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.Degree.Definitions
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.Algebra.Polynomial.Monic
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.Tactic.Abel
#align_import ring_theory.polynomial.pochhammer from "leanprover-community/mathlib"@"53b216bcc1146df1c4a0a868778... | Mathlib/RingTheory/Polynomial/Pochhammer.lean | 295 | 295 | theorem descPochhammer_zero_eval_zero : (descPochhammer R 0).eval 0 = 1 := by | simp
| 1,962 |
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.Algebra.Polynomial.Monic
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.Tactic.Abel
#align_import ring_theory.polynomial.pochhammer from "leanprover-community/mathlib"@"53b216bcc1146df1c4a0a868778... | Mathlib/RingTheory/Polynomial/Pochhammer.lean | 298 | 299 | theorem descPochhammer_ne_zero_eval_zero {n : β} (h : n β 0) : (descPochhammer R n).eval 0 = 0 := by |
simp [descPochhammer_eval_zero, h]
| 1,962 |
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.Algebra.Polynomial.Monic
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.Tactic.Abel
#align_import ring_theory.polynomial.pochhammer from "leanprover-community/mathlib"@"53b216bcc1146df1c4a0a868778... | Mathlib/RingTheory/Polynomial/Pochhammer.lean | 301 | 312 | theorem descPochhammer_succ_right (n : β) :
descPochhammer R (n + 1) = descPochhammer R n * (X - (n : R[X])) := by |
suffices h : descPochhammer β€ (n + 1) = descPochhammer β€ n * (X - (n : β€[X])) by
apply_fun Polynomial.map (algebraMap β€ R) at h
simpa [descPochhammer_map, Polynomial.map_mul, Polynomial.map_add, map_X,
Polynomial.map_intCast] using h
induction' n with n ih
Β· simp [descPochhammer]
Β· conv_lhs =>
... | 1,962 |
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.Algebra.Polynomial.Monic
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.Tactic.Abel
#align_import ring_theory.polynomial.pochhammer from "leanprover-community/mathlib"@"53b216bcc1146df1c4a0a868778... | Mathlib/RingTheory/Polynomial/Pochhammer.lean | 315 | 324 | theorem descPochhammer_natDegree (n : β) [NoZeroDivisors R] [Nontrivial R] :
(descPochhammer R n).natDegree = n := by |
induction' n with n hn
Β· simp
Β· have : natDegree (X - (n : R[X])) = 1 := natDegree_X_sub_C (n : R)
rw [descPochhammer_succ_right,
natDegree_mul _ (ne_zero_of_natDegree_gt <| this.symm βΈ Nat.zero_lt_one), hn, this]
cases n
Β· simp
Β· refine ne_zero_of_natDegree_gt <| hn.symm βΈ Nat.add_one_po... | 1,962 |
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.Algebra.Polynomial.Monic
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.Tactic.Abel
#align_import ring_theory.polynomial.pochhammer from "leanprover-community/mathlib"@"53b216bcc1146df1c4a0a868778... | Mathlib/RingTheory/Polynomial/Pochhammer.lean | 326 | 329 | theorem descPochhammer_succ_eval {S : Type*} [Ring S] (n : β) (k : S) :
(descPochhammer S (n + 1)).eval k = (descPochhammer S n).eval k * (k - n) := by |
rw [descPochhammer_succ_right, mul_sub, eval_sub, eval_mul_X, β Nat.cast_comm, β C_eq_natCast,
eval_C_mul, Nat.cast_comm, β mul_sub]
| 1,962 |
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.Algebra.Polynomial.Monic
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.Tactic.Abel
#align_import ring_theory.polynomial.pochhammer from "leanprover-community/mathlib"@"53b216bcc1146df1c4a0a868778... | Mathlib/RingTheory/Polynomial/Pochhammer.lean | 331 | 339 | theorem descPochhammer_succ_comp_X_sub_one (n : β) :
(descPochhammer R (n + 1)).comp (X - 1) =
descPochhammer R (n + 1) - (n + (1 : R[X])) β’ (descPochhammer R n).comp (X - 1) := by |
suffices (descPochhammer β€ (n + 1)).comp (X - 1) =
descPochhammer β€ (n + 1) - (n + 1) * (descPochhammer β€ n).comp (X - 1)
by simpa [map_comp] using congr_arg (Polynomial.map (Int.castRingHom R)) this
nth_rw 2 [descPochhammer_succ_left]
rw [β sub_mul, descPochhammer_succ_right β€ n, mul_comp, mul_comm, s... | 1,962 |
import Mathlib.Algebra.Polynomial.Smeval
import Mathlib.GroupTheory.GroupAction.Ring
import Mathlib.RingTheory.Polynomial.Pochhammer
section Multichoose
open Function Polynomial
class BinomialRing (R : Type*) [AddCommMonoid R] [Pow R β] where
nsmul_right_injective (n : β) (h : n β 0) : Injective (n β’ Β· : R β... | Mathlib/RingTheory/Binomial.lean | 90 | 97 | theorem ascPochhammer_smeval_cast (R : Type*) [Semiring R] {S : Type*} [NonAssocSemiring S]
[Pow S β] [Module R S] [IsScalarTower R S S] [NatPowAssoc S]
(x : S) (n : β) : (ascPochhammer R n).smeval x = (ascPochhammer β n).smeval x := by |
induction' n with n hn
Β· simp only [Nat.zero_eq, ascPochhammer_zero, smeval_one, one_smul]
Β· simp only [ascPochhammer_succ_right, mul_add, smeval_add, smeval_mul_X, β Nat.cast_comm]
simp only [β C_eq_natCast, smeval_C_mul, hn, β nsmul_eq_smul_cast R n]
exact rfl
| 1,963 |
import Mathlib.Algebra.Polynomial.Smeval
import Mathlib.GroupTheory.GroupAction.Ring
import Mathlib.RingTheory.Polynomial.Pochhammer
section Multichoose
open Function Polynomial
class BinomialRing (R : Type*) [AddCommMonoid R] [Pow R β] where
nsmul_right_injective (n : β) (h : n β 0) : Injective (n β’ Β· : R β... | Mathlib/RingTheory/Binomial.lean | 101 | 103 | theorem ascPochhammer_smeval_eq_eval [Semiring R] (r : R) (n : β) :
(ascPochhammer β n).smeval r = (ascPochhammer R n).eval r := by |
rw [eval_eq_smeval, ascPochhammer_smeval_cast R]
| 1,963 |
import Mathlib.Algebra.Polynomial.Smeval
import Mathlib.GroupTheory.GroupAction.Ring
import Mathlib.RingTheory.Polynomial.Pochhammer
section Multichoose
open Function Polynomial
class BinomialRing (R : Type*) [AddCommMonoid R] [Pow R β] where
nsmul_right_injective (n : β) (h : n β 0) : Injective (n β’ Β· : R β... | Mathlib/RingTheory/Binomial.lean | 107 | 115 | theorem descPochhammer_smeval_eq_ascPochhammer (r : R) (n : β) :
(descPochhammer β€ n).smeval r = (ascPochhammer β n).smeval (r - n + 1) := by |
induction n with
| zero => simp only [descPochhammer_zero, ascPochhammer_zero, smeval_one, npow_zero]
| succ n ih =>
rw [Nat.cast_succ, sub_add, add_sub_cancel_right, descPochhammer_succ_right, smeval_mul, ih,
ascPochhammer_succ_left, X_mul, smeval_mul_X, smeval_comp, smeval_sub, β C_eq_natCast,
... | 1,963 |
import Mathlib.Algebra.Polynomial.Smeval
import Mathlib.GroupTheory.GroupAction.Ring
import Mathlib.RingTheory.Polynomial.Pochhammer
section Multichoose
open Function Polynomial
class BinomialRing (R : Type*) [AddCommMonoid R] [Pow R β] where
nsmul_right_injective (n : β) (h : n β 0) : Injective (n β’ Β· : R β... | Mathlib/RingTheory/Binomial.lean | 117 | 127 | theorem descPochhammer_smeval_eq_descFactorial (n k : β) :
(descPochhammer β€ k).smeval (n : R) = n.descFactorial k := by |
induction k with
| zero =>
rw [descPochhammer_zero, Nat.descFactorial_zero, Nat.cast_one, smeval_one, npow_zero, one_smul]
| succ k ih =>
rw [descPochhammer_succ_right, Nat.descFactorial_succ, smeval_mul, ih, mul_comm, Nat.cast_mul,
