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 |
|---|---|---|---|---|---|
import Mathlib.Tactic.NormNum
import Mathlib.Tactic.TryThis
import Mathlib.Util.AtomM
set_option autoImplicit true
namespace Mathlib.Tactic.Abel
open Lean Elab Meta Tactic Qq
initialize registerTraceClass `abel
initialize registerTraceClass `abel.detail
structure Context where
α : Expr
univ :... | Mathlib/Tactic/Abel.lean | 144 | 146 | theorem term_add_term {α} [AddCommMonoid α] (n₁ x a₁ n₂ a₂ n' a') (h₁ : n₁ + n₂ = n')
(h₂ : a₁ + a₂ = a') : @term α _ n₁ x a₁ + @term α _ n₂ x a₂ = term n' x a' := by |
simp [h₁.symm, h₂.symm, term, add_nsmul, add_assoc, add_left_comm]
|
import Mathlib.Analysis.BoxIntegral.Partition.Filter
import Mathlib.Analysis.BoxIntegral.Partition.Measure
import Mathlib.Topology.UniformSpace.Compact
import Mathlib.Init.Data.Bool.Lemmas
#align_import analysis.box_integral.basic from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open... | Mathlib/Analysis/BoxIntegral/Basic.lean | 455 | 459 | theorem dist_integralSum_integral_le_of_memBaseSet (h : Integrable I l f vol) (h₀ : 0 < ε)
(hπ : l.MemBaseSet I c (h.convergenceR ε c) π) (hπp : π.IsPartition) :
dist (integralSum f vol π) (integral I l f vol) ≤ ε := by |
rw [convergenceR, dif_pos h₀] at hπ
exact (hasIntegral_iff.1 h.hasIntegral ε h₀).choose_spec.2 c _ hπ hπp
|
import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.Topology.Algebra.Module.Basic
#align_import analysis.normed_space.basic from "leanprover-community/mathlib"@"bc91ed7093bf098d253401e69df601fc33dde156"
open Metric Set Function Filter
open scoped NNReal Topology
instance Real.punctured_nhds_module_neBot {E ... | Mathlib/Analysis/NormedSpace/Real.lean | 124 | 128 | theorem exists_norm_eq {c : ℝ} (hc : 0 ≤ c) : ∃ x : E, ‖x‖ = c := by |
rcases exists_ne (0 : E) with ⟨x, hx⟩
rw [← norm_ne_zero_iff] at hx
use c • ‖x‖⁻¹ • x
simp [norm_smul, Real.norm_of_nonneg hc, abs_of_nonneg hc, inv_mul_cancel hx]
|
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 | 318 | 323 | theorem perm_option_to_list {o₁ o₂ : Option α} : o₁.toList ~ o₂.toList ↔ o₁ = o₂ := by |
refine ⟨fun p => ?_, fun e => e ▸ Perm.refl _⟩
cases' o₁ with a <;> cases' o₂ with b; · rfl
· cases p.length_eq
· cases p.length_eq
· exact Option.mem_toList.1 (p.symm.subset <| by simp)
|
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Smooth.Basic
import Mathlib.RingTheory.Unramified.Basic
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
-- Porting note: added to make the syntax work below.
open scoped TensorProdu... | Mathlib/RingTheory/Etale/Basic.lean | 66 | 69 | theorem iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by |
rw [formallyUnramified_iff, formallySmooth_iff, formallyEtale_iff]
simp_rw [← forall_and, Function.Bijective]
|
import Mathlib.Analysis.BoxIntegral.Partition.Filter
import Mathlib.Analysis.BoxIntegral.Partition.Measure
import Mathlib.Topology.UniformSpace.Compact
import Mathlib.Init.Data.Bool.Lemmas
#align_import analysis.box_integral.basic from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open... | Mathlib/Analysis/BoxIntegral/Basic.lean | 348 | 351 | theorem HasIntegral.smul (hf : HasIntegral I l f vol y) (c : ℝ) :
HasIntegral I l (c • f) vol (c • y) := by |
simpa only [HasIntegral, ← integralSum_smul] using
(tendsto_const_nhds : Tendsto _ _ (𝓝 c)).smul hf
|
import Mathlib.Algebra.IsPrimePow
import Mathlib.Algebra.Squarefree.Basic
import Mathlib.Order.Hom.Bounded
import Mathlib.Algebra.GCDMonoid.Basic
#align_import ring_theory.chain_of_divisors from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
variable {M : Type*} [CancelCommMonoidWithZero... | Mathlib/RingTheory/ChainOfDivisors.lean | 111 | 132 | theorem eq_second_of_chain_of_prime_dvd {p q r : Associates M} {n : ℕ} (hn : n ≠ 0)
{c : Fin (n + 1) → Associates M} (h₁ : StrictMono c)
(h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ i, r = c i) (hp : Prime p) (hr : r ∣ q) (hp' : p ∣ r) :
p = c 1 := by |
cases' n with n
· contradiction
obtain ⟨i, rfl⟩ := h₂.1 (dvd_trans hp' hr)
refine congr_arg c (eq_of_ge_of_not_gt ?_ fun hi => ?_)
· rw [Fin.le_iff_val_le_val, Fin.val_one, Nat.succ_le_iff, ← Fin.val_zero' (n.succ + 1), ←
Fin.lt_iff_val_lt_val, Fin.pos_iff_ne_zero]
rintro rfl
exact hp.not_unit ... |
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing
import Mathlib.CategoryTheory.Adjunction.Evaluation
import Mathlib.CategoryTheory.Limits.Types
import Mathlib.CategoryTheory.Limits.Preserves.Filtered
import Mathlib.Cat... | Mathlib/Topology/Sheaves/Stalks.lean | 150 | 153 | theorem stalkPushforward_germ (f : X ⟶ Y) (F : X.Presheaf C) (U : Opens Y)
(x : (Opens.map f).obj U) :
(f _* F).germ ⟨(f : X → Y) (x : X), x.2⟩ ≫ F.stalkPushforward C f x = F.germ x := by |
simp [germ, stalkPushforward]
|
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.MeasureTheory.Group.Pointwise
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Measure.Haar.Basic
import Mathlib.MeasureTheory.Measure.Doubling
import Mathlib.MeasureTheory.Constructions.BorelSpace.Metric
#align_import measu... | Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean | 603 | 608 | theorem _root_.AlternatingMap.measure_parallelepiped (ω : G [⋀^Fin n]→ₗ[ℝ] ℝ)
(v : Fin n → G) : ω.measure (parallelepiped v) = ENNReal.ofReal |ω v| := by |
conv_rhs => rw [ω.eq_smul_basis_det (finBasisOfFinrankEq ℝ G _i.out)]
simp only [addHaar_parallelepiped, AlternatingMap.measure, coe_nnreal_smul_apply,
AlternatingMap.smul_apply, Algebra.id.smul_eq_mul, abs_mul, ENNReal.ofReal_mul (abs_nonneg _),
Real.ennnorm_eq_ofReal_abs]
|
import Mathlib.Algebra.Polynomial.Mirror
import Mathlib.Analysis.Complex.Polynomial
#align_import data.polynomial.unit_trinomial from "leanprover-community/mathlib"@"302eab4f46abb63de520828de78c04cb0f9b5836"
namespace Polynomial
open scoped Polynomial
open Finset
section Semiring
variable {R : Type*} [Semirin... | Mathlib/Algebra/Polynomial/UnitTrinomial.lean | 61 | 64 | theorem trinomial_trailing_coeff' (hkm : k < m) (hmn : m < n) :
(trinomial k m n u v w).coeff k = u := by |
rw [trinomial_def, coeff_add, coeff_add, coeff_C_mul_X_pow, coeff_C_mul_X_pow, coeff_C_mul_X_pow,
if_pos rfl, if_neg hkm.ne, if_neg (hkm.trans hmn).ne, add_zero, add_zero]
|
import Mathlib.Geometry.Manifold.MFDeriv.Defs
#align_import geometry.manifold.mfderiv from "leanprover-community/mathlib"@"e473c3198bb41f68560cab68a0529c854b618833"
noncomputable section
open scoped Topology Manifold
open Set Bundle
section DerivativesProperties
variable
{𝕜 : Type*} [NontriviallyNormedFiel... | Mathlib/Geometry/Manifold/MFDeriv/Basic.lean | 337 | 342 | theorem mdifferentiableOn_of_locally_mdifferentiableOn
(h : ∀ x ∈ s, ∃ u, IsOpen u ∧ x ∈ u ∧ MDifferentiableOn I I' f (s ∩ u)) :
MDifferentiableOn I I' f s := by |
intro x xs
rcases h x xs with ⟨t, t_open, xt, ht⟩
exact (mdifferentiableWithinAt_inter (t_open.mem_nhds xt)).1 (ht x ⟨xs, xt⟩)
|
