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 |
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import Mathlib.Algebra.MvPolynomial.Basic
import Mathlib.RingTheory.Polynomial.Basic
import Mathlib.RingTheory.PrincipalIdealDomain
#align_import ring_theory.adjoin.fg from "leanprover-community/mathlib"@"c4658a649d216f57e99621708b09dcb3dcccbd23"
universe u v w
open Subsemiring Ring Submodule
open Pointwise
na... | Mathlib/RingTheory/Adjoin/FG.lean | 170 | 179 | theorem induction_on_adjoin [IsNoetherian R A] (P : Subalgebra R A → Prop) (base : P ⊥)
(ih : ∀ (S : Subalgebra R A) (x : A), P S → P (Algebra.adjoin R (insert x S)))
(S : Subalgebra R A) : P S := by |
classical
obtain ⟨t, rfl⟩ := S.fg_of_noetherian
refine Finset.induction_on t ?_ ?_
· simpa using base
intro x t _ h
rw [Finset.coe_insert]
simpa only [Algebra.adjoin_insert_adjoin] using ih _ x h
| 7 |
import Mathlib.Combinatorics.Enumerative.DoubleCounting
import Mathlib.Combinatorics.SimpleGraph.AdjMatrix
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathlib.Data.Set.Finite
#align_import combinatorics.simple_graph.strongly_regular from "leanprover-community/mathlib"@"2b35fc7bea4640cb75e477e83f32fbd5389208... | Mathlib/Combinatorics/SimpleGraph/StronglyRegular.lean | 125 | 134 | theorem sdiff_compl_neighborFinset_inter_eq {v w : V} (h : G.Adj v w) :
((G.neighborFinset v)ᶜ ∩ (G.neighborFinset w)ᶜ) \ ({w} ∪ {v}) =
(G.neighborFinset v)ᶜ ∩ (G.neighborFinset w)ᶜ := by |
ext
simp only [and_imp, mem_union, mem_sdiff, mem_compl, and_iff_left_iff_imp, mem_neighborFinset,
mem_inter, mem_singleton]
rintro hnv hnw (rfl | rfl)
· exact hnv h
· apply hnw
rwa [adj_comm]
| 7 |
import Mathlib.RingTheory.Algebraic
import Mathlib.RingTheory.Localization.AtPrime
import Mathlib.RingTheory.Localization.Integral
#align_import ring_theory.ideal.over from "leanprover-community/mathlib"@"198cb64d5c961e1a8d0d3e219feb7058d5353861"
variable {R : Type*} [CommRing R]
namespace Ideal
open Polynomial... | Mathlib/RingTheory/Ideal/Over.lean | 77 | 89 | theorem injective_quotient_le_comap_map (P : Ideal R[X]) :
Function.Injective <|
Ideal.quotientMap
(Ideal.map (Polynomial.mapRingHom (Quotient.mk (P.comap (C : R →+* R[X])))) P)
(Polynomial.mapRingHom (Ideal.Quotient.mk (P.comap (C : R →+* R[X]))))
le_comap_map := by |
refine quotientMap_injective' (le_of_eq ?_)
rw [comap_map_of_surjective (mapRingHom (Ideal.Quotient.mk (P.comap (C : R →+* R[X]))))
(map_surjective (Ideal.Quotient.mk (P.comap (C : R →+* R[X]))) Ideal.Quotient.mk_surjective)]
refine le_antisymm (sup_le le_rfl ?_) (le_sup_of_le_left le_rfl)
refine fun p h... | 7 |
import Mathlib.CategoryTheory.Groupoid
import Mathlib.Combinatorics.Quiver.Basic
#align_import category_theory.groupoid.basic from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a485d0456fc271482da"
namespace CategoryTheory
namespace Groupoid
variable (C : Type*) [Groupoid C]
section Thin
| Mathlib/CategoryTheory/Groupoid/Basic.lean | 23 | 30 | theorem isThin_iff : Quiver.IsThin C ↔ ∀ c : C, Subsingleton (c ⟶ c) := by |
refine ⟨fun h c => h c c, fun h c d => Subsingleton.intro fun f g => ?_⟩
haveI := h d
calc
f = f ≫ inv g ≫ g := by simp only [inv_eq_inv, IsIso.inv_hom_id, Category.comp_id]
_ = f ≫ inv f ≫ g := by congr 1
simp only [inv_eq_inv, IsIso.inv_hom_id, eq_iff_true_of_subsingleton]
... | 7 |
import Mathlib.Algebra.Exact
import Mathlib.RingTheory.TensorProduct.Basic
section Modules
open TensorProduct LinearMap
section Semiring
variable {R : Type*} [CommSemiring R] {M N P Q: Type*}
[AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [AddCommMonoid Q]
[Module R M] [Module R N] [Module R P] [... | Mathlib/LinearAlgebra/TensorProduct/RightExactness.lean | 124 | 133 | theorem LinearMap.lTensor_range :
range (lTensor Q g) =
range (lTensor Q (Submodule.subtype (range g))) := by |
have : g = (Submodule.subtype _).comp g.rangeRestrict := rfl
nth_rewrite 1 [this]
rw [lTensor_comp]
apply range_comp_of_range_eq_top
rw [range_eq_top]
apply lTensor_surjective
rw [← range_eq_top, range_rangeRestrict]
| 7 |
import Mathlib.Data.Finset.Fin
import Mathlib.Data.Int.Order.Units
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.GroupTheory.Perm.Support
import Mathlib.Logic.Equiv.Fintype
#align_import group_theory.perm.sign from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
universe u v
o... | Mathlib/GroupTheory/Perm/Finite.lean | 57 | 65 | theorem perm_inv_on_of_perm_on_finset {s : Finset α} {f : Perm α} (h : ∀ x ∈ s, f x ∈ s) {y : α}
(hy : y ∈ s) : f⁻¹ y ∈ s := by |
have h0 : ∀ y ∈ s, ∃ (x : _) (hx : x ∈ s), y = (fun i (_ : i ∈ s) => f i) x hx :=
Finset.surj_on_of_inj_on_of_card_le (fun x hx => (fun i _ => f i) x hx) (fun a ha => h a ha)
(fun a₁ a₂ ha₁ ha₂ heq => (Equiv.apply_eq_iff_eq f).mp heq) rfl.ge
obtain ⟨y2, hy2, heq⟩ := h0 y hy
convert hy2
rw [heq]
sim... | 7 |
import Mathlib.RingTheory.RootsOfUnity.Basic
import Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed
import Mathlib.Algebra.GCDMonoid.IntegrallyClosed
import Mathlib.FieldTheory.Finite.Basic
#align_import ring_theory.roots_of_unity.minpoly from "leanprover-community/mathlib"@"7fdeecc0d03cd40f7a165e6cf00a4d2286db599f"
... | Mathlib/RingTheory/RootsOfUnity/Minpoly.lean | 95 | 104 | theorem minpoly_dvd_pow_mod {p : ℕ} [hprime : Fact p.Prime] (hdiv : ¬p ∣ n) :
map (Int.castRingHom (ZMod p)) (minpoly ℤ μ) ∣
map (Int.castRingHom (ZMod p)) (minpoly ℤ (μ ^ p)) ^ p := by |
set Q := minpoly ℤ (μ ^ p)
have hfrob :
map (Int.castRingHom (ZMod p)) Q ^ p = map (Int.castRingHom (ZMod p)) (expand ℤ p Q) := by
rw [← ZMod.expand_card, map_expand]
rw [hfrob]
apply RingHom.map_dvd (mapRingHom (Int.castRingHom (ZMod p)))
exact minpoly_dvd_expand h hdiv
| 7 |
import Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho
import Mathlib.LinearAlgebra.Orientation
#align_import analysis.inner_product_space.orientation from "leanprover-community/mathlib"@"bd65478311e4dfd41f48bf38c7e3b02fb75d0163"
noncomputable section
variable {E : Type*} [NormedAddCommGroup E] [InnerProduct... | Mathlib/Analysis/InnerProductSpace/Orientation.lean | 76 | 84 | theorem same_orientation_iff_det_eq_det :
e.toBasis.det = f.toBasis.det ↔ e.toBasis.orientation = f.toBasis.orientation := by |
constructor
· intro h
dsimp [Basis.orientation]
congr
· intro h
rw [e.toBasis.det.eq_smul_basis_det f.toBasis]
simp [e.det_to_matrix_orthonormalBasis_of_same_orientation f h]
| 7 |
import Mathlib.Algebra.PUnitInstances
import Mathlib.Tactic.Abel
import Mathlib.Tactic.Ring
import Mathlib.Order.Hom.Lattice
#align_import algebra.ring.boolean_ring from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
open scoped symmDiff
variable {α β γ : Type*}
class BooleanRing (α) ... | Mathlib/Algebra/Ring/BooleanRing.lean | 90 | 97 | theorem mul_add_mul : a * b + b * a = 0 := by |
have : a + b = a + b + (a * b + b * a) :=
calc
a + b = (a + b) * (a + b) := by rw [mul_self]
_ = a * a + a * b + (b * a + b * b) := by rw [add_mul, mul_add, mul_add]
_ = a + a * b + (b * a + b) := by simp only [mul_self]
_ = a + b + (a * b + b * a) := by abel
rwa [self_eq_add_right] at ... | 7 |
import Mathlib.MeasureTheory.Integral.Lebesgue
open Set hiding restrict restrict_apply
open Filter ENNReal NNReal MeasureTheory.Measure
namespace MeasureTheory
variable {α : Type*} {m0 : MeasurableSpace α} {μ : Measure α}
noncomputable
def Measure.withDensity {m : MeasurableSpace α} (μ : Measure α) (f : α → ℝ≥... | Mathlib/MeasureTheory/Measure/WithDensity.lean | 44 | 52 | theorem withDensity_apply_le (f : α → ℝ≥0∞) (s : Set α) :
∫⁻ a in s, f a ∂μ ≤ μ.withDensity f s := by |
let t := toMeasurable (μ.withDensity f) s
calc
∫⁻ a in s, f a ∂μ ≤ ∫⁻ a in t, f a ∂μ :=
lintegral_mono_set (subset_toMeasurable (withDensity μ f) s)
_ = μ.withDensity f t :=
(withDensity_apply f (measurableSet_toMeasurable (withDensity μ f) s)).symm
_ = μ.withDensity f s := measure_toMeasurable s
| 7 |
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Int.GCD
import Mathlib.RingTheory.Coprime.Basic
#align_import ring_theory.coprime.lemmas from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226"
universe u v
section IsCoprime
variable {R : Type ... | Mathlib/RingTheory/Coprime/Lemmas.lean | 33 | 40 | theorem Int.isCoprime_iff_gcd_eq_one {m n : ℤ} : IsCoprime m n ↔ Int.gcd m n = 1 := by |
constructor
· rintro ⟨a, b, h⟩
have : 1 = m * a + n * b := by rwa [mul_comm m, mul_comm n, eq_comm]
exact Nat.dvd_one.mp (Int.gcd_dvd_iff.mpr ⟨a, b, this⟩)
· rw [← Int.ofNat_inj, IsCoprime, Int.gcd_eq_gcd_ab, mul_comm m, mul_comm n, Nat.cast_one]
intro h
exact ⟨_, _, h⟩
| 7 |
import Mathlib.Algebra.Module.Defs
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.FreeModule.Basic
import Mathlib.LinearAlgebra.TensorProduct.Tower
#align_import algebra.module.projective from "leanprover-community/mathlib"@"405ea5cee7a7070ff8fb8dcb4cfb003532e34bce"
universe u v
open LinearMap ... | Mathlib/Algebra/Module/Projective.lean | 156 | 163 | theorem Projective.of_basis {ι : Type*} (b : Basis ι R P) : Projective R P := by |
-- need P →ₗ (P →₀ R) for definition of projective.
