Context stringlengths 57 6.04k | file_name stringlengths 21 79 | start int64 14 1.49k | end int64 18 1.5k | theorem stringlengths 25 1.55k | proof stringlengths 5 7.36k | num_lines int64 1 150 | complexity_score float64 2.72 139,370,958,066,637,970,000,000,000,000,000,000,000,000,000,000,000,000,000B | diff_level int64 0 2 | file_diff_level float64 0 2 | theorem_same_file int64 1 32 | rank_file int64 0 2.51k |
|---|---|---|---|---|---|---|---|---|---|---|---|
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds
#align_import data.real.pi.bounds from "leanprover-community/mathlib"@"402f8982dddc1864bd703da2d6e2ee304a866973"
-- Porting note: needed to add a lot of type ascriptions for lean to interpret numbers as reals.
open scoped Real
namespace Real
theorem ... | Mathlib/Data/Real/Pi/Bounds.lean | 77 | 82 | theorem pi_lower_bound_start (n : β) {a}
(h : sqrtTwoAddSeries ((0 : β) / (1 : β)) n β€ (2 : β) - (a / (2 : β) ^ (n + 1)) ^ 2) :
a < Ο := by |
refine lt_of_le_of_lt ?_ (pi_gt_sqrtTwoAddSeries n); rw [mul_comm]
refine (div_le_iff (pow_pos (by norm_num) _ : (0 : β) < _)).mp (le_sqrt_of_sq_le ?_)
rwa [le_sub_comm, show (0 : β) = (0 : β) / (1 : β) by rw [Nat.cast_zero, zero_div]]
| 3 | 20.085537 | 1 | 1.833333 | 6 | 1,911 |
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds
#align_import data.real.pi.bounds from "leanprover-community/mathlib"@"402f8982dddc1864bd703da2d6e2ee304a866973"
-- Porting note: needed to add a lot of type ascriptions for lean to interpret numbers as reals.
open scoped Real
namespace Real
theorem ... | Mathlib/Data/Real/Pi/Bounds.lean | 85 | 93 | theorem sqrtTwoAddSeries_step_up (c d : β) {a b n : β} {z : β} (hz : sqrtTwoAddSeries (c / d) n β€ z)
(hb : 0 < b) (hd : 0 < d) (h : (2 * b + a) * d ^ 2 β€ c ^ 2 * b) :
sqrtTwoAddSeries (a / b) (n + 1) β€ z := by |
refine le_trans ?_ hz; rw [sqrtTwoAddSeries_succ]; apply sqrtTwoAddSeries_monotone_left
have hb' : 0 < (b : β) := Nat.cast_pos.2 hb
have hd' : 0 < (d : β) := Nat.cast_pos.2 hd
rw [sqrt_le_left (div_nonneg c.cast_nonneg d.cast_nonneg), div_pow,
add_div_eq_mul_add_div _ _ (ne_of_gt hb'), div_le_div_iff hb' (... | 6 | 403.428793 | 2 | 1.833333 | 6 | 1,911 |
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds
#align_import data.real.pi.bounds from "leanprover-community/mathlib"@"402f8982dddc1864bd703da2d6e2ee304a866973"
-- Porting note: needed to add a lot of type ascriptions for lean to interpret numbers as reals.
open scoped Real
namespace Real
theorem ... | Mathlib/Data/Real/Pi/Bounds.lean | 128 | 136 | theorem pi_upper_bound_start (n : β) {a}
(h : (2 : β) - ((a - 1 / (4 : β) ^ n) / (2 : β) ^ (n + 1)) ^ 2 β€
sqrtTwoAddSeries ((0 : β) / (1 : β)) n)
(hβ : (1 : β) / (4 : β) ^ n β€ a) : Ο < a := by |
refine lt_of_lt_of_le (pi_lt_sqrtTwoAddSeries n) ?_
rw [β le_sub_iff_add_le, β le_div_iff', sqrt_le_left, sub_le_comm]
Β· rwa [Nat.cast_zero, zero_div] at h
Β· exact div_nonneg (sub_nonneg.2 hβ) (pow_nonneg (le_of_lt zero_lt_two) _)
Β· exact pow_pos zero_lt_two _
| 5 | 148.413159 | 2 | 1.833333 | 6 | 1,911 |
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds
#align_import data.real.pi.bounds from "leanprover-community/mathlib"@"402f8982dddc1864bd703da2d6e2ee304a866973"
-- Porting note: needed to add a lot of type ascriptions for lean to interpret numbers as reals.
open scoped Real
namespace Real
theorem ... | Mathlib/Data/Real/Pi/Bounds.lean | 139 | 147 | theorem sqrtTwoAddSeries_step_down (a b : β) {c d n : β} {z : β}
(hz : z β€ sqrtTwoAddSeries (a / b) n) (hb : 0 < b) (hd : 0 < d)
(h : a ^ 2 * d β€ (2 * d + c) * b ^ 2) : z β€ sqrtTwoAddSeries (c / d) (n + 1) := by |
apply le_trans hz; rw [sqrtTwoAddSeries_succ]; apply sqrtTwoAddSeries_monotone_left
apply le_sqrt_of_sq_le
have hb' : 0 < (b : β) := Nat.cast_pos.2 hb
have hd' : 0 < (d : β) := Nat.cast_pos.2 hd
rw [div_pow, add_div_eq_mul_add_div _ _ (ne_of_gt hd'), div_le_div_iff (pow_pos hb' _) hd']
exact mod_cast h
| 6 | 403.428793 | 2 | 1.833333 | 6 | 1,911 |
import Mathlib.Algebra.MvPolynomial.Supported
import Mathlib.RingTheory.WittVector.Truncated
#align_import ring_theory.witt_vector.mul_coeff from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
noncomputable section
namespace WittVector
variable (p : β) [hp : Fact p.Prime]
variable {k ... | Mathlib/RingTheory/WittVector/MulCoeff.lean | 56 | 61 | theorem wittPolyProd_vars (n : β) : (wittPolyProd p n).vars β univ ΓΛ’ range (n + 1) := by |
rw [wittPolyProd]
apply Subset.trans (vars_mul _ _)
refine union_subset ?_ ?_ <;>
Β· refine Subset.trans (vars_rename _ _) ?_
simp [wittPolynomial_vars, image_subset_iff]
| 5 | 148.413159 | 2 | 1.833333 | 6 | 1,912 |
import Mathlib.Algebra.MvPolynomial.Supported
import Mathlib.RingTheory.WittVector.Truncated
#align_import ring_theory.witt_vector.mul_coeff from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
noncomputable section
namespace WittVector
variable (p : β) [hp : Fact p.Prime]
variable {k ... | Mathlib/RingTheory/WittVector/MulCoeff.lean | 69 | 85 | theorem wittPolyProdRemainder_vars (n : β) :
(wittPolyProdRemainder p n).vars β univ ΓΛ’ range n := by |
rw [wittPolyProdRemainder]
refine Subset.trans (vars_sum_subset _ _) ?_
rw [biUnion_subset]
intro x hx
apply Subset.trans (vars_mul _ _)
refine union_subset ?_ ?_
Β· apply Subset.trans (vars_pow _ _)
have : (p : π) = C (p : β€) := by simp only [Int.cast_natCast, eq_intCast]
rw [this, vars_C]
a... | 15 | 3,269,017.372472 | 2 | 1.833333 | 6 | 1,912 |
import Mathlib.Algebra.MvPolynomial.Supported
import Mathlib.RingTheory.WittVector.Truncated
#align_import ring_theory.witt_vector.mul_coeff from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
noncomputable section
namespace WittVector
variable (p : β) [hp : Fact p.Prime]
variable {k ... | Mathlib/RingTheory/WittVector/MulCoeff.lean | 99 | 110 | theorem remainder_vars (n : β) : (remainder p n).vars β univ ΓΛ’ range (n + 1) := by |
rw [remainder]
apply Subset.trans (vars_mul _ _)
refine union_subset ?_ ?_ <;>
Β· refine Subset.trans (vars_sum_subset _ _) ?_
rw [biUnion_subset]
intro x hx
rw [rename_monomial, vars_monomial, Finsupp.mapDomain_single]
Β· apply Subset.trans Finsupp.support_single_subset
simpa using mem_ran... | 11 | 59,874.141715 | 2 | 1.833333 | 6 | 1,912 |
import Mathlib.Algebra.MvPolynomial.Supported
import Mathlib.RingTheory.WittVector.Truncated
#align_import ring_theory.witt_vector.mul_coeff from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
noncomputable section
namespace WittVector
variable (p : β) [hp : Fact p.Prime]
variable {k ... | Mathlib/RingTheory/WittVector/MulCoeff.lean | 120 | 135 | theorem mul_polyOfInterest_aux1 (n : β) :
β i β range (n + 1), (p : π) ^ i * wittMul p i ^ p ^ (n - i) = wittPolyProd p n := by |
simp only [wittPolyProd]
convert wittStructureInt_prop p (X (0 : Fin 2) * X 1) n using 1
Β· simp only [wittPolynomial, wittMul]
rw [AlgHom.map_sum]
congr 1 with i
congr 1
have hsupp : (Finsupp.single i (p ^ (n - i))).support = {i} := by
rw [Finsupp.support_eq_singleton]
simp only [and_... | 14 | 1,202,604.284165 | 2 | 1.833333 | 6 | 1,912 |
import Mathlib.Algebra.MvPolynomial.Supported
import Mathlib.RingTheory.WittVector.Truncated
#align_import ring_theory.witt_vector.mul_coeff from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
noncomputable section
namespace WittVector
variable (p : β) [hp : Fact p.Prime]
variable {k ... | Mathlib/RingTheory/WittVector/MulCoeff.lean | 138 | 142 | theorem mul_polyOfInterest_aux2 (n : β) :
(p : π) ^ n * wittMul p n + wittPolyProdRemainder p n = wittPolyProd p n := by |
convert mul_polyOfInterest_aux1 p n
rw [sum_range_succ, add_comm, Nat.sub_self, pow_zero, pow_one]
rfl
| 3 | 20.085537 | 1 | 1.833333 | 6 | 1,912 |
import Mathlib.Algebra.MvPolynomial.Supported
import Mathlib.RingTheory.WittVector.Truncated
#align_import ring_theory.witt_vector.mul_coeff from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
noncomputable section
namespace WittVector
variable (p : β) [hp : Fact p.Prime]
variable {k ... | Mathlib/RingTheory/WittVector/MulCoeff.lean | 145 | 176 | theorem mul_polyOfInterest_aux3 (n : β) : wittPolyProd p (n + 1) =
-((p : π) ^ (n + 1) * X (0, n + 1)) * ((p : π) ^ (n + 1) * X (1, n + 1)) +
(p : π) ^ (n + 1) * X (0, n + 1) * rename (Prod.mk (1 : Fin 2)) (wittPolynomial p β€ (n + 1)) +
(p : π) ^ (n + 1) * X (1, n + 1) * rename (Prod.mk (0 : Fin 2)) (wi... |
-- a useful auxiliary fact
have mvpz : (p : π) ^ (n + 1) = MvPolynomial.C ((p : β€) ^ (n + 1)) := by norm_cast
-- Porting note: the original proof applies `sum_range_succ` through a non-`conv` rewrite,
-- but this does not work in Lean 4; the whole proof also times out very badly. The proof has been