import Mathlib.Algebra.Lie.Submodule
#align_import algebra.lie.ideal_operations from "leanprover-community/mathlib"@"8983bec7cdf6cb2dd1f21315c8a34ab00d7b2f6d"
universe u v w w₁ w₂
namespace LieSubmodule
variable {R : Type u} {L : Type v} {M : Type w} {M₂ : Type w₁}
variable [CommRing R] [LieRing L] [LieAlgebra ... | Mathlib/Algebra/Lie/IdealOperations.lean | 221 | 223 | theorem map_comap_eq (hf : N₂ ≤ f.range) : map f (comap f N₂) = N₂ := by |
rw [SetLike.ext'_iff]
exact Set.image_preimage_eq_of_subset hf
|
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.Prod
import Mathlib.SetTheory.Cardinal.Basic
import Mathlib.Tactic.FinCases
import Mathlib.Tactic.LinearCombination
import Mathlib.Lean.Expr.ExtraRecognizers
import Mathlib.Data.Set.Subsingleton
#align_import lin... | Mathlib/LinearAlgebra/LinearIndependent.lean | 307 | 315 | theorem LinearIndependent.map_of_surjective_injective {R' : Type*} {M' : Type*}
[Semiring R'] [AddCommMonoid M'] [Module R' M'] (hv : LinearIndependent R v)
(i : ZeroHom R R') (j : M →+ M') (hi : Surjective i) (hj : ∀ m, j m = 0 → m = 0)
(hc : ∀ (r : R) (m : M), j (r • m) = i r • j m) : LinearIndependent R'... |
obtain ⟨i', hi'⟩ := hi.hasRightInverse
refine hv.map_of_injective_injective i' j (fun _ h ↦ ?_) hj fun r m ↦ ?_
· apply_fun i at h
rwa [hi', i.map_zero] at h
rw [hc (i' r) m, hi']
|
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.RingTheory.Ideal.Quotient
#align_import linear_algebra.smodeq from "leanprover-community/mathlib"@"146d3d1fa59c091fedaad8a4afa09d6802886d24"
open Submodule
open Polynomial
variable {R : Type*} [Ring R]
variable {A : Type*} [CommRing A]
variable {M : Type*} [... | Mathlib/LinearAlgebra/SModEq.lean | 114 | 119 | theorem eval {R : Type*} [CommRing R] {I : Ideal R} {x y : R} (h : x ≡ y [SMOD I]) (f : R[X]) :
f.eval x ≡ f.eval y [SMOD I] := by |
rw [SModEq.def] at h ⊢
show Ideal.Quotient.mk I (f.eval x) = Ideal.Quotient.mk I (f.eval y)
replace h : Ideal.Quotient.mk I x = Ideal.Quotient.mk I y := h
rw [← Polynomial.eval₂_at_apply, ← Polynomial.eval₂_at_apply, h]
|
import Mathlib.Logic.Pairwise
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#align_import data.set.lattice from "leanprover-community/mathlib"@"b86832321b586c6ac23ef8cdef6a7a27e42b13bd"
open Function Set
universe u
variable {α β γ : Type*} {ι ι' ι... | Mathlib/Data/Set/Lattice.lean | 1,293 | 1,295 | theorem iUnion_singleton_eq_range {α β : Type*} (f : α → β) : ⋃ x : α, {f x} = range f := by |
ext x
simp [@eq_comm _ x]
|
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
import Mathlib.MeasureTheory.Decomposition.Jordan
import Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure
noncomputable section
open scoped Classical MeasureTheory NNReal ENNReal
open Set
variable {α β : Type*... | Mathlib/MeasureTheory/Decomposition/SignedLebesgue.lean | 417 | 423 | theorem rnDeriv_neg (s : SignedMeasure α) (μ : Measure α) [s.HaveLebesgueDecomposition μ] :
(-s).rnDeriv μ =ᵐ[μ] -s.rnDeriv μ := by |
refine
Integrable.ae_eq_of_withDensityᵥ_eq (integrable_rnDeriv _ _) (integrable_rnDeriv _ _).neg ?_
rw [withDensityᵥ_neg, ← add_right_inj ((-s).singularPart μ),
singularPart_add_withDensity_rnDeriv_eq, singularPart_neg, ← neg_add,
singularPart_add_withDensity_rnDeriv_eq]
|
import Mathlib.Algebra.BigOperators.Ring.List
import Mathlib.Data.Nat.ModEq
import Mathlib.Data.Nat.GCD.BigOperators
namespace Nat
variable {ι : Type*}
lemma modEq_list_prod_iff {a b} {l : List ℕ} (co : l.Pairwise Coprime) :
a ≡ b [MOD l.prod] ↔ ∀ i, a ≡ b [MOD l.get i] := by
induction' l with m l ih
· si... | Mathlib/Data/Nat/ChineseRemainder.lean | 145 | 151 | theorem chineseRemainderOfMultiset_lt_prod {m : Multiset ι}
(nod : m.Nodup) (hs : ∀ i ∈ m, s i ≠ 0) (pp : Set.Pairwise {x | x ∈ m} (Coprime on s)) :
chineseRemainderOfMultiset a s nod hs pp < (m.map s).prod := by |
induction' m using Quot.ind with l
unfold chineseRemainderOfMultiset; simp
exact chineseRemainderOfList_lt_prod a s l
(List.Nodup.pairwise_of_forall_ne nod pp) (by simpa using hs)
|
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import Mathlib.Analysis.Calculus.Deriv.Comp
import Mathlib.Analysis.Calculus.LocalExtr.Rolle
import Mathlib.Analysis.Convex.Normed
import Mathlib.Analysis.RCLike.Basic
#align_import... | Mathlib/Analysis/Calculus/MeanValue.lean | 337 | 350 | theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ}
(hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x)
(bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by |
let g x := f x - f a
have hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const
have hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x := by
intro x hx
simpa using (hf' x hx).sub (hasDerivWithinAt_const _ _ _)
let B x := C * (x - a)
have hB : ∀ x, HasDerivAt B C x := by
intro x
si... |
import Mathlib.Analysis.Convex.Basic
import Mathlib.Analysis.InnerProductSpace.Orthogonal
import Mathlib.Analysis.InnerProductSpace.Symmetric
import Mathlib.Analysis.NormedSpace.RCLike
import Mathlib.Analysis.RCLike.Lemmas
import Mathlib.Algebra.DirectSum.Decomposition
#align_import analysis.inner_product_space.proje... | Mathlib/Analysis/InnerProductSpace/Projection.lean | 551 | 553 | theorem orthogonalProjection_mem_subspace_eq_self (v : K) : orthogonalProjection K v = v := by |
ext
apply eq_orthogonalProjection_of_mem_of_inner_eq_zero <;> simp
|
import Mathlib.RingTheory.Polynomial.Cyclotomic.Basic
import Mathlib.RingTheory.RootsOfUnity.Minpoly
#align_import ring_theory.polynomial.cyclotomic.roots from "leanprover-community/mathlib"@"7fdeecc0d03cd40f7a165e6cf00a4d2286db599f"
namespace Polynomial
variable {R : Type*} [CommRing R] {n : ℕ}
theorem isRoot_... | Mathlib/RingTheory/Polynomial/Cyclotomic/Roots.lean | 107 | 113 | theorem roots_cyclotomic_nodup [NeZero (n : R)] : (cyclotomic n R).roots.Nodup := by |
obtain h | ⟨ζ, hζ⟩ := (cyclotomic n R).roots.empty_or_exists_mem
· exact h.symm ▸ Multiset.nodup_zero
rw [mem_roots <| cyclotomic_ne_zero n R, isRoot_cyclotomic_iff] at hζ
refine Multiset.nodup_of_le
(roots.le_of_dvd (X_pow_sub_C_ne_zero (NeZero.pos_of_neZero_natCast R) 1) <|
cyclotomic.dvd_X_pow_sub... |
import Mathlib.Topology.Instances.Real
import Mathlib.Order.Filter.Archimedean
#align_import analysis.subadditive from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
open Set Filter Topology
def Subadditive (u : ℕ → ℝ) : Prop :=
∀ m n, u (m + n) ≤ u m + u n
#al... | Mathlib/Analysis/Subadditive.lean | 45 | 48 | theorem lim_le_div (hbdd : BddBelow (range fun n => u n / n)) {n : ℕ} (hn : n ≠ 0) :
h.lim ≤ u n / n := by |
rw [Subadditive.lim]
exact csInf_le (hbdd.mono <| image_subset_range _ _) ⟨n, hn.bot_lt, rfl⟩
|
import Mathlib.Algebra.Polynomial.Splits
#align_import algebra.cubic_discriminant from "leanprover-community/mathlib"@"930133160e24036d5242039fe4972407cd4f1222"
noncomputable section
@[ext]
structure Cubic (R : Type*) where
(a b c d : R)
#align cubic Cubic
