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/-
Copyright (c) 2020 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Sébastien Gouëzel
-/
import Mathlib.Analysis.NormedSpace.IndicatorFunction
import Mathlib.Data.Fintype.Order
import Mathlib.MeasureTheory.Function.AEEqFun
import Mathlib.Me... | eLpNorm (fun _ : α => c) p μ = ‖c‖ₑ * μ Set.univ ^ (1 / ENNReal.toReal p) := by
by_cases h_top : p = ∞
| Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean | 242 | 243 |
/-
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, Heather Macbeth
-/
import Mathlib.MeasureTheory.Constructions.Pi
/-!
# Marginals of multivariate functions
In this file, we define a convenient way to compute int... | (hst : Disjoint s t) : ∫⋯∫⁻_s ∪ t, f ∂μ = ∫⋯∫⁻_s, ∫⋯∫⁻_t, f ∂μ ∂μ := by
ext1 x
let e := MeasurableEquiv.piFinsetUnion X hst
| Mathlib/MeasureTheory/Integral/Marginal.lean | 137 | 139 |
/-
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.Analysis.SpecificLimits.Basic
import Mathlib.Topology.MetricSpace.IsometricSMul
/-!
# Hausdorff distance
The Hausdorff distance on subsets of a... | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | 852 | 856 | |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Eric Wieser
-/
import Mathlib.Algebra.Quaternion
import Mathlib.Analysis.InnerProductSpace.Continuous
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.To... | Mathlib/Analysis/Quaternion.lean | 136 | 136 | |
/-
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.Algebra.Lie.Basic
import Mathlib.Algebra.Lie.Subalgebra
import Mathlib.Algebra.Lie.Submodule
import Mathlib.Algebra.Algebra.Subalgebra.Basic
/-!
# Lie algeb... |
lemma LieSubalgebra.coe_ad_pow (H : LieSubalgebra R L) (x y : H) (n : ℕ) :
((ad R H x ^ n) y : L) = (ad R L x ^ n) y :=
| Mathlib/Algebra/Lie/OfAssociative.lean | 256 | 258 |
/-
Copyright (c) 2023 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.Ring.CharZero
import Mathlib.Algebra.Ring.Int.Units
import Mathlib.GroupTheory.Coprod.Basic
import Mathlib.GroupTheory.Complement
/-!
## HNN Exte... | (of (φ a : G) : HNNExtension G A B φ) = t * of (a : G) * t⁻¹ := by
rw [t_mul_of]; simp
| Mathlib/GroupTheory/HNNExtension.lean | 77 | 79 |
/-
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.Complex
import Qq
/-! # P... | theorem rpow_sum_of_pos {ι : Type*} {a : ℝ} (ha : 0 < a) (f : ι → ℝ) (s : Finset ι) :
(a ^ ∑ x ∈ s, f x) = ∏ x ∈ s, a ^ f x :=
map_sum (⟨⟨fun (x : ℝ) => (a ^ x : ℝ), rpow_zero a⟩, rpow_add ha⟩ : ℝ →+ (Additive ℝ)) f s
theorem rpow_sum_of_nonneg {ι : Type*} {a : ℝ} (ha : 0 ≤ a) {s : Finset ι} {f : ι → ℝ}
| Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | 217 | 221 |
/-
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.Multiset.FinsetOps
import Mathlib.Data.Multiset.Fold
/-!
# Lattice operations on multisets
-/
namespace Multiset
variable {α : Type*}
/-! ###... |
end Sup
| Mathlib/Data/Multiset/Lattice.lean | 84 | 85 |
/-
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.CharP.Invertible
import Mathlib.Algebra.Order.Interval.Set.Group
import Mathlib.Analysis.Convex.Basic
import Mathlib.Analysis.Convex.Segment
import... | /-- The point `y` is weakly between `x` and `z`. -/
def Wbtw (x y z : P) : Prop :=
y ∈ affineSegment R x z
| Mathlib/Analysis/Convex/Between.lean | 122 | 124 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.Data.ENNReal.Basic
/-!
# Maps between real and extended non-negative real numbers
This file focuses on the functions `ENNReal.toReal... | Mathlib/Data/ENNReal/Real.lean | 651 | 652 | |
/-
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... | refine ⟨fun h => (h.1.trans h.2.symm).subset, fun h => ?_⟩
exact ⟨(h.trans interior_subset).antisymm subset_closure,
interior_subset.antisymm (subset_closure.trans h)⟩
@[simp] alias ⟨IsClopen.frontier_eq, _⟩ := isClopen_iff_frontier_eq_empty
| Mathlib/Topology/Clopen.lean | 30 | 34 |
/-
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... | haveI := mulRightMono_of_mulRightStrictMono M
⟨fun H => not_le.mp fun k => H.not_le <| Right.one_le_pow_of_le k, Right.pow_lt_one_of_lt hn⟩
| Mathlib/Algebra/Order/Monoid/Unbundled/Pow.lean | 290 | 291 |
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Mario Carneiro
-/
import Mathlib.Algebra.Order.Group.OrderIso
import Mathlib.Algebra.Ring.Int.Defs
import Mathlib.Data.Nat.Find
import Mathlib.Order.Bounds.Defs
/-! # Le... |
/--
If `P : ℤ → Prop` is a predicate such that the set `{m : P m}` is bounded below and nonempty,
then this set has the least element. This lemma uses classical logic to avoid assumption
`[DecidablePred P]`. See `Int.leastOfBdd` for a constructive counterpart. -/
theorem exists_least_of_bdd
{P : ℤ → Prop}
(Hbd... | Mathlib/Data/Int/LeastGreatest.lean | 61 | 68 |
/-
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, Benjamin Davidson
-/
import Mathlib.Algebra.QuadraticDiscriminant
import Mathlib.Analysis.SpecialFunctions.Pow.Complex
/-!
