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 | rank int64 0 2.4k |
|---|---|---|---|---|---|---|
import Mathlib.Algebra.Polynomial.Basic
#align_import data.polynomial.monomial from "leanprover-community/mathlib"@"220f71ba506c8958c9b41bd82226b3d06b0991e8"
noncomputable section
namespace Polynomial
open Polynomial
universe u
variable {R : Type u} {a b : R} {m n : β}
variable [Semiring R] {p q r : R[X]}
| Mathlib/Algebra/Polynomial/Monomial.lean | 28 | 32 | theorem monomial_one_eq_iff [Nontrivial R] {i j : β} :
(monomial i 1 : R[X]) = monomial j 1 β i = j := by |
-- Porting note: `ofFinsupp.injEq` is required.
simp_rw [β ofFinsupp_single, ofFinsupp.injEq]
exact AddMonoidAlgebra.of_injective.eq_iff
| 1,506 |
import Mathlib.Algebra.Polynomial.Basic
#align_import data.polynomial.monomial from "leanprover-community/mathlib"@"220f71ba506c8958c9b41bd82226b3d06b0991e8"
noncomputable section
namespace Polynomial
open Polynomial
universe u
variable {R : Type u} {a b : R} {m n : β}
variable [Semiring R] {p q r : R[X]}
th... | Mathlib/Algebra/Polynomial/Monomial.lean | 39 | 56 | theorem card_support_le_one_iff_monomial {f : R[X]} :
Finset.card f.support β€ 1 β β n a, f = monomial n a := by |
constructor
Β· intro H
rw [Finset.card_le_one_iff_subset_singleton] at H
rcases H with β¨n, hnβ©
refine β¨n, f.coeff n, ?_β©
ext i
by_cases hi : i = n
Β· simp [hi, coeff_monomial]
Β· have : f.coeff i = 0 := by
rw [β not_mem_support_iff]
exact fun hi' => hi (Finset.mem_singleton... | 1,506 |
import Mathlib.Algebra.Polynomial.Basic
#align_import data.polynomial.monomial from "leanprover-community/mathlib"@"220f71ba506c8958c9b41bd82226b3d06b0991e8"
noncomputable section
namespace Polynomial
open Polynomial
universe u
variable {R : Type u} {a b : R} {m n : β}
variable [Semiring R] {p q r : R[X]}
th... | Mathlib/Algebra/Polynomial/Monomial.lean | 59 | 80 | theorem ringHom_ext {S} [Semiring S] {f g : R[X] β+* S} (hβ : β a, f (C a) = g (C a))
(hβ : f X = g X) : f = g := by |
set f' := f.comp (toFinsuppIso R).symm.toRingHom with hf'
set g' := g.comp (toFinsuppIso R).symm.toRingHom with hg'
have A : f' = g' := by
-- Porting note: Was `ext; simp [..]; simpa [..] using hβ`.
ext : 1
Β· ext
simp [f', g', hβ, RingEquiv.toRingHom_eq_coe]
Β· refine MonoidHom.ext_mnat ?_
... | 1,506 |
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.RingTheory.Ideal.Basic
#align_import data.polynomial.induction from "leanprover-community/mathlib"@"63417e01fbc711beaf25fa73b6edb395c0cfddd0"
noncomputable section
open Finsupp Finset
namespace Polynomial
open Polynomial
universe u v w x y z
variable {R ... | Mathlib/Algebra/Polynomial/Induction.lean | 75 | 78 | theorem span_le_of_C_coeff_mem (cf : β i : β, C (f.coeff i) β I) :
Ideal.span { g | β i, g = C (f.coeff i) } β€ I := by |
simp only [@eq_comm _ _ (C _)]
exact (Ideal.span_le.trans range_subset_iff).mpr cf
| 1,507 |
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.RingTheory.Ideal.Basic
#align_import data.polynomial.induction from "leanprover-community/mathlib"@"63417e01fbc711beaf25fa73b6edb395c0cfddd0"
noncomputable section
open Finsupp Finset
namespace Polynomial
open Polynomial
universe u v w x y z
variable {R ... | Mathlib/Algebra/Polynomial/Induction.lean | 82 | 94 | theorem mem_span_C_coeff : f β Ideal.span { g : R[X] | β i : β, g = C (coeff f i) } := by |
let p := Ideal.span { g : R[X] | β i : β, g = C (coeff f i) }
nth_rw 1 [(sum_C_mul_X_pow_eq f).symm]
refine Submodule.sum_mem _ fun n _hn => ?_
dsimp
have : C (coeff f n) β p := by
apply subset_span
rw [mem_setOf_eq]
use n
have : monomial n (1 : R) β’ C (coeff f n) β p := p.smul_mem _ this
con... | 1,507 |
import Mathlib.Algebra.MonoidAlgebra.Support
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.Algebra.Regular.Basic
import Mathlib.Data.Nat.Choose.Sum
#align_import data.polynomial.coeff from "leanprover-community/mathlib"@"2651125b48fc5c170ab1111afd0817c903b1fc6c"
set_option linter.uppercaseLean3 false
no... | Mathlib/Algebra/Polynomial/Coeff.lean | 40 | 44 | theorem coeff_add (p q : R[X]) (n : β) : coeff (p + q) n = coeff p n + coeff q n := by |
rcases p with β¨β©
rcases q with β¨β©
simp_rw [β ofFinsupp_add, coeff]
exact Finsupp.add_apply _ _ _
| 1,508 |
import Mathlib.Algebra.MonoidAlgebra.Support
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.Algebra.Regular.Basic
import Mathlib.Data.Nat.Choose.Sum
#align_import data.polynomial.coeff from "leanprover-community/mathlib"@"2651125b48fc5c170ab1111afd0817c903b1fc6c"
set_option linter.uppercaseLean3 false
no... | Mathlib/Algebra/Polynomial/Coeff.lean | 49 | 49 | theorem coeff_bit0 (p : R[X]) (n : β) : coeff (bit0 p) n = bit0 (coeff p n) := by | simp [bit0]
| 1,508 |
import Mathlib.Algebra.MonoidAlgebra.Support
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.Algebra.Regular.Basic
import Mathlib.Data.Nat.Choose.Sum
#align_import data.polynomial.coeff from "leanprover-community/mathlib"@"2651125b48fc5c170ab1111afd0817c903b1fc6c"
set_option linter.uppercaseLean3 false
no... | Mathlib/Algebra/Polynomial/Coeff.lean | 53 | 57 | theorem coeff_smul [SMulZeroClass S R] (r : S) (p : R[X]) (n : β) :
coeff (r β’ p) n = r β’ coeff p n := by |
rcases p with β¨β©
simp_rw [β ofFinsupp_smul, coeff]
exact Finsupp.smul_apply _ _ _
| 1,508 |
import Mathlib.Algebra.MonoidAlgebra.Support
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.Algebra.Regular.Basic
import Mathlib.Data.Nat.Choose.Sum
#align_import data.polynomial.coeff from "leanprover-community/mathlib"@"2651125b48fc5c170ab1111afd0817c903b1fc6c"
set_option linter.uppercaseLean3 false
no... | Mathlib/Algebra/Polynomial/Coeff.lean | 60 | 65 | theorem support_smul [SMulZeroClass S R] (r : S) (p : R[X]) :
support (r β’ p) β support p := by |
intro i hi
simp? [mem_support_iff] at hi β’ says simp only [mem_support_iff, coeff_smul, ne_eq] at hi β’
contrapose! hi
simp [hi]
| 1,508 |
import Mathlib.Algebra.MonoidAlgebra.Support
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.Algebra.Regular.Basic
import Mathlib.Data.Nat.Choose.Sum
#align_import data.polynomial.coeff from "leanprover-community/mathlib"@"2651125b48fc5c170ab1111afd0817c903b1fc6c"
set_option linter.uppercaseLean3 false
no... | Mathlib/Algebra/Polynomial/Coeff.lean | 69 | 74 | theorem card_support_mul_le : (p * q).support.card β€ p.support.card * q.support.card := by |
calc (p * q).support.card
_ = (p.toFinsupp * q.toFinsupp).support.card := by rw [β support_toFinsupp, toFinsupp_mul]
_ β€ (p.toFinsupp.support + q.toFinsupp.support).card :=
Finset.card_le_card (AddMonoidAlgebra.support_mul p.toFinsupp q.toFinsupp)
_ β€ p.support.card * q.support.card := Finset.card_image... | 1,508 |
import Mathlib.Algebra.MonoidAlgebra.Support
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.Algebra.Regular.Basic
import Mathlib.Data.Nat.Choose.Sum
#align_import data.polynomial.coeff from "leanprover-community/mathlib"@"2651125b48fc5c170ab1111afd0817c903b1fc6c"
set_option linter.uppercaseLean3 false
no... | Mathlib/Algebra/Polynomial/Coeff.lean | 120 | 124 | theorem coeff_sum [Semiring S] (n : β) (f : β β R β S[X]) :
coeff (p.sum f) n = p.sum fun a b => coeff (f a b) n := by |
rcases p with β¨β©
-- porting note (#10745): was `simp [Polynomial.sum, support, coeff]`.