smeval_sub, smeval_X, smeval_natCast, npow_one, npow_zero, nsmul_o... | 1,963 |
import Mathlib.Algebra.Polynomial.Smeval
import Mathlib.GroupTheory.GroupAction.Ring
import Mathlib.RingTheory.Polynomial.Pochhammer
section Multichoose
open Function Polynomial
class BinomialRing (R : Type*) [AddCommMonoid R] [Pow R β] where
nsmul_right_injective (n : β) (h : n β 0) : Injective (n β’ Β· : R β... | Mathlib/RingTheory/Binomial.lean | 129 | 138 | theorem ascPochhammer_smeval_neg_eq_descPochhammer (r : R) (k : β) :
(ascPochhammer β k).smeval (-r) = (-1)^k * (descPochhammer β€ k).smeval r := by |
induction k with
| zero => simp only [ascPochhammer_zero, descPochhammer_zero, smeval_one, npow_zero, one_mul]
| succ k ih =>
simp only [ascPochhammer_succ_right, smeval_mul, ih, descPochhammer_succ_right, sub_eq_add_neg]
have h : (X + (k : β[X])).smeval (-r) = - (X + (-k : β€[X])).smeval r := by
si... | 1,963 |
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.Polynomial.Monic
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.LinearAlgebra.Vandermonde
import Mathlib.RingTheory.Polynomial.Pochhammer
namespace Nat
def superFactorial : β β β
| 0 => 1
| succ n => factorial n.succ * superFactoria... | Mathlib/Data/Nat/Factorial/SuperFactorial.lean | 75 | 86 | theorem det_vandermonde_id_eq_superFactorial (n : β) :
(Matrix.vandermonde (fun (i : Fin (n + 1)) β¦ (i : R))).det = Nat.superFactorial n := by |
induction' n with n hn
Β· simp [Matrix.det_vandermonde]
Β· rw [Nat.superFactorial, Matrix.det_vandermonde, Fin.prod_univ_succAbove _ 0]
push_cast
congr
Β· simp only [Fin.val_zero, Nat.cast_zero, sub_zero]
norm_cast
simp [Fin.prod_univ_eq_prod_range (fun i β¦ (βi + 1)) (n + 1)]
Β· rw [Matri... | 1,964 |
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.Polynomial.Monic
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.LinearAlgebra.Vandermonde
import Mathlib.RingTheory.Polynomial.Pochhammer
namespace Nat
def superFactorial : β β β
| 0 => 1
| succ n => factorial n.succ * superFactoria... | Mathlib/Data/Nat/Factorial/SuperFactorial.lean | 96 | 102 | theorem superFactorial_four_mul (n : β) :
sf (4 * n) = ((β i β range (2 * n), (2 * i + 1) !) * 2 ^ n) ^ 2 * (2 * n) ! :=
calc
sf (4 * n) = (β i β range (2 * n), (2 * i + 1) !) ^ 2 * 2 ^ (2 * n) * (2 * n) ! := by |
rw [β superFactorial_two_mul, β mul_assoc, Nat.mul_two]
_ = ((β i β range (2 * n), (2 * i + 1) !) * 2 ^ n) ^ 2 * (2 * n) ! := by
rw [pow_mul', mul_pow]
| 1,964 |
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.Polynomial.Monic
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.LinearAlgebra.Vandermonde
import Mathlib.RingTheory.Polynomial.Pochhammer
namespace Nat
def superFactorial : β β β
| 0 => 1
| succ n => factorial n.succ * superFactoria... | Mathlib/Data/Nat/Factorial/SuperFactorial.lean | 114 | 125 | theorem superFactorial_dvd_vandermonde_det {n : β} (v : Fin (n + 1) β β€) :
β(Nat.superFactorial n) β£ (Matrix.vandermonde v).det := by |
let m := inf' univ β¨0, mem_univ _β© v
let w' := fun i β¦ (v i - m).toNat
have hw' : β i, (w' i : β€) = v i - m := fun i β¦ Int.toNat_sub_of_le (inf'_le _ (mem_univ _))
have h := Matrix.det_eval_matrixOfPolynomials_eq_det_vandermonde (fun i β¦ β(w' i))
(fun i => descPochhammer β€ i)
(fun i => descPochhamm... | 1,964 |
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.RingTheory.Polynomial.Nilpotent
open scoped Classical Polynomial
open Polynomial
noncomputable section
| Mathlib/RingTheory/Polynomial/IrreducibleRing.lean | 37 | 61 | theorem Polynomial.Monic.irreducible_of_irreducible_map_of_isPrime_nilradical
{R S : Type*} [CommRing R] [(nilradical R).IsPrime] [CommRing S] [IsDomain S]
(Ο : R β+* S) (f : R[X]) (hm : f.Monic) (hi : Irreducible (f.map Ο)) : Irreducible f := by |
let R' := R β§Έ nilradical R
let Ο : R' β+* S := Ideal.Quotient.lift (nilradical R) Ο
(haveI := RingHom.ker_isPrime Ο; nilradical_le_prime (RingHom.ker Ο))
let ΞΉ := algebraMap R R'
rw [show Ο = Ο.comp ΞΉ from rfl, β map_map] at hi
replace hi := hm.map ΞΉ |>.irreducible_of_irreducible_map _ _ hi
refine β¨fun... | 1,965 |
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
namespace Polynomial
open Polynomial
section Semiring
variable {R : Type*} [Semiring R] (p q : R... | Mathlib/Algebra/Polynomial/Mirror.lean | 44 | 44 | theorem mirror_zero : (0 : R[X]).mirror = 0 := by | simp [mirror]
| 1,966 |
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
namespace Polynomial
open Polynomial
section Semiring
variable {R : Type*} [Semiring R] (p q : R... | Mathlib/Algebra/Polynomial/Mirror.lean | 47 | 53 | theorem mirror_monomial (n : β) (a : R) : (monomial n a).mirror = monomial n a := by |
classical
by_cases ha : a = 0
Β· rw [ha, monomial_zero_right, mirror_zero]
Β· rw [mirror, reverse, natDegree_monomial n a, if_neg ha, natTrailingDegree_monomial ha, β
C_mul_X_pow_eq_monomial, reflect_C_mul_X_pow, revAt_le (le_refl n), tsub_self, pow_zero,
mul_one]
| 1,966 |
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
namespace Polynomial
open Polynomial
section Semiring
variable {R : Type*} [Semiring R] (p q : R... | Mathlib/Algebra/Polynomial/Mirror.lean | 66 | 72 | theorem mirror_natDegree : p.mirror.natDegree = p.natDegree := by |
by_cases hp : p = 0
Β· rw [hp, mirror_zero]
nontriviality R
rw [mirror, natDegree_mul', reverse_natDegree, natDegree_X_pow,
tsub_add_cancel_of_le p.natTrailingDegree_le_natDegree]
rwa [leadingCoeff_X_pow, mul_one, reverse_leadingCoeff, Ne, trailingCoeff_eq_zero]
| 1,966 |
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
namespace Polynomial
open Polynomial
section Semiring
variable {R : Type*} [Semiring R] (p q : R... | Mathlib/Algebra/Polynomial/Mirror.lean | 75 | 79 | theorem mirror_natTrailingDegree : p.mirror.natTrailingDegree = p.natTrailingDegree := by |
by_cases hp : p = 0
Β· rw [hp, mirror_zero]
Β· rw [mirror, natTrailingDegree_mul_X_pow ((mt reverse_eq_zero.mp) hp),
natTrailingDegree_reverse, zero_add]
| 1,966 |
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
namespace Polynomial
open Polynomial
section Semiring
variable {R : Type*} [Semiring R] (p q : R... | Mathlib/Algebra/Polynomial/Mirror.lean | 82 | 97 | theorem coeff_mirror (n : β) :
p.mirror.coeff n = p.coeff (revAt (p.natDegree + p.natTrailingDegree) n) := by |
by_cases h2 : p.natDegree < n
Β· rw [coeff_eq_zero_of_natDegree_lt (by rwa [mirror_natDegree])]
by_cases h1 : n β€ p.natDegree + p.natTrailingDegree
Β· rw [revAt_le h1, coeff_eq_zero_of_lt_natTrailingDegree]
exact (tsub_lt_iff_left h1).mpr (Nat.add_lt_add_right h2 _)
Β· rw [β revAtFun_eq, revAtFun, i... | 1,966 |
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
namespace Polynomial
open Polynomial
section Semiring
variable {R : Type*} [Semiring R] (p q : R... | Mathlib/Algebra/Polynomial/Mirror.lean | 101 | 120 | theorem mirror_eval_one : p.mirror.eval 1 = p.eval 1 := by |
