (f : R⟦X⟧) : R⟦X⟧ := mk fun n ↦ coeff R (n + 1) f * (n + 1)
theorem coeff_derivativeFun (f : R⟦X⟧) (n : ℕ) :
coeff R n f.derivativeFun = coeff R (n + 1) f * (n + 1) := by
rw [derivativeFun, coeff_mk]
theorem derivativeFun_coe (f : R[X]) : (f : R⟦X⟧).derivativeFun = derivative f := by
ext
rw [coeff_derivativeFun, coeff_coe, coeff_coe, coeff_derivative]
theorem derivativeFun_add (f g : R⟦X⟧) :
derivativeFun (f + g) = derivativeFun f + derivativeFun g := by
ext
rw [coeff_derivativeFun, map_add, map_add, coeff_derivativeFun,
coeff_derivativeFun, add_mul]
theorem derivativeFun_C (r : R) : derivativeFun (C R r) = 0 := by
ext n
-- Note that `map_zero` didn't get picked up, apparently due to a missing `FunLike.coe`
rw [coeff_derivativeFun, coeff_succ_C, zero_mul, (coeff R n).map_zero]
theorem trunc_derivativeFun (f : R⟦X⟧) (n : ℕ) :
trunc n f.derivativeFun = derivative (trunc (n + 1) f) := by
ext d
rw [coeff_trunc]
split_ifs with h
· have : d + 1 < n + 1 := succ_lt_succ_iff.2 h
rw [coeff_derivativeFun, coeff_derivative, coeff_trunc, if_pos this]
· have : ¬d + 1 < n + 1 := by rwa [succ_lt_succ_iff]
rw [coeff_derivative, coeff_trunc, if_neg this, zero_mul]
--A special case of `derivativeFun_mul`, used in its proof.
private theorem derivativeFun_coe_mul_coe (f g : R[X]) : derivativeFun (f * g : R⟦X⟧) =
f * derivative g + g * derivative f := by
rw [← coe_mul, derivativeFun_coe, derivative_mul,
add_comm, mul_comm _ g, ← coe_mul, ← coe_mul, Polynomial.coe_add]
theorem derivativeFun_mul (f g : R⟦X⟧) :
derivativeFun (f * g) = f • g.derivativeFun + g • f.derivativeFun := by
ext n
have h₁ : n < n + 1 := lt_succ_self n
have h₂ : n < n + 1 + 1 := Nat.lt_add_right _ h₁
rw [coeff_derivativeFun, map_add, coeff_mul_eq_coeff_trunc_mul_trunc _ _ (lt_succ_self _),
smul_eq_mul, smul_eq_mul, coeff_mul_eq_coeff_trunc_mul_trunc₂ g f.derivativeFun h₂ h₁,
coeff_mul_eq_coeff_trunc_mul_trunc₂ f g.derivativeFun h₂ h₁, trunc_derivativeFun,
trunc_derivativeFun, ← map_add, ← derivativeFun_coe_mul_coe, coeff_derivativeFun]
| Mathlib/RingTheory/PowerSeries/Derivative.lean | 87 | 88 | theorem derivativeFun_one : derivativeFun (1 : R⟦X⟧) = 0 := by |
rw [← map_one (C R), derivativeFun_C (1 : R)]
| 1 | 2.718282 | 0 | 1.111111 | 9 | 1,200 |
import Mathlib.RingTheory.PowerSeries.Trunc
import Mathlib.RingTheory.PowerSeries.Inverse
import Mathlib.RingTheory.Derivation.Basic
namespace PowerSeries
open Polynomial Derivation Nat
section CommutativeSemiring
variable {R} [CommSemiring R]
noncomputable def derivativeFun (f : R⟦X⟧) : R⟦X⟧ := mk fun n ↦ coeff R (n + 1) f * (n + 1)
theorem coeff_derivativeFun (f : R⟦X⟧) (n : ℕ) :
coeff R n f.derivativeFun = coeff R (n + 1) f * (n + 1) := by
rw [derivativeFun, coeff_mk]
theorem derivativeFun_coe (f : R[X]) : (f : R⟦X⟧).derivativeFun = derivative f := by
ext
rw [coeff_derivativeFun, coeff_coe, coeff_coe, coeff_derivative]
theorem derivativeFun_add (f g : R⟦X⟧) :
derivativeFun (f + g) = derivativeFun f + derivativeFun g := by
ext
rw [coeff_derivativeFun, map_add, map_add, coeff_derivativeFun,
coeff_derivativeFun, add_mul]
theorem derivativeFun_C (r : R) : derivativeFun (C R r) = 0 := by
ext n
-- Note that `map_zero` didn't get picked up, apparently due to a missing `FunLike.coe`
rw [coeff_derivativeFun, coeff_succ_C, zero_mul, (coeff R n).map_zero]
theorem trunc_derivativeFun (f : R⟦X⟧) (n : ℕ) :
trunc n f.derivativeFun = derivative (trunc (n + 1) f) := by
ext d
rw [coeff_trunc]
split_ifs with h
· have : d + 1 < n + 1 := succ_lt_succ_iff.2 h
rw [coeff_derivativeFun, coeff_derivative, coeff_trunc, if_pos this]
· have : ¬d + 1 < n + 1 := by rwa [succ_lt_succ_iff]
rw [coeff_derivative, coeff_trunc, if_neg this, zero_mul]
--A special case of `derivativeFun_mul`, used in its proof.