simp [Polynomial.sum, support_ofFinsupp, coeff_ofFinsupp]
| 1,508 |
import Mathlib.Algebra.MonoidAlgebra.Support
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.Algebra.Regular.Basic
import Mathlib.Data.Nat.Choose.Sum
#align_import data.polynomial.coeff from "leanprover-community/mathlib"@"2651125b48fc5c170ab1111afd0817c903b1fc6c"
set_option linter.uppercaseLean3 false
no... | Mathlib/Algebra/Polynomial/Coeff.lean | 130 | 134 | theorem coeff_mul (p q : R[X]) (n : β) :
coeff (p * q) n = β x β antidiagonal n, coeff p x.1 * coeff q x.2 := by |
rcases p with β¨pβ©; rcases q with β¨qβ©
simp_rw [β ofFinsupp_mul, coeff]
exact AddMonoidAlgebra.mul_apply_antidiagonal p q n _ Finset.mem_antidiagonal
| 1,508 |
import Mathlib.Algebra.MonoidAlgebra.Support
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.Algebra.Regular.Basic
import Mathlib.Data.Nat.Choose.Sum
#align_import data.polynomial.coeff from "leanprover-community/mathlib"@"2651125b48fc5c170ab1111afd0817c903b1fc6c"
set_option linter.uppercaseLean3 false
no... | Mathlib/Algebra/Polynomial/Coeff.lean | 138 | 138 | theorem mul_coeff_zero (p q : R[X]) : coeff (p * q) 0 = coeff p 0 * coeff q 0 := by | simp [coeff_mul]
| 1,508 |
import Mathlib.Algebra.Polynomial.Coeff
import Mathlib.Data.Nat.Choose.Basic
#align_import data.nat.choose.vandermonde from "leanprover-community/mathlib"@"d6814c584384ddf2825ff038e868451a7c956f31"
open Polynomial Finset Finset.Nat
| Mathlib/Data/Nat/Choose/Vandermonde.lean | 27 | 34 | theorem Nat.add_choose_eq (m n k : β) :
(m + n).choose k = β ij β antidiagonal k, m.choose ij.1 * n.choose ij.2 := by |
calc
(m + n).choose k = ((X + 1) ^ (m + n)).coeff k := by rw [coeff_X_add_one_pow, Nat.cast_id]
_ = ((X + 1) ^ m * (X + 1) ^ n).coeff k := by rw [pow_add]
_ = β ij β antidiagonal k, m.choose ij.1 * n.choose ij.2 := by
rw [coeff_mul, Finset.sum_congr rfl]
simp only [coeff_X_add_one_pow, Nat.ca... | 1,509 |
import Mathlib.Algebra.MonoidAlgebra.Degree
import Mathlib.Algebra.Polynomial.Coeff
import Mathlib.Algebra.Polynomial.Monomial
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Nat.WithBot
import Mathlib.Data.Nat.Cast.WithTop
import Mathlib.Data.Nat.SuccPred
#align_import data.polynomial.degree.definitions... | Mathlib/Algebra/Polynomial/Degree/Definitions.lean | 123 | 124 | theorem degree_of_subsingleton [Subsingleton R] : degree p = β₯ := by |
rw [Subsingleton.elim p 0, degree_zero]
| 1,510 |
import Mathlib.Algebra.MonoidAlgebra.Degree
import Mathlib.Algebra.Polynomial.Coeff
import Mathlib.Algebra.Polynomial.Monomial
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Nat.WithBot
import Mathlib.Data.Nat.Cast.WithTop
import Mathlib.Data.Nat.SuccPred
#align_import data.polynomial.degree.definitions... | Mathlib/Algebra/Polynomial/Degree/Definitions.lean | 128 | 129 | theorem natDegree_of_subsingleton [Subsingleton R] : natDegree p = 0 := by |
rw [Subsingleton.elim p 0, natDegree_zero]
| 1,510 |
import Mathlib.Algebra.MonoidAlgebra.Degree
import Mathlib.Algebra.Polynomial.Coeff
import Mathlib.Algebra.Polynomial.Monomial
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Nat.WithBot
import Mathlib.Data.Nat.Cast.WithTop
import Mathlib.Data.Nat.SuccPred
#align_import data.polynomial.degree.definitions... | Mathlib/Algebra/Polynomial/Degree/Definitions.lean | 132 | 135 | theorem degree_eq_natDegree (hp : p β 0) : degree p = (natDegree p : WithBot β) := by |
let β¨n, hnβ© := not_forall.1 (mt Option.eq_none_iff_forall_not_mem.2 (mt degree_eq_bot.1 hp))
have hn : degree p = some n := Classical.not_not.1 hn
rw [natDegree, hn]; rfl
| 1,510 |
import Mathlib.Algebra.MonoidAlgebra.Degree
import Mathlib.Algebra.Polynomial.Coeff
import Mathlib.Algebra.Polynomial.Monomial
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Nat.WithBot
import Mathlib.Data.Nat.Cast.WithTop
import Mathlib.Data.Nat.SuccPred
#align_import data.polynomial.degree.definitions... | Mathlib/Algebra/Polynomial/Degree/Definitions.lean | 138 | 144 | theorem supDegree_eq_natDegree (p : R[X]) : p.toFinsupp.supDegree id = p.natDegree := by |
obtain rfl|h := eq_or_ne p 0
Β· simp
apply WithBot.coe_injective
rw [β AddMonoidAlgebra.supDegree_withBot_some_comp, Function.comp_id, supDegree_eq_degree,
degree_eq_natDegree h, Nat.cast_withBot]
rwa [support_toFinsupp, nonempty_iff_ne_empty, Ne, support_eq_empty]
| 1,510 |
import Mathlib.Algebra.MonoidAlgebra.Degree
import Mathlib.Algebra.Polynomial.Coeff
import Mathlib.Algebra.Polynomial.Monomial
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Nat.WithBot
import Mathlib.Data.Nat.Cast.WithTop
import Mathlib.Data.Nat.SuccPred
#align_import data.polynomial.degree.definitions... | Mathlib/Algebra/Polynomial/Degree/Definitions.lean | 146 | 147 | theorem degree_eq_iff_natDegree_eq {p : R[X]} {n : β} (hp : p β 0) :
p.degree = n β p.natDegree = n := by | rw [degree_eq_natDegree hp]; exact WithBot.coe_eq_coe
| 1,510 |
import Mathlib.Algebra.MonoidAlgebra.Degree
import Mathlib.Algebra.Polynomial.Coeff
import Mathlib.Algebra.Polynomial.Monomial
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Nat.WithBot
import Mathlib.Data.Nat.Cast.WithTop
import Mathlib.Data.Nat.SuccPred
#align_import data.polynomial.degree.definitions... | Mathlib/Algebra/Polynomial/Degree/Definitions.lean | 150 | 154 | theorem degree_eq_iff_natDegree_eq_of_pos {p : R[X]} {n : β} (hn : 0 < n) :
p.degree = n β p.natDegree = n := by |
obtain rfl|h := eq_or_ne p 0
Β· simp [hn.ne]
Β· exact degree_eq_iff_natDegree_eq h
| 1,510 |
import Mathlib.Algebra.MonoidAlgebra.Degree
import Mathlib.Algebra.Polynomial.Coeff
import Mathlib.Algebra.Polynomial.Monomial
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Nat.WithBot
import Mathlib.Data.Nat.Cast.WithTop
import Mathlib.Data.Nat.SuccPred
#align_import data.polynomial.degree.definitions... | Mathlib/Algebra/Polynomial/Degree/Definitions.lean | 157 | 159 | theorem natDegree_eq_of_degree_eq_some {p : R[X]} {n : β} (h : degree p = n) : natDegree p = n := by |
-- Porting note: `Nat.cast_withBot` is required.
