Context stringlengths 227 76.5k | target stringlengths 0 11.6k | file_name stringlengths 21 79 | start int64 14 3.67k | end int64 16 3.69k |
|---|---|---|---|---|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl, Patrick Massot
-/
import Mathlib.Data.Set.Image
import Mathlib.Data.SProd
/-!
# Sets in product and pi types
This file proves basic properties of prod... | theorem univ_pi_ite (s : Set ι) [DecidablePred (· ∈ s)] (t : ∀ i, Set (α i)) :
(pi univ fun i => if i ∈ s then t i else univ) = s.pi t := by
ext
simp_rw [mem_univ_pi]
| Mathlib/Data/Set/Prod.lean | 889 | 892 |
/-
Copyright (c) 2023 Emily Witt. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Emily Witt, Kim Morrison, Jake Levinson, Sam van Gool
-/
import Mathlib.Algebra.Category.ModuleCat.Colimits
import Mathlib.Algebra.Category.ModuleCat.Projective
import Mathlib.CategoryTheo... | rw [← Ideal.radical_le_radical_iff] at hL ⊢
exact hJK.symm.trans_le hL
-- TODO generalize this to the equivalence of full categories for any `iff`.
/-- The equivalence of categories `SelfLERadical J ≌ SelfLERadical K`
when `J.radical = K.radical`. -/
def SelfLERadical.castEquivalence (hJK : J.radical = K.radic... | Mathlib/Algebra/Homology/LocalCohomology.lean | 231 | 237 |
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Analysis.Convex.Basic
import Mathlib.Topology.Algebra.Group.Pointwise
import Mathlib.Topology.Order.Basic
/-!
# Strictly convex sets
This file defines st... | fun h x hx y hy hxy a b ha hb hab ↦ by
convert h hx hy hxy ha hb <;> rw [hab, div_one]⟩
theorem StrictConvex.mem_smul_of_zero_mem (hs : StrictConvex 𝕜 s) (zero_mem : (0 : E) ∈ s)
(hx : x ∈ s) (hx₀ : x ≠ 0) {t : 𝕜} (ht : 1 < t) : x ∈ t • interior s := by
rw [mem_smul_set_iff_inv_smul_mem₀ (by positivity... | Mathlib/Analysis/Convex/Strict.lean | 366 | 371 |
/-
Copyright (c) 2023 Antoine Chambert-Loir and María Inés de Frutos-Fernández. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Chambert-Loir, María Inés de Frutos-Fernández, Bhavik Mehta, Eric Wieser
-/
import Mathlib.Algebra.Order.Monoid.Canonical.Defs
import ... |
theorem swap_mem_antidiagonal [AddCommMonoid A] [HasAntidiagonal A] {n : A} {xy : A × A} :
xy.swap ∈ antidiagonal n ↔ xy ∈ antidiagonal n := by
| Mathlib/Algebra/Order/Antidiag/Prod.lean | 79 | 81 |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Alex Kontorovich, Heather Macbeth
-/
import Mathlib.MeasureTheory.Integral.IntervalIntegral.Periodic
deprecated_module (since := "2025-04-13")
| Mathlib/MeasureTheory/Integral/Periodic.lean | 181 | 192 | |
/-
Copyright (c) 2018 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.BigOperators.Expect
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Algebra.Order.Field.Canonical
import Mathlib.Algebra.O... | Mathlib/Data/NNReal/Basic.lean | 1,094 | 1,096 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kim Morrison
-/
import Mathlib.Algebra.BigOperators.Finsupp.Basic
import Mathlib.Algebra.BigOperators.Group.Finset.Preimage
import Mathlib.Algebra.Module.Defs
import Ma... | Mathlib/Data/Finsupp/Basic.lean | 1,598 | 1,604 | |
/-
Copyright (c) 2022 Eric Rodriguez. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Rodriguez
-/
import Mathlib.Algebra.GroupWithZero.Units.Lemmas
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Ring.Cast
import Mathlib.Data.Fi... | -/
| Mathlib/Data/Sign.lean | 219 | 219 |
/-
Copyright (c) 2018 Andreas Swerdlow. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andreas Swerdlow, Kexing Ying
-/
import Mathlib.Algebra.Algebra.Tower
import Mathlib.LinearAlgebra.BilinearMap
/-!
# Bilinear form
This file defines a bilinear form over a module. ... | Mathlib/LinearAlgebra/BilinearForm/Basic.lean | 192 | 195 | |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.CharP.Frobenius
import Mathlib.Algebra.CharP.Pi
import Mathlib.Algebra.CharP.Quotient
import Mathlib.Algebra.CharP.Subring
import Mathlib.Analysis.Specia... | if x = 0 then 0 else v (algebraMap O K x.out)
| Mathlib/RingTheory/Perfection.lean | 361 | 362 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jeremy Avigad
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Notation.Pi
import Mathlib.Data.Set.Lattice
import Mathlib.Order.Filter.Defs
/-!
# Theory of... | Mathlib/Order/Filter/Basic.lean | 1,317 | 1,319 | |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
-/
import Mathlib.Data.Stream.Defs
import Mathlib.Logic.Function.Basic
import Mathlib.Data.List.Defs
import Mathlib.Data.Nat.Basic
import Mathlib.Tactic.Common... |
theorem cycle_eq : ∀ (l : List α) (h : l ≠ []), cycle l h = l ++ₛ cycle l h
| [], h => absurd rfl h
| List.cons a l, _ =>
have gen : ∀ l' a', corec Stream'.cycleF Stream'.cycleG (a', l', a, l) =
(a'::l') ++ₛ corec Stream'.cycleF Stream'.cycleG (a, l, a, l) := by
intro l'
induction' l' with ... | Mathlib/Data/Stream/Init.lean | 615 | 622 |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.FormalMultilinearSeries
import Mathlib.Analysis.SpecificLimits.Normed
import Mathlib.Logic.Equiv.Fin.Basic
imp... | let ⟨_, hrf⟩ := hf
hrf.coeff_zero v
/-!
### Analytic functions
-/
@[simp] theorem analyticOn_empty : AnalyticOn 𝕜 f ∅ := by intro; simp
@[simp] theorem analyticOnNhd_empty : AnalyticOnNhd 𝕜 f ∅ := by intro; simp
| Mathlib/Analysis/Analytic/Basic.lean | 719 | 729 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Group.Indicator
import Mathlib.Data.Int.Cast.Pi
import Mathlib.Data.Nat.Cast.Basic
import Mathlib.MeasureTheory.MeasurableSpace... | Mathlib/MeasureTheory/MeasurableSpace/Basic.lean | 562 | 566 | |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Data.Set.Piecewise
import Mathlib.Logic.Equiv.Defs
import Mathlib.Tactic.Core
import Mathlib.Tactic.Attr.Core
/-!