namespace Cubic
open Cubic Polynomial
open Polynom... | Mathlib/Algebra/CubicDiscriminant.lean | 145 | 146 | theorem of_b_eq_zero (ha : P.a = 0) (hb : P.b = 0) : P.toPoly = C P.c * X + C P.d := by |
rw [of_a_eq_zero ha, hb, C_0, zero_mul, zero_add]
|
import Mathlib.Analysis.Convex.Topology
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.Analysis.Seminorm
import Mathlib.Analysis.LocallyConvex.Bounded
import Mathlib.Analysis.RCLike.Basic
#align_import analysis.convex.gauge from "leanprover-community/mathlib"@"373b03b5b9d0486534edbe94747f23cb3712f93d"
... | Mathlib/Analysis/Convex/Gauge.lean | 103 | 110 | theorem gauge_zero' : gauge (0 : Set E) = 0 := by |
ext x
rw [gauge_def']
obtain rfl | hx := eq_or_ne x 0
· simp only [csInf_Ioi, mem_zero, Pi.zero_apply, eq_self_iff_true, sep_true, smul_zero]
· simp only [mem_zero, Pi.zero_apply, inv_eq_zero, smul_eq_zero]
convert Real.sInf_empty
exact eq_empty_iff_forall_not_mem.2 fun r hr => hr.2.elim (ne_of_gt hr... |
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
import Mathlib.CategoryTheory.Limits.Preserves.Basic
#align_import category_theory.limits.preserves.shapes.binary_products from "leanprover-community/mathlib"@"024a4231815538ac739f52d08dd20a55da0d6b23"
noncomputable section
universe v₁ v₂ u₁ u₂
open Cat... | Mathlib/CategoryTheory/Limits/Preserves/Shapes/BinaryProducts.lean | 100 | 103 | theorem PreservesLimitPair.iso_inv_fst :
(PreservesLimitPair.iso G X Y).inv ≫ G.map prod.fst = prod.fst := by |
rw [← Iso.cancel_iso_hom_left (PreservesLimitPair.iso G X Y), ← Category.assoc, Iso.hom_inv_id]
simp
|
import Mathlib.Algebra.Group.Equiv.TypeTags
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Module.LinearMap.Basic
import Mathlib.Algebra.MonoidAlgebra.Basic
import Mathlib.LinearAlgebra.Dual
import Mathlib.LinearAlgebra.Contraction
import Mathlib.RingTheory.TensorProduct.Basic
#align_import representation_... | Mathlib/RepresentationTheory/Basic.lean | 221 | 234 | theorem ofModule_asAlgebraHom_apply_apply (r : MonoidAlgebra k G)
(m : RestrictScalars k (MonoidAlgebra k G) M) :
((ofModule M).asAlgebraHom r) m =
(RestrictScalars.addEquiv _ _ _).symm (r • RestrictScalars.addEquiv _ _ _ m) := by |
apply MonoidAlgebra.induction_on r
· intro g
simp only [one_smul, MonoidAlgebra.lift_symm_apply, MonoidAlgebra.of_apply,
Representation.asAlgebraHom_single, Representation.ofModule, AddEquiv.apply_eq_iff_eq,
RestrictScalars.lsmul_apply_apply]
· intro f g fw gw
simp only [fw, gw, map_add, add_... |
import Mathlib.Algebra.Group.Equiv.Basic
import Mathlib.Data.ENat.Lattice
import Mathlib.Data.Part
import Mathlib.Tactic.NormNum
#align_import data.nat.part_enat from "leanprover-community/mathlib"@"3ff3f2d6a3118b8711063de7111a0d77a53219a8"
open Part hiding some
def PartENat : Type :=
Part ℕ
#align part_enat ... | Mathlib/Data/Nat/PartENat.lean | 594 | 595 | theorem toWithTop_top' {h : Decidable (⊤ : PartENat).Dom} : toWithTop ⊤ = ⊤ := by |
convert toWithTop_top
|
import Mathlib.Combinatorics.SimpleGraph.Subgraph
import Mathlib.Data.List.Rotate
#align_import combinatorics.simple_graph.connectivity from "leanprover-community/mathlib"@"b99e2d58a5e6861833fa8de11e51a81144258db4"
open Function
universe u v w
namespace SimpleGraph
variable {V : Type u} {V' : Type v} {V'' : Typ... | Mathlib/Combinatorics/SimpleGraph/Connectivity.lean | 258 | 264 | theorem append_assoc {u v w x : V} (p : G.Walk u v) (q : G.Walk v w) (r : G.Walk w x) :
p.append (q.append r) = (p.append q).append r := by |
induction p with
| nil => rfl
| cons h p' ih =>
dsimp only [append]
rw [ih]
|
import Mathlib.Data.List.Forall2
#align_import data.list.zip from "leanprover-community/mathlib"@"134625f523e737f650a6ea7f0c82a6177e45e622"
-- Make sure we don't import algebra
assert_not_exists Monoid
universe u
open Nat
namespace List
variable {α : Type u} {β γ δ ε : Type*}
#align list.zip_with_cons_cons Li... | Mathlib/Data/List/Zip.lean | 245 | 246 | theorem revzip_map_fst (l : List α) : (revzip l).map Prod.fst = l := by |
rw [← unzip_left, unzip_revzip]
|
import Mathlib.Algebra.GradedMonoid
import Mathlib.Algebra.Order.Monoid.Canonical.Defs
import Mathlib.Algebra.MvPolynomial.Basic
#align_import ring_theory.mv_polynomial.weighted_homogeneous from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
noncomputable section
open Set Function Fins... | Mathlib/RingTheory/MvPolynomial/WeightedHomogeneous.lean | 196 | 204 | theorem isWeightedHomogeneous_monomial (w : σ → M) (d : σ →₀ ℕ) (r : R) {m : M}
(hm : weightedDegree w d = m) : IsWeightedHomogeneous w (monomial d r) m := by |
classical
intro c hc
rw [coeff_monomial] at hc
split_ifs at hc with h
· subst c
exact hm
· contradiction
|
import Mathlib.LinearAlgebra.Contraction
#align_import linear_algebra.coevaluation from "leanprover-community/mathlib"@"d6814c584384ddf2825ff038e868451a7c956f31"
noncomputable section
section coevaluation
open TensorProduct FiniteDimensional
open TensorProduct
universe u v
variable (K : Type u) [Field K]
var... | Mathlib/LinearAlgebra/Coevaluation.lean | 61 | 76 | theorem contractLeft_assoc_coevaluation :
(contractLeft K V).rTensor _ ∘ₗ
(TensorProduct.assoc K _ _ _).symm.toLinearMap ∘ₗ
(coevaluation K V).lTensor (Module.Dual K V) =
(TensorProduct.lid K _).symm.toLinearMap ∘ₗ (TensorProduct.rid K _).toLinearMap := by |
letI := Classical.decEq (Basis.ofVectorSpaceIndex K V)
apply TensorProduct.ext
apply (Basis.ofVectorSpace K V).dualBasis.ext; intro j; apply LinearMap.ext_ring
rw [LinearMap.compr₂_apply, LinearMap.compr₂_apply, TensorProduct.mk_apply]
simp only [LinearMap.coe_comp, Function.comp_apply, LinearEquiv.coe_toLin... |
import Mathlib.MeasureTheory.Measure.NullMeasurable
import Mathlib.MeasureTheory.MeasurableSpace.Basic
import Mathlib.Topology.Algebra.Order.LiminfLimsup
#align_import measure_theory.measure.measure_space from "leanprover-community/mathlib"@"343e80208d29d2d15f8050b929aa50fe4ce71b55"
noncomputable section
open Set... | Mathlib/MeasureTheory/Measure/MeasureSpace.lean | 460 | 465 | theorem nonempty_inter_of_measure_lt_add' {m : MeasurableSpace α} (μ : Measure α) {s t u : Set α}
(hs : MeasurableSet s) (h's : s ⊆ u) (h't : t ⊆ u) (h : μ u < μ s + μ t) :
(s ∩ t).Nonempty := by |
rw [add_comm] at h
rw [inter_comm]
exact nonempty_inter_of_measure_lt_add μ hs h't h's h
|
import Mathlib.Combinatorics.SimpleGraph.Subgraph
import Mathlib.Data.List.Rotate
#align_import combinatorics.simple_graph.connectivity from "leanprover-community/mathlib"@"b99e2d58a5e6861833fa8de11e51a81144258db4"
open Function
universe u v w
namespace SimpleGraph
variable {V : Type u} {V' : Type v} {V'' : Typ... | Mathlib/Combinatorics/SimpleGraph/Connectivity.lean | 1,216 | 1,225 | theorem count_support_takeUntil_eq_one {u v w : V} (p : G.Walk v w) (h : u ∈ p.support) :
(p.takeUntil u h).support.count u = 1 := by |
induction p
· rw [mem_support_nil_iff] at h
subst u
simp!
· cases h
· simp!