... | apply or_congr <;>
field_simp [sin_eq_zero_iff, (by norm_num : -(2 : ℂ) ≠ 0), eq_sub_iff_add_eq',
| Mathlib/Analysis/SpecialFunctions/Trigonometric/Complex.lean | 87 | 88 |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.FieldTheory.Minpoly.Field
import Mathlib.LinearAlgebra.SModEq
import Mathlib.RingTheory.Ideal.BigOperators
/-!
# Power basis
This file defines a structure ... | Mathlib/RingTheory/PowerBasis.lean | 507 | 511 | |
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.CategoryTheory.Comma.Arrow
import Mathlib.Order.CompleteBooleanAlgebra
/-!
# Properties of morphisms
We provide the basic framework for talking about prope... | rfl
/-- The inverse image of a `MorphismProperty D` by a functor `C ⥤ D` -/
def inverseImage (P : MorphismProperty D) (F : C ⥤ D) : MorphismProperty C := fun _ _ f =>
| Mathlib/CategoryTheory/MorphismProperty/Basic.lean | 99 | 102 |
/-
Copyright (c) 2019 Calle Sönne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Analysis.Normed.Group.AddCircle
import Mathlib.Algebra.CharZero.Quotient
import Mathlib.Topology... | ⟨neg_lt_neg (lt_of_le_of_ne h.2 hnegpi), (neg_nonpos.2 h.1).trans Real.pi_pos.le⟩,
abs_neg, abs_eq_self.2 h.1]
| Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | 536 | 537 |
/-
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 | 1,548 | 1,552 | |
/-
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.Algebra.Lie.Solvable
import Mathlib.Algebra.Lie.Quotient
import Mathlib.Algebra.Lie.Normalizer
import Mathlib.Algebra.Order.Archimedean.Basic
import Mathlib.... | exact LieSubmodule.lie_mem_lie (LieSubmodule.mem_top x) ih
theorem iterate_toEnd_mem_lowerCentralSeries₂ (x y : L) (m : M) (k : ℕ) :
(toEnd R L M x ∘ₗ toEnd R L M y)^[k] m ∈
lowerCentralSeries R L M (2 * k) := by
induction k with
| zero => simp
| Mathlib/Algebra/Lie/Nilpotent.lean | 208 | 214 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Logic.Function.Basic
/-!
# Semiconjugate and commuting maps
We define the following predicates:
* `Function.Semiconj`: `f : α → β` semiconjugates ... | Mathlib/Logic/Function/Conjugate.lean | 194 | 196 | |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Option.NAry
import Mathlib.Data.Seq.Computation
import Mathlib.Tactic.ApplyFun
import Mathlib.Data.List.Basic
/-!
# Possibly infinite lists
This... |
@[simp]
theorem destruct_cons (a : α) : ∀ s, destruct (cons a s) = some (a, s)
| ⟨f, al⟩ => by
unfold cons destruct Functor.map
apply congr_arg fun s => some (a, s)
apply Subtype.eq; dsimp [tail]
| Mathlib/Data/Seq/Seq.lean | 207 | 214 |
/-
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 | 1,210 | 1,212 | |
/-
Copyright (c) 2021 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Data.Prod.Lex
import Mathlib.Data.Sigma.Lex
import Mathlib.Order.RelIso.Set
import Mathlib.Order.WellQuasiOrder
import Mathlib.Tactic.TFAE
/-!
# Well-... | theorem _root_.IsAntichain.partiallyWellOrderedOn_iff (hs : IsAntichain r s) :
s.PartiallyWellOrderedOn r ↔ s.Finite :=
⟨hs.finite_of_partiallyWellOrderedOn, Finite.partiallyWellOrderedOn⟩
@[simp]
theorem partiallyWellOrderedOn_singleton (a : α) : PartiallyWellOrderedOn {a} r :=
| Mathlib/Order/WellFoundedSet.lean | 313 | 318 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Batteries.Data.Nat.Gcd
import Mathlib.Algebra.Group.Nat.Units
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.GroupWi... | theorem lcm_mul_left {m n k : ℕ} : (m * n).lcm (m * k) = m * n.lcm k := by
apply dvd_antisymm
| Mathlib/Data/Nat/GCD/Basic.lean | 75 | 76 |
/-
Copyright (c) 2015, 2017 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Robert Y. Lewis, Johannes Hölzl, Mario Carneiro, Sébastien Gouëzel
-/
import Mathlib.Topology.MetricSpace.Pseudo.Lemmas
import Mathlib.Topology.EMetricSpace.Basic
... |
end CauchySeq
| Mathlib/Topology/MetricSpace/Cauchy.lean | 159 | 168 |
/-
Copyright (c) 2018 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
/-!
# Equalizers and coequalizers
This file defines (co)equalizers a... |
/-- Every cofork is isomorphic to one of the form `Cofork.ofπ _ _`. -/
| Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean | 656 | 657 |
/-
Copyright (c) 2023 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis
-/
import Mathlib.Computability.AkraBazzi.GrowsPolynomially
import Mathlib.Analysis.Calculus.Deriv.Inv
import Mathlib.Analysis.SpecialFunctions.Pow.Deriv
/-!