# Partial equivalences
This f... | (PartialEquiv.pi ei).symm = .pi fun i ↦ (ei i).symm :=
rfl
| Mathlib/Logic/Equiv/PartialEquiv.lean | 861 | 862 |
/-
Copyright (c) 2024 Christian Merten. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christian Merten
-/
import Mathlib.CategoryTheory.Galois.Basic
import Mathlib.CategoryTheory.Limits.FintypeCat
import Mathlib.CategoryTheory.Limits.Preserves.Limits
import Mathlib.Ca... | lemma evaluationEquivOfIsGalois_apply (A : C) [IsGalois A] (a : F.obj A) (φ : Aut A) :
evaluationEquivOfIsGalois F A a φ = F.map φ.hom a :=
rfl
@[simp]
lemma evaluationEquivOfIsGalois_symm_fiber (A : C) [IsGalois A] (a b : F.obj A) :
F.map ((evaluationEquivOfIsGalois F A a).symm b).hom a = b := by
change (... | Mathlib/CategoryTheory/Galois/GaloisObjects.lean | 123 | 135 |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.ShortComplex.QuasiIso
import Mathlib.CategoryTheory.Limits.Preserves.Finite
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Kernels
/-!
... | variable [hl₁.IsPreservedBy F] [hl₂.IsPreservedBy F]
lemma map_cyclesMap' : F.map (ShortComplex.cyclesMap' φ hl₁ hl₂) =
ShortComplex.cyclesMap' (F.mapShortComplex.map φ) (hl₁.map F) (hl₂.map F) := by
| Mathlib/Algebra/Homology/ShortComplex/PreservesHomology.lean | 360 | 363 |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Complex.Circle
import Mathlib.Analysis.SpecialFunctions.Complex.Log
/-!
# Maps on the unit circle
In this file we prove some basic lemmas ... | (@scaled_exp_map_periodic T).lift
theorem toCircle_apply_mk (x : ℝ) : @toCircle T x = Circle.exp (2 * π / T * x) :=
rfl
| Mathlib/Analysis/SpecialFunctions/Complex/Circle.lean | 135 | 138 |
/-
Copyright (c) 2022 Rémi Bottinelli. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémi Bottinelli, Junyan Xu
-/
import Mathlib.Algebra.Group.Subgroup.Defs
import Mathlib.CategoryTheory.Groupoid.VertexGroup
import Mathlib.CategoryTheory.Groupoid.Basic
import Mathlib... | Mathlib/CategoryTheory/Groupoid/Subgroupoid.lean | 692 | 696 | |
/-
Copyright (c) 2014 Parikshit Khanna. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro
-/
import Mathlib.Control.Basic
import Mathlib.Data.Nat.Basic
import Mathlib.Data.Option.Basic
im... | Mathlib/Data/List/Basic.lean | 1,404 | 1,416 | |
/-
Copyright (c) 2021 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer
-/
import Mathlib.Tactic.CategoryTheory.Monoidal.Basic
import Mathlib.CategoryTheory.Closed.Monoidal
import Mathlib.Tactic.ApplyFun
/-!
# Rigid (autonomous) monoida... | @[simp]
theorem tensorLeftHomEquiv_whiskerRight_comp_evaluation {X Y : C} [HasLeftDual X] [HasLeftDual Y]
(f : X ⟶ Y) : (tensorLeftHomEquiv _ _ _ _) (f ▷ _ ≫ ε_ _ _) = (ᘁf) ≫ (ρ_ _).inv := by
dsimp [tensorLeftHomEquiv, leftAdjointMate]
simp
| Mathlib/CategoryTheory/Monoidal/Rigid/Basic.lean | 430 | 434 |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Analytic.CPolynomial
import Mathlib.Analysis.Analytic.Inverse
import Mathlib.Analysis.Analytic.Within
import Mathlib.Analysis.Calculus.Deriv... |
/-- If a function is analytic on an open set `s`, so is its derivative. -/
theorem AnalyticOnNhd.deriv_of_isOpen (h : AnalyticOnNhd 𝕜 f s) (hs : IsOpen s) :
| Mathlib/Analysis/Calculus/FDeriv/Analytic.lean | 432 | 434 |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Data.Finset.Sort
import Mathlib.Data.Fin.VecNotation
import Mathlib.Data.Sign
import Mathlib.LinearAlgebra.AffineSpace.Combination
import Mathlib.LinearAlg... | · rcases h with ⟨r, hr⟩
refine ⟨r, fun i hi => ?_⟩
rw [s.affineCombination_vsub, ← s.weightedVSub_const_smul, ← sub_eq_zero, ← map_sub] at hr
have hw' : (∑ j ∈ s, (r • (w₁ - w₂) - w) j) = 0 := by
simp_rw [Pi.sub_apply, Pi.smul_apply, Pi.sub_apply, smul_sub, Finset.sum_sub_distrib, ←
| Mathlib/LinearAlgebra/AffineSpace/Independent.lean | 511 | 515 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.BoolIndicator
import Mathlib.Topology.ContinuousOn
/-!
# Clopen sets
A clopen set is a set that is both clo... | theorem isClopen_range_inl : IsClopen (range (Sum.inl : X → X ⊕ Y)) :=
⟨isClosed_range_inl, isOpen_range_inl⟩
theorem isClopen_range_inr : IsClopen (range (Sum.inr : Y → X ⊕ Y)) :=
⟨isClosed_range_inr, isOpen_range_inr⟩
theorem isClopen_range_sigmaMk {X : ι → Type*} [∀ i, TopologicalSpace (X i)] {i : ι} :
IsC... | Mathlib/Topology/Clopen.lean | 113 | 120 |
/-
Copyright (c) 2023 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Algebra.MonoidAlgebra.Ideal
import Mathlib.Algebra.MvPolynomial.Division
/-!
# Lemmas about ideals of `MvPolynomial`
Notably this contains results about mo... | rw [Set.image_image] at this
refine this.trans ?_
simp [Nat.one_le_iff_ne_zero]
end MvPolynomial
| Mathlib/RingTheory/MvPolynomial/Ideal.lean | 48 | 54 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Data.Finset.Pi
import Mathlib.Data.Finset.Sigma
import Mathlib.Data.Finset.Sum
import Mathlib.Data.Set.Finite.Basic
/-!
# Preimage of ... | rw [sigma_preimage_mk, filter_true_of_mem <| image_subset_iff.1 ht]
theorem sigma_image_fst_preimage_mk {β : α → Type*} [DecidableEq α] (s : Finset (Σ a, β a)) :
((s.image Sigma.fst).sigma fun a => s.preimage (Sigma.mk a) sigma_mk_injective.injOn) =
| Mathlib/Data/Finset/Preimage.lean | 126 | 129 |
/-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Data.Set.Prod
/-!