rw [natDegree, h, Nat.cast_withBot, WithBot.unbot'_coe]
| 1,510 |
import Mathlib.Algebra.MonoidAlgebra.Degree
import Mathlib.Algebra.Polynomial.Coeff
import Mathlib.Algebra.Polynomial.Monomial
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Nat.WithBot
import Mathlib.Data.Nat.Cast.WithTop
import Mathlib.Data.Nat.SuccPred
#align_import data.polynomial.degree.definitions... | Mathlib/Algebra/Polynomial/Degree/Definitions.lean | 537 | 538 | theorem coeff_mul_X_sub_C {p : R[X]} {r : R} {a : β} :
coeff (p * (X - C r)) (a + 1) = coeff p a - coeff p (a + 1) * r := by | simp [mul_sub]
| 1,510 |
import Mathlib.Algebra.Polynomial.Degree.Definitions
#align_import ring_theory.polynomial.opposites from "leanprover-community/mathlib"@"63417e01fbc711beaf25fa73b6edb395c0cfddd0"
open Polynomial
open Polynomial MulOpposite
variable {R : Type*} [Semiring R]
noncomputable section
namespace Polynomial
def opRi... | Mathlib/RingTheory/Polynomial/Opposites.lean | 38 | 42 | theorem opRingEquiv_op_monomial (n : β) (r : R) :
opRingEquiv R (op (monomial n r : R[X])) = monomial n (op r) := by |
simp only [opRingEquiv, RingEquiv.coe_trans, Function.comp_apply,
AddMonoidAlgebra.opRingEquiv_apply, RingEquiv.op_apply_apply, toFinsuppIso_apply, unop_op,
toFinsupp_monomial, Finsupp.mapRange_single, toFinsuppIso_symm_apply, ofFinsupp_single]
| 1,511 |
import Mathlib.Algebra.Polynomial.Degree.Definitions
#align_import ring_theory.polynomial.opposites from "leanprover-community/mathlib"@"63417e01fbc711beaf25fa73b6edb395c0cfddd0"
open Polynomial
open Polynomial MulOpposite
variable {R : Type*} [Semiring R]
noncomputable section
namespace Polynomial
def opRi... | Mathlib/RingTheory/Polynomial/Opposites.lean | 57 | 59 | theorem opRingEquiv_op_C_mul_X_pow (r : R) (n : β) :
opRingEquiv R (op (C r * X ^ n : R[X])) = C (op r) * X ^ n := by |
simp only [X_pow_mul, op_mul, op_pow, map_mul, map_pow, opRingEquiv_op_X, opRingEquiv_op_C]
| 1,511 |
import Mathlib.Algebra.Polynomial.Degree.Definitions
#align_import ring_theory.polynomial.opposites from "leanprover-community/mathlib"@"63417e01fbc711beaf25fa73b6edb395c0cfddd0"
open Polynomial
open Polynomial MulOpposite
variable {R : Type*} [Semiring R]
noncomputable section
namespace Polynomial
def opRi... | Mathlib/RingTheory/Polynomial/Opposites.lean | 85 | 87 | theorem opRingEquiv_symm_C_mul_X_pow (r : Rα΅α΅α΅) (n : β) :
(opRingEquiv R).symm (C r * X ^ n : Rα΅α΅α΅[X]) = op (C (unop r) * X ^ n) := by |
rw [C_mul_X_pow_eq_monomial, opRingEquiv_symm_monomial, C_mul_X_pow_eq_monomial]
| 1,511 |
import Mathlib.Algebra.Polynomial.Degree.Definitions
#align_import ring_theory.polynomial.opposites from "leanprover-community/mathlib"@"63417e01fbc711beaf25fa73b6edb395c0cfddd0"
open Polynomial
open Polynomial MulOpposite
variable {R : Type*} [Semiring R]
noncomputable section
namespace Polynomial
def opRi... | Mathlib/RingTheory/Polynomial/Opposites.lean | 95 | 99 | theorem coeff_opRingEquiv (p : R[X]α΅α΅α΅) (n : β) :
(opRingEquiv R p).coeff n = op ((unop p).coeff n) := by |
induction' p using MulOpposite.rec' with p
cases p
rfl
| 1,511 |
import Mathlib.Algebra.Polynomial.Degree.Definitions
#align_import ring_theory.polynomial.opposites from "leanprover-community/mathlib"@"63417e01fbc711beaf25fa73b6edb395c0cfddd0"
open Polynomial
open Polynomial MulOpposite
variable {R : Type*} [Semiring R]
noncomputable section
namespace Polynomial
def opRi... | Mathlib/RingTheory/Polynomial/Opposites.lean | 103 | 106 | theorem support_opRingEquiv (p : R[X]α΅α΅α΅) : (opRingEquiv R p).support = (unop p).support := by |
induction' p using MulOpposite.rec' with p
cases p
exact Finsupp.support_mapRange_of_injective (map_zero _) _ op_injective
| 1,511 |
import Mathlib.Algebra.Polynomial.Degree.Definitions
#align_import ring_theory.polynomial.opposites from "leanprover-community/mathlib"@"63417e01fbc711beaf25fa73b6edb395c0cfddd0"
open Polynomial
open Polynomial MulOpposite
variable {R : Type*} [Semiring R]
noncomputable section
namespace Polynomial
def opRi... | Mathlib/RingTheory/Polynomial/Opposites.lean | 110 | 114 | theorem natDegree_opRingEquiv (p : R[X]α΅α΅α΅) : (opRingEquiv R p).natDegree = (unop p).natDegree := by |
by_cases p0 : p = 0
Β· simp only [p0, _root_.map_zero, natDegree_zero, unop_zero]
Β· simp only [p0, natDegree_eq_support_max', Ne, AddEquivClass.map_eq_zero_iff, not_false_iff,
support_opRingEquiv, unop_eq_zero_iff]
| 1,511 |
import Mathlib.Algebra.Polynomial.Degree.Definitions
#align_import ring_theory.polynomial.opposites from "leanprover-community/mathlib"@"63417e01fbc711beaf25fa73b6edb395c0cfddd0"
open Polynomial
open Polynomial MulOpposite
variable {R : Type*} [Semiring R]
noncomputable section
namespace Polynomial
def opRi... | Mathlib/RingTheory/Polynomial/Opposites.lean | 118 | 120 | theorem leadingCoeff_opRingEquiv (p : R[X]α΅α΅α΅) :
(opRingEquiv R p).leadingCoeff = op (unop p).leadingCoeff := by |
rw [leadingCoeff, coeff_opRingEquiv, natDegree_opRingEquiv, leadingCoeff]
| 1,511 |
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Data.ENat.Basic
#align_import data.polynomial.degree.trailing_degree from "leanprover-community/mathlib"@"302eab4f46abb63de520828de78c04cb0f9b5836"
noncomputable section
open Function Polynomial Finsupp Finset
open scoped Polynomial
namespace ... | Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean | 102 | 108 | theorem trailingDegree_eq_natTrailingDegree (hp : p β 0) :
trailingDegree p = (natTrailingDegree p : ββ) := by |
let β¨n, hnβ© :=
not_forall.1 (mt Option.eq_none_iff_forall_not_mem.2 (mt trailingDegree_eq_top.1 hp))
have hn : trailingDegree p = n := Classical.not_not.1 hn
rw [natTrailingDegree, hn]
rfl
| 1,512 |
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Data.ENat.Basic
#align_import data.polynomial.degree.trailing_degree from "leanprover-community/mathlib"@"302eab4f46abb63de520828de78c04cb0f9b5836"
noncomputable section
open Function Polynomial Finsupp Finset
open scoped Polynomial
namespace ... | Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean | 111 | 114 | theorem trailingDegree_eq_iff_natTrailingDegree_eq {p : R[X]} {n : β} (hp : p β 0) :
p.trailingDegree = n β p.natTrailingDegree = n := by |
rw [trailingDegree_eq_natTrailingDegree hp]
exact WithTop.coe_eq_coe
| 1,512 |
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Data.ENat.Basic
#align_import data.polynomial.degree.trailing_degree from "leanprover-community/mathlib"@"302eab4f46abb63de520828de78c04cb0f9b5836"
noncomputable section
open Function Polynomial Finsupp Finset
open scoped Polynomial
namespace ... | Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean | 117 | 130 | theorem trailingDegree_eq_iff_natTrailingDegree_eq_of_pos {p : R[X]} {n : β} (hn : 0 < n) :
p.trailingDegree = n β p.natTrailingDegree = n := by |
constructor
Β· intro H
rwa [β trailingDegree_eq_iff_natTrailingDegree_eq]
rintro rfl
rw [trailingDegree_zero] at H
exact Option.noConfusion H
Β· intro H
rwa [trailingDegree_eq_iff_natTrailingDegree_eq]
rintro rfl
rw [natTrailingDegree_zero] at H
rw [H] at hn
exact lt_irrefl _ hn... | 1,512 |
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Data.ENat.Basic
#align_import data.polynomial.degree.trailing_degree from "leanprover-community/mathlib"@"302eab4f46abb63de520828de78c04cb0f9b5836"
noncomputable section
open Function Polynomial Finsupp Finset
open scoped Polynomial
namespace ... | Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean | 141 | 145 | theorem natTrailingDegree_le_trailingDegree : β(natTrailingDegree p) β€ trailingDegree p := by |
by_cases hp : p = 0;
Β· rw [hp, trailingDegree_zero]
exact le_top
rw [trailingDegree_eq_natTrailingDegree hp]
| 1,512 |
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Data.ENat.Basic
#align_import data.polynomial.degree.trailing_degree from "leanprover-community/mathlib"@"302eab4f46abb63de520828de78c04cb0f9b5836"