simp_rw [eval_eq_sum_range, one_pow, mul_one, mirror_natDegree]
refine Finset.sum_bij_ne_zero ?_ ?_ ?_ ?_ ?_
Β· exact fun n _ _ => revAt (p.natDegree + p.natTrailingDegree) n
Β· intro n hn hp
rw [Finset.mem_range_succ_iff] at *
rw [revAt_le (hn.trans (Nat.le_add_right _ _))]
rw [tsub_le_iff_tsub_le, ... | 1,966 |
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
namespace Polynomial
open Polynomial
section Semiring
variable {R : Type*} [Semiring R] (p q : R... | Mathlib/Algebra/Polynomial/Mirror.lean | 151 | 153 | theorem mirror_trailingCoeff : p.mirror.trailingCoeff = p.leadingCoeff := by |
rw [leadingCoeff, trailingCoeff, mirror_natTrailingDegree, coeff_mirror,
revAt_le (Nat.le_add_left _ _), add_tsub_cancel_right]
| 1,966 |
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.Algebra.Polynomial.RingDivision
#align_import field_theory.ratfunc from "leanprover-community/mathlib"@"bf9bbbcf0c1c1ead18280b0d010e417b10abb1b6"
noncomputable section
open scoped Classical
open scoped nonZeroDivisors Polynomial
universe u v
va... | Mathlib/FieldTheory/RatFunc/Defs.lean | 123 | 127 | theorem liftOn_ofFractionRing_mk {P : Sort v} (n : K[X]) (d : K[X]β°) (f : K[X] β K[X] β P)
(H : β {p q p' q'} (_hq : q β K[X]β°) (_hq' : q' β K[X]β°), q' * p = q * p' β f p q = f p' q') :
RatFunc.liftOn (ofFractionRing (Localization.mk n d)) f @H = f n d := by |
rw [RatFunc.liftOn]
exact Localization.liftOn_mk _ _ _ _
| 1,967 |
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.Algebra.Polynomial.RingDivision
#align_import field_theory.ratfunc from "leanprover-community/mathlib"@"bf9bbbcf0c1c1ead18280b0d010e417b10abb1b6"
noncomputable section
open scoped Classical
open scoped nonZeroDivisors Polynomial
universe u v
va... | Mathlib/FieldTheory/RatFunc/Defs.lean | 130 | 136 | theorem liftOn_condition_of_liftOn'_condition {P : Sort v} {f : K[X] β K[X] β P}
(H : β {p q a} (hq : q β 0) (_ha : a β 0), f (a * p) (a * q) = f p q) β¦p q p' q' : K[X]β¦
(hq : q β 0) (hq' : q' β 0) (h : q' * p = q * p') : f p q = f p' q' :=
calc
f p q = f (q' * p) (q' * q) := (H hq hq').symm
_ = f (q ... | rw [h, mul_comm q']
_ = f p' q' := H hq' hq
| 1,967 |
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.Algebra.Polynomial.RingDivision
#align_import field_theory.ratfunc from "leanprover-community/mathlib"@"bf9bbbcf0c1c1ead18280b0d010e417b10abb1b6"
noncomputable section
open scoped Classical
open scoped nonZeroDivisors Polynomial
universe u v
va... | Mathlib/FieldTheory/RatFunc/Defs.lean | 154 | 155 | theorem mk_eq_div' (p q : K[X]) :
RatFunc.mk p q = ofFractionRing (algebraMap _ _ p / algebraMap _ _ q) := by | rw [RatFunc.mk]
| 1,967 |
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.Algebra.Polynomial.RingDivision
#align_import field_theory.ratfunc from "leanprover-community/mathlib"@"bf9bbbcf0c1c1ead18280b0d010e417b10abb1b6"
noncomputable section
open scoped Classical
open scoped nonZeroDivisors Polynomial
universe u v
va... | Mathlib/FieldTheory/RatFunc/Defs.lean | 158 | 159 | theorem mk_zero (p : K[X]) : RatFunc.mk p 0 = ofFractionRing (0 : FractionRing K[X]) := by |
rw [mk_eq_div', RingHom.map_zero, div_zero]
| 1,967 |
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.Algebra.Polynomial.RingDivision
#align_import field_theory.ratfunc from "leanprover-community/mathlib"@"bf9bbbcf0c1c1ead18280b0d010e417b10abb1b6"
noncomputable section
open scoped Classical
open scoped nonZeroDivisors Polynomial
universe u v
va... | Mathlib/FieldTheory/RatFunc/Defs.lean | 162 | 165 | theorem mk_coe_def (p : K[X]) (q : K[X]β°) :
-- Porting note: filled in `(FractionRing K[X])` that was an underscore.
RatFunc.mk p q = ofFractionRing (IsLocalization.mk' (FractionRing K[X]) p q) := by |
simp only [mk_eq_div', β Localization.mk_eq_mk', FractionRing.mk_eq_div]
| 1,967 |
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.Algebra.Polynomial.RingDivision
#align_import field_theory.ratfunc from "leanprover-community/mathlib"@"bf9bbbcf0c1c1ead18280b0d010e417b10abb1b6"
noncomputable section
open scoped Classical
open scoped nonZeroDivisors Polynomial
universe u v
va... | Mathlib/FieldTheory/RatFunc/Defs.lean | 168 | 171 | theorem mk_def_of_mem (p : K[X]) {q} (hq : q β K[X]β°) :
RatFunc.mk p q = ofFractionRing (IsLocalization.mk' (FractionRing K[X]) p β¨q, hqβ©) := by |
-- Porting note: there was an `[anonymous]` in the simp set
simp only [β mk_coe_def]
| 1,967 |
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.Algebra.Polynomial.RingDivision
#align_import field_theory.ratfunc from "leanprover-community/mathlib"@"bf9bbbcf0c1c1ead18280b0d010e417b10abb1b6"
noncomputable section
open scoped Classical
open scoped nonZeroDivisors Polynomial
universe u v
va... | Mathlib/FieldTheory/RatFunc/Defs.lean | 181 | 185 | theorem mk_eq_localization_mk (p : K[X]) {q : K[X]} (hq : q β 0) :
RatFunc.mk p q =
ofFractionRing (Localization.mk p β¨q, mem_nonZeroDivisors_iff_ne_zero.mpr hqβ©) := by |
-- Porting note: the original proof, did not need to pass `hq`
rw [mk_def_of_ne _ hq, Localization.mk_eq_mk']
| 1,967 |
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.Algebra.Polynomial.RingDivision
#align_import field_theory.ratfunc from "leanprover-community/mathlib"@"bf9bbbcf0c1c1ead18280b0d010e417b10abb1b6"
noncomputable section
open scoped Classical
open scoped nonZeroDivisors Polynomial
universe u v
va... | Mathlib/FieldTheory/RatFunc/Defs.lean | 189 | 192 | theorem mk_one' (p : K[X]) :
RatFunc.mk p 1 = ofFractionRing (algebraMap K[X] (FractionRing K[X]) p) := by |
-- Porting note: had to hint `M := K[X]β°` below
rw [β IsLocalization.mk'_one (M := K[X]β°) (FractionRing K[X]) p, β mk_coe_def, Submonoid.coe_one]
| 1,967 |
import Mathlib.FieldTheory.RatFunc.Defs
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Polynomial.Content
#align_import field_theory.ratfunc from "leanprover-community/mathlib"@"bf9bbbcf0c1c1ead18280b0d010e417b10abb1b6"
universe u v
noncompu... | Mathlib/FieldTheory/RatFunc/Basic.lean | 75 | 76 | theorem ofFractionRing_zero : (ofFractionRing 0 : RatFunc K) = 0 := by |
simp only [Zero.zero, OfNat.ofNat, RatFunc.zero]
| 1,968 |
import Mathlib.FieldTheory.RatFunc.Defs
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Polynomial.Content
#align_import field_theory.ratfunc from "leanprover-community/mathlib"@"bf9bbbcf0c1c1ead18280b0d010e417b10abb1b6"
universe u v
noncompu... | Mathlib/FieldTheory/RatFunc/Basic.lean | 89 | 91 | theorem ofFractionRing_add (p q : FractionRing K[X]) :
ofFractionRing (p + q) = ofFractionRing p + ofFractionRing q := by |
simp only [HAdd.hAdd, Add.add, RatFunc.add]
| 1,968 |
import Mathlib.FieldTheory.RatFunc.Defs
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Polynomial.Content
#align_import field_theory.ratfunc from "leanprover-community/mathlib"@"bf9bbbcf0c1c1ead18280b0d010e417b10abb1b6"
universe u v
