Context stringlengths 57 85k | file_name stringlengths 21 79 | start int64 14 2.42k | end int64 18 2.43k | theorem stringlengths 25 2.71k | proof stringlengths 5 10.6k |
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import Mathlib.RingTheory.Polynomial.Basic
import Mathlib.RingTheory.Ideal.LocalRing
#align_import data.polynomial.expand from "leanprover-community/mathlib"@"bbeb185db4ccee8ed07dc48449414ebfa39cb821"
universe u v w
open Polynomial
open Finset
namespace Polynomial
section CommSemiring
variable (R : Type u) [... | Mathlib/Algebra/Polynomial/Expand.lean | 212 | 221 | theorem coeff_contract {p : ℕ} (hp : p ≠ 0) (f : R[X]) (n : ℕ) :
(contract p f).coeff n = f.coeff (n * p) := by |
simp only [contract, coeff_monomial, sum_ite_eq', finset_sum_coeff, mem_range, not_lt,
ite_eq_left_iff]
intro hn
apply (coeff_eq_zero_of_natDegree_lt _).symm
calc
f.natDegree < f.natDegree + 1 := Nat.lt_succ_self _
_ ≤ n * 1 := by simpa only [mul_one] using hn
_ ≤ n * p := mul_le_mul_of_nonneg_... |
import Mathlib.Data.Matrix.Basic
variable {l m n o : Type*}
universe u v w
variable {R : Type*} {α : Type v} {β : Type w}
namespace Matrix
def col (w : m → α) : Matrix m Unit α :=
of fun x _ => w x
#align matrix.col Matrix.col
-- TODO: set as an equation lemma for `col`, see mathlib4#3024
@[simp]
theorem col... | Mathlib/Data/Matrix/RowCol.lean | 67 | 69 | theorem col_smul [SMul R α] (x : R) (v : m → α) : col (x • v) = x • col v := by |
ext
rfl
|
import Mathlib.Topology.Algebra.InfiniteSum.Order
import Mathlib.Topology.Algebra.InfiniteSum.Ring
import Mathlib.Topology.Instances.Real
import Mathlib.Topology.MetricSpace.Isometry
#align_import topology.instances.nnreal from "leanprover-community/mathlib"@"32253a1a1071173b33dc7d6a218cf722c6feb514"
noncomputabl... | Mathlib/Topology/Instances/NNReal.lean | 179 | 182 | theorem summable_coe {f : α → ℝ≥0} : (Summable fun a => (f a : ℝ)) ↔ Summable f := by |
constructor
· exact fun ⟨a, ha⟩ => ⟨⟨a, ha.nonneg fun x => (f x).2⟩, hasSum_coe.1 ha⟩
· exact fun ⟨a, ha⟩ => ⟨a.1, hasSum_coe.2 ha⟩
|
import Mathlib.Algebra.Group.Defs
import Mathlib.Init.Logic
import Mathlib.Tactic.Cases
#align_import algebra.group.semiconj from "leanprover-community/mathlib"@"a148d797a1094ab554ad4183a4ad6f130358ef64"
assert_not_exists MonoidWithZero
assert_not_exists DenselyOrdered
variable {S M G : Type*}
@[to_additive "`x... | Mathlib/Algebra/Group/Semiconj/Defs.lean | 97 | 97 | theorem one_right (a : M) : SemiconjBy a 1 1 := by | rw [SemiconjBy, mul_one, one_mul]
|
import Mathlib.Data.List.Basic
namespace List
variable {α β : Type*}
@[simp]
| Mathlib/Data/List/ReduceOption.lean | 19 | 21 | theorem reduceOption_cons_of_some (x : α) (l : List (Option α)) :
reduceOption (some x :: l) = x :: l.reduceOption := by |
simp only [reduceOption, filterMap, id, eq_self_iff_true, and_self_iff]
|
import Mathlib.RingTheory.JacobsonIdeal
#align_import ring_theory.nakayama from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
variable {R M : Type*} [CommRing R] [AddCommGroup M] [Module R M]
open Ideal
namespace Submodule
theorem eq_smul_of_le_smul_of_le_jacobson {I J : Ideal R} {... | Mathlib/RingTheory/Nakayama.lean | 137 | 140 | theorem le_of_le_smul_of_le_jacobson_bot {R M} [CommRing R] [AddCommGroup M] [Module R M]
{I : Ideal R} {N N' : Submodule R M} (hN' : N'.FG)
(hIJ : I ≤ jacobson ⊥) (hNN : N' ≤ N ⊔ I • N') : N' ≤ N := by |
rw [← sup_eq_left, sup_eq_sup_smul_of_le_smul_of_le_jacobson hN' hIJ hNN, bot_smul, sup_bot_eq]
|
import Mathlib.Algebra.MvPolynomial.Variables
#align_import data.mv_polynomial.supported from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
universe u v w
namespace MvPolynomial
variable {σ τ : Type*} {R : Type u} {S : Type v} {r : R} {e : ℕ} {n m : σ}
section CommSemiring
variable... | Mathlib/Algebra/MvPolynomial/Supported.lean | 135 | 141 | theorem exists_restrict_to_vars (R : Type*) [CommRing R] {F : MvPolynomial σ ℤ}
(hF : ↑F.vars ⊆ s) : ∃ f : (s → R) → R, ∀ x : σ → R, f (x ∘ (↑) : s → R) = aeval x F := by |
rw [← mem_supported, supported_eq_range_rename, AlgHom.mem_range] at hF
cases' hF with F' hF'
use fun z ↦ aeval z F'
intro x
simp only [← hF', aeval_rename]
|
import Mathlib.Data.Nat.Prime
import Mathlib.Data.PNat.Basic
#align_import data.pnat.prime from "leanprover-community/mathlib"@"09597669f02422ed388036273d8848119699c22f"
namespace PNat
open Nat
def gcd (n m : ℕ+) : ℕ+ :=
⟨Nat.gcd (n : ℕ) (m : ℕ), Nat.gcd_pos_of_pos_left (m : ℕ) n.pos⟩
#align pnat.gcd PNat.gc... | Mathlib/Data/PNat/Prime.lean | 103 | 106 | theorem eq_one_of_lt_two {n : ℕ+} : n < 2 → n = 1 := by |
intro h; apply le_antisymm; swap
· apply PNat.one_le
· exact PNat.lt_add_one_iff.1 h
|
import Mathlib.Geometry.Manifold.SmoothManifoldWithCorners
import Mathlib.Geometry.Manifold.LocalInvariantProperties
#align_import geometry.manifold.cont_mdiff from "leanprover-community/mathlib"@"e5ab837fc252451f3eb9124ae6e7b6f57455e7b9"
open Set Function Filter ChartedSpace SmoothManifoldWithCorners
open scope... | Mathlib/Geometry/Manifold/ContMDiff/Defs.lean | 157 | 161 | theorem contDiffWithinAtProp_mono_of_mem (n : ℕ∞) ⦃s x t⦄ ⦃f : H → H'⦄ (hts : s ∈ 𝓝[t] x)
(h : ContDiffWithinAtProp I I' n f s x) : ContDiffWithinAtProp I I' n f t x := by |
refine h.mono_of_mem ?_
refine inter_mem ?_ (mem_of_superset self_mem_nhdsWithin inter_subset_right)
rwa [← Filter.mem_map, ← I.image_eq, I.symm_map_nhdsWithin_image]
|
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology MeasureTheory
names... | Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean | 106 | 106 | theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 := by | rw [condexp, dif_neg hm_not]
|
import Mathlib.Algebra.Order.ZeroLEOne
import Mathlib.Data.List.InsertNth
import Mathlib.Logic.Relation
import Mathlib.Logic.Small.Defs
import Mathlib.Order.GameAdd
#align_import set_theory.game.pgame from "leanprover-community/mathlib"@"8900d545017cd21961daa2a1734bb658ef52c618"
set_option autoImplicit true
names... | Mathlib/SetTheory/Game/PGame.lean | 988 | 990 | theorem lf_iff_lt_or_fuzzy {x y : PGame} : x ⧏ y ↔ x < y ∨ x ‖ y := by |
simp only [lt_iff_le_and_lf, Fuzzy, ← PGame.not_le]
tauto
|
import Mathlib.Tactic.TFAE
import Mathlib.Topology.ContinuousOn
#align_import topology.inseparable from "leanprover-community/mathlib"@"bcfa726826abd57587355b4b5b7e78ad6527b7e4"
open Set Filter Function Topology List
variable {X Y Z α ι : Type*} {π : ι → Type*} [TopologicalSpace X] [TopologicalSpace Y]
[Topolo... | Mathlib/Topology/Inseparable.lean | 282 | 284 | theorem inseparable_iff_closure_eq : (x ~ᵢ y) ↔ closure ({x} : Set X) = closure {y} := by |
simp only [inseparable_iff_specializes_and, specializes_iff_closure_subset, ← subset_antisymm_iff,
eq_comm]
|
import Mathlib.Logic.Equiv.Fin
import Mathlib.Topology.DenseEmbedding
import Mathlib.Topology.Support
import Mathlib.Topology.Connected.LocallyConnected
#align_import topology.homeomorph from "leanprover-community/mathlib"@"4c3e1721c58ef9087bbc2c8c38b540f70eda2e53"
open Set Filter
open Topology
variable {X : Typ... | Mathlib/Topology/Homeomorph.lean | 442 | 444 | theorem map_punctured_nhds_eq (h : X ≃ₜ Y) (x : X) : map h (𝓝[≠] x) = 𝓝[≠] (h x) := by |
convert h.embedding.map_nhdsWithin_eq ({x}ᶜ) x
rw [h.image_compl, Set.image_singleton]
|
import Mathlib.RepresentationTheory.FdRep
import Mathlib.LinearAlgebra.Trace