noncomputable section
open Function Polynomial Finsupp Finset
open scoped Polynomial
namespace ... | Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean | 148 | 151 | theorem natTrailingDegree_eq_of_trailingDegree_eq [Semiring S] {q : S[X]}
(h : trailingDegree p = trailingDegree q) : natTrailingDegree p = natTrailingDegree q := by |
unfold natTrailingDegree
rw [h]
| 1,512 |
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Data.ENat.Basic
#align_import data.polynomial.degree.trailing_degree from "leanprover-community/mathlib"@"302eab4f46abb63de520828de78c04cb0f9b5836"
noncomputable section
open Function Polynomial Finsupp Finset
open scoped Polynomial
namespace ... | Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean | 158 | 164 | theorem natTrailingDegree_le_of_ne_zero (h : coeff p n β 0) : natTrailingDegree p β€ n := by |
have : WithTop.some (natTrailingDegree p) = Nat.cast (natTrailingDegree p) := rfl
rw [β WithTop.coe_le_coe, this, β trailingDegree_eq_natTrailingDegree]
Β· exact trailingDegree_le_of_ne_zero h
Β· intro h
subst h
exact h rfl
| 1,512 |
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Induction
#align_import data.polynomial.eval from "leanprover-community/mathlib"@"728baa2f54e6062c5879a3e397ac6bac323e506f"
set_option linter.uppercaseLean3 false
noncomputable section
open Finset AddMonoidAlgebra
open Polyn... | Mathlib/Algebra/Polynomial/Eval.lean | 48 | 49 | theorem evalβ_eq_sum {f : R β+* S} {x : S} : p.evalβ f x = p.sum fun e a => f a * x ^ e := by |
rw [evalβ_def]
| 1,513 |
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Induction
#align_import data.polynomial.eval from "leanprover-community/mathlib"@"728baa2f54e6062c5879a3e397ac6bac323e506f"
set_option linter.uppercaseLean3 false
noncomputable section
open Finset AddMonoidAlgebra
open Polyn... | Mathlib/Algebra/Polynomial/Eval.lean | 52 | 54 | theorem evalβ_congr {R S : Type*} [Semiring R] [Semiring S] {f g : R β+* S} {s t : S}
{Ο Ο : R[X]} : f = g β s = t β Ο = Ο β evalβ f s Ο = evalβ g t Ο := by |
rintro rfl rfl rfl; rfl
| 1,513 |
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Induction
#align_import data.polynomial.eval from "leanprover-community/mathlib"@"728baa2f54e6062c5879a3e397ac6bac323e506f"
set_option linter.uppercaseLean3 false
noncomputable section
open Finset AddMonoidAlgebra
open Polyn... | Mathlib/Algebra/Polynomial/Eval.lean | 58 | 61 | theorem evalβ_at_zero : p.evalβ f 0 = f (coeff p 0) := by |
simp (config := { contextual := true }) only [evalβ_eq_sum, zero_pow_eq, mul_ite, mul_zero,
mul_one, sum, Classical.not_not, mem_support_iff, sum_ite_eq', ite_eq_left_iff,
RingHom.map_zero, imp_true_iff, eq_self_iff_true]
| 1,513 |
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Induction
#align_import data.polynomial.eval from "leanprover-community/mathlib"@"728baa2f54e6062c5879a3e397ac6bac323e506f"
set_option linter.uppercaseLean3 false
noncomputable section
open Finset AddMonoidAlgebra
open Polyn... | Mathlib/Algebra/Polynomial/Eval.lean | 65 | 65 | theorem evalβ_zero : (0 : R[X]).evalβ f x = 0 := by | simp [evalβ_eq_sum]
| 1,513 |
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Induction
#align_import data.polynomial.eval from "leanprover-community/mathlib"@"728baa2f54e6062c5879a3e397ac6bac323e506f"
set_option linter.uppercaseLean3 false
noncomputable section
open Finset AddMonoidAlgebra
open Polyn... | Mathlib/Algebra/Polynomial/Eval.lean | 69 | 69 | theorem evalβ_C : (C a).evalβ f x = f a := by | simp [evalβ_eq_sum]
| 1,513 |
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Induction
#align_import data.polynomial.eval from "leanprover-community/mathlib"@"728baa2f54e6062c5879a3e397ac6bac323e506f"
set_option linter.uppercaseLean3 false
noncomputable section
open Finset AddMonoidAlgebra
open Polyn... | Mathlib/Algebra/Polynomial/Eval.lean | 73 | 73 | theorem evalβ_X : X.evalβ f x = x := by | simp [evalβ_eq_sum]
| 1,513 |
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Induction
#align_import data.polynomial.eval from "leanprover-community/mathlib"@"728baa2f54e6062c5879a3e397ac6bac323e506f"
set_option linter.uppercaseLean3 false
noncomputable section
open Finset AddMonoidAlgebra
open Polyn... | Mathlib/Algebra/Polynomial/Eval.lean | 77 | 78 | theorem evalβ_monomial {n : β} {r : R} : (monomial n r).evalβ f x = f r * x ^ n := by |
simp [evalβ_eq_sum]
| 1,513 |
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Induction
#align_import data.polynomial.eval from "leanprover-community/mathlib"@"728baa2f54e6062c5879a3e397ac6bac323e506f"
set_option linter.uppercaseLean3 false
noncomputable section
open Finset AddMonoidAlgebra
open Polyn... | Mathlib/Algebra/Polynomial/Eval.lean | 82 | 85 | theorem evalβ_X_pow {n : β} : (X ^ n).evalβ f x = x ^ n := by |
rw [X_pow_eq_monomial]
convert evalβ_monomial f x (n := n) (r := 1)
simp
| 1,513 |
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Induction
#align_import data.polynomial.eval from "leanprover-community/mathlib"@"728baa2f54e6062c5879a3e397ac6bac323e506f"
set_option linter.uppercaseLean3 false
noncomputable section
open Finset AddMonoidAlgebra
open Polyn... | Mathlib/Algebra/Polynomial/Eval.lean | 89 | 91 | theorem evalβ_add : (p + q).evalβ f x = p.evalβ f x + q.evalβ f x := by |
simp only [evalβ_eq_sum]
apply sum_add_index <;> simp [add_mul]
| 1,513 |
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Induction
#align_import data.polynomial.eval from "leanprover-community/mathlib"@"728baa2f54e6062c5879a3e397ac6bac323e506f"
set_option linter.uppercaseLean3 false
noncomputable section
open Finset AddMonoidAlgebra
open Polyn... | Mathlib/Algebra/Polynomial/Eval.lean | 95 | 95 | theorem evalβ_one : (1 : R[X]).evalβ f x = 1 := by | rw [β C_1, evalβ_C, f.map_one]
| 1,513 |
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Induction
#align_import data.polynomial.eval from "leanprover-community/mathlib"@"728baa2f54e6062c5879a3e397ac6bac323e506f"
set_option linter.uppercaseLean3 false
noncomputable section
open Finset AddMonoidAlgebra
open Polyn... | Mathlib/Algebra/Polynomial/Eval.lean | 100 | 100 | theorem evalβ_bit0 : (bit0 p).evalβ f x = bit0 (p.evalβ f x) := by | rw [bit0, evalβ_add, bit0]
| 1,513 |
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Induction
#align_import data.polynomial.eval from "leanprover-community/mathlib"@"728baa2f54e6062c5879a3e397ac6bac323e506f"
set_option linter.uppercaseLean3 false
noncomputable section
open Finset AddMonoidAlgebra
open Polyn... | Mathlib/Algebra/Polynomial/Eval.lean | 105 | 106 | theorem evalβ_bit1 : (bit1 p).evalβ f x = bit1 (p.evalβ f x) := by |
rw [bit1, evalβ_add, evalβ_bit0, evalβ_one, bit1]
| 1,513 |
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Induction
#align_import data.polynomial.eval from "leanprover-community/mathlib"@"728baa2f54e6062c5879a3e397ac6bac323e506f"
set_option linter.uppercaseLean3 false
noncomputable section
open Finset AddMonoidAlgebra
open Polyn... | Mathlib/Algebra/Polynomial/Eval.lean | 110 | 115 | theorem evalβ_smul (g : R β+* S) (p : R[X]) (x : S) {s : R} :
evalβ g x (s β’ p) = g s * evalβ g x p := by |
have A : p.natDegree < p.natDegree.succ := Nat.lt_succ_self _
have B : (s β’ p).natDegree < p.natDegree.succ := (natDegree_smul_le _ _).trans_lt A
rw [evalβ_eq_sum, evalβ_eq_sum, sum_over_range' _ _ _ A, sum_over_range' _ _ _ B] <;>
simp [mul_sum, mul_assoc]
| 1,513 |
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Induction
#align_import data.polynomial.eval from "leanprover-community/mathlib"@"728baa2f54e6062c5879a3e397ac6bac323e506f"
set_option linter.uppercaseLean3 false
noncomputable section
open Finset AddMonoidAlgebra
open Polyn... | Mathlib/Algebra/Polynomial/Eval.lean | 135 | 139 | theorem evalβ_natCast (n : β) : (n : R[X]).evalβ f x = n := by |
induction' n with n ih
-- Porting note: `Nat.zero_eq` is required.