noncompu... | Mathlib/FieldTheory/RatFunc/Basic.lean | 104 | 106 | theorem ofFractionRing_sub (p q : FractionRing K[X]) :
ofFractionRing (p - q) = ofFractionRing p - ofFractionRing q := by |
simp only [Sub.sub, HSub.hSub, RatFunc.sub]
| 1,968 |
import Mathlib.FieldTheory.RatFunc.Defs
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Polynomial.Content
#align_import field_theory.ratfunc from "leanprover-community/mathlib"@"bf9bbbcf0c1c1ead18280b0d010e417b10abb1b6"
universe u v
noncompu... | Mathlib/FieldTheory/RatFunc/Basic.lean | 117 | 118 | theorem ofFractionRing_neg (p : FractionRing K[X]) :
ofFractionRing (-p) = -ofFractionRing p := by | simp only [Neg.neg, RatFunc.neg]
| 1,968 |
import Mathlib.FieldTheory.RatFunc.Defs
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Polynomial.Content
#align_import field_theory.ratfunc from "leanprover-community/mathlib"@"bf9bbbcf0c1c1ead18280b0d010e417b10abb1b6"
universe u v
noncompu... | Mathlib/FieldTheory/RatFunc/Basic.lean | 131 | 132 | theorem ofFractionRing_one : (ofFractionRing 1 : RatFunc K) = 1 := by |
simp only [One.one, OfNat.ofNat, RatFunc.one]
| 1,968 |
import Mathlib.FieldTheory.RatFunc.Defs
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Polynomial.Content
#align_import field_theory.ratfunc from "leanprover-community/mathlib"@"bf9bbbcf0c1c1ead18280b0d010e417b10abb1b6"
universe u v
noncompu... | Mathlib/FieldTheory/RatFunc/Basic.lean | 145 | 147 | theorem ofFractionRing_mul (p q : FractionRing K[X]) :
ofFractionRing (p * q) = ofFractionRing p * ofFractionRing q := by |
simp only [Mul.mul, HMul.hMul, RatFunc.mul]
| 1,968 |
import Mathlib.FieldTheory.RatFunc.Defs
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Polynomial.Content
#align_import field_theory.ratfunc from "leanprover-community/mathlib"@"bf9bbbcf0c1c1ead18280b0d010e417b10abb1b6"
universe u v
noncompu... | Mathlib/FieldTheory/RatFunc/Basic.lean | 164 | 166 | theorem ofFractionRing_div (p q : FractionRing K[X]) :
ofFractionRing (p / q) = ofFractionRing p / ofFractionRing q := by |
simp only [Div.div, HDiv.hDiv, RatFunc.div]
| 1,968 |
import Mathlib.FieldTheory.RatFunc.Defs
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Polynomial.Content
#align_import field_theory.ratfunc from "leanprover-community/mathlib"@"bf9bbbcf0c1c1ead18280b0d010e417b10abb1b6"
universe u v
noncompu... | Mathlib/FieldTheory/RatFunc/Basic.lean | 177 | 179 | theorem ofFractionRing_inv (p : FractionRing K[X]) :
ofFractionRing pβ»ΒΉ = (ofFractionRing p)β»ΒΉ := by |
simp only [Inv.inv, RatFunc.inv]
| 1,968 |
import Mathlib.FieldTheory.RatFunc.Defs
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Polynomial.Content
#align_import field_theory.ratfunc from "leanprover-community/mathlib"@"bf9bbbcf0c1c1ead18280b0d010e417b10abb1b6"
universe u v
noncompu... | Mathlib/FieldTheory/RatFunc/Basic.lean | 209 | 211 | theorem ofFractionRing_smul [SMul R (FractionRing K[X])] (c : R) (p : FractionRing K[X]) :
ofFractionRing (c β’ p) = c β’ ofFractionRing p := by |
simp only [SMul.smul, HSMul.hSMul, RatFunc.smul]
| 1,968 |
import Mathlib.FieldTheory.RatFunc.Defs
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Polynomial.Content
#align_import field_theory.ratfunc from "leanprover-community/mathlib"@"bf9bbbcf0c1c1ead18280b0d010e417b10abb1b6"
universe u v
noncompu... | Mathlib/FieldTheory/RatFunc/Basic.lean | 214 | 217 | theorem toFractionRing_smul [SMul R (FractionRing K[X])] (c : R) (p : RatFunc K) :
toFractionRing (c β’ p) = c β’ toFractionRing p := by |
cases p
rw [β ofFractionRing_smul]
| 1,968 |
import Mathlib.FieldTheory.RatFunc.Defs
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Polynomial.Content
#align_import field_theory.ratfunc from "leanprover-community/mathlib"@"bf9bbbcf0c1c1ead18280b0d010e417b10abb1b6"
universe u v
noncompu... | Mathlib/FieldTheory/RatFunc/Basic.lean | 220 | 225 | theorem smul_eq_C_smul (x : RatFunc K) (r : K) : r β’ x = Polynomial.C r β’ x := by |
cases' x with x
-- Porting note: had to specify the induction principle manually
induction x using Localization.induction_on
rw [β ofFractionRing_smul, β ofFractionRing_smul, Localization.smul_mk,
Localization.smul_mk, smul_eq_mul, Polynomial.smul_eq_C_mul]
| 1,968 |
import Mathlib.FieldTheory.RatFunc.Defs
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Polynomial.Content
#align_import field_theory.ratfunc from "leanprover-community/mathlib"@"bf9bbbcf0c1c1ead18280b0d010e417b10abb1b6"
universe u v
noncompu... | Mathlib/FieldTheory/RatFunc/Basic.lean | 235 | 239 | theorem mk_smul (c : R) (p q : K[X]) : RatFunc.mk (c β’ p) q = c β’ RatFunc.mk p q := by |
by_cases hq : q = 0
Β· rw [hq, mk_zero, mk_zero, β ofFractionRing_smul, smul_zero]
Β· rw [mk_eq_localization_mk _ hq, mk_eq_localization_mk _ hq, β Localization.smul_mk, β
ofFractionRing_smul]
| 1,968 |
import Mathlib.FieldTheory.RatFunc.Basic
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Polynomial.Content
noncomputable section
universe u
variable {K : Type u}
namespace RatFunc
section Eval
open scoped Classical
open scoped nonZeroDiv... | Mathlib/FieldTheory/RatFunc/AsPolynomial.lean | 61 | 62 | theorem smul_eq_C_mul (r : K) (x : RatFunc K) : r β’ x = C r * x := by |
rw [Algebra.smul_def, algebraMap_eq_C]
| 1,969 |
import Mathlib.Data.Int.Interval
import Mathlib.RingTheory.Binomial
import Mathlib.RingTheory.HahnSeries.PowerSeries
import Mathlib.RingTheory.HahnSeries.Summable
import Mathlib.FieldTheory.RatFunc.AsPolynomial
import Mathlib.RingTheory.Localization.FractionRing
#align_import ring_theory.laurent_series from "leanprov... | Mathlib/RingTheory/LaurentSeries.lean | 87 | 89 | theorem coeff_coe_powerSeries (x : PowerSeries R) (n : β) :
HahnSeries.coeff (x : LaurentSeries R) n = PowerSeries.coeff R n x := by |
rw [ofPowerSeries_apply_coeff]
| 1,970 |
import Mathlib.Data.Int.Interval
import Mathlib.RingTheory.Binomial
import Mathlib.RingTheory.HahnSeries.PowerSeries
import Mathlib.RingTheory.HahnSeries.Summable
import Mathlib.FieldTheory.RatFunc.AsPolynomial
import Mathlib.RingTheory.Localization.FractionRing
#align_import ring_theory.laurent_series from "leanprov... | Mathlib/RingTheory/LaurentSeries.lean | 106 | 108 | theorem powerSeriesPart_zero : powerSeriesPart (0 : LaurentSeries R) = 0 := by |
ext
simp [(PowerSeries.coeff _ _).map_zero] -- Note: this doesn't get picked up any more
| 1,970 |
import Mathlib.Data.Int.Interval
import Mathlib.RingTheory.Binomial
import Mathlib.RingTheory.HahnSeries.PowerSeries
import Mathlib.RingTheory.HahnSeries.Summable
import Mathlib.FieldTheory.RatFunc.AsPolynomial
import Mathlib.RingTheory.Localization.FractionRing
#align_import ring_theory.laurent_series from "leanprov... | Mathlib/RingTheory/LaurentSeries.lean | 112 | 121 | theorem powerSeriesPart_eq_zero (x : LaurentSeries R) : x.powerSeriesPart = 0 β x = 0 := by |
constructor
Β· contrapose!