# N-ary images of sets
This file defines `Set.image2`, the binary image of sets.
This is mostly useful to define pointwise oper... | Mathlib/Data/Set/NAry.lean | 396 | 399 | |
/-
Copyright (c) 2024 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.LinearAlgebra.DFinsupp
import Mathlib.RingTheory.Finiteness.Basic
import Mathlib.LinearAlgebra.TensorProduct.Basic
/-!
# Some finiteness results of tensor product
T... |
/-- Variation of `TensorProduct.exists_finite_submodule_left_of_finite` where `M` and `N` are
already submodules. -/
theorem exists_finite_submodule_left_of_finite' (s : Set (M₁ ⊗[R] N₁)) (hs : s.Finite) :
∃ (M' : Submodule R M) (hM : M' ≤ M₁), Module.Finite R M' ∧
s ⊆ LinearMap.range ((inclusion hM).rTensor... | Mathlib/LinearAlgebra/TensorProduct/Finiteness.lean | 156 | 162 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.Group.Action.End
import Mathlib.Algebra.Group.Action.Pointwise.Set.Basic
import Mathlib.Algebra.Group.Action.Prod
import Mathlib.Algebra.Group.Subg... | Mathlib/GroupTheory/GroupAction/Basic.lean | 371 | 372 | |
/-
Copyright (c) 2014 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Field.Defs
import Mathlib.Algebra.Ring.Commute
import Mathlib.Algebra.Ring.Invertible
impo... | -- See note [reducible non-instances]
noncomputable abbrev Field.ofIsUnitOrEqZero [CommRing R] (h : ∀ a : R, IsUnit a ∨ a = 0) :
Field R where
| Mathlib/Algebra/Field/Basic.lean | 198 | 200 |
/-
Copyright (c) 2015 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Robert Y. Lewis, Yury Kudryashov
-/
import Mathlib.Algebra.Order.Monoid.Unbundled.Basic
import Mathlib.Algebra.Order.Monoid.Unbundled.OrderDual
import Mathlib.Tactic.Lift... |
@[to_additive nsmul_left_monotone]
theorem pow_right_monotone (ha : 1 ≤ a) : Monotone fun n : ℕ ↦ a ^ n :=
monotone_nat_of_le_succ fun n ↦ by rw [pow_succ]; exact le_mul_of_one_le_right' ha
| Mathlib/Algebra/Order/Monoid/Unbundled/Pow.lean | 56 | 60 |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Anne Baanen
-/
import Mathlib.Algebra.Algebra.Equiv
import Mathlib.LinearAlgebra.Span.Basic
/-!
# Towers of algebras
In this file we prove basic facts about towers of algebra.
... | variable {A} in
theorem of_algebraMap_smul [SMul R M] (h : ∀ (r : R) (x : M), algebraMap R A r • x = r • x) :
IsScalarTower R A M where
| Mathlib/Algebra/Algebra/Tower.lean | 88 | 90 |
/-
Copyright (c) 2019 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Group.Submonoid.Operations
import Mathlib.Data.DFinsupp.Sigma
import Mathlib.Data.DFinsupp.Submonoid
/-!
# Direct sum
This file defines the direct sum ... | obtain rfl | h := Decidable.eq_or_ne i j
· rw [DirectSum.of_eq_same, if_pos rfl]
· rw [DirectSum.of_eq_of_ne _ _ _ h, if_neg h, ZeroMemClass.coe_zero, ZeroMemClass.coe_zero]
/-- The `DirectSum` formed by a collection of additive submonoids (or subgroups, or submodules) of
`M` is said to be internal if the canoni... | Mathlib/Algebra/DirectSum/Basic.lean | 376 | 381 |
/-
Copyright (c) 2019 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Algebra.Order.AbsoluteValue.Basic
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mat... | 0 < ∏ i ∈ s, f i ↔ (∀ i ∈ s, (0 : R) < f i) :=
CanonicallyOrderedAdd.multiset_prod_pos.trans Multiset.forall_mem_map_iff
/-- If `g, h ≤ f` and `g i + h i ≤ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `CanonicallyOrderedAdd`.
-/
lemma prod_a... | Mathlib/Algebra/Order/BigOperators/Ring/Finset.lean | 159 | 172 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Kevin Buzzard, Yury Kudryashov, Eric Wieser
-/
import Mathlib.Algebra.Algebra.Prod
import Mathlib.Algebra.Group.Graph
import Mathlib.LinearAlgebra.Span.... | Mathlib/LinearAlgebra/Prod.lean | 1,058 | 1,061 | |
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Yaël Dillies
-/
import Mathlib.Data.List.Iterate
import Mathlib.GroupTheory.Perm.Cycle.Basic
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Tactic.Group
/-!
# ... | simp only [Subgroup.coe_subtype]
exact Disjoint.mono (disjoint_ofSubtype_of_memFixedPoints_self u)
le_rfl (support_zpowers_of_mem_cycleFactorsFinset_le (v c))
lemma commute_ofSubtype_noncommPiCoprod (u : Perm (Function.fixedPoints f))
(v : (c : { x // x ∈ f.cycleFactorsFinset }) → (Subgroup.zpowers (... | Mathlib/GroupTheory/Perm/Cycle/Factors.lean | 618 | 633 |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Polynomial.Expand
import Mathlib.Algebra.Polynomial.Splits
import Mathlib.Algebra.Squarefree.Basic
import Mathlib.FieldTheory.IntermediateField.Basic
imp... | exact IsCoprime.of_mul_right_left (IsCoprime.of_add_mul_left_right this)
| Mathlib/FieldTheory/Separable.lean | 82 | 83 |
/-
Copyright (c) 2018 Ellen Arlt. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ellen Arlt, Blair Shi, Sean Leather, Mario Carneiro, Johan Commelin, Lu-Ming Zhang
-/
import Mathlib.Algebra.Algebra.Opposite
import Mathlib.Algebra.Algebra.Pi
import Mathlib.Algebra.BigOp... | Mathlib/Data/Matrix/Basic.lean | 2,420 | 2,422 | |
/-
Copyright (c) 2023 Sophie Morel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sophie Morel
-/
import Mathlib.Analysis.Analytic.Constructions
import Mathlib.Analysis.Analytic.CPolynomialDef
/-! # Properties of continuously polynomial functions
We expand the API a... | Mathlib/Analysis/Analytic/CPolynomial.lean | 199 | 202 | |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.BoxIntegral.Partition.Filter
import Mathlib.Analysis.BoxIntegral.Partition.Measure
import Mathlib.Analysis.Oscillation
import Mathlib.Data.B... | rcases exists_pos_mul_lt ε0 (2 * μ.toBoxAdditive I) with ⟨ε₁, ε₁0, hε₁⟩
rcases hb with ⟨C, hC⟩
have C0 : 0 ≤ C := by
obtain ⟨x, hx⟩ := BoxIntegral.Box.nonempty_coe I
exact le_trans (norm_nonneg (f x)) <| hC x (I.coe_subset_Icc hx)
rcases exists_pos_mul_lt ε0 (4 * C) with ⟨ε₂, ε₂0, hε₂⟩
have ε₂0' : ENN... | Mathlib/Analysis/BoxIntegral/Basic.lean | 631 | 638 |
/-
Copyright (c) 2017 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Stephen Morgan, Kim Morrison
-/
import Mathlib.CategoryTheory.Equivalence
/-!