Β· simp only [evalβ_zero, Nat.cast_zero, Nat.zero_eq]
Β· rw [n.cast_succ, evalβ_add, ih, evalβ_one, n.cast_succ]
| 1,513 |
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Induction
#align_import data.polynomial.eval from "leanprover-community/mathlib"@"728baa2f54e6062c5879a3e397ac6bac323e506f"
set_option linter.uppercaseLean3 false
noncomputable section
open Finset AddMonoidAlgebra
open Polyn... | Mathlib/Algebra/Polynomial/Eval.lean | 153 | 161 | theorem evalβ_sum (p : T[X]) (g : β β T β R[X]) (x : S) :
(p.sum g).evalβ f x = p.sum fun n a => (g n a).evalβ f x := by |
let T : R[X] β+ S :=
{ toFun := evalβ f x
map_zero' := evalβ_zero _ _
map_add' := fun p q => evalβ_add _ _ }
have A : β y, evalβ f x y = T y := fun y => rfl
simp only [A]
rw [sum, map_sum, sum]
| 1,513 |
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.Analysis.SpecialFunctions.Exp
open Filter Topology Real
namespace Polynomial
| Mathlib/Analysis/SpecialFunctions/PolynomialExp.lean | 27 | 31 | theorem tendsto_div_exp_atTop (p : β[X]) : Tendsto (fun x β¦ p.eval x / exp x) atTop (π 0) := by |
induction p using Polynomial.induction_on' with
| h_monomial n c => simpa [exp_neg, div_eq_mul_inv, mul_assoc]
using tendsto_const_nhds.mul (tendsto_pow_mul_exp_neg_atTop_nhds_zero n)
| h_add p q hp hq => simpa [add_div] using hp.add hq
| 1,514 |
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.RingTheory.Ideal.Quotient
#align_import linear_algebra.smodeq from "leanprover-community/mathlib"@"146d3d1fa59c091fedaad8a4afa09d6802886d24"
open Submodule
open Polynomial
variable {R : Type*} [Ring R]
variable {A : Type*} [CommRing A]
variable {M : Type*} [... | Mathlib/LinearAlgebra/SModEq.lean | 44 | 44 | theorem sub_mem : x β‘ y [SMOD U] β x - y β U := by | rw [SModEq.def, Submodule.Quotient.eq]
| 1,515 |
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.RingTheory.Ideal.Quotient
#align_import linear_algebra.smodeq from "leanprover-community/mathlib"@"146d3d1fa59c091fedaad8a4afa09d6802886d24"
open Submodule
open Polynomial
variable {R : Type*} [Ring R]
variable {A : Type*} [CommRing A]
variable {M : Type*} [... | Mathlib/LinearAlgebra/SModEq.lean | 53 | 54 | theorem bot : x β‘ y [SMOD (β₯ : Submodule R M)] β x = y := by |
rw [SModEq.def, Submodule.Quotient.eq, mem_bot, sub_eq_zero]
| 1,515 |
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.RingTheory.Ideal.Quotient
#align_import linear_algebra.smodeq from "leanprover-community/mathlib"@"146d3d1fa59c091fedaad8a4afa09d6802886d24"
open Submodule
open Polynomial
variable {R : Type*} [Ring R]
variable {A : Type*} [CommRing A]
variable {M : Type*} [... | Mathlib/LinearAlgebra/SModEq.lean | 87 | 89 | theorem add (hxyβ : xβ β‘ yβ [SMOD U]) (hxyβ : xβ β‘ yβ [SMOD U]) : xβ + xβ β‘ yβ + yβ [SMOD U] := by |
rw [SModEq.def] at hxyβ hxyβ β’
simp_rw [Quotient.mk_add, hxyβ, hxyβ]
| 1,515 |
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.RingTheory.Ideal.Quotient
#align_import linear_algebra.smodeq from "leanprover-community/mathlib"@"146d3d1fa59c091fedaad8a4afa09d6802886d24"
open Submodule
open Polynomial
variable {R : Type*} [Ring R]
variable {A : Type*} [CommRing A]
variable {M : Type*} [... | Mathlib/LinearAlgebra/SModEq.lean | 92 | 94 | theorem smul (hxy : x β‘ y [SMOD U]) (c : R) : c β’ x β‘ c β’ y [SMOD U] := by |
rw [SModEq.def] at hxy β’
simp_rw [Quotient.mk_smul, hxy]
| 1,515 |
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.RingTheory.Ideal.Quotient
#align_import linear_algebra.smodeq from "leanprover-community/mathlib"@"146d3d1fa59c091fedaad8a4afa09d6802886d24"
open Submodule
open Polynomial
variable {R : Type*} [Ring R]
variable {A : Type*} [CommRing A]
variable {M : Type*} [... | Mathlib/LinearAlgebra/SModEq.lean | 97 | 100 | theorem mul {I : Ideal A} {xβ xβ yβ yβ : A} (hxyβ : xβ β‘ yβ [SMOD I])
(hxyβ : xβ β‘ yβ [SMOD I]) : xβ * xβ β‘ yβ * yβ [SMOD I] := by |
simp only [SModEq.def, Ideal.Quotient.mk_eq_mk, map_mul] at hxyβ hxyβ β’
rw [hxyβ, hxyβ]
| 1,515 |
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.RingTheory.Ideal.Quotient
#align_import linear_algebra.smodeq from "leanprover-community/mathlib"@"146d3d1fa59c091fedaad8a4afa09d6802886d24"
open Submodule
open Polynomial
variable {R : Type*} [Ring R]
variable {A : Type*} [CommRing A]
variable {M : Type*} [... | Mathlib/LinearAlgebra/SModEq.lean | 102 | 102 | theorem zero : x β‘ 0 [SMOD U] β x β U := by | rw [SModEq.def, Submodule.Quotient.eq, sub_zero]
| 1,515 |
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.RingTheory.Ideal.Quotient
#align_import linear_algebra.smodeq from "leanprover-community/mathlib"@"146d3d1fa59c091fedaad8a4afa09d6802886d24"
open Submodule
open Polynomial
variable {R : Type*} [Ring R]
variable {A : Type*} [CommRing A]
variable {M : Type*} [... | Mathlib/LinearAlgebra/SModEq.lean | 114 | 119 | theorem eval {R : Type*} [CommRing R] {I : Ideal R} {x y : R} (h : x β‘ y [SMOD I]) (f : R[X]) :
f.eval x β‘ f.eval y [SMOD I] := by |
rw [SModEq.def] at h β’
show Ideal.Quotient.mk I (f.eval x) = Ideal.Quotient.mk I (f.eval y)
replace h : Ideal.Quotient.mk I x = Ideal.Quotient.mk I y := h
rw [β Polynomial.evalβ_at_apply, β Polynomial.evalβ_at_apply, h]
| 1,515 |
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.GroupTheory.GroupAction.Ring
#align_import data.polynomial.derivative from "leanprover-community/mathlib"@"bbeb185db4ccee8ed07dc48449414ebfa39cb821"
noncomputable section
open Finset
open Polynomial
namespace Pol... | Mathlib/Algebra/Polynomial/Derivative.lean | 57 | 73 | theorem coeff_derivative (p : R[X]) (n : β) :
coeff (derivative p) n = coeff p (n + 1) * (n + 1) := by |
rw [derivative_apply]
simp only [coeff_X_pow, coeff_sum, coeff_C_mul]
rw [sum, Finset.sum_eq_single (n + 1)]
Β· simp only [Nat.add_succ_sub_one, add_zero, mul_one, if_true, eq_self_iff_true]; norm_cast
Β· intro b
cases b
Β· intros
rw [Nat.cast_zero, mul_zero, zero_mul]
Β· intro _ H
rw [Na... | 1,516 |
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.GroupTheory.GroupAction.Ring
#align_import data.polynomial.derivative from "leanprover-community/mathlib"@"bbeb185db4ccee8ed07dc48449414ebfa39cb821"
noncomputable section
open Finset
open Polynomial
namespace Pol... | Mathlib/Algebra/Polynomial/Derivative.lean | 86 | 89 | theorem derivative_monomial (a : R) (n : β) :
derivative (monomial n a) = monomial (n - 1) (a * n) := by |