-- nearl... | 27 | 532,048,240,601.79865 | 2 | 1.833333 | 6 | 1,912 |
import Mathlib.CategoryTheory.Sites.IsSheafFor
import Mathlib.CategoryTheory.Limits.Shapes.Types
import Mathlib.Tactic.ApplyFun
#align_import category_theory.sites.sheaf_of_types from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe w v u
namespace CategoryTheory
open Opposite ... | Mathlib/CategoryTheory/Sites/EqualizerSheafCondition.lean | 133 | 135 | theorem w : forkMap P (S : Presieve X) β« firstMap P S = forkMap P S β« secondMap P S := by |
ext
simp [firstMap, secondMap, forkMap]
| 2 | 7.389056 | 1 | 1.833333 | 6 | 1,913 |
import Mathlib.CategoryTheory.Sites.IsSheafFor
import Mathlib.CategoryTheory.Limits.Shapes.Types
import Mathlib.Tactic.ApplyFun
#align_import category_theory.sites.sheaf_of_types from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe w v u
namespace CategoryTheory
open Opposite ... | Mathlib/CategoryTheory/Sites/EqualizerSheafCondition.lean | 142 | 152 | theorem compatible_iff (x : FirstObj P S) :
((firstObjEqFamily P S).hom x).Compatible β firstMap P S x = secondMap P S x := by |
rw [Presieve.compatible_iff_sieveCompatible]
constructor
Β· intro t
apply SecondObj.ext
intros Y Z g f hf
simpa [firstMap, secondMap] using t _ g hf
Β· intro t Y Z f g hf
rw [Types.limit_ext_iff'] at t
simpa [firstMap, secondMap] using t β¨β¨Y, Z, g, f, hfβ©β©
| 9 | 8,103.083928 | 2 | 1.833333 | 6 | 1,913 |
import Mathlib.CategoryTheory.Sites.IsSheafFor
import Mathlib.CategoryTheory.Limits.Shapes.Types
import Mathlib.Tactic.ApplyFun
#align_import category_theory.sites.sheaf_of_types from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe w v u
namespace CategoryTheory
open Opposite ... | Mathlib/CategoryTheory/Sites/EqualizerSheafCondition.lean | 156 | 174 | theorem equalizer_sheaf_condition :
Presieve.IsSheafFor P (S : Presieve X) β Nonempty (IsLimit (Fork.ofΞΉ _ (w P S))) := by |
rw [Types.type_equalizer_iff_unique,
β Equiv.forall_congr_left (firstObjEqFamily P (S : Presieve X)).toEquiv.symm]
simp_rw [β compatible_iff]
simp only [inv_hom_id_apply, Iso.toEquiv_symm_fun]
apply forallβ_congr
intro x _
apply exists_unique_congr
intro t
rw [β Iso.toEquiv_symm_fun]
rw [Equiv.eq... | 17 | 24,154,952.753575 | 2 | 1.833333 | 6 | 1,913 |
import Mathlib.CategoryTheory.Sites.IsSheafFor
import Mathlib.CategoryTheory.Limits.Shapes.Types
import Mathlib.Tactic.ApplyFun
#align_import category_theory.sites.sheaf_of_types from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe w v u
namespace CategoryTheory
open Opposite ... | Mathlib/CategoryTheory/Sites/EqualizerSheafCondition.lean | 216 | 223 | theorem w : forkMap P R β« firstMap P R = forkMap P R β« secondMap P R := by |
dsimp
ext fg
simp only [firstMap, secondMap, forkMap]
simp only [limit.lift_Ο, limit.lift_Ο_assoc, assoc, Fan.mk_Ο_app]
haveI := Presieve.hasPullbacks.has_pullbacks fg.1.2.2 fg.2.2.2
rw [β P.map_comp, β op_comp, pullback.condition]
simp
| 7 | 1,096.633158 | 2 | 1.833333 | 6 | 1,913 |
import Mathlib.CategoryTheory.Sites.IsSheafFor
import Mathlib.CategoryTheory.Limits.Shapes.Types
import Mathlib.Tactic.ApplyFun
#align_import category_theory.sites.sheaf_of_types from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe w v u
namespace CategoryTheory
open Opposite ... | Mathlib/CategoryTheory/Sites/EqualizerSheafCondition.lean | 230 | 240 | theorem compatible_iff (x : FirstObj P R) :
((firstObjEqFamily P R).hom x).Compatible β firstMap P R x = secondMap P R x := by |
rw [Presieve.pullbackCompatible_iff]
constructor
Β· intro t
apply Limits.Types.limit_ext
rintro β¨β¨Y, f, hfβ©, Z, g, hgβ©
simpa [firstMap, secondMap] using t hf hg
Β· intro t Y Z f g hf hg
rw [Types.limit_ext_iff'] at t
simpa [firstMap, secondMap] using t β¨β¨β¨Y, f, hfβ©, Z, g, hgβ©β©
| 9 | 8,103.083928 | 2 | 1.833333 | 6 | 1,913 |
import Mathlib.CategoryTheory.Sites.IsSheafFor
import Mathlib.CategoryTheory.Limits.Shapes.Types
import Mathlib.Tactic.ApplyFun
#align_import category_theory.sites.sheaf_of_types from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe w v u
namespace CategoryTheory
open Opposite ... | Mathlib/CategoryTheory/Sites/EqualizerSheafCondition.lean | 246 | 261 | theorem sheaf_condition : R.IsSheafFor P β Nonempty (IsLimit (Fork.ofΞΉ _ (w P R))) := by |
rw [Types.type_equalizer_iff_unique]
erw [β Equiv.forall_congr_left (firstObjEqFamily P R).toEquiv.symm]
simp_rw [β compatible_iff, β Iso.toEquiv_fun, Equiv.apply_symm_apply]
apply forallβ_congr
intro x _
apply exists_unique_congr
intro t
rw [Equiv.eq_symm_apply]
constructor
Β· intro q
funext Y ... | 15 | 3,269,017.372472 | 2 | 1.833333 | 6 | 1,913 |
import Mathlib.LinearAlgebra.QuadraticForm.IsometryEquiv
#align_import linear_algebra.quadratic_form.prod from "leanprover-community/mathlib"@"9b2755b951bc323c962bd072cd447b375cf58101"
universe u v w
variable {ΞΉ : Type*} {R : Type*} {Mβ Mβ Nβ Nβ : Type*} {Mα΅’ Nα΅’ : ΞΉ β Type*}
namespace QuadraticForm
section Pro... | Mathlib/LinearAlgebra/QuadraticForm/Prod.lean | 137 | 147 | theorem anisotropic_of_prod {R} [OrderedCommRing R] [Module R Mβ] [Module R Mβ]
{Qβ : QuadraticForm R Mβ} {Qβ : QuadraticForm R Mβ} (h : (Qβ.prod Qβ).Anisotropic) :
Qβ.Anisotropic β§ Qβ.Anisotropic := by |
simp_rw [Anisotropic, prod_apply, Prod.forall, Prod.mk_eq_zero] at h
constructor
Β· intro x hx
refine (h x 0 ?_).1
rw [hx, zero_add, map_zero]
Β· intro x hx
refine (h 0 x ?_).2
rw [hx, add_zero, map_zero]
| 8 | 2,980.957987 | 2 | 1.833333 | 6 | 1,914 |
import Mathlib.LinearAlgebra.QuadraticForm.IsometryEquiv
#align_import linear_algebra.quadratic_form.prod from "leanprover-community/mathlib"@"9b2755b951bc323c962bd072cd447b375cf58101"
universe u v w
variable {ΞΉ : Type*} {R : Type*} {Mβ Mβ Nβ Nβ : Type*} {Mα΅’ Nα΅’ : ΞΉ β Type*}
namespace QuadraticForm
section Pro... | Mathlib/LinearAlgebra/QuadraticForm/Prod.lean | 150 | 160 | theorem nonneg_prod_iff {R} [OrderedCommRing R] [Module R Mβ] [Module R Mβ]
{Qβ : QuadraticForm R Mβ} {Qβ : QuadraticForm R Mβ} :
(β x, 0 β€ (Qβ.prod Qβ) x) β (β x, 0 β€ Qβ x) β§ β x, 0 β€ Qβ x := by |
simp_rw [Prod.forall, prod_apply]
constructor
Β· intro h
constructor
Β· intro x; simpa only [add_zero, map_zero] using h x 0
Β· intro x; simpa only [zero_add, map_zero] using h 0 x
Β· rintro β¨hβ, hββ© xβ xβ
exact add_nonneg (hβ xβ) (hβ xβ)
| 8 | 2,980.957987 | 2 | 1.833333 | 6 | 1,914 |
import Mathlib.LinearAlgebra.QuadraticForm.IsometryEquiv
#align_import linear_algebra.quadratic_form.prod from "leanprover-community/mathlib"@"9b2755b951bc323c962bd072cd447b375cf58101"
universe u v w
variable {ΞΉ : Type*} {R : Type*} {Mβ Mβ Nβ Nβ : Type*} {Mα΅’ Nα΅’ : ΞΉ β Type*}
namespace QuadraticForm
section Pro... | Mathlib/LinearAlgebra/QuadraticForm/Prod.lean | 257 | 261 | theorem pi_apply_single [Fintype ΞΉ] [DecidableEq ΞΉ]
(Q : β i, QuadraticForm R (Mα΅’ i)) (i : ΞΉ) (m : Mα΅’ i) :
pi Q (Pi.single i m) = Q i m := by |
rw [pi_apply, Fintype.sum_eq_single i fun j hj => ?_, Pi.single_eq_same]
rw [Pi.single_eq_of_ne hj, map_zero]
| 2 | 7.389056 | 1 | 1.833333 | 6 | 1,914 |
import Mathlib.LinearAlgebra.QuadraticForm.IsometryEquiv
#align_import linear_algebra.quadratic_form.prod from "leanprover-community/mathlib"@"9b2755b951bc323c962bd072cd447b375cf58101"
universe u v w
variable {ΞΉ : Type*} {R : Type*} {Mβ Mβ Nβ Nβ : Type*} {Mα΅’ Nα΅’ : ΞΉ β Type*}
namespace QuadraticForm
section Pro... | Mathlib/LinearAlgebra/QuadraticForm/Prod.lean | 313 | 325 | theorem anisotropic_of_pi [Fintype ΞΉ] {R} [OrderedCommRing R] [β i, Module R (Mα΅’ i)]
{Q : β i, QuadraticForm R (Mα΅’ i)} (h : (pi Q).Anisotropic) : β i, (Q i).Anisotropic := by |
simp_rw [Anisotropic, pi_apply, Function.funext_iff, Pi.zero_apply] at h
intro i x hx
classical
have := h (Pi.single i x) ?_ i
Β· rw [Pi.single_eq_same] at this
exact this
apply Finset.sum_eq_zero
intro j _
by_cases hji : j = i
Β· subst hji; rw [Pi.single_eq_same, hx]
Β· rw [Pi.single_eq_of_ne hji... | 11 | 59,874.141715 | 2 | 1.833333 | 6 | 1,914 |
import Mathlib.LinearAlgebra.QuadraticForm.IsometryEquiv
#align_import linear_algebra.quadratic_form.prod from "leanprover-community/mathlib"@"9b2755b951bc323c962bd072cd447b375cf58101"
universe u v w
variable {ΞΉ : Type*} {R : Type*} {Mβ Mβ Nβ Nβ : Type*} {Mα΅’ Nα΅’ : ΞΉ β Type*}
namespace QuadraticForm
section Pro... | Mathlib/LinearAlgebra/QuadraticForm/Prod.lean | 328 | 339 | theorem nonneg_pi_iff [Fintype ΞΉ] {R} [OrderedCommRing R] [β i, Module R (Mα΅’ i)]
{Q : β i, QuadraticForm R (Mα΅’ i)} : (β x, 0 β€ pi Q x) β β i x, 0 β€ Q i x := by |
simp_rw [pi, sum_apply, comp_apply, LinearMap.proj_apply]
constructor
-- TODO: does this generalize to a useful lemma independent of `QuadraticForm`?