# Opposite categories
We provide a category instance on `Cᵒᵖ`.
The morphisms `X ⟶ Y` are defined to be the... |
namespace Functor
| Mathlib/CategoryTheory/Opposites.lean | 153 | 154 |
/-
Copyright (c) 2017 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Keeley Hoek
-/
import Mathlib.Algebra.NeZero
import Mathlib.Data.Int.DivMod
import Mathlib.Logic.Embedding.Basic
import Mathlib.Logic.Equiv.Set
import Mathlib.Tactic.... | Mathlib/Data/Fin/Basic.lean | 1,579 | 1,581 | |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Algebra.ModEq
import Mathlib.Algebra.Order.Archimedean.Basic
import Mathlib.Algebra.Ring.Periodic
import Mathlib.Data.Int.SuccPred
import Mathlib.Order.Cir... | Mathlib/Algebra/Order/ToIntervalMod.lean | 1,030 | 1,031 | |
/-
Copyright (c) 2022 Rémi Bottinelli. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémi Bottinelli, Junyan Xu
-/
import Mathlib.Algebra.Group.Subgroup.Defs
import Mathlib.CategoryTheory.Groupoid.VertexGroup
import Mathlib.CategoryTheory.Groupoid.Basic
import Mathlib... | Mathlib/CategoryTheory/Groupoid/Subgroupoid.lean | 681 | 683 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mitchell Lee
-/
import Mathlib.Algebra.BigOperators.Group.Finset.Indicator
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Topology.Algebra.InfiniteSum.Defs
imp... |
@[to_additive]
theorem HasProd.compl_mul {s : Set β} (ha : HasProd (f ∘ (↑) : (sᶜ : Set β) → α) a)
(hb : HasProd (f ∘ (↑) : s → α) b) : HasProd f (a * b) :=
ha.mul_isCompl isCompl_compl.symm hb
| Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | 312 | 317 |
/-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson
-/
import Mathlib.Computability.Language
import Mathlib.Tactic.AdaptationNote
/-!
# Regular Expressions
This file contains the formal definition for regular expressions and ... | refine ⟨t, u, h.2, ?_, hQ⟩
rw [rmatch] at hP
convert hP
exact h.1
theorem star_rmatch_iff (P : RegularExpression α) :
∀ x : List α, (star P).rmatch x ↔ ∃ S : List (List α), x
| Mathlib/Computability/RegularExpressions.lean | 245 | 251 |
/-
Copyright (c) 2022 Hanting Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Hanting Zhang
-/
import Mathlib.Topology.MetricSpace.Antilipschitz
import Mathlib.Topology.MetricSpace.Isometry
import Mathlib.Topology.MetricSpace.Lipschitz
import Mathlib.Data.FunLike... | section PseudoMetricDilation
| Mathlib/Topology/MetricSpace/Dilation.lean | 448 | 449 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Kevin Buzzard, Yury Kudryashov
-/
import Mathlib.Algebra.Group.Subgroup.Lattice
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra... | rw [← SetLike.mem_coe, iInf_coe, Set.mem_iInter]; rfl
@[simp]
theorem mem_finset_inf {ι} {s : Finset ι} {p : ι → Submodule R M} {x : M} :
x ∈ s.inf p ↔ ∀ i ∈ s, x ∈ p i := by
simp only [← SetLike.mem_coe, finset_inf_coe, Set.mem_iInter]
| Mathlib/Algebra/Module/Submodule/Lattice.lean | 242 | 248 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Order.Minimal
import Mathlib.Order.Zorn
import Mathlib.Topology.ContinuousOn
/-!
# Irreducibility in topological space... | theorem Subtype.irreducibleSpace (h : IsIrreducible s) : IrreducibleSpace s where
isPreirreducible_univ :=
(Subtype.preirreducibleSpace h.isPreirreducible).isPreirreducible_univ
toNonempty := h.nonempty.to_subtype
/-- An infinite type with cofinite topology is an irreducible topological space. -/
instance (pri... | Mathlib/Topology/Irreducible.lean | 202 | 217 |
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Constructions.Polish.Basic
import Mathlib.MeasureTheory.Function.UniformIntegrable
import Mathlib.Probability.Martingale.Upcrossing
/-!
# Mar... | ∀ᵐ ω ∂μ.trim (sSup_le fun _ ⟨_, hn⟩ => hn ▸ ℱ.le _ : ⨆ n, ℱ n ≤ m0),
∃ c, Tendsto (fun n => f n ω) atTop (𝓝 c) := by
letI := (⨆ n, ℱ n)
rw [ae_iff, trim_measurableSet_eq]
· exact hf.exists_ae_tendsto_of_bdd hbdd
| Mathlib/Probability/Martingale/Convergence.lean | 194 | 198 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Data.Fintype.Lattice
import Mathlib.RingTheory.Coprime.Lemmas
import Mathlib... | Submodule.smul_assoc _ _ _
theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M)
(H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by
suffices LinearMap.range (LinearMap.toSpanSingleton R M x) ≤ M' by
rw [← LinearMap.toSpanSingleton_one R M x]
exact this (Linear... | Mathlib/RingTheory/Ideal/Operations.lean | 82 | 96 |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.FieldTheory.Extension
import Mathlib.FieldTheory.Normal.Defs
import Mathlib.FieldTheory.Perfect
import Mathlib.RingTheory.Localization.Integral
/-!