rw [derivative_apply, sum_monomial_index, C_mul_X_pow_eq_monomial]
simp
| 1,516 |
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.GroupTheory.GroupAction.Ring
#align_import data.polynomial.derivative from "leanprover-community/mathlib"@"bbeb185db4ccee8ed07dc48449414ebfa39cb821"
noncomputable section
open Finset
open Polynomial
namespace Pol... | Mathlib/Algebra/Polynomial/Derivative.lean | 92 | 93 | theorem derivative_C_mul_X (a : R) : derivative (C a * X) = C a := by |
simp [C_mul_X_eq_monomial, derivative_monomial, Nat.cast_one, mul_one]
| 1,516 |
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.GroupTheory.GroupAction.Ring
#align_import data.polynomial.derivative from "leanprover-community/mathlib"@"bbeb185db4ccee8ed07dc48449414ebfa39cb821"
noncomputable section
open Finset
open Polynomial
namespace Pol... | Mathlib/Algebra/Polynomial/Derivative.lean | 97 | 99 | theorem derivative_C_mul_X_pow (a : R) (n : β) :
derivative (C a * X ^ n) = C (a * n) * X ^ (n - 1) := by |
rw [C_mul_X_pow_eq_monomial, C_mul_X_pow_eq_monomial, derivative_monomial]
| 1,516 |
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.GroupTheory.GroupAction.Ring
#align_import data.polynomial.derivative from "leanprover-community/mathlib"@"bbeb185db4ccee8ed07dc48449414ebfa39cb821"
noncomputable section
open Finset
open Polynomial
namespace Pol... | Mathlib/Algebra/Polynomial/Derivative.lean | 103 | 104 | theorem derivative_C_mul_X_sq (a : R) : derivative (C a * X ^ 2) = C (a * 2) * X := by |
rw [derivative_C_mul_X_pow, Nat.cast_two, pow_one]
| 1,516 |
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.GroupTheory.GroupAction.Ring
#align_import data.polynomial.derivative from "leanprover-community/mathlib"@"bbeb185db4ccee8ed07dc48449414ebfa39cb821"
noncomputable section
open Finset
open Polynomial
namespace Pol... | Mathlib/Algebra/Polynomial/Derivative.lean | 109 | 110 | theorem derivative_X_pow (n : β) : derivative (X ^ n : R[X]) = C (n : R) * X ^ (n - 1) := by |
convert derivative_C_mul_X_pow (1 : R) n <;> simp
| 1,516 |
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.GroupTheory.GroupAction.Ring
#align_import data.polynomial.derivative from "leanprover-community/mathlib"@"bbeb185db4ccee8ed07dc48449414ebfa39cb821"
noncomputable section
open Finset
open Polynomial
namespace Pol... | Mathlib/Algebra/Polynomial/Derivative.lean | 115 | 116 | theorem derivative_X_sq : derivative (X ^ 2 : R[X]) = C 2 * X := by |
rw [derivative_X_pow, Nat.cast_two, pow_one]
| 1,516 |
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.GroupTheory.GroupAction.Ring
#align_import data.polynomial.derivative from "leanprover-community/mathlib"@"bbeb185db4ccee8ed07dc48449414ebfa39cb821"
noncomputable section
open Finset
open Polynomial
namespace Pol... | Mathlib/Algebra/Polynomial/Derivative.lean | 121 | 121 | theorem derivative_C {a : R} : derivative (C a) = 0 := by | simp [derivative_apply]
| 1,516 |
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.GroupTheory.GroupAction.Ring
#align_import data.polynomial.derivative from "leanprover-community/mathlib"@"bbeb185db4ccee8ed07dc48449414ebfa39cb821"
noncomputable section
open Finset
open Polynomial
namespace Pol... | Mathlib/Algebra/Polynomial/Derivative.lean | 125 | 126 | theorem derivative_of_natDegree_zero {p : R[X]} (hp : p.natDegree = 0) : derivative p = 0 := by |
rw [eq_C_of_natDegree_eq_zero hp, derivative_C]
| 1,516 |
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.GroupTheory.GroupAction.Ring
#align_import data.polynomial.derivative from "leanprover-community/mathlib"@"bbeb185db4ccee8ed07dc48449414ebfa39cb821"
noncomputable section
open Finset
open Polynomial
namespace Pol... | Mathlib/Algebra/Polynomial/Derivative.lean | 149 | 150 | theorem derivative_X_add_C (c : R) : derivative (X + C c) = 1 := by |
rw [derivative_add, derivative_X, derivative_C, add_zero]
| 1,516 |
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev from "leanprover-community/mathlib"@"d774451114d6045faeb6751c396bea1eb9058946"
namespace Polynomial.Chebyshev
set_option linter.uppercaseLean3 false -- `T` `U` `X`
open Polynomial
v... | Mathlib/RingTheory/Polynomial/Chebyshev.lean | 90 | 91 | theorem T_add_one (n : β€) : T R (n + 1) = 2 * X * T R n - T R (n - 1) := by |
linear_combination (norm := ring_nf) T_add_two R (n - 1)
| 1,517 |
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev from "leanprover-community/mathlib"@"d774451114d6045faeb6751c396bea1eb9058946"
namespace Polynomial.Chebyshev
set_option linter.uppercaseLean3 false -- `T` `U` `X`
open Polynomial
v... | Mathlib/RingTheory/Polynomial/Chebyshev.lean | 93 | 94 | theorem T_sub_two (n : β€) : T R (n - 2) = 2 * X * T R (n - 1) - T R n := by |
linear_combination (norm := ring_nf) T_add_two R (n - 2)
| 1,517 |
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev from "leanprover-community/mathlib"@"d774451114d6045faeb6751c396bea1eb9058946"
namespace Polynomial.Chebyshev
set_option linter.uppercaseLean3 false -- `T` `U` `X`
open Polynomial
v... | Mathlib/RingTheory/Polynomial/Chebyshev.lean | 96 | 97 | theorem T_sub_one (n : β€) : T R (n - 1) = 2 * X * T R n - T R (n + 1) := by |
linear_combination (norm := ring_nf) T_add_two R (n - 1)
| 1,517 |
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev from "leanprover-community/mathlib"@"d774451114d6045faeb6751c396bea1eb9058946"
namespace Polynomial.Chebyshev
set_option linter.uppercaseLean3 false -- `T` `U` `X`
open Polynomial
v... | Mathlib/RingTheory/Polynomial/Chebyshev.lean | 99 | 100 | theorem T_eq (n : β€) : T R n = 2 * X * T R (n - 1) - T R (n - 2) := by |
linear_combination (norm := ring_nf) T_add_two R (n - 2)
| 1,517 |
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev from "leanprover-community/mathlib"@"d774451114d6045faeb6751c396bea1eb9058946"
namespace Polynomial.Chebyshev
set_option linter.uppercaseLean3 false -- `T` `U` `X`
open Polynomial
v... | Mathlib/RingTheory/Polynomial/Chebyshev.lean | 113 | 114 | theorem T_two : T R 2 = 2 * X ^ 2 - 1 := by |
simpa [pow_two, mul_assoc] using T_add_two R 0
| 1,517 |
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev from "leanprover-community/mathlib"@"d774451114d6045faeb6751c396bea1eb9058946"
namespace Polynomial.Chebyshev
set_option linter.uppercaseLean3 false -- `T` `U` `X`
open Polynomial
v... | Mathlib/RingTheory/Polynomial/Chebyshev.lean | 118 | 129 | theorem T_neg (n : β€) : T R (-n) = T R n := by |