Β· intro h i x
classical
convert h (Pi.single i x) using 1
rw [Finset.sum_eq_single_of_mem i (Finset.mem_univ _) fun j _ hji => ?_, Pi.single_eq_same]... | 10 | 22,026.465795 | 2 | 1.833333 | 6 | 1,914 |
import Mathlib.LinearAlgebra.QuadraticForm.IsometryEquiv
#align_import linear_algebra.quadratic_form.prod from "leanprover-community/mathlib"@"9b2755b951bc323c962bd072cd447b375cf58101"
universe u v w
variable {ΞΉ : Type*} {R : Type*} {Mβ Mβ Nβ Nβ : Type*} {Mα΅’ Nα΅’ : ΞΉ β Type*}
namespace QuadraticForm
section Pro... | Mathlib/LinearAlgebra/QuadraticForm/Prod.lean | 342 | 353 | theorem posDef_pi_iff [Fintype ΞΉ] {R} [OrderedCommRing R] [β i, Module R (Mα΅’ i)]
{Q : β i, QuadraticForm R (Mα΅’ i)} : (pi Q).PosDef β β i, (Q i).PosDef := by |
simp_rw [posDef_iff_nonneg, nonneg_pi_iff]
constructor
Β· rintro β¨hle, haβ©
intro i
exact β¨hle i, anisotropic_of_pi ha iβ©
Β· intro h
refine β¨fun i => (h i).1, fun x hx => funext fun i => (h i).2 _ ?_β©
rw [pi_apply, Finset.sum_eq_zero_iff_of_nonneg fun j _ => ?_] at hx
Β· exact hx _ (Finset.mem_... | 10 | 22,026.465795 | 2 | 1.833333 | 6 | 1,914 |
import Mathlib.Data.Finsupp.Lex
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Order.GameAdd
#align_import logic.hydra from "leanprover-community/mathlib"@"48085f140e684306f9e7da907cd5932056d1aded"
namespace Relation
open Multiset Prod
variable {Ξ± : Type*}
def CutExpand (r : Ξ± β Ξ± β Prop) (s' s : Multise... | Mathlib/Logic/Hydra.lean | 62 | 74 | theorem cutExpand_le_invImage_lex [DecidableEq Ξ±] [IsIrrefl Ξ± r] :
CutExpand r β€ InvImage (Finsupp.Lex (rαΆ β (Β· β Β·)) (Β· < Β·)) toFinsupp := by |
rintro s t β¨u, a, hr, heβ©
replace hr := fun a' β¦ mt (hr a')
classical
refine β¨a, fun b h β¦ ?_, ?_β© <;> simp_rw [toFinsupp_apply]
Β· apply_fun count b at he
simpa only [count_add, count_singleton, if_neg h.2, add_zero, count_eq_zero.2 (hr b h.1)]
using he
Β· apply_fun count a at he
simp only [co... | 11 | 59,874.141715 | 2 | 1.833333 | 6 | 1,915 |
import Mathlib.Data.Finsupp.Lex
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Order.GameAdd
#align_import logic.hydra from "leanprover-community/mathlib"@"48085f140e684306f9e7da907cd5932056d1aded"
namespace Relation
open Multiset Prod
variable {Ξ± : Type*}
def CutExpand (r : Ξ± β Ξ± β Prop) (s' s : Multise... | Mathlib/Logic/Hydra.lean | 89 | 98 | theorem cutExpand_iff [DecidableEq Ξ±] [IsIrrefl Ξ± r] {s' s : Multiset Ξ±} :
CutExpand r s' s β
β (t : Multiset Ξ±) (a : Ξ±), (β a' β t, r a' a) β§ a β s β§ s' = s.erase a + t := by |
simp_rw [CutExpand, add_singleton_eq_iff]
refine existsβ_congr fun t a β¦ β¨?_, ?_β©
Β· rintro β¨ht, ha, rflβ©
obtain h | h := mem_add.1 ha
exacts [β¨ht, h, erase_add_left_pos t hβ©, (@irrefl Ξ± r _ a (ht a h)).elim]
Β· rintro β¨ht, h, rflβ©
exact β¨ht, mem_add.2 (Or.inl h), (erase_add_left_pos t h).symmβ©
| 7 | 1,096.633158 | 2 | 1.833333 | 6 | 1,915 |
import Mathlib.Data.Finsupp.Lex
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Order.GameAdd
#align_import logic.hydra from "leanprover-community/mathlib"@"48085f140e684306f9e7da907cd5932056d1aded"
namespace Relation
open Multiset Prod
variable {Ξ± : Type*}
def CutExpand (r : Ξ± β Ξ± β Prop) (s' s : Multise... | Mathlib/Logic/Hydra.lean | 101 | 104 | theorem not_cutExpand_zero [IsIrrefl Ξ± r] (s) : Β¬CutExpand r s 0 := by |
classical
rw [cutExpand_iff]
rintro β¨_, _, _, β¨β©, _β©
| 3 | 20.085537 | 1 | 1.833333 | 6 | 1,915 |
import Mathlib.Data.Finsupp.Lex
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Order.GameAdd
#align_import logic.hydra from "leanprover-community/mathlib"@"48085f140e684306f9e7da907cd5932056d1aded"
namespace Relation
open Multiset Prod
variable {Ξ± : Type*}
def CutExpand (r : Ξ± β Ξ± β Prop) (s' s : Multise... | Mathlib/Logic/Hydra.lean | 109 | 121 | theorem cutExpand_fibration (r : Ξ± β Ξ± β Prop) :
Fibration (GameAdd (CutExpand r) (CutExpand r)) (CutExpand r) fun s β¦ s.1 + s.2 := by |
rintro β¨sβ, sββ© s β¨t, a, hr, heβ©; dsimp at he β’
classical
obtain β¨ha, rflβ© := add_singleton_eq_iff.1 he
rw [add_assoc, mem_add] at ha
obtain h | h := ha
Β· refine β¨(sβ.erase a + t, sβ), GameAdd.fst β¨t, a, hr, ?_β©, ?_β©
Β· rw [add_comm, β add_assoc, singleton_add, cons_erase h]
Β· rw [add_assoc sβ, eras... | 11 | 59,874.141715 | 2 | 1.833333 | 6 | 1,915 |
import Mathlib.Data.Finsupp.Lex
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Order.GameAdd
#align_import logic.hydra from "leanprover-community/mathlib"@"48085f140e684306f9e7da907cd5932056d1aded"
namespace Relation
open Multiset Prod
variable {Ξ± : Type*}
def CutExpand (r : Ξ± β Ξ± β Prop) (s' s : Multise... | Mathlib/Logic/Hydra.lean | 126 | 133 | theorem acc_of_singleton [IsIrrefl Ξ± r] {s : Multiset Ξ±} (hs : β a β s, Acc (CutExpand r) {a}) :
Acc (CutExpand r) s := by |
induction s using Multiset.induction with
| empty => exact Acc.intro 0 fun s h β¦ (not_cutExpand_zero s h).elim
| cons a s ihs =>
rw [β s.singleton_add a]
rw [forall_mem_cons] at hs
exact (hs.1.prod_gameAdd <| ihs fun a ha β¦ hs.2 a ha).of_fibration _ (cutExpand_fibration r)
| 6 | 403.428793 | 2 | 1.833333 | 6 | 1,915 |
import Mathlib.Data.Finsupp.Lex
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Order.GameAdd
#align_import logic.hydra from "leanprover-community/mathlib"@"48085f140e684306f9e7da907cd5932056d1aded"
namespace Relation
open Multiset Prod
variable {Ξ± : Type*}
def CutExpand (r : Ξ± β Ξ± β Prop) (s' s : Multise... | Mathlib/Logic/Hydra.lean | 138 | 146 | theorem _root_.Acc.cutExpand [IsIrrefl Ξ± r] {a : Ξ±} (hacc : Acc r a) : Acc (CutExpand r) {a} := by |
induction' hacc with a h ih
refine Acc.intro _ fun s β¦ ?_
classical
simp only [cutExpand_iff, mem_singleton]
rintro β¨t, a, hr, rfl, rflβ©
refine acc_of_singleton fun a' β¦ ?_
rw [erase_singleton, zero_add]
exact ih a' β hr a'
| 8 | 2,980.957987 | 2 | 1.833333 | 6 | 1,915 |
import Mathlib.NumberTheory.Padics.PadicIntegers
import Mathlib.RingTheory.ZMod
#align_import number_theory.padics.ring_homs from "leanprover-community/mathlib"@"565eb991e264d0db702722b4bde52ee5173c9950"
noncomputable section
open scoped Classical
open Nat LocalRing Padic
namespace PadicInt
variable {p : β} [h... | Mathlib/NumberTheory/Padics/RingHoms.lean | 72 | 75 | theorem modPart_lt_p : modPart p r < p := by |
convert Int.emod_lt _ _
Β· simp
Β· exact mod_cast hp_prime.1.ne_zero
| 3 | 20.085537 | 1 | 1.833333 | 12 | 1,916 |
import Mathlib.NumberTheory.Padics.PadicIntegers
import Mathlib.RingTheory.ZMod
#align_import number_theory.padics.ring_homs from "leanprover-community/mathlib"@"565eb991e264d0db702722b4bde52ee5173c9950"
noncomputable section
open scoped Classical
open Nat LocalRing Padic
namespace PadicInt
variable {p : β} [h... | Mathlib/NumberTheory/Padics/RingHoms.lean | 82 | 101 | theorem isUnit_den (r : β) (h : β(r : β_[p])β β€ 1) : IsUnit (r.den : β€_[p]) := by |
rw [isUnit_iff]
apply le_antisymm (r.den : β€_[p]).2
rw [β not_lt, coe_natCast]
intro norm_denom_lt
have hr : β(r * r.den : β_[p])β = β(r.num : β_[p])β := by
congr
rw_mod_cast [@Rat.mul_den_eq_num r]
rw [padicNormE.mul] at hr
have key : β(r.num : β_[p])β < 1 := by
calc
_ = _ := hr.symm
... | 18 | 65,659,969.137331 | 2 | 1.833333 | 12 | 1,916 |
import Mathlib.NumberTheory.Padics.PadicIntegers
import Mathlib.RingTheory.ZMod
#align_import number_theory.padics.ring_homs from "leanprover-community/mathlib"@"565eb991e264d0db702722b4bde52ee5173c9950"
noncomputable section
open scoped Classical
open Nat LocalRing Padic
namespace PadicInt
variable {p : β} [h... | Mathlib/NumberTheory/Padics/RingHoms.lean | 104 | 121 | theorem norm_sub_modPart_aux (r : β) (h : β(r : β_[p])β β€ 1) :
βp β£ r.num - r.num * r.den.gcdA p % p * βr.den := by |
rw [β ZMod.intCast_zmod_eq_zero_iff_dvd]
simp only [Int.cast_natCast, ZMod.natCast_mod, Int.cast_mul, Int.cast_sub]
have := congr_arg (fun x => x % p : β€ β ZMod p) (gcd_eq_gcd_ab r.den p)
simp only [Int.cast_natCast, CharP.cast_eq_zero, EuclideanDomain.mod_zero, Int.cast_add,
Int.cast_mul, zero_mul, add_ze... | 16 | 8,886,110.520508 | 2 | 1.833333 | 12 | 1,916 |
import Mathlib.NumberTheory.Padics.PadicIntegers
import Mathlib.RingTheory.ZMod
#align_import number_theory.padics.ring_homs from "leanprover-community/mathlib"@"565eb991e264d0db702722b4bde52ee5173c9950"
noncomputable section
open scoped Classical
open Nat LocalRing Padic
namespace PadicInt
variable {p : β} [h... | Mathlib/NumberTheory/Padics/RingHoms.lean | 124 | 134 | theorem norm_sub_modPart (h : β(r : β_[p])β β€ 1) : β(β¨r, hβ© - modPart p r : β€_[p])β < 1 := by |
let n := modPart p r
rw [norm_lt_one_iff_dvd, β (isUnit_den r h).dvd_mul_right]
suffices βp β£ r.num - n * r.den by
convert (Int.castRingHom β€_[p]).map_dvd this
simp only [sub_mul, Int.cast_natCast, eq_intCast, Int.cast_mul, sub_left_inj, Int.cast_sub]
apply Subtype.coe_injective
simp only [coe_mu... | 10 | 22,026.465795 | 2 | 1.833333 | 12 | 1,916 |
import Mathlib.NumberTheory.Padics.PadicIntegers
import Mathlib.RingTheory.ZMod
#align_import number_theory.padics.ring_homs from "leanprover-community/mathlib"@"565eb991e264d0db702722b4bde52ee5173c9950"
noncomputable section
open scoped Classical
open Nat LocalRing Padic
namespace PadicInt
variable {p : β} [h... | Mathlib/NumberTheory/Padics/RingHoms.lean | 142 | 150 | theorem zmod_congr_of_sub_mem_span_aux (n : β) (x : β€_[p]) (a b : β€)
(ha : x - a β (Ideal.span {(p : β€_[p]) ^ n}))
(hb : x - b β (Ideal.span {(p : β€_[p]) ^ n})) : (a : ZMod (p ^ n)) = b := by |
rw [Ideal.mem_span_singleton] at ha hb
rw [β sub_eq_zero, β Int.cast_sub, ZMod.intCast_zmod_eq_zero_iff_dvd, Int.natCast_pow]
rw [β dvd_neg, neg_sub] at ha
have := dvd_add ha hb
rwa [sub_eq_add_neg, sub_eq_add_neg, add_assoc, neg_add_cancel_left, β sub_eq_add_neg, β
Int.cast_sub, pow_p_dvd_int_iff] at th... | 6 | 403.428793 | 2 | 1.833333 | 12 | 1,916 |
import Mathlib.NumberTheory.Padics.PadicIntegers
import Mathlib.RingTheory.ZMod
#align_import number_theory.padics.ring_homs from "leanprover-community/mathlib"@"565eb991e264d0db702722b4bde52ee5173c9950"
noncomputable section
open scoped Classical
open Nat LocalRing Padic
namespace PadicInt
variable {p : β} [h... | Mathlib/NumberTheory/Padics/RingHoms.lean | 498 | 500 | theorem nthHom_zero : nthHom f 0 = 0 := by |
simp (config := { unfoldPartialApp := true }) [nthHom]
rfl
| 2 | 7.389056 | 1 | 1.833333 | 12 | 1,916 |
import Mathlib.NumberTheory.Padics.PadicIntegers
import Mathlib.RingTheory.ZMod
#align_import number_theory.padics.ring_homs from "leanprover-community/mathlib"@"565eb991e264d0db702722b4bde52ee5173c9950"
noncomputable section
open scoped Classical
open Nat LocalRing Padic
namespace PadicInt
variable {p : β} [h... | Mathlib/NumberTheory/Padics/RingHoms.lean | 505 | 511 | theorem pow_dvd_nthHom_sub (r : R) (i j : β) (h : i β€ j) :
(p : β€) ^ i β£ nthHom f r j - nthHom f r i := by |
specialize f_compat i j h
rw [β Int.natCast_pow, β ZMod.intCast_zmod_eq_zero_iff_dvd, Int.cast_sub]
dsimp [nthHom]
rw [β f_compat, RingHom.comp_apply]
simp only [ZMod.cast_id, ZMod.castHom_apply, sub_self, ZMod.natCast_val, ZMod.intCast_cast]
| 5 | 148.413159 | 2 | 1.833333 | 12 | 1,916 |
import Mathlib.NumberTheory.Padics.PadicIntegers
import Mathlib.RingTheory.ZMod
#align_import number_theory.padics.ring_homs from "leanprover-community/mathlib"@"565eb991e264d0db702722b4bde52ee5173c9950"
noncomputable section
open scoped Classical
open Nat LocalRing Padic
namespace PadicInt
variable {p : β} [h... | Mathlib/NumberTheory/Padics/RingHoms.lean | 514 | 525 | theorem isCauSeq_nthHom (r : R) : IsCauSeq (padicNorm p) fun n => nthHom f r n := by |
intro Ξ΅ hΞ΅
obtain β¨k, hkβ© : β k : β, (p : β) ^ (-((k : β) : β€)) < Ξ΅ := exists_pow_neg_lt_rat p hΞ΅
use k
intro j hj
refine lt_of_le_of_lt ?_ hk
-- Need to do beta reduction first, as `norm_cast` doesn't.
