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) 2021 Aaron Anderson, Jesse Michael Han, Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jesse Michael Han, Floris van Doorn
-/
import Mathlib.Data.Finset.Basic
import Mathlib.ModelTheory.Syntax
import Mathlib.Data.List.... | theorem realize_rel₁ {R : L.Relations 1} {t : L.Term _} :
| Mathlib/ModelTheory/Semantics.lean | 277 | 277 |
/-
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... |
lemma rpow_natCast_mul (x : ℝ≥0∞) (n : ℕ) (z : ℝ) : x ^ (n * z) = (x ^ n) ^ z := by
| Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean | 914 | 915 |
/-
Copyright (c) 2020 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Topology.Algebra.Algebra
import Mathlib.Analysis.InnerProductSpace.Convex
import Mathlib.Algebra.Module.LinearMap.Rat
import Mathlib.Tactic.Module
/... |
private theorem add_left_aux7 (y z : E) :
‖(I : 𝕜) • y + 2 • z‖ * ‖(I : 𝕜) • y + 2 • z‖ + ‖(I : 𝕜) • y‖ * ‖(I : 𝕜) • y‖ =
2 * (‖(I : 𝕜) • y + z‖ * ‖(I : 𝕜) • y + z‖ + ‖z‖ * ‖z‖) := by
| Mathlib/Analysis/InnerProductSpace/OfNorm.lean | 165 | 168 |
/-
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.Algebra.GroupWithZero.Pointwise.Set.Basic
import Mathlib.Algebra.Ring.Pointwise.Set
import Mathlib.Topology.MetricSpace.Isometry
import Mathlib.Topol... | instance Multiplicative.isIsIsometricVAdd'' [Add M] [PseudoEMetricSpace M]
[IsIsometricVAdd Mᵃᵒᵖ M] : IsIsometricSMul (Multiplicative M)ᵐᵒᵖ (Multiplicative M) :=
| Mathlib/Topology/MetricSpace/IsometricSMul.lean | 446 | 447 |
/-
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
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.Order.CauSeq.BigOperators
import Mathlib.Algebra.Order.Star.Basic
import Mathlib.Data.C... | Mathlib/Data/Complex/Exponential.lean | 1,671 | 1,674 | |
/-
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.Bool.Basic
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.CoreAttrs
/-!
# (Semi-)lattices
Semilatti... |
theorem inf_def [Min α] [Min β] (p q : α × β) : p ⊓ q = (p.fst ⊓ q.fst, p.snd ⊓ q.snd) :=
rfl
| Mathlib/Order/Lattice.lean | 1,112 | 1,114 |
/-
Copyright (c) 2018 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Johannes Hölzl, Yaël Dillies
-/
import Mathlib.Analysis.Normed.Group.Seminorm
import Mathlib.Data.NNReal.Basic
import Mathlib.Topology.Algebra.Support
import Mathlib.To... | convert NormedCommGroup.nhds_basis_norm_lt (1 : E)
| Mathlib/Analysis/Normed/Group/Basic.lean | 669 | 669 |
/-
Copyright (c) 2018 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Bhavik Mehta
-/
import Mathlib.CategoryTheory.Limits.HasLimits
import Mathlib.CategoryTheory.Discrete.Basic
/-!
# Categorical (co)products
This file defines (co)products ... | hasLimitEquivalenceComp _
have α : Discrete.functor g ≅ (Discrete.equivalence e).functor ⋙ Discrete.functor f :=
Discrete.natIso (fun ⟨j⟩ => iso j)
exact hasLimit_of_iso α.symm
/-- Make a fan `f` into a limit fan by providing `lift`, `fac`, and `uniq` --
just a convenience lemma to avoid having to go thr... | Mathlib/CategoryTheory/Limits/Shapes/Products.lean | 105 | 111 |
/-
Copyright (c) 2024 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated
import Mathlib.Probability.Process.Filtration
/-!