import Mathlib.RepresentationTheory.Invariants
#align_import representation_theory.character from "leanprover-community/mathlib"@"55b3f8206b8596db8bb1804d8a92814a0b6670c9"
noncomputable section
universe u
open CategoryTheory LinearMap ... | Mathlib/RepresentationTheory/Character.lean | 64 | 65 | theorem char_tensor (V W : FdRep k G) : (V ⊗ W).character = V.character * W.character := by |
ext g; convert trace_tensorProduct' (V.ρ g) (W.ρ g)
|
import Mathlib.Order.Filter.Basic
import Mathlib.Topology.Bases
import Mathlib.Data.Set.Accumulate
import Mathlib.Topology.Bornology.Basic
import Mathlib.Topology.LocallyFinite
open Set Filter Topology TopologicalSpace Classical Function
universe u v
variable {X : Type u} {Y : Type v} {ι : Type*}
variable [Topolog... | Mathlib/Topology/Compactness/Compact.lean | 706 | 707 | theorem compl_mem_coclosedCompact : sᶜ ∈ coclosedCompact X ↔ IsCompact (closure s) := by |
rw [mem_coclosedCompact_iff, compl_compl]
|
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.Data.Nat.GCD.Basic
import Mathlib.Tactic.Ring
import Mathlib.Tactic.Linarith
#align_import data.nat.choose.central from "leanprover-community/mathlib"@"0a0ec35061ed9960bf0e7ffb0335f44447b58977"
namespace Nat
def centralBinom (n : ℕ) :=
(2 * n).choose n
#alig... | Mathlib/Data/Nat/Choose/Central.lean | 131 | 138 | theorem succ_dvd_centralBinom (n : ℕ) : n + 1 ∣ n.centralBinom := by |
have h_s : (n + 1).Coprime (2 * n + 1) := by
rw [two_mul, add_assoc, coprime_add_self_right, coprime_self_add_left]
exact coprime_one_left n
apply h_s.dvd_of_dvd_mul_left
apply Nat.dvd_of_mul_dvd_mul_left zero_lt_two
rw [← mul_assoc, ← succ_mul_centralBinom_succ, mul_comm]
exact mul_dvd_mul_left _ (t... |
import Mathlib.Topology.Separation
import Mathlib.Topology.UniformSpace.Basic
import Mathlib.Topology.UniformSpace.Cauchy
#align_import topology.uniform_space.uniform_convergence from "leanprover-community/mathlib"@"2705404e701abc6b3127da906f40bae062a169c9"
noncomputable section
open Topology Uniformity Filter S... | Mathlib/Topology/UniformSpace/UniformConvergence.lean | 142 | 144 | theorem tendstoUniformly_iff_tendstoUniformlyOnFilter :
TendstoUniformly F f p ↔ TendstoUniformlyOnFilter F f p ⊤ := by |
rw [← tendstoUniformlyOn_univ, tendstoUniformlyOn_iff_tendstoUniformlyOnFilter, principal_univ]
|
import Mathlib.MeasureTheory.Measure.Typeclasses
import Mathlib.Analysis.Complex.Basic
#align_import measure_theory.measure.vector_measure from "leanprover-community/mathlib"@"70a4f2197832bceab57d7f41379b2592d1110570"
noncomputable section
open scoped Classical
open NNReal ENNReal MeasureTheory
namespace Measur... | Mathlib/MeasureTheory/Measure/VectorMeasure.lean | 197 | 200 | theorem of_diff {M : Type*} [AddCommGroup M] [TopologicalSpace M] [T2Space M]
{v : VectorMeasure α M} {A B : Set α} (hA : MeasurableSet A) (hB : MeasurableSet B)
(h : A ⊆ B) : v (B \ A) = v B - v A := by |
rw [← of_add_of_diff hA hB h, add_sub_cancel_left]
|
import Mathlib.Data.ENNReal.Real
import Mathlib.Order.Interval.Finset.Nat
import Mathlib.Topology.UniformSpace.Pi
import Mathlib.Topology.UniformSpace.UniformConvergence
import Mathlib.Topology.UniformSpace.UniformEmbedding
#align_import topology.metric_space.emetric_space from "leanprover-community/mathlib"@"c8f3055... | Mathlib/Topology/EMetricSpace/Basic.lean | 575 | 576 | theorem mem_ball_self (h : 0 < ε) : x ∈ ball x ε := by |
rwa [mem_ball, edist_self]
|
import Mathlib.Analysis.InnerProductSpace.Adjoint
#align_import analysis.inner_product_space.positive from "leanprover-community/mathlib"@"caa58cbf5bfb7f81ccbaca4e8b8ac4bc2b39cc1c"
open InnerProductSpace RCLike ContinuousLinearMap
open scoped InnerProduct ComplexConjugate
namespace ContinuousLinearMap
variable... | Mathlib/Analysis/InnerProductSpace/Positive.lean | 71 | 74 | theorem isPositive_zero : IsPositive (0 : E →L[𝕜] E) := by |
refine ⟨isSelfAdjoint_zero _, fun x => ?_⟩
change 0 ≤ re ⟪_, _⟫
rw [zero_apply, inner_zero_left, ZeroHomClass.map_zero]
|
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Data.Nat.GCD.BigOperators
import Mathlib.Data.Nat.PrimeFin
import Mathlib.NumberTheory.Padics.PadicVal
import Mathlib.Order.Interval.Finset.Nat
#align_import data.nat.factorization.basic from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
... | Mathlib/Data/Nat/Factorization/Basic.lean | 409 | 419 | theorem factorization_le_iff_dvd {d n : ℕ} (hd : d ≠ 0) (hn : n ≠ 0) :
d.factorization ≤ n.factorization ↔ d ∣ n := by |
constructor
· intro hdn
set K := n.factorization - d.factorization with hK
use K.prod (· ^ ·)
rw [← factorization_prod_pow_eq_self hn, ← factorization_prod_pow_eq_self hd,
← Finsupp.prod_add_index' pow_zero pow_add, hK, add_tsub_cancel_of_le hdn]
· rintro ⟨c, rfl⟩
rw [factorization_mul hd... |
import Mathlib.Analysis.Calculus.SmoothSeries
import Mathlib.Analysis.Calculus.BumpFunction.InnerProduct
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.InnerProductSpace.EuclideanDist
import Mathlib.Data.Set.Pointwise.Support
import Mathlib.MeasureTheory.Measure.Haar.NormedSpace
import Mathlib.MeasureTheo... | Mathlib/Analysis/Calculus/BumpFunction/FiniteDimension.lean | 421 | 461 | theorem y_pos_of_mem_ball {D : ℝ} {x : E} (Dpos : 0 < D) (D_lt_one : D < 1)
(hx : x ∈ ball (0 : E) (1 + D)) : 0 < y D x := by |
simp only [mem_ball_zero_iff] at hx
refine (integral_pos_iff_support_of_nonneg (w_mul_φ_nonneg D x) ?_).2 ?_
· have F_comp : HasCompactSupport (w D) := w_compact_support E Dpos
have B : LocallyIntegrable (φ : E → ℝ) μ :=
(locallyIntegrable_const _).indicator measurableSet_closedBall
have C : Contin... |
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Zip
import Mathlib.Data.Nat.Defs
import Mathlib.Data.List.Infix
#align_import data.list.rotate from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
universe u
variable {α : Type u}
open Nat Function
namespace List
theorem rotate... | Mathlib/Data/List/Rotate.lean | 41 | 41 | theorem rotate_nil (n : ℕ) : ([] : List α).rotate n = [] := by | simp [rotate]
|
import Mathlib.Tactic.Qify
import Mathlib.Data.ZMod.Basic
import Mathlib.NumberTheory.DiophantineApproximation
import Mathlib.NumberTheory.Zsqrtd.Basic
#align_import number_theory.pell from "leanprover-community/mathlib"@"7ad820c4997738e2f542f8a20f32911f52020e26"
namespace Pell
open Zsqrtd
theorem is_pell_s... | Mathlib/NumberTheory/Pell.lean | 329 | 336 | theorem sign_y_zpow_eq_sign_of_x_pos_of_y_pos {a : Solution₁ d} (hax : 0 < a.x) (hay : 0 < a.y)
(n : ℤ) : (a ^ n).y.sign = n.sign := by |
rcases n with ((_ | n) | n)
· rfl
· rw [Int.ofNat_eq_coe, zpow_natCast]
exact Int.sign_eq_one_of_pos (y_pow_succ_pos hax hay n)
· rw [zpow_negSucc]
exact Int.sign_eq_neg_one_of_neg (neg_neg_of_pos (y_pow_succ_pos hax hay n))
|
import Mathlib.Topology.UniformSpace.UniformConvergenceTopology
#align_import topology.uniform_space.equicontinuity from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
section
open UniformSpace Filter Set Uniformity Topology UniformConvergence Function
variable {ι κ X X' Y Z α α' β β'... | Mathlib/Topology/UniformSpace/Equicontinuity.lean | 663 | 670 | theorem Filter.HasBasis.equicontinuousAt_iff {κ₁ κ₂ : Type*} {p₁ : κ₁ → Prop} {s₁ : κ₁ → Set X}
{p₂ : κ₂ → Prop} {s₂ : κ₂ → Set (α × α)} {F : ι → X → α} {x₀ : X} (hX : (𝓝 x₀).HasBasis p₁ s₁)
(hα : (𝓤 α).HasBasis p₂ s₂) :
EquicontinuousAt F x₀ ↔