· simp! only
split_ifs with h' <;> rw [eq_comm] at h' <;> subst_vars <;> simp! [*, List.count_cons]
|
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Data.Set.NAry
import Mathlib.Order.Directed
#align_import order.bounds.basic from "leanprover-community/mathlib"@"b1abe23ae96fef89ad30d9f4362c307f72a55010"
open Function Set
open OrderDual (toDual ofDual)
universe u v w x
variable {α : Type u} {β : Type v}... | Mathlib/Order/Bounds/Basic.lean | 778 | 779 | theorem isLUB_Ioo {a b : γ} (hab : a < b) : IsLUB (Ioo a b) b := by |
simpa only [dual_Ioo] using isGLB_Ioo hab.dual
|
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Data.Set.UnionLift
#align_import algebra.algebra.subalgebra.basic from "leanprover-community/mathlib"@"b915e9392ecb2a861e1e766f0e1df6ac481188ca"
namespace Subalgebra
open Algebra
variable {R A B : Type*} [CommSemiring R] [Semiring A] [Algebra R A] [... | Mathlib/Algebra/Algebra/Subalgebra/Directed.lean | 78 | 81 | theorem iSupLift_inclusion {i : ι} (x : K i) (h : K i ≤ T) :
iSupLift K dir f hf T hT (inclusion h x) = f i x := by |
dsimp [iSupLift, inclusion]
rw [Set.iUnionLift_inclusion]
|
import Mathlib.Topology.Compactness.SigmaCompact
import Mathlib.Topology.Connected.TotallyDisconnected
import Mathlib.Topology.Inseparable
#align_import topology.separation from "leanprover-community/mathlib"@"d91e7f7a7f1c7e9f0e18fdb6bde4f652004c735d"
open Function Set Filter Topology TopologicalSpace
open scoped... | Mathlib/Topology/Separation.lean | 901 | 906 | theorem PreconnectedSpace.trivial_of_discrete [PreconnectedSpace X] [DiscreteTopology X] :
Subsingleton X := by |
rw [← not_nontrivial_iff_subsingleton]
rintro ⟨x, y, hxy⟩
rw [Ne, ← mem_singleton_iff, (isClopen_discrete _).eq_univ <| singleton_nonempty y] at hxy
exact hxy (mem_univ x)
|
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
#align_import analysis.special_functions.pow.asymptotics from "leanprover-community/mathlib"@"0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8"
set_option linter.uppercaseLean3 false
noncomputable section
open scoped Classical
open Real Topology NNReal ENNReal Filter C... | Mathlib/Analysis/SpecialFunctions/Pow/Asymptotics.lean | 36 | 46 | theorem tendsto_rpow_atTop {y : ℝ} (hy : 0 < y) : Tendsto (fun x : ℝ => x ^ y) atTop atTop := by |
rw [tendsto_atTop_atTop]
intro b
use max b 0 ^ (1 / y)
intro x hx
exact
le_of_max_le_left
(by
convert rpow_le_rpow (rpow_nonneg (le_max_right b 0) (1 / y)) hx (le_of_lt hy)
using 1
rw [← rpow_mul (le_max_right b 0), (eq_div_iff (ne_of_gt hy)).mp rfl, Real.rpow_one])
|
import Mathlib.Order.Filter.SmallSets
import Mathlib.Tactic.Monotonicity
import Mathlib.Topology.Compactness.Compact
import Mathlib.Topology.NhdsSet
import Mathlib.Algebra.Group.Defs
#align_import topology.uniform_space.basic from "leanprover-community/mathlib"@"195fcd60ff2bfe392543bceb0ec2adcdb472db4c"
open Set F... | Mathlib/Topology/UniformSpace/Basic.lean | 746 | 749 | theorem nhds_basis_uniformity' {p : ι → Prop} {s : ι → Set (α × α)} (h : (𝓤 α).HasBasis p s)
{x : α} : (𝓝 x).HasBasis p fun i => ball x (s i) := by |
rw [nhds_eq_comap_uniformity]
exact h.comap (Prod.mk x)
|
import Mathlib.Logic.Encodable.Lattice
import Mathlib.MeasureTheory.MeasurableSpace.Defs
#align_import measure_theory.pi_system from "leanprover-community/mathlib"@"98e83c3d541c77cdb7da20d79611a780ff8e7d90"
open MeasurableSpace Set
open scoped Classical
open MeasureTheory
def IsPiSystem {α} (C : Set (Set α)) :... | Mathlib/MeasureTheory/PiSystem.lean | 149 | 151 | theorem isPiSystem_image_Iic (s : Set α) : IsPiSystem (Iic '' s) := by |
rintro _ ⟨a, ha, rfl⟩ _ ⟨b, hb, rfl⟩ -
exact ⟨a ⊓ b, inf_ind a b ha hb, Iic_inter_Iic.symm⟩
|
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.Data.Nat.Factorial.BigOperators
import Mathlib.Data.Fin.VecNotation
import Mathlib.Data.Finset.Sym
import Mathlib.Data.Finsupp.Multiset
#align_import data.nat.choose.multinomial from "leanprover-community/mathlib"@"2738d2ca56cbc... | Mathlib/Data/Nat/Choose/Multinomial.lean | 112 | 114 | theorem binomial_spec [DecidableEq α] (hab : a ≠ b) :
(f a)! * (f b)! * multinomial {a, b} f = (f a + f b)! := by |
simpa [Finset.sum_pair hab, Finset.prod_pair hab] using multinomial_spec {a, b} f
|
import Mathlib.LinearAlgebra.Dimension.Finite
import Mathlib.LinearAlgebra.Dimension.Constructions
open Cardinal Submodule Set FiniteDimensional
universe u v
section Module
variable {K : Type u} {V : Type v} [Ring K] [StrongRankCondition K] [AddCommGroup V] [Module K V]
noncomputable def Basis.ofRankEqZero [Mo... | Mathlib/LinearAlgebra/Dimension/FreeAndStrongRankCondition.lean | 192 | 198 | theorem finrank_eq_one_iff [Module.Free K V] (ι : Type*) [Unique ι] :
finrank K V = 1 ↔ Nonempty (Basis ι K V) := by |
constructor
· intro h
exact ⟨basisUnique ι h⟩
· rintro ⟨b⟩
simpa using finrank_eq_card_basis b
|
import Mathlib.LinearAlgebra.Dual
open Function Module
variable (R M N : Type*) [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N]
structure PerfectPairing :=
toLin : M →ₗ[R] N →ₗ[R] R
bijectiveLeft : Bijective toLin
bijectiveRight : Bijective toLin.flip
attribute [nolint docBlame] P... | Mathlib/LinearAlgebra/PerfectPairing.lean | 96 | 100 | theorem toDualRight_symm_toDualLeft (x : M) :
p.toDualRight.symm.dualMap (p.toDualLeft x) = Dual.eval R M x := by |
ext f
simp only [LinearEquiv.dualMap_apply, Dual.eval_apply]
exact toDualLeft_of_toDualRight_symm p x f
|
import Mathlib.Topology.UniformSpace.Basic
import Mathlib.Topology.Separation
import Mathlib.Order.Filter.CountableInter
#align_import topology.G_delta from "leanprover-community/mathlib"@"b9e46fe101fc897fb2e7edaf0bf1f09ea49eb81a"
noncomputable section
open Topology TopologicalSpace Filter Encodable Set
open sco... | Mathlib/Topology/GDelta.lean | 136 | 138 | theorem IsGδ.inter {s t : Set X} (hs : IsGδ s) (ht : IsGδ t) : IsGδ (s ∩ t) := by |
rw [inter_eq_iInter]
exact .iInter (Bool.forall_bool.2 ⟨ht, hs⟩)
|
import Mathlib.Algebra.Divisibility.Basic
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Ring.Defs
#align_import algebra.euclidean_domain.defs from "leanprover-community/mathlib"@"ee7b9f9a9ac2a8d9f04ea39bbfe6b1a3be053b38"
universe u
class EuclideanDomain (R : Type u) extends CommRing R, Nontrivial R ... | Mathlib/Algebra/EuclideanDomain/Defs.lean | 131 | 133 | theorem mod_add_div' (m k : R) : m % k + m / k * k = m := by |
rw [mul_comm]
exact mod_add_div _ _
|
import Mathlib.Algebra.Lie.Abelian
import Mathlib.Algebra.Lie.IdealOperations
import Mathlib.Algebra.Lie.Quotient
#align_import algebra.lie.normalizer from "leanprover-community/mathlib"@"938fead7abdc0cbbca8eba7a1052865a169dc102"
variable {R L M M' : Type*}
variable [CommRing R] [LieRing L] [LieAlgebra R L]
varia... | Mathlib/Algebra/Lie/Normalizer.lean | 86 | 87 | theorem top_lie_le_iff_le_normalizer (N' : LieSubmodule R L M) :
⁅(⊤ : LieIdeal R L), N⁆ ≤ N' ↔ N ≤ N'.normalizer := by | rw [lie_le_iff]; tauto
|
import Mathlib.RingTheory.DedekindDomain.Ideal
import Mathlib.RingTheory.IsAdjoinRoot
#align_import number_theory.kummer_dedekind from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
variable (R : Type*) {S : Type*} [CommRing R] [CommRing S] [Algebra R S]
open Ideal Polynomial DoubleQuo... | Mathlib/NumberTheory/KummerDedekind.lean | 274 | 282 | theorem multiplicity_factors_map_eq_multiplicity (hI : IsMaximal I) (hI' : I ≠ ⊥)
(hx : (conductor R x).comap (algebraMap R S) ⊔ I = ⊤) (hx' : IsIntegral R x) {J : Ideal S}
(hJ : J ∈ normalizedFactors (I.map (algebraMap R S))) :
multiplicity J (I.map (algebraMap R S)) =
multiplicity (↑(normalizedFacto... |
rw [normalizedFactorsMapEquivNormalizedFactorsMinPolyMk, Equiv.coe_trans, Function.comp_apply,
multiplicity_normalizedFactorsEquivSpanNormalizedFactors_symm_eq_multiplicity,
normalizedFactorsEquivOfQuotEquiv_multiplicity_eq_multiplicity]
|
import Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv
import Mathlib.Analysis.Calculus.FDeriv.Add
import Mathlib.Analysis.Calculus.FDeriv.Prod
import Mathlib.Analysis.NormedSpace.Complemented
#align_import analysis.calculus.implicit from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a21598... | Mathlib/Analysis/Calculus/Implicit.lean | 201 | 214 | theorem implicitFunction_hasStrictFDerivAt (g'inv : G →L[𝕜] E)
(hg'inv : φ.rightDeriv.comp g'inv = ContinuousLinearMap.id 𝕜 G)
(hg'invf : φ.leftDeriv.comp g'inv = 0) :
HasStrictFDerivAt (φ.implicitFunction (φ.leftFun φ.pt)) g'inv (φ.rightFun φ.pt) := by |
have := φ.hasStrictFDerivAt.to_localInverse
simp only [prodFun] at this
convert this.comp (φ.rightFun φ.pt) ((hasStrictFDerivAt_const _ _).prod (hasStrictFDerivAt_id _))
-- Porting note: added parentheses to help `simp`
simp only [ContinuousLinearMap.ext_iff, (ContinuousLinearMap.comp_apply)] at hg'inv hg'in... |