-- get it from `ι → (P →₀ R)` coming from `b`.
use b.constr ℕ fun i => Finsupp.single (b i) (1 : R)
intro m
simp only [b.constr_apply, mul_one, id, Finsupp.smul_single', Finsupp.total_single,
map_finsupp_sum]
exact b.total_repr m
| 7 |
import Mathlib.Topology.Algebra.UniformConvergence
#align_import topology.algebra.equicontinuity from "leanprover-community/mathlib"@"01ad394a11bf06b950232720cf7e8fc6b22f0d6a"
open Function
open UniformConvergence
@[to_additive]
| Mathlib/Topology/Algebra/Equicontinuity.lean | 20 | 31 | theorem equicontinuous_of_equicontinuousAt_one {ι G M hom : Type*} [TopologicalSpace G]
[UniformSpace M] [Group G] [Group M] [TopologicalGroup G] [UniformGroup M]
[FunLike hom G M] [MonoidHomClass hom G M] (F : ι → hom)
(hf : EquicontinuousAt ((↑) ∘ F) (1 : G)) :
Equicontinuous ((↑) ∘ F) := by |
rw [equicontinuous_iff_continuous]
rw [equicontinuousAt_iff_continuousAt] at hf
let φ : G →* (ι →ᵤ M) :=
{ toFun := swap ((↑) ∘ F)
map_one' := by dsimp [UniformFun]; ext; exact map_one _
map_mul' := fun a b => by dsimp [UniformFun]; ext; exact map_mul _ _ _ }
exact continuous_of_continuousAt_on... | 7 |
import Mathlib.Analysis.MeanInequalities
import Mathlib.Analysis.MeanInequalitiesPow
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Data.Set.Image
import Mathlib.Topology.Algebra.Order.LiminfLimsup
#align_import analysis.normed_space.lp_space from "leanprover-community/mathlib"@"de83b43717abe3... | Mathlib/Analysis/NormedSpace/lpSpace.lean | 160 | 167 | theorem neg {f : ∀ i, E i} (hf : Memℓp f p) : Memℓp (-f) p := by |
rcases p.trichotomy with (rfl | rfl | hp)
· apply memℓp_zero
simp [hf.finite_dsupport]
· apply memℓp_infty
simpa using hf.bddAbove
· apply memℓp_gen
simpa using hf.summable hp
| 7 |
import Mathlib.Analysis.Convex.Hull
import Mathlib.LinearAlgebra.AffineSpace.Independent
#align_import analysis.convex.simplicial_complex.basic from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
open Finset Set
variable (𝕜 E : Type*) {ι : Type*} [OrderedRing 𝕜] [AddCommGroup E] [Mod... | Mathlib/Analysis/Convex/SimplicialComplex/Basic.lean | 110 | 119 | theorem disjoint_or_exists_inter_eq_convexHull (hs : s ∈ K.faces) (ht : t ∈ K.faces) :
Disjoint (convexHull 𝕜 (s : Set E)) (convexHull 𝕜 ↑t) ∨
∃ u ∈ K.faces, convexHull 𝕜 (s : Set E) ∩ convexHull 𝕜 ↑t = convexHull 𝕜 ↑u := by |
classical
by_contra! h
refine h.2 (s ∩ t) (K.down_closed hs inter_subset_left fun hst => h.1 <|
disjoint_iff_inf_le.mpr <| (K.inter_subset_convexHull hs ht).trans ?_) ?_
· rw [← coe_inter, hst, coe_empty, convexHull_empty]
rfl
· rw [coe_inter, convexHull_inter_convexHull hs ht]
| 7 |
import Mathlib.NumberTheory.LegendreSymbol.Basic
import Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum
#align_import number_theory.legendre_symbol.quadratic_reciprocity from "leanprover-community/mathlib"@"5b2fe80501ff327b9109fb09b7cc8c325cd0d7d9"
open Nat
section Values
variable {p : ℕ} [Fact p.Pri... | Mathlib/NumberTheory/LegendreSymbol/QuadraticReciprocity.lean | 89 | 96 | theorem exists_sq_eq_neg_two_iff : IsSquare (-2 : ZMod p) ↔ p % 8 = 1 ∨ p % 8 = 3 := by |
rw [FiniteField.isSquare_neg_two_iff, card p]
have h₁ := Prime.mod_two_eq_one_iff_ne_two.mpr hp
rw [← mod_mod_of_dvd p (by decide : 2 ∣ 8)] at h₁
have h₂ := mod_lt p (by norm_num : 0 < 8)
revert h₂ h₁
generalize p % 8 = m; clear! p
intros; interval_cases m <;> simp_all -- Porting note (#11043): was `deci... | 7 |
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.PUnitInstances
import Mathlib.GroupTheory.Congruence.Basic
open FreeMonoid Function List Set
namespace Monoid
@[to_additive "The minimal additive congruence relation `c` on `FreeAddMonoid (M ⊕ N)`... | Mathlib/GroupTheory/Coprod/Basic.lean | 189 | 199 | theorem induction_on' {C : M ∗ N → Prop} (m : M ∗ N)
(one : C 1)
(inl_mul : ∀ m x, C x → C (inl m * x))
(inr_mul : ∀ n x, C x → C (inr n * x)) : C m := by |
rcases mk_surjective m with ⟨x, rfl⟩
induction x using FreeMonoid.recOn with
| h0 => exact one
| ih x xs ih =>
cases x with
| inl m => simpa using inl_mul m _ ih
| inr n => simpa using inr_mul n _ ih
| 7 |
import Mathlib.Logic.Function.Conjugate
#align_import logic.function.iterate from "leanprover-community/mathlib"@"792a2a264169d64986541c6f8f7e3bbb6acb6295"
universe u v
variable {α : Type u} {β : Type v}
def Nat.iterate {α : Sort u} (op : α → α) : ℕ → α → α
| 0, a => a
| succ k, a => iterate op k (op a)
#a... | Mathlib/Logic/Function/Iterate.lean | 121 | 129 | theorem iterate_left {g : ℕ → α → α} (H : ∀ n, Semiconj f (g n) (g <| n + 1)) (n k : ℕ) :
Semiconj f^[n] (g k) (g <| n + k) := by |
induction n generalizing k with
| zero =>
rw [Nat.zero_add]
exact id_left
| succ n ihn =>
rw [Nat.add_right_comm, Nat.add_assoc]
exact (H k).trans (ihn (k + 1))
| 7 |
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.GroupTheory.Perm.Basic
#align_import group_theory.perm.support from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Equiv Finset
namespace Equiv.Perm
variable {α : Type*}
section Disjoint
... | Mathlib/GroupTheory/Perm/Support.lean | 144 | 152 | theorem nodup_of_pairwise_disjoint {l : List (Perm α)} (h1 : (1 : Perm α) ∉ l)
(h2 : l.Pairwise Disjoint) : l.Nodup := by |
refine List.Pairwise.imp_of_mem ?_ h2
intro τ σ h_mem _ h_disjoint _
subst τ
suffices (σ : Perm α) = 1 by
rw [this] at h_mem
exact h1 h_mem
exact ext fun a => or_self_iff.mp (h_disjoint a)
| 7 |
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Algebra.Subalgebra.Prod
import Mathlib.Algebra.Algebra.Subalgebra.Tower
import Mathlib.LinearAlgebra.Basis
import Mathlib.LinearAlgebra.Prod
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.Prod
#align_import ring_theory.adjoin.basic fr... | Mathlib/RingTheory/Adjoin/Basic.lean | 99 | 113 | theorem adjoin_induction₂ {p : A → A → Prop} {a b : A} (ha : a ∈ adjoin R s) (hb : b ∈ adjoin R s)
(Hs : ∀ x ∈ s, ∀ y ∈ s, p x y) (Halg : ∀ r₁ r₂, p (algebraMap R A r₁) (algebraMap R A r₂))
(Halg_left : ∀ (r), ∀ x ∈ s, p (algebraMap R A r) x)
(Halg_right : ∀ (r), ∀ x ∈ s, p x (algebraMap R A r))
(Hadd_l... |
refine adjoin_induction hb ?_ (fun r => ?_) (Hadd_right a) (Hmul_right a)
· exact adjoin_induction ha Hs Halg_left
(fun x y Hx Hy z hz => Hadd_left x y z (Hx z hz) (Hy z hz))
fun x y Hx Hy z hz => Hmul_left x y z (Hx z hz) (Hy z hz)
· exact adjoin_induction ha (Halg_right r) (fun r' => Halg r' r)
... | 7 |
import Mathlib.Data.Complex.Basic
import Mathlib.MeasureTheory.Integral.CircleIntegral
#align_import measure_theory.integral.circle_transform from "leanprover-community/mathlib"@"d11893b411025250c8e61ff2f12ccbd7ee35ab15"
open Set MeasureTheory Metric Filter Function
open scoped Interval Real
noncomputable secti... | Mathlib/MeasureTheory/Integral/CircleTransform.lean | 109 | 117 | theorem continuousOn_abs_circleTransformBoundingFunction {R r : ℝ} (hr : r < R) (z : ℂ) :
ContinuousOn (abs ∘ circleTransformBoundingFunction R z) (closedBall z r ×ˢ univ) := by |
have : ContinuousOn (circleTransformBoundingFunction R z) (closedBall z r ×ˢ univ) := by
apply_rules [ContinuousOn.smul, continuousOn_const]
· simp only [deriv_circleMap]
apply_rules [ContinuousOn.mul, (continuous_circleMap 0 R).comp_continuousOn continuousOn_snd,
continuousOn_const]
· simp... | 7 |
import Mathlib.Geometry.Euclidean.Sphere.Basic
#align_import geometry.euclidean.sphere.second_inter from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open RealInnerProductSpace
namespace EuclideanGeometry
variable {V : Type*} {P : Type*} [NormedAddCommGroup V]... | Mathlib/Geometry/Euclidean/Sphere/SecondInter.lean | 70 | 78 | theorem Sphere.secondInter_eq_self_iff {s : Sphere P} {p : P} {v : V} :
s.secondInter p v = p ↔ ⟪v, p -ᵥ s.center⟫ = 0 := by |
refine ⟨fun hp => ?_, fun hp => ?_⟩
· by_cases hv : v = 0
· simp [hv]
rwa [Sphere.secondInter, eq_comm, eq_vadd_iff_vsub_eq, vsub_self, eq_comm, smul_eq_zero,
or_iff_left hv, div_eq_zero_iff, inner_self_eq_zero, or_iff_left hv, mul_eq_zero,
or_iff_right (by norm_num : (-2 : ℝ) ≠ 0)] at hp
· r... | 7 |
import Mathlib.Topology.Bases
import Mathlib.Topology.DenseEmbedding
#align_import topology.stone_cech from "leanprover-community/mathlib"@"0a0ec35061ed9960bf0e7ffb0335f44447b58977"
noncomputable section
open Filter Set
open Topology
universe u v
section Ultrafilter
def ultrafilterBasis (α : Type u) : Set ... | Mathlib/Topology/StoneCech.lean | 110 | 117 | theorem ultrafilter_comap_pure_nhds (b : Ultrafilter α) : comap pure (𝓝 b) ≤ b := by |
rw [TopologicalSpace.nhds_generateFrom]
simp only [comap_iInf, comap_principal]
intro s hs
rw [← le_principal_iff]
refine iInf_le_of_le { u | s ∈ u } ?_
refine iInf_le_of_le ⟨hs, ⟨s, rfl⟩⟩ ?_
exact principal_mono.2 fun a => id
| 7 |
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Algebra.Order.Invertible
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.Algebra.Order.Group.Instances
import Mathlib.LinearAlgebra.AffineSpace.Slope
import Mathlib.LinearAlgebra.AffineSpace.Midpoint
import Mathlib.Tactic.FieldSimp