-- Added to adapt to leanprover/lean4#2734.
beta_reduce
norm_cast
rw [β padicNorm.dvd_iff_norm_l... | 11 | 59,874.141715 | 2 | 1.833333 | 12 | 1,916 |
import Mathlib.NumberTheory.Padics.PadicIntegers
import Mathlib.RingTheory.ZMod
#align_import number_theory.padics.ring_homs from "leanprover-community/mathlib"@"565eb991e264d0db702722b4bde52ee5173c9950"
noncomputable section
open scoped Classical
open Nat LocalRing Padic
namespace PadicInt
variable {p : β} [h... | Mathlib/NumberTheory/Padics/RingHoms.lean | 537 | 544 | theorem nthHomSeq_one : nthHomSeq f_compat 1 β 1 := by |
intro Ξ΅ hΞ΅
change _ < _ at hΞ΅
use 1
intro j hj
haveI : Fact (1 < p ^ j) := β¨Nat.one_lt_pow (by omega) hp_prime.1.one_ltβ©
suffices (ZMod.cast (1 : ZMod (p ^ j)) : β) = 1 by simp [nthHomSeq, nthHom, this, hΞ΅]
rw [ZMod.cast_eq_val, ZMod.val_one, Nat.cast_one]
| 7 | 1,096.633158 | 2 | 1.833333 | 12 | 1,916 |
import Mathlib.NumberTheory.Padics.PadicIntegers
import Mathlib.RingTheory.ZMod
#align_import number_theory.padics.ring_homs from "leanprover-community/mathlib"@"565eb991e264d0db702722b4bde52ee5173c9950"
noncomputable section
open scoped Classical
open Nat LocalRing Padic
namespace PadicInt
variable {p : β} [h... | Mathlib/NumberTheory/Padics/RingHoms.lean | 547 | 560 | theorem nthHomSeq_add (r s : R) :
nthHomSeq f_compat (r + s) β nthHomSeq f_compat r + nthHomSeq f_compat s := by |
intro Ξ΅ hΞ΅
obtain β¨n, hnβ© := exists_pow_neg_lt_rat p hΞ΅
use n
intro j hj
dsimp [nthHomSeq]
apply lt_of_le_of_lt _ hn
rw [β Int.cast_add, β Int.cast_sub, β padicNorm.dvd_iff_norm_le, β
ZMod.intCast_zmod_eq_zero_iff_dvd]
dsimp [nthHom]
simp only [ZMod.natCast_val, RingHom.map_add, Int.cast_sub, ZMo... | 12 | 162,754.791419 | 2 | 1.833333 | 12 | 1,916 |
import Mathlib.NumberTheory.Padics.PadicIntegers
import Mathlib.RingTheory.ZMod
#align_import number_theory.padics.ring_homs from "leanprover-community/mathlib"@"565eb991e264d0db702722b4bde52ee5173c9950"
noncomputable section
open scoped Classical
open Nat LocalRing Padic
namespace PadicInt
variable {p : β} [h... | Mathlib/NumberTheory/Padics/RingHoms.lean | 563 | 575 | theorem nthHomSeq_mul (r s : R) :
nthHomSeq f_compat (r * s) β nthHomSeq f_compat r * nthHomSeq f_compat s := by |
intro Ξ΅ hΞ΅
obtain β¨n, hnβ© := exists_pow_neg_lt_rat p hΞ΅
use n
intro j hj
dsimp [nthHomSeq]
apply lt_of_le_of_lt _ hn
rw [β Int.cast_mul, β Int.cast_sub, β padicNorm.dvd_iff_norm_le, β
ZMod.intCast_zmod_eq_zero_iff_dvd]
dsimp [nthHom]
simp only [ZMod.natCast_val, RingHom.map_mul, Int.cast_sub, ZMo... | 11 | 59,874.141715 | 2 | 1.833333 | 12 | 1,916 |
import Mathlib.NumberTheory.Padics.PadicIntegers
import Mathlib.RingTheory.ZMod
#align_import number_theory.padics.ring_homs from "leanprover-community/mathlib"@"565eb991e264d0db702722b4bde52ee5173c9950"
noncomputable section
open scoped Classical
open Nat LocalRing Padic
namespace PadicInt
variable {p : β} [h... | Mathlib/NumberTheory/Padics/RingHoms.lean | 586 | 597 | theorem limNthHom_spec (r : R) :
β Ξ΅ : β, 0 < Ξ΅ β β N : β, β n β₯ N, βlimNthHom f_compat r - nthHom f r nβ < Ξ΅ := by |
intro Ξ΅ hΞ΅
obtain β¨Ξ΅', hΞ΅'0, hΞ΅'β© : β v : β, (0 : β) < v β§ βv < Ξ΅ := exists_rat_btwn hΞ΅
norm_cast at hΞ΅'0
obtain β¨N, hNβ© := padicNormE.defn (nthHomSeq f_compat r) hΞ΅'0
use N
intro n hn
apply _root_.lt_trans _ hΞ΅'
change (padicNormE _ : β) < _
norm_cast
exact hN _ hn
| 10 | 22,026.465795 | 2 | 1.833333 | 12 | 1,916 |
import Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic
import Mathlib.NumberTheory.GaussSum
#align_import number_theory.legendre_symbol.quadratic_char.gauss_sum from "leanprover-community/mathlib"@"5b2fe80501ff327b9109fb09b7cc8c325cd0d7d9"
section SpecialValues
open ZMod MulChar
variable {F : Type*} ... | Mathlib/NumberTheory/LegendreSymbol/QuadraticChar/GaussSum.lean | 42 | 61 | theorem FiniteField.isSquare_two_iff :
IsSquare (2 : F) β Fintype.card F % 8 β 3 β§ Fintype.card F % 8 β 5 := by |
classical
by_cases hF : ringChar F = 2
focus
have h := FiniteField.even_card_of_char_two hF
simp only [FiniteField.isSquare_of_char_two hF, true_iff_iff]
rotate_left
focus
have h := FiniteField.odd_card_of_char_ne_two hF
rw [β quadraticChar_one_iff_isSquare (Ring.two_ne_zero hF), quadraticCha... | 18 | 65,659,969.137331 | 2 | 1.833333 | 6 | 1,917 |
import Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic
import Mathlib.NumberTheory.GaussSum
#align_import number_theory.legendre_symbol.quadratic_char.gauss_sum from "leanprover-community/mathlib"@"5b2fe80501ff327b9109fb09b7cc8c325cd0d7d9"
section SpecialValues
open ZMod MulChar
variable {F : Type*} ... | Mathlib/NumberTheory/LegendreSymbol/QuadraticChar/GaussSum.lean | 65 | 68 | theorem quadraticChar_neg_two [DecidableEq F] (hF : ringChar F β 2) :
quadraticChar F (-2) = Οβ' (Fintype.card F) := by |
rw [(by norm_num : (-2 : F) = -1 * 2), map_mul, Οβ'_eq_Οβ_mul_Οβ, quadraticChar_neg_one hF,
quadraticChar_two hF, @cast_natCast _ (ZMod 4) _ _ _ (by decide : 4 β£ 8)]
| 2 | 7.389056 | 1 | 1.833333 | 6 | 1,917 |
import Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic
import Mathlib.NumberTheory.GaussSum
#align_import number_theory.legendre_symbol.quadratic_char.gauss_sum from "leanprover-community/mathlib"@"5b2fe80501ff327b9109fb09b7cc8c325cd0d7d9"
section SpecialValues
open ZMod MulChar
variable {F : Type*} ... | Mathlib/NumberTheory/LegendreSymbol/QuadraticChar/GaussSum.lean | 72 | 91 | theorem FiniteField.isSquare_neg_two_iff :
IsSquare (-2 : F) β Fintype.card F % 8 β 5 β§ Fintype.card F % 8 β 7 := by |
classical
by_cases hF : ringChar F = 2
focus
have h := FiniteField.even_card_of_char_two hF
simp only [FiniteField.isSquare_of_char_two hF, true_iff_iff]
rotate_left
focus
have h := FiniteField.odd_card_of_char_ne_two hF
rw [β quadraticChar_one_iff_isSquare (neg_ne_zero.mpr (Ring.two_ne_zero ... | 18 | 65,659,969.137331 | 2 | 1.833333 | 6 | 1,917 |
import Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic
import Mathlib.NumberTheory.GaussSum
#align_import number_theory.legendre_symbol.quadratic_char.gauss_sum from "leanprover-community/mathlib"@"5b2fe80501ff327b9109fb09b7cc8c325cd0d7d9"
section SpecialValues
open ZMod MulChar
variable {F : Type*} ... | Mathlib/NumberTheory/LegendreSymbol/QuadraticChar/GaussSum.lean | 97 | 115 | theorem quadraticChar_card_card [DecidableEq F] (hF : ringChar F β 2) {F' : Type*} [Field F']
[Fintype F'] [DecidableEq F'] (hF' : ringChar F' β 2) (h : ringChar F' β ringChar F) :
quadraticChar F (Fintype.card F') =
quadraticChar F' (quadraticChar F (-1) * Fintype.card F) := by |
let Ο := (quadraticChar F).ringHomComp (algebraMap β€ F')
have hΟβ : Ο.IsNontrivial := by
obtain β¨a, haβ© := quadraticChar_exists_neg_one hF
have hu : IsUnit a := by
contrapose ha
exact ne_of_eq_of_ne (map_nonunit (quadraticChar F) ha) (mt zero_eq_neg.mp one_ne_zero)
use hu.unit
simp only... | 15 | 3,269,017.372472 | 2 | 1.833333 | 6 | 1,917 |
import Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic
import Mathlib.NumberTheory.GaussSum
#align_import number_theory.legendre_symbol.quadratic_char.gauss_sum from "leanprover-community/mathlib"@"5b2fe80501ff327b9109fb09b7cc8c325cd0d7d9"
section SpecialValues
open ZMod MulChar
variable {F : Type*} ... | Mathlib/NumberTheory/LegendreSymbol/QuadraticChar/GaussSum.lean | 119 | 125 | theorem quadraticChar_odd_prime [DecidableEq F] (hF : ringChar F β 2) {p : β} [Fact p.Prime]
(hpβ : p β 2) (hpβ : ringChar F β p) :
quadraticChar F p = quadraticChar (ZMod p) (Οβ (Fintype.card F) * Fintype.card F) := by |
rw [β quadraticChar_neg_one hF]
have h := quadraticChar_card_card hF (ne_of_eq_of_ne (ringChar_zmod_n p) hpβ)
(ne_of_eq_of_ne (ringChar_zmod_n p) hpβ.symm)
rwa [card p] at h
| 4 | 54.59815 | 2 | 1.833333 | 6 | 1,917 |
import Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic
import Mathlib.NumberTheory.GaussSum
#align_import number_theory.legendre_symbol.quadratic_char.gauss_sum from "leanprover-community/mathlib"@"5b2fe80501ff327b9109fb09b7cc8c325cd0d7d9"
section SpecialValues
open ZMod MulChar
variable {F : Type*} ... | Mathlib/NumberTheory/LegendreSymbol/QuadraticChar/GaussSum.lean | 130 | 143 | theorem FiniteField.isSquare_odd_prime_iff (hF : ringChar F β 2) {p : β} [Fact p.Prime]
(hp : p β 2) :
IsSquare (p : F) β quadraticChar (ZMod p) (Οβ (Fintype.card F) * Fintype.card F) β -1 := by |
classical
by_cases hFp : ringChar F = p
Β· rw [show (p : F) = 0 by rw [β hFp]; exact ringChar.Nat.cast_ringChar]
simp only [isSquare_zero, Ne, true_iff_iff, map_mul]
obtain β¨n, _, hcβ© := FiniteField.card F (ringChar F)
have hchar : ringChar F = ringChar (ZMod p) := by rw [hFp]; exact (ringChar_zmod_n ... | 11 | 59,874.141715 | 2 | 1.833333 | 6 | 1,917 |
import Mathlib.MeasureTheory.Function.LpSeminorm.Basic
import Mathlib.MeasureTheory.Integral.MeanInequalities
#align_import measure_theory.function.lp_seminorm from "leanprover-community/mathlib"@"c4015acc0a223449d44061e27ddac1835a3852b9"
open Filter
open scoped ENNReal Topology
namespace MeasureTheory
section S... | Mathlib/MeasureTheory/Function/LpSeminorm/CompareExp.lean | 26 | 45 | theorem snorm'_le_snorm'_mul_rpow_measure_univ {p q : β} (hp0_lt : 0 < p) (hpq : p β€ q)
(hf : AEStronglyMeasurable f ΞΌ) :
snorm' f p ΞΌ β€ snorm' f q ΞΌ * ΞΌ Set.univ ^ (1 / p - 1 / q) := by |
have hq0_lt : 0 < q := lt_of_lt_of_le hp0_lt hpq
by_cases hpq_eq : p = q
Β· rw [hpq_eq, sub_self, ENNReal.rpow_zero, mul_one]
have hpq : p < q := lt_of_le_of_ne hpq hpq_eq
let g := fun _ : Ξ± => (1 : ββ₯0β)
have h_rw : (β«β» a, (βf aββ : ββ₯0β) ^ p βΞΌ) = β«β» a, ((βf aββ : ββ₯0β) * g a) ^ p βΞΌ :=
lintegral_cong... | 17 | 24,154,952.753575 | 2 | 1.833333 | 6 | 1,918 |
import Mathlib.MeasureTheory.Function.LpSeminorm.Basic
import Mathlib.MeasureTheory.Integral.MeanInequalities
#align_import measure_theory.function.lp_seminorm from "leanprover-community/mathlib"@"c4015acc0a223449d44061e27ddac1835a3852b9"
open Filter
open scoped ENNReal Topology
namespace MeasureTheory