# Algebraica... | apply IsAlgClosed.of_exists_root
intro p hmp hp
have hpe : degree (p.map e.symm.toRingHom) ≠ 0 := by
rw [degree_map]
exact ne_of_gt (degree_pos_of_irreducible hp)
rcases IsAlgClosed.exists_root (k := k) (p.map e.symm.toRingHom) hpe with ⟨x, hx⟩
use e x
rw [IsRoot] at hx
apply e.symm.injective
rw... | Mathlib/FieldTheory/IsAlgClosed/Basic.lean | 149 | 162 |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Geometry.Euclidean.Projection
import Mathlib.Geometry.Euclidean.Sphere.Basic
import Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional
import Mathlib.Tact... | Mathlib/Geometry/Euclidean/Circumcenter.lean | 844 | 849 | |
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Yaël Dillies
-/
import Mathlib.Algebra.Module.BigOperators
import Mathlib.GroupTheory.Perm.Basic
import Mathlib.GroupTheory.Perm.Finite
import Mathlib.GroupTheory.Perm.Lis... | Set.Subset.antisymm (Set.range_subset_iff.2 fun _ => h.1.mapsTo.perm_pow _ ha) fun _ =>
h.exists_pow_eq' hs ha
theorem IsCycleOn.range_zpow (h : f.IsCycleOn s) (ha : a ∈ s) :
| Mathlib/GroupTheory/Perm/Cycle/Basic.lean | 814 | 817 |
/-
Copyright (c) 2023 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.SimpleGraph.Triangle.Basic
/-!
# Construct a tripartite graph from its triangles
This file contains the constru... | @[simp] lemma in₂₁_iff : (graph t).Adj (in₂ c) (in₁ b) ↔ ∃ a, (a, b, c) ∈ t :=
⟨by rintro ⟨⟩; exact ⟨_, ‹_›⟩, fun ⟨_, h⟩ ↦ in₂₁ h⟩
| Mathlib/Combinatorics/SimpleGraph/Triangle/Tripartite.lean | 82 | 83 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.MeasureTheory.Measure.Typeclasses.Finite
import Mathlib.MeasureTheory.Measure.Typeclasses.NoAtoms
import Mathlib.MeasureTheory.Measure.... | Mathlib/MeasureTheory/Measure/Typeclasses.lean | 1,150 | 1,159 | |
/-
Copyright (c) 2021 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot
-/
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.Topology.Algebra.Nonarchimedean.Bases
import Mathlib.Topology.Algebra.UniformRing
/-!
# Adic topology
Given a co... | variable (I : Ideal R) (M : Type*) [AddCommGroup M] [Module R M]
theorem adic_module_basis :
I.ringFilterBasis.SubmodulesBasis fun n : ℕ => I ^ n • (⊤ : Submodule R M) :=
{ inter := fun i j =>
⟨max i j,
| Mathlib/Topology/Algebra/Nonarchimedean/AdicTopology.lean | 106 | 111 |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.HomologicalComplexBiprod
import Mathlib.Algebra.Homology.Homotopy
import Mathlib.CategoryTheory.MorphismProperty.IsInvertedBy
/-! The homotopy ... | @[reassoc]
lemma inlX_d (i j k : ι) (hij : c.Rel i j) (hjk : c.Rel j k) :
inlX φ j i hij ≫ d φ i j = -F.d j k ≫ inlX φ k j hjk + φ.f j ≫ inrX φ j := by
apply ext_to_X φ j k hjk
· dsimp
simp [d_fstX φ _ _ _ hij hjk]
· simp [d_sndX φ _ _ hij]
| Mathlib/Algebra/Homology/HomotopyCofiber.lean | 185 | 191 |
/-
Copyright (c) 2022 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Analysis.Calculus.BumpFunction.Basic
import Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap
import Mathlib.MeasureTheory.Measure.Lebesgu... | end ContDiffBump
| Mathlib/Analysis/Calculus/BumpFunction/Normed.lean | 148 | 161 |
/-
Copyright (c) 2024 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Algebra.Lie.Weights.Killing
import Mathlib.LinearAlgebra.RootSystem.Basic
import Mathlib.LinearAlgebra.RootSystem.Reduced
import Mathlib.LinearAlgebra.RootSy... | show 2 • (2⁻¹ : K) • (α : H → K) = α
rw [← Nat.cast_smul_eq_nsmul K, smul_smul]; simp
apply α.genWeightSpace_ne_bot
have := rootSpace_two_smul W (fun (e : (W : H → K) = 0) ↦ hα <| by
apply_fun (2 • ·) at e; simpa [hW] using e)
rwa [hW] at this
lemma eq_neg_one_or_eq_zero_or_eq_one_of_eq_smul
(hα ... | Mathlib/Algebra/Lie/Weights/RootSystem.lean | 301 | 335 |
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Algebra.Module.BigOperators
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.Nat.Squarefree
imp... | invOf := μ
invOf_mul_self := coe_moebius_mul_coe_zeta
mul_invOf_self := coe_zeta_mul_coe_moebius
/-- A unit in `ArithmeticFunction R` that evaluates to `ζ`, with inverse `μ`. -/
def zetaUnit : (ArithmeticFunction R)ˣ :=
| Mathlib/NumberTheory/ArithmeticFunction.lean | 1,099 | 1,104 |
/-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Topology
import Mathlib.Analysis.SpecialFunctions.Arsinh
import Mathlib.Geometry.Euclidean.Inversion.Basic
/-!
# Met... |
theorem dist_le_dist_coe_div_sqrt (z w : ℍ) : dist z w ≤ dist (z : ℂ) w / √(z.im * w.im) := by
rw [dist_le_iff_le_sinh, ← div_mul_eq_div_div_swap, self_le_sinh_iff]
positivity
| Mathlib/Analysis/Complex/UpperHalfPlane/Metric.lean | 101 | 105 |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Multiset.Bind
/-!
# Sections of a multiset
-/
assert_not_exists Ring
namespace Multiset
variable {α : Type*}
section Sections
/-- The sectio... | Multiset.induction_on s (by simp) (by simp +contextual)
end Sections
end Multiset
| Mathlib/Data/Multiset/Sections.lean | 60 | 64 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Sébastien Gouëzel,
Rémy Degenne, David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
/-!
# Power function... | theorem rpow_lt_one {x : ℝ≥0∞} {z : ℝ} (hx : x < 1) (hz : 0 < z) : x ^ z < 1 := by
lift x to ℝ≥0 using ne_of_lt (lt_of_lt_of_le hx le_top)
simp only [coe_lt_one_iff] at hx
simp [← coe_rpow_of_nonneg _ (le_of_lt hz), NNReal.rpow_lt_one hx hz]
| Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean | 821 | 824 |
/-
Copyright (c) 2021 Hunter Monroe. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Hunter Monroe, Kyle Miller
-/
import Mathlib.Combinatorics.SimpleGraph.Dart
import Mathlib.Data.FunLike.Fintype
import Mathlib.Logic.Embedding.Set
/-!
# Maps between graphs
This file ... | ext; rfl
| Mathlib/Combinatorics/SimpleGraph/Maps.lean | 161 | 162 |
/-
Copyright (c) 2022 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Algebra.Ring.Parity
import Mathlib.Combinatorics.SimpleGraph.Path
/-!