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex
#align_import analysis.special_functions.trigonometric.arctan from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
namespace Real
open Set Filter
open scoped Topology Real
theorem tan_add {x y : ℝ}
... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Arctan.lean | 225 | 227 | theorem arctan_inv_of_neg {x : ℝ} (h : x < 0) : arctan x⁻¹ = -(π / 2) - arctan x := by |
have := arctan_inv_of_pos (neg_pos.mpr h)
rwa [inv_neg, arctan_neg, neg_eq_iff_eq_neg, neg_sub', arctan_neg, neg_neg] at this
|
import Mathlib.Data.Finsupp.Encodable
import Mathlib.LinearAlgebra.Pi
import Mathlib.LinearAlgebra.Span
import Mathlib.Data.Set.Countable
#align_import linear_algebra.finsupp from "leanprover-community/mathlib"@"9d684a893c52e1d6692a504a118bfccbae04feeb"
noncomputable section
open Set LinearMap Submodule
namespa... | Mathlib/LinearAlgebra/Finsupp.lean | 430 | 436 | theorem disjoint_supported_supported_iff [Nontrivial M] {s t : Set α} :
Disjoint (supported M R s) (supported M R t) ↔ Disjoint s t := by |
refine ⟨fun h => Set.disjoint_left.mpr fun x hx1 hx2 => ?_, disjoint_supported_supported⟩
rcases exists_ne (0 : M) with ⟨y, hy⟩
have := h.le_bot ⟨single_mem_supported R y hx1, single_mem_supported R y hx2⟩
rw [mem_bot, single_eq_zero] at this
exact hy this
|
import Mathlib.Init.Function
import Mathlib.Init.Order.Defs
#align_import data.bool.basic from "leanprover-community/mathlib"@"c4658a649d216f57e99621708b09dcb3dcccbd23"
namespace Bool
@[deprecated (since := "2024-06-07")] alias decide_True := decide_true_eq_true
#align bool.to_bool_true decide_true_eq_true
@[dep... | Mathlib/Data/Bool/Basic.lean | 185 | 185 | theorem xor_iff_ne : ∀ {x y : Bool}, xor x y = true ↔ x ≠ y := by | decide
|
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Iterate
import Mathlib.Order.SemiconjSup
import Mathlib.Tactic.Monotonicity
import Mathlib.Topology.Order.MonotoneContinuity
#align_import dynamics.circle.rotation_number.translation_number from "leanprover-... | Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean | 875 | 878 | theorem map_lt_add_translationNumber_add_one (x : ℝ) : f x < x + τ f + 1 :=
calc
f x < x + ⌊τ f⌋ + 1 := f.map_lt_add_floor_translationNumber_add_one x
_ ≤ x + τ f + 1 := by | gcongr; apply floor_le
|
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.integral.average from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open ENNReal MeasureTheory MeasureTheory.Measure Metric Set Filter TopologicalSpace Function
open scoped Topology ENNReal Convex
variable... | Mathlib/MeasureTheory/Integral/Average.lean | 205 | 215 | theorem laverage_union_mem_segment (hd : AEDisjoint μ s t) (ht : NullMeasurableSet t μ)
(hsμ : μ s ≠ ∞) (htμ : μ t ≠ ∞) :
⨍⁻ x in s ∪ t, f x ∂μ ∈ [⨍⁻ x in s, f x ∂μ -[ℝ≥0∞] ⨍⁻ x in t, f x ∂μ] := by |
by_cases hs₀ : μ s = 0
· rw [← ae_eq_empty] at hs₀
rw [restrict_congr_set (hs₀.union EventuallyEq.rfl), empty_union]
exact right_mem_segment _ _ _
· refine
⟨μ s / (μ s + μ t), μ t / (μ s + μ t), zero_le _, zero_le _, ?_, (laverage_union hd ht).symm⟩
rw [← ENNReal.add_div,
ENNReal.div_self... |
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 | 111 | 122 | theorem LinearMap.lTensor_surjective (hg : Function.Surjective g) :
Function.Surjective (lTensor Q g) := by |
intro z
induction z using TensorProduct.induction_on with
| zero => exact ⟨0, map_zero _⟩
| tmul q p =>
obtain ⟨n, rfl⟩ := hg p
exact ⟨q ⊗ₜ[R] n, rfl⟩
| add x y hx hy =>
obtain ⟨x, rfl⟩ := hx
obtain ⟨y, rfl⟩ := hy
exact ⟨x + y, map_add _ _ _⟩
|
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 | 107 | 107 | theorem birthday_one : birthday 1 = 1 := by | rw [birthday_def]; simp
|
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 | 178 | 181 | theorem exists_sum_eq_one_iff_pairwise_coprime' [Fintype I] [Nonempty I] [DecidableEq I] :
(∃ μ : I → R, (∑ i : I, μ i * ∏ j ∈ {i}ᶜ, s j) = 1) ↔ Pairwise (IsCoprime on s) := by |
convert exists_sum_eq_one_iff_pairwise_coprime Finset.univ_nonempty (s := s) using 1
simp only [Function.onFun, pairwise_subtype_iff_pairwise_finset', coe_univ, Set.pairwise_univ]
|
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Analysis.Normed.Group.AddCircle
import Mathlib.Algebra.CharZero.Quotient
import Mathlib.Topology.Instances.Sign
#align_import analysis.special_functions.trigonometric.angle from "leanprover-community/mathlib"@"213b0cff7bc5ab6696ee07cceec80829... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | 826 | 828 | theorem tan_eq_of_two_zsmul_eq {θ ψ : Angle} (h : (2 : ℤ) • θ = (2 : ℤ) • ψ) : tan θ = tan ψ := by |
simp_rw [two_zsmul, ← two_nsmul] at h
exact tan_eq_of_two_nsmul_eq h
|
import Mathlib.Probability.Kernel.CondDistrib
#align_import probability.kernel.condexp from "leanprover-community/mathlib"@"00abe0695d8767201e6d008afa22393978bb324d"
open MeasureTheory Set Filter TopologicalSpace
open scoped ENNReal MeasureTheory ProbabilityTheory
namespace ProbabilityTheory
section AuxLemmas
... | Mathlib/Probability/Kernel/Condexp.lean | 52 | 57 | theorem _root_.MeasureTheory.Integrable.comp_snd_map_prod_id [NormedAddCommGroup F] (hm : m ≤ mΩ)
(hf : Integrable f μ) : Integrable (fun x : Ω × Ω => f x.2)
(@Measure.map Ω (Ω × Ω) (m.prod mΩ) mΩ (fun ω => (id ω, id ω)) μ) := by |
rw [← integrable_comp_snd_map_prod_mk_iff (measurable_id'' hm)] at hf
simp_rw [id] at hf ⊢
exact hf
|
import Mathlib.Algebra.Algebra.Spectrum
import Mathlib.LinearAlgebra.GeneralLinearGroup
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.RingTheory.Nilpotent.Basic
#align_import linear_algebra.eigenspace.basic from "leanprover-community/mathlib"@"6b0169218d01f2837d79ea2784882009a0da1aa1"
universe u v... | Mathlib/LinearAlgebra/Eigenspace/Basic.lean | 98 | 101 | theorem hasEigenvalue_of_hasEigenvector {f : End R M} {μ : R} {x : M} (h : HasEigenvector f μ x) :
HasEigenvalue f μ := by |
rw [HasEigenvalue, Submodule.ne_bot_iff]
use x; exact h
|
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.Calculus.Deriv.Linear
import Mathlib.Analysis.Complex.Conformal
import Mathlib.Analysis.Calculus.Conformal.NormedSpace
#align_import analysis.complex.real_deriv from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
se... | Mathlib/Analysis/Complex/RealDeriv.lean | 135 | 136 | theorem HasDerivAt.comp_ofReal (hf : HasDerivAt e e' ↑z) : HasDerivAt (fun y : ℝ => e ↑y) e' z := by |
simpa only [ofRealCLM_apply, ofReal_one, mul_one] using hf.comp z ofRealCLM.hasDerivAt
|
import Mathlib.Topology.Separation
import Mathlib.Topology.Bases
#align_import topology.dense_embedding from "leanprover-community/mathlib"@"148aefbd371a25f1cff33c85f20c661ce3155def"
noncomputable section
open Set Filter
open scoped Topology
variable {α : Type*} {β : Type*} {γ : Type*} {δ : Type*}
structure D... | Mathlib/Topology/DenseEmbedding.lean | 83 | 90 | theorem interior_compact_eq_empty [T2Space β] (di : DenseInducing i) (hd : Dense (range i)ᶜ)
{s : Set α} (hs : IsCompact s) : interior s = ∅ := by |
refine eq_empty_iff_forall_not_mem.2 fun x hx => ?_
rw [mem_interior_iff_mem_nhds] at hx
have := di.closure_image_mem_nhds hx
rw [(hs.image di.continuous).isClosed.closure_eq] at this
rcases hd.inter_nhds_nonempty this with ⟨y, hyi, hys⟩
exact hyi (image_subset_range _ _ hys)
|
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Iterate
import Mathlib.Order.SemiconjSup
import Mathlib.Tactic.Monotonicity
import Mathlib.Topology.Order.MonotoneContinuity
#align_import dynamics.circle.rotation_number.translation_number from "leanprover-... | Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean | 519 | 521 | theorem dist_map_map_zero_lt : dist (f 0 + g 0) (f (g 0)) < 1 := by |
rw [dist_comm, Real.dist_eq, abs_lt, lt_sub_iff_add_lt', sub_lt_iff_lt_add', ← sub_eq_add_neg]
exact ⟨f.lt_map_map_zero g, f.map_map_zero_lt g⟩
|
import Mathlib.SetTheory.Game.Basic
import Mathlib.SetTheory.Ordinal.NaturalOps
#align_import set_theory.game.ordinal from "leanprover-community/mathlib"@"b90e72c7eebbe8de7c8293a80208ea2ba135c834"
universe u
open SetTheory PGame
open scoped NaturalOps PGame
namespace Ordinal
noncomputable def toPGame : Ordin... | Mathlib/SetTheory/Game/Ordinal.lean | 134 | 137 | theorem toPGame_le {a b : Ordinal} (h : a ≤ b) : a.toPGame ≤ b.toPGame := by |
refine le_iff_forall_lf.2 ⟨fun i => ?_, isEmptyElim⟩
rw [toPGame_moveLeft']
exact toPGame_lf ((toLeftMovesToPGame_symm_lt i).trans_le h)
|
import Mathlib.CategoryTheory.Comma.Over
import Mathlib.CategoryTheory.DiscreteCategory
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.Shapes.Terminal
#align_import category_theory.limits.shapes.binary_products from "leanprover-community/mathlib"@"fec1d95fc61c750c1ddbb5b1f7f48b8e811a80d7"
... | Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean | 843 | 845 | theorem coprod.desc_comp_assoc {C : Type u} [Category C] {V W X Y : C}