#align_import li... | Mathlib/LinearAlgebra/AffineSpace/Ordered.lean | 240 | 248 | theorem map_le_lineMap_iff_slope_le_slope_right (h : 0 < (1 - r) * (b - a)) :
f c ≤ lineMap (f a) (f b) r ↔ slope f a b ≤ slope f c b := by |
rw [← lineMap_apply_one_sub, ← lineMap_apply_one_sub _ _ r]
revert h; generalize 1 - r = r'; clear! r; intro h
simp_rw [lineMap_apply, slope, vsub_eq_sub, vadd_eq_add, smul_eq_mul]
rw [sub_add_eq_sub_sub_swap, sub_self, zero_sub, neg_mul_eq_mul_neg, neg_sub,
le_inv_smul_iff_of_pos h, smul_smul, mul_inv_can... | 7 |
import Mathlib.CategoryTheory.Comma.Over
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Yoneda
import Mathlib.Data.Set.Lattice
import Mathlib.Order.CompleteLattice
#align_import category_theory.sites.sieves from "leanprover-community/mathlib"@"239d882c4fb58361ee8b3b39fb2091320edef... | Mathlib/CategoryTheory/Sites/Sieves.lean | 141 | 148 | theorem ofArrows_pUnit : (ofArrows _ fun _ : PUnit => f) = singleton f := by |
funext Y
ext g
constructor
· rintro ⟨_⟩
apply singleton.mk
· rintro ⟨_⟩
exact ofArrows.mk PUnit.unit
| 7 |
import Mathlib.RingTheory.WittVector.IsPoly
#align_import ring_theory.witt_vector.mul_p from "leanprover-community/mathlib"@"7abfbc92eec87190fba3ed3d5ec58e7c167e7144"
namespace WittVector
variable {p : ℕ} {R : Type*} [hp : Fact p.Prime] [CommRing R]
local notation "𝕎" => WittVector p -- type as `\bbW`
open Mv... | Mathlib/RingTheory/WittVector/MulP.lean | 72 | 80 | theorem bind₁_wittMulN_wittPolynomial (n k : ℕ) :
bind₁ (wittMulN p n) (wittPolynomial p ℤ k) = n * wittPolynomial p ℤ k := by |
induction' n with n ih
· simp [wittMulN, Nat.cast_zero, zero_mul, bind₁_zero_wittPolynomial]
· rw [wittMulN, ← bind₁_bind₁, wittAdd, wittStructureInt_prop]
simp only [AlgHom.map_add, Nat.cast_succ, bind₁_X_right]
rw [add_mul, one_mul, bind₁_rename, bind₁_rename]
simp only [ih, Function.uncurry, Funct... | 7 |
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 ≫ firstMap P R = forkMap P R ≫ secondMap P R := by |
dsimp
ext fg
simp only [firstMap, secondMap, forkMap]
simp only [limit.lift_π, limit.lift_π_assoc, assoc, Fan.mk_π_app]
haveI := Presieve.hasPullbacks.has_pullbacks fg.1.2.2 fg.2.2.2
rw [← P.map_comp, ← op_comp, pullback.condition]
simp
| 7 |
import Mathlib.Order.CompleteLattice
import Mathlib.Data.Finset.Lattice
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Category.Preorder
import Mathlib.CategoryTheory.Limits.Shapes.Products
import Mathlib.CategoryTheory.Limits.Shapes.FiniteLimits
#align_import category_theory.limi... | Mathlib/CategoryTheory/Limits/Lattice.lean | 99 | 107 | theorem finite_coproduct_eq_finset_sup [SemilatticeSup α] [OrderBot α] {ι : Type u} [Fintype ι]
(f : ι → α) : ∐ f = Fintype.elems.sup f := by |
trans
· exact
(IsColimit.coconePointUniqueUpToIso (colimit.isColimit _)
(finiteColimitCocone (Discrete.functor f)).isColimit).to_eq
change Finset.univ.sup (f ∘ discreteEquiv.toEmbedding) = Fintype.elems.sup f
simp only [← Finset.sup_map, Finset.univ_map_equiv_to_embedding]
rfl
| 7 |
import Mathlib.Algebra.Algebra.Prod
import Mathlib.LinearAlgebra.Basic
import Mathlib.LinearAlgebra.Span
import Mathlib.Order.PartialSups
#align_import linear_algebra.prod from "leanprover-community/mathlib"@"cd391184c85986113f8c00844cfe6dda1d34be3d"
universe u v w x y z u' v' w' y'
variable {R : Type u} {K : Ty... | Mathlib/LinearAlgebra/Prod.lean | 148 | 155 | theorem range_inl : range (inl R M M₂) = ker (snd R M M₂) := by |
ext x
simp only [mem_ker, mem_range]
constructor
· rintro ⟨y, rfl⟩
rfl
· intro h
exact ⟨x.fst, Prod.ext rfl h.symm⟩
| 7 |
import Mathlib.Data.Sigma.Lex
import Mathlib.Order.BoundedOrder
import Mathlib.Mathport.Notation
import Mathlib.Data.Sigma.Basic
#align_import data.sigma.order from "leanprover-community/mathlib"@"1fc36cc9c8264e6e81253f88be7fb2cb6c92d76a"
namespace Sigma
variable {ι : Type*} {α : ι → Type*}
-- Porting note: I... | Mathlib/Data/Sigma/Order.lean | 89 | 96 | theorem lt_def [∀ i, LT (α i)] {a b : Σi, α i} : a < b ↔ ∃ h : a.1 = b.1, h.rec a.2 < b.2 := by |
constructor
· rintro ⟨i, a, b, h⟩
exact ⟨rfl, h⟩
· obtain ⟨i, a⟩ := a
obtain ⟨j, b⟩ := b
rintro ⟨rfl : i = j, h⟩
exact lt.fiber _ _ _ h
| 7 |
import Mathlib.Topology.Defs.Induced
import Mathlib.Topology.Basic
#align_import topology.order from "leanprover-community/mathlib"@"bcfa726826abd57587355b4b5b7e78ad6527b7e4"
open Function Set Filter Topology
universe u v w
namespace TopologicalSpace
variable {α : Type u}
inductive GenerateOpen (g : Set (Set ... | Mathlib/Topology/Order.lean | 129 | 138 | theorem nhds_mkOfNhds_single [DecidableEq α] {a₀ : α} {l : Filter α} (h : pure a₀ ≤ l) (b : α) :
@nhds α (TopologicalSpace.mkOfNhds (update pure a₀ l)) b =
(update pure a₀ l : α → Filter α) b := by |
refine nhds_mkOfNhds _ _ (le_update_iff.mpr ⟨h, fun _ _ => le_rfl⟩) fun a s hs => ?_
rcases eq_or_ne a a₀ with (rfl | ha)
· filter_upwards [hs] with b hb
rcases eq_or_ne b a with (rfl | hb)
· exact hs
· rwa [update_noteq hb]
· simpa only [update_noteq ha, mem_pure, eventually_pure] using hs
| 7 |
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv
import Mathlib.Analysis.Calculus.FDeriv.Extend
import Mathlib.Analysis.Calculus.Deriv.Prod
import Mathlib.Analysis.SpecialFunctions.Log.Deriv
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv
#... | Mathlib/Analysis/SpecialFunctions/Pow/Deriv.lean | 305 | 313 | theorem contDiffAt_rpow_of_ne (p : ℝ × ℝ) (hp : p.1 ≠ 0) {n : ℕ∞} :
ContDiffAt ℝ n (fun p : ℝ × ℝ => p.1 ^ p.2) p := by |
cases' hp.lt_or_lt with hneg hpos
exacts
[(((contDiffAt_fst.log hneg.ne).mul contDiffAt_snd).exp.mul
(contDiffAt_snd.mul contDiffAt_const).cos).congr_of_eventuallyEq
((continuousAt_fst.eventually (gt_mem_nhds hneg)).mono fun p hp => rpow_def_of_neg hp _),
((contDiffAt_fst.log hpos.ne').mul ... | 7 |
import Mathlib.CategoryTheory.Closed.Cartesian
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.BinaryProducts
import Mathlib.CategoryTheory.Adjunction.FullyFaithful
#align_import category_theory.closed.functor from "leanprover-community/mathlib"@"cea27692b3fdeb328a2ddba6aabf181754543184"
noncomputable secti... | Mathlib/CategoryTheory/Closed/Functor.lean | 107 | 116 | theorem expComparison_whiskerLeft {A A' : C} (f : A' ⟶ A) :
expComparison F A ≫ whiskerLeft _ (pre (F.map f)) =
whiskerRight (pre f) _ ≫ expComparison F A' := by |
ext B
dsimp
apply uncurry_injective
rw [uncurry_natural_left, uncurry_natural_left, uncurry_expComparison, uncurry_pre,
prod.map_swap_assoc, ← F.map_id, expComparison_ev, ← F.map_id, ←
prodComparison_inv_natural_assoc, ← prodComparison_inv_natural_assoc, ← F.map_comp, ←
F.map_comp, prod_map_pre_app... | 7 |
import Mathlib.Analysis.Calculus.Deriv.Inv
import Mathlib.Analysis.Calculus.Deriv.Polynomial
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
import Mathlib.Analysis.SpecialFunctions.PolynomialExp
#align_import analysis.calculus.bump_function_inner from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9... | Mathlib/Analysis/SpecialFunctions/SmoothTransition.lean | 127 | 135 | theorem contDiff_polynomial_eval_inv_mul {n : ℕ∞} (p : ℝ[X]) :
ContDiff ℝ n (fun x ↦ p.eval x⁻¹ * expNegInvGlue x) := by |
apply contDiff_all_iff_nat.2 (fun m => ?_) n
induction m generalizing p with
| zero => exact contDiff_zero.2 <| continuous_polynomial_eval_inv_mul _
| succ m ihm =>
refine contDiff_succ_iff_deriv.2 ⟨differentiable_polynomial_eval_inv_mul _, ?_⟩
convert ihm (X ^ 2 * (p - derivative (R := ℝ) p)) using 2
... | 7 |
import Mathlib.Algebra.Lie.Abelian
import Mathlib.Algebra.Lie.IdealOperations
import Mathlib.Order.Hom.Basic
#align_import algebra.lie.solvable from "leanprover-community/mathlib"@"a50170a88a47570ed186b809ca754110590f9476"
universe u v w w₁ w₂
variable (R : Type u) (L : Type v) (M : Type w) {L' : Type w₁}
variab... | Mathlib/Algebra/Lie/Solvable.lean | 116 | 124 | theorem derivedSeriesOfIdeal_add_le_add (J : LieIdeal R L) (k l : ℕ) :
D (k + l) (I + J) ≤ D k I + D l J := by |
let D₁ : LieIdeal R L →o LieIdeal R L :=
{ toFun := fun I => ⁅I, I⁆
monotone' := fun I J h => LieSubmodule.mono_lie I J I J h h }
have h₁ : ∀ I J : LieIdeal R L, D₁ (I ⊔ J) ≤ D₁ I ⊔ J := by
simp [D₁, LieSubmodule.lie_le_right, LieSubmodule.lie_le_left, le_sup_of_le_right]
rw [← D₁.iterate_sup_le_su... | 7 |
import Mathlib.Algebra.Category.ModuleCat.EpiMono
import Mathlib.Algebra.Category.ModuleCat.Kernels
import Mathlib.CategoryTheory.Subobject.WellPowered
import Mathlib.CategoryTheory.Subobject.Limits
#align_import algebra.category.Module.subobject from "leanprover-community/mathlib"@"6d584f1709bedbed9175bd9350df46599b... | Mathlib/Algebra/Category/ModuleCat/Subobject.lean | 111 | 120 | theorem cokernel_π_imageSubobject_ext {L M N : ModuleCat.{v} R} (f : L ⟶ M) [HasImage f]
(g : (imageSubobject f : ModuleCat.{v} R) ⟶ N) [HasCokernel g] {x y : N} (l : L)
(w : x = y + g (factorThruImageSubobject f l)) : cokernel.π g x = cokernel.π g y := by |
subst w
-- Porting note: The proof from here used to just be `simp`.