section S... | Mathlib/MeasureTheory/Function/LpSeminorm/CompareExp.lean | 48 | 58 | theorem snorm'_le_snormEssSup_mul_rpow_measure_univ {q : β} (hq_pos : 0 < q) :
snorm' f q ΞΌ β€ snormEssSup f ΞΌ * ΞΌ Set.univ ^ (1 / q) := by |
have h_le : (β«β» a : Ξ±, (βf aββ : ββ₯0β) ^ q βΞΌ) β€ β«β» _ : Ξ±, snormEssSup f ΞΌ ^ q βΞΌ := by
refine lintegral_mono_ae ?_
have h_nnnorm_le_snorm_ess_sup := coe_nnnorm_ae_le_snormEssSup f ΞΌ
exact h_nnnorm_le_snorm_ess_sup.mono fun x hx => by gcongr
rw [snorm', β ENNReal.rpow_one (snormEssSup f ΞΌ)]
nth_rw 2 ... | 9 | 8,103.083928 | 2 | 1.833333 | 6 | 1,918 |
import Mathlib.MeasureTheory.Function.LpSeminorm.Basic
import Mathlib.MeasureTheory.Integral.MeanInequalities
#align_import measure_theory.function.lp_seminorm from "leanprover-community/mathlib"@"c4015acc0a223449d44061e27ddac1835a3852b9"
open Filter
open scoped ENNReal Topology
namespace MeasureTheory
section S... | Mathlib/MeasureTheory/Function/LpSeminorm/CompareExp.lean | 61 | 85 | theorem snorm_le_snorm_mul_rpow_measure_univ {p q : ββ₯0β} (hpq : p β€ q)
(hf : AEStronglyMeasurable f ΞΌ) :
snorm f p ΞΌ β€ snorm f q ΞΌ * ΞΌ Set.univ ^ (1 / p.toReal - 1 / q.toReal) := by |
by_cases hp0 : p = 0
Β· simp [hp0, zero_le]
rw [β Ne] at hp0
have hp0_lt : 0 < p := lt_of_le_of_ne (zero_le _) hp0.symm
have hq0_lt : 0 < q := lt_of_lt_of_le hp0_lt hpq
by_cases hq_top : q = β
Β· simp only [hq_top, _root_.div_zero, one_div, ENNReal.top_toReal, sub_zero, snorm_exponent_top,
GroupWithZ... | 22 | 3,584,912,846.131591 | 2 | 1.833333 | 6 | 1,918 |
import Mathlib.MeasureTheory.Function.LpSeminorm.Basic
import Mathlib.MeasureTheory.Integral.MeanInequalities
#align_import measure_theory.function.lp_seminorm from "leanprover-community/mathlib"@"c4015acc0a223449d44061e27ddac1835a3852b9"
open Filter
open scoped ENNReal Topology
namespace MeasureTheory
section S... | Mathlib/MeasureTheory/Function/LpSeminorm/CompareExp.lean | 88 | 92 | theorem snorm'_le_snorm'_of_exponent_le {p q : β} (hp0_lt : 0 < p)
(hpq : p β€ q) (ΞΌ : Measure Ξ±) [IsProbabilityMeasure ΞΌ] (hf : AEStronglyMeasurable f ΞΌ) :
snorm' f p ΞΌ β€ snorm' f q ΞΌ := by |
have h_le_ΞΌ := snorm'_le_snorm'_mul_rpow_measure_univ hp0_lt hpq hf
rwa [measure_univ, ENNReal.one_rpow, mul_one] at h_le_ΞΌ
| 2 | 7.389056 | 1 | 1.833333 | 6 | 1,918 |
import Mathlib.MeasureTheory.Function.LpSeminorm.Basic
import Mathlib.MeasureTheory.Integral.MeanInequalities
#align_import measure_theory.function.lp_seminorm from "leanprover-community/mathlib"@"c4015acc0a223449d44061e27ddac1835a3852b9"
open Filter
open scoped ENNReal Topology
namespace MeasureTheory
section B... | Mathlib/MeasureTheory/Function/LpSeminorm/CompareExp.lean | 158 | 196 | theorem snorm_le_snorm_top_mul_snorm (p : ββ₯0β) (f : Ξ± β E) {g : Ξ± β F}
(hg : AEStronglyMeasurable g ΞΌ) (b : E β F β G)
(h : βα΅ x βΞΌ, βb (f x) (g x)ββ β€ βf xββ * βg xββ) :
snorm (fun x => b (f x) (g x)) p ΞΌ β€ snorm f β ΞΌ * snorm g p ΞΌ := by |
by_cases hp_top : p = β
Β· simp_rw [hp_top, snorm_exponent_top]
refine le_trans (essSup_mono_ae <| h.mono fun a ha => ?_) (ENNReal.essSup_mul_le _ _)
simp_rw [Pi.mul_apply, β ENNReal.coe_mul, ENNReal.coe_le_coe]
exact ha
by_cases hp_zero : p = 0
Β· simp only [hp_zero, snorm_exponent_zero, mul_zero, l... | 35 | 1,586,013,452,313,430.8 | 2 | 1.833333 | 6 | 1,918 |
import Mathlib.MeasureTheory.Function.LpSeminorm.Basic
import Mathlib.MeasureTheory.Integral.MeanInequalities
#align_import measure_theory.function.lp_seminorm from "leanprover-community/mathlib"@"c4015acc0a223449d44061e27ddac1835a3852b9"
open Filter
open scoped ENNReal Topology
namespace MeasureTheory
section B... | Mathlib/MeasureTheory/Function/LpSeminorm/CompareExp.lean | 208 | 222 | theorem snorm'_le_snorm'_mul_snorm' {p q r : β} (hf : AEStronglyMeasurable f ΞΌ)
(hg : AEStronglyMeasurable g ΞΌ) (b : E β F β G)
(h : βα΅ x βΞΌ, βb (f x) (g x)ββ β€ βf xββ * βg xββ) (hp0_lt : 0 < p) (hpq : p < q)
(hpqr : 1 / p = 1 / q + 1 / r) :
snorm' (fun x => b (f x) (g x)) p ΞΌ β€ snorm' f q ΞΌ * snorm' g ... |
rw [snorm']
calc
(β«β» a : Ξ±, ββb (f a) (g a)ββ ^ p βΞΌ) ^ (1 / p) β€
(β«β» a : Ξ±, β(βf aββ * βg aββ) ^ p βΞΌ) ^ (1 / p) :=
(ENNReal.rpow_le_rpow_iff <| one_div_pos.mpr hp0_lt).mpr <|
lintegral_mono_ae <|
h.mono fun a ha => (ENNReal.rpow_le_rpow_iff hp0_lt).mpr <| ENNReal.coe_le_coe.mp... | 10 | 22,026.465795 | 2 | 1.833333 | 6 | 1,918 |
import Mathlib.Analysis.Convex.Normed
import Mathlib.Analysis.Convex.Strict
import Mathlib.Analysis.Normed.Order.Basic
import Mathlib.Analysis.NormedSpace.AddTorsor
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.Analysis.NormedSpace.Ray
#align_import analysis.convex.strict_convex_space from "leanprover-... | Mathlib/Analysis/Convex/StrictConvexSpace.lean | 76 | 81 | theorem strictConvex_closedBall [StrictConvexSpace π E] (x : E) (r : β) :
StrictConvex π (closedBall x r) := by |
rcases le_or_lt r 0 with hr | hr
Β· exact (subsingleton_closedBall x hr).strictConvex
rw [β vadd_closedBall_zero]
exact (StrictConvexSpace.strictConvex_closedBall r hr).vadd _
| 4 | 54.59815 | 2 | 1.833333 | 6 | 1,919 |
import Mathlib.Analysis.Convex.Normed
import Mathlib.Analysis.Convex.Strict
import Mathlib.Analysis.Normed.Order.Basic
import Mathlib.Analysis.NormedSpace.AddTorsor
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.Analysis.NormedSpace.Ray
#align_import analysis.convex.strict_convex_space from "leanprover-... | Mathlib/Analysis/Convex/StrictConvexSpace.lean | 95 | 106 | theorem StrictConvexSpace.of_norm_combo_lt_one
(h : β x y : E, βxβ = 1 β βyβ = 1 β x β y β β a b : β, a + b = 1 β§ βa β’ x + b β’ yβ < 1) :
StrictConvexSpace β E := by |
refine
StrictConvexSpace.of_strictConvex_closed_unit_ball β
((convex_closedBall _ _).strictConvex' fun x hx y hy hne => ?_)
rw [interior_closedBall (0 : E) one_ne_zero, closedBall_diff_ball,
mem_sphere_zero_iff_norm] at hx hy
rcases h x y hx hy hne with β¨a, b, hab, hltβ©
use b
rwa [AffineMap.lin... | 9 | 8,103.083928 | 2 | 1.833333 | 6 | 1,919 |
import Mathlib.Analysis.Convex.Normed
import Mathlib.Analysis.Convex.Strict
import Mathlib.Analysis.Normed.Order.Basic
import Mathlib.Analysis.NormedSpace.AddTorsor
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.Analysis.NormedSpace.Ray
#align_import analysis.convex.strict_convex_space from "leanprover-... | Mathlib/Analysis/Convex/StrictConvexSpace.lean | 109 | 120 | theorem StrictConvexSpace.of_norm_combo_ne_one
(h :
β x y : E,
βxβ = 1 β βyβ = 1 β x β y β β a b : β, 0 β€ a β§ 0 β€ b β§ a + b = 1 β§ βa β’ x + b β’ yβ β 1) :
StrictConvexSpace β E := by |
refine StrictConvexSpace.of_strictConvex_closed_unit_ball β
((convex_closedBall _ _).strictConvex ?_)
simp only [interior_closedBall _ one_ne_zero, closedBall_diff_ball, Set.Pairwise,
frontier_closedBall _ one_ne_zero, mem_sphere_zero_iff_norm]
intro x hx y hy hne
rcases h x y hx hy hne with β¨a, b, ha,... | 7 | 1,096.633158 | 2 | 1.833333 | 6 | 1,919 |
import Mathlib.Analysis.Convex.Normed
import Mathlib.Analysis.Convex.Strict
import Mathlib.Analysis.Normed.Order.Basic
import Mathlib.Analysis.NormedSpace.AddTorsor
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.Analysis.NormedSpace.Ray
#align_import analysis.convex.strict_convex_space from "leanprover-... | Mathlib/Analysis/Convex/StrictConvexSpace.lean | 123 | 130 | theorem StrictConvexSpace.of_norm_add_ne_two
(h : β β¦x y : Eβ¦, βxβ = 1 β βyβ = 1 β x β y β βx + yβ β 2) : StrictConvexSpace β E := by |
refine
StrictConvexSpace.of_norm_combo_ne_one fun x y hx hy hne =>
β¨1 / 2, 1 / 2, one_half_pos.le, one_half_pos.le, add_halves _, ?_β©
rw [β smul_add, norm_smul, Real.norm_of_nonneg one_half_pos.le, one_div, β div_eq_inv_mul, Ne,
div_eq_one_iff_eq (two_ne_zero' β)]
exact h hx hy hne
| 6 | 403.428793 | 2 | 1.833333 | 6 | 1,919 |
import Mathlib.Analysis.Convex.Normed
import Mathlib.Analysis.Convex.Strict
import Mathlib.Analysis.Normed.Order.Basic
import Mathlib.Analysis.NormedSpace.AddTorsor
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.Analysis.NormedSpace.Ray
#align_import analysis.convex.strict_convex_space from "leanprover-... | Mathlib/Analysis/Convex/StrictConvexSpace.lean | 141 | 145 | theorem StrictConvexSpace.of_norm_add
(h : β x y : E, βxβ = 1 β βyβ = 1 β βx + yβ = 2 β SameRay β x y) : StrictConvexSpace β E := by |
refine StrictConvexSpace.of_pairwise_sphere_norm_ne_two fun x hx y hy => mt fun hβ => ?_
rw [mem_sphere_zero_iff_norm] at hx hy
exact (sameRay_iff_of_norm_eq (hx.trans hy.symm)).1 (h x y hx hy hβ)
| 3 | 20.085537 | 1 | 1.833333 | 6 | 1,919 |
import Mathlib.Analysis.Convex.Normed
import Mathlib.Analysis.Convex.Strict
import Mathlib.Analysis.Normed.Order.Basic
import Mathlib.Analysis.NormedSpace.AddTorsor
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.Analysis.NormedSpace.Ray
#align_import analysis.convex.strict_convex_space from "leanprover-... | Mathlib/Analysis/Convex/StrictConvexSpace.lean | 152 | 158 | theorem combo_mem_ball_of_ne (hx : x β closedBall z r) (hy : y β closedBall z r) (hne : x β y)
(ha : 0 < a) (hb : 0 < b) (hab : a + b = 1) : a β’ x + b β’ y β ball z r := by |
rcases eq_or_ne r 0 with (rfl | hr)
Β· rw [closedBall_zero, mem_singleton_iff] at hx hy
exact (hne (hx.trans hy.symm)).elim
Β· simp only [β interior_closedBall _ hr] at hx hy β’
exact strictConvex_closedBall β z r hx hy hne ha hb hab
| 5 | 148.413159 | 2 | 1.833333 | 6 | 1,919 |
import Mathlib.RingTheory.FiniteType
import Mathlib.RingTheory.Localization.AtPrime
import Mathlib.RingTheory.Localization.Away.Basic
import Mathlib.RingTheory.Localization.Integer
import Mathlib.RingTheory.Localization.Submodule
import Mathlib.RingTheory.Nilpotent.Lemmas
import Mathlib.RingTheory.RingHomProperties
im... | Mathlib/RingTheory/LocalProperties.lean | 153 | 163 | theorem RingHom.ofLocalizationSpan_iff_finite :
RingHom.OfLocalizationSpan @P β RingHom.OfLocalizationFiniteSpan @P := by |
delta RingHom.OfLocalizationSpan RingHom.OfLocalizationFiniteSpan
apply forallβ
_congr
-- TODO: Using `refine` here breaks `resetI`.