# Divid... | refine sub_nonneg_of_le <|
(strictAntiOn_smoothingFn.le_iff_le ?n_gt_one ?bn_gt_one).mpr ?le
case n_gt_one =>
rwa [Set.mem_Ioi, Nat.one_lt_cast]
case bn_gt_one =>
calc 1 = b i * (b i)⁻¹ := by rw [mul_inv_ca... | Mathlib/Computability/AkraBazzi/AkraBazzi.lean | 1,048 | 1,142 |
/-
Copyright (c) 2021 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.Data.Set.Finite.Lattice
import Mathlib.Order.ConditionallyCompleteLattice.Indexed
import Mathlib.Order.Interval.Finset.Nat
import Mathlib.Order.SuccPred.Ba... |
end SemilatticeSup
section DistribLattice
/-!
| Mathlib/Order/PartialSups.lean | 165 | 170 |
/-
Copyright (c) 2020 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.Computation.CorrectnessTerminating
import Mathlib.D... | · simp [IntFractPair.stream, IH.symm, v_eq_q, stream_q_nth_eq, fr_zero]
· replace IH : some (IntFractPair.mk b (fr : K)) = IntFractPair.stream (↑q) n := by
rwa [stream_q_nth_eq] at IH
have : (fr : K)⁻¹ = ((fr⁻¹ : ℚ) : K) := by norm_cast
have coe_of_fr := coe_of_rat_eq this
... | Mathlib/Algebra/ContinuedFractions/Computation/TerminatesIffRat.lean | 174 | 194 |
/-
Copyright (c) 2023 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Order.SupClosed
/-!
# Sublattices
This file defines sublattices.
## TODO
Subsemilattices, if people care about them.
## Tags
sublattice
-/
open Func... | lemma map_inf (L M : Sublattice α) (f : LatticeHom α β) (hf : Injective f) :
map f (L ⊓ M) = map f L ⊓ map f M := by
| Mathlib/Order/Sublattice.lean | 288 | 289 |
/-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Yaël Dillies
-/
import Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap
/-!
# Integral average of a function
In this file we define `MeasureTheory.average... | theorem laverage_eq_lintegral [IsProbabilityMeasure μ] (f : α → ℝ≥0∞) :
| Mathlib/MeasureTheory/Integral/Average.lean | 112 | 112 |
/-
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, Jeremy Avigad
-/
import Mathlib.Data.Set.Finite.Basic
import Mathlib.Data.Set.Finite.Range
import Mathlib.Data.Set.Lattice
import Mathlib.Topology.Defs.... | Mathlib/Topology/Basic.lean | 727 | 727 | |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad
-/
import Mathlib.Data.Finset.Basic
import Mathlib.Data.Finset.Image
/-!
# Cardinality of a finite set
This defines the cardinality of a `Fins... | theorem card_filter_le (s : Finset α) (p : α → Prop) [DecidablePred p] :
#(s.filter p) ≤ #s :=
| Mathlib/Data/Finset/Card.lean | 260 | 261 |
/-
Copyright (c) 2023 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang
-/
import Mathlib.Algebra.Colimit.Module
import Mathlib.LinearAlgebra.TensorProduct.Basic
/-!
# Tensor product and direct limits commute with each other.
Given a family o... | simp [directLimitRight, congr_apply_of]
@[simp] lemma directLimitRight_symm_of_tmul {i : ι} (m : M) (g : G i) :
| Mathlib/LinearAlgebra/TensorProduct/DirectLimit.lean | 106 | 108 |
/-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Topology.MetricSpace.Dilation
/-!
# Dilation equivalence
In this file we define `DilationEquiv X Y`, a type of bundled equivalences between `X` and... |
/-- Reinterpret a `DilationEquiv` as a homeomorphism. -/
def toHomeomorph (e : X ≃ᵈ Y) : X ≃ₜ Y where
| Mathlib/Topology/MetricSpace/DilationEquiv.lean | 205 | 207 |
/-
Copyright (c) 2022 Alex Kontorovich and Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Kontorovich, Heather Macbeth
-/
import Mathlib.Algebra.Group.Opposite
import Mathlib.MeasureTheory.Constructions.Polish.Basic
import Mathlib.MeasureTheory.Gr... | function `f` on `G` times the lift to `G` of a function `g` on the quotient `G ⧸ Γ` with respect
to a right-invariant measure `μ` on `G`, is equal to the integral over the quotient of the
automorphization of `f` times `g`. -/
lemma QuotientGroup.integral_mul_eq_integral_automorphize_mul {K : Type*} [NormedField K... | Mathlib/MeasureTheory/Measure/Haar/Quotient.lean | 400 | 429 |
/-
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... |
section Functor
variable {E : Type u₃} [Category.{v₃} E] (G : D ⥤ E)
/--
If `R` is a sieve, then the `CategoryTheory.Presieve.functorPullback` of `R` is actually a sieve.
-/
@[simps]
def functorPullback (R : Sieve (F.obj X)) : Sieve X where
arrows := Presieve.functorPullback F R
| Mathlib/CategoryTheory/Sites/Sieves.lean | 630 | 640 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Mario Carneiro
-/
import Mathlib.Data.Subtype
import Mathlib.Order.Defs.LinearOrder
import Mathlib.Order.Notation
import Mathlib.Tactic.GCongr.Core
import Mathlib.Tactic.... |
lemma ltByCases_rec {h₁ : x < y → P} {h₂ : x = y → P} {h₃ : y < x → P} (p : P)
(hlt : (h : x < y) → h₁ h = p) (heq : (h : x = y) → h₂ h = p)
(hgt : (h : y < x) → h₃ h = p) :
ltByCases x y h₁ h₂ h₃ = p :=
ltByCases x y
| Mathlib/Order/Basic.lean | 437 | 442 |
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Algebra.GroupWithZero.Indicator
import Mathlib.Topology.Piecewise
import Mathlib.Topology.Instances.ENNReal.Lemmas
/-!