simp only [map_add, add_right_eq_self]
change ((cokernel.π g) ∘ (g) ∘ (factorThruImageSubobject f)) l = 0
rw [← coe_comp, ← coe_comp, Category.assoc]
simp only [cokernel.condition, comp_zero]
rfl
| 7 |
import Mathlib.Data.List.Chain
import Mathlib.Data.List.Enum
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Pairwise
import Mathlib.Data.List.Zip
#align_import data.list.range from "leanprover-community/mathlib"@"7b78d1776212a91ecc94cf601f83bdcc46b04213"
set_option autoImplicit true
universe u
open Nat... | Mathlib/Data/List/Range.lean | 104 | 112 | theorem chain'_range_succ (r : ℕ → ℕ → Prop) (n : ℕ) :
Chain' r (range n.succ) ↔ ∀ m < n, r m m.succ := by |
rw [range_succ]
induction' n with n hn
· simp
· rw [range_succ]
simp only [append_assoc, singleton_append, chain'_append_cons_cons, chain'_singleton,
and_true_iff]
rw [hn, forall_lt_succ]
| 7 |
import Mathlib.Algebra.Lie.Abelian
import Mathlib.Algebra.Lie.IdealOperations
import Mathlib.Order.Hom.Basic
#align_import algebra.lie.solvable from "leanprover-community/mathlib"@"a50170a88a47570ed186b809ca754110590f9476"
universe u v w w₁ w₂
variable (R : Type u) (L : Type v) (M : Type w) {L' : Type w₁}
variab... | Mathlib/Algebra/Lie/Solvable.lean | 89 | 97 | theorem derivedSeriesOfIdeal_le {I J : LieIdeal R L} {k l : ℕ} (h₁ : I ≤ J) (h₂ : l ≤ k) :
D k I ≤ D l J := by |
revert l; induction' k with k ih <;> intro l h₂
· rw [le_zero_iff] at h₂; rw [h₂, derivedSeriesOfIdeal_zero]; exact h₁
· have h : l = k.succ ∨ l ≤ k := by rwa [le_iff_eq_or_lt, Nat.lt_succ_iff] at h₂
cases' h with h h
· rw [h, derivedSeriesOfIdeal_succ, derivedSeriesOfIdeal_succ]
exact LieSubmodule... | 7 |
import Mathlib.CategoryTheory.Sites.Grothendieck
import Mathlib.CategoryTheory.Sites.Pretopology
import Mathlib.CategoryTheory.Limits.Lattice
import Mathlib.Topology.Sets.Opens
#align_import category_theory.sites.spaces from "leanprover-community/mathlib"@"b6fa3beb29f035598cf0434d919694c5e98091eb"
universe u
nam... | Mathlib/CategoryTheory/Sites/Spaces.lean | 78 | 86 | theorem pretopology_ofGrothendieck :
Pretopology.ofGrothendieck _ (Opens.grothendieckTopology T) = Opens.pretopology T := by |
apply le_antisymm
· intro X R hR x hx
rcases hR x hx with ⟨U, f, ⟨V, g₁, g₂, hg₂, _⟩, hU⟩
exact ⟨V, g₂, hg₂, g₁.le hU⟩
· intro X R hR x hx
rcases hR x hx with ⟨U, f, hf, hU⟩
exact ⟨U, f, Sieve.le_generate R U hf, hU⟩
| 7 |
import Mathlib.Algebra.Lie.CartanSubalgebra
import Mathlib.Algebra.Lie.Weights.Basic
suppress_compilation
open Set
variable {R L : Type*} [CommRing R] [LieRing L] [LieAlgebra R L]
(H : LieSubalgebra R L) [LieAlgebra.IsNilpotent R H]
{M : Type*} [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L ... | Mathlib/Algebra/Lie/Weights/Cartan.lean | 61 | 69 | theorem lie_mem_weightSpace_of_mem_weightSpace {χ₁ χ₂ : H → R} {x : L} {m : M}
(hx : x ∈ rootSpace H χ₁) (hm : m ∈ weightSpace M χ₂) : ⁅x, m⁆ ∈ weightSpace M (χ₁ + χ₂) := by |
rw [weightSpace, LieSubmodule.mem_iInf]
intro y
replace hx : x ∈ weightSpaceOf L (χ₁ y) y := by
rw [rootSpace, weightSpace, LieSubmodule.mem_iInf] at hx; exact hx y
replace hm : m ∈ weightSpaceOf M (χ₂ y) y := by
rw [weightSpace, LieSubmodule.mem_iInf] at hm; exact hm y
exact lie_mem_maxGenEigenspace... | 7 |
import Mathlib.Data.ENNReal.Basic
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.MetricSpace.Thickening
#align_import topology.metric_space.thickened_indicator from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open scoped Classical
open NNReal ENNReal Topol... | Mathlib/Topology/MetricSpace/ThickenedIndicator.lean | 58 | 66 | theorem continuous_thickenedIndicatorAux {δ : ℝ} (δ_pos : 0 < δ) (E : Set α) :
Continuous (thickenedIndicatorAux δ E) := by |
unfold thickenedIndicatorAux
let f := fun x : α => (⟨1, infEdist x E / ENNReal.ofReal δ⟩ : ℝ≥0 × ℝ≥0∞)
let sub := fun p : ℝ≥0 × ℝ≥0∞ => (p.1 : ℝ≥0∞) - p.2
rw [show (fun x : α => (1 : ℝ≥0∞) - infEdist x E / ENNReal.ofReal δ) = sub ∘ f by rfl]
apply (@ENNReal.continuous_nnreal_sub 1).comp
apply (ENNReal.cont... | 7 |
import Mathlib.MeasureTheory.Covering.DensityTheorem
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
#align_import measure_theory.covering.one_dim from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
open Set MeasureTheory IsUnifLocDoublingMeasure Filter
open scoped Topology
names... | Mathlib/MeasureTheory/Covering/OneDim.lean | 33 | 41 | theorem tendsto_Icc_vitaliFamily_right (x : ℝ) :
Tendsto (fun y => Icc x y) (𝓝[>] x) ((vitaliFamily (volume : Measure ℝ) 1).filterAt x) := by |
refine (VitaliFamily.tendsto_filterAt_iff _).2 ⟨?_, ?_⟩
· filter_upwards [self_mem_nhdsWithin] with y hy using Icc_mem_vitaliFamily_at_right hy
· intro ε εpos
have : x ∈ Ico x (x + ε) := ⟨le_refl _, by linarith⟩
filter_upwards [Icc_mem_nhdsWithin_Ioi this] with y hy
rw [closedBall_eq_Icc]
exact I... | 7 |
import Mathlib.MeasureTheory.Measure.Dirac
set_option autoImplicit true
open Set
open scoped ENNReal Classical
variable [MeasurableSpace α] [MeasurableSpace β] {s : Set α}
noncomputable section
namespace MeasureTheory.Measure
def count : Measure α :=
sum dirac
#align measure_theory.measure.count MeasureTheo... | Mathlib/MeasureTheory/Measure/Count.lean | 73 | 80 | theorem count_apply_infinite (hs : s.Infinite) : count s = ∞ := by |
refine top_unique (le_of_tendsto' ENNReal.tendsto_nat_nhds_top fun n => ?_)
rcases hs.exists_subset_card_eq n with ⟨t, ht, rfl⟩
calc
(t.card : ℝ≥0∞) = ∑ i ∈ t, 1 := by simp
_ = ∑' i : (t : Set α), 1 := (t.tsum_subtype 1).symm
_ ≤ count (t : Set α) := le_count_apply
_ ≤ count s := measure_mono ht
| 7 |
import Mathlib.SetTheory.Ordinal.Arithmetic
import Mathlib.SetTheory.Ordinal.Exponential
#align_import set_theory.ordinal.cantor_normal_form from "leanprover-community/mathlib"@"991ff3b5269848f6dd942ae8e9dd3c946035dc8b"
noncomputable section
universe u
open List
namespace Ordinal
@[elab_as_elim]
noncomputabl... | Mathlib/SetTheory/Ordinal/CantorNormalForm.lean | 121 | 129 | theorem CNF_fst_le_log {b o : Ordinal.{u}} {x : Ordinal × Ordinal} :
x ∈ CNF b o → x.1 ≤ log b o := by |
refine CNFRec b ?_ (fun o ho H ↦ ?_) o
· rw [CNF_zero]
intro contra; contradiction
· rw [CNF_ne_zero ho, mem_cons]
rintro (rfl | h)
· exact le_rfl
· exact (H h).trans (log_mono_right _ (mod_opow_log_lt_self b ho).le)
| 7 |
import Mathlib.LinearAlgebra.FiniteDimensional
#align_import linear_algebra.projective_space.basic from "leanprover-community/mathlib"@"c4658a649d216f57e99621708b09dcb3dcccbd23"
variable (K V : Type*) [DivisionRing K] [AddCommGroup V] [Module K V]
def projectivizationSetoid : Setoid { v : V // v ≠ 0 } :=
(MulA... | Mathlib/LinearAlgebra/Projectivization/Basic.lean | 108 | 116 | theorem mk_eq_mk_iff' (v w : V) (hv : v ≠ 0) (hw : w ≠ 0) :
mk K v hv = mk K w hw ↔ ∃ a : K, a • w = v := by |
rw [mk_eq_mk_iff K v w hv hw]
constructor
· rintro ⟨a, ha⟩
exact ⟨a, ha⟩
· rintro ⟨a, ha⟩
refine ⟨Units.mk0 a fun c => hv.symm ?_, ha⟩
rwa [c, zero_smul] at ha
| 7 |
import Mathlib.RingTheory.RootsOfUnity.Basic
import Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed
import Mathlib.Algebra.GCDMonoid.IntegrallyClosed
import Mathlib.FieldTheory.Finite.Basic
#align_import ring_theory.roots_of_unity.minpoly from "leanprover-community/mathlib"@"7fdeecc0d03cd40f7a165e6cf00a4d2286db599f"
... | Mathlib/RingTheory/RootsOfUnity/Minpoly.lean | 82 | 90 | theorem minpoly_dvd_expand {p : ℕ} (hdiv : ¬p ∣ n) :
minpoly ℤ μ ∣ expand ℤ p (minpoly ℤ (μ ^ p)) := by |
rcases n.eq_zero_or_pos with (rfl | hpos)
· simp_all
letI : IsIntegrallyClosed ℤ := GCDMonoid.toIsIntegrallyClosed
refine minpoly.isIntegrallyClosed_dvd (h.isIntegral hpos) ?_
rw [aeval_def, coe_expand, ← comp, eval₂_eq_eval_map, map_comp, Polynomial.map_pow, map_X,
eval_comp, eval_pow, eval_X, ← eval₂_e... | 7 |
import Mathlib.Analysis.NormedSpace.AddTorsor
import Mathlib.LinearAlgebra.AffineSpace.Ordered
import Mathlib.Topology.ContinuousFunction.Basic
import Mathlib.Topology.GDelta
import Mathlib.Analysis.NormedSpace.FunctionSeries
import Mathlib.Analysis.SpecificLimits.Basic
#align_import topology.urysohns_lemma from "lea... | Mathlib/Topology/UrysohnsLemma.lean | 185 | 192 | theorem approx_le_one (c : CU P) (n : ℕ) (x : X) : c.approx n x ≤ 1 := by |
induction' n with n ihn generalizing c
· exact indicator_apply_le' (fun _ => le_rfl) fun _ => zero_le_one
· simp only [approx, midpoint_eq_smul_add, invOf_eq_inv, smul_eq_mul, ← div_eq_inv_mul]
have := add_le_add (ihn (left c)) (ihn (right c))
set_option tactic.skipAssignedInstances false in
norm_num... | 7 |
import Mathlib.Data.Fintype.Basic
#align_import data.fintype.quotient from "leanprover-community/mathlib"@"d78597269638367c3863d40d45108f52207e03cf"
def Quotient.finChoiceAux {ι : Type*} [DecidableEq ι] {α : ι → Type*} [S : ∀ i, Setoid (α i)] :
∀ l : List ι, (∀ i ∈ l, Quotient (S i)) → @Quotient (∀ i ∈ l, α ... | Mathlib/Data/Fintype/Quotient.lean | 76 | 84 | theorem Quotient.finChoice_eq {ι : Type*} [DecidableEq ι] [Fintype ι] {α : ι → Type*}
[∀ i, Setoid (α i)] (f : ∀ i, α i) : (Quotient.finChoice fun i => ⟦f i⟧) = ⟦f⟧ := by |
dsimp only [Quotient.finChoice]
conv_lhs =>
enter [1]
tactic =>
change _ = ⟦fun i _ => f i⟧
exact Quotient.inductionOn (@Finset.univ ι _).1 fun l => Quotient.finChoiceAux_eq _ _
rfl
| 7 |
import Mathlib.Topology.Order.IsLUB
open Set Filter TopologicalSpace Topology Function
open OrderDual (toDual ofDual)
variable {α β γ : Type*}
section DenselyOrdered
variable [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] {a b : α}
{s : Set α}
theorem closure_Ioi' {a : α} (h : (Io... | Mathlib/Topology/Order/DenselyOrdered.lean | 136 | 143 | theorem Ioc_subset_closure_interior (a b : α) : Ioc a b ⊆ closure (interior (Ioc a b)) := by |
rcases eq_or_ne a b with (rfl | h)
· simp
· calc
Ioc a b ⊆ Icc a b := Ioc_subset_Icc_self
_ = closure (Ioo a b) := (closure_Ioo h).symm
_ ⊆ closure (interior (Ioc a b)) :=
closure_mono (interior_maximal Ioo_subset_Ioc_self isOpen_Ioo)
| 7 |
import Mathlib.MeasureTheory.Group.GeometryOfNumbers
import Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls
import Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic
#align_import number_theory.number_field.canonical_embedding from "leanprover-community/mathlib"@"60da01b41bbe4206f05d34fd70c8dd7498717a30"
... | Mathlib/NumberTheory/NumberField/CanonicalEmbedding/ConvexBody.lean | 326 | 333 | theorem norm_le_convexBodySumFun (x : E K) : ‖x‖ ≤ convexBodySumFun x := by |
rw [norm_eq_sup'_normAtPlace]
refine (Finset.sup'_le_iff _ _).mpr fun w _ ↦ ?_
rw [convexBodySumFun_apply, ← Finset.univ.add_sum_erase _ (Finset.mem_univ w)]
refine le_add_of_le_of_nonneg ?_ ?_
· exact le_mul_of_one_le_left (normAtPlace_nonneg w x) one_le_mult
· exact Finset.sum_nonneg (fun _ _ => mul_non... | 7 |
import Mathlib.Order.Ideal
import Mathlib.Order.PFilter
#align_import order.prime_ideal from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a485d0456fc271482da"
open Order.PFilter
namespace Order
variable {P : Type*}
namespace Ideal
-- Porting note(#5171): this linter isn't ported yet.