simp only [ne_eq]
intro h
rw [PowerSeries.ext_iff, not_forall]
refine β¨0, ?_β©
simp [coeff_order_ne_zero h]
Β· rintro rfl
simp
| 1,970 |
import Mathlib.Data.Int.Interval
import Mathlib.RingTheory.Binomial
import Mathlib.RingTheory.HahnSeries.PowerSeries
import Mathlib.RingTheory.HahnSeries.Summable
import Mathlib.FieldTheory.RatFunc.AsPolynomial
import Mathlib.RingTheory.Localization.FractionRing
#align_import ring_theory.laurent_series from "leanprov... | Mathlib/RingTheory/LaurentSeries.lean | 125 | 140 | theorem single_order_mul_powerSeriesPart (x : LaurentSeries R) :
(single x.order 1 : LaurentSeries R) * x.powerSeriesPart = x := by |
ext n
rw [β sub_add_cancel n x.order, single_mul_coeff_add, sub_add_cancel, one_mul]
by_cases h : x.order β€ n
Β· rw [Int.eq_natAbs_of_zero_le (sub_nonneg_of_le h), coeff_coe_powerSeries,
powerSeriesPart_coeff, β Int.eq_natAbs_of_zero_le (sub_nonneg_of_le h),
add_sub_cancel]
Β· rw [ofPowerSeries_app... | 1,970 |
import Mathlib.Data.Int.Interval
import Mathlib.RingTheory.Binomial
import Mathlib.RingTheory.HahnSeries.PowerSeries
import Mathlib.RingTheory.HahnSeries.Summable
import Mathlib.FieldTheory.RatFunc.AsPolynomial
import Mathlib.RingTheory.Localization.FractionRing
#align_import ring_theory.laurent_series from "leanprov... | Mathlib/RingTheory/LaurentSeries.lean | 143 | 146 | theorem ofPowerSeries_powerSeriesPart (x : LaurentSeries R) :
ofPowerSeries β€ R x.powerSeriesPart = single (-x.order) 1 * x := by |
refine Eq.trans ?_ (congr rfl x.single_order_mul_powerSeriesPart)
rw [β mul_assoc, single_mul_single, neg_add_self, mul_one, β C_apply, C_one, one_mul]
| 1,970 |
import Mathlib.FieldTheory.RatFunc.AsPolynomial
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Polynomial.Content
noncomputable section
universe u
variable {K : Type u}
namespace RatFunc
section IntDegree
open Polynomial
variable [Field... | Mathlib/FieldTheory/RatFunc/Degree.lean | 44 | 45 | theorem intDegree_zero : intDegree (0 : RatFunc K) = 0 := by |
rw [intDegree, num_zero, natDegree_zero, denom_zero, natDegree_one, sub_self]
| 1,971 |
import Mathlib.FieldTheory.RatFunc.AsPolynomial
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Polynomial.Content
noncomputable section
universe u
variable {K : Type u}
namespace RatFunc
section IntDegree
open Polynomial
variable [Field... | Mathlib/FieldTheory/RatFunc/Degree.lean | 49 | 50 | theorem intDegree_one : intDegree (1 : RatFunc K) = 0 := by |
rw [intDegree, num_one, denom_one, sub_self]
| 1,971 |
import Mathlib.FieldTheory.RatFunc.AsPolynomial
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Polynomial.Content
noncomputable section
universe u
variable {K : Type u}
namespace RatFunc
section IntDegree
open Polynomial
variable [Field... | Mathlib/FieldTheory/RatFunc/Degree.lean | 54 | 55 | theorem intDegree_C (k : K) : intDegree (C k) = 0 := by |
rw [intDegree, num_C, natDegree_C, denom_C, natDegree_one, sub_self]
| 1,971 |
import Mathlib.FieldTheory.RatFunc.AsPolynomial
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Polynomial.Content
noncomputable section
universe u
variable {K : Type u}
namespace RatFunc
section IntDegree
open Polynomial
variable [Field... | Mathlib/FieldTheory/RatFunc/Degree.lean | 59 | 61 | theorem intDegree_X : intDegree (X : RatFunc K) = 1 := by |
rw [intDegree, num_X, Polynomial.natDegree_X, denom_X, Polynomial.natDegree_one,
Int.ofNat_one, Int.ofNat_zero, sub_zero]
| 1,971 |
import Mathlib.FieldTheory.RatFunc.AsPolynomial
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Polynomial.Content
noncomputable section
universe u
variable {K : Type u}
namespace RatFunc
section IntDegree
open Polynomial
variable [Field... | Mathlib/FieldTheory/RatFunc/Degree.lean | 65 | 68 | theorem intDegree_polynomial {p : K[X]} :
intDegree (algebraMap K[X] (RatFunc K) p) = natDegree p := by |
rw [intDegree, RatFunc.num_algebraMap, RatFunc.denom_algebraMap, Polynomial.natDegree_one,
Int.ofNat_zero, sub_zero]
| 1,971 |
import Mathlib.FieldTheory.RatFunc.AsPolynomial
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Polynomial.Content
noncomputable section
universe u
variable {K : Type u}
namespace RatFunc
section IntDegree
open Polynomial
variable [Field... | Mathlib/FieldTheory/RatFunc/Degree.lean | 71 | 81 | theorem intDegree_mul {x y : RatFunc K} (hx : x β 0) (hy : y β 0) :
intDegree (x * y) = intDegree x + intDegree y := by |
simp only [intDegree, add_sub, sub_add, sub_sub_eq_add_sub, sub_sub, sub_eq_sub_iff_add_eq_add]
norm_cast
rw [β Polynomial.natDegree_mul x.denom_ne_zero y.denom_ne_zero, β
Polynomial.natDegree_mul (RatFunc.num_ne_zero (mul_ne_zero hx hy))
(mul_ne_zero x.denom_ne_zero y.denom_ne_zero),
β Polynomial.... | 1,971 |
import Mathlib.FieldTheory.RatFunc.AsPolynomial
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Polynomial.Content
noncomputable section
universe u
variable {K : Type u}
namespace RatFunc
section IntDegree
open Polynomial
variable [Field... | Mathlib/FieldTheory/RatFunc/Degree.lean | 85 | 91 | theorem intDegree_neg (x : RatFunc K) : intDegree (-x) = intDegree x := by |
by_cases hx : x = 0
Β· rw [hx, neg_zero]
Β· rw [intDegree, intDegree, β natDegree_neg x.num]
exact
natDegree_sub_eq_of_prod_eq (num_ne_zero (neg_ne_zero.mpr hx)) (denom_ne_zero (-x))
(neg_ne_zero.mpr (num_ne_zero hx)) (denom_ne_zero x) (num_denom_neg x)
| 1,971 |
import Mathlib.FieldTheory.RatFunc.AsPolynomial
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Polynomial.Content
noncomputable section
universe u
variable {K : Type u}
namespace RatFunc
section IntDegree
open Polynomial
variable [Field... | Mathlib/FieldTheory/RatFunc/Degree.lean | 102 | 107 | theorem natDegree_num_mul_right_sub_natDegree_denom_mul_left_eq_intDegree {x : RatFunc K}
(hx : x β 0) {s : K[X]} (hs : s β 0) :
((x.num * s).natDegree : β€) - (s * x.denom).natDegree = x.intDegree := by |
apply natDegree_sub_eq_of_prod_eq (mul_ne_zero (num_ne_zero hx) hs)
(mul_ne_zero hs x.denom_ne_zero) (num_ne_zero hx) x.denom_ne_zero
rw [mul_assoc]
| 1,971 |
import Mathlib.FieldTheory.RatFunc.AsPolynomial
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Polynomial.Content
noncomputable section
universe u
variable {K : Type u}
namespace RatFunc
section IntDegree
open Polynomial
variable [Field... | Mathlib/FieldTheory/RatFunc/Degree.lean | 110 | 121 | theorem intDegree_add_le {x y : RatFunc K} (hy : y β 0) (hxy : x + y β 0) :
intDegree (x + y) β€ max (intDegree x) (intDegree y) := by |
by_cases hx : x = 0
Β· simp only [hx, zero_add, ne_eq] at hxy
simp [hx, hxy]
rw [intDegree_add hxy, β
natDegree_num_mul_right_sub_natDegree_denom_mul_left_eq_intDegree hx y.denom_ne_zero,
mul_comm y.denom, β
natDegree_num_mul_right_sub_natDegree_denom_mul_left_eq_intDegree hy x.denom_ne_zero,
... | 1,971 |
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.MvPolynomial.Degrees
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.LinearAlgebra.FinsuppVectorSpace
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
#align_import ring_theory.mv_polynomial.basic from "leanprover-community/mathlib"@"2f5b500a507... | Mathlib/RingTheory/MvPolynomial/Basic.lean | 68 | 72 | theorem mapRange_eq_map {R S : Type*} [CommSemiring R] [CommSemiring S] (p : MvPolynomial Ο R)
(f : R β+* S) : Finsupp.mapRange f f.map_zero p = map f p := by |
rw [p.as_sum, Finsupp.mapRange_finset_sum, map_sum (map f)]
refine Finset.sum_congr rfl fun n _ => ?_
rw [map_monomial, β single_eq_monomial, Finsupp.mapRange_single, single_eq_monomial]
| 1,972 |
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.MvPolynomial.Degrees
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.LinearAlgebra.FinsuppVectorSpace