∀ k₂, p₂ k₂ → ∃ k₁, p₁ k₁ ∧ ∀ x ∈ s₁ k₁, ∀ i, (F i x₀, ... |
rw [equicontinuousAt_iff_continuousAt, ContinuousAt,
hX.tendsto_iff (UniformFun.hasBasis_nhds_of_basis ι α _ hα)]
rfl
|
import Mathlib.AlgebraicGeometry.Morphisms.QuasiCompact
import Mathlib.Topology.QuasiSeparated
#align_import algebraic_geometry.morphisms.quasi_separated from "leanprover-community/mathlib"@"1a51edf13debfcbe223fa06b1cb353b9ed9751cc"
noncomputable section
open CategoryTheory CategoryTheory.Limits Opposite Topolog... | Mathlib/AlgebraicGeometry/Morphisms/QuasiSeparated.lean | 210 | 213 | theorem quasiSeparated_over_affine_iff {X Y : Scheme} (f : X ⟶ Y) [IsAffine Y] :
QuasiSeparated f ↔ QuasiSeparatedSpace X.carrier := by |
rw [quasiSeparated_eq_affineProperty,
QuasiSeparated.affineProperty_isLocal.affine_target_iff f, QuasiSeparated.affineProperty]
|
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Analysis.Normed.Group.Basic
import Mathlib.Topology.Instances.NNReal
#align_import analysis.normed.group.infinite_sum from "leanprover-community/mathlib"@"9a59dcb7a2d06bf55da57b9030169219980660cd"
open Topology NNReal
open Finset Filter Metric
variabl... | Mathlib/Analysis/Normed/Group/InfiniteSum.lean | 78 | 89 | theorem cauchySeq_range_of_norm_bounded {f : ℕ → E} (g : ℕ → ℝ)
(hg : CauchySeq fun n => ∑ i ∈ range n, g i) (hf : ∀ i, ‖f i‖ ≤ g i) :
CauchySeq fun n => ∑ i ∈ range n, f i := by |
refine Metric.cauchySeq_iff'.2 fun ε hε => ?_
refine (Metric.cauchySeq_iff'.1 hg ε hε).imp fun N hg n hn => ?_
specialize hg n hn
rw [dist_eq_norm, ← sum_Ico_eq_sub _ hn] at hg ⊢
calc
‖∑ k ∈ Ico N n, f k‖ ≤ ∑ k ∈ _, ‖f k‖ := norm_sum_le _ _
_ ≤ ∑ k ∈ _, g k := sum_le_sum fun x _ => hf x
_ ≤ ‖∑ k ... |
import Mathlib.Algebra.BigOperators.Group.Multiset
import Mathlib.Data.Multiset.Dedup
#align_import data.multiset.bind from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
assert_not_exists MonoidWithZero
assert_not_exists MulAction
universe v
variable {α : Type*} {β : Type v} {γ δ : Ty... | Mathlib/Data/Multiset/Bind.lean | 130 | 130 | theorem singleton_bind : bind {a} f = f a := by | simp [bind]
|
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Interval.Set.IsoIoo
import Mathlib.Topology.Order.MonotoneContinuity
import Mathlib.Topology.UrysohnsBounded
#align_import topology.tietze_extension from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
variable {X Y : T... | Mathlib/Topology/TietzeExtension.lean | 314 | 417 | theorem exists_extension_forall_exists_le_ge_of_closedEmbedding [Nonempty X] (f : X →ᵇ ℝ)
{e : X → Y} (he : ClosedEmbedding e) :
∃ g : Y →ᵇ ℝ, (∀ y, ∃ x₁ x₂, g y ∈ Icc (f x₁) (f x₂)) ∧ g ∘ e = f := by |
inhabit X
-- Put `a = ⨅ x, f x` and `b = ⨆ x, f x`
obtain ⟨a, ha⟩ : ∃ a, IsGLB (range f) a := ⟨_, isGLB_ciInf f.isBounded_range.bddBelow⟩
obtain ⟨b, hb⟩ : ∃ b, IsLUB (range f) b := ⟨_, isLUB_ciSup f.isBounded_range.bddAbove⟩
-- Then `f x ∈ [a, b]` for all `x`
have hmem : ∀ x, f x ∈ Icc a b := fun x => ⟨ha.... |
import Mathlib.Order.Interval.Set.UnorderedInterval
import Mathlib.Algebra.Order.Interval.Set.Monoid
import Mathlib.Data.Set.Pointwise.Basic
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Group.MinMax
#align_import data.set.pointwise.interval from "leanprover-community/mathlib"@"2196ab363eb097c... | Mathlib/Data/Set/Pointwise/Interval.lean | 335 | 336 | theorem preimage_const_sub_Ioc : (fun x => a - x) ⁻¹' Ioc b c = Ico (a - c) (a - b) := by |
simp [← Ioi_inter_Iic, ← Ici_inter_Iio, inter_comm]
|
import Mathlib.Data.List.Count
import Mathlib.Data.List.Dedup
import Mathlib.Data.List.InsertNth
import Mathlib.Data.List.Lattice
import Mathlib.Data.List.Permutation
import Mathlib.Data.Nat.Factorial.Basic
#align_import data.list.perm from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
... | Mathlib/Data/List/Perm.lean | 142 | 146 | theorem perm_comp_perm : (Perm ∘r Perm : List α → List α → Prop) = Perm := by |
funext a c; apply propext
constructor
· exact fun ⟨b, hab, hba⟩ => Perm.trans hab hba
· exact fun h => ⟨a, Perm.refl a, h⟩
|
import Mathlib.Data.Nat.Multiplicity
import Mathlib.Data.ZMod.Algebra
import Mathlib.RingTheory.WittVector.Basic
import Mathlib.RingTheory.WittVector.IsPoly
import Mathlib.FieldTheory.Perfect
#align_import ring_theory.witt_vector.frobenius from "leanprover-community/mathlib"@"0723536a0522d24fc2f159a096fb3304bef77472"... | Mathlib/RingTheory/WittVector/Frobenius.lean | 218 | 220 | theorem coeff_frobeniusFun (x : 𝕎 R) (n : ℕ) :
coeff (frobeniusFun x) n = MvPolynomial.aeval x.coeff (frobeniusPoly p n) := by |
rw [frobeniusFun, coeff_mk]
|
import Mathlib.Analysis.Normed.Field.Basic
import Mathlib.LinearAlgebra.SesquilinearForm
import Mathlib.Topology.Algebra.Module.WeakDual
#align_import analysis.locally_convex.polar from "leanprover-community/mathlib"@"bcfa726826abd57587355b4b5b7e78ad6527b7e4"
variable {𝕜 E F : Type*}
open Topology
namespace Li... | Mathlib/Analysis/LocallyConvex/Polar.lean | 106 | 109 | theorem polar_zero : B.polar ({0} : Set E) = Set.univ := by |
refine Set.eq_univ_iff_forall.mpr fun y x hx => ?_
rw [Set.mem_singleton_iff.mp hx, map_zero, LinearMap.zero_apply, norm_zero]
exact zero_le_one
|
import Mathlib.SetTheory.Cardinal.Finite
#align_import data.set.ncard from "leanprover-community/mathlib"@"74c2af38a828107941029b03839882c5c6f87a04"
namespace Set
variable {α β : Type*} {s t : Set α}
noncomputable def encard (s : Set α) : ℕ∞ := PartENat.withTopEquiv (PartENat.card s)
@[simp] theorem encard_uni... | Mathlib/Data/Set/Card.lean | 184 | 187 | theorem encard_lt_encard_iff_encard_diff_lt_encard_diff (h : (s ∩ t).Finite) :
s.encard < t.encard ↔ (s \ t).encard < (t \ s).encard := by |
rw [← encard_diff_add_encard_inter s t, ← encard_diff_add_encard_inter t s, inter_comm t s,
WithTop.add_lt_add_iff_right h.encard_lt_top.ne]
|
import Mathlib.Algebra.Algebra.Defs
import Mathlib.Algebra.CharP.ExpChar
import Mathlib.FieldTheory.Separable
#align_import field_theory.separable_degree from "leanprover-community/mathlib"@"d11893b411025250c8e61ff2f12ccbd7ee35ab15"
noncomputable section
namespace Polynomial
open scoped Classical
open Polynomial... | Mathlib/RingTheory/Polynomial/SeparableDegree.lean | 96 | 99 | theorem HasSeparableContraction.eq_degree {f : F[X]} (hf : HasSeparableContraction 1 f) :
hf.degree = f.natDegree := by |
let ⟨a, ha⟩ := hf.dvd_degree'
rw [← ha, one_pow a, mul_one]
|
import Mathlib.Order.Antichain
import Mathlib.Order.UpperLower.Basic
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Order.RelIso.Set
#align_import order.minimal from "leanprover-community/mathlib"@"59694bd07f0a39c5beccba34bd9f413a160782bf"
open Function Set
variable {α : Type*} (r r₁ r₂ : α → α → Prop) (s... | Mathlib/Order/Minimal.lean | 117 | 119 | theorem mem_maximals_iff_forall_lt_not_mem' (rlt : α → α → Prop) [IsNonstrictStrictOrder α r rlt] :
x ∈ maximals r s ↔ x ∈ s ∧ ∀ ⦃y⦄, rlt x y → y ∉ s := by |
simp [maximals, right_iff_left_not_left_of r rlt, not_imp_not, imp.swap (a := _ ∈ _)]
|
import Mathlib.SetTheory.Cardinal.Finite
#align_import data.set.ncard from "leanprover-community/mathlib"@"74c2af38a828107941029b03839882c5c6f87a04"
namespace Set
variable {α β : Type*} {s t : Set α}
noncomputable def encard (s : Set α) : ℕ∞ := PartENat.withTopEquiv (PartENat.card s)
@[simp] theorem encard_uni... | Mathlib/Data/Set/Card.lean | 225 | 226 | theorem tsub_encard_le_encard_diff (s t : Set α) : s.encard - t.encard ≤ (s \ t).encard := by |