-- @[nolint has_... | Mathlib/Order/PrimeIdeal.lean | 131 | 139 | theorem IsPrime.of_mem_or_mem [IsProper I] (hI : ∀ {x y : P}, x ⊓ y ∈ I → x ∈ I ∨ y ∈ I) :
IsPrime I := by |
rw [isPrime_iff]
use ‹_›
refine .of_def ?_ ?_ ?_
· exact Set.nonempty_compl.2 (I.isProper_iff.1 ‹_›)
· intro x hx y hy
exact ⟨x ⊓ y, fun h => (hI h).elim hx hy, inf_le_left, inf_le_right⟩
· exact @mem_compl_of_ge _ _ _
| 7 |
import Mathlib.Algebra.Module.Submodule.Localization
import Mathlib.LinearAlgebra.Dimension.DivisionRing
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.OreLocalization.OreSet
open Cardinal nonZeroDivisors
section CommRing
universe u u' v v'
variable {R : Type u} (S : Type u') {M : T... | Mathlib/LinearAlgebra/Dimension/Localization.lean | 85 | 93 | theorem exists_set_linearIndependent_of_isDomain [IsDomain R] :
∃ s : Set M, #s = Module.rank R M ∧ LinearIndependent (ι := s) R Subtype.val := by |
obtain ⟨w, hw⟩ :=
IsLocalizedModule.linearIndependent_lift R⁰ (LocalizedModule.mkLinearMap R⁰ M) le_rfl
(Module.Free.chooseBasis (FractionRing R) (LocalizedModule R⁰ M)).linearIndependent
refine ⟨Set.range w, ?_, (linearIndependent_subtype_range hw.injective).mpr hw⟩
apply Cardinal.lift_injective.{max ... | 7 |
import Mathlib.Data.Set.Image
#align_import data.nat.set from "leanprover-community/mathlib"@"cf9386b56953fb40904843af98b7a80757bbe7f9"
namespace Nat
section Set
open Set
theorem zero_union_range_succ : {0} ∪ range succ = univ := by
ext n
cases n <;> simp
#align nat.zero_union_range_succ Nat.zero_union_ran... | Mathlib/Data/Nat/Set.lean | 37 | 46 | theorem range_rec {α : Type*} (x : α) (f : ℕ → α → α) :
(Set.range fun n => Nat.rec x f n : Set α) =
{x} ∪ Set.range fun n => Nat.rec (f 0 x) (f ∘ succ) n := by |
convert (range_of_succ (fun n => Nat.rec x f n : ℕ → α)).symm using 4
dsimp
rename_i n
induction' n with n ihn
· rfl
· dsimp at ihn ⊢
rw [ihn]
| 7 |
import Mathlib.Algebra.Polynomial.Degree.TrailingDegree
import Mathlib.Algebra.Polynomial.EraseLead
import Mathlib.Algebra.Polynomial.Eval
#align_import data.polynomial.reverse from "leanprover-community/mathlib"@"44de64f183393284a16016dfb2a48ac97382f2bd"
namespace Polynomial
open Polynomial Finsupp Finset
open... | Mathlib/Algebra/Polynomial/Reverse.lean | 40 | 47 | theorem revAtFun_invol {N i : ℕ} : revAtFun N (revAtFun N i) = i := by |
unfold revAtFun
split_ifs with h j
· exact tsub_tsub_cancel_of_le h
· exfalso
apply j
exact Nat.sub_le N i
· rfl
| 7 |
import Batteries.Tactic.SeqFocus
import Batteries.Data.List.Lemmas
import Batteries.Data.List.Init.Attach
namespace Std.Range
def numElems (r : Range) : Nat :=
if r.step = 0 then
-- This is a very weird choice, but it is chosen to coincide with the `forIn` impl
if r.stop ≤ r.start then 0 else r.stop
els... | .lake/packages/batteries/Batteries/Data/Range/Lemmas.lean | 40 | 47 | theorem mem_range'_elems (r : Range) (h : x ∈ List.range' r.start r.numElems r.step) : x ∈ r := by |
obtain ⟨i, h', rfl⟩ := List.mem_range'.1 h
refine ⟨Nat.le_add_right .., ?_⟩
unfold numElems at h'; split at h'
· split at h' <;> [cases h'; simp_all]
· next step0 =>
refine Nat.not_le.1 fun h =>
Nat.not_le.2 h' <| (numElems_le_iff (Nat.pos_of_ne_zero step0)).2 h
| 7 |
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
variable ... | Mathlib/MeasureTheory/Function/LpSeminorm/TriangleInequality.lean | 87 | 94 | theorem LpAddConst_lt_top (p : ℝ≥0∞) : LpAddConst p < ∞ := by |
rw [LpAddConst]
split_ifs with h
· apply ENNReal.rpow_lt_top_of_nonneg _ ENNReal.two_ne_top
simp only [one_div, sub_nonneg]
apply one_le_inv (ENNReal.toReal_pos h.1.ne' (h.2.trans ENNReal.one_lt_top).ne)
simpa using ENNReal.toReal_mono ENNReal.one_ne_top h.2.le
· exact ENNReal.one_lt_top
| 7 |
import Mathlib.Algebra.Algebra.Prod
import Mathlib.LinearAlgebra.Basic
import Mathlib.LinearAlgebra.Span
import Mathlib.Order.PartialSups
#align_import linear_algebra.prod from "leanprover-community/mathlib"@"cd391184c85986113f8c00844cfe6dda1d34be3d"
universe u v w x y z u' v' w' y'
variable {R : Type u} {K : Ty... | Mathlib/LinearAlgebra/Prod.lean | 162 | 169 | theorem range_inr : range (inr R M M₂) = ker (fst R M M₂) := by |
ext x
simp only [mem_ker, mem_range]
constructor
· rintro ⟨y, rfl⟩
rfl
· intro h
exact ⟨x.snd, Prod.ext h.symm rfl⟩
| 7 |
import Mathlib.Data.List.Cycle
import Mathlib.GroupTheory.Perm.Cycle.Type
import Mathlib.GroupTheory.Perm.List
#align_import group_theory.perm.cycle.concrete from "leanprover-community/mathlib"@"00638177efd1b2534fc5269363ebf42a7871df9a"
open Equiv Equiv.Perm List
variable {α : Type*}
namespace Cycle
variable [... | Mathlib/GroupTheory/Perm/Cycle/Concrete.lean | 149 | 156 | theorem formPerm_subsingleton (s : Cycle α) (h : Subsingleton s) : formPerm s h.nodup = 1 := by |
induction' s using Quot.inductionOn with s
simp only [formPerm_coe, mk_eq_coe]
simp only [length_subsingleton_iff, length_coe, mk_eq_coe] at h
cases' s with hd tl
· simp
· simp only [length_eq_zero, add_le_iff_nonpos_left, List.length, nonpos_iff_eq_zero] at h
simp [h]
| 7 |
import Mathlib.Topology.Algebra.FilterBasis
import Mathlib.Topology.Algebra.UniformGroup
#align_import topology.algebra.uniform_filter_basis from "leanprover-community/mathlib"@"531db2ef0fdddf8b3c8dcdcd87138fe969e1a81a"
open uniformity Filter
open Filter
namespace AddGroupFilterBasis
variable {G : Type*} [AddC... | Mathlib/Topology/Algebra/UniformFilterBasis.lean | 42 | 51 | theorem cauchy_iff {F : Filter G} :
@Cauchy G B.uniformSpace F ↔
F.NeBot ∧ ∀ U ∈ B, ∃ M ∈ F, ∀ᵉ (x ∈ M) (y ∈ M), y - x ∈ U := by |
letI := B.uniformSpace
haveI := B.uniformAddGroup
suffices F ×ˢ F ≤ uniformity G ↔ ∀ U ∈ B, ∃ M ∈ F, ∀ᵉ (x ∈ M) (y ∈ M), y - x ∈ U by
constructor <;> rintro ⟨h', h⟩ <;> refine ⟨h', ?_⟩ <;> [rwa [← this]; rwa [this]]
rw [uniformity_eq_comap_nhds_zero G, ← map_le_iff_le_comap]
change Tendsto _ _ _ ↔ _
si... | 7 |
import Mathlib.Data.Int.Order.Units
import Mathlib.Data.ZMod.IntUnitsPower
import Mathlib.RingTheory.TensorProduct.Basic
import Mathlib.LinearAlgebra.DirectSum.TensorProduct
import Mathlib.Algebra.DirectSum.Algebra
suppress_compilation
open scoped TensorProduct DirectSum
variable {R ι A B : Type*}
namespace Tens... | Mathlib/LinearAlgebra/TensorProduct/Graded/External.lean | 137 | 145 | theorem gradedComm_of_zero_tmul (a : 𝒜 0) (b : ⨁ i, ℬ i) :
gradedComm R 𝒜 ℬ (lof R _ 𝒜 0 a ⊗ₜ b) = b ⊗ₜ lof R _ 𝒜 _ a := by |
suffices
(gradedComm R 𝒜 ℬ).toLinearMap ∘ₗ (TensorProduct.mk R (⨁ i, 𝒜 i) (⨁ i, ℬ i)) (lof R _ 𝒜 0 a) =
(TensorProduct.mk R _ _).flip (lof R _ 𝒜 0 a) from
DFunLike.congr_fun this b
ext i b
dsimp
rw [gradedComm_of_tmul_of, mul_zero, uzpow_zero, one_smul]
| 7 |
import Mathlib.Analysis.Calculus.Deriv.Inv
import Mathlib.Analysis.Calculus.Deriv.Polynomial
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
import Mathlib.Analysis.SpecialFunctions.PolynomialExp
#align_import analysis.calculus.bump_function_inner from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9... | Mathlib/Analysis/SpecialFunctions/SmoothTransition.lean | 91 | 99 | theorem tendsto_polynomial_inv_mul_zero (p : ℝ[X]) :
Tendsto (fun x ↦ p.eval x⁻¹ * expNegInvGlue x) (𝓝 0) (𝓝 0) := by |
simp only [expNegInvGlue, mul_ite, mul_zero]
refine tendsto_const_nhds.if ?_
simp only [not_le]
have : Tendsto (fun x ↦ p.eval x⁻¹ / exp x⁻¹) (𝓝[>] 0) (𝓝 0) :=
p.tendsto_div_exp_atTop.comp tendsto_inv_zero_atTop
refine this.congr' <| mem_of_superset self_mem_nhdsWithin fun x hx ↦ ?_
simp [expNegInvGl... | 7 |
import Mathlib.AlgebraicTopology.SplitSimplicialObject
import Mathlib.AlgebraicTopology.DoldKan.Degeneracies
import Mathlib.AlgebraicTopology.DoldKan.FunctorN
#align_import algebraic_topology.dold_kan.split_simplicial_object from "leanprover-community/mathlib"@"32a7e535287f9c73f2e4d2aef306a39190f0b504"
open Categ... | Mathlib/AlgebraicTopology/DoldKan/SplitSimplicialObject.lean | 61 | 69 | theorem decomposition_id (Δ : SimplexCategoryᵒᵖ) :
𝟙 (X.obj Δ) = ∑ A : IndexSet Δ, s.πSummand A ≫ (s.cofan Δ).inj A := by |
apply s.hom_ext'
intro A
dsimp
erw [comp_id, comp_sum, Finset.sum_eq_single A, cofan_inj_πSummand_eq_id_assoc]
· intro B _ h₂
rw [s.cofan_inj_πSummand_eq_zero_assoc _ _ h₂, zero_comp]
· simp
| 7 |
import Mathlib.Topology.Algebra.UniformConvergence
#align_import topology.algebra.equicontinuity from "leanprover-community/mathlib"@"01ad394a11bf06b950232720cf7e8fc6b22f0d6a"
open Function
open UniformConvergence
@[to_additive]
theorem equicontinuous_of_equicontinuousAt_one {ι G M hom : Type*} [TopologicalSpac... | Mathlib/Topology/Algebra/Equicontinuity.lean | 36 | 47 | theorem uniformEquicontinuous_of_equicontinuousAt_one {ι G M hom : Type*} [UniformSpace G]
[UniformSpace M] [Group G] [Group M] [UniformGroup G] [UniformGroup M]
[FunLike hom G M] [MonoidHomClass hom G M]
(F : ι → hom) (hf : EquicontinuousAt ((↑) ∘ F) (1 : G)) :
UniformEquicontinuous ((↑) ∘ F) := by |
rw [uniformEquicontinuous_iff_uniformContinuous]
rw [equicontinuousAt_iff_continuousAt] at hf
let φ : G →* (ι →ᵤ M) :=
{ toFun := swap ((↑) ∘ F)
map_one' := by dsimp [UniformFun]; ext; exact map_one _
map_mul' := fun a b => by dsimp [UniformFun]; ext; exact map_mul _ _ _ }