# Semicontinuous maps
A ... | upperSemicontinuousOn_iInf fun i => upperSemicontinuousOn_iInf fun hi => h i hi
theorem upperSemicontinuous_ciInf {f : ι → α → δ'} (bdd : ∀ x, BddBelow (range fun i => f i x))
(h : ∀ i, UpperSemicontinuous (f i)) : UpperSemicontinuous fun x' => ⨅ i, f i x' := fun x =>
upperSemicontinuousAt_ciInf (Eventually.of... | Mathlib/Topology/Semicontinuous.lean | 1,048 | 1,053 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Yury Kudryashov, Neil Strickland
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Opposites
import Mathlib.Algebra.GroupWit... | theorem leftDistribClass [Mul R] [Add R] [LeftDistribClass R] (add : ∀ x y, f (x + y) = f x + f y)
(mul : ∀ x y, f (x * y) = f x * f y) : LeftDistribClass S where
left_distrib x y z := hf <| by simp only [*, left_distrib]
| Mathlib/Algebra/Ring/InjSurj.lean | 23 | 25 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Pairwise
import Mathlib.Data.Set.BooleanAlgebra
/-!
# The set lattice
This file is a collectio... | Mathlib/Data/Set/Lattice.lean | 2,265 | 2,266 | |
/-
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.RingTheory.Ideal.Operations
/-!
# Maps on modules and ideals
Main definitions include `Ideal.map`, `Ideal.comap`, `RingHom.ker`, `Module.annihilator`
and `Subm... |
variable {G : Type*} [FunLike G S R]
theorem map_le_comap_of_inv_on [RingHomClass G S R] (g : G) (I : Ideal R)
(hf : Set.LeftInvOn g f I) :
I.map f ≤ I.comap g := by
| Mathlib/RingTheory/Ideal/Maps.lean | 90 | 95 |
/-
Copyright (c) 2021 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.... | rcases (natDegree_det_X_add_C_le 1 A).eq_or_lt with h | h
· simp only [RingHom.map_one, Matrix.map_one, C_eq_zero] at h
rw [h]
· -- contradiction. we have a hypothesis that the degree is less than |n|
-- but we know that coeff _ n = 1
have H := coeff_eq_zero_of_natDegree_lt h
rw [coeff_det_X_add_C... | Mathlib/LinearAlgebra/Matrix/Polynomial.lean | 89 | 102 |
/-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
import Mathlib.SetTheory.Nimber.Basic
/-!
# Nim and the Sprague-Grundy theore... | theorem grundyValue_le_of_forall_moveLeft {G : PGame} {o : Nimber}
(h : ∀ i, grundyValue (G.moveLeft i) ≠ o) : G.grundyValue ≤ o := by
| Mathlib/SetTheory/Game/Nim.lean | 255 | 256 |
/-
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.Asymptotics
/-!
# Continu... | exact fun p hp => (ofReal_cpow_of_nonpos (le_of_lt hp.1) p.2).symm
have A : ContinuousAt (fun p => ⟨-↑p.1, p.2⟩ : ℝ × ℂ → ℂ × ℂ) (x, y) := by fun_prop
apply ContinuousAt.mul
· refine (continuousAt_cpow (Or.inl ?_)).comp A
rwa [neg_re, ofReal_re, neg_pos]
· exact (continuous_exp.comp (continu... | Mathlib/Analysis/SpecialFunctions/Pow/Continuity.lean | 355 | 361 |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Sébastien Gouëzel
-/
import Mathlib.Analysis.Normed.Module.Basic
import Mathlib.MeasureTheory.Function.SimpleFuncDense
/-!
# Strongly measurable and finitely strongly meas... | Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean | 1,841 | 1,843 | |
/-
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.... | obtain ⟨y, rfl⟩ := exists_succ_eq.2 hi; exact predAbove_zero_succ
lemma predAbove_zero [NeZero n] {i : Fin (n + 1)} :
predAbove (0 : Fin n) i = if hi : i = 0 then 0 else i.pred hi := by
split_ifs with hi
| Mathlib/Data/Fin/Basic.lean | 1,252 | 1,256 |
/-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis
-/
import Mathlib.RingTheory.Valuation.Basic
import Mathlib.NumberTheory.Padics.PadicNorm
import Mathlib.Analysis.Normed.Field.Lemmas
import Mathlib.Tactic.Peel
import... | theorem norm_nonneg (f : PadicSeq p) : 0 ≤ f.norm := by
classical exact if hf : f ≈ 0 then by simp [hf, norm] else by simp [norm, hf, padicNorm.nonneg]
/-- An auxiliary lemma for manipulating sequence indices. -/
theorem lift_index_left_left {f : PadicSeq p} (hf : ¬f ≈ 0) (v2 v3 : ℕ) :
| Mathlib/NumberTheory/Padics/PadicNumbers.lean | 155 | 159 |
/-
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.Analysis.SpecialFunctions.Trigonometric.Arctan
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
/-!
# Right-angled triangles
This file proves ba... | namespace EuclideanGeometry
open InnerProductGeometry
variable {V : Type*} {P : Type*} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P]
| Mathlib/Geometry/Euclidean/Angle/Unoriented/RightAngle.lean | 316 | 320 |
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Tactic.Attr.Register
import Mathlib.Tactic.Basic
import Batteries.Logic
import Batteries.Tactic.Trans
import Batteries.Util.LibraryNot... | Mathlib/Logic/Basic.lean | 1,179 | 1,180 | |
/-
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 | 1,327 | 1,330 | |
/-
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.Topology.Category.TopCat.Limits.Pullbacks
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
/-!