private theorem derivativeFun_coe_mul_coe (f g : R[X]) : derivativeFun (f * g : R⟦X⟧) =
f * derivative g + g * derivative f := by
rw [← coe_mul, derivativeFun_coe, derivative_mul,
add_comm, mul_comm _ g, ← coe_mul, ← coe_mul, Polynomial.coe_add]
theorem derivativeFun_mul (f g : R⟦X⟧) :
derivativeFun (f * g) = f • g.derivativeFun + g • f.derivativeFun := by
ext n
have h₁ : n < n + 1 := lt_succ_self n
have h₂ : n < n + 1 + 1 := Nat.lt_add_right _ h₁
rw [coeff_derivativeFun, map_add, coeff_mul_eq_coeff_trunc_mul_trunc _ _ (lt_succ_self _),
smul_eq_mul, smul_eq_mul, coeff_mul_eq_coeff_trunc_mul_trunc₂ g f.derivativeFun h₂ h₁,
coeff_mul_eq_coeff_trunc_mul_trunc₂ f g.derivativeFun h₂ h₁, trunc_derivativeFun,
trunc_derivativeFun, ← map_add, ← derivativeFun_coe_mul_coe, coeff_derivativeFun]
theorem derivativeFun_one : derivativeFun (1 : R⟦X⟧) = 0 := by
rw [← map_one (C R), derivativeFun_C (1 : R)]
| Mathlib/RingTheory/PowerSeries/Derivative.lean | 90 | 92 | theorem derivativeFun_smul (r : R) (f : R⟦X⟧) : derivativeFun (r • f) = r • derivativeFun f := by |
rw [smul_eq_C_mul, smul_eq_C_mul, derivativeFun_mul, derivativeFun_C, smul_zero, add_zero,
smul_eq_mul]
| 2 | 7.389056 | 1 | 1.111111 | 9 | 1,200 |
import Mathlib.RingTheory.PowerSeries.Trunc
import Mathlib.RingTheory.PowerSeries.Inverse
import Mathlib.RingTheory.Derivation.Basic
namespace PowerSeries
open Polynomial Derivation Nat
section CommutativeSemiring
variable {R} [CommSemiring R]
noncomputable def derivativeFun (f : R⟦X⟧) : R⟦X⟧ := mk fun n ↦ coeff R (n + 1) f * (n + 1)
theorem coeff_derivativeFun (f : R⟦X⟧) (n : ℕ) :
coeff R n f.derivativeFun = coeff R (n + 1) f * (n + 1) := by
rw [derivativeFun, coeff_mk]
theorem derivativeFun_coe (f : R[X]) : (f : R⟦X⟧).derivativeFun = derivative f := by
ext
rw [coeff_derivativeFun, coeff_coe, coeff_coe, coeff_derivative]
theorem derivativeFun_add (f g : R⟦X⟧) :
derivativeFun (f + g) = derivativeFun f + derivativeFun g := by
ext
rw [coeff_derivativeFun, map_add, map_add, coeff_derivativeFun,
coeff_derivativeFun, add_mul]
theorem derivativeFun_C (r : R) : derivativeFun (C R r) = 0 := by
ext n
-- Note that `map_zero` didn't get picked up, apparently due to a missing `FunLike.coe`
rw [coeff_derivativeFun, coeff_succ_C, zero_mul, (coeff R n).map_zero]
theorem trunc_derivativeFun (f : R⟦X⟧) (n : ℕ) :
trunc n f.derivativeFun = derivative (trunc (n + 1) f) := by
ext d
rw [coeff_trunc]
split_ifs with h
· have : d + 1 < n + 1 := succ_lt_succ_iff.2 h
rw [coeff_derivativeFun, coeff_derivative, coeff_trunc, if_pos this]
· have : ¬d + 1 < n + 1 := by rwa [succ_lt_succ_iff]
rw [coeff_derivative, coeff_trunc, if_neg this, zero_mul]
--A special case of `derivativeFun_mul`, used in its proof.