# Filtration built from the finite partitions of a coun... | lemma measurable_memPartitionSet_subtype (ht : ∀ n, MeasurableSet (t n)) (n : ℕ)
(m : MeasurableSpace (memPartition t n)) :
@Measurable α (memPartition t n) (partitionFiltration ht n) m
(fun a ↦ ⟨memPartitionSet t n a, memPartitionSet_mem t n a⟩) := by
refine @measurable_to_countable' (memPartition t n)... | Mathlib/Probability/Process/PartitionFiltration.lean | 64 | 76 |
/-
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.Order.Group.Pointwise.Interval
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Data.Rat.Cardinal
import Mathlib.SetTheory.Cardi... |
theorem cantorFunctionAux_succ (f : ℕ → Bool) :
| Mathlib/Data/Real/Cardinality.lean | 78 | 79 |
/-
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.Field.IsField
import Mathlib.Algebra.Polynomial.Inductions
import Mathlib.Algebra.Polynomial.Monic
im... | lemma mul_self_modByMonic (hq : q.Monic) : (p * q) %ₘ q = 0 := by
rw [modByMonic_eq_zero_iff_dvd hq]
exact dvd_mul_left q p
| Mathlib/Algebra/Polynomial/Div.lean | 659 | 661 |
/-
Copyright (c) 2024 Jeremy Tan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Tan
-/
import Mathlib.Analysis.SpecialFunctions.Complex.LogBounds
/-!
# Complex arctangent
This file defines the complex arctangent `Complex.arctan` as
$$\arctan z = -\frac i2 \lo... | /-- `cos z` is nonzero when the bounds in `arctan_tan` are met (`z` lies in the vertical strip
`-π / 2 < z.re < π / 2` and `z ≠ π / 2`). -/
lemma cos_ne_zero_of_arctan_bounds {z : ℂ} (h₀ : z ≠ π / 2) (h₁ : -(π / 2) < z.re)
(h₂ : z.re ≤ π / 2) : cos z ≠ 0 := by
refine cos_ne_zero_iff.mpr (fun k ↦ ?_)
rw [ne_eq, ... | Mathlib/Analysis/SpecialFunctions/Complex/Arctan.lean | 48 | 62 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Field.IsField
import Mathlib.Algebra.GroupWithZero.N... | Mathlib/RingTheory/Localization/Basic.lean | 571 | 574 | |
/-
Copyright (c) 2021 Aaron Anderson, Jesse Michael Han, Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jesse Michael Han, Floris van Doorn
-/
import Mathlib.Data.Finset.Basic
import Mathlib.ModelTheory.Syntax
import Mathlib.Data.List.... |
theorem realize_restrictVarLeft [DecidableEq α] {γ : Type*} {t : L.Term (α ⊕ γ)}
{f : t.varFinsetLeft → β}
{xs : β ⊕ γ → M} (xs' : α → M) (hxs' : ∀ a, xs (Sum.inl (f a)) = xs' a) :
(t.restrictVarLeft f).realize xs = t.realize (Sum.elim xs' (xs ∘ Sum.inr)) := by
induction t with
| Mathlib/ModelTheory/Semantics.lean | 138 | 143 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Sean Leather
-/
import Batteries.Data.List.Perm
import Mathlib.Data.List.Pairwise
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Lookmap
import Mathlib.Data.Si... | | Or.inr h => mem_dlookup_kunion.mpr <| Or.inr ⟨mt mem_keys_kunion.mp (not_or.mpr ⟨h.1, h₂⟩), h.2⟩
end List
| Mathlib/Data/List/Sigma.lean | 732 | 739 |
/-
Copyright (c) 2024 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll
-/
import Mathlib.Algebra.BigOperators.Field
import Mathlib.NumberTheory.LSeries.Basic
/-!