# Open immersions of structured spaces
We say that a m... | simp only [← isoRestrict_hom_ofRestrict f, Iso.inv_hom_id_assoc]
/-- For an open immersion `f : X ⟶ Y` and an open set `U ⊆ X`, we have the map `X(U) ⟶ Y(U)`. -/
noncomputable def invApp (U : Opens X) :
X.presheaf.obj (op U) ⟶ Y.presheaf.obj (op (opensFunctor f |>.obj U)) :=
PresheafedSpace.IsOpenImmersion.invA... | Mathlib/Geometry/RingedSpace/OpenImmersion.lean | 1,184 | 1,194 |
/-
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.Logic.Encodable.Pi
import Mathlib.SetTheory.Cardinal.Basic
import Mathlib.Topology.MetricSpace.Closeds
import Mathlib.Topology.MetricSpace.Comple... | variable {X : Type u} [MetricSpace X] [CompactSpace X] [Nonempty X] {Y : Type v} [MetricSpace Y]
[CompactSpace Y] [Nonempty Y]
/-- If there are subsets which are `ε₁`-dense and `ε₃`-dense in two spaces, and
isometric up to `ε₂`, then the Gromov-Hausdorff distance between the spaces is bounded by
`ε₁ + ε₂/2 + ε₃`. -/... | Mathlib/Topology/MetricSpace/GromovHausdorff.lean | 540 | 548 |
/-
Copyright (c) 2021 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn, Joachim Breitner
-/
import Mathlib.Algebra.Group.Action.End
import Mathlib.Algebra.Group.Action.Pointwise.Set.Basic
import Mathlib.Algebra.Group.Submonoid.Membership
import Mat... | /-- The empty reduced word. -/
@[simps]
def empty : Word M where
toList := []
ne_one := by simp
chain_ne := List.chain'_nil
instance : Inhabited (Word M) :=
⟨empty⟩
| Mathlib/GroupTheory/CoprodI.lean | 269 | 277 |
/-
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... | section Field
| Mathlib/Algebra/Field/Basic.lean | 154 | 155 |
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang, Yury Kudryashov
-/
import Mathlib.Order.UpperLower.Closure
import Mathlib.Order.UpperLower.Fibration
import Mathlib.Tactic.TFAE
import Mathlib.Topology.ContinuousOn
import Ma... |
lemma Union_closure_singleton_eq_iff {s : Set X} :
(⋃ x ∈ s, closure {x}) = s ↔ StableUnderSpecialization s :=
| Mathlib/Topology/Inseparable.lean | 281 | 283 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl
-/
import Mathlib.MeasureTheory.Integral.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Lebesgue.Countable
import Mathlib.MeasureTheory.Integral.Le... | Mathlib/MeasureTheory/Integral/Lebesgue.lean | 1,677 | 1,679 | |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.Algebra.MvPolynomial.PDeriv
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Algebra.Polynomial.Eva... | simp only [Nat.succ_eq_add_one, add_le_add_iff_right] at h
simp only [Fin.val_last, Fin.init_def]
dsimp
apply not_mem_span_of_apply_not_mem_span_image (@Polynomial.derivative ℚ _ ^ (n - k))
-- Note: https://github.com/leanprover-community/mathlib4/pull/8386 had to change `span_image` into ... | Mathlib/RingTheory/Polynomial/Bernstein.lean | 234 | 238 |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2
import Mathlib.MeasureTheory.Measure.Real
/-! # Conditional expectation in L1
This file contains ... |
end CondexpIndL1
variable (G)
| Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean | 274 | 278 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.Order.Group.Finset
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Algebra.Polynomial.Eva... | theorem monic_map_iff [Semiring S] [Nontrivial S] {f : R →+* S} {p : R[X]} :
(p.map f).Monic ↔ p.Monic := by
rw [Monic, leadingCoeff_map, ← f.map_one, Function.Injective.eq_iff f.injective, Monic]
end DivisionRing
section Field
variable [Field R] {p q : R[X]}
| Mathlib/Algebra/Polynomial/FieldDivision.lean | 275 | 283 |
/-
Copyright (c) 2019 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Yury Kudryashov, Yaël Dillies
-/
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.Algebra.Or... | forall₂_congr fun _ => starConvex_iff_openSegment_subset
theorem convex_iff_forall_pos :
Convex 𝕜 s ↔
| Mathlib/Analysis/Convex/Basic.lean | 131 | 134 |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jan-David Salchow, Sébastien Gouëzel, Jean Lo, Yury Kudryashov, Frédéric Dupuis,
Heather Macbeth
-/
import Mathlib.Algebra.Module.Opposite
import Mathlib.Topology.Algebra.Group.Qu... | Mathlib/Topology/Algebra/Module/Basic.lean | 2,200 | 2,202 | |
/-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.GCDMonoid.Multiset
import Mathlib.Algebra.GCDMonoid.Nat
import Mathlib.Algebra.Group.TypeTags.Finite
import Mathlib.Combinatorics.Enumerative... |
theorem two_dvd_card_support {σ : Perm α} (hσ : σ ^ 2 = 1) : 2 ∣ #σ.support :=
(congr_arg (Dvd.dvd 2) σ.sum_cycleType).mp
(Multiset.dvd_sum fun n hn => by
rw [_root_.le_antisymm
(Nat.le_of_dvd zero_lt_two <|
(dvd_of_mem_cycleType hn).trans <| orderOf_dvd_of_pow_eq_one hσ)
| Mathlib/GroupTheory/Perm/Cycle/Type.lean | 195 | 201 |
/-
Copyright (c) 2023 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.FieldTheory.SplittingField.Construction