# Trails and Eulerian trails
This module contains additional theory about trails, in... | convert_to _ = ({u, v} : Finset V).card
· simp [hn]
· congr
ext x
simp [hn, imp_iff_not_or]
theorem IsEulerian.card_odd_degree [Fintype V] [DecidableRel G.Adj] {u v : V} {p : G.Walk u v}
(ht : p.IsEulerian) : Fintype.card { v : V | Odd (G.degree v) } = 0 ∨
Fintype.card { v : V | Odd (... | Mathlib/Combinatorics/SimpleGraph/Trails.lean | 145 | 160 |
/-
Copyright (c) 2023 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Data.Finset.Pi
import Mathlib.Logic.Function.DependsOn
/-!
# Update a function on a set of values
This file defines `Function.updateFinset`, the ... |
theorem update_eq_updateFinset {i y} :
Function.update x i y = updateFinset x {i} (uniqueElim y) := by
congr with j
by_cases hj : j = i
· cases hj
simp only [dif_pos, Finset.mem_singleton, update_self, updateFinset]
exact uniqueElim_default (α := fun j : ({i} : Finset ι) => π j) y
| Mathlib/Data/Finset/Update.lean | 43 | 50 |
/-
Copyright (c) 2022 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.NumberTheory.BernoulliPolynomials
import Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic
import Mathlib.Analysis.Calculus.Deriv.Polynomial
import... | simp only [Nat.cast_one, tsub_self, neg_mul, one_mul, eq_self_iff_true, if_true,
Nat.factorial_one, pow_one, inv_I, mul_neg]
rw [bernoulli_zero_fourier_coeff hn, sub_zero, mul_one, div_neg, neg_div]
· rw [bernoulliFourierCoeff_recurrence (k + 1) hn, Nat.add_sub_cancel k 1]
| Mathlib/NumberTheory/ZetaValues.lean | 126 | 129 |
/-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.ImproperIntegrals
import Mathlib.Analysis.Calculus.ParametricIntegral
import Mathlib.MeasureTheory.Measure.Haar.NormedSpace
... | gcongr
exact he t ((le_max_left e 1).trans_lt ht).le
refine (HasFiniteIntegral.mul_const ?_ _).mono' this
exact (integrableOn_Ioi_rpow_of_lt (by linarith) he').hasFiniteIntegral
/-- If `f` is a locally integrable real-valued function which is `O(x ^ (-b))` at `0`, then for any
`b < s`, its Mellin t... | Mathlib/Analysis/MellinTransform.lean | 210 | 231 |
/-
Copyright (c) 2019 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Eric Wieser
-/
import Mathlib.Data.Matrix.ConjTranspose
/-!
# Row and column matrices
This file provides results about row and column matrices.
## Main definitions
* `Mat... |
@[simp]
theorem updateCol_subsingleton [Subsingleton n] (A : Matrix m n R) (i : n) (b : m → R) :
A.updateCol i b = (replicateCol (Fin 1) b).submatrix id (Function.const n 0) := by
ext x y
| Mathlib/Data/Matrix/RowCol.lean | 274 | 278 |
/-
Copyright (c) 2022 Eric Rodriguez. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Rodriguez
-/
import Mathlib.Algebra.GroupWithZero.Units.Lemmas
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Ring.Cast
import Mathlib.Data.Fi... | def SignType.sign : α →o SignType :=
⟨fun a => if 0 < a then 1 else if a < 0 then -1 else 0, fun a b h => by
| Mathlib/Data/Sign.lean | 300 | 301 |
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.LinearAlgebra.AffineSpace.Basis
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
/-!
# Matrix results for barycentric co-ordinates
Results about the ... | ext j
change (∑ i, w₁ i • b.coord j (p i)) = ∑ i, w₂ i • b.coord j (p i)
rw [← Finset.univ.affineCombination_eq_linear_combination _ _ hw₁,
← Finset.univ.affineCombination_eq_linear_combination _ _ hw₂,
← Function.comp_def (b.coord j) p, ← Finset.univ.map_affineCombination p w₁ hw₁,
← Fins... | Mathlib/LinearAlgebra/AffineSpace/Matrix.lean | 61 | 76 |
/-
Copyright (c) 2021 Luke Kershaw. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Luke Kershaw, Joël Riou
-/
import Mathlib.Algebra.Homology.ShortComplex.Basic
import Mathlib.CategoryTheory.Limits.Constructions.FiniteProductsOfBinaryProducts
import Mathlib.CategoryThe... | isZero₁_iff_isIso₂ _ (rot_of_distTriang _ hT)
lemma isZero₃_iff_isIso₁ : IsZero T.obj₃ ↔ IsIso T.mor₁ := by
refine Iff.trans ?_ (Triangle.isZero₁_iff_isIso₂ _ (inv_rot_of_distTriang _ hT))
dsimp
simp only [IsZero.iff_id_eq_zero, ← Functor.map_id, Functor.map_eq_zero_iff]
lemma isZero₁_of_isIso₂ (h : IsIso T.m... | Mathlib/CategoryTheory/Triangulated/Pretriangulated.lean | 366 | 376 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Group.Action.Pi
import Mathlib.Algebra.Order.AbsoluteValue.Basic
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Group.Mi... | have := lt_of_le_of_lt (abv_add abv _ _) (add_lt_add (hi j ij k jk) hj)
rwa [sub_add_cancel, add_halves] at this
| Mathlib/Algebra/Order/CauSeq/Basic.lean | 445 | 447 |
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.AlgebraicGeometry.Scheme
import Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq
import Mathlib.Categor... |
instance toSchemeHom_isOpenImmersion : AlgebraicGeometry.IsOpenImmersion (toSchemeHom Y f) :=
H
theorem scheme_eq_of_locallyRingedSpace_eq {X Y : Scheme.{u}}
(H : X.toLocallyRingedSpace = Y.toLocallyRingedSpace) : X = Y := by
cases X; cases Y; congr
theorem scheme_toScheme {X Y : Scheme.{u}} (f : X ⟶ Y) [Alg... | Mathlib/AlgebraicGeometry/OpenImmersion.lean | 304 | 316 |
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Finset.Lattice.Fold
import Mathlib.Data.Fintype.Vector
import Mathlib.Data.Multiset.Sym
/-!
# Symmetric powers of a finset
This file defines the sym... |
theorem sym2_toFinset [DecidableEq α] (m : Multiset α) :
| Mathlib/Data/Finset/Sym.lean | 96 | 97 |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Data.Set.Piecewise
import Mathlib.Logic.Equiv.Defs
import Mathlib.Tactic.Core
import Mathlib.Tactic.Attr.Core
/-!