[HasBinaryCoproduct X Y] (f : V ⟶ W) (g : X ⟶ V) (h : Y ⟶ V) {Z : C} (l : W ⟶ Z) :
coprod.desc g h ≫ f ≫ l = coprod.desc (g ≫ f) (h ≫ f) ≫ l := by | simp
|
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 | 535 | 551 | theorem exists_subset_nhds_of_isCompact' [Nonempty ι] {V : ι → Set X}
(hV : Directed (· ⊇ ·) V) (hV_cpct : ∀ i, IsCompact (V i)) (hV_closed : ∀ i, IsClosed (V i))
{U : Set X} (hU : ∀ x ∈ ⋂ i, V i, U ∈ 𝓝 x) : ∃ i, V i ⊆ U := by |
obtain ⟨W, hsubW, W_op, hWU⟩ := exists_open_set_nhds hU
suffices ∃ i, V i ⊆ W from this.imp fun i hi => hi.trans hWU
by_contra! H
replace H : ∀ i, (V i ∩ Wᶜ).Nonempty := fun i => Set.inter_compl_nonempty_iff.mpr (H i)
have : (⋂ i, V i ∩ Wᶜ).Nonempty := by
refine
IsCompact.nonempty_iInter_of_directe... |
import Mathlib.Data.PFunctor.Multivariate.W
import Mathlib.Data.QPF.Multivariate.Basic
#align_import data.qpf.multivariate.constructions.fix from "leanprover-community/mathlib"@"28aa996fc6fb4317f0083c4e6daf79878d81be33"
universe u v
namespace MvQPF
open TypeVec
open MvFunctor (LiftP LiftR)
open MvFunctor
var... | Mathlib/Data/QPF/Multivariate/Constructions/Fix.lean | 64 | 67 | theorem recF_eq {α : TypeVec n} {β : Type u} (g : F (α.append1 β) → β) (a : q.P.A)
(f' : q.P.drop.B a ⟹ α) (f : q.P.last.B a → q.P.W α) :
recF g (q.P.wMk a f' f) = g (abs ⟨a, splitFun f' (recF g ∘ f)⟩) := by |
rw [recF, MvPFunctor.wRec_eq]; rfl
|
import Mathlib.FieldTheory.Galois
import Mathlib.Topology.Algebra.FilterBasis
import Mathlib.Topology.Algebra.OpenSubgroup
import Mathlib.Tactic.ByContra
#align_import field_theory.krull_topology from "leanprover-community/mathlib"@"039a089d2a4b93c761b234f3e5f5aeb752bac60f"
open scoped Classical Pointwise
theore... | Mathlib/FieldTheory/KrullTopology.lean | 93 | 100 | theorem IntermediateField.fixingSubgroup.bot {K L : Type*} [Field K] [Field L] [Algebra K L] :
IntermediateField.fixingSubgroup (⊥ : IntermediateField K L) = ⊤ := by |
ext f
refine ⟨fun _ => Subgroup.mem_top _, fun _ => ?_⟩
rintro ⟨x, hx : x ∈ (⊥ : IntermediateField K L)⟩
rw [IntermediateField.mem_bot] at hx
rcases hx with ⟨y, rfl⟩
exact f.commutes y
|
import Mathlib.Combinatorics.SimpleGraph.Subgraph
import Mathlib.Data.List.Rotate
#align_import combinatorics.simple_graph.connectivity from "leanprover-community/mathlib"@"b99e2d58a5e6861833fa8de11e51a81144258db4"
open Function
universe u v w
namespace SimpleGraph
variable {V : Type u} {V' : Type v} {V'' : Typ... | Mathlib/Combinatorics/SimpleGraph/Connectivity.lean | 1,037 | 1,038 | theorem IsTrail.reverse {u v : V} (p : G.Walk u v) (h : p.IsTrail) : p.reverse.IsTrail := by |
simpa [isTrail_def] using h
|
import Mathlib.Analysis.InnerProductSpace.TwoDim
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
#align_import geometry.euclidean.angle.oriented.basic from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
noncomputable section
open FiniteDimensional Complex
open scoped Real Rea... | Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean | 743 | 751 | theorem oangle_eq_neg_angle_of_sign_eq_neg_one {x y : V} (h : (o.oangle x y).sign = -1) :
o.oangle x y = -InnerProductGeometry.angle x y := by |
by_cases hx : x = 0; · exfalso; simp [hx] at h
by_cases hy : y = 0; · exfalso; simp [hy] at h
refine (o.oangle_eq_angle_or_eq_neg_angle hx hy).resolve_left ?_
intro hxy
rw [hxy, ← SignType.neg_iff, ← not_le] at h
exact h (Real.Angle.sign_coe_nonneg_of_nonneg_of_le_pi (InnerProductGeometry.angle_nonneg _ _)... |
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Data.Nat.Totient
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.GroupTheory.Subgroup.Simple
import Mathlib.Tactic.Group
import Mathlib.GroupTheory.Exponent
#align_import group_theory.specific_groups.cyclic from "leanprover-community/mathli... | Mathlib/GroupTheory/SpecificGroups/Cyclic.lean | 170 | 173 | theorem isCyclic_of_prime_card {α : Type u} [Group α] [Fintype α] {p : ℕ} [hp : Fact p.Prime]
(h : Fintype.card α = p) : IsCyclic α := by |
obtain ⟨g, hg⟩ : ∃ g, g ≠ 1 := Fintype.exists_ne_of_one_lt_card (h.symm ▸ hp.1.one_lt) 1
exact ⟨g, fun g' ↦ mem_zpowers_of_prime_card h hg⟩
|
import Mathlib.Logic.Relation
import Mathlib.Data.Option.Basic
import Mathlib.Data.Seq.Seq
#align_import data.seq.wseq from "leanprover-community/mathlib"@"a7e36e48519ab281320c4d192da6a7b348ce40ad"
namespace Stream'
open Function
universe u v w
def WSeq (α) :=
Seq (Option α)
#align stream.wseq Stream'.WSeq
... | Mathlib/Data/Seq/WSeq.lean | 804 | 806 | theorem destruct_tail (s : WSeq α) : destruct (tail s) = destruct s >>= tail.aux := by |
simp only [tail, destruct_flatten, tail.aux]; rw [← bind_pure_comp, LawfulMonad.bind_assoc]
apply congr_arg; ext1 (_ | ⟨a, s⟩) <;> apply (@pure_bind Computation _ _ _ _ _ _).trans _ <;> simp
|
import Mathlib.Algebra.Group.Subgroup.Finite
import Mathlib.Algebra.Group.Subgroup.Pointwise
import Mathlib.GroupTheory.Congruence.Basic
import Mathlib.GroupTheory.Coset
#align_import group_theory.quotient_group from "leanprover-community/mathlib"@"59694bd07f0a39c5beccba34bd9f413a160782bf"
open Function
open scope... | Mathlib/GroupTheory/QuotientGroup.lean | 149 | 152 | theorem eq_iff_div_mem {N : Subgroup G} [nN : N.Normal] {x y : G} :
(x : G ⧸ N) = y ↔ x / y ∈ N := by |
refine eq_comm.trans (QuotientGroup.eq.trans ?_)
rw [nN.mem_comm_iff, div_eq_mul_inv]
|
import Mathlib.MeasureTheory.Measure.Content
import Mathlib.MeasureTheory.Group.Prod
import Mathlib.Topology.Algebra.Group.Compact
#align_import measure_theory.measure.haar.basic from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
noncomputable section
open Set Inv Function Topological... | Mathlib/MeasureTheory/Measure/Haar/Basic.lean | 273 | 283 | theorem mul_left_index_le {K : Set G} (hK : IsCompact K) {V : Set G} (hV : (interior V).Nonempty)
(g : G) : index ((fun h => g * h) '' K) V ≤ index K V := by |
rcases index_elim hK hV with ⟨s, h1s, h2s⟩; rw [← h2s]
apply Nat.sInf_le; rw [mem_image]
refine ⟨s.map (Equiv.mulRight g⁻¹).toEmbedding, ?_, Finset.card_map _⟩
simp only [mem_setOf_eq]; refine Subset.trans (image_subset _ h1s) ?_
rintro _ ⟨g₁, ⟨_, ⟨g₂, rfl⟩, ⟨_, ⟨hg₂, rfl⟩, hg₁⟩⟩, rfl⟩
simp only [mem_preim... |
import Mathlib.Combinatorics.SetFamily.Shadow
#align_import combinatorics.set_family.compression.uv from "leanprover-community/mathlib"@"6f8ab7de1c4b78a68ab8cf7dd83d549eb78a68a1"
open Finset
variable {α : Type*}
theorem sup_sdiff_injOn [GeneralizedBooleanAlgebra α] (u v : α) :
{ x | Disjoint u x ∧ v ≤ x }.... | Mathlib/Combinatorics/SetFamily/Compression/UV.lean | 107 | 110 | theorem compress_sdiff_sdiff (a b : α) : compress (a \ b) (b \ a) b = a := by |
refine (compress_of_disjoint_of_le disjoint_sdiff_self_left sdiff_le).trans ?_
rw [sup_sdiff_self_right, sup_sdiff, disjoint_sdiff_self_right.sdiff_eq_left, sup_eq_right]
exact sdiff_sdiff_le
|
import Mathlib.Probability.Variance
#align_import probability.moments from "leanprover-community/mathlib"@"85453a2a14be8da64caf15ca50930cf4c6e5d8de"
open MeasureTheory Filter Finset Real
noncomputable section
open scoped MeasureTheory ProbabilityTheory ENNReal NNReal
namespace ProbabilityTheory
variable {Ω ι ... | Mathlib/Probability/Moments.lean | 156 | 157 | theorem mgf_zero' : mgf X μ 0 = (μ Set.univ).toReal := by |
simp only [mgf, zero_mul, exp_zero, integral_const, smul_eq_mul, mul_one]
|
import Mathlib.Logic.Pairwise
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#align_import data.set.lattice from "leanprover-community/mathlib"@"b86832321b586c6ac23ef8cdef6a7a27e42b13bd"
open Function Set
universe u
variable {α β γ : Type*} {ι ι' ι... | Mathlib/Data/Set/Lattice.lean | 1,506 | 1,512 | theorem surjective_iff_surjective_of_iUnion_eq_univ :
Surjective f ↔ ∀ i, Surjective ((U i).restrictPreimage f) := by |
refine ⟨fun H i => (U i).restrictPreimage_surjective H, fun H x => ?_⟩
obtain ⟨i, hi⟩ :=
Set.mem_iUnion.mp
(show x ∈ Set.iUnion U by rw [hU]; trivial)
exact ⟨_, congr_arg Subtype.val (H i ⟨x, hi⟩).choose_spec⟩
|
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Algebra.Order.Group.Indicator
import Mathlib.Order.LiminfLimsup
import Mathlib.Order.Filter.Archimedean
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.Algebra.Group.Basic
import Mathlib.Data.Set.La... | Mathlib/Topology/Algebra/Order/LiminfLimsup.lean | 487 | 498 | theorem iUnion_Ici_eq_Ioi_of_lt_of_tendsto (x : R) {as : ι → R} (x_lt : ∀ i, x < as i)
{F : Filter ι} [Filter.NeBot F] (as_lim : Filter.Tendsto as F (𝓝 x)) :
⋃ i : ι, Ici (as i) = Ioi x := by |
have obs : x ∉ range as := by
intro maybe_x_is
rcases mem_range.mp maybe_x_is with ⟨i, hi⟩
simpa only [hi, lt_self_iff_false] using x_lt i
-- Porting note: `rw at *` was too destructive. Let's only rewrite `obs` and the goal.