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.Separable
import Mathlib.FieldTheory.Normal.... | exact (minpoly.irreducible halg.isIntegral).natSepDegree_dvd_natDegree
have : finrank F F⟮α⟯ = 0 := finrank_of_infinite_dimensional fun _ ↦
halg ((AdjoinSimple.isIntegral_gen F α).1 (IsIntegral.of_finite F _)).isAlgebraic
rw [this]
exact dvd_zero _
/-- The separable degree of `F⟮α⟯ / F` is smaller than t... | Mathlib/FieldTheory/SeparableDegree.lean | 699 | 722 |
/-
Copyright (c) 2021 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Order.Lattice
import Mathlib.Data.List.Sort
import Mathlib.Logic.Equiv.Fin.Basic
import Mathlib.Logic.Equiv.Functor
import Mathlib.Data.Fintype.Pigeonhole
... |
protected theorem smash {s₁ s₂ t₁ t₂ : CompositionSeries X}
(hs : s₁.last = s₂.head) (ht : t₁.last = t₂.head)
(h₁ : Equivalent s₁ t₁) (h₂ : Equivalent s₂ t₂) :
Equivalent (smash s₁ s₂ hs) (smash t₁ t₂ ht) :=
let e : Fin (s₁.length + s₂.length) ≃ Fin (t₁.length + t₂.length) :=
calc
Fin (s₁.lengt... | Mathlib/Order/JordanHolder.lean | 268 | 279 |
/-
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... | · rw [Simplex.orthogonalProjectionSpan] at hp
rw [hp₁, hp₂, ← hp]
simp only [true_or, eq_self_iff_true, smul_zero, vsub_self]
· have hz : ⟪p -ᵥ orthogonalProjection span_s p, p -ᵥ orthogonalProjection span_s p⟫ ≠ 0 := by
simpa only [Ne, vsub_eq_zero_iff_eq, inner_self_eq_zero] using hp
rw [mul_lef... | Mathlib/Geometry/Euclidean/Circumcenter.lean | 802 | 818 |
/-
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.Data.Set.Function
import Mathlib.Logic.Pairwise
import Mathlib.Logic.Relation
/-!
# Relations holding pairwise
This file develops pairwise relations ... | hij <| Prod.ext (hs.elim_set hi hj _ hai haj) <| ht.elim_set hi' hj' _ hbi hbj
theorem pairwiseDisjoint_pi {ι' α : ι → Type*} {s : ∀ i, Set (ι' i)} {f : ∀ i, ι' i → Set (α i)}
(hs : ∀ i, (s i).PairwiseDisjoint (f i)) :
((univ : Set ι).pi s).PairwiseDisjoint fun I => (univ : Set ι).pi fun i => f _ (I i) :=
... | Mathlib/Data/Set/Pairwise/Basic.lean | 343 | 349 |
/-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz, Nikolas Kuhn
-/
import Mathlib.CategoryTheory.Sites.Coherent.CoherentSheaves
/-!
# Description of the covering sieves of the coherent topology
This file characterises the cov... | theorem EffectiveEpiFamily.transitive_of_finite {α : Type} [Finite α] {Y : α → C}
(π : (a : α) → (Y a ⟶ X)) (h : EffectiveEpiFamily Y π) {β : α → Type} [∀ (a : α), Finite (β a)]
{Y_n : (a : α) → β a → C} (π_n : (a : α) → (b : β a) → (Y_n a b ⟶ Y a))
(H : ∀ a, EffectiveEpiFamily (Y_n a) (π_n a)) :
Effect... | Mathlib/CategoryTheory/Sites/Coherent/CoherentTopology.lean | 44 | 68 |
/-
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 | 1,924 | 1,926 | |
/-
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 | 651 | 652 | |
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.FinitePresentation
import Mathlib.RingTheory.FiniteStability
import Mathlib.RingTheory.Ideal.Cotangent
import Mathlib.RingTheory.Ideal.Quotient.Ni... | ext x
exact (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x).symm
exact ⟨l.comp g, by rw [← AlgHom.comp_assoc, ← this, AlgHom.comp_assoc, hg, AlgHom.comp_id]⟩
variable (hf : Function.Surjective f)
include hf
/-- Let `P →ₐ[R] A` be a surjection with kernel `J`, and `P` a formally smooth `R`-algebra,
then `A` ... | Mathlib/RingTheory/Smooth/Basic.lean | 208 | 226 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Johan Commelin, Andrew Yang, Joël Riou
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
import Mathlib.CategoryTheory.Monoid... | (shiftFunctorAdd C a₂ a₃).inv.app (X⟦a₁⟧) ≫
(shiftFunctorAdd' C a₁ (a₂ + a₃) (a₁ + a₂ + a₃) (add_assoc _ _ _).symm).inv.app X := by
simpa using NatTrans.congr_app (congr_arg Iso.inv (shiftFunctorAdd_assoc C a₁ a₂ a₃)) X
end Defs
section AddMonoid
variable [AddMonoid A] [HasShift C A] (X Y : C) (f : X ⟶ Y... | Mathlib/CategoryTheory/Shift/Basic.lean | 338 | 346 |
/-
Copyright (c) 2022 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Reverse
import Mathlib.Algebra.Polynomial.Inductions
import Mathlib.RingTheory.Localizati... | section ExactDegrees
@[simp]
theorem degree_C_mul_T (n : ℤ) (a : R) (a0 : a ≠ 0) : degree (C a * T n) = n := by
rw [degree, support_C_mul_T_of_ne_zero a0 n]
exact Finset.max_singleton
| Mathlib/Algebra/Polynomial/Laurent.lean | 431 | 437 |
/-
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.Analysis.Analytic.Within
import Mathlib.Analysis.Calculus.FDeriv.Analytic
import Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries
/-!
# Higher d... | Mathlib/Analysis/Calculus/ContDiff/Defs.lean | 1,322 | 1,327 | |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Frédéric Dupuis
-/
import Mathlib.Analysis.Convex.Cone.Basic
import Mathlib.Data.Real.Archimedean
import Mathlib.LinearAlgebra.LinearPMap
/-!