# Partial equivalences
This f... | Mathlib/Logic/Equiv/PartialEquiv.lean | 1,096 | 1,099 | |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Mario Carneiro
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Data.FunLike.Equiv
import Mathlib.Data.Quot
import Mathlib.Data.Subtype
import Mathlib.Logic.U... | (psigmaCongrRight F).symm = psigmaCongrRight fun a => (F a).symm := rfl
| Mathlib/Logic/Equiv/Defs.lean | 601 | 602 |
/-
Copyright (c) 2020 Kenji Nakagawa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenji Nakagawa, Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.Algebra.Subalgebra.Pointwise
import Mathlib.Algebra.Polynomial.FieldDivision
import Mathlib.RingTheory.Spect... | ∃ x ∈ I ^ e, x ∉ I ^ (e + 1) :=
SetLike.exists_of_lt (I.pow_right_strictAnti hI0 hI1 e.lt_succ_self)
| Mathlib/RingTheory/DedekindDomain/Ideal.lean | 708 | 710 |
/-
Copyright (c) 2024 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Data.Set.Functor
import Mathlib.Order.Sublattice
import Mathlib.Order.Hom.CompleteLattice
/-!
# Complete Sublattices
This file defines complete sublattices... | /-- The pull back of a complete sublattice under a complete lattice hom is a complete sublattice. -/
@[simps] def comap (L : CompleteSublattice β) : CompleteSublattice α where
carrier := f ⁻¹' L
supClosed' := L.supClosed.preimage f
| Mathlib/Order/CompleteSublattice.lean | 130 | 133 |
/-
Copyright (c) 2021 Kevin Kappelmann. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Kappelmann
-/
import Mathlib.Algebra.ContinuedFractions.Computation.Approximations
import Mathlib.Algebra.ContinuedFractions.ConvergentsEquiv
import Mathlib.Algebra.Order.Archi... | have nB_pos : 0 < nB := nB_ineq.trans_lt' <| mod_cast fib_pos.2 <| succ_pos _
have zero_lt_mul_conts : 0 < B * nB := by positivity
suffices 1 < ε * (B * nB) from (div_lt_iff₀ zero_lt_mul_conts).mpr this
-- use that `N' ≥ n` was obtained from the archimedean property to show the following
calc 1 < ε ... | Mathlib/Algebra/ContinuedFractions/Computation/ApproximationCorollaries.lean | 98 | 141 |
/-
Copyright (c) 2022 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
-/
import Mathlib.Algebra.IsPrimePow
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.Nat.Prime.Pow
import Mathlib.NumberTheory.Divisors
/-!
# Prime powers a... | · rintro ⟨hp, hk, hn⟩
exact ⟨hk, by rw [← hn, Nat.Prime.factorization_pow hp]⟩
· rintro ⟨hk, hn⟩
have hn0 : n ≠ 0 := by
rintro rfl
simp_all only [Finsupp.single_eq_zero, eq_comm, Nat.factorization_zero, hk.ne']
rw [Nat.eq_pow_of_factorization_eq_single hn0 hn]
exact ⟨Nat.prime_of_mem_pri... | Mathlib/Data/Nat/Factorization/PrimePow.lean | 41 | 54 |
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov, Sébastien Gouëzel
-/
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
import Mathlib.MeasureTheory.Measure.Typeclasses.Probability
import... |
theorem outer_le_length (s : Set ℝ) : f.outer s ≤ f.length s :=
OuterMeasure.ofFunction_le _
/-- If a compact interval `[a, b]` is covered by a union of open interval `(c i, d i)`, then
`f b - f a ≤ ∑ f (d i) - f (c i)`. This is an auxiliary technical statement to prove the same
statement for half-open intervals, t... | Mathlib/MeasureTheory/Measure/Stieltjes.lean | 200 | 208 |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn
-/
import Mathlib.Analysis.Calculus.ContDiff.Operations
import Mathlib.Data.Finset.Sym
import Mathlib.Data.Nat.Choose.Cast
import Mathlib.Data.Na... | simp_rw [← iteratedFDerivWithin_univ]
exact B.norm_iteratedFDerivWithin_le_of_bilinear hf.contDiffOn hg.contDiffOn uniqueDiffOn_univ
(mem_univ x) hn
/-- Bounding the norm of the iterated derivative of `B (f x) (g x)` within a set in terms of the
iterated derivatives of `f` and `g` when `B` is bilinear of norm ... | Mathlib/Analysis/Calculus/ContDiff/Bounds.lean | 211 | 218 |
/-
Copyright (c) 2020 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov
-/
import Mathlib.Algebra.Group.Indicator
import Mathlib.Algebra.GroupWithZero.Basic
/-!
# Indicator functions and support of a function in groups with z... | fun x hfg hg => hfg <| by simp only [hg, mul_zero]
| Mathlib/Algebra/GroupWithZero/Indicator.lean | 101 | 102 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Patrick Stevens
-/
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.BigOperators.Ring.Finset
import ... | (∑ m ∈ x.powerset, (-1 : ℤ) ^ #m) = if x = ∅ then 1 else 0 := by
rw [sum_powerset_apply_card]
simp only [nsmul_eq_mul', ← card_eq_zero, Int.alternating_sum_range_choose]
theorem sum_powerset_neg_one_pow_card_of_nonempty {α : Type*} {x : Finset α} (h0 : x.Nonempty) :
(∑ m ∈ x.powerset, (-1 : ℤ) ^ #m) = 0 :=... | Mathlib/Data/Nat/Choose/Sum.lean | 180 | 185 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Algebra.Ring.Pointwise.Set
import Mathlib.Order.Filter.AtTopBot.CompleteLattice
import Mathlib.Order.Filter.AtTopBot.G... | · refine (Equiv.divLeft a).iInf_congr fun x => ?_; simp [Ioi]
· refine (Equiv.divRight a).iInf_congr fun x => ?_; simp [Iio]
@[to_additive]
| Mathlib/Topology/Order/LeftRightNhds.lean | 376 | 379 |
/-
Copyright (c) 2019 Neil Strickland. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Neil Strickland
-/
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Algebra.Group.NatPowAssoc
import Mathlib.Algebra.Order.... | ← tsub_add_eq_tsub_tsub, add_comm 1 i]
have : i + 1 + (n - (i + 1)) = n := add_tsub_cancel_of_le (mem_range.mp hi)
rw [add_comm (i + 1)] at this
| Mathlib/Algebra/GeomSum.lean | 98 | 100 |
/-
Copyright (c) 2022 Rémi Bottinelli. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémi Bottinelli, Junyan Xu
-/
import Mathlib.Algebra.Group.Subgroup.Defs
import Mathlib.CategoryTheory.Groupoid.VertexGroup
import Mathlib.CategoryTheory.Groupoid.Basic
import Mathlib... |
variable (D : Set C)
/-- The full subgroupoid on a set `D : Set C` -/
def full : Subgroupoid C where
arrows c d := {_f | c ∈ D ∧ d ∈ D}
inv := by rintro _ _ _ ⟨⟩; constructor <;> assumption
mul := by rintro _ _ _ _ ⟨⟩ _ ⟨⟩; constructor <;> assumption
theorem full_objs : (full D).objs = D :=
Set.ext fun _ => ... | Mathlib/CategoryTheory/Groupoid/Subgroupoid.lean | 572 | 591 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Logic.Nontrivial.Basic
import Mathlib.Order.TypeTags
import Mathlib.Data.Option.NAry
import Mathlib.Tactic.Contrapose
import Mathlib.Tactic.Lift
import... | wf := .intro fun
| ⊥ => ⟨_, by simp⟩
| (a : α) => (wellFounded_lt.1 a).rec fun _ _ ih ↦ .intro _ fun
| ⊥, _ => ⟨_, by simp⟩
| (b : α), hlt => ih _ (coe_lt_coe.1 hlt)
| Mathlib/Order/WithBot.lean | 422 | 426 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Sigma.Basic
import Mathlib.Algebra.Order.Ring.Nat
/-!