induction n using Polynomial.Chebyshev.induct with
| zero => rfl
| one => show 2 * X * 1 - X = X; ring
| add_two n ih1 ih2 =>
have hβ := T_add_two R n
have hβ := T_sub_two R (-n)
linear_combination (norm := ring_nf) (2 * (X:R[X])) * ih1 - ih2 - hβ + hβ
| neg_add_one n ih1 ih2 =>
have hβ := T_... | 1,517 |
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev from "leanprover-community/mathlib"@"d774451114d6045faeb6751c396bea1eb9058946"
namespace Polynomial.Chebyshev
set_option linter.uppercaseLean3 false -- `T` `U` `X`
open Polynomial
v... | Mathlib/RingTheory/Polynomial/Chebyshev.lean | 131 | 132 | theorem T_natAbs (n : β€) : T R n.natAbs = T R n := by |
obtain h | h := Int.natAbs_eq n <;> nth_rw 2 [h]; simp
| 1,517 |
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev from "leanprover-community/mathlib"@"d774451114d6045faeb6751c396bea1eb9058946"
namespace Polynomial.Chebyshev
set_option linter.uppercaseLean3 false -- `T` `U` `X`
open Polynomial
v... | Mathlib/RingTheory/Polynomial/Chebyshev.lean | 134 | 134 | theorem T_neg_two : T R (-2) = 2 * X ^ 2 - 1 := by | simp [T_two]
| 1,517 |
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev from "leanprover-community/mathlib"@"d774451114d6045faeb6751c396bea1eb9058946"
namespace Polynomial.Chebyshev
set_option linter.uppercaseLean3 false -- `T` `U` `X`
open Polynomial
v... | Mathlib/RingTheory/Polynomial/Chebyshev.lean | 153 | 154 | theorem U_add_one (n : β€) : U R (n + 1) = 2 * X * U R n - U R (n - 1) := by |
linear_combination (norm := ring_nf) U_add_two R (n - 1)
| 1,517 |
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev from "leanprover-community/mathlib"@"d774451114d6045faeb6751c396bea1eb9058946"
namespace Polynomial.Chebyshev
set_option linter.uppercaseLean3 false -- `T` `U` `X`
open Polynomial
v... | Mathlib/RingTheory/Polynomial/Chebyshev.lean | 156 | 157 | theorem U_sub_two (n : β€) : U R (n - 2) = 2 * X * U R (n - 1) - U R n := by |
linear_combination (norm := ring_nf) U_add_two R (n - 2)
| 1,517 |
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev from "leanprover-community/mathlib"@"d774451114d6045faeb6751c396bea1eb9058946"
namespace Polynomial.Chebyshev
set_option linter.uppercaseLean3 false -- `T` `U` `X`
open Polynomial
v... | Mathlib/RingTheory/Polynomial/Chebyshev.lean | 159 | 160 | theorem U_sub_one (n : β€) : U R (n - 1) = 2 * X * U R n - U R (n + 1) := by |
linear_combination (norm := ring_nf) U_add_two R (n - 1)
| 1,517 |
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev from "leanprover-community/mathlib"@"d774451114d6045faeb6751c396bea1eb9058946"
namespace Polynomial.Chebyshev
set_option linter.uppercaseLean3 false -- `T` `U` `X`
open Polynomial
v... | Mathlib/RingTheory/Polynomial/Chebyshev.lean | 162 | 163 | theorem U_eq (n : β€) : U R n = 2 * X * U R (n - 1) - U R (n - 2) := by |
linear_combination (norm := ring_nf) U_add_two R (n - 2)
| 1,517 |
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 P... | Mathlib/RingTheory/Polynomial/Hermite/Basic.lean | 55 | 56 | theorem hermite_succ (n : β) : hermite (n + 1) = X * hermite n - derivative (hermite n) := by |
rw [hermite]
| 1,518 |
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 P... | Mathlib/RingTheory/Polynomial/Hermite/Basic.lean | 59 | 62 | theorem hermite_eq_iterate (n : β) : hermite n = (fun p => X * p - derivative p)^[n] 1 := by |
induction' n with n ih
Β· rfl
Β· rw [Function.iterate_succ_apply', β ih, hermite_succ]
| 1,518 |
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 P... | Mathlib/RingTheory/Polynomial/Hermite/Basic.lean | 72 | 74 | theorem hermite_one : hermite 1 = X := by |
rw [hermite_succ, hermite_zero]
simp only [map_one, mul_one, derivative_one, sub_zero]
| 1,518 |
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 P... | Mathlib/RingTheory/Polynomial/Hermite/Basic.lean | 82 | 83 | theorem coeff_hermite_succ_zero (n : β) : coeff (hermite (n + 1)) 0 = -coeff (hermite n) 1 := by |
simp [coeff_derivative]
| 1,518 |
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 P... | Mathlib/RingTheory/Polynomial/Hermite/Basic.lean | 86 | 89 | theorem coeff_hermite_succ_succ (n k : β) : 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
| 1,518 |
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 P... | Mathlib/RingTheory/Polynomial/Hermite/Basic.lean | 92 | 99 | theorem coeff_hermite_of_lt {n k : β} (hnk : n < k) : coeff (hermite n) k = 0 := by |
obtain β¨k, rflβ© := Nat.exists_eq_add_of_lt hnk
clear hnk
induction' n with n ih generalizing k
Β· apply coeff_C
Β· 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]
| 1,518 |
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 P... | Mathlib/RingTheory/Polynomial/Hermite/Basic.lean | 103 | 107 | theorem coeff_hermite_self (n : β) : coeff (hermite n) n = 1 := by |
induction' n with n ih
Β· apply coeff_C
Β· rw [coeff_hermite_succ_succ, ih, coeff_hermite_of_lt, mul_zero, sub_zero]
simp
| 1,518 |
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 P... | Mathlib/RingTheory/Polynomial/Hermite/Basic.lean | 111 | 116 | theorem degree_hermite (n : β) : (hermite n).degree = n := by |
rw [degree_eq_of_le_of_coeff_ne_zero]
Β· simp_rw [degree_le_iff_coeff_zero, Nat.cast_lt]
rintro m hnm
exact coeff_hermite_of_lt hnm
Β· simp [coeff_hermite_self n]
| 1,518 |
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 P... | Mathlib/RingTheory/Polynomial/Hermite/Basic.lean | 125 | 126 | theorem leadingCoeff_hermite (n : β) : (hermite n).leadingCoeff = 1 := by |
rw [β coeff_natDegree, natDegree_hermite, coeff_hermite_self]
| 1,518 |
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 P... | Mathlib/RingTheory/Polynomial/Hermite/Basic.lean | 133 | 143 | theorem coeff_hermite_of_odd_add {n k : β} (hnk : Odd (n + k)) : coeff (hermite n) k = 0 := by |
induction' n with n ih generalizing k
Β· rw [zero_add k] at hnk
exact coeff_hermite_of_lt hnk.pos
Β· cases' k with k
Β· rw [Nat.succ_add_eq_add_succ] at hnk
rw [coeff_hermite_succ_zero, ih hnk, neg_zero]
Β· rw [coeff_hermite_succ_succ, ih, ih, mul_zero, sub_zero]
Β· rwa [Nat.succ_add_eq_add_su... | 1,518 |
import Mathlib.Algebra.Polynomial.Eval
#align_import data.polynomial.degree.lemmas from "leanprover-community/mathlib"@"728baa2f54e6062c5879a3e397ac6bac323e506f"
noncomputable section
open Polynomial
open Finsupp Finset
namespace Polynomial