intros
constructor
Β· intro h s; exact h s
Β· intro h s hs hs'
obtain β¨s', hβ, hββ© := (Ideal.span_eq_top_iff_finite s).mp hs
exact h s' hβ fun x => hs' β¨_, hβ x.propβ©
| 9 | 8,103.083928 | 2 | 1.833333 | 6 | 1,920 |
import Mathlib.RingTheory.FiniteType
import Mathlib.RingTheory.Localization.AtPrime
import Mathlib.RingTheory.Localization.Away.Basic
import Mathlib.RingTheory.Localization.Integer
import Mathlib.RingTheory.Localization.Submodule
import Mathlib.RingTheory.Nilpotent.Lemmas
import Mathlib.RingTheory.RingHomProperties
im... | Mathlib/RingTheory/LocalProperties.lean | 166 | 176 | theorem RingHom.ofLocalizationSpanTarget_iff_finite :
RingHom.OfLocalizationSpanTarget @P β RingHom.OfLocalizationFiniteSpanTarget @P := by |
delta RingHom.OfLocalizationSpanTarget RingHom.OfLocalizationFiniteSpanTarget
apply forallβ
_congr
-- TODO: Using `refine` here breaks `resetI`.
intros
constructor
Β· intro h s; exact h s
Β· intro h s hs hs'
obtain β¨s', hβ, hββ© := (Ideal.span_eq_top_iff_finite s).mp hs
exact h s' hβ fun x => hs' β¨_,... | 9 | 8,103.083928 | 2 | 1.833333 | 6 | 1,920 |
import Mathlib.RingTheory.FiniteType
import Mathlib.RingTheory.Localization.AtPrime
import Mathlib.RingTheory.Localization.Away.Basic
import Mathlib.RingTheory.Localization.Integer
import Mathlib.RingTheory.Localization.Submodule
import Mathlib.RingTheory.Nilpotent.Lemmas
import Mathlib.RingTheory.RingHomProperties
im... | Mathlib/RingTheory/LocalProperties.lean | 181 | 189 | theorem RingHom.PropertyIsLocal.respectsIso (hP : RingHom.PropertyIsLocal @P) :
RingHom.RespectsIso @P := by |
apply hP.StableUnderComposition.respectsIso
introv
letI := e.toRingHom.toAlgebra
-- Porting note: was `apply_with hP.holds_for_localization_away { instances := ff }`
have : IsLocalization.Away (1 : R) S := by
apply IsLocalization.away_of_isUnit_of_bijective _ isUnit_one e.bijective
exact RingHom.Proper... | 7 | 1,096.633158 | 2 | 1.833333 | 6 | 1,920 |
import Mathlib.RingTheory.FiniteType
import Mathlib.RingTheory.Localization.AtPrime
import Mathlib.RingTheory.Localization.Away.Basic
import Mathlib.RingTheory.Localization.Integer
import Mathlib.RingTheory.Localization.Submodule
import Mathlib.RingTheory.Nilpotent.Lemmas
import Mathlib.RingTheory.RingHomProperties
im... | Mathlib/RingTheory/LocalProperties.lean | 193 | 197 | theorem RingHom.LocalizationPreserves.away (H : RingHom.LocalizationPreserves @P) (r : R)
[IsLocalization.Away r R'] [IsLocalization.Away (f r) S'] (hf : P f) :
P (IsLocalization.Away.map R' S' f r) := by |
have : IsLocalization ((Submonoid.powers r).map f) S' := by rw [Submonoid.map_powers]; assumption
exact H f (Submonoid.powers r) R' S' hf
| 2 | 7.389056 | 1 | 1.833333 | 6 | 1,920 |
import Mathlib.RingTheory.FiniteType
import Mathlib.RingTheory.Localization.AtPrime
import Mathlib.RingTheory.Localization.Away.Basic
import Mathlib.RingTheory.Localization.Integer
import Mathlib.RingTheory.Localization.Submodule
import Mathlib.RingTheory.Nilpotent.Lemmas
import Mathlib.RingTheory.RingHomProperties
im... | Mathlib/RingTheory/LocalProperties.lean | 236 | 255 | theorem Ideal.le_of_localization_maximal {I J : Ideal R}
(h : β (P : Ideal R) (hP : P.IsMaximal),
Ideal.map (algebraMap R (Localization.AtPrime P)) I β€
Ideal.map (algebraMap R (Localization.AtPrime P)) J) :
I β€ J := by |
intro x hx
suffices J.colon (Ideal.span {x}) = β€ by
simpa using Submodule.mem_colon.mp
(show (1 : R) β J.colon (Ideal.span {x}) from this.symm βΈ Submodule.mem_top) x
(Ideal.mem_span_singleton_self x)
refine Not.imp_symm (J.colon (Ideal.span {x})).exists_le_maximal ?_
push_neg
intro P hP le
... | 15 | 3,269,017.372472 | 2 | 1.833333 | 6 | 1,920 |
import Mathlib.RingTheory.FiniteType
import Mathlib.RingTheory.Localization.AtPrime
import Mathlib.RingTheory.Localization.Away.Basic
import Mathlib.RingTheory.Localization.Integer
import Mathlib.RingTheory.Localization.Submodule
import Mathlib.RingTheory.Nilpotent.Lemmas
import Mathlib.RingTheory.RingHomProperties
im... | Mathlib/RingTheory/LocalProperties.lean | 290 | 300 | theorem eq_zero_of_localization (r : R)
(h : β (J : Ideal R) (hJ : J.IsMaximal), algebraMap R (Localization.AtPrime J) r = 0) :
r = 0 := by |
rw [β Ideal.span_singleton_eq_bot]
apply ideal_eq_bot_of_localization
intro J hJ
delta IsLocalization.coeSubmodule
erw [Submodule.map_span, Submodule.span_eq_bot]
rintro _ β¨_, h', rflβ©
cases Set.mem_singleton_iff.mpr h'
exact h J hJ
| 8 | 2,980.957987 | 2 | 1.833333 | 6 | 1,920 |
import Mathlib.FieldTheory.Separable
import Mathlib.RingTheory.IntegralDomain
import Mathlib.Algebra.CharP.Reduced
import Mathlib.Tactic.ApplyFun
#align_import field_theory.finite.basic from "leanprover-community/mathlib"@"12a85fac627bea918960da036049d611b1a3ee43"
variable {K : Type*} {R : Type*}
local notation ... | Mathlib/FieldTheory/Finite/Basic.lean | 76 | 98 | theorem exists_root_sum_quadratic [Fintype R] {f g : R[X]} (hf2 : degree f = 2) (hg2 : degree g = 2)
(hR : Fintype.card R % 2 = 1) : β a b, f.eval a + g.eval b = 0 :=
letI := Classical.decEq R
suffices Β¬Disjoint (univ.image fun x : R => eval x f)
(univ.image fun x : R => eval x (-g)) by
simp only [disjo... |
rw [card_union_of_disjoint hd];
simp [natDegree_eq_of_degree_eq_some hf2, natDegree_eq_of_degree_eq_some hg2, mul_add]
| 2 | 7.389056 | 1 | 1.833333 | 6 | 1,921 |
import Mathlib.FieldTheory.Separable
import Mathlib.RingTheory.IntegralDomain
import Mathlib.Algebra.CharP.Reduced
import Mathlib.Tactic.ApplyFun
#align_import field_theory.finite.basic from "leanprover-community/mathlib"@"12a85fac627bea918960da036049d611b1a3ee43"
variable {K : Type*} {R : Type*}
local notation ... | Mathlib/FieldTheory/Finite/Basic.lean | 104 | 111 | theorem prod_univ_units_id_eq_neg_one [CommRing K] [IsDomain K] [Fintype KΛ£] :
β x : KΛ£, x = (-1 : KΛ£) := by |
classical
have : (β x β (@univ KΛ£ _).erase (-1), x) = 1 :=
prod_involution (fun x _ => xβ»ΒΉ) (by simp)
(fun a => by simp (config := { contextual := true }) [Units.inv_eq_self_iff])
(fun a => by simp [@inv_eq_iff_eq_inv _ _ a]) (by simp)
rw [β insert_erase (mem_univ (-1 : KΛ£)), prod_inser... | 6 | 403.428793 | 2 | 1.833333 | 6 | 1,921 |
import Mathlib.FieldTheory.Separable
import Mathlib.RingTheory.IntegralDomain
import Mathlib.Algebra.CharP.Reduced
import Mathlib.Tactic.ApplyFun
#align_import field_theory.finite.basic from "leanprover-community/mathlib"@"12a85fac627bea918960da036049d611b1a3ee43"
variable {K : Type*} {R : Type*}
local notation ... | Mathlib/FieldTheory/Finite/Basic.lean | 115 | 139 | theorem card_cast_subgroup_card_ne_zero [Ring K] [NoZeroDivisors K] [Nontrivial K]
(G : Subgroup KΛ£) [Fintype G] : (Fintype.card G : K) β 0 := by |
let n := Fintype.card G
intro nzero
have β¨p, char_pβ© := CharP.exists K
have hd : p β£ n := (CharP.cast_eq_zero_iff K p n).mp nzero
cases CharP.char_is_prime_or_zero K p with
| inr pzero =>
exact (Fintype.card_pos).ne' <| Nat.eq_zero_of_zero_dvd <| pzero βΈ hd
| inl pprime =>
have fact_pprime := Fac... | 23 | 9,744,803,446.248903 | 2 | 1.833333 | 6 | 1,921 |
import Mathlib.FieldTheory.Separable
import Mathlib.RingTheory.IntegralDomain
import Mathlib.Algebra.CharP.Reduced
import Mathlib.Tactic.ApplyFun
#align_import field_theory.finite.basic from "leanprover-community/mathlib"@"12a85fac627bea918960da036049d611b1a3ee43"
variable {K : Type*} {R : Type*}
local notation ... | Mathlib/FieldTheory/Finite/Basic.lean | 142 | 164 | theorem sum_subgroup_units_eq_zero [Ring K] [NoZeroDivisors K]
{G : Subgroup KΛ£} [Fintype G] (hg : G β β₯) :
β x : G, (x.val : K) = 0 := by |
rw [Subgroup.ne_bot_iff_exists_ne_one] at hg
rcases hg with β¨a, haβ©
-- The action of a on G as an embedding
let a_mul_emb : G βͺ G := mulLeftEmbedding a
-- ... and leaves G unchanged
have h_unchanged : Finset.univ.map a_mul_emb = Finset.univ := by simp
-- Therefore the sum of x over a G is the sum of a x ... | 20 | 485,165,195.40979 | 2 | 1.833333 | 6 | 1,921 |
import Mathlib.FieldTheory.Separable
import Mathlib.RingTheory.IntegralDomain
import Mathlib.Algebra.CharP.Reduced
import Mathlib.Tactic.ApplyFun
#align_import field_theory.finite.basic from "leanprover-community/mathlib"@"12a85fac627bea918960da036049d611b1a3ee43"
variable {K : Type*} {R : Type*}
local notation ... | Mathlib/FieldTheory/Finite/Basic.lean | 168 | 176 | theorem sum_subgroup_units [Ring K] [NoZeroDivisors K]
{G : Subgroup KΛ£} [Fintype G] [Decidable (G = β₯)] :
β x : G, (x.val : K) = if G = β₯ then 1 else 0 := by |
by_cases G_bot : G = β₯
Β· subst G_bot
simp only [ite_true, Subgroup.mem_bot, Fintype.card_ofSubsingleton, Nat.cast_ite, Nat.cast_one,
Nat.cast_zero, univ_unique, Set.default_coe_singleton, sum_singleton, Units.val_one]
Β· simp only [G_bot, ite_false]
exact sum_subgroup_units_eq_zero G_bot
| 6 | 403.428793 | 2 | 1.833333 | 6 | 1,921 |
import Mathlib.FieldTheory.Separable
import Mathlib.RingTheory.IntegralDomain
import Mathlib.Algebra.CharP.Reduced
import Mathlib.Tactic.ApplyFun
#align_import field_theory.finite.basic from "leanprover-community/mathlib"@"12a85fac627bea918960da036049d611b1a3ee43"
variable {K : Type*} {R : Type*}
local notation ... | Mathlib/FieldTheory/Finite/Basic.lean | 179 | 210 | theorem sum_subgroup_pow_eq_zero [CommRing K] [NoZeroDivisors K]
{G : Subgroup KΛ£} [Fintype G] {k : β} (k_pos : k β 0) (k_lt_card_G : k < Fintype.card G) :
β x : G, ((x : KΛ£) : K) ^ k = 0 := by |
nontriviality K
have := NoZeroDivisors.to_isDomain K
rcases (exists_pow_ne_one_of_isCyclic k_pos k_lt_card_G) with β¨a, haβ©
rw [Finset.sum_eq_multiset_sum]
have h_multiset_map :
Finset.univ.val.map (fun x : G => ((x : KΛ£) : K) ^ k) =
Finset.univ.val.map (fun x : G => ((x : KΛ£) : K) ^ k * ((a : KΛ£) :... | 29 | 3,931,334,297,144.042 | 2 | 1.833333 | 6 | 1,921 |
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
import Mathlib.Analysis.Analytic.IsolatedZeros
import Mathlib.Analysis.Complex.CauchyIntegral
#align_import analysis.special_functions.gamma.beta from "l... | Mathlib/Analysis/SpecialFunctions/Gamma/Beta.lean | 63 | 76 | theorem betaIntegral_convergent_left {u : β} (hu : 0 < re u) (v : β) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 (1 / 2) := by |
apply IntervalIntegrable.mul_continuousOn
Β· refine intervalIntegral.intervalIntegrable_cpow' ?_
rwa [sub_re, one_re, β zero_sub, sub_lt_sub_iff_right]
Β· apply ContinuousAt.continuousOn
intro x hx
rw [uIcc_of_le (by positivity : (0 : β) β€ 1 / 2)] at hx
apply ContinuousAt.cpow
Β· exact (continuo... | 11 | 59,874.141715 | 2 | 1.857143 | 7 | 1,923 |
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
import Mathlib.Analysis.Analytic.IsolatedZeros
import Mathlib.Analysis.Complex.CauchyIntegral
#align_import analysis.special_functions.gamma.beta from "l... | Mathlib/Analysis/SpecialFunctions/Gamma/Beta.lean | 80 | 90 | theorem betaIntegral_convergent {u v : β} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 1 := by |
refine (betaIntegral_convergent_left hu v).trans ?_
rw [IntervalIntegrable.iff_comp_neg]
convert ((betaIntegral_convergent_left hv u).comp_add_right 1).symm using 1
Β· ext1 x
conv_lhs => rw [mul_comm]
congr 2 <;> Β· push_cast; ring
Β· norm_num
Β· norm_num
| 8 | 2,980.957987 | 2 | 1.857143 | 7 | 1,923 |
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
import Mathlib.Analysis.Analytic.IsolatedZeros
import Mathlib.Analysis.Complex.CauchyIntegral
#align_import analysis.special_functions.gamma.beta from "l... | Mathlib/Analysis/SpecialFunctions/Gamma/Beta.lean | 93 | 102 | theorem betaIntegral_symm (u v : β) : betaIntegral v u = betaIntegral u v := by |
rw [betaIntegral, betaIntegral]
have := intervalIntegral.integral_comp_mul_add (a := 0) (b := 1) (c := -1)