# Extension theorems
W... | obtain ⟨y, -, hy⟩ : ∃ y ∈ ⊤, y ∉ f.domain := SetLike.exists_of_lt (lt_top_iff_ne_top.2 hdom)
obtain ⟨c, le_c, c_le⟩ :
∃ c, (∀ x : f.domain, -(x : E) - y ∈ s → f x ≤ c) ∧
∀ x : f.domain, (x : E) + y ∈ s → c ≤ f x := by
set Sp := f '' { x : f.domain | (x : E) + y ∈ s }
set Sn := f '' { x : f.dom... | Mathlib/Analysis/Convex/Cone/Extension.lean | 64 | 112 |
/-
Copyright (c) 2020 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Grading
import Mathlib.Algebra.Module.Opposite
/-!
# Conjugations
This file defines the grade reversal and grade involution f... | theorem reverse_involute :
∀ a : CliffordAlgebra Q, reverse (involute a) = involute (reverse a) :=
reverse_involute_commute
end Reverse
/-!
### Statements about conjugations of products of lists
-/
| Mathlib/LinearAlgebra/CliffordAlgebra/Conjugation.lean | 152 | 161 |
/-
Copyright (c) 2015, 2017 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Robert Y. Lewis, Johannes Hölzl, Mario Carneiro, Sébastien Gouëzel
-/
import Mathlib.Data.ENNReal.Real
import Mathlib.Tactic.Bound.Attribute
import Mathlib.Topology... | closedBall x ε = {x} := by
ext x'
| Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | 425 | 426 |
/-
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 | 925 | 926 | |
/-
Copyright (c) 2024 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.MellinTransform
/-!
# Abstract functional equations for Mellin transforms
This file formalises a general version of an argument used to pro... | have hc3 : mellin (fun x ↦ f_modif P x - f P x + P.f₀) s =
(1 / s) • P.f₀ + (P.ε / (↑P.k - s)) • P.g₀ := P.f_modif_aux2 hs
have := (hasMellin_sub hc2.1 hc1).2
simp_rw [← sub_add, hc3, eq_sub_iff_add_eq, ← eq_sub_iff_add_eq', ← sub_sub] at this
exact this
| Mathlib/NumberTheory/LSeries/AbstractFuncEq.lean | 427 | 432 |
/-
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... | theorem tprod_range {g : γ → β} (f : β → α) (hg : Injective g) :
∏' x : Set.range g, f x = ∏' x, f (g x) := by
rw [← Set.image_univ, tprod_image f hg.injOn]
simp_rw [← comp_apply (g := g), tprod_univ (f ∘ g)]
/-- If `f b = 1` for all `b ∈ t`, then the product of `f a` with `a ∈ s` is the same as the
product of... | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | 491 | 506 |
/-
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 | 958 | 958 | |
/-
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.Defs
import Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno
import Mathlib.Analysis.Calculus.FDeriv.Ad... |
section deriv
/-!
### One dimension
All results up to now have been expressed in terms of the general Fréchet derivative `fderiv`. For
maps defined on the field, the one-dimensional derivative `deriv` is often easier to use. In this
paragraph, we reformulate some higher smoothness results in terms of `deriv`.
| Mathlib/Analysis/Calculus/ContDiff/Basic.lean | 1,231 | 1,239 |
/-
Copyright (c) 2023 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot
-/
import Mathlib.Order.Filter.Germ.Basic
import Mathlib.Topology.NhdsSet
import Mathlib.Topology.LocallyConstant.Basic
import Mathlib.Analysis.Normed.Module.Basic
/-! ... | show (f ∘ i) (⟨x, x_in⟩ : s) = (f ∘ i) (⟨x', x'_in⟩ : s)
have : PreconnectedSpace s := Subtype.preconnectedSpace hs
exact eq_of_germ_isConstant (fun y ↦ Germ.isConstant_comp_subtype (h y y.2)) _ _
@[to_additive (attr := simp)]
theorem Germ.coe_prod {α : Type*} (l : Filter α) (R : Type*) [CommMonoid R] {ι} (f : ι... | Mathlib/Topology/Germ.lean | 169 | 174 |
/-
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.IsPrimePow
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Ring.Int
import Mathlib.Algebra.Ring.CharZero
im... |
@[simp]
theorem divisorsAntidiagonal_zero : divisorsAntidiagonal 0 = ∅ := by
ext
simp
| Mathlib/NumberTheory/Divisors.lean | 299 | 304 |
/-
Copyright (c) 2024 Lean FRO. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.Data.List.InsertIdx
/-!
This is a stub file for importing `Mathlib.Data.List.InsertNth`,
which has been renamed to `Mathlib.Data.List.InsertIdx`.
This file c... | Mathlib/Data/List/InsertNth.lean | 83 | 91 | |
/-
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.Algebra.Subalgebra.Prod
import Mathlib.Algebra.Algebra.Subalgebra.Tower
import Mathlib.LinearAlgebra.Basis.Basic
import Mathlib.LinearAlgebra.Prod
/-!
# ... | Mathlib/RingTheory/Adjoin/Basic.lean | 224 | 230 | |
/-
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.Group.Subgroup.Ker
import Mathlib.Algebra.BigOperators.Group.List.Basic
/-!