# Linearity of the L-series of `f` as a function of `f`
We show that the `LSer... | @[simp]
lemma LSeries_add {f g : ℕ → ℂ} {s : ℂ} (hf : LSeriesSummable f s) (hg : LSeriesSummable g s) :
LSeries (f + g) s = LSeries f s + LSeries g s := by
simpa [LSeries, term_add] using hf.tsum_add hg
| Mathlib/NumberTheory/LSeries/Linearity.lean | 40 | 43 |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Robert Y. Lewis
-/
import Mathlib.FieldTheory.Finite.Polynomial
import Mathlib.NumberTheory.Basic
import Mathlib.RingTheory.WittVector.WittPolynomial
/-!
# Witt struct... | apply MvPolynomial.map_injective (Int.castRingHom ℚ) Int.cast_injective
simp only [map_bind₁, map_rename, map_expand, rename_expand, map_wittPolynomial]
have key := (wittStructureRat_prop p (map (Int.castRingHom ℚ) Φ) n).symm
apply_fun expand p at key
simp only [expand_bind₁] at key
rw [key]; clear key
ap... | Mathlib/RingTheory/WittVector/StructurePolynomial.lean | 210 | 227 |
/-
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, Simon Hudon, Mario Carneiro
-/
import Aesop
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Nat.Init
import Mathlib.Data.Int.Init
import Mathlib.... | (h_mul : ∀ a, P a → P (g * a)) (h_inv : ∀ a, P a → P (g⁻¹ * a)) (n : ℤ) : P (g ^ n) := by
| Mathlib/Algebra/Group/Basic.lean | 890 | 890 |
/-
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.... | classical
simpa only [natSepDegree_eq_of_isAlgClosed (AlgebraicClosure F), aroots_def, map_expand,
Fintype.card_coe] using Fintype.card_eq.2
⟨(f.map (algebraMap F (AlgebraicClosure F))).rootsExpandPowEquivRoots q n⟩
theorem natSepDegree_X_pow_char_pow_sub_C (q : ℕ) [ExpChar F q] (n : ℕ) (y : F) :
(X ... | Mathlib/FieldTheory/SeparableDegree.lean | 479 | 488 |
/-
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.NumberTheory.LSeries.AbstractFuncEq
import Mathlib.NumberTheory.ModularForms.JacobiTheta.Bounds
import Mathlib.NumberTheory.LSeries.MellinEqDirichlet
i... | refine this.congr_fun fun n ↦ ?_
simp_rw [Int.cast_neg, neg_sq, mul_neg, ofReal_mul, Int.cast_natCast, ofReal_natCast,
ofReal_ofNat, ← add_mul, ofReal_sin, Complex.sin]
push_cast
congr 1
rw [← mul_div_assoc, ← div_mul_eq_mul_div, ← div_mul_eq_mul_div, div_self two_ne_zero, one_mul,
neg_mul, neg_mul,... | Mathlib/NumberTheory/LSeries/HurwitzZetaOdd.lean | 244 | 258 |
/-
Copyright (c) 2022 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang, Jujian Zhang
-/
import Mathlib.RingTheory.IsTensorProduct
import Mathlib.RingTheory.Localization.Module
import Mathlib.LinearAlgebra.DirectSum.Finsupp
import Mathlib.Algebra... | given by `m/s ↦ (1/s) ⊗ₜ m`.
-/
theorem isLocalizedModule_iff_isBaseChange : IsLocalizedModule S f ↔ IsBaseChange A f := by
refine ⟨fun _ ↦ IsLocalizedModule.isBaseChange S A f, fun h ↦ ?_⟩
have : IsBaseChange A (LocalizedModule.mkLinearMap S M) := IsLocalizedModule.isBaseChange S A _
let e := (this.equiv.symm.tr... | Mathlib/RingTheory/Localization/BaseChange.lean | 41 | 49 |
/-
Copyright (c) 2024 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.MvPolynomial.Monad
import Mathlib.LinearAlgebra.Charpoly.ToMatrix
import Mathlib.LinearAlgebra.FreeModule.StrongRankCondition
import Mathlib.Li... | (polyCharpolyAux φ b bₘ).map (MvPolynomial.eval (b.repr x)) = (φ x).charpoly := by
nontriviality R
rw [polyCharpolyAux_map_eq_toMatrix_charpoly, LinearMap.charpoly_toMatrix]
@[simp]
lemma polyCharpolyAux_coeff_eval [Module.Finite R M] [Module.Free R M] (x : L) (i : ℕ) :
| Mathlib/Algebra/Module/LinearMap/Polynomial.lean | 295 | 300 |
/-
Copyright (c) 2021 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.GroupTheory.Index
/-!