rw [tsub_le_iff_left, add_comm]; apply encard_le_encard_diff_add_encard
|
import Mathlib.Order.UpperLower.Basic
import Mathlib.Data.Finset.Preimage
#align_import combinatorics.young.young_diagram from "leanprover-community/mathlib"@"59694bd07f0a39c5beccba34bd9f413a160782bf"
open Function
@[ext]
structure YoungDiagram where
cells : Finset (ℕ × ℕ)
isLowerSet : IsLowerSet (cel... | Mathlib/Combinatorics/Young/YoungDiagram.lean | 307 | 310 | theorem mem_iff_lt_rowLen {μ : YoungDiagram} {i j : ℕ} : (i, j) ∈ μ ↔ j < μ.rowLen i := by |
rw [rowLen, Nat.lt_find_iff]
push_neg
exact ⟨fun h _ hmj => μ.up_left_mem (by rfl) hmj h, fun h => h _ (by rfl)⟩
|
import Mathlib.Data.ZMod.Basic
import Mathlib.GroupTheory.Exponent
#align_import group_theory.specific_groups.dihedral from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
inductive DihedralGroup (n : ℕ) : Type
| r : ZMod n → DihedralGroup n
| sr : ZMod n → DihedralGroup n
derivin... | Mathlib/GroupTheory/SpecificGroups/Dihedral.lean | 159 | 164 | theorem orderOf_sr (i : ZMod n) : orderOf (sr i) = 2 := by |
apply orderOf_eq_prime
· rw [sq, sr_mul_self]
· -- Porting note: Previous proof was `decide`
revert n
simp_rw [one_def, ne_eq, forall_const, not_false_eq_true]
|
import Mathlib.Topology.MetricSpace.PseudoMetric
#align_import topology.metric_space.basic from "leanprover-community/mathlib"@"c8f305514e0d47dfaa710f5a52f0d21b588e6328"
open Set Filter Bornology
open scoped NNReal Uniformity
universe u v w
variable {α : Type u} {β : Type v} {X ι : Type*}
variable [PseudoMetricS... | Mathlib/Topology/MetricSpace/Basic.lean | 87 | 88 | theorem dist_pos {x y : γ} : 0 < dist x y ↔ x ≠ y := by |
simpa only [not_le] using not_congr dist_le_zero
|
import Mathlib.Order.Interval.Set.UnorderedInterval
import Mathlib.Algebra.Order.Interval.Set.Monoid
import Mathlib.Data.Set.Pointwise.Basic
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Group.MinMax
#align_import data.set.pointwise.interval from "leanprover-community/mathlib"@"2196ab363eb097c... | Mathlib/Data/Set/Pointwise/Interval.lean | 340 | 341 | theorem preimage_const_sub_Ioo : (fun x => a - x) ⁻¹' Ioo b c = Ioo (a - c) (a - b) := by |
simp [← Ioi_inter_Iio, inter_comm]
|
import Mathlib.Topology.Algebra.Algebra
import Mathlib.Analysis.InnerProductSpace.Basic
#align_import analysis.inner_product_space.of_norm from "leanprover-community/mathlib"@"baa88307f3e699fa7054ef04ec79fa4f056169cb"
open RCLike
open scoped ComplexConjugate
variable {𝕜 : Type*} [RCLike 𝕜] (E : Type*) [Normed... | Mathlib/Analysis/InnerProductSpace/OfNorm.lean | 305 | 309 | theorem innerProp (r : 𝕜) : innerProp' E r := by |
intro x y
rw [← re_add_im r, add_smul, add_left, real_prop _ x, ← smul_smul, real_prop _ _ y, I_prop,
map_add, map_mul, conj_ofReal, conj_ofReal, conj_I]
ring
|
import Mathlib.CategoryTheory.Sites.Spaces
import Mathlib.Topology.Sheaves.Sheaf
import Mathlib.CategoryTheory.Sites.DenseSubsite
#align_import topology.sheaves.sheaf_condition.sites from "leanprover-community/mathlib"@"d39590fc8728fbf6743249802486f8c91ffe07bc"
noncomputable section
set_option linter.uppercaseLe... | Mathlib/Topology/Sheaves/SheafCondition/Sites.lean | 103 | 107 | theorem mem_grothendieckTopology :
Sieve.generate (presieveOfCovering U) ∈ Opens.grothendieckTopology X (iSup U) := by |
intro x hx
obtain ⟨i, hxi⟩ := Opens.mem_iSup.mp hx
exact ⟨U i, Opens.leSupr U i, ⟨U i, 𝟙 _, Opens.leSupr U i, ⟨i, rfl⟩, Category.id_comp _⟩, hxi⟩
|
import Mathlib.LinearAlgebra.Matrix.Basis
import Mathlib.LinearAlgebra.Matrix.Nondegenerate
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.LinearAlgebra.Matrix.ToLinearEquiv
import Mathlib.LinearAlgebra.BilinearForm.Properties
import Mathlib.LinearAlgebra.Matrix.SesquilinearForm
#align_import l... | Mathlib/LinearAlgebra/Matrix/BilinearForm.lean | 88 | 93 | theorem toBilin'Aux_toMatrixAux [DecidableEq n] (B₂ : BilinForm R₂ (n → R₂)) :
-- Porting note: had to hint the base ring even though it should be clear from context...
Matrix.toBilin'Aux (BilinForm.toMatrixAux (R₂ := R₂)
(fun j => stdBasis R₂ (fun _ => R₂) j 1) B₂) = B₂ := by |
rw [BilinForm.toMatrixAux, Matrix.toBilin'Aux,
toLinearMap₂'Aux_toMatrix₂Aux]
|
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.RingTheory.Localization.FractionRing
#align_import data.polynomial.ring_division from "leanprover-community/mathlib"@"8efcf8022aac8e01df8d302dcebdbc25d6a886c8"
noncomputable section
namespace Polynomial
universe u v w z
variable {R : Type u} {S : Ty... | Mathlib/Algebra/Polynomial/Roots.lean | 55 | 61 | theorem roots_def [DecidableEq R] (p : R[X]) [Decidable (p = 0)] :
p.roots = if h : p = 0 then ∅ else Classical.choose (exists_multiset_roots h) := by |
-- porting noteL `‹_›` doesn't work for instance arguments
rename_i iR ip0
obtain rfl := Subsingleton.elim iR (Classical.decEq R)
obtain rfl := Subsingleton.elim ip0 (Classical.dec (p = 0))
rfl
|
import Mathlib.Algebra.Order.Monoid.Unbundled.Basic
#align_import algebra.order.monoid.min_max from "leanprover-community/mathlib"@"de87d5053a9fe5cbde723172c0fb7e27e7436473"
open Function
variable {α β : Type*}
section CovariantClassMulLe
variable [LinearOrder α]
section Mul
variable [Mul α]
@[to_additive... | Mathlib/Algebra/Order/Monoid/Unbundled/MinMax.lean | 117 | 121 | theorem le_or_le_of_mul_le_mul [CovariantClass α α (· * ·) (· < ·)]
[CovariantClass α α (Function.swap (· * ·)) (· < ·)] {a₁ a₂ b₁ b₂ : α} :
a₁ * b₁ ≤ a₂ * b₂ → a₁ ≤ a₂ ∨ b₁ ≤ b₂ := by |
contrapose!
exact fun h => mul_lt_mul_of_lt_of_lt h.1 h.2
|
import Mathlib.SetTheory.Ordinal.Basic
import Mathlib.Data.Nat.SuccPred
#align_import set_theory.ordinal.arithmetic from "leanprover-community/mathlib"@"31b269b60935483943542d547a6dd83a66b37dc7"
assert_not_exists Field
assert_not_exists Module
noncomputable section
open Function Cardinal Set Equiv Order
open sc... | Mathlib/SetTheory/Ordinal/Arithmetic.lean | 1,464 | 1,467 | theorem bsup_eq_bsup {ι : Type u} (r r' : ι → ι → Prop) [IsWellOrder ι r] [IsWellOrder ι r']
(f : ι → Ordinal.{max u v}) :
bsup.{_, v} _ (bfamilyOfFamily' r f) = bsup.{_, v} _ (bfamilyOfFamily' r' f) := by |
rw [bsup_eq_sup', bsup_eq_sup']
|
import Mathlib.Analysis.Complex.Basic
import Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle
#align_import analysis.complex.re_im_topology from "leanprover-community/mathlib"@"468b141b14016d54b479eb7a0fff1e360b7e3cf6"
open Set
noncomputable section
namespace Complex
theorem isHomeomorphicTrivialFiber... | Mathlib/Analysis/Complex/ReImTopology.lean | 144 | 145 | theorem frontier_setOf_le_re (a : ℝ) : frontier { z : ℂ | a ≤ z.re } = { z | z.re = a } := by |
simpa only [frontier_Ici] using frontier_preimage_re (Ici a)
|
import Mathlib.MeasureTheory.Measure.Typeclasses
import Mathlib.Analysis.Complex.Basic
#align_import measure_theory.measure.vector_measure from "leanprover-community/mathlib"@"70a4f2197832bceab57d7f41379b2592d1110570"
noncomputable section
open scoped Classical
open NNReal ENNReal MeasureTheory
namespace Measur... | Mathlib/MeasureTheory/Measure/VectorMeasure.lean | 234 | 238 | theorem of_nonneg_disjoint_union_eq_zero {s : SignedMeasure α} {A B : Set α} (h : Disjoint A B)
(hA₁ : MeasurableSet A) (hB₁ : MeasurableSet B) (hA₂ : 0 ≤ s A) (hB₂ : 0 ≤ s B)
(hAB : s (A ∪ B) = 0) : s A = 0 := by |