exact uniformContinuo... | 7 |
import Mathlib.Algebra.Polynomial.Splits
import Mathlib.RingTheory.MvPolynomial.Symmetric
#align_import ring_theory.polynomial.vieta from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
open Polynomial
namespace Multiset
open Polynomial
section Ring
variable {R : Type*} [CommRing R]
... | Mathlib/RingTheory/Polynomial/Vieta.lean | 94 | 101 | theorem esymm_neg (s : Multiset R) (k : ℕ) : (map Neg.neg s).esymm k = (-1) ^ k * esymm s k := by |
rw [esymm, esymm, ← Multiset.sum_map_mul_left, Multiset.powersetCard_map, Multiset.map_map,
map_congr rfl]
intro x hx
rw [(mem_powersetCard.mp hx).right.symm, ← prod_replicate, ← Multiset.map_const]
nth_rw 3 [← map_id' x]
rw [← prod_map_mul, map_congr rfl, Function.comp_apply]
exact fun z _ => neg_one_... | 7 |
import Mathlib.Data.Finsupp.Defs
#align_import data.list.to_finsupp from "leanprover-community/mathlib"@"06a655b5fcfbda03502f9158bbf6c0f1400886f9"
namespace List
variable {M : Type*} [Zero M] (l : List M) [DecidablePred (getD l · 0 ≠ 0)] (n : ℕ)
def toFinsupp : ℕ →₀ M where
toFun i := getD l i 0
support := ... | Mathlib/Data/List/ToFinsupp.lean | 147 | 156 | theorem toFinsupp_eq_sum_map_enum_single {R : Type*} [AddMonoid R] (l : List R)
[DecidablePred (getD l · 0 ≠ 0)] :
toFinsupp l = (l.enum.map fun nr : ℕ × R => Finsupp.single nr.1 nr.2).sum := by |
/- Porting note (#11215): TODO: `induction` fails to substitute `l = []` in
`[DecidablePred (getD l · 0 ≠ 0)]`, so we manually do some `revert`/`intro` as a workaround -/
revert l; intro l
induction l using List.reverseRecOn with
| nil => exact toFinsupp_nil
| append_singleton x xs ih =>
classical simp... | 7 |
import Mathlib.Topology.Separation
#align_import topology.shrinking_lemma from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99c438514"
open Set Function
open scoped Classical
noncomputable section
variable {ι X : Type*} [TopologicalSpace X] [NormalSpace X]
namespace ShrinkingLemma
-- the tr... | Mathlib/Topology/ShrinkingLemma.lean | 124 | 132 | theorem mem_find_carrier_iff {c : Set (PartialRefinement u s)} {i : ι} (ne : c.Nonempty) :
i ∈ (find c ne i).carrier ↔ i ∈ chainSupCarrier c := by |
rw [find]
split_ifs with h
· have := h.choose_spec
exact iff_of_true this.2 (mem_iUnion₂.2 ⟨_, this.1, this.2⟩)
· push_neg at h
refine iff_of_false (h _ ne.some_mem) ?_
simpa only [chainSupCarrier, mem_iUnion₂, not_exists]
| 7 |
import Mathlib.Data.ZMod.Basic
import Mathlib.GroupTheory.Coxeter.Basic
namespace CoxeterSystem
open List Matrix Function Classical
variable {B : Type*}
variable {W : Type*} [Group W]
variable {M : CoxeterMatrix B} (cs : CoxeterSystem M W)
local prefix:100 "s" => cs.simple
local prefix:100 "π" => cs.wordProd
... | Mathlib/GroupTheory/Coxeter/Length.lean | 152 | 159 | theorem length_eq_one_iff {w : W} : ℓ w = 1 ↔ ∃ i : B, w = s i := by |
constructor
· intro h
rcases cs.exists_reduced_word w with ⟨ω, hω, rfl⟩
rcases List.length_eq_one.mp (hω.trans h) with ⟨i, rfl⟩
exact ⟨i, cs.wordProd_singleton i⟩
· rintro ⟨i, rfl⟩
exact cs.length_simple i
| 7 |
import Mathlib.RingTheory.AdjoinRoot
import Mathlib.FieldTheory.Minpoly.Field
import Mathlib.RingTheory.Polynomial.GaussLemma
#align_import field_theory.minpoly.is_integrally_closed from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
open scoped Classical Polynomial
open Polynomial Set... | Mathlib/FieldTheory/Minpoly/IsIntegrallyClosed.lean | 125 | 135 | theorem _root_.IsIntegrallyClosed.minpoly.unique {s : S} {P : R[X]} (hmo : P.Monic)
(hP : Polynomial.aeval s P = 0)
(Pmin : ∀ Q : R[X], Q.Monic → Polynomial.aeval s Q = 0 → degree P ≤ degree Q) :
P = minpoly R s := by |
have hs : IsIntegral R s := ⟨P, hmo, hP⟩
symm; apply eq_of_sub_eq_zero
by_contra hnz
refine IsIntegrallyClosed.degree_le_of_ne_zero hs hnz (by simp [hP]) |>.not_lt ?_
refine degree_sub_lt ?_ (ne_zero hs) ?_
· exact le_antisymm (min R s hmo hP) (Pmin (minpoly R s) (monic hs) (aeval R s))
· rw [(monic hs).... | 7 |
import Mathlib.Analysis.InnerProductSpace.Rayleigh
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Algebra.DirectSum.Decomposition
import Mathlib.LinearAlgebra.Eigenspace.Minpoly
#align_import analysis.inner_product_space.spectrum from "leanprover-community/mathlib"@"6b0169218d01f2837d79ea2784882009a0da... | Mathlib/Analysis/InnerProductSpace/Spectrum.lean | 83 | 91 | theorem orthogonalFamily_eigenspaces :
OrthogonalFamily 𝕜 (fun μ => eigenspace T μ) fun μ => (eigenspace T μ).subtypeₗᵢ := by |
rintro μ ν hμν ⟨v, hv⟩ ⟨w, hw⟩
by_cases hv' : v = 0
· simp [hv']
have H := hT.conj_eigenvalue_eq_self (hasEigenvalue_of_hasEigenvector ⟨hv, hv'⟩)
rw [mem_eigenspace_iff] at hv hw
refine Or.resolve_left ?_ hμν.symm
simpa [inner_smul_left, inner_smul_right, hv, hw, H] using (hT v w).symm
| 7 |
import Mathlib.Data.Finsupp.Basic
import Mathlib.Data.Finsupp.Order
#align_import data.finsupp.multiset from "leanprover-community/mathlib"@"59694bd07f0a39c5beccba34bd9f413a160782bf"
open Finset
variable {α β ι : Type*}
namespace Finsupp
def toMultiset : (α →₀ ℕ) →+ Multiset α where
toFun f := Finsupp.sum f... | Mathlib/Data/Finsupp/Multiset.lean | 105 | 114 | theorem count_toMultiset [DecidableEq α] (f : α →₀ ℕ) (a : α) : (toMultiset f).count a = f a :=
calc
(toMultiset f).count a = Finsupp.sum f (fun x n => (n • {x} : Multiset α).count a) := by |
rw [toMultiset_apply]; exact map_sum (Multiset.countAddMonoidHom a) _ f.support
_ = f.sum fun x n => n * ({x} : Multiset α).count a := by simp only [Multiset.count_nsmul]
_ = f a * ({a} : Multiset α).count a :=
sum_eq_single _
(fun a' _ H => by simp only [Multiset.count_singleton, if_false,... | 7 |
import Mathlib.Combinatorics.Hall.Finite
import Mathlib.CategoryTheory.CofilteredSystem
import Mathlib.Data.Rel
#align_import combinatorics.hall.basic from "leanprover-community/mathlib"@"8195826f5c428fc283510bc67303dd4472d78498"
open Finset CategoryTheory
universe u v
def hallMatchingsOn {ι : Type u} {α : Typ... | Mathlib/Combinatorics/Hall/Basic.lean | 77 | 86 | theorem hallMatchingsOn.nonempty {ι : Type u} {α : Type v} [DecidableEq α] (t : ι → Finset α)
(h : ∀ s : Finset ι, s.card ≤ (s.biUnion t).card) (ι' : Finset ι) :
Nonempty (hallMatchingsOn t ι') := by |
classical
refine ⟨Classical.indefiniteDescription _ ?_⟩
apply (all_card_le_biUnion_card_iff_existsInjective' fun i : ι' => t i).mp
intro s'
convert h (s'.image (↑)) using 1
· simp only [card_image_of_injective s' Subtype.coe_injective]
· rw [image_biUnion]
| 7 |
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integral
#align_import ring_theory.integrally_closed from "leanprover-community/mathlib"@"d35b4ff446f1421bd551fafa4b8efd98ac3ac408"
open scoped nonZeroDivisors Polynomial
open Polynomial
abbrev IsIntegrallyClosedIn (R A : Type*) [... | Mathlib/RingTheory/IntegrallyClosed.lean | 110 | 120 | theorem isIntegrallyClosedIn_iff {R A : Type*} [CommRing R] [CommRing A] [Algebra R A] :
IsIntegrallyClosedIn R A ↔
Function.Injective (algebraMap R A) ∧
∀ {x : A}, IsIntegral R x → ∃ y, algebraMap R A y = x := by |
constructor
· rintro ⟨_, cl⟩
aesop
· rintro ⟨inj, cl⟩
refine ⟨inj, by aesop, ?_⟩
rintro ⟨y, rfl⟩
apply isIntegral_algebraMap
| 7 |
import Mathlib.Analysis.NormedSpace.Spectrum
import Mathlib.Topology.ContinuousFunction.NonUnitalFunctionalCalculus
import Mathlib.Topology.ContinuousFunction.StoneWeierstrass
section UniqueUnital
section NNReal
open NNReal
variable {X : Type*} [TopologicalSpace X]
variable {A : Type*} [TopologicalSpace A] [Ring... | Mathlib/Topology/ContinuousFunction/UniqueCFC.lean | 207 | 218 | theorem RCLike.uniqueNonUnitalContinuousFunctionalCalculus_of_compactSpace_quasispectrum
[TopologicalSpace A] [T2Space A] [NonUnitalRing A] [StarRing A] [Module 𝕜 A]
[IsScalarTower 𝕜 A A] [SMulCommClass 𝕜 A A] [h : ∀ a : A, CompactSpace (quasispectrum 𝕜 a)] :
UniqueNonUnitalContinuousFunctionalCalculus ... |
rw [DFunLike.ext'_iff, ← Set.eqOn_univ, ← (ContinuousMapZero.adjoin_id_dense h0).closure_eq]
refine Set.EqOn.closure (fun f hf ↦ ?_) hφ hψ
rw [← NonUnitalStarAlgHom.mem_equalizer]
apply adjoin_le ?_ hf
rw [Set.singleton_subset_iff]
exact h
compactSpace_quasispectrum := h
| 7 |
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv
import Mathlib.Analysis.Calculus.FDeriv.Extend
import Mathlib.Analysis.Calculus.Deriv.Prod
import Mathlib.Analysis.SpecialFunctions.Log.Deriv
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv
#... | Mathlib/Analysis/SpecialFunctions/Pow/Deriv.lean | 52 | 60 | theorem hasStrictDerivAt_const_cpow {x y : ℂ} (h : x ≠ 0 ∨ y ≠ 0) :
HasStrictDerivAt (fun y => x ^ y) (x ^ y * log x) y := by |
rcases em (x = 0) with (rfl | hx)
· replace h := h.neg_resolve_left rfl
rw [log_zero, mul_zero]
refine (hasStrictDerivAt_const _ 0).congr_of_eventuallyEq ?_
exact (isOpen_ne.eventually_mem h).mono fun y hy => (zero_cpow hy).symm
· simpa only [cpow_def_of_ne_zero hx, mul_one] using
((hasStrictDe... | 7 |
import Mathlib.Dynamics.Ergodic.AddCircle
import Mathlib.MeasureTheory.Covering.LiminfLimsup
#align_import number_theory.well_approximable from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
open Set Filter Function Metric MeasureTheory