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
#align_import ring_theory.mv_polynomial.basic from "leanprover-community/mathlib"@"2f5b500a507... | Mathlib/RingTheory/MvPolynomial/Basic.lean | 107 | 110 | theorem mem_restrictTotalDegree (p : MvPolynomial Ο R) :
p β restrictTotalDegree Ο R m β p.totalDegree β€ m := by |
rw [totalDegree, Finset.sup_le_iff]
rfl
| 1,972 |
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.MvPolynomial.Degrees
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.LinearAlgebra.FinsuppVectorSpace
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
#align_import ring_theory.mv_polynomial.basic from "leanprover-community/mathlib"@"2f5b500a507... | Mathlib/RingTheory/MvPolynomial/Basic.lean | 113 | 116 | theorem mem_restrictDegree (p : MvPolynomial Ο R) (n : β) :
p β restrictDegree Ο R n β β s β p.support, β i, (s : Ο ββ β) i β€ n := by |
rw [restrictDegree, restrictSupport, Finsupp.mem_supported]
rfl
| 1,972 |
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.MvPolynomial.Degrees
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.LinearAlgebra.FinsuppVectorSpace
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
#align_import ring_theory.mv_polynomial.basic from "leanprover-community/mathlib"@"2f5b500a507... | Mathlib/RingTheory/MvPolynomial/Basic.lean | 119 | 123 | theorem mem_restrictDegree_iff_sup [DecidableEq Ο] (p : MvPolynomial Ο R) (n : β) :
p β restrictDegree Ο R n β β i, p.degrees.count i β€ n := by |
simp only [mem_restrictDegree, degrees_def, Multiset.count_finset_sup, Finsupp.count_toMultiset,
Finset.sup_le_iff]
exact β¨fun h n s hs => h s hs n, fun h s hs n => h n s hsβ©
| 1,972 |
import Mathlib.Algebra.MvPolynomial.Equiv
import Mathlib.Algebra.MvPolynomial.Supported
import Mathlib.LinearAlgebra.LinearIndependent
import Mathlib.RingTheory.Adjoin.Basic
import Mathlib.RingTheory.Algebraic
import Mathlib.RingTheory.MvPolynomial.Basic
#align_import ring_theory.algebraic_independent from "leanprove... | Mathlib/RingTheory/AlgebraicIndependent.lean | 90 | 96 | theorem algebraicIndependent_empty_type_iff [IsEmpty ΞΉ] :
AlgebraicIndependent R x β Injective (algebraMap R A) := by |
have : aeval x = (Algebra.ofId R A).comp (@isEmptyAlgEquiv R ΞΉ _ _).toAlgHom := by
ext i
exact IsEmpty.elim' βΉIsEmpty ΞΉβΊ i
rw [AlgebraicIndependent, this, β Injective.of_comp_iff' _ (@isEmptyAlgEquiv R ΞΉ _ _).bijective]
rfl
| 1,973 |
import Mathlib.Algebra.MvPolynomial.Equiv
import Mathlib.Algebra.MvPolynomial.Supported
import Mathlib.LinearAlgebra.LinearIndependent
import Mathlib.RingTheory.Adjoin.Basic
import Mathlib.RingTheory.Algebraic
import Mathlib.RingTheory.MvPolynomial.Basic
#align_import ring_theory.algebraic_independent from "leanprove... | Mathlib/RingTheory/AlgebraicIndependent.lean | 103 | 106 | theorem algebraMap_injective : Injective (algebraMap R A) := by |
simpa [Function.comp] using
(Injective.of_comp_iff (algebraicIndependent_iff_injective_aeval.1 hx) MvPolynomial.C).2
(MvPolynomial.C_injective _ _)
| 1,973 |
import Mathlib.Algebra.MvPolynomial.Equiv
import Mathlib.Algebra.MvPolynomial.Supported
import Mathlib.LinearAlgebra.LinearIndependent
import Mathlib.RingTheory.Adjoin.Basic
import Mathlib.RingTheory.Algebraic
import Mathlib.RingTheory.MvPolynomial.Basic
#align_import ring_theory.algebraic_independent from "leanprove... | Mathlib/RingTheory/AlgebraicIndependent.lean | 109 | 118 | theorem linearIndependent : LinearIndependent R x := by |
rw [linearIndependent_iff_injective_total]
have : Finsupp.total ΞΉ A R x =
(MvPolynomial.aeval x).toLinearMap.comp (Finsupp.total ΞΉ _ R X) := by
ext
simp
rw [this]
refine hx.comp ?_
rw [β linearIndependent_iff_injective_total]
exact linearIndependent_X _ _
| 1,973 |
import Mathlib.Algebra.MvPolynomial.Equiv
import Mathlib.Algebra.MvPolynomial.Supported
import Mathlib.LinearAlgebra.LinearIndependent
import Mathlib.RingTheory.Adjoin.Basic
import Mathlib.RingTheory.Algebraic
import Mathlib.RingTheory.MvPolynomial.Basic
#align_import ring_theory.algebraic_independent from "leanprove... | Mathlib/RingTheory/AlgebraicIndependent.lean | 129 | 131 | theorem comp (f : ΞΉ' β ΞΉ) (hf : Function.Injective f) : AlgebraicIndependent R (x β f) := by |
intro p q
simpa [aeval_rename, (rename_injective f hf).eq_iff] using @hx (rename f p) (rename f q)
| 1,973 |
import Mathlib.Algebra.MvPolynomial.Equiv
import Mathlib.Algebra.MvPolynomial.Supported
import Mathlib.LinearAlgebra.LinearIndependent
import Mathlib.RingTheory.Adjoin.Basic
import Mathlib.RingTheory.Algebraic
import Mathlib.RingTheory.MvPolynomial.Basic
#align_import ring_theory.algebraic_independent from "leanprove... | Mathlib/RingTheory/AlgebraicIndependent.lean | 134 | 135 | theorem coe_range : AlgebraicIndependent R ((β) : range x β A) := by |
simpa using hx.comp _ (rangeSplitting_injective x)
| 1,973 |
import Mathlib.Algebra.MvPolynomial.Equiv
import Mathlib.Algebra.MvPolynomial.Supported
import Mathlib.LinearAlgebra.LinearIndependent
import Mathlib.RingTheory.Adjoin.Basic
import Mathlib.RingTheory.Algebraic
import Mathlib.RingTheory.MvPolynomial.Basic
#align_import ring_theory.algebraic_independent from "leanprove... | Mathlib/RingTheory/AlgebraicIndependent.lean | 138 | 149 | theorem map {f : A ββ[R] A'} (hf_inj : Set.InjOn f (adjoin R (range x))) :
AlgebraicIndependent R (f β x) := by |
have : aeval (f β x) = f.comp (aeval x) := by ext; simp
have h : β p : MvPolynomial ΞΉ R, aeval x p β (@aeval R _ _ _ _ _ ((β) : range x β A)).range := by
intro p
rw [AlgHom.mem_range]
refine β¨MvPolynomial.rename (codRestrict x (range x) mem_range_self) p, ?_β©
simp [Function.comp, aeval_rename]
in... | 1,973 |
import Mathlib.Algebra.MvPolynomial.Equiv
import Mathlib.Algebra.MvPolynomial.Supported
import Mathlib.LinearAlgebra.LinearIndependent
import Mathlib.RingTheory.Adjoin.Basic
import Mathlib.RingTheory.Algebraic
import Mathlib.RingTheory.MvPolynomial.Basic
#align_import ring_theory.algebraic_independent from "leanprove... | Mathlib/RingTheory/AlgebraicIndependent.lean | 156 | 160 | theorem of_comp (f : A ββ[R] A') (hfv : AlgebraicIndependent R (f β x)) :
AlgebraicIndependent R x := by |
have : aeval (f β x) = f.comp (aeval x) := by ext; simp
rw [AlgebraicIndependent, this, AlgHom.coe_comp] at hfv
exact hfv.of_comp
| 1,973 |
import Mathlib.Algebra.MvPolynomial.CommRing
import Mathlib.LinearAlgebra.Dimension.StrongRankCondition
import Mathlib.RingTheory.MvPolynomial.Basic
#align_import field_theory.mv_polynomial from "leanprover-community/mathlib"@"039a089d2a4b93c761b234f3e5f5aeb752bac60f"
noncomputable section
open scoped Classical
... | Mathlib/FieldTheory/MvPolynomial.lean | 34 | 40 | theorem quotient_mk_comp_C_injective (I : Ideal (MvPolynomial Ο K)) (hI : I β β€) :
Function.Injective ((Ideal.Quotient.mk I).comp MvPolynomial.C) := by |
refine (injective_iff_map_eq_zero _).2 fun x hx => ?_
rw [RingHom.comp_apply, Ideal.Quotient.eq_zero_iff_mem] at hx
refine _root_.by_contradiction fun hx0 => absurd (I.eq_top_iff_one.2 ?_) hI
have := I.mul_mem_left (MvPolynomial.C xβ»ΒΉ) hx
rwa [β MvPolynomial.C.map_mul, inv_mul_cancel hx0, MvPolynomial.C_1] a... | 1,974 |
import Mathlib.Algebra.MvPolynomial.CommRing
import Mathlib.LinearAlgebra.Dimension.StrongRankCondition
import Mathlib.RingTheory.MvPolynomial.Basic
#align_import field_theory.mv_polynomial from "leanprover-community/mathlib"@"039a089d2a4b93c761b234f3e5f5aeb752bac60f"
noncomputable section
open scoped Classical
... | Mathlib/FieldTheory/MvPolynomial.lean | 54 | 55 | theorem rank_mvPolynomial : Module.rank K (MvPolynomial Ο K) = Cardinal.mk (Ο ββ β) := by |
rw [β Cardinal.lift_inj, β (basisMonomials Ο K).mk_eq_rank]