rw [of_union h hA₁ hB₁] at hAB
linarith
|
import Mathlib.MeasureTheory.OuterMeasure.Caratheodory
#align_import measure_theory.measure.outer_measure from "leanprover-community/mathlib"@"343e80208d29d2d15f8050b929aa50fe4ce71b55"
noncomputable section
open Set Function Filter
open scoped Classical NNReal Topology ENNReal
namespace MeasureTheory
open Outer... | Mathlib/MeasureTheory/OuterMeasure/Induced.lean | 49 | 49 | theorem extend_eq {s : α} (h : P s) : extend m s = m s h := by | simp [extend, h]
|
import Mathlib.Data.Stream.Init
import Mathlib.Tactic.ApplyFun
import Mathlib.Control.Fix
import Mathlib.Order.OmegaCompletePartialOrder
#align_import control.lawful_fix from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7"
universe u v
open scoped Classical
variable {α : Type*} {β : α →... | Mathlib/Control/LawfulFix.lean | 57 | 60 | theorem approx_mono' {i : ℕ} : Fix.approx f i ≤ Fix.approx f (succ i) := by |
induction i with
| zero => dsimp [approx]; apply @bot_le _ _ _ (f ⊥)
| succ _ i_ih => intro; apply f.monotone; apply i_ih
|
import Mathlib.Algebra.ModEq
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Order.Archimedean
import Mathlib.Algebra.Periodic
import Mathlib.Data.Int.SuccPred
import Mathlib.GroupTheory.QuotientGroup
import Mathlib.Order.Circular
import Mathlib.Data.List.TFAE
import Mathlib.Data.Set.Lattice
#align_import a... | Mathlib/Algebra/Order/ToIntervalMod.lean | 377 | 379 | theorem toIocDiv_sub_eq_toIocDiv_add' (a b c : α) :
toIocDiv hp (a - c) b = toIocDiv hp a (b + c) := by |
rw [← sub_neg_eq_add, toIocDiv_sub_eq_toIocDiv_add, sub_eq_add_neg]
|
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Nat.ModEq
import Mathlib.Data.Set.Finite
#align_import combinatorics.pigeonhole from "leanprover-community/mathlib"@"d6fad0e5bf2d6f48da9175d25c3dc5706b3834ce"
un... | Mathlib/Combinatorics/Pigeonhole.lean | 228 | 231 | theorem exists_lt_card_fiber_of_nsmul_lt_card_of_maps_to (hf : ∀ a ∈ s, f a ∈ t)
(ht : t.card • b < s.card) : ∃ y ∈ t, b < (s.filter fun x => f x = y).card := by |
simp_rw [cast_card] at ht ⊢
exact exists_lt_sum_fiber_of_maps_to_of_nsmul_lt_sum hf ht
|
import Mathlib.Init.Core
import Mathlib.LinearAlgebra.AffineSpace.Basis
import Mathlib.LinearAlgebra.FiniteDimensional
#align_import linear_algebra.affine_space.finite_dimensional from "leanprover-community/mathlib"@"67e606eaea14c7854bdc556bd53d98aefdf76ec0"
noncomputable section
open Affine
section AffineSpace... | Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean | 517 | 520 | theorem collinear_iff_not_affineIndependent {p : Fin 3 → P} :
Collinear k (Set.range p) ↔ ¬AffineIndependent k p := by |
rw [collinear_iff_finrank_le_one,
finrank_vectorSpan_le_iff_not_affineIndependent k p (Fintype.card_fin 3)]
|
import Mathlib.Algebra.Module.DedekindDomain
import Mathlib.LinearAlgebra.FreeModule.PID
import Mathlib.Algebra.Module.Projective
import Mathlib.Algebra.Category.ModuleCat.Biproducts
import Mathlib.RingTheory.SimpleModule
#align_import algebra.module.pid from "leanprover-community/mathlib"@"cdc34484a07418af43daf8198b... | Mathlib/Algebra/Module/PID.lean | 89 | 98 | theorem Submodule.exists_isInternal_prime_power_torsion_of_pid [Module.Finite R M]
(hM : Module.IsTorsion R M) :
∃ (ι : Type u) (_ : Fintype ι) (_ : DecidableEq ι) (p : ι → R) (_ : ∀ i, Irreducible <| p i)
(e : ι → ℕ), DirectSum.IsInternal fun i => torsionBy R M <| p i ^ e i := by |
refine ⟨_, ?_, _, _, ?_, _, Submodule.isInternal_prime_power_torsion_of_pid hM⟩
· exact Finset.fintypeCoeSort _
· rintro ⟨p, hp⟩
have hP := prime_of_factor p (Multiset.mem_toFinset.mp hp)
haveI := Ideal.isPrime_of_prime hP
exact (IsPrincipal.prime_generator_of_isPrime p hP.ne_zero).irreducible
|
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.Module.Defs
import Mathlib.Tactic.Abel
namespace Finset
variable {R M : Type*} [Ring R] [AddCommGroup M] [Module R M] (f : ℕ → R) (g : ℕ → M) {m n : ℕ}
-- The partial sum of `g`, starting from zero
local notation "G " n:80 => ∑ i ∈ range n, g i
... | Mathlib/Algebra/BigOperators/Module.lean | 63 | 69 | theorem sum_range_by_parts :
∑ i ∈ range n, f i • g i =
f (n - 1) • G n - ∑ i ∈ range (n - 1), (f (i + 1) - f i) • G (i + 1) := by |
by_cases hn : n = 0
· simp [hn]
· rw [range_eq_Ico, sum_Ico_by_parts f g (Nat.pos_of_ne_zero hn), sum_range_zero, smul_zero,
sub_zero, range_eq_Ico]
|
import Mathlib.Algebra.MvPolynomial.Rename
#align_import data.mv_polynomial.comap from "leanprover-community/mathlib"@"aba31c938d3243cc671be7091b28a1e0814647ee"
namespace MvPolynomial
variable {σ : Type*} {τ : Type*} {υ : Type*} {R : Type*} [CommSemiring R]
noncomputable def comap (f : MvPolynomial σ R →ₐ[R] M... | Mathlib/Algebra/MvPolynomial/Comap.lean | 90 | 92 | theorem comap_rename (f : σ → τ) (x : τ → R) : comap (rename f) x = x ∘ f := by |
funext
simp [rename_X, comap_apply, aeval_X]
|
import Mathlib.Algebra.Ring.Equiv
#align_import algebra.ring.comp_typeclasses from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
variable {R₁ : Type*} {R₂ : Type*} {R₃ : Type*}
variable [Semiring R₁] [Semiring R₂] [Semiring R₃]
-- This at first seems not very useful. However we need ... | Mathlib/Algebra/Ring/CompTypeclasses.lean | 100 | 102 | theorem comp_apply_eq {x : R₁} : σ' (σ x) = x := by |
rw [← RingHom.comp_apply, comp_eq]
simp
|
import Mathlib.Order.Filter.Prod
#align_import order.filter.n_ary from "leanprover-community/mathlib"@"78f647f8517f021d839a7553d5dc97e79b508dea"
open Function Set
open Filter
namespace Filter
variable {α α' β β' γ γ' δ δ' ε ε' : Type*} {m : α → β → γ} {f f₁ f₂ : Filter α}
{g g₁ g₂ : Filter β} {h h₁ h₂ : Filt... | Mathlib/Order/Filter/NAry.lean | 137 | 138 | theorem map₂_pure_left : map₂ m (pure a) g = g.map (m a) := by |
rw [← map_prod_eq_map₂, pure_prod, map_map]; rfl
|
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Reverse
import Mathlib.Algebra.Polynomial.Inductions
import Mathlib.RingTheory.Localization.Basic
#align_import data.polynomial.laurent from "leanprover-community/mathlib"@"831c494092374cfe9f50591ed0ac81a25efc5b86"
open Polynomial Func... | Mathlib/Algebra/Polynomial/Laurent.lean | 498 | 504 | theorem degree_eq_bot_iff {f : R[T;T⁻¹]} : f.degree = ⊥ ↔ f = 0 := by |
refine ⟨fun h => ?_, fun h => by rw [h, degree_zero]⟩
rw [degree, Finset.max_eq_sup_withBot] at h
ext n
refine not_not.mp fun f0 => ?_
simp_rw [Finset.sup_eq_bot_iff, Finsupp.mem_support_iff, Ne, WithBot.coe_ne_bot] at h
exact h n f0
|
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Algebra.Order.Interval.Set.Group
import Mathlib.Analysis.Convex.Segment
import Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional
import Mathlib.Tactic.FieldSimp
#align_import analysis.convex.between from "leanprover-community/mathlib"@"571e13cacbed7bf042fd3058c... | Mathlib/Analysis/Convex/Between.lean | 257 | 265 | theorem Sbtw.mem_image_Ioo {x y z : P} (h : Sbtw R x y z) :
y ∈ lineMap x z '' Set.Ioo (0 : R) 1 := by |
rcases h with ⟨⟨t, ht, rfl⟩, hyx, hyz⟩
rcases Set.eq_endpoints_or_mem_Ioo_of_mem_Icc ht with (rfl | rfl | ho)
· exfalso
exact hyx (lineMap_apply_zero _ _)
· exfalso
exact hyz (lineMap_apply_one _ _)
· exact ⟨t, ho, rfl⟩
|
import Mathlib.Tactic.Ring.Basic
import Mathlib.Tactic.TryThis
import Mathlib.Tactic.Conv
import Mathlib.Util.Qq
set_option autoImplicit true
-- In this file we would like to be able to use multi-character auto-implicits.