# Free groups
This file defines free groups over a type. Furthermore, it is... |
@[to_additive (attr := simp)]
theorem prod.of {x : α} : prod (of x) = x :=
lift.of
@[to_additive]
| Mathlib/GroupTheory/FreeGroup/Basic.lean | 766 | 771 |
/-
Copyright (c) 2021 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn, Joachim Breitner
-/
import Mathlib.Algebra.Group.Action.End
import Mathlib.Algebra.Group.Action.Pointwise.Set.Basic
import Mathlib.Algebra.Group.Submonoid.Membership
import Mat... | @[simp]
theorem iSup_mrange_of : ⨆ i, MonoidHom.mrange (of : M i →* CoprodI M) = ⊤ := by
simp [← mrange_eq_iSup]
| Mathlib/GroupTheory/CoprodI.lean | 194 | 196 |
/-
Copyright (c) 2019 Calle Sönne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Analysis.Normed.Group.AddCircle
import Mathlib.Algebra.CharZero.Quotient
import Mathlib.Topology... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | 985 | 988 | |
/-
Copyright (c) 2020 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Algebra.Order.Group.Nat
import Mathlib.Algebra.Order.Sub.Basic
/-!
# `Nat.Upto`
`Nat.Upto p`, with `p` a predicate on `ℕ`, is a subtype of elements `n : ℕ`... | rw [this]
exact (measure _).wf
ext ⟨a, ha⟩ ⟨b, _⟩
dsimp [InvImage, Upto.GT]
rw [tsub_lt_tsub_iff_left_of_le (le_of_not_lt fun h' => ha _ h' h)]
/-- Zero is always a member of `Nat.Upto p` because it has no predecessors. -/
def zero : Nat.Upto p :=
| Mathlib/Data/Nat/Upto.lean | 52 | 59 |
/-
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 | 3,073 | 3,077 | |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Multiset.ZeroCons
/-!
# Basic results on multisets
-/
-- No algebra should be required
assert_not_exists Monoid
universe v
open List S... | Mathlib/Data/Multiset/Basic.lean | 3,054 | 3,055 | |
/-
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 | 2,274 | 2,305 | |
/-
Copyright (c) 2023 Newell Jensen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Newell Jensen
-/
import Mathlib.GroupTheory.SpecificGroups.Cyclic
import Mathlib.GroupTheory.SpecificGroups.Dihedral
/-!
# Klein Four Group
The Klein (Vierergruppe) four-group is a no... | @[to_additive]
lemma eq_finset_univ [Fintype G] [DecidableEq G]
{x y : G} (hx : x ≠ 1) (hy : y ≠ 1) (hxy : x ≠ y) : {x * y, x, y, (1 : G)} = Finset.univ := by
apply Finset.eq_univ_of_card
rw [card_four']
repeat rw [card_insert_of_not_mem]
on_goal 4 => simpa using mul_not_mem_of_exponent_two (by simp) hx hy ... | Mathlib/GroupTheory/SpecificGroups/KleinFour.lean | 109 | 116 |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing
/-!
# Stalks
For a presheaf `F` on a topol... | ∃ (W : Opens X) (_m : x ∈ W) (iU : W ⟶ U) (iV : W ⟶ V), F.map iU.op s = F.map iV.op t := by
obtain ⟨W, iU, iV, e⟩ :=
(Types.FilteredColimit.isColimit_eq_iff.{v, v} _
(isColimitOfPreserves (forget C) (colimit.isColimit ((OpenNhds.inclusion x).op ⋙ F)))).mp
h
exact ⟨(unop W).1, (unop W).2, i... | Mathlib/Topology/Sheaves/Stalks.lean | 419 | 425 |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Logic.Basic
import Mathlib.Tactic.Positivity.Basic
/-!
# Algebraic order homomorphism classes
This file defines hom classes for common properties at the ... |
@[to_additive existing]
theorem le_map_add_map_div [Group α] [AddCommMagma β] [LE β] [MulLEAddHomClass F α β] (f : F)
| Mathlib/Algebra/Order/Hom/Basic.lean | 124 | 126 |
/-
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.Filter.Tendsto
import Mathlib.Data.Set.Accumulate
import Mathlib.Topology.Bornology.Basic
import Mathlib.Topolog... | and `y` is a unique `MapClusterPt` for `f` along `l` in `s`,
then `f` tends to `𝓝 y` along `l`. -/
lemma IsCompact.tendsto_nhds_of_unique_mapClusterPt {Y} {l : Filter Y} {y : X} {f : Y → X}
(hs : IsCompact s) (hmem : ∀ᶠ x in l, f x ∈ s) (h : ∀ x ∈ s, MapClusterPt x l f → x = y) :
Tendsto f l (𝓝 y) :=
hs.le_... | Mathlib/Topology/Compactness/Compact.lean | 151 | 158 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Algebra.MonoidAlgebra.Degree
import Mathlib.Algebra.MvPolynomial.Rename
/-!
# Degrees of polynomials
This file establ... | rw [Finset.mem_sdiff]
exact ⟨hb₁, hb⟩
rw [totalDegree_eq, hb₂, totalDegree_eq]
exact Finset.le_sup (f := fun m => Multiset.card (Finsupp.toMultiset m)) hbb
theorem totalDegree_add_eq_right_of_totalDegree_lt {p q : MvPolynomial σ R}
(h : q.totalDegree < p.totalDegree) : (q + p).totalDegree = p.t... | Mathlib/Algebra/MvPolynomial/Degrees.lean | 409 | 429 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.MonoidAlgebra.Degree
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.Polynomial.Basi... | Mathlib/Algebra/Polynomial/Degree/Definitions.lean | 1,351 | 1,352 | |
/-
Copyright (c) 2022 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
import Mathlib.Topology.UniformSpace.UniformConvergenceTopology
/-!
# Equicontinuity of a family of functions
Let `X` be a topological space and `α` a `UniformS... | rfl
/-!
| Mathlib/Topology/UniformSpace/Equicontinuity.lean | 203 | 205 |
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