# Complements
In this file we define the complement of a subgroup.
## Main definitions
- `Subgroup.IsComplement S T` where ... | · rintro ⟨g, rfl⟩
exact isComplement_univ_singleton
@[to_additive]
theorem isComplement_univ_right : IsComplement S univ ↔ ∃ g : G, S = {g} := by
| Mathlib/GroupTheory/Complement.lean | 124 | 128 |
/-
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.Algebra.Homology.Linear
import Mathlib.Algebra.Homology.ShortComplex.HomologicalComplex
import Mathlib.Tactic.Abel
/-!
# Chain homotopies
We define chain... | variable {f g}
namespace Homotopy
/-- `f` is homotopic to `g` iff `f - g` is homotopic to `0`.
-/
def equivSubZero : Homotopy f g ≃ Homotopy (f - g) 0 where
| Mathlib/Algebra/Homology/Homotopy.lean | 124 | 130 |
/-
Copyright (c) 2022 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Heather Macbeth
-/
import Mathlib.Data.Nat.Cast.WithTop
import Mathlib.FieldTheory.IsAlgClosed.Basic
import Mathlib.RingTheory.WittVector.DiscreteValuationRing
/-!
#... | have :=
succNthVal_spec' p n a₁ a₂ (fun i : Fin (n + 1) => frobeniusRotationCoeff p ha₁ ha₂ i.val)
ha₁ ha₂
simp only [frobeniusRotationCoeff, Fin.val_zero] at this
convert this using 3
| Mathlib/RingTheory/WittVector/FrobeniusFractionField.lean | 212 | 216 |
/-
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.Limits.Opposites
import Mathlib.CategoryTheory.Limits.Preserves.Limits
import Mathlib.CategoryTheory.Limits.Yoneda
/-!
# Ind- and ... | coyoneda.obj (op <| colimit F) ≅ limit (F.rightOp ⋙ coyoneda) :=
coyoneda.mapIso (limitRightOpIsoOpColimit F).symm ≪≫ preservesLimitIso coyoneda F.rightOp
@[reassoc (attr := simp)]
lemma coyonedaOpColimitIsoLimitCoyoneda'_hom_comp_π (i : I) :
(coyonedaOpColimitIsoLimitCoyoneda' F).hom ≫ limit.π (F.rightOp ⋙ ... | Mathlib/CategoryTheory/Limits/IndYoneda.lean | 92 | 98 |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, Patrick Massot
-/
import Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic
import Mathlib.MeasureTheory.Measure.Real
import Mathlib.Order.Filter.Indi... | theorem continuousOn_primitive_Icc (h_int : IntegrableOn f (Icc a b) μ) :
ContinuousOn (fun x => ∫ t in Icc a x, f t ∂μ) (Icc a b) := by
have aux : (fun x => ∫ t in Icc a x, f t ∂μ) = fun x => ∫ t in Ioc a x, f t ∂μ := by
ext x
exact integral_Icc_eq_integral_Ioc
rw [aux]
exact continuousOn_primitive h... | Mathlib/MeasureTheory/Integral/DominatedConvergence.lean | 477 | 483 |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.Data.List.Lattice
import Mathlib.Data.Bool.Basic
import Mathlib.Order.Lattice
/-!