set_option relaxedAutoImplicit true
namespace Mathlib.Tactic
open Lean hiding Rat
open Qq Me... | Mathlib/Tactic/Ring/RingNF.lean | 126 | 127 | theorem rat_rawCast_neg {R} [DivisionRing R] :
(Rat.rawCast (.negOfNat n) d : R) = Int.rawCast (.negOfNat n) / Nat.rawCast d := by | simp
|
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Data.Finset.NatAntidiagonal
import Mathlib.Data.Nat.GCD.Basic
import Mathlib.Init.Data.Nat.Lemmas
import Mathlib.Logic.Function.Iterate
import Mathlib.Tactic.Ring
import Mathlib.Tactic.Zify
#align_import data.nat.fib from "leanprover-community/mathlib"@"... | Mathlib/Data/Nat/Fib/Basic.lean | 200 | 201 | theorem fib_bit0 (n : ℕ) : fib (bit0 n) = fib n * (2 * fib (n + 1) - fib n) := by |
rw [bit0_eq_two_mul, fib_two_mul]
|
import Mathlib.Data.List.Basic
#align_import data.list.join from "leanprover-community/mathlib"@"18a5306c091183ac90884daa9373fa3b178e8607"
-- Make sure we don't import algebra
assert_not_exists Monoid
variable {α β : Type*}
namespace List
attribute [simp] join
-- Porting note (#10618): simp can prove this
-- @... | Mathlib/Data/List/Join.lean | 196 | 200 | theorem reverse_join (L : List (List α)) :
L.join.reverse = (L.map reverse).reverse.join := by |
induction' L with _ _ ih
· rfl
· rw [join, reverse_append, ih, map_cons, reverse_cons', join_concat]
|
import Mathlib.Order.Filter.Basic
import Mathlib.Order.Filter.CountableInter
import Mathlib.SetTheory.Cardinal.Ordinal
import Mathlib.SetTheory.Cardinal.Cofinality
open Set Filter Cardinal
universe u
variable {ι : Type u} {α β : Type u} {c : Cardinal.{u}}
class CardinalInterFilter (l : Filter α) (c : Cardinal.{... | Mathlib/Order/Filter/CardinalInter.lean | 145 | 149 | theorem EventuallyLE.cardinal_bInter {S : Set ι} (hS : #S < c)
{s t : ∀ i ∈ S, Set α} (h : ∀ i hi, s i hi ≤ᶠ[l] t i hi) :
⋂ i ∈ S, s i ‹_› ≤ᶠ[l] ⋂ i ∈ S, t i ‹_› := by |
simp only [biInter_eq_iInter]
exact EventuallyLE.cardinal_iInter hS fun i => h i i.2
|
import Mathlib.SetTheory.Ordinal.Basic
import Mathlib.Data.Nat.SuccPred
#align_import set_theory.ordinal.arithmetic from "leanprover-community/mathlib"@"31b269b60935483943542d547a6dd83a66b37dc7"
assert_not_exists Field
assert_not_exists Module
noncomputable section
open Function Cardinal Set Equiv Order
open sc... | Mathlib/SetTheory/Ordinal/Arithmetic.lean | 1,815 | 1,817 | theorem lt_blsub_iff {o : Ordinal.{u}} {f : ∀ b < o, Ordinal.{max u v}} {a} :
a < blsub.{_, v} o f ↔ ∃ i hi, a ≤ f i hi := by |
simpa only [not_forall, not_lt, not_le] using not_congr (@blsub_le_iff.{_, v} _ f a)
|
import Mathlib.Data.Set.Subsingleton
import Mathlib.Logic.Equiv.Defs
import Mathlib.Algebra.Group.Defs
#align_import data.part from "leanprover-community/mathlib"@"80c43012d26f63026d362c3aba28f3c3bafb07e6"
open Function
structure Part.{u} (α : Type u) : Type u where
Dom : Prop
get : Dom → α
#align part... | Mathlib/Data/Part.lean | 592 | 593 | theorem bind_some_right (x : Part α) : x.bind some = x := by |
erw [bind_some_eq_map]; simp [map_id']
|
import Mathlib.Algebra.ContinuedFractions.Translations
#align_import algebra.continued_fractions.continuants_recurrence from "leanprover-community/mathlib"@"5f11361a98ae4acd77f5c1837686f6f0102cdc25"
namespace GeneralizedContinuedFraction
variable {K : Type*} {g : GeneralizedContinuedFraction K} {n : ℕ} [Division... | Mathlib/Algebra/ContinuedFractions/ContinuantsRecurrence.lean | 42 | 46 | theorem continuants_recurrence {gp ppred pred : Pair K} (succ_nth_s_eq : g.s.get? (n + 1) = some gp)
(nth_conts_eq : g.continuants n = ppred) (succ_nth_conts_eq : g.continuants (n + 1) = pred) :
g.continuants (n + 2) = ⟨gp.b * pred.a + gp.a * ppred.a, gp.b * pred.b + gp.a * ppred.b⟩ := by |
rw [nth_cont_eq_succ_nth_cont_aux] at nth_conts_eq succ_nth_conts_eq
exact continuants_recurrenceAux succ_nth_s_eq nth_conts_eq succ_nth_conts_eq
|
import Mathlib.Algebra.Group.Subgroup.Finite
import Mathlib.Data.Finset.Fin
import Mathlib.Data.Finset.Sort
import Mathlib.Data.Int.Order.Units
import Mathlib.GroupTheory.Perm.Support
import Mathlib.Logic.Equiv.Fin
import Mathlib.Tactic.NormNum.Ineq
#align_import group_theory.perm.sign from "leanprover-community/math... | Mathlib/GroupTheory/Perm/Sign.lean | 575 | 578 | theorem sign_prodCongrLeft (σ : α → Perm β) : sign (prodCongrLeft σ) = ∏ k, sign (σ k) := by |
refine (sign_eq_sign_of_equiv _ _ (prodComm β α) ?_).trans (sign_prodCongrRight σ)
rintro ⟨b, α⟩
rfl
|
import Mathlib.Algebra.IsPrimePow
import Mathlib.NumberTheory.ArithmeticFunction
import Mathlib.Analysis.SpecialFunctions.Log.Basic
#align_import number_theory.von_mangoldt from "leanprover-community/mathlib"@"c946d6097a6925ad16d7ec55677bbc977f9846de"
namespace ArithmeticFunction
open Finset Nat
open scoped Arit... | Mathlib/NumberTheory/VonMangoldt.lean | 111 | 122 | theorem vonMangoldt_sum {n : ℕ} : ∑ i ∈ n.divisors, Λ i = Real.log n := by |
refine recOnPrimeCoprime ?_ ?_ ?_ n
· simp
· intro p k hp
rw [sum_divisors_prime_pow hp, cast_pow, Real.log_pow, Finset.sum_range_succ', Nat.pow_zero,
vonMangoldt_apply_one]
simp [vonMangoldt_apply_pow (Nat.succ_ne_zero _), vonMangoldt_apply_prime hp]
intro a b ha' hb' hab ha hb
simp only [vonM... |
import Mathlib.Probability.Kernel.MeasurableIntegral
#align_import probability.kernel.composition from "leanprover-community/mathlib"@"3b92d54a05ee592aa2c6181a4e76b1bb7cc45d0b"
open MeasureTheory
open scoped ENNReal
namespace ProbabilityTheory
namespace kernel
variable {α β ι : Type*} {mα : MeasurableSpace α}... | Mathlib/Probability/Kernel/Composition.lean | 710 | 715 | theorem sum_comap_seq (κ : kernel α β) [IsSFiniteKernel κ] (hg : Measurable g) :
(kernel.sum fun n => comap (seq κ n) g hg) = comap κ g hg := by |
ext a s hs
rw [kernel.sum_apply, comap_apply' κ hg a s, Measure.sum_apply _ hs, ← measure_sum_seq κ,
Measure.sum_apply _ hs]
simp_rw [comap_apply' _ hg _ s]
|
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Analysis.Convex.Hull
import Mathlib.LinearAlgebra.AffineSpace.Basis
#align_import analysis.convex.combination from "leanprover-community/mathlib"@"92bd7b1ffeb306a89f450bee126ddd8a284c259d"
open Set Function
open scoped Classical
open Pointwise
... | Mathlib/Analysis/Convex/Combination.lean | 128 | 133 | theorem Finset.centerMass_subset {t' : Finset ι} (ht : t ⊆ t') (h : ∀ i ∈ t', i ∉ t → w i = 0) :
t.centerMass w z = t'.centerMass w z := by |
rw [centerMass, sum_subset ht h, smul_sum, centerMass, smul_sum]
apply sum_subset ht
intro i hit' hit
rw [h i hit' hit, zero_smul, smul_zero]
|
import Mathlib.MeasureTheory.Function.LpSeminorm.Basic
#align_import measure_theory.function.lp_seminorm from "leanprover-community/mathlib"@"c4015acc0a223449d44061e27ddac1835a3852b9"
namespace MeasureTheory
open Filter
open scoped ENNReal
variable {α E : Type*} {m m0 : MeasurableSpace α} {p : ℝ≥0∞} {q : ℝ} {μ :... | Mathlib/MeasureTheory/Function/LpSeminorm/Trim.lean | 59 | 65 | theorem snorm_trim (hm : m ≤ m0) {f : α → E} (hf : StronglyMeasurable[m] f) :
snorm f p (μ.trim hm) = snorm f p μ := by |
by_cases h0 : p = 0
· simp [h0]
by_cases h_top : p = ∞
· simpa only [h_top, snorm_exponent_top] using snormEssSup_trim hm hf
simpa only [snorm_eq_snorm' h0 h_top] using snorm'_trim hm hf
|
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed
import Mathlib.RingTheory.PowerBasis
#align_import ring_theory.is_adjoin_root from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c"
open scoped Polynomial
open Polynomial
noncomputable sec... | Mathlib/RingTheory/IsAdjoinRoot.lean | 158 | 158 | theorem aeval_root (h : IsAdjoinRoot S f) : aeval h.root f = 0 := by | rw [aeval_eq, map_self]
|
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib.CategoryTheory.Bicategory.LocallyDiscrete
#align_import category_theory.bicategory.coherence from "leanprover-community/mathlib"@"f187f1074fa1857c94589cc653c786ca... | Mathlib/CategoryTheory/Bicategory/Coherence.lean | 148 | 157 | theorem normalizeAux_congr {a b c : B} (p : Path a b) {f g : Hom b c} (η : f ⟶ g) :
normalizeAux p f = normalizeAux p g := by |
rcases η with ⟨η'⟩
apply @congr_fun _ _ fun p => normalizeAux p f
clear p η
induction η' with
| vcomp _ _ _ _ => apply Eq.trans <;> assumption
| whisker_left _ _ ih => funext; apply congr_fun ih
| whisker_right _ _ ih => funext; apply congr_arg₂ _ (congr_fun ih _) rfl
| _ => funext; rfl
|
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Data.Finset.Sym
import Mathlib.Data.Nat.Choose.Cast
import Mathlib.Data.Nat.Choose.Multinomial
#align_import analysis.calculus.cont_diff from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
noncomputable section
open scoped C... | Mathlib/Analysis/Calculus/ContDiff/Bounds.lean | 360 | 468 | theorem norm_iteratedFDerivWithin_comp_le_aux {Fu Gu : Type u} [NormedAddCommGroup Fu]
[NormedSpace 𝕜 Fu] [NormedAddCommGroup Gu] [NormedSpace 𝕜 Gu] {g : Fu → Gu} {f : E → Fu} {n : ℕ}
{s : Set E} {t : Set Fu} {x : E} (hg : ContDiffOn 𝕜 n g t) (hf : ContDiffOn 𝕜 n f s)
(ht : UniqueDiffOn 𝕜 t) (hs : Uniq... |
/- We argue by induction on `n`, using that `D^(n+1) (g ∘ f) = D^n (g ' ∘ f ⬝ f')`. The successive
derivatives of `g' ∘ f` are controlled thanks to the inductive assumption, and those of `f'` are
controlled by assumption.