(fun x : β => (x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1)) neg_one_lt_zero.ne 1
rw [inv_neg, inv_one, neg_one_smul, β intervalIntegral.integral_symm] at this
simp? at this says
simp only [neg_mul,... | 9 | 8,103.083928 | 2 | 1.857143 | 7 | 1,923 |
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
import Mathlib.Analysis.Analytic.IsolatedZeros
import Mathlib.Analysis.Complex.CauchyIntegral
#align_import analysis.special_functions.gamma.beta from "l... | Mathlib/Analysis/SpecialFunctions/Gamma/Beta.lean | 105 | 111 | theorem betaIntegral_eval_one_right {u : β} (hu : 0 < re u) : betaIntegral u 1 = 1 / u := by |
simp_rw [betaIntegral, sub_self, cpow_zero, mul_one]
rw [integral_cpow (Or.inl _)]
Β· rw [ofReal_zero, ofReal_one, one_cpow, zero_cpow, sub_zero, sub_add_cancel]
rw [sub_add_cancel]
contrapose! hu; rw [hu, zero_re]
Β· rwa [sub_re, one_re, β sub_pos, sub_neg_eq_add, sub_add_cancel]
| 6 | 403.428793 | 2 | 1.857143 | 7 | 1,923 |
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
import Mathlib.Analysis.Analytic.IsolatedZeros
import Mathlib.Analysis.Complex.CauchyIntegral
#align_import analysis.special_functions.gamma.beta from "l... | Mathlib/Analysis/SpecialFunctions/Gamma/Beta.lean | 114 | 132 | theorem betaIntegral_scaled (s t : β) {a : β} (ha : 0 < a) :
β« x in (0)..a, (x : β) ^ (s - 1) * ((a : β) - x) ^ (t - 1) =
(a : β) ^ (s + t - 1) * betaIntegral s t := by |
have ha' : (a : β) β 0 := ofReal_ne_zero.mpr ha.ne'
rw [betaIntegral]
have A : (a : β) ^ (s + t - 1) = a * ((a : β) ^ (s - 1) * (a : β) ^ (t - 1)) := by
rw [(by abel : s + t - 1 = 1 + (s - 1) + (t - 1)), cpow_add _ _ ha', cpow_add 1 _ ha', cpow_one,
mul_assoc]
rw [A, mul_assoc, β intervalIntegral.int... | 16 | 8,886,110.520508 | 2 | 1.857143 | 7 | 1,923 |
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
import Mathlib.Analysis.Analytic.IsolatedZeros
import Mathlib.Analysis.Complex.CauchyIntegral
#align_import analysis.special_functions.gamma.beta from "l... | Mathlib/Analysis/SpecialFunctions/Gamma/Beta.lean | 136 | 151 | theorem Gamma_mul_Gamma_eq_betaIntegral {s t : β} (hs : 0 < re s) (ht : 0 < re t) :
Gamma s * Gamma t = Gamma (s + t) * betaIntegral s t := by |
-- Note that we haven't proved (yet) that the Gamma function has no zeroes, so we can't formulate
-- this as a formula for the Beta function.
have conv_int := integral_posConvolution
(GammaIntegral_convergent hs) (GammaIntegral_convergent ht) (ContinuousLinearMap.mul β β)
simp_rw [ContinuousLinearMap.mul_a... | 14 | 1,202,604.284165 | 2 | 1.857143 | 7 | 1,923 |
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
import Mathlib.Analysis.Analytic.IsolatedZeros
import Mathlib.Analysis.Complex.CauchyIntegral
#align_import analysis.special_functions.gamma.beta from "l... | Mathlib/Analysis/SpecialFunctions/Gamma/Beta.lean | 530 | 534 | theorem one_div_Gamma_eq_self_mul_one_div_Gamma_add_one (s : β) :
(Gamma s)β»ΒΉ = s * (Gamma (s + 1))β»ΒΉ := by |
rcases ne_or_eq s 0 with (h | rfl)
Β· rw [Gamma_add_one s h, mul_inv, mul_inv_cancel_leftβ h]
Β· rw [zero_add, Gamma_zero, inv_zero, zero_mul]
| 3 | 20.085537 | 1 | 1.857143 | 7 | 1,923 |
import Mathlib.Analysis.InnerProductSpace.Orientation
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
#align_import measure_theory.measure.haar.inner_product_space from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
open FiniteDimensional MeasureTheory MeasureTheory.Measure Set
var... | Mathlib/MeasureTheory/Measure/Haar/InnerProductSpace.lean | 34 | 43 | theorem Orientation.measure_orthonormalBasis (o : Orientation β F (Fin n))
(b : OrthonormalBasis ΞΉ β F) : o.volumeForm.measure (parallelepiped b) = 1 := by |
have e : ΞΉ β Fin n := by
refine Fintype.equivFinOfCardEq ?_
rw [β _i.out, finrank_eq_card_basis b.toBasis]
have A : βb = b.reindex e β e := by
ext x
simp only [OrthonormalBasis.coe_reindex, Function.comp_apply, Equiv.symm_apply_apply]
rw [A, parallelepiped_comp_equiv, AlternatingMap.measure_paral... | 8 | 2,980.957987 | 2 | 1.857143 | 7 | 1,924 |
import Mathlib.Analysis.InnerProductSpace.Orientation
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
#align_import measure_theory.measure.haar.inner_product_space from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
open FiniteDimensional MeasureTheory MeasureTheory.Measure Set
var... | Mathlib/MeasureTheory/Measure/Haar/InnerProductSpace.lean | 48 | 54 | theorem Orientation.measure_eq_volume (o : Orientation β F (Fin n)) :
o.volumeForm.measure = volume := by |
have A : o.volumeForm.measure (stdOrthonormalBasis β F).toBasis.parallelepiped = 1 :=
Orientation.measure_orthonormalBasis o (stdOrthonormalBasis β F)
rw [addHaarMeasure_unique o.volumeForm.measure
(stdOrthonormalBasis β F).toBasis.parallelepiped, A, one_smul]
simp only [volume, Basis.addHaar]
| 5 | 148.413159 | 2 | 1.857143 | 7 | 1,924 |
import Mathlib.Analysis.InnerProductSpace.Orientation
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
#align_import measure_theory.measure.haar.inner_product_space from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
open FiniteDimensional MeasureTheory MeasureTheory.Measure Set
var... | Mathlib/MeasureTheory/Measure/Haar/InnerProductSpace.lean | 61 | 66 | theorem OrthonormalBasis.volume_parallelepiped (b : OrthonormalBasis ΞΉ β F) :
volume (parallelepiped b) = 1 := by |
haveI : Fact (finrank β F = finrank β F) := β¨rflβ©
let o := (stdOrthonormalBasis β F).toBasis.orientation
rw [β o.measure_eq_volume]
exact o.measure_orthonormalBasis b
| 4 | 54.59815 | 2 | 1.857143 | 7 | 1,924 |
import Mathlib.Analysis.InnerProductSpace.Orientation
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
#align_import measure_theory.measure.haar.inner_product_space from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
open FiniteDimensional MeasureTheory MeasureTheory.Measure Set
var... | Mathlib/MeasureTheory/Measure/Haar/InnerProductSpace.lean | 71 | 76 | theorem OrthonormalBasis.addHaar_eq_volume {ΞΉ F : Type*} [Fintype ΞΉ] [NormedAddCommGroup F]
[InnerProductSpace β F] [FiniteDimensional β F] [MeasurableSpace F] [BorelSpace F]
(b : OrthonormalBasis ΞΉ β F) :
b.toBasis.addHaar = volume := by |
rw [Basis.addHaar_eq_iff]
exact b.volume_parallelepiped
| 2 | 7.389056 | 1 | 1.857143 | 7 | 1,924 |
import Mathlib.Analysis.InnerProductSpace.Orientation
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
#align_import measure_theory.measure.haar.inner_product_space from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
open FiniteDimensional MeasureTheory MeasureTheory.Measure Set
var... | Mathlib/MeasureTheory/Measure/Haar/InnerProductSpace.lean | 84 | 89 | theorem OrthonormalBasis.measurePreserving_measurableEquiv (b : OrthonormalBasis ΞΉ β F) :
MeasurePreserving b.measurableEquiv volume volume := by |
convert (b.measurableEquiv.symm.measurable.measurePreserving _).symm
rw [β (EuclideanSpace.basisFun ΞΉ β).addHaar_eq_volume]
erw [MeasurableEquiv.coe_toEquiv_symm, Basis.map_addHaar _ b.repr.symm.toContinuousLinearEquiv]
exact b.addHaar_eq_volume.symm
| 4 | 54.59815 | 2 | 1.857143 | 7 | 1,924 |
import Mathlib.Analysis.InnerProductSpace.Orientation
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
#align_import measure_theory.measure.haar.inner_product_space from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
open FiniteDimensional MeasureTheory MeasureTheory.Measure Set
var... | Mathlib/MeasureTheory/Measure/Haar/InnerProductSpace.lean | 102 | 108 | theorem EuclideanSpace.volume_preserving_measurableEquiv :
MeasurePreserving (EuclideanSpace.measurableEquiv ΞΉ) := by |
suffices volume = map (EuclideanSpace.measurableEquiv ΞΉ).symm volume by
convert ((EuclideanSpace.measurableEquiv ΞΉ).symm.measurable.measurePreserving _).symm
rw [β addHaarMeasure_eq_volume_pi, β Basis.parallelepiped_basisFun, β Basis.addHaar_def,
coe_measurableEquiv_symm, β PiLp.continuousLinearEquiv_symm_... | 5 | 148.413159 | 2 | 1.857143 | 7 | 1,924 |
import Mathlib.Analysis.InnerProductSpace.Orientation
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
#align_import measure_theory.measure.haar.inner_product_space from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
open FiniteDimensional MeasureTheory MeasureTheory.Measure Set
var... | Mathlib/MeasureTheory/Measure/Haar/InnerProductSpace.lean | 138 | 143 | theorem measurePreserving : MeasurePreserving f := by |
refine β¨f.continuous.measurable, ?_β©
rcases exists_orthonormalBasis β E with β¨w, b, _hwβ©
erw [β OrthonormalBasis.addHaar_eq_volume b, β OrthonormalBasis.addHaar_eq_volume (b.map f),
Basis.map_addHaar _ f.toContinuousLinearEquiv]
congr
| 5 | 148.413159 | 2 | 1.857143 | 7 | 1,924 |
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Analytic.CPolynomial
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
import Mathlib.Analysis.Calculus.FDeriv.Add
#align_import analysis.calculus.fderiv_analytic from "leanprover-community/mathlib"@"3bce8d800a6f2... | Mathlib/Analysis/Calculus/FDeriv/Analytic.lean | 39 | 44 | theorem HasFPowerSeriesAt.hasStrictFDerivAt (h : HasFPowerSeriesAt f p x) :
HasStrictFDerivAt f (continuousMultilinearCurryFin1 π E F (p 1)) x := by |
refine h.isBigO_image_sub_norm_mul_norm_sub.trans_isLittleO (IsLittleO.of_norm_right ?_)
refine isLittleO_iff_exists_eq_mul.2 β¨fun y => βy - (x, x)β, ?_, EventuallyEq.rflβ©
refine (continuous_id.sub continuous_const).norm.tendsto' _ _ ?_
rw [_root_.id, sub_self, norm_zero]
| 4 | 54.59815 | 2 | 1.857143 | 7 | 1,925 |
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Analytic.CPolynomial
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
import Mathlib.Analysis.Calculus.FDeriv.Add
#align_import analysis.calculus.fderiv_analytic from "leanprover-community/mathlib"@"3bce8d800a6f2... | Mathlib/Analysis/Calculus/FDeriv/Analytic.lean | 91 | 101 | theorem HasFPowerSeriesOnBall.fderiv [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesOnBall (fderiv π f) p.derivSeries x r := by |
refine .congr (f := fun z β¦ continuousMultilinearCurryFin1 π E F (p.changeOrigin (z - x) 1)) ?_
fun z hz β¦ ?_
Β· refine continuousMultilinearCurryFin1 π E F
|>.toContinuousLinearEquiv.toContinuousLinearMap.comp_hasFPowerSeriesOnBall ?_
simpa using ((p.hasFPowerSeriesOnBall_changeOrigin 1
(h.r_... | 9 | 8,103.083928 | 2 | 1.857143 | 7 | 1,925 |
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Analytic.CPolynomial
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
import Mathlib.Analysis.Calculus.FDeriv.Add
#align_import analysis.calculus.fderiv_analytic from "leanprover-community/mathlib"@"3bce8d800a6f2... | Mathlib/Analysis/Calculus/FDeriv/Analytic.lean | 105 | 109 | theorem AnalyticOn.fderiv [CompleteSpace F] (h : AnalyticOn π f s) :
AnalyticOn π (fderiv π f) s := by |
intro y hy
rcases h y hy with β¨p, r, hpβ©
exact hp.fderiv.analyticAt
| 3 | 20.085537 | 1 | 1.857143 | 7 | 1,925 |
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Analytic.CPolynomial
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
import Mathlib.Analysis.Calculus.FDeriv.Add
#align_import analysis.calculus.fderiv_analytic from "leanprover-community/mathlib"@"3bce8d800a6f2... | Mathlib/Analysis/Calculus/FDeriv/Analytic.lean | 113 | 122 | theorem AnalyticOn.iteratedFDeriv [CompleteSpace F] (h : AnalyticOn π f s) (n : β) :
AnalyticOn π (iteratedFDeriv π n f) s := by |
induction' n with n IH
Β· rw [iteratedFDeriv_zero_eq_comp]
exact ((continuousMultilinearCurryFin0 π E F).symm : F βL[π] E[Γ0]βL[π] F).comp_analyticOn h
Β· rw [iteratedFDeriv_succ_eq_comp_left]
-- Porting note: for reasons that I do not understand at all, `?g` cannot be inlined.