universe u v w
variable {R : Type u} {S : Type v} {ΞΉ : Type w} {a b ... | Mathlib/Algebra/Polynomial/Degree/Lemmas.lean | 37 | 61 | theorem natDegree_comp_le : natDegree (p.comp q) β€ natDegree p * natDegree q :=
letI := Classical.decEq R
if h0 : p.comp q = 0 then by rw [h0, natDegree_zero]; exact Nat.zero_le _
else
WithBot.coe_le_coe.1 <|
calc
β(natDegree (p.comp q)) = degree (p.comp q) := (degree_eq_natDegree h0).symm
... |
rw [natDegree_C, Nat.cast_zero, zero_add, nsmul_eq_mul];
simp
_ β€ (natDegree p * natDegree q : β) :=
WithBot.coe_le_coe.2 <|
mul_le_mul_of_nonneg_right (le_natDegree_of_ne_zero (mem_support_iff.1 hn))
(Nat.zero_le _)
| 1,519 |
import Mathlib.Algebra.Polynomial.Eval
#align_import data.polynomial.degree.lemmas from "leanprover-community/mathlib"@"728baa2f54e6062c5879a3e397ac6bac323e506f"
noncomputable section
open Polynomial
open Finsupp Finset
namespace Polynomial
universe u v w
variable {R : Type u} {S : Type v} {ΞΉ : Type w} {a b ... | Mathlib/Algebra/Polynomial/Degree/Lemmas.lean | 72 | 73 | theorem natDegree_le_iff_coeff_eq_zero : p.natDegree β€ n β β N : β, n < N β p.coeff N = 0 := by |
simp_rw [natDegree_le_iff_degree_le, degree_le_iff_coeff_zero, Nat.cast_lt]
| 1,519 |
import Mathlib.Algebra.Polynomial.Eval
#align_import data.polynomial.degree.lemmas from "leanprover-community/mathlib"@"728baa2f54e6062c5879a3e397ac6bac323e506f"
noncomputable section
open Polynomial
open Finsupp Finset
namespace Polynomial
universe u v w
variable {R : Type u} {S : Type v} {ΞΉ : Type w} {a b ... | Mathlib/Algebra/Polynomial/Degree/Lemmas.lean | 76 | 81 | theorem natDegree_add_le_iff_left {n : β} (p q : R[X]) (qn : q.natDegree β€ n) :
(p + q).natDegree β€ n β p.natDegree β€ n := by |
refine β¨fun h => ?_, fun h => natDegree_add_le_of_degree_le h qnβ©
refine natDegree_le_iff_coeff_eq_zero.mpr fun m hm => ?_
convert natDegree_le_iff_coeff_eq_zero.mp h m hm using 1
rw [coeff_add, natDegree_le_iff_coeff_eq_zero.mp qn _ hm, add_zero]
| 1,519 |
import Mathlib.Algebra.Polynomial.Eval
#align_import data.polynomial.degree.lemmas from "leanprover-community/mathlib"@"728baa2f54e6062c5879a3e397ac6bac323e506f"
noncomputable section
open Polynomial
open Finsupp Finset
namespace Polynomial
universe u v w
variable {R : Type u} {S : Type v} {ΞΉ : Type w} {a b ... | Mathlib/Algebra/Polynomial/Degree/Lemmas.lean | 84 | 87 | theorem natDegree_add_le_iff_right {n : β} (p q : R[X]) (pn : p.natDegree β€ n) :
(p + q).natDegree β€ n β q.natDegree β€ n := by |
rw [add_comm]
exact natDegree_add_le_iff_left _ _ pn
| 1,519 |
import Mathlib.Algebra.Polynomial.Eval
#align_import data.polynomial.degree.lemmas from "leanprover-community/mathlib"@"728baa2f54e6062c5879a3e397ac6bac323e506f"
noncomputable section
open Polynomial
open Finsupp Finset
namespace Polynomial
universe u v w
variable {R : Type u} {S : Type v} {ΞΉ : Type w} {a b ... | Mathlib/Algebra/Polynomial/Degree/Lemmas.lean | 90 | 94 | theorem natDegree_C_mul_le (a : R) (f : R[X]) : (C a * f).natDegree β€ f.natDegree :=
calc
(C a * f).natDegree β€ (C a).natDegree + f.natDegree := natDegree_mul_le
_ = 0 + f.natDegree := by | rw [natDegree_C a]
_ = f.natDegree := zero_add _
| 1,519 |
import Mathlib.Algebra.Polynomial.Eval
#align_import data.polynomial.degree.lemmas from "leanprover-community/mathlib"@"728baa2f54e6062c5879a3e397ac6bac323e506f"
noncomputable section
open Polynomial
open Finsupp Finset
namespace Polynomial
universe u v w
variable {R : Type u} {S : Type v} {ΞΉ : Type w} {a b ... | Mathlib/Algebra/Polynomial/Degree/Lemmas.lean | 98 | 102 | theorem natDegree_mul_C_le (f : R[X]) (a : R) : (f * C a).natDegree β€ f.natDegree :=
calc
(f * C a).natDegree β€ f.natDegree + (C a).natDegree := natDegree_mul_le
_ = f.natDegree + 0 := by | rw [natDegree_C a]
_ = f.natDegree := add_zero _
| 1,519 |
import Mathlib.Algebra.Polynomial.Eval
#align_import data.polynomial.degree.lemmas from "leanprover-community/mathlib"@"728baa2f54e6062c5879a3e397ac6bac323e506f"
noncomputable section
open Polynomial
open Finsupp Finset
namespace Polynomial
universe u v w
variable {R : Type u} {S : Type v} {ΞΉ : Type w} {a b ... | Mathlib/Algebra/Polynomial/Degree/Lemmas.lean | 111 | 117 | theorem natDegree_C_mul_eq_of_mul_eq_one {ai : R} (au : ai * a = 1) :
(C a * p).natDegree = p.natDegree :=
le_antisymm (natDegree_C_mul_le a p)
(calc
p.natDegree = (1 * p).natDegree := by | nth_rw 1 [β one_mul p]
_ = (C ai * (C a * p)).natDegree := by rw [β C_1, β au, RingHom.map_mul, β mul_assoc]
_ β€ (C a * p).natDegree := natDegree_C_mul_le ai (C a * p))
| 1,519 |
import Mathlib.Algebra.Polynomial.Eval
#align_import data.polynomial.degree.lemmas from "leanprover-community/mathlib"@"728baa2f54e6062c5879a3e397ac6bac323e506f"
noncomputable section
open Polynomial
open Finsupp Finset
namespace Polynomial
universe u v w
variable {R : Type u} {S : Type v} {ΞΉ : Type w} {a b ... | Mathlib/Algebra/Polynomial/Degree/Lemmas.lean | 121 | 127 | theorem natDegree_mul_C_eq_of_mul_eq_one {ai : R} (au : a * ai = 1) :
(p * C a).natDegree = p.natDegree :=
le_antisymm (natDegree_mul_C_le p a)
(calc
p.natDegree = (p * 1).natDegree := by | nth_rw 1 [β mul_one p]
_ = (p * C a * C ai).natDegree := by rw [β C_1, β au, RingHom.map_mul, β mul_assoc]
_ β€ (p * C a).natDegree := natDegree_mul_C_le (p * C a) ai)
| 1,519 |
import Mathlib.Algebra.Polynomial.Eval
#align_import data.polynomial.degree.lemmas from "leanprover-community/mathlib"@"728baa2f54e6062c5879a3e397ac6bac323e506f"
noncomputable section
open Polynomial
open Finsupp Finset
namespace Polynomial
universe u v w
variable {R : Type u} {S : Type v} {ΞΉ : Type w} {a b ... | Mathlib/Algebra/Polynomial/Degree/Lemmas.lean | 356 | 357 | theorem degree_mul_C (a0 : a β 0) : (p * C a).degree = p.degree := by |
rw [degree_mul, degree_C a0, add_zero]
| 1,519 |
import Mathlib.Algebra.Polynomial.Eval
#align_import data.polynomial.degree.lemmas from "leanprover-community/mathlib"@"728baa2f54e6062c5879a3e397ac6bac323e506f"
noncomputable section
open Polynomial
open Finsupp Finset
namespace Polynomial
universe u v w
variable {R : Type u} {S : Type v} {ΞΉ : Type w} {a b ... | Mathlib/Algebra/Polynomial/Degree/Lemmas.lean | 361 | 362 | theorem degree_C_mul (a0 : a β 0) : (C a * p).degree = p.degree := by |
rw [degree_mul, degree_C a0, zero_add]
| 1,519 |
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