As composition of linear maps is a bilinear map, one may use
`ContinuousLinearM... |
import Mathlib.Data.Fin.VecNotation
import Mathlib.SetTheory.Cardinal.Basic
#align_import model_theory.basic from "leanprover-community/mathlib"@"369525b73f229ccd76a6ec0e0e0bf2be57599768"
set_option autoImplicit true
universe u v u' v' w w'
open Cardinal
open Cardinal
namespace FirstOrder
-- intended to b... | Mathlib/ModelTheory/Basic.lean | 267 | 271 | theorem card_sum :
(L.sum L').card = Cardinal.lift.{max u' v'} L.card + Cardinal.lift.{max u v} L'.card := by |
simp only [card_eq_card_functions_add_card_relations, card_functions_sum, card_relations_sum,
sum_add_distrib', lift_add, lift_sum, lift_lift]
simp only [add_assoc, add_comm (Cardinal.sum fun i => (#(L'.Functions i)).lift)]
|
import Mathlib.Geometry.Manifold.ContMDiff.Product
import Mathlib.Analysis.NormedSpace.OperatorNorm.Prod
open Set ChartedSpace SmoothManifoldWithCorners
open scoped Topology Manifold
variable {𝕜 : Type*} [NontriviallyNormedField 𝕜]
-- declare a smooth manifold `M` over the pair `(E, H)`.
{E : Type*}
[Norme... | Mathlib/Geometry/Manifold/ContMDiff/NormedSpace.lean | 81 | 82 | theorem contMDiff_iff_contDiff {f : E → E'} : ContMDiff 𝓘(𝕜, E) 𝓘(𝕜, E') n f ↔ ContDiff 𝕜 n f := by |
rw [← contDiffOn_univ, ← contMDiffOn_univ, contMDiffOn_iff_contDiffOn]
|
import Mathlib.Algebra.Order.Ring.Rat
import Mathlib.Tactic.NormNum.Inv
import Mathlib.Tactic.NormNum.Pow
import Mathlib.Util.AtomM
set_option autoImplicit true
namespace Mathlib.Tactic
namespace Ring
open Mathlib.Meta Qq NormNum Lean.Meta AtomM
open Lean (MetaM Expr mkRawNatLit)
def instCommSemiringNat : CommSe... | Mathlib/Tactic/Ring/Basic.lean | 663 | 663 | theorem pow_one (a : R) : a ^ nat_lit 1 = a := by | simp
|
import Mathlib.Algebra.CharZero.Lemmas
import Mathlib.Order.Interval.Finset.Basic
#align_import data.int.interval from "leanprover-community/mathlib"@"1d29de43a5ba4662dd33b5cfeecfc2a27a5a8a29"
open Finset Int
namespace Int
instance instLocallyFiniteOrder : LocallyFiniteOrder ℤ where
finsetIcc a b :=
(Fins... | Mathlib/Data/Int/Interval.lean | 181 | 182 | theorem card_fintype_Ioc_of_le (h : a ≤ b) : (Fintype.card (Set.Ioc a b) : ℤ) = b - a := by |
rw [card_fintype_Ioc, toNat_sub_of_le h]
|
import Mathlib.FieldTheory.Adjoin
open Polynomial
namespace IntermediateField
variable (F E K : Type*) [Field F] [Field E] [Field K] [Algebra F E] [Algebra F K] {S : Set E}
structure Lifts where
carrier : IntermediateField F E
emb : carrier →ₐ[F] K
#align intermediate_field.lifts IntermediateField.Lif... | Mathlib/FieldTheory/Extension.lean | 175 | 182 | theorem exists_algHom_adjoin_of_splits_of_aeval : ∃ φ : adjoin F S →ₐ[F] K, φ ⟨x, hx⟩ = y := by |
have := isAlgebraic_adjoin (fun s hs ↦ (hK s hs).1)
have ix : IsAlgebraic F _ := Algebra.IsAlgebraic.isAlgebraic (⟨x, hx⟩ : adjoin F S)
rw [isAlgebraic_iff_isIntegral, isIntegral_iff] at ix
obtain ⟨φ, hφ⟩ := exists_algHom_adjoin_of_splits hK ((algHomAdjoinIntegralEquiv F ix).symm
⟨y, mem_aroots.mpr ⟨minpol... |
import Mathlib.Data.List.Defs
import Mathlib.Data.Option.Basic
import Mathlib.Data.Nat.Defs
import Mathlib.Init.Data.List.Basic
import Mathlib.Util.AssertExists
-- Make sure we haven't imported `Data.Nat.Order.Basic`
assert_not_exists OrderedSub
namespace List
universe u v
variable {α : Type u} {β : Type v} (l :... | Mathlib/Data/List/GetD.lean | 38 | 44 | theorem getD_eq_get {n : ℕ} (hn : n < l.length) : l.getD n d = l.get ⟨n, hn⟩ := by |
induction l generalizing n with
| nil => simp at hn
| cons head tail ih =>
cases n
· exact getD_cons_zero
· exact ih _
|
import Mathlib.Analysis.Calculus.TangentCone
import Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics
#align_import analysis.calculus.fderiv.basic from "leanprover-community/mathlib"@"41bef4ae1254365bc190aee63b947674d2977f01"
open Filter Asymptotics ContinuousLinearMap Set Metric
open scoped Classical
open To... | Mathlib/Analysis/Calculus/FDeriv/Basic.lean | 336 | 346 | theorem HasFDerivAt.le_of_lip' {f : E → F} {f' : E →L[𝕜] F} {x₀ : E} (hf : HasFDerivAt f f' x₀)
{C : ℝ} (hC₀ : 0 ≤ C) (hlip : ∀ᶠ x in 𝓝 x₀, ‖f x - f x₀‖ ≤ C * ‖x - x₀‖) : ‖f'‖ ≤ C := by |
refine le_of_forall_pos_le_add fun ε ε0 => opNorm_le_of_nhds_zero ?_ ?_
· exact add_nonneg hC₀ ε0.le
rw [← map_add_left_nhds_zero x₀, eventually_map] at hlip
filter_upwards [isLittleO_iff.1 (hasFDerivAt_iff_isLittleO_nhds_zero.1 hf) ε0, hlip] with y hy hyC
rw [add_sub_cancel_left] at hyC
calc
‖f' y‖ ≤ ... |
import Mathlib.CategoryTheory.Limits.IsLimit
import Mathlib.CategoryTheory.Category.ULift
import Mathlib.CategoryTheory.EssentiallySmall
import Mathlib.Logic.Equiv.Basic
#align_import category_theory.limits.has_limits from "leanprover-community/mathlib"@"2738d2ca56cbc63be80c3bd48e9ed90ad94e947d"
noncomputable sec... | Mathlib/CategoryTheory/Limits/HasLimits.lean | 374 | 380 | theorem HasLimit.isoOfEquivalence_hom_π {F : J ⥤ C} [HasLimit F] {G : K ⥤ C} [HasLimit G]
(e : J ≌ K) (w : e.functor ⋙ G ≅ F) (k : K) :
(HasLimit.isoOfEquivalence e w).hom ≫ limit.π G k =
limit.π F (e.inverse.obj k) ≫ w.inv.app (e.inverse.obj k) ≫ G.map (e.counit.app k) := by |
simp only [HasLimit.isoOfEquivalence, IsLimit.conePointsIsoOfEquivalence_hom]
dsimp
simp
|
import Mathlib.CategoryTheory.GlueData
import Mathlib.Topology.Category.TopCat.Limits.Pullbacks
import Mathlib.Topology.Category.TopCat.Opens
import Mathlib.Tactic.Generalize
import Mathlib.CategoryTheory.Elementwise
#align_import topology.gluing from "leanprover-community/mathlib"@"178a32653e369dce2da68dc6b2694e385d... | Mathlib/Topology/Gluing.lean | 533 | 544 | theorem range_fromOpenSubsetsGlue : Set.range (fromOpenSubsetsGlue U) = ⋃ i, (U i : Set α) := by |
ext
constructor
· rintro ⟨x, rfl⟩
obtain ⟨i, ⟨x, hx'⟩, rfl⟩ := (ofOpenSubsets U).ι_jointly_surjective x
-- Porting note: another `rw ↦ erw`
-- See above
erw [ι_fromOpenSubsetsGlue_apply]
exact Set.subset_iUnion _ i hx'
· rintro ⟨_, ⟨i, rfl⟩, hx⟩
rename_i x
exact ⟨(ofOpenSubsets U).t... |
import Mathlib.Analysis.Normed.Group.SemiNormedGroupCat
import Mathlib.Analysis.Normed.Group.Quotient
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
#align_import analysis.normed.group.SemiNormedGroup.kernels from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
open CategoryTheory C... | Mathlib/Analysis/Normed/Group/SemiNormedGroupCat/Kernels.lean | 273 | 280 | theorem explicitCokernelDesc_unique {X Y Z : SemiNormedGroupCat.{u}} {f : X ⟶ Y} {g : Y ⟶ Z}
(w : f ≫ g = 0) (e : explicitCokernel f ⟶ Z) (he : explicitCokernelπ f ≫ e = g) :
e = explicitCokernelDesc w := by |
apply (isColimitCokernelCocone f).uniq (Cofork.ofπ g (by simp [w]))
rintro (_ | _)
· convert w.symm
simp
· exact he
|
import Mathlib.Algebra.BigOperators.WithTop
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Data.ENNReal.Basic
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open Set NNReal ENNReal
namespace ENNReal
variable {a b c d : ℝ≥0∞} {r p q... | Mathlib/Data/ENNReal/Operations.lean | 246 | 252 | theorem mul_lt_top_iff {a b : ℝ≥0∞} : a * b < ∞ ↔ a < ∞ ∧ b < ∞ ∨ a = 0 ∨ b = 0 := by |
constructor
· intro h
rw [← or_assoc, or_iff_not_imp_right, or_iff_not_imp_right]
intro hb ha
exact ⟨lt_top_of_mul_ne_top_left h.ne hb, lt_top_of_mul_ne_top_right h.ne ha⟩
· rintro (⟨ha, hb⟩ | rfl | rfl) <;> [exact mul_lt_top ha.ne hb.ne; simp; simp]
|
import Mathlib.RingTheory.Localization.LocalizationLocalization
import Mathlib.RingTheory.Localization.Submodule
import Mathlib.RingTheory.DiscreteValuationRing.TFAE