# Intervals in ℕ
This file defines intervals of naturals. `List.Ico m n... | @[simp]
theorem mem {n m l : ℕ} : l ∈ Ico n m ↔ n ≤ l ∧ l < m := by
suffices n ≤ l ∧ l < n + (m - n) ↔ n ≤ l ∧ l < m by simp [Ico, this]
| Mathlib/Data/List/Intervals.lean | 56 | 58 |
/-
Copyright (c) 2021 Ruben Van de Velde. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ruben Van de Velde
-/
import Mathlib.Algebra.BigOperators.Group.Multiset.Defs
import Mathlib.Algebra.Order.BigOperators.Ring.List
/-!
# Big operators on a multiset in ordered ring... | (hp_mul : ∀ a b, p a → p b → p (a * b)) (s : Multiset α) (hps : ∀ a, a ∈ s → p a) :
f s.prod ≤ (s.map f).prod := by
revert s
refine Multiset.induction (by simp [h_one]) ?_
intro a s hs hpsa
by_cases hs0 : s = ∅
| Mathlib/Algebra/Order/BigOperators/Ring/Multiset.lean | 32 | 37 |
/-
Copyright (c) 2022 Jake Levinson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jake Levinson
-/
import Mathlib.Data.Finset.Preimage
import Mathlib.Data.Finset.Prod
import Mathlib.Data.SetLike.Basic
import Mathlib.Order.UpperLower.Basic
/-!
# Young diagrams
A You... |
@[simp]
| Mathlib/Combinatorics/Young/YoungDiagram.lean | 214 | 215 |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Robert Y. Lewis
-/
import Mathlib.Algebra.MvPolynomial.Funext
import Mathlib.Algebra.Ring.ULift
import Mathlib.RingTheory.WittVector.Basic
/-!
# The `IsPoly` predicate
... | apply MvPolynomial.map_injective (Int.castRingHom ℚ) Int.cast_injective
rw [← funext_iff] at h
replace h :=
congr_arg (fun fam => bind₁ (MvPolynomial.map (Int.castRingHom ℚ) ∘ fam) (xInTermsOfW p ℚ n)) h
simpa only [Function.comp_def, map_bind₁, map_wittPolynomial, ← bind₁_bind₁,
bind₁_wittPolynomial_xI... | Mathlib/RingTheory/WittVector/IsPoly.lean | 125 | 133 |
/-
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... |
end PadicSeq
section
open PadicSeq
-- Porting note: Commented out `padic_index_simp` tactic
/-
| Mathlib/NumberTheory/Padics/PadicNumbers.lean | 222 | 230 |
/-
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.RingTheory.Artinian.Module
/-!
# Lie subalgebras
This file defines Lie subalgebras of a Lie algebra and provides basic rel... | rfl
end LieHom
theorem Submodule.exists_lieSubalgebra_coe_eq_iff (p : Submodule R L) :
| Mathlib/Algebra/Lie/Subalgebra.lean | 329 | 333 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Violeta Hernández Palacios
-/
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Data.Nat.SuccPred
import Mathlib.Order.SuccPred.Initial... | @[simp]
theorem zero_mod (b : Ordinal) : 0 % b = 0 := by simp only [mod_def, zero_div, mul_zero, sub_self]
theorem div_add_mod (a b : Ordinal) : b * (a / b) + a % b = a :=
Ordinal.add_sub_cancel_of_le <| mul_div_le _ _
| Mathlib/SetTheory/Ordinal/Arithmetic.lean | 985 | 989 |
/-
Copyright (c) 2021 Stuart Presnell. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Stuart Presnell
-/
import Mathlib.Data.Nat.PrimeFin
import Mathlib.Data.Nat.Factorization.Defs
import Mathlib.Data.Nat.GCD.BigOperators
import Mathlib.Order.Interval.Finset.Nat
import... | Mathlib/Data/Nat/Factorization/Basic.lean | 719 | 726 | |
/-
Copyright (c) 2019 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Analytic.Constructions
import Mathlib.Analysis.Calculus.FDeriv.Analytic
import Mathlib.Analysis.Calculus.FDe... | (u.map fun i ↦ ((u.erase i).map (g · x)).prod • g' i).sum x := by
simp only [← Multiset.attach_map_val u, Multiset.map_map]
exact .congr_fderiv
(hasFDerivAt_multiset_prod.comp x <| hasFDerivAt_pi.mpr fun i ↦ h (Subtype.val i) i.prop :)
(by ext; simp [Finset.sum_multiset_map_count, u.erase_attach_map (... | Mathlib/Analysis/Calculus/FDeriv/Mul.lean | 735 | 744 |
/-
Copyright (c) 2019 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot
-/
import Mathlib.RingTheory.SimpleRing.Basic
import Mathlib.Topology.Algebra.Field
import Mathlib.Topology.Algebra.UniformRing
/-!