| 1,974 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.HasseDeriv
#align_import data.polynomial.taylor from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
noncomputable section
namespace Polynomial
open Polynomial... | Mathlib/Algebra/Polynomial/Taylor.lean | 46 | 46 | theorem taylor_X : taylor r X = X + C r := by | simp only [taylor_apply, X_comp]
| 1,975 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.HasseDeriv
#align_import data.polynomial.taylor from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
noncomputable section
namespace Polynomial
open Polynomial... | Mathlib/Algebra/Polynomial/Taylor.lean | 51 | 51 | theorem taylor_C (x : R) : taylor r (C x) = C x := by | simp only [taylor_apply, C_comp]
| 1,975 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.HasseDeriv
#align_import data.polynomial.taylor from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
noncomputable section
namespace Polynomial
open Polynomial... | Mathlib/Algebra/Polynomial/Taylor.lean | 56 | 59 | theorem taylor_zero' : taylor (0 : R) = LinearMap.id := by |
ext
simp only [taylor_apply, add_zero, comp_X, _root_.map_zero, LinearMap.id_comp,
Function.comp_apply, LinearMap.coe_comp]
| 1,975 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.HasseDeriv
#align_import data.polynomial.taylor from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
noncomputable section
namespace Polynomial
open Polynomial... | Mathlib/Algebra/Polynomial/Taylor.lean | 62 | 62 | theorem taylor_zero (f : R[X]) : taylor 0 f = f := by | rw [taylor_zero', LinearMap.id_apply]
| 1,975 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.HasseDeriv
#align_import data.polynomial.taylor from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
noncomputable section
namespace Polynomial
open Polynomial... | Mathlib/Algebra/Polynomial/Taylor.lean | 66 | 66 | theorem taylor_one : taylor r (1 : R[X]) = C 1 := by | rw [β C_1, taylor_C]
| 1,975 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.HasseDeriv
#align_import data.polynomial.taylor from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
noncomputable section
namespace Polynomial
open Polynomial... | Mathlib/Algebra/Polynomial/Taylor.lean | 70 | 71 | theorem taylor_monomial (i : β) (k : R) : taylor r (monomial i k) = C k * (X + C r) ^ i := by |
simp [taylor_apply]
| 1,975 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.HasseDeriv
#align_import data.polynomial.taylor from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
noncomputable section
namespace Polynomial
open Polynomial... | Mathlib/Algebra/Polynomial/Taylor.lean | 88 | 89 | theorem taylor_coeff_zero : (taylor r f).coeff 0 = f.eval r := by |
rw [taylor_coeff, hasseDeriv_zero, LinearMap.id_apply]
| 1,975 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.HasseDeriv
#align_import data.polynomial.taylor from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
noncomputable section
namespace Polynomial
open Polynomial... | Mathlib/Algebra/Polynomial/Taylor.lean | 93 | 94 | theorem taylor_coeff_one : (taylor r f).coeff 1 = f.derivative.eval r := by |
rw [taylor_coeff, hasseDeriv_one]
| 1,975 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.HasseDeriv
#align_import data.polynomial.taylor from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
noncomputable section
namespace Polynomial
open Polynomial... | Mathlib/Algebra/Polynomial/Taylor.lean | 98 | 102 | theorem natDegree_taylor (p : R[X]) (r : R) : natDegree (taylor r p) = natDegree p := by |
refine map_natDegree_eq_natDegree _ ?_
nontriviality R
intro n c c0
simp [taylor_monomial, natDegree_C_mul_eq_of_mul_ne_zero, natDegree_pow_X_add_C, c0]
| 1,975 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.HasseDeriv
#align_import data.polynomial.taylor from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
noncomputable section
namespace Polynomial
open Polynomial... | Mathlib/Algebra/Polynomial/Taylor.lean | 106 | 107 | theorem taylor_mul {R} [CommSemiring R] (r : R) (p q : R[X]) :
taylor r (p * q) = taylor r p * taylor r q := by | simp only [taylor_apply, mul_comp]
| 1,975 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.HasseDeriv
#align_import data.polynomial.taylor from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
noncomputable section
namespace Polynomial
open Polynomial... | Mathlib/Algebra/Polynomial/Taylor.lean | 116 | 118 | theorem taylor_taylor {R} [CommSemiring R] (f : R[X]) (r s : R) :
taylor r (taylor s f) = taylor (r + s) f := by |
simp only [taylor_apply, comp_assoc, map_add, add_comp, X_comp, C_comp, C_add, add_assoc]
| 1,975 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.HasseDeriv
#align_import data.polynomial.taylor from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
noncomputable section
namespace Polynomial
open Polynomial... | Mathlib/Algebra/Polynomial/Taylor.lean | 121 | 123 | theorem taylor_eval {R} [CommSemiring R] (r : R) (f : R[X]) (s : R) :
(taylor r f).eval s = f.eval (s + r) := by |
simp only [taylor_apply, eval_comp, eval_C, eval_X, eval_add]
| 1,975 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.HasseDeriv
#align_import data.polynomial.taylor from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
noncomputable section
namespace Polynomial
open Polynomial... | Mathlib/Algebra/Polynomial/Taylor.lean | 126 | 127 | theorem taylor_eval_sub {R} [CommRing R] (r : R) (f : R[X]) (s : R) :
(taylor r f).eval (s - r) = f.eval s := by | rw [taylor_eval, sub_add_cancel]
| 1,975 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.HasseDeriv
#align_import data.polynomial.taylor from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
noncomputable section
namespace Polynomial
open Polynomial... | Mathlib/Algebra/Polynomial/Taylor.lean | 130 | 134 | theorem taylor_injective {R} [CommRing R] (r : R) : Function.Injective (taylor r) := by |
intro f g h
apply_fun taylor (-r) at h
simpa only [taylor_apply, comp_assoc, add_comp, X_comp, C_comp, C_neg, neg_add_cancel_right,
comp_X] using h
| 1,975 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.HasseDeriv
#align_import data.polynomial.taylor from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
noncomputable section
namespace Polynomial
open Polynomial... | Mathlib/Algebra/Polynomial/Taylor.lean | 137 | 142 | theorem eq_zero_of_hasseDeriv_eq_zero {R} [CommRing R] (f : R[X]) (r : R)
(h : β k, (hasseDeriv k f).eval r = 0) : f = 0 := by |
apply taylor_injective r
rw [LinearMap.map_zero]
ext k
simp only [taylor_coeff, h, coeff_zero]
| 1,975 |
import Mathlib.Algebra.Polynomial.Taylor
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.AdicCompletion.Basic
#align_import ring_theory.henselian from "leanprover-community/mathlib"@"d1accf4f9cddb3666c6e8e4da0ac2d19c4ed73f0"
noncomputable section
universe u v
open Polynomial LocalRing Polyno... | Mathlib/RingTheory/Henselian.lean | 65 | 82 | theorem isLocalRingHom_of_le_jacobson_bot {R : Type*} [CommRing R] (I : Ideal R)
(h : I β€ Ideal.jacobson β₯) : IsLocalRingHom (Ideal.Quotient.mk I) := by |
constructor
intro a h
have : IsUnit (Ideal.Quotient.mk (Ideal.jacobson β₯) a) := by
rw [isUnit_iff_exists_inv] at *
obtain β¨b, hbβ© := h
obtain β¨b, rflβ© := Ideal.Quotient.mk_surjective b
use Ideal.Quotient.mk _ b
rw [β (Ideal.Quotient.mk _).map_one, β (Ideal.Quotient.mk _).map_mul, Ideal.Quotie... | 1,976 |
import Mathlib.Algebra.Polynomial.Taylor
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.AdicCompletion.Basic
#align_import ring_theory.henselian from "leanprover-community/mathlib"@"d1accf4f9cddb3666c6e8e4da0ac2d19c4ed73f0"
noncomputable section
universe u v