private theorem derivativeFun_coe_mul_coe (f g : R[X]) : derivativeFun (f * g : R⟦X⟧) =
f * derivative g + g * derivative f := by
rw [← coe_mul, derivativeFun_coe, derivative_mul,
add_comm, mul_comm _ g, ← coe_mul, ← coe_mul, Polynomial.coe_add]
theorem derivativeFun_mul (f g : R⟦X⟧) :
derivativeFun (f * g) = f • g.derivativeFun + g • f.derivativeFun := by
ext n
have h₁ : n < n + 1 := lt_succ_self n
have h₂ : n < n + 1 + 1 := Nat.lt_add_right _ h₁
rw [coeff_derivativeFun, map_add, coeff_mul_eq_coeff_trunc_mul_trunc _ _ (lt_succ_self _),
smul_eq_mul, smul_eq_mul, coeff_mul_eq_coeff_trunc_mul_trunc₂ g f.derivativeFun h₂ h₁,
coeff_mul_eq_coeff_trunc_mul_trunc₂ f g.derivativeFun h₂ h₁, trunc_derivativeFun,
trunc_derivativeFun, ← map_add, ← derivativeFun_coe_mul_coe, coeff_derivativeFun]
theorem derivativeFun_one : derivativeFun (1 : R⟦X⟧) = 0 := by
rw [← map_one (C R), derivativeFun_C (1 : R)]
theorem derivativeFun_smul (r : R) (f : R⟦X⟧) : derivativeFun (r • f) = r • derivativeFun f := by
rw [smul_eq_C_mul, smul_eq_C_mul, derivativeFun_mul, derivativeFun_C, smul_zero, add_zero,
smul_eq_mul]
variable (R)
noncomputable def derivative : Derivation R R⟦X⟧ R⟦X⟧ where
toFun := derivativeFun
map_add' := derivativeFun_add
map_smul' := derivativeFun_smul
map_one_eq_zero' := derivativeFun_one
leibniz' := derivativeFun_mul
scoped notation "d⁄dX" => derivative
variable {R}
@[simp] theorem derivative_C (r : R) : d⁄dX R (C R r) = 0 := derivativeFun_C r
theorem coeff_derivative (f : R⟦X⟧) (n : ℕ) :
coeff R n (d⁄dX R f) = coeff R (n + 1) f * (n + 1) := coeff_derivativeFun f n
theorem derivative_coe (f : R[X]) : d⁄dX R f = Polynomial.derivative f := derivativeFun_coe f
@[simp] theorem derivative_X : d⁄dX R (X : R⟦X⟧) = 1 := by
ext
rw [coeff_derivative, coeff_one, coeff_X, boole_mul]
simp_rw [add_left_eq_self]
split_ifs with h
· rw [h, cast_zero, zero_add]
· rfl
theorem trunc_derivative (f : R⟦X⟧) (n : ℕ) :
trunc n (d⁄dX R f) = Polynomial.derivative (trunc (n + 1) f) :=
trunc_derivativeFun ..
| Mathlib/RingTheory/PowerSeries/Derivative.lean | 127 | 133 | theorem trunc_derivative' (f : R⟦X⟧) (n : ℕ) :
trunc (n-1) (d⁄dX R f) = Polynomial.derivative (trunc n f) := by |
cases n with
| zero =>
simp
| succ n =>
rw [succ_sub_one, trunc_derivative]
| 5 | 148.413159 | 2 | 1.111111 | 9 | 1,200 |
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.LinearAlgebra.Matrix.Block
#align_import analysis.inner_product_space.gram_schmidt_ortho from "leanprover-community/mathlib"@"1a4df69ca1a9a0e5e26bfe12e2b92814216016d0"
open Finset Submodule FiniteDimensional
variable (𝕜 : Type*) {E : Type*} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E]
variable {ι : Type*} [LinearOrder ι] [LocallyFiniteOrderBot ι] [IsWellOrder ι (· < ·)]
attribute [local instance] IsWellOrder.toHasWellFounded
local notation "⟪" x ", " y "⟫" => @inner 𝕜 _ _ x y
noncomputable def gramSchmidt [IsWellOrder ι (· < ·)] (f : ι → E) (n : ι) : E :=
f n - ∑ i : Iio n, orthogonalProjection (𝕜 ∙ gramSchmidt f i) (f n)
termination_by n
decreasing_by exact mem_Iio.1 i.2
#align gram_schmidt gramSchmidt
| Mathlib/Analysis/InnerProductSpace/GramSchmidtOrtho.lean | 58 | 60 | theorem gramSchmidt_def (f : ι → E) (n : ι) :