# Completion of topological fields
... | rw [mem_compl_singleton_iff] at y_ne
apply CompleteSpace.complete
have : (fun (x : K) => (↑x⁻¹ : hat K)) =
((fun (y : K) => (↑y : hat K))∘(fun (x : K) => (x⁻¹ : K))) := by
unfold Function.comp
simp
rw [this, ← Filter.map_map]
apply Cauchy.map _ (Completion.uniformContinuous_coe K)
apply Comple... | Mathlib/Topology/Algebra/UniformField.lean | 72 | 93 |
/-
Copyright (c) 2023 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash, Deepro Choudhury, Scott Carnahan
-/
import Mathlib.LinearAlgebra.PerfectPairing.Basic
import Mathlib.LinearAlgebra.Reflection
/-!
# Root data and root systems
This file con... | @[deprecated (since := "2025-04-20")]
alias toLin_toPerfectPairing := toLinearMap_eq_toPerfectPairing
| Mathlib/LinearAlgebra/RootSystem/Defs.lean | 123 | 124 |
/-
Copyright (c) 2020 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Halting
import Mathlib.Computability.TuringMachine
import Mathlib.Data.Num.Lemmas
import Mathlib.Tactic.DeriveFintype
import Mathlib.Comp... | rcases e with ⟨e₁, e₂⟩
rw [e₁, e₂]
· refine TransGen.head rfl ?_
rw [tr]; simp only [pop', TM2.step, Option.mem_def, TM2.stepAux, Option.elim]
rcases e₁ : S k with - | ⟨a', Sk⟩ <;> rw [e₁, splitAtPred] at e
· cases e
cases e₂ : p a' <;> simp only [e₂, cond] at e
swap
· cases e
rcas... | Mathlib/Computability/TMToPartrec.lean | 626 | 696 |
/-
Copyright (c) 2023 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll, Yaël Dillies
-/
import Mathlib.Analysis.Calculus.Deriv.Add
import Mathlib.Analysis.Calculus.Deriv.Comp
/-!
### Invariance of the derivative under translation
We show th... |
/-- Translation in the domain does not change the derivative. -/
lemma HasDerivAt.comp_add_const (x a : 𝕜) (hf : HasDerivAt f f' (x + a)) :
HasDerivAt (fun x ↦ f (x + a)) f' x := by
simpa [Function.comp_def] using HasDerivAt.scomp (𝕜 := 𝕜) x hf <| hasDerivAt_id' x |>.add_const a
| Mathlib/Analysis/Calculus/Deriv/Shift.lean | 23 | 28 |
/-
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.ContDiff.Operations
import Mathlib.Analysis.Normed.Module.FiniteDimension
/-!