open scoped MeasureTheory Topology Pointwise
@[... | Mathlib/NumberTheory/WellApproximable.lean | 121 | 129 | theorem image_pow_subset (n : ℕ) (hm : 0 < m) :
(fun (y : A) => y ^ m) '' approxOrderOf A (n * m) δ ⊆ approxOrderOf A n (m * δ) := by |
rintro - ⟨a, ha, rfl⟩
obtain ⟨b, hb : orderOf b = n * m, hab : a ∈ ball b δ⟩ := mem_approxOrderOf_iff.mp ha
replace hb : b ^ m ∈ {y : A | orderOf y = n} := by
rw [mem_setOf_eq, orderOf_pow' b hm.ne', hb, Nat.gcd_mul_left_left, n.mul_div_cancel hm]
apply ball_subset_thickening hb (m * δ)
convert pow_mem_b... | 7 |
import Mathlib.RingTheory.PrincipalIdealDomain
#align_import ring_theory.bezout from "leanprover-community/mathlib"@"6623e6af705e97002a9054c1c05a980180276fc1"
universe u v
variable {R : Type u} [CommRing R]
namespace IsBezout
theorem iff_span_pair_isPrincipal :
IsBezout R ↔ ∀ x y : R, (Ideal.span {x, y} : ... | Mathlib/RingTheory/Bezout.lean | 42 | 50 | theorem _root_.Function.Surjective.isBezout {S : Type v} [CommRing S] (f : R →+* S)
(hf : Function.Surjective f) [IsBezout R] : IsBezout S := by |
rw [iff_span_pair_isPrincipal]
intro x y
obtain ⟨⟨x, rfl⟩, ⟨y, rfl⟩⟩ := hf x, hf y
use f (gcd x y)
trans Ideal.map f (Ideal.span {gcd x y})
· rw [span_gcd, Ideal.map_span, Set.image_insert_eq, Set.image_singleton]
· rw [Ideal.map_span, Set.image_singleton]; rfl
| 7 |
import Mathlib.ModelTheory.Substructures
#align_import model_theory.finitely_generated from "leanprover-community/mathlib"@"0602c59878ff3d5f71dea69c2d32ccf2e93e5398"
open FirstOrder Set
namespace FirstOrder
namespace Language
open Structure
variable {L : Language} {M : Type*} [L.Structure M]
namespace Substru... | Mathlib/ModelTheory/FinitelyGenerated.lean | 52 | 60 | theorem fg_iff_exists_fin_generating_family {N : L.Substructure M} :
N.FG ↔ ∃ (n : ℕ) (s : Fin n → M), closure L (range s) = N := by |
rw [fg_def]
constructor
· rintro ⟨S, Sfin, hS⟩
obtain ⟨n, f, rfl⟩ := Sfin.fin_embedding
exact ⟨n, f, hS⟩
· rintro ⟨n, s, hs⟩
exact ⟨range s, finite_range s, hs⟩
| 7 |
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 :
∀ x y : E,
‖x‖ = 1 → ‖y‖ = 1 → x ≠ y → ∃ a b : ℝ, 0 ≤ a ∧ 0 ≤ b ∧ a + b = 1 ∧ ‖a • x + b • y‖ ≠ 1) :
StrictConvexSpace ℝ E := by |
refine StrictConvexSpace.of_strictConvex_closed_unit_ball ℝ
((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 ⟨a, b, ha,... | 7 |
import Mathlib.Dynamics.Ergodic.AddCircle
import Mathlib.MeasureTheory.Covering.LiminfLimsup
#align_import number_theory.well_approximable from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
open Set Filter Function Metric MeasureTheory
open scoped MeasureTheory Topology Pointwise
@[... | Mathlib/NumberTheory/WellApproximable.lean | 134 | 142 | theorem smul_subset_of_coprime (han : (orderOf a).Coprime n) :
a • approxOrderOf A n δ ⊆ approxOrderOf A (orderOf a * n) δ := by |
simp_rw [approxOrderOf, thickening_eq_biUnion_ball, ← image_smul, image_iUnion₂, image_smul,
smul_ball'', smul_eq_mul, mem_setOf_eq]
refine iUnion₂_subset_iff.mpr fun b hb c hc => ?_
simp only [mem_iUnion, exists_prop]
refine ⟨a * b, ?_, hc⟩
rw [← hb] at han ⊢
exact (Commute.all a b).orderOf_mul_eq_mul... | 7 |
import Mathlib.AlgebraicGeometry.Properties
#align_import algebraic_geometry.function_field from "leanprover-community/mathlib"@"d39590fc8728fbf6743249802486f8c91ffe07bc"
-- Explicit universe annotations were used in this file to improve perfomance #12737
set_option linter.uppercaseLean3 false
universe u v
open... | Mathlib/AlgebraicGeometry/FunctionField.lean | 67 | 75 | theorem germ_injective_of_isIntegral [IsIntegral X] {U : Opens X.carrier} (x : U) :
Function.Injective (X.presheaf.germ x) := by |
rw [injective_iff_map_eq_zero]
intro y hy
rw [← (X.presheaf.germ x).map_zero] at hy
obtain ⟨W, hW, iU, iV, e⟩ := X.presheaf.germ_eq _ x.prop x.prop _ _ hy
cases Subsingleton.elim iU iV
haveI : Nonempty W := ⟨⟨_, hW⟩⟩
exact map_injective_of_isIntegral X iU e
| 7 |
import Mathlib.Algebra.Module.LinearMap.Basic
import Mathlib.RingTheory.HahnSeries.Basic
#align_import ring_theory.hahn_series from "leanprover-community/mathlib"@"a484a7d0eade4e1268f4fb402859b6686037f965"
set_option linter.uppercaseLean3 false
open Finset Function
open scoped Classical
noncomputable section
v... | Mathlib/RingTheory/HahnSeries/Addition.lean | 81 | 89 | theorem min_order_le_order_add {Γ} [Zero Γ] [LinearOrder Γ] {x y : HahnSeries Γ R}
(hxy : x + y ≠ 0) : min x.order y.order ≤ (x + y).order := by |
by_cases hx : x = 0; · simp [hx]
by_cases hy : y = 0; · simp [hy]
rw [order_of_ne hx, order_of_ne hy, order_of_ne hxy]
apply le_of_eq_of_le _ (Set.IsWF.min_le_min_of_subset (support_add_subset (x := x) (y := y)))
· simp
· simp [hy]
· exact (Set.IsWF.min_union _ _ _ _).symm
| 7 |
import Mathlib.RingTheory.AdjoinRoot
import Mathlib.FieldTheory.Minpoly.Field
import Mathlib.RingTheory.Polynomial.GaussLemma
#align_import field_theory.minpoly.is_integrally_closed from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
open scoped Classical Polynomial
open Polynomial Set... | Mathlib/FieldTheory/Minpoly/IsIntegrallyClosed.lean | 138 | 145 | theorem prime_of_isIntegrallyClosed {x : S} (hx : IsIntegral R x) : Prime (minpoly R x) := by |
refine
⟨(minpoly.monic hx).ne_zero,
⟨fun h_contra => (ne_of_lt (minpoly.degree_pos hx)) (degree_eq_zero_of_isUnit h_contra).symm,
fun a b h => or_iff_not_imp_left.mpr fun h' => ?_⟩⟩
rw [← minpoly.isIntegrallyClosed_dvd_iff hx] at h' h ⊢
rw [aeval_mul] at h
exact eq_zero_of_ne_zero_of_mul_left... | 7 |
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.BinaryProducts
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Products
import Mathlib.CategoryTheory.Limits.ConcreteCategory
import Mathlib.CategoryTheory.Limits.Shapes.Types
import Mathlib.CategoryTheory.Limits.Shapes.Multiequalizer
import Mathlib.CategoryT... | Mathlib/CategoryTheory/Limits/Shapes/ConcreteCategory.lean | 324 | 333 | theorem widePushout_exists_rep {B : C} {α : Type _} {X : α → C} (f : ∀ j : α, B ⟶ X j)
[HasWidePushout.{v} B X f] [PreservesColimit (wideSpan B X f) (forget C)]
(x : ↑(widePushout B X f)) : (∃ y : B, head f y = x) ∨ ∃ (i : α) (y : X i), ι f i y = x := by |
obtain ⟨_ | j, y, rfl⟩ := Concrete.colimit_exists_rep _ x
· left
use y
rfl
· right
use j, y
rfl
| 7 |
import Mathlib.SetTheory.Game.Ordinal
import Mathlib.SetTheory.Ordinal.NaturalOps
#align_import set_theory.game.birthday from "leanprover-community/mathlib"@"a347076985674932c0e91da09b9961ed0a79508c"
universe u
open Ordinal
namespace SetTheory
open scoped NaturalOps PGame
namespace PGame
noncomputable def b... | Mathlib/SetTheory/Game/Birthday.lean | 122 | 129 | theorem toPGame_birthday (o : Ordinal) : o.toPGame.birthday = o := by |
induction' o using Ordinal.induction with o IH
rw [toPGame_def, PGame.birthday]
simp only [lsub_empty, max_zero_right]
-- Porting note: was `nth_rw 1 [← lsub_typein o]`
conv_rhs => rw [← lsub_typein o]
congr with x
exact IH _ (typein_lt_self x)
| 7 |
import Mathlib.Analysis.InnerProductSpace.Basic
import Mathlib.Analysis.SpecialFunctions.Complex.Arg
#align_import analysis.complex.arg from "leanprover-community/mathlib"@"45a46f4f03f8ae41491bf3605e8e0e363ba192fd"
variable {x y : ℂ}
namespace Complex
| Mathlib/Analysis/Complex/Arg.lean | 31 | 38 | theorem sameRay_iff : SameRay ℝ x y ↔ x = 0 ∨ y = 0 ∨ x.arg = y.arg := by |
rcases eq_or_ne x 0 with (rfl | hx)
· simp
rcases eq_or_ne y 0 with (rfl | hy)
· simp
simp only [hx, hy, false_or_iff, sameRay_iff_norm_smul_eq, arg_eq_arg_iff hx hy]
field_simp [hx, hy]
rw [mul_comm, eq_comm]
| 7 |
import Mathlib.Topology.Category.TopCat.Adjunctions
#align_import topology.category.Top.epi_mono from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe u
open CategoryTheory
open TopCat
namespace TopCat
| Mathlib/Topology/Category/TopCat/EpiMono.lean | 27 | 34 | theorem epi_iff_surjective {X Y : TopCat.{u}} (f : X ⟶ Y) : Epi f ↔ Function.Surjective f := by |
suffices Epi f ↔ Epi ((forget TopCat).map f) by
rw [this, CategoryTheory.epi_iff_surjective]
rfl
constructor
· intro
infer_instance
· apply Functor.epi_of_epi_map
| 7 |
import Mathlib.CategoryTheory.EqToHom
import Mathlib.CategoryTheory.Quotient
import Mathlib.Combinatorics.Quiver.Path
#align_import category_theory.path_category from "leanprover-community/mathlib"@"c6dd521ebdce53bb372c527569dd7c25de53a08b"
universe v₁ v₂ u₁ u₂
namespace CategoryTheory
section
def Paths (V : ... | Mathlib/CategoryTheory/PathCategory.lean | 93 | 100 | theorem lift_spec {C} [Category C] (φ : V ⥤q C) : of ⋙q (lift φ).toPrefunctor = φ := by |
fapply Prefunctor.ext
· rintro X
rfl
· rintro X Y f
rcases φ with ⟨φo, φm⟩
dsimp [lift, Quiver.Hom.toPath]
simp only [Category.id_comp]
| 7 |
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.LinearAlgebra.Vandermonde
import Mathlib.RingTheory.Polynomial.Basic
#align_import linear_algebra.lagrange from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
open Polynomial
section PolynomialDetermination
namespace Poly... | Mathlib/LinearAlgebra/Lagrange.lean | 44 | 52 | theorem eq_zero_of_degree_lt_of_eval_finset_eq_zero (degree_f_lt : f.degree < s.card)
(eval_f : ∀ x ∈ s, f.eval x = 0) : f = 0 := by |