#align_import ring_theory.dedekind_domain.dvr from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
variable (R A K : Type*... | Mathlib/RingTheory/DedekindDomain/Dvr.lean | 134 | 144 | theorem IsLocalization.AtPrime.discreteValuationRing_of_dedekind_domain [IsDedekindDomain A]
{P : Ideal A} (hP : P ≠ ⊥) [pP : P.IsPrime] (Aₘ : Type*) [CommRing Aₘ] [IsDomain Aₘ]
[Algebra A Aₘ] [IsLocalization.AtPrime Aₘ P] : DiscreteValuationRing Aₘ := by |
classical
letI : IsNoetherianRing Aₘ :=
IsLocalization.isNoetherianRing P.primeCompl _ IsDedekindRing.toIsNoetherian
letI : LocalRing Aₘ := IsLocalization.AtPrime.localRing Aₘ P
have hnf := IsLocalization.AtPrime.not_isField A hP Aₘ
exact
((DiscreteValuationRing.TFAE Aₘ hnf).out 0 2).mpr
(IsLoc... |
import Mathlib.MeasureTheory.Integral.IntervalIntegral
import Mathlib.Analysis.Calculus.Deriv.ZPow
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.Analysis.SpecialFunctions.NonIntegrable
import Mathlib.Analysis.Analytic.Basic
#align_import measure_theory.integral.circle_integral from "leanprover-communit... | Mathlib/MeasureTheory/Integral/CircleIntegral.lean | 171 | 174 | theorem hasDerivAt_circleMap (c : ℂ) (R : ℝ) (θ : ℝ) :
HasDerivAt (circleMap c R) (circleMap 0 R θ * I) θ := by |
simpa only [mul_assoc, one_mul, ofRealCLM_apply, circleMap, ofReal_one, zero_add]
using (((ofRealCLM.hasDerivAt (x := θ)).mul_const I).cexp.const_mul (R : ℂ)).const_add c
|
import Mathlib.Data.Multiset.Basic
import Mathlib.Data.Vector.Basic
import Mathlib.Data.Setoid.Basic
import Mathlib.Tactic.ApplyFun
#align_import data.sym.basic from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226"
assert_not_exists MonoidWithZero
set_option autoImplicit true
open Funct... | Mathlib/Data/Sym/Basic.lean | 554 | 556 | theorem mem_fill_iff {a b : α} {i : Fin (n + 1)} {s : Sym α (n - i)} :
a ∈ Sym.fill b i s ↔ (i : ℕ) ≠ 0 ∧ a = b ∨ a ∈ s := by |
rw [fill, mem_cast, mem_append_iff, or_comm, mem_replicate]
|
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.GroupTheory.GroupAction.Ring
#align_import data.polynomial.derivative from "leanprover-community/mathlib"@"bbeb185db4ccee8ed07dc48449414ebfa39cb821"
noncomputable section
open Finset
open Polynomial
namespace Pol... | Mathlib/Algebra/Polynomial/Derivative.lean | 213 | 219 | theorem natDegree_iterate_derivative (p : R[X]) (k : ℕ) :
(derivative^[k] p).natDegree ≤ p.natDegree - k := by |
induction k with
| zero => rw [Function.iterate_zero_apply, Nat.sub_zero]
| succ d hd =>
rw [Function.iterate_succ_apply', Nat.sub_succ']
exact (natDegree_derivative_le _).trans <| Nat.sub_le_sub_right hd 1
|
import Mathlib.Data.Real.Basic
import Mathlib.Data.ENNReal.Real
import Mathlib.Data.Sign
#align_import data.real.ereal from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
open Function ENNReal NNReal Set
noncomputable section
def EReal := WithBot (WithTop ℝ)
deriving Bot, Zero, One,... | Mathlib/Data/Real/EReal.lean | 1,123 | 1,125 | theorem bot_mul_of_neg {x : EReal} (h : x < 0) : ⊥ * x = ⊤ := by |
rw [EReal.mul_comm]
exact mul_bot_of_neg h
|
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 | 128 | 136 | theorem toFinsupp_cons_eq_single_add_embDomain {R : Type*} [AddZeroClass R] (x : R) (xs : List R)
[DecidablePred (getD (x::xs) · 0 ≠ 0)] [DecidablePred (getD xs · 0 ≠ 0)] :
toFinsupp (x::xs) =
Finsupp.single 0 x + (toFinsupp xs).embDomain ⟨Nat.succ, Nat.succ_injective⟩ := by |
classical
convert toFinsupp_append [x] xs using 3
· exact (toFinsupp_singleton x).symm
· ext n
exact add_comm n 1
|
import Mathlib.Algebra.FreeMonoid.Basic
#align_import algebra.free_monoid.count from "leanprover-community/mathlib"@"a2d2e18906e2b62627646b5d5be856e6a642062f"
variable {α : Type*} (p : α → Prop) [DecidablePred p]
namespace FreeAddMonoid
def countP : FreeAddMonoid α →+ ℕ where
toFun := List.countP p
map_zero... | Mathlib/Algebra/FreeMonoid/Count.lean | 43 | 45 | theorem count_of [DecidableEq α] (x y : α) : count x (of y) = (Pi.single x 1 : α → ℕ) y := by |
simp [Pi.single, Function.update, count, countP, List.countP, List.countP.go,
Bool.beq_eq_decide_eq]
|
import Mathlib.Order.Filter.AtTopBot
import Mathlib.Order.Filter.Pi
#align_import order.filter.cofinite from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1"
open Set Function
variable {ι α β : Type*} {l : Filter α}
namespace Filter
def cofinite : Filter α :=
comk Set.Finite finite_e... | Mathlib/Order/Filter/Cofinite.lean | 63 | 65 | theorem frequently_cofinite_iff_infinite {p : α → Prop} :
(∃ᶠ x in cofinite, p x) ↔ Set.Infinite { x | p x } := by |
simp only [Filter.Frequently, eventually_cofinite, not_not, Set.Infinite]
|
import Mathlib.Algebra.BigOperators.Ring.List
import Mathlib.Data.Nat.Prime
import Mathlib.Data.List.Prime
import Mathlib.Data.List.Sort
import Mathlib.Data.List.Chain
#align_import data.nat.factors from "leanprover-community/mathlib"@"008205aa645b3f194c1da47025c5f110c8406eab"
open Bool Subtype
open Nat
namespac... | Mathlib/Data/Nat/Factors.lean | 166 | 170 | theorem le_of_mem_factors {n p : ℕ} (h : p ∈ n.factors) : p ≤ n := by |
rcases n.eq_zero_or_pos with (rfl | hn)
· rw [factors_zero] at h
cases h
· exact le_of_dvd hn (dvd_of_mem_factors h)
|
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Data.Nat.GCD.BigOperators
import Mathlib.Data.Nat.PrimeFin
import Mathlib.NumberTheory.Padics.PadicVal
import Mathlib.Order.Interval.Finset.Nat
#align_import data.nat.factorization.basic from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
... | Mathlib/Data/Nat/Factorization/Basic.lean | 422 | 427 | theorem factorization_prime_le_iff_dvd {d n : ℕ} (hd : d ≠ 0) (hn : n ≠ 0) :
(∀ p : ℕ, p.Prime → d.factorization p ≤ n.factorization p) ↔ d ∣ n := by |
rw [← factorization_le_iff_dvd hd hn]
refine ⟨fun h p => (em p.Prime).elim (h p) fun hp => ?_, fun h p _ => h p⟩
simp_rw [factorization_eq_zero_of_non_prime _ hp]
rfl
|
import Mathlib.RepresentationTheory.FdRep
import Mathlib.LinearAlgebra.Trace
import Mathlib.RepresentationTheory.Invariants
#align_import representation_theory.character from "leanprover-community/mathlib"@"55b3f8206b8596db8bb1804d8a92814a0b6670c9"
noncomputable section
universe u
open CategoryTheory LinearMap ... | Mathlib/RepresentationTheory/Character.lean | 59 | 60 | theorem char_one (V : FdRep k G) : V.character 1 = FiniteDimensional.finrank k V := by |
simp only [character, map_one, trace_one]
|
import Mathlib.Topology.UniformSpace.UniformConvergence
import Mathlib.Topology.UniformSpace.UniformEmbedding
import Mathlib.Topology.UniformSpace.CompleteSeparated
import Mathlib.Topology.UniformSpace.Compact
import Mathlib.Topology.Algebra.Group.Basic
import Mathlib.Topology.DiscreteSubset
import Mathlib.Tactic.Abel... | Mathlib/Topology/Algebra/UniformGroup.lean | 89 | 92 | theorem UniformContinuous.inv [UniformSpace β] {f : β → α} (hf : UniformContinuous f) :
UniformContinuous fun x => (f x)⁻¹ := by |
have : UniformContinuous fun x => 1 / f x := uniformContinuous_const.div hf
simp_all
|
import Mathlib.Algebra.BigOperators.WithTop
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Data.ENNReal.Basic
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open Set NNReal ENNReal
namespace ENNReal
variable {a b c d : ℝ≥0∞} {r p q... | Mathlib/Data/ENNReal/Operations.lean | 193 | 197 | theorem toNNReal_add {r₁ r₂ : ℝ≥0∞} (h₁ : r₁ ≠ ∞) (h₂ : r₂ ≠ ∞) :
(r₁ + r₂).toNNReal = r₁.toNNReal + r₂.toNNReal := by |
lift r₁ to ℝ≥0 using h₁
lift r₂ to ℝ≥0 using h₂
rfl
|
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