open Polynomial LocalRing Polyno... | Mathlib/RingTheory/Henselian.lean | 121 | 155 | theorem HenselianLocalRing.TFAE (R : Type u) [CommRing R] [LocalRing R] :
TFAE
[HenselianLocalRing R,
β f : R[X], f.Monic β β aβ : ResidueField R, aeval aβ f = 0 β
aeval aβ (derivative f) β 0 β β a : R, f.IsRoot a β§ residue R a = aβ,
β {K : Type u} [Field K],
β (Ο : R β+* K... |
tfae_have 3 β 2
Β· intro H
exact H (residue R) Ideal.Quotient.mk_surjective
tfae_have 2 β 1
Β· intro H
constructor
intro f hf aβ hβ hβ
specialize H f hf (residue R aβ)
have aux := flip mem_nonunits_iff.mp hβ
simp only [aeval_def, ResidueField.algebraMap_eq, evalβ_at_apply, β
Ideal.Q... | 1,976 |
import Mathlib.Algebra.MvPolynomial.PDeriv
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.LinearIndependent
import Mathlib.RingTheory.Polynomial.Pochhammer
#align_import ring_theory.polynomial.bernstein from "le... | Mathlib/RingTheory/Polynomial/Bernstein.lean | 61 | 62 | theorem eq_zero_of_lt {n Ξ½ : β} (h : n < Ξ½) : bernsteinPolynomial R n Ξ½ = 0 := by |
simp [bernsteinPolynomial, Nat.choose_eq_zero_of_lt h]
| 1,977 |
import Mathlib.Algebra.MvPolynomial.PDeriv
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.LinearIndependent
import Mathlib.RingTheory.Polynomial.Pochhammer
#align_import ring_theory.polynomial.bernstein from "le... | Mathlib/RingTheory/Polynomial/Bernstein.lean | 70 | 71 | theorem map (f : R β+* S) (n Ξ½ : β) :
(bernsteinPolynomial R n Ξ½).map f = bernsteinPolynomial S n Ξ½ := by | simp [bernsteinPolynomial]
| 1,977 |
import Mathlib.Algebra.MvPolynomial.PDeriv
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.LinearIndependent
import Mathlib.RingTheory.Polynomial.Pochhammer
#align_import ring_theory.polynomial.bernstein from "le... | Mathlib/RingTheory/Polynomial/Bernstein.lean | 76 | 78 | theorem flip (n Ξ½ : β) (h : Ξ½ β€ n) :
(bernsteinPolynomial R n Ξ½).comp (1 - X) = bernsteinPolynomial R n (n - Ξ½) := by |
simp [bernsteinPolynomial, h, tsub_tsub_assoc, mul_right_comm]
| 1,977 |
import Mathlib.Algebra.MvPolynomial.PDeriv
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.LinearIndependent
import Mathlib.RingTheory.Polynomial.Pochhammer
#align_import ring_theory.polynomial.bernstein from "le... | Mathlib/RingTheory/Polynomial/Bernstein.lean | 81 | 83 | theorem flip' (n Ξ½ : β) (h : Ξ½ β€ n) :
bernsteinPolynomial R n Ξ½ = (bernsteinPolynomial R n (n - Ξ½)).comp (1 - X) := by |
simp [β flip _ _ _ h, Polynomial.comp_assoc]
| 1,977 |
import Mathlib.Algebra.MvPolynomial.PDeriv
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.LinearIndependent
import Mathlib.RingTheory.Polynomial.Pochhammer
#align_import ring_theory.polynomial.bernstein from "le... | Mathlib/RingTheory/Polynomial/Bernstein.lean | 86 | 90 | theorem eval_at_0 (n Ξ½ : β) : (bernsteinPolynomial R n Ξ½).eval 0 = if Ξ½ = 0 then 1 else 0 := by |
rw [bernsteinPolynomial]
split_ifs with h
Β· subst h; simp
Β· simp [zero_pow h]
| 1,977 |
import Mathlib.Algebra.MvPolynomial.PDeriv
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.LinearIndependent
import Mathlib.RingTheory.Polynomial.Pochhammer
#align_import ring_theory.polynomial.bernstein from "le... | Mathlib/RingTheory/Polynomial/Bernstein.lean | 93 | 99 | theorem eval_at_1 (n Ξ½ : β) : (bernsteinPolynomial R n Ξ½).eval 1 = if Ξ½ = n then 1 else 0 := by |
rw [bernsteinPolynomial]
split_ifs with h
Β· subst h; simp
Β· obtain hΞ½n | hnΞ½ := Ne.lt_or_lt h
Β· simp [zero_pow $ Nat.sub_ne_zero_of_lt hΞ½n]
Β· simp [Nat.choose_eq_zero_of_lt hnΞ½]
| 1,977 |
import Mathlib.Algebra.MvPolynomial.PDeriv
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.LinearIndependent
import Mathlib.RingTheory.Polynomial.Pochhammer
#align_import ring_theory.polynomial.bernstein from "le... | Mathlib/RingTheory/Polynomial/Bernstein.lean | 102 | 131 | theorem derivative_succ_aux (n Ξ½ : β) :
Polynomial.derivative (bernsteinPolynomial R (n + 1) (Ξ½ + 1)) =
(n + 1) * (bernsteinPolynomial R n Ξ½ - bernsteinPolynomial R n (Ξ½ + 1)) := by |
rw [bernsteinPolynomial]
suffices ((n + 1).choose (Ξ½ + 1) : R[X]) * ((β(Ξ½ + 1 : β) : R[X]) * X ^ Ξ½) * (1 - X) ^ (n - Ξ½) -
((n + 1).choose (Ξ½ + 1) : R[X]) * X ^ (Ξ½ + 1) * ((β(n - Ξ½) : R[X]) * (1 - X) ^ (n - Ξ½ - 1)) =
(β(n + 1) : R[X]) * ((n.choose Ξ½ : R[X]) * X ^ Ξ½ * (1 - X) ^ (n - Ξ½) -
(n.choos... | 1,977 |
import Mathlib.Algebra.MvPolynomial.PDeriv
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.LinearIndependent
import Mathlib.RingTheory.Polynomial.Pochhammer
#align_import ring_theory.polynomial.bernstein from "le... | Mathlib/RingTheory/Polynomial/Bernstein.lean | 134 | 138 | theorem derivative_succ (n Ξ½ : β) : Polynomial.derivative (bernsteinPolynomial R n (Ξ½ + 1)) =
n * (bernsteinPolynomial R (n - 1) Ξ½ - bernsteinPolynomial R (n - 1) (Ξ½ + 1)) := by |
cases n
Β· simp [bernsteinPolynomial]
Β· rw [Nat.cast_succ]; apply derivative_succ_aux
| 1,977 |
import Mathlib.Algebra.MvPolynomial.PDeriv
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.LinearIndependent
import Mathlib.RingTheory.Polynomial.Pochhammer
#align_import ring_theory.polynomial.bernstein from "le... | Mathlib/RingTheory/Polynomial/Bernstein.lean | 141 | 143 | theorem derivative_zero (n : β) :
Polynomial.derivative (bernsteinPolynomial R n 0) = -n * bernsteinPolynomial R (n - 1) 0 := by |
simp [bernsteinPolynomial, Polynomial.derivative_pow]
| 1,977 |
import Mathlib.Algebra.MvPolynomial.PDeriv
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.LinearIndependent
import Mathlib.RingTheory.Polynomial.Pochhammer
#align_import ring_theory.polynomial.bernstein from "le... | Mathlib/RingTheory/Polynomial/Bernstein.lean | 146 | 161 | theorem iterate_derivative_at_0_eq_zero_of_lt (n : β) {Ξ½ k : β} :
k < Ξ½ β (Polynomial.derivative^[k] (bernsteinPolynomial R n Ξ½)).eval 0 = 0 := by |
cases' Ξ½ with Ξ½
Β· rintro β¨β©
Β· rw [Nat.lt_succ_iff]
induction' k with k ih generalizing n Ξ½
Β· simp [eval_at_0]
Β· simp only [derivative_succ, Int.natCast_eq_zero, mul_eq_zero, Function.comp_apply,
Function.iterate_succ, Polynomial.iterate_derivative_sub,
Polynomial.iterate_derivative_na... | 1,977 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Monic
#align_import data.polynomial.lifts from "leanprover-community/mathlib"@"63417e01fbc711beaf25fa73b6edb395c0cfddd0"
open Polynomial
noncomputable section
namespace Polynomial
universe u v w
section Semiring
variable {R : Type... | Mathlib/Algebra/Polynomial/Lifts.lean | 61 | 62 | theorem mem_lifts (p : S[X]) : p β lifts f β β q : R[X], map f q = p := by |
simp only [coe_mapRingHom, lifts, RingHom.mem_rangeS]
| 1,978 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Monic
#align_import data.polynomial.lifts from "leanprover-community/mathlib"@"63417e01fbc711beaf25fa73b6edb395c0cfddd0"
open Polynomial
noncomputable section
namespace Polynomial
universe u v w
section Semiring
variable {R : Type... | Mathlib/Algebra/Polynomial/Lifts.lean | 65 | 66 | theorem lifts_iff_set_range (p : S[X]) : p β lifts f β p β Set.range (map f) := by |
simp only [coe_mapRingHom, lifts, Set.mem_range, RingHom.mem_rangeS]
| 1,978 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Monic
#align_import data.polynomial.lifts from "leanprover-community/mathlib"@"63417e01fbc711beaf25fa73b6edb395c0cfddd0"
open Polynomial
noncomputable section
namespace Polynomial
universe u v w
section Semiring
variable {R : Type... | Mathlib/Algebra/Polynomial/Lifts.lean | 69 | 70 | theorem lifts_iff_ringHom_rangeS (p : S[X]) : p β lifts f β p β (mapRingHom f).rangeS := by |
simp only [coe_mapRingHom, lifts, Set.mem_range, RingHom.mem_rangeS]
| 1,978 |
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