rw [← mem_degreeLT] at degree_f_lt
simp_rw [eval_eq_sum_degreeLTEquiv degree_f_lt] at eval_f
rw [← degreeLTEquiv_eq_zero_iff_eq_zero degree_f_lt]
exact
Matrix.eq_zero_of_forall_index_sum_mul_pow_eq_zero
(Injective.comp (Embedding.subtype _).inj' (equivFinOfCardEq (card_coe _)).symm.injective)
f... | 7 |
import Mathlib.Algebra.Polynomial.Coeff
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.RingTheory.PowerSeries.Basic
#align_import ring_theory.power_series.basic from "leanprover-community/mathlib"@"2d5739b61641ee4e7e53eca5688a08f66f2e6a60"
noncomputable section
open Polynomial
open Finset (antid... | Mathlib/RingTheory/PowerSeries/Trunc.lean | 99 | 106 | theorem degree_trunc_lt (f : R⟦X⟧) (n) : (trunc n f).degree < n := by |
rw [degree_lt_iff_coeff_zero]
intros
rw [coeff_trunc]
split_ifs with h
· rw [← not_le] at h
contradiction
· rfl
| 7 |
import Mathlib.Data.Real.Basic
#align_import data.real.sign from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
namespace Real
noncomputable def sign (r : ℝ) : ℝ :=
if r < 0 then -1 else if 0 < r then 1 else 0
#align real.sign Real.sign
theorem sign_of_neg {r : ℝ} (hr : r < 0) : si... | Mathlib/Data/Real/Sign.lean | 64 | 71 | theorem sign_eq_zero_iff {r : ℝ} : sign r = 0 ↔ r = 0 := by |
refine ⟨fun h => ?_, fun h => h.symm ▸ sign_zero⟩
obtain hn | rfl | hp := lt_trichotomy r (0 : ℝ)
· rw [sign_of_neg hn, neg_eq_zero] at h
exact (one_ne_zero h).elim
· rfl
· rw [sign_of_pos hp] at h
exact (one_ne_zero h).elim
| 7 |
import Mathlib.RingTheory.PowerSeries.Trunc
import Mathlib.RingTheory.PowerSeries.Inverse
import Mathlib.RingTheory.Derivation.Basic
namespace PowerSeries
open Polynomial Derivation Nat
section CommutativeSemiring
variable {R} [CommSemiring R]
noncomputable def derivativeFun (f : R⟦X⟧) : R⟦X⟧ := mk fun n ↦ coef... | Mathlib/RingTheory/PowerSeries/Derivative.lean | 60 | 68 | theorem trunc_derivativeFun (f : R⟦X⟧) (n : ℕ) :
trunc n f.derivativeFun = derivative (trunc (n + 1) f) := by |
ext d
rw [coeff_trunc]
split_ifs with h
· have : d + 1 < n + 1 := succ_lt_succ_iff.2 h
rw [coeff_derivativeFun, coeff_derivative, coeff_trunc, if_pos this]
· have : ¬d + 1 < n + 1 := by rwa [succ_lt_succ_iff]
rw [coeff_derivative, coeff_trunc, if_neg this, zero_mul]
| 7 |
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
import Mathlib.MeasureTheory.Measure.Haar.Quotient
import Mathlib.MeasureTheory.Constructions.Polish
import Mathlib.MeasureTheory.Integral.IntervalIntegral
import Mathlib.Topology.Algebra.Order.Floor
#align_import measure_theory.integral.periodic from "leanprover-c... | Mathlib/MeasureTheory/Integral/Periodic.lean | 39 | 46 | theorem isAddFundamentalDomain_Ioc {T : ℝ} (hT : 0 < T) (t : ℝ)
(μ : Measure ℝ := by | volume_tac) :
IsAddFundamentalDomain (AddSubgroup.zmultiples T) (Ioc t (t + T)) μ := by
refine IsAddFundamentalDomain.mk' measurableSet_Ioc.nullMeasurableSet fun x => ?_
have : Bijective (codRestrict (fun n : ℤ => n • T) (AddSubgroup.zmultiples T) _) :=
(Equiv.ofInjective (fun n : ℤ => n • T) (zsmul_strict... | 7 |
import Mathlib.RingTheory.DedekindDomain.Ideal
#align_import ring_theory.dedekind_domain.factorization from "leanprover-community/mathlib"@"2f588be38bb5bec02f218ba14f82fc82eb663f87"
noncomputable section
open scoped Classical nonZeroDivisors
open Set Function UniqueFactorizationMonoid IsDedekindDomain IsDedekind... | Mathlib/RingTheory/DedekindDomain/Factorization.lean | 81 | 90 | theorem Associates.finite_factors {I : Ideal R} (hI : I ≠ 0) :
∀ᶠ v : HeightOneSpectrum R in Filter.cofinite,
((Associates.mk v.asIdeal).count (Associates.mk I).factors : ℤ) = 0 := by |
have h_supp : {v : HeightOneSpectrum R | ¬((Associates.mk v.asIdeal).count
(Associates.mk I).factors : ℤ) = 0} = {v : HeightOneSpectrum R | v.asIdeal ∣ I} := by
ext v
simp_rw [Int.natCast_eq_zero]
exact Associates.count_ne_zero_iff_dvd hI v.irreducible
rw [Filter.eventually_cofinite, h_supp]
ex... | 7 |
import Mathlib.Algebra.CharP.Defs
import Mathlib.RingTheory.Multiplicity
import Mathlib.RingTheory.PowerSeries.Basic
#align_import ring_theory.power_series.basic from "leanprover-community/mathlib"@"2d5739b61641ee4e7e53eca5688a08f66f2e6a60"
noncomputable section
open Polynomial
open Finset (antidiagonal mem_anti... | Mathlib/RingTheory/PowerSeries/Order.lean | 121 | 129 | theorem le_order (φ : R⟦X⟧) (n : PartENat) (h : ∀ i : ℕ, ↑i < n → coeff R i φ = 0) :
n ≤ order φ := by |
induction n using PartENat.casesOn
· show _ ≤ _
rw [top_le_iff, order_eq_top]
ext i
exact h _ (PartENat.natCast_lt_top i)
· apply nat_le_order
simpa only [PartENat.coe_lt_coe] using h
| 7 |
import Mathlib.Analysis.SpecialFunctions.Integrals
#align_import data.real.pi.wallis from "leanprover-community/mathlib"@"980755c33b9168bc82f774f665eaa27878140fac"
open scoped Real Topology Nat
open Filter Finset intervalIntegral
namespace Real
namespace Wallis
set_option linter.uppercaseLean3 false
noncomp... | Mathlib/Data/Real/Pi/Wallis.lean | 91 | 98 | theorem le_W (k : ℕ) : ((2 : ℝ) * k + 1) / (2 * k + 2) * (π / 2) ≤ W k := by |
rw [← le_div_iff pi_div_two_pos, div_eq_inv_mul (W k) _]
rw [W_eq_integral_sin_pow_div_integral_sin_pow, le_div_iff (integral_sin_pow_pos _)]
convert integral_sin_pow_succ_le (2 * k + 1)
rw [integral_sin_pow (2 * k)]
simp only [sin_zero, ne_eq, add_eq_zero, and_false, not_false_eq_true, zero_pow, cos_zero,
... | 7 |
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Measure.Haar.Unique
#align_import measure_theory.measure.lebesgue.integral from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
open Set Filter MeasureTheory... | Mathlib/MeasureTheory/Measure/Lebesgue/Integral.lean | 22 | 31 | theorem volume_regionBetween_eq_integral' [SigmaFinite μ] (f_int : IntegrableOn f s μ)
(g_int : IntegrableOn g s μ) (hs : MeasurableSet s) (hfg : f ≤ᵐ[μ.restrict s] g) :
μ.prod volume (regionBetween f g s) = ENNReal.ofReal (∫ y in s, (g - f) y ∂μ) := by |
have h : g - f =ᵐ[μ.restrict s] fun x => Real.toNNReal (g x - f x) :=
hfg.mono fun x hx => (Real.coe_toNNReal _ <| sub_nonneg.2 hx).symm
rw [volume_regionBetween_eq_lintegral f_int.aemeasurable g_int.aemeasurable hs,
integral_congr_ae h, lintegral_congr_ae,
lintegral_coe_eq_integral _ ((integrable_cong... | 7 |
import Mathlib.Data.Finset.Lattice
#align_import combinatorics.set_family.compression.down from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
variable {α : Type*} [DecidableEq α] {𝒜 ℬ : Finset (Finset α)} {s : Finset α} {a : α}
open Finset
-- The namespace is here to distinguish fro... | Mathlib/Combinatorics/SetFamily/Compression/Down.lean | 241 | 248 | theorem mem_compression : s ∈ 𝓓 a 𝒜 ↔ s ∈ 𝒜 ∧ s.erase a ∈ 𝒜 ∨ s ∉ 𝒜 ∧ insert a s ∈ 𝒜 := by |
simp_rw [compression, mem_disjUnion, mem_filter, mem_image, and_comm (a := (¬ s ∈ 𝒜))]
refine
or_congr_right
(and_congr_left fun hs =>
⟨?_, fun h => ⟨_, h, erase_insert <| insert_ne_self.1 <| ne_of_mem_of_not_mem h hs⟩⟩)
rintro ⟨t, ht, rfl⟩
rwa [insert_erase (erase_ne_self.1 (ne_of_mem_of_no... | 7 |
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.Tactic.ComputeDegree
#align_import linear_algebra.matrix.polynomial from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
set_... | Mathlib/LinearAlgebra/Matrix/Polynomial.lean | 62 | 70 | theorem coeff_det_X_add_C_zero (A B : Matrix n n α) :
coeff (det ((X : α[X]) • A.map C + B.map C)) 0 = det B := by |
rw [det_apply, finset_sum_coeff, det_apply]
refine Finset.sum_congr rfl ?_
rintro g -
convert coeff_smul (R := α) (sign g) _ 0
rw [coeff_zero_prod]
refine Finset.prod_congr rfl ?_
simp
| 7 |
import Mathlib.Algebra.Order.Floor
import Mathlib.Algebra.Order.Field.Power
import Mathlib.Data.Nat.Log
#align_import data.int.log from "leanprover-community/mathlib"@"1f0096e6caa61e9c849ec2adbd227e960e9dff58"
variable {R : Type*} [LinearOrderedSemifield R] [FloorSemiring R]
namespace Int
def log (b : ℕ) (r : ... | Mathlib/Data/Int/Log.lean | 138 | 145 | theorem log_zpow {b : ℕ} (hb : 1 < b) (z : ℤ) : log b ((b : R) ^ z : R) = z := by |
obtain ⟨n, rfl | rfl⟩ := Int.eq_nat_or_neg z
· rw [log_of_one_le_right _ (one_le_zpow_of_nonneg _ <| Int.natCast_nonneg _), zpow_natCast, ←
Nat.cast_pow, Nat.floor_natCast, Nat.log_pow hb]
exact mod_cast hb.le
· rw [log_of_right_le_one _ (zpow_le_one_of_nonpos _ <| neg_nonpos.mpr (Int.natCast_nonneg _)... | 7 |
import Mathlib.Data.Finsupp.Basic
import Mathlib.Data.Finsupp.Order
#align_import data.finsupp.multiset from "leanprover-community/mathlib"@"59694bd07f0a39c5beccba34bd9f413a160782bf"
open Finset
variable {α β ι : Type*}
namespace Finsupp
def toMultiset : (α →₀ ℕ) →+ Multiset α where
toFun f := Finsupp.sum f... | Mathlib/Data/Finsupp/Multiset.lean | 71 | 79 | theorem toMultiset_map (f : α →₀ ℕ) (g : α → β) :
f.toMultiset.map g = toMultiset (f.mapDomain g) := by |
refine f.induction ?_ ?_
· rw [toMultiset_zero, Multiset.map_zero, mapDomain_zero, toMultiset_zero]
· intro a n f _ _ ih
rw [toMultiset_add, Multiset.map_add, ih, mapDomain_add, mapDomain_single,
toMultiset_single, toMultiset_add, toMultiset_single, ← Multiset.coe_mapAddMonoidHom,
(Multiset.mapAd... | 7 |
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