convert ContinuousLin... | 8 | 2,980.957987 | 2 | 1.857143 | 7 | 1,925 |
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Analytic.CPolynomial
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
import Mathlib.Analysis.Calculus.FDeriv.Add
#align_import analysis.calculus.fderiv_analytic from "leanprover-community/mathlib"@"3bce8d800a6f2... | Mathlib/Analysis/Calculus/FDeriv/Analytic.lean | 314 | 346 | theorem changeOrigin_toFormalMultilinearSeries [DecidableEq ΞΉ] :
continuousMultilinearCurryFin1 π (β i, E i) F (f.toFormalMultilinearSeries.changeOrigin x 1) =
f.linearDeriv x := by |
ext y
rw [continuousMultilinearCurryFin1_apply, linearDeriv_apply,
changeOrigin, FormalMultilinearSeries.sum]
cases isEmpty_or_nonempty ΞΉ
Β· have (l) : 1 + l β Fintype.card ΞΉ := by
rw [add_comm, Fintype.card_eq_zero]; exact Nat.succ_ne_zero _
simp_rw [Fintype.sum_empty, changeOriginSeries_suppor... | 30 | 10,686,474,581,524.463 | 2 | 1.857143 | 7 | 1,925 |
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Analytic.CPolynomial
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
import Mathlib.Analysis.Calculus.FDeriv.Add
#align_import analysis.calculus.fderiv_analytic from "leanprover-community/mathlib"@"3bce8d800a6f2... | Mathlib/Analysis/Calculus/FDeriv/Analytic.lean | 449 | 458 | theorem derivSeries_apply_diag (n : β) (x : E) :
derivSeries p n (fun _ β¦ x) x = (n + 1) β’ p (n + 1) fun _ β¦ x := by |
simp only [derivSeries, compFormalMultilinearSeries_apply, changeOriginSeries,
compContinuousMultilinearMap_coe, ContinuousLinearEquiv.coe_coe, LinearIsometryEquiv.coe_coe,
Function.comp_apply, ContinuousMultilinearMap.sum_apply, map_sum, coe_sum', Finset.sum_apply,
continuousMultilinearCurryFin1_apply, ... | 8 | 2,980.957987 | 2 | 1.857143 | 7 | 1,925 |
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Analytic.CPolynomial
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
import Mathlib.Analysis.Calculus.FDeriv.Add
#align_import analysis.calculus.fderiv_analytic from "leanprover-community/mathlib"@"3bce8d800a6f2... | Mathlib/Analysis/Calculus/FDeriv/Analytic.lean | 469 | 474 | theorem iteratedFDeriv_zero_apply_diag : iteratedFDeriv π 0 f x = p 0 := by |
ext
convert (h.hasSum <| EMetric.mem_ball_self h.r_pos).tsum_eq.symm
Β· rw [iteratedFDeriv_zero_apply, add_zero]
Β· rw [tsum_eq_single 0 fun n hn β¦ by haveI := NeZero.mk hn; exact (p n).map_zero]
exact congr(p 0 $(Subsingleton.elim _ _))
| 5 | 148.413159 | 2 | 1.857143 | 7 | 1,925 |
import Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics
#align_import analysis.special_functions.pow.continuity from "leanprover-community/mathlib"@"0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8"
noncomputable section
open scoped Classical
open Real Topology NNReal ENNReal Filter ComplexConjugate
open Filter Finset... | Mathlib/Analysis/SpecialFunctions/Pow/Continuity.lean | 36 | 41 | theorem zero_cpow_eq_nhds {b : β} (hb : b β 0) : (fun x : β => (0 : β) ^ x) =αΆ [π b] 0 := by |
suffices βαΆ x : β in π b, x β 0 from
this.mono fun x hx β¦ by
dsimp only
rw [zero_cpow hx, Pi.zero_apply]
exact IsOpen.eventually_mem isOpen_ne hb
| 5 | 148.413159 | 2 | 1.857143 | 7 | 1,926 |
import Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics
#align_import analysis.special_functions.pow.continuity from "leanprover-community/mathlib"@"0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8"
noncomputable section
open scoped Classical
open Real Topology NNReal ENNReal Filter ComplexConjugate
open Filter Finset... | Mathlib/Analysis/SpecialFunctions/Pow/Continuity.lean | 44 | 50 | theorem cpow_eq_nhds {a b : β} (ha : a β 0) :
(fun x => x ^ b) =αΆ [π a] fun x => exp (log x * b) := by |
suffices βαΆ x : β in π a, x β 0 from
this.mono fun x hx β¦ by
dsimp only
rw [cpow_def_of_ne_zero hx]
exact IsOpen.eventually_mem isOpen_ne ha
| 5 | 148.413159 | 2 | 1.857143 | 7 | 1,926 |
import Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics
#align_import analysis.special_functions.pow.continuity from "leanprover-community/mathlib"@"0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8"
noncomputable section
open scoped Classical
open Real Topology NNReal ENNReal Filter ComplexConjugate
open Filter Finset... | Mathlib/Analysis/SpecialFunctions/Pow/Continuity.lean | 53 | 62 | theorem cpow_eq_nhds' {p : β Γ β} (hp_fst : p.fst β 0) :
(fun x => x.1 ^ x.2) =αΆ [π p] fun x => exp (log x.1 * x.2) := by |
suffices βαΆ x : β Γ β in π p, x.1 β 0 from
this.mono fun x hx β¦ by
dsimp only
rw [cpow_def_of_ne_zero hx]
refine IsOpen.eventually_mem ?_ hp_fst
change IsOpen { x : β Γ β | x.1 = 0 }αΆ
rw [isOpen_compl_iff]
exact isClosed_eq continuous_fst continuous_const
| 8 | 2,980.957987 | 2 | 1.857143 | 7 | 1,926 |
import Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics
#align_import analysis.special_functions.pow.continuity from "leanprover-community/mathlib"@"0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8"
noncomputable section
open scoped Classical
open Real Topology NNReal ENNReal Filter ComplexConjugate
open Filter Finset... | Mathlib/Analysis/SpecialFunctions/Pow/Continuity.lean | 66 | 71 | theorem continuousAt_const_cpow {a b : β} (ha : a β 0) : ContinuousAt (fun x : β => a ^ x) b := by |
have cpow_eq : (fun x : β => a ^ x) = fun x => exp (log a * x) := by
ext1 b
rw [cpow_def_of_ne_zero ha]
rw [cpow_eq]
exact continuous_exp.continuousAt.comp (ContinuousAt.mul continuousAt_const continuousAt_id)
| 5 | 148.413159 | 2 | 1.857143 | 7 | 1,926 |
import Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics
#align_import analysis.special_functions.pow.continuity from "leanprover-community/mathlib"@"0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8"
noncomputable section
open scoped Classical
open Real Topology NNReal ENNReal Filter ComplexConjugate
open Filter Finset... | Mathlib/Analysis/SpecialFunctions/Pow/Continuity.lean | 74 | 78 | theorem continuousAt_const_cpow' {a b : β} (h : b β 0) : ContinuousAt (fun x : β => a ^ x) b := by |
by_cases ha : a = 0
Β· rw [ha, continuousAt_congr (zero_cpow_eq_nhds h)]
exact continuousAt_const
Β· exact continuousAt_const_cpow ha
| 4 | 54.59815 | 2 | 1.857143 | 7 | 1,926 |
import Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics
#align_import analysis.special_functions.pow.continuity from "leanprover-community/mathlib"@"0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8"
noncomputable section
open scoped Classical
open Real Topology NNReal ENNReal Filter ComplexConjugate
open Filter Finset... | Mathlib/Analysis/SpecialFunctions/Pow/Continuity.lean | 84 | 91 | theorem continuousAt_cpow {p : β Γ β} (hp_fst : p.fst β slitPlane) :
ContinuousAt (fun x : β Γ β => x.1 ^ x.2) p := by |
rw [continuousAt_congr (cpow_eq_nhds' <| slitPlane_ne_zero hp_fst)]
refine continuous_exp.continuousAt.comp ?_
exact
ContinuousAt.mul
(ContinuousAt.comp (continuousAt_clog hp_fst) continuous_fst.continuousAt)
continuous_snd.continuousAt
| 6 | 403.428793 | 2 | 1.857143 | 7 | 1,926 |
import Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics
#align_import analysis.special_functions.pow.continuity from "leanprover-community/mathlib"@"0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8"
noncomputable section
open scoped Classical
open Real Topology NNReal ENNReal Filter ComplexConjugate
open Filter Finset... | Mathlib/Analysis/SpecialFunctions/Pow/Continuity.lean | 105 | 109 | theorem Filter.Tendsto.const_cpow {l : Filter Ξ±} {f : Ξ± β β} {a b : β} (hf : Tendsto f l (π b))
(h : a β 0 β¨ b β 0) : Tendsto (fun x => a ^ f x) l (π (a ^ b)) := by |
cases h with
| inl h => exact (continuousAt_const_cpow h).tendsto.comp hf
| inr h => exact (continuousAt_const_cpow' h).tendsto.comp hf
| 3 | 20.085537 | 1 | 1.857143 | 7 | 1,926 |
import Mathlib.Probability.Notation
import Mathlib.Probability.Integration
import Mathlib.MeasureTheory.Function.L2Space
#align_import probability.variance from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
open MeasureTheory Filter Finset
noncomputable section
open scoped MeasureThe... | Mathlib/Probability/Variance.lean | 65 | 72 | theorem _root_.MeasureTheory.Memβp.evariance_lt_top [IsFiniteMeasure ΞΌ] (hX : Memβp X 2 ΞΌ) :
evariance X ΞΌ < β := by |
have := ENNReal.pow_lt_top (hX.sub <| memβp_const <| ΞΌ[X]).2 2
rw [snorm_eq_lintegral_rpow_nnnorm two_ne_zero ENNReal.two_ne_top, β ENNReal.rpow_two] at this
simp only [coe_two, Pi.sub_apply, ENNReal.one_toReal, one_div] at this
rw [β ENNReal.rpow_mul, inv_mul_cancel (two_ne_zero : (2 : β) β 0), ENNReal.rpow_o... | 6 | 403.428793 | 2 | 1.857143 | 7 | 1,927 |
import Mathlib.Probability.Notation
import Mathlib.Probability.Integration
import Mathlib.MeasureTheory.Function.L2Space
#align_import probability.variance from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
open MeasureTheory Filter Finset
noncomputable section
open scoped MeasureThe... | Mathlib/Probability/Variance.lean | 75 | 89 | theorem evariance_eq_top [IsFiniteMeasure ΞΌ] (hXm : AEStronglyMeasurable X ΞΌ) (hX : Β¬Memβp X 2 ΞΌ) :
evariance X ΞΌ = β := by |
by_contra h
rw [β Ne, β lt_top_iff_ne_top] at h
have : Memβp (fun Ο => X Ο - ΞΌ[X]) 2 ΞΌ := by
refine β¨hXm.sub aestronglyMeasurable_const, ?_β©
rw [snorm_eq_lintegral_rpow_nnnorm two_ne_zero ENNReal.two_ne_top]
simp only [coe_two, ENNReal.one_toReal, ENNReal.rpow_two, Ne]
exact ENNReal.rpow_lt_top_o... | 13 | 442,413.392009 | 2 | 1.857143 | 7 | 1,927 |
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