have := iInf_eq_of_forall_le_of_tendsto (fun i ↦ (x_lt i).le) as_lim
rw [... |
import Mathlib.Algebra.MvPolynomial.Supported
import Mathlib.RingTheory.WittVector.Truncated
#align_import ring_theory.witt_vector.mul_coeff from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
noncomputable section
namespace WittVector
variable (p : ℕ) [hp : Fact p.Prime]
variable {k ... | Mathlib/RingTheory/WittVector/MulCoeff.lean | 69 | 85 | theorem wittPolyProdRemainder_vars (n : ℕ) :
(wittPolyProdRemainder p n).vars ⊆ univ ×ˢ range n := by |
rw [wittPolyProdRemainder]
refine Subset.trans (vars_sum_subset _ _) ?_
rw [biUnion_subset]
intro x hx
apply Subset.trans (vars_mul _ _)
refine union_subset ?_ ?_
· apply Subset.trans (vars_pow _ _)
have : (p : 𝕄) = C (p : ℤ) := by simp only [Int.cast_natCast, eq_intCast]
rw [this, vars_C]
a... |
import Mathlib.Data.Nat.Cast.Basic
import Mathlib.Algebra.CharZero.Defs
import Mathlib.Algebra.Order.Group.Abs
import Mathlib.Data.Nat.Cast.NeZero
import Mathlib.Algebra.Order.Ring.Nat
#align_import data.nat.cast.basic from "leanprover-community/mathlib"@"acebd8d49928f6ed8920e502a6c90674e75bd441"
variable {α β : T... | Mathlib/Data/Nat/Cast/Order.lean | 134 | 134 | theorem one_lt_cast : 1 < (n : α) ↔ 1 < n := by | rw [← cast_one, cast_lt]
|
import Mathlib.Data.List.Join
#align_import data.list.permutation from "leanprover-community/mathlib"@"dd71334db81d0bd444af1ee339a29298bef40734"
-- Make sure we don't import algebra
assert_not_exists Monoid
open Nat
variable {α β : Type*}
namespace List
theorem permutationsAux2_fst (t : α) (ts : List α) (r : L... | Mathlib/Data/List/Permutation.lean | 267 | 269 | theorem permutations_append (is ts : List α) :
permutations (is ++ ts) = (permutations is).map (· ++ ts) ++ permutationsAux ts is.reverse := by |
simp [permutations, permutationsAux_append]
|
import Mathlib.Order.RelClasses
import Mathlib.Order.Interval.Set.Basic
#align_import order.bounded from "leanprover-community/mathlib"@"aba57d4d3dae35460225919dcd82fe91355162f9"
namespace Set
variable {α : Type*} {r : α → α → Prop} {s t : Set α}
theorem Bounded.mono (hst : s ⊆ t) (hs : Bounded r t) : Bounde... | Mathlib/Order/Bounded.lean | 108 | 113 | theorem bounded_le_iff_bounded_lt [Preorder α] [NoMaxOrder α] :
Bounded (· ≤ ·) s ↔ Bounded (· < ·) s := by |
refine ⟨fun h => ?_, bounded_le_of_bounded_lt⟩
cases' h with a ha
cases' exists_gt a with b hb
exact ⟨b, fun c hc => lt_of_le_of_lt (ha c hc) hb⟩
|
import Mathlib.Algebra.Associated
import Mathlib.Algebra.GeomSum
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.Data.Nat.Lattice
import Mathlib.RingTheory.Nilpotent.Defs
#align_import ring_th... | Mathlib/RingTheory/Nilpotent/Basic.lean | 58 | 62 | theorem IsNilpotent.isUnit_sub_one [Ring R] {r : R} (hnil : IsNilpotent r) : IsUnit (r - 1) := by |
obtain ⟨n, hn⟩ := hnil
refine ⟨⟨r - 1, -∑ i ∈ Finset.range n, r ^ i, ?_, ?_⟩, rfl⟩
· simp [mul_geom_sum, hn]
· simp [geom_sum_mul, hn]
|
import Mathlib.CategoryTheory.NatIso
import Mathlib.CategoryTheory.EqToHom
#align_import category_theory.quotient from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a485d0456fc271482da"
def HomRel (C) [Quiver C] :=
∀ ⦃X Y : C⦄, (X ⟶ Y) → (X ⟶ Y) → Prop
#align hom_rel HomRel
-- Porting Note: `deriving I... | Mathlib/CategoryTheory/Quotient.lean | 177 | 183 | theorem lift_spec : functor r ⋙ lift r F H = F := by |
apply Functor.ext; rotate_left
· rintro X
rfl
· rintro X Y f
dsimp [lift, functor]
simp
|
import Mathlib.RingTheory.IntegrallyClosed
import Mathlib.RingTheory.Trace
import Mathlib.RingTheory.Norm
#align_import ring_theory.discriminant from "leanprover-community/mathlib"@"3e068ece210655b7b9a9477c3aff38a492400aa1"
universe u v w z
open scoped Matrix
open Matrix FiniteDimensional Fintype Polynomial Fin... | Mathlib/RingTheory/Discriminant.lean | 215 | 258 | theorem discr_powerBasis_eq_norm [IsSeparable K L] :
discr K pb.basis =
(-1) ^ (n * (n - 1) / 2) *
norm K (aeval pb.gen (derivative (R := K) (minpoly K pb.gen))) := by |
let E := AlgebraicClosure L
letI := fun a b : E => Classical.propDecidable (Eq a b)
have e : Fin pb.dim ≃ (L →ₐ[K] E) := by
refine equivOfCardEq ?_
rw [Fintype.card_fin, AlgHom.card]
exact (PowerBasis.finrank pb).symm
have hnodup : ((minpoly K pb.gen).aroots E).Nodup :=
nodup_roots (Separable.m... |
import Mathlib.CategoryTheory.Monoidal.Category
import Mathlib.CategoryTheory.Adjunction.FullyFaithful
import Mathlib.CategoryTheory.Products.Basic
#align_import category_theory.monoidal.functor from "leanprover-community/mathlib"@"3d7987cda72abc473c7cdbbb075170e9ac620042"
open CategoryTheory
universe v₁ v₂ v₃ u... | Mathlib/CategoryTheory/Monoidal/Functor.lean | 372 | 373 | theorem map_whiskerLeft (X : C) {Y Z : C} (f : Y ⟶ Z) :
F.map (X ◁ f) = inv (F.μ X Y) ≫ F.obj X ◁ F.map f ≫ F.μ X Z := by | simp
|
import Mathlib.GroupTheory.GroupAction.BigOperators
import Mathlib.Logic.Equiv.Fin
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.Algebra.Module.Prod
import Mathlib.Algebra.Module.Submodule.Ker
#align_import linear_algebra.pi from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
un... | Mathlib/LinearAlgebra/Pi.lean | 69 | 69 | theorem pi_zero : pi (fun i => 0 : (i : ι) → M₂ →ₗ[R] φ i) = 0 := by | ext; rfl
|
import Mathlib.Algebra.Ring.Int
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.Size
#align_import data.int.bitwise from "leanprover-community/mathlib"@"0743cc5d9d86bcd1bba10f480e948a257d65056f"
#align_import init.data.int.bitwise from "leanprover-community/lean"@"855e5b74e3a52a40552e8f067169d747d48743fd"
... | Mathlib/Data/Int/Bitwise.lean | 173 | 179 | theorem bodd_add (m n : ℤ) : bodd (m + n) = xor (bodd m) (bodd n) := by |
cases' m with m m <;>
cases' n with n n <;>
simp only [ofNat_eq_coe, ofNat_add_negSucc, negSucc_add_ofNat,
negSucc_add_negSucc, bodd_subNatNat] <;>
simp only [negSucc_coe, bodd_neg, bodd_coe, ← Nat.bodd_add, Bool.xor_comm, ← Nat.cast_add]
rw [← Nat.succ_add, add_assoc]
|
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.special_functions.log.base from "leanprover-community/mathlib"@"f23a09ce6d3f367220dc3cecad6b7eb69eb01690"
open Set Filter Function
open Topology
noncomputable section
namespace Real
variable {b x y : ℝ}