gramSchmidt 𝕜 f n = f n - ∑ i ∈ Iio n, orthogonalProjection (𝕜 ∙ gramSchmidt 𝕜 f i) (f n) := by |
rw [← sum_attach, attach_eq_univ, gramSchmidt]
| 1 | 2.718282 | 0 | 1.125 | 8 | 1,201 |
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.LinearAlgebra.Matrix.Block
#align_import analysis.inner_product_space.gram_schmidt_ortho from "leanprover-community/mathlib"@"1a4df69ca1a9a0e5e26bfe12e2b92814216016d0"
open Finset Submodule FiniteDimensional
variable (𝕜 : Type*) {E : Type*} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E]
variable {ι : Type*} [LinearOrder ι] [LocallyFiniteOrderBot ι] [IsWellOrder ι (· < ·)]
attribute [local instance] IsWellOrder.toHasWellFounded
local notation "⟪" x ", " y "⟫" => @inner 𝕜 _ _ x y
noncomputable def gramSchmidt [IsWellOrder ι (· < ·)] (f : ι → E) (n : ι) : E :=
f n - ∑ i : Iio n, orthogonalProjection (𝕜 ∙ gramSchmidt f i) (f n)
termination_by n
decreasing_by exact mem_Iio.1 i.2
#align gram_schmidt gramSchmidt
theorem gramSchmidt_def (f : ι → E) (n : ι) :
gramSchmidt 𝕜 f n = f n - ∑ i ∈ Iio n, orthogonalProjection (𝕜 ∙ gramSchmidt 𝕜 f i) (f n) := by
rw [← sum_attach, attach_eq_univ, gramSchmidt]
#align gram_schmidt_def gramSchmidt_def
| Mathlib/Analysis/InnerProductSpace/GramSchmidtOrtho.lean | 63 | 65 | theorem gramSchmidt_def' (f : ι → E) (n : ι) :
f n = gramSchmidt 𝕜 f n + ∑ i ∈ Iio n, orthogonalProjection (𝕜 ∙ gramSchmidt 𝕜 f i) (f n) := by |
rw [gramSchmidt_def, sub_add_cancel]
| 1 | 2.718282 | 0 | 1.125 | 8 | 1,201 |
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.LinearAlgebra.Matrix.Block
#align_import analysis.inner_product_space.gram_schmidt_ortho from "leanprover-community/mathlib"@"1a4df69ca1a9a0e5e26bfe12e2b92814216016d0"
open Finset Submodule FiniteDimensional
variable (𝕜 : Type*) {E : Type*} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E]
variable {ι : Type*} [LinearOrder ι] [LocallyFiniteOrderBot ι] [IsWellOrder ι (· < ·)]
attribute [local instance] IsWellOrder.toHasWellFounded
local notation "⟪" x ", " y "⟫" => @inner 𝕜 _ _ x y
noncomputable def gramSchmidt [IsWellOrder ι (· < ·)] (f : ι → E) (n : ι) : E :=
f n - ∑ i : Iio n, orthogonalProjection (𝕜 ∙ gramSchmidt f i) (f n)
termination_by n
decreasing_by exact mem_Iio.1 i.2
#align gram_schmidt gramSchmidt
theorem gramSchmidt_def (f : ι → E) (n : ι) :
gramSchmidt 𝕜 f n = f n - ∑ i ∈ Iio n, orthogonalProjection (𝕜 ∙ gramSchmidt 𝕜 f i) (f n) := by
rw [← sum_attach, attach_eq_univ, gramSchmidt]
#align gram_schmidt_def gramSchmidt_def
theorem gramSchmidt_def' (f : ι → E) (n : ι) :
f n = gramSchmidt 𝕜 f n + ∑ i ∈ Iio n, orthogonalProjection (𝕜 ∙ gramSchmidt 𝕜 f i) (f n) := by
rw [gramSchmidt_def, sub_add_cancel]
#align gram_schmidt_def' gramSchmidt_def'
| Mathlib/Analysis/InnerProductSpace/GramSchmidtOrtho.lean | 68 | 72 | theorem gramSchmidt_def'' (f : ι → E) (n : ι) :
f n = gramSchmidt 𝕜 f n + ∑ i ∈ Iio n,
(⟪gramSchmidt 𝕜 f i, f n⟫ / (‖gramSchmidt 𝕜 f i‖ : 𝕜) ^ 2) • gramSchmidt 𝕜 f i := by |
convert gramSchmidt_def' 𝕜 f n
rw [orthogonalProjection_singleton, RCLike.ofReal_pow]
| 2 | 7.389056 | 1 | 1.125 | 8 | 1,201 |
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