# A computable model of ZFA without infinity
In this file we define finite hereditary list... | rcases h with (rfl | h)
· exact ⟨_, m, e⟩
· exact mem_of_subset' s h
theorem subset_def {l₁ l₂ : Lists' α true} : l₁ ⊆ l₂ ↔ ∀ a ∈ l₁.toList, a ∈ l₂ :=
| Mathlib/SetTheory/Lists.lean | 172 | 176 |
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Edward Ayers
-/
import Mathlib.CategoryTheory.Limits.Shapes.Pullback.HasPullback
import Mathlib.Data.Set.BooleanAlgebra
/-!
# Theory of sieves
- For an object `X` of a ca... | theorem functorPushforward_comp (R : Presieve X) :
R.functorPushforward (F ⋙ G) = (R.functorPushforward F).functorPushforward G := by
funext x
ext f
constructor
· rintro ⟨X, f₁, g₁, h₁, rfl⟩
exact ⟨F.obj X, F.map f₁, g₁, ⟨X, f₁, 𝟙 _, h₁, by simp⟩, rfl⟩
· rintro ⟨X, f₁, g₁, ⟨X', f₂, g₂, h₁, rfl⟩, rfl⟩... | Mathlib/CategoryTheory/Sites/Sieves.lean | 246 | 254 |
/-
Copyright (c) 2022 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston
-/
import Mathlib.Algebra.Category.ModuleCat.Projective
import Mathlib.AlgebraicTopology.ExtraDegeneracy
import Mathlib.CategoryTheory.Abelian.Ext
import Mathlib.G... | Mathlib/RepresentationTheory/GroupCohomology/Resolution.lean | 672 | 679 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot, Yury Kudryashov, Rémy Degenne
-/
import Mathlib.Data.Set.Subsingleton
import Mathlib.Order.Interval.Set.Defs
/-!
# Intervals
In any pr... | Mathlib/Order/Interval/Set/Basic.lean | 1,190 | 1,200 | |
/-
Copyright (c) 2022 Julian Berman. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Julian Berman
-/
import Mathlib.GroupTheory.PGroup
import Mathlib.LinearAlgebra.Quotient.Defs
/-!
# Torsion groups
This file defines torsion groups, i.e. groups where all elements hav... | section AddCommGroup
| Mathlib/GroupTheory/Torsion.lean | 412 | 413 |
/-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Logic.Encodable.Pi
import Mathlib.MeasureTheory.Group.Measure
import Mathlib.MeasureTheory.MeasurableSpace.... | Finset.prod_eq_zero_iff, piPremeasure]
· simp [h, piPremeasure]
theorem piPremeasure_pi_mono {s t : Set (∀ i, α i)} (h : s ⊆ t) :
piPremeasure m s ≤ piPremeasure m t :=
Finset.prod_le_prod' fun _ _ => measure_mono (image_subset _ h)
theorem piPremeasure_pi_eval {s : Set (∀ i, α i)} :
piPremeasure m ... | Mathlib/MeasureTheory/Constructions/Pi.lean | 87 | 95 |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Algebra.ModEq
import Mathlib.Algebra.Order.Archimedean.Basic
import Mathlib.Algebra.Ring.Periodic
import Mathlib.Data.Int.SuccPred
import Mathlib.Order.Cir... | exact (h₁₂₃.2.trans_le (toIcoMod_le_toIocMod _ x₃ x₂)).trans h₂₃₄.2
namespace QuotientAddGroup
| Mathlib/Algebra/Order/ToIntervalMod.lean | 788 | 790 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.CharP.Basic
import Mathlib.Algebra.Module.End
import Mathlib.Algebra.Ring.Prod
import Mathlib.Data.Fintype.Units
import Mathlib.GroupTheory.GroupAc... | cast_one dvd_rfl
@[simp]
theorem cast_add' (a b : ZMod n) : (cast (a + b : ZMod n) : R) = cast a + cast b :=
cast_add dvd_rfl a b
@[simp]
theorem cast_mul' (a b : ZMod n) : (cast (a * b : ZMod n) : R) = cast a * cast b :=
| Mathlib/Data/ZMod/Basic.lean | 349 | 356 |
/-
Copyright (c) 2022 Vincent Beffara. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Vincent Beffara
-/
import Mathlib.Analysis.Analytic.IsolatedZeros
import Mathlib.Analysis.Complex.CauchyIntegral
import Mathlib.Analysis.Complex.AbsMax
import Mathlib.Topology.MetricS... | /- The function `f` is analytic in a neighborhood of `z₀`; by the isolated zeros principle, if `f`
is not constant in a neighborhood of `z₀`, then it is nonzero, and therefore bounded below, on
every small enough circle around `z₀` and then `DiffContOnCl.ball_subset_image_closedBall`
provides an explicit ... | Mathlib/Analysis/Complex/OpenMapping.lean | 77 | 106 |
/-
Copyright (c) 2024 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Comma.StructuredArrow.Basic
import Mathlib.CategoryTheory.Limits.Shapes.Equivalence
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Terminal... | section
variable {L : C ⥤ D} {F₁ F₂ : C ⥤ H}
/-- When two left extensions `α₁ : LeftExtension L F₁` and `α₂ : LeftExtension L F₂`
are essentially the same via an isomorphism of functors `F₁ ≅ F₂`,
then `α₁` is universal iff `α₂` is. -/
noncomputable def LeftExtension.isUniversalEquivOfIso₂
| Mathlib/CategoryTheory/Functor/KanExtension/Basic.lean | 499 | 506 |
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