-- @... | Mathlib/Analysis/SpecialFunctions/Log/Base.lean | 97 | 98 | theorem logb_mul_base {a b : ℝ} (h₁ : a ≠ 0) (h₂ : b ≠ 0) (c : ℝ) :
logb (a * b) c = ((logb a c)⁻¹ + (logb b c)⁻¹)⁻¹ := by | rw [← inv_logb_mul_base h₁ h₂ c, inv_inv]
|
import Mathlib.Algebra.Homology.Homotopy
import Mathlib.AlgebraicTopology.DoldKan.Notations
#align_import algebraic_topology.dold_kan.homotopies from "leanprover-community/mathlib"@"b12099d3b7febf4209824444dd836ef5ad96db55"
open CategoryTheory CategoryTheory.Category CategoryTheory.Limits CategoryTheory.Preadditi... | Mathlib/AlgebraicTopology/DoldKan/Homotopies.lean | 156 | 166 | theorem hσ'_naturality (q : ℕ) (n m : ℕ) (hnm : c.Rel m n) {X Y : SimplicialObject C} (f : X ⟶ Y) :
f.app (op [n]) ≫ hσ' q n m hnm = hσ' q n m hnm ≫ f.app (op [m]) := by |
have h : n + 1 = m := hnm
subst h
simp only [hσ', eqToHom_refl, comp_id]
unfold hσ
split_ifs
· rw [zero_comp, comp_zero]
· simp only [zsmul_comp, comp_zsmul]
erw [f.naturality]
rfl
|
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.Calculus.Deriv.Mul
import Mathlib.Analysis.Calculus.Deriv.Shift
import Mathlib.Analysis.Calculus.IteratedDeriv.Defs
variable
{𝕜 : Type*} [NontriviallyNormedField 𝕜]
{F : Type*} [NormedAddCommGroup F] [NormedSpace 𝕜 F]
{R : Type*} [Semi... | Mathlib/Analysis/Calculus/IteratedDeriv/Lemmas.lean | 48 | 56 | theorem iteratedDerivWithin_const_neg (hn : 0 < n) (c : F) :
iteratedDerivWithin n (fun z => c - f z) s x = iteratedDerivWithin n (fun z => -f z) s x := by |
obtain ⟨n, rfl⟩ := n.exists_eq_succ_of_ne_zero hn.ne'
rw [iteratedDerivWithin_succ' h hx, iteratedDerivWithin_succ' h hx]
refine iteratedDerivWithin_congr h ?_ hx
intro y hy
have : UniqueDiffWithinAt 𝕜 s y := h.uniqueDiffWithinAt hy
rw [derivWithin.neg this]
exact derivWithin_const_sub this _
|
import Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral
import Mathlib.Analysis.Complex.CauchyIntegral
import Mathlib.MeasureTheory.Integral.Pi
import Mathlib.Analysis.Fourier.FourierTransform
open Real Set MeasureTheory Filter Asymptotics intervalIntegral
open scoped Real Topology FourierTransform Re... | Mathlib/Analysis/SpecialFunctions/Gaussian/FourierTransform.lean | 383 | 386 | theorem _root_.fourierIntegral_gaussian_innerProductSpace (hb : 0 < b.re) (w : V) :
𝓕 (fun v ↦ cexp (- b * ‖v‖^2)) w =
(π / b) ^ (FiniteDimensional.finrank ℝ V / 2 : ℂ) * cexp (-π ^ 2 * ‖w‖^2 / b) := by |
simpa using fourierIntegral_gaussian_innerProductSpace' hb 0 w
|
import Mathlib.Topology.Algebra.Module.WeakDual
import Mathlib.Algebra.Algebra.Spectrum
import Mathlib.Topology.ContinuousFunction.Algebra
import Mathlib.Data.Set.Lattice
#align_import topology.algebra.module.character_space from "leanprover-community/mathlib"@"a148d797a1094ab554ad4183a4ad6f130358ef64"
namespace ... | Mathlib/Topology/Algebra/Module/CharacterSpace.lean | 128 | 134 | theorem union_zero_isClosed [T2Space 𝕜] [ContinuousMul 𝕜] :
IsClosed (characterSpace 𝕜 A ∪ {0}) := by |
simp only [union_zero, Set.setOf_forall]
exact
isClosed_iInter fun x =>
isClosed_iInter fun y =>
isClosed_eq (eval_continuous _) <| (eval_continuous _).mul (eval_continuous _)
|
import Mathlib.Data.Matrix.Basis
import Mathlib.Data.Matrix.DMatrix
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.LinearAlgebra.Matrix.Reindex
import Mathlib.Tactic.FieldSimp
#align_import linear_algebra.matrix.transvection from "leanprover-community/mathlib"@"0e2aab2b0d521f060f62a14d2cf2e2c54e... | Mathlib/LinearAlgebra/Matrix/Transvection.lean | 535 | 544 | theorem isTwoBlockDiagonal_listTransvecCol_mul_mul_listTransvecRow
(hM : M (inr unit) (inr unit) ≠ 0) :
IsTwoBlockDiagonal ((listTransvecCol M).prod * M * (listTransvecRow M).prod) := by |
constructor
· ext i j
have : j = unit := by simp only [eq_iff_true_of_subsingleton]
simp [toBlocks₁₂, this, listTransvecCol_mul_mul_listTransvecRow_last_row M hM]
· ext i j
have : i = unit := by simp only [eq_iff_true_of_subsingleton]
simp [toBlocks₂₁, this, listTransvecCol_mul_mul_listTransvecRo... |
import Mathlib.RingTheory.Localization.Basic
#align_import ring_theory.localization.integer from "leanprover-community/mathlib"@"9556784a5b84697562e9c6acb40500d4a82e675a"
variable {R : Type*} [CommSemiring R] {M : Submonoid R} {S : Type*} [CommSemiring S]
variable [Algebra R S] {P : Type*} [CommSemiring P]
open ... | Mathlib/RingTheory/Localization/Integer.lean | 63 | 66 | theorem isInteger_smul {a : R} {b : S} (hb : IsInteger R b) : IsInteger R (a • b) := by |
rcases hb with ⟨b', hb⟩
use a * b'
rw [← hb, (algebraMap R S).map_mul, Algebra.smul_def]
|
import Mathlib.Order.ConditionallyCompleteLattice.Finset
import Mathlib.Order.Interval.Finset.Nat
#align_import data.nat.lattice from "leanprover-community/mathlib"@"52fa514ec337dd970d71d8de8d0fd68b455a1e54"
assert_not_exists MonoidWithZero
open Set
namespace Nat
open scoped Classical
noncomputable instance : ... | Mathlib/Data/Nat/Lattice.lean | 191 | 192 | theorem iSup_lt_succ (u : ℕ → α) (n : ℕ) : ⨆ k < n + 1, u k = (⨆ k < n, u k) ⊔ u n := by |
simp [Nat.lt_succ_iff_lt_or_eq, iSup_or, iSup_sup_eq]
|
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Data.ZMod.Basic
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Polynomial.Chebyshev
import Mathlib.RingTheory.Ideal.LocalRing
#align_import ring_theory.polynomial.dickson from "leanprover-community/mathlib"@"70fd9563a21e7b963887c936... | Mathlib/RingTheory/Polynomial/Dickson.lean | 82 | 83 | theorem dickson_add_two (n : ℕ) :
dickson k a (n + 2) = X * dickson k a (n + 1) - C a * dickson k a n := by | rw [dickson]
|
import Mathlib.Geometry.Manifold.MFDeriv.UniqueDifferential
import Mathlib.Geometry.Manifold.ContMDiffMap
#align_import geometry.manifold.cont_mdiff_mfderiv from "leanprover-community/mathlib"@"e473c3198bb41f68560cab68a0529c854b618833"
open Set Function Filter ChartedSpace SmoothManifoldWithCorners Bundle
open sc... | Mathlib/Geometry/Manifold/ContMDiffMFDeriv.lean | 533 | 537 | theorem ContMDiff.contMDiff_tangentMap (hf : ContMDiff I I' n f) (hmn : m + 1 ≤ n) :
ContMDiff I.tangent I'.tangent m (tangentMap I I' f) := by |
rw [← contMDiffOn_univ] at hf ⊢
convert hf.contMDiffOn_tangentMapWithin hmn uniqueMDiffOn_univ
rw [tangentMapWithin_univ]
|
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