# Infinitely smooth "bump" func... | ((f x).rOut / (f x).rIn, (f x).rIn⁻¹ • (g x - c x))) s x
refine (((someContDiffBumpBase E).smooth.contDiffAt ?_).of_le
| Mathlib/Analysis/Calculus/BumpFunction/Basic.lean | 179 | 180 |
/-
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, Kexing Ying
-/
import Mathlib.Topology.Semicontinuous
import Mathlib.MeasureTheory.Function.AEMeasurableSequence
import Mathlib.MeasureTheory.Order.Lat... | theorem Measurable.isLUB {ι} [Countable ι] {f : ι → δ → α} {g : δ → α} (hf : ∀ i, Measurable (f i))
(hg : ∀ b, IsLUB { a | ∃ i, f i b = a } (g b)) : Measurable g := by
change ∀ b, IsLUB (range fun i => f i b) (g b) at hg
rw [‹BorelSpace α›.measurable_eq, borel_eq_generateFrom_Ioi α]
| Mathlib/MeasureTheory/Constructions/BorelSpace/Order.lean | 540 | 543 |
/-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap
import Mathlib.MeasureTheory.Covering.Bes... | ∀ A : E →L[ℝ] E, ∃ δ : ℝ≥0, 0 < δ ∧
∀ (t : Set E), ApproximatesLinearOn f A t δ →
μ (f '' t) ≤ (Real.toNNReal |A.det| + 1 : ℝ≥0) * μ t := by
intro A
let m : ℝ≥0 := Real.toNNReal |A.det| + 1
have I : ENNReal.ofReal |A.det| < m := by
simp only [m, ENNReal.ofReal, lt_add_iff_pos_r... | Mathlib/MeasureTheory/Function/Jacobian.lean | 559 | 595 |
/-
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... | open Pointwise
open Cardinal
open scoped Function -- required for scoped `on` notation
| Mathlib/GroupTheory/CoprodI.lean | 806 | 810 |
/-
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 Batteries.Data.List.Perm
import Mathlib.Tactic.Common
/-!
# Counting in lists
This fi... | Mathlib/Data/List/Count.lean | 144 | 146 | |
/-
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.Nat.Totient
import Mathlib.Data.ZMod.Aut
import Mathlib.Data.ZMod.QuotientGroup
import Mathlib.GroupTheory.Exponent
import Mathlib.GroupTheory.Sub... | @[to_additive isAddCyclic_of_card_nsmul_eq_zero_le, stacks 09HX "This theorem is stronger than \
09HX. It removes the abelian condition, and requires only `≤` instead of `=`."]
theorem isCyclic_of_card_pow_eq_one_le : IsCyclic α :=
| Mathlib/GroupTheory/SpecificGroups/Cyclic.lean | 516 | 518 |
/-
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.Group.Indicator
import Mathlib.Algebra.Group.InjSurj
import Mathlib.Data.Set.Finite.Basic
import Mathlib.Tactic.FastInstance
impo... |
variable [Zero M] [Zero N]
| Mathlib/Data/Finsupp/Defs.lean | 361 | 362 |
/-
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... | theorem degree_le_degree (h : coeff q (natDegree p) ≠ 0) : degree p ≤ degree q := by
by_cases hp : p = 0
| Mathlib/Algebra/Polynomial/Degree/Definitions.lean | 128 | 129 |
/-
Copyright (c) 2023 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.Deriv.Comp
import Mathlib.Analysis.Calculus.Deriv.Add
import Mathlib.Analysis.Calculus.Deriv.Mul
import Mathlib.Analysis.Calcul... | hf.le_of_lipschitzOn univ_mem (lipschitzOnWith_univ.2 hlip)
variable (𝕜)
/-- Converse to the mean value inequality: if `f` is `C`-lipschitz
on a neighborhood of `x₀` then its line derivative at `x₀` in the direction `v` has norm
bounded by `C * ‖v‖`. This version only assumes that `‖f x - f x₀‖ ≤ C * ‖x - x₀‖` in ... | Mathlib/Analysis/Calculus/LineDeriv/Basic.lean | 421 | 429 |
/-
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.Ordering.Basic
import Mathlib.Order.Synonym
/-!
# Comparison
This file provides basic results about orderings and comparison in linear orders.
... | | eq, h, _, _ => h
| gt, h, hab, _ => (not_le_of_lt h hab).elim
theorem Compares.inj [Preorder α] {o₁} :
| Mathlib/Order/Compare.lean | 82 | 85 |
/-
Copyright (c) 2022 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.Order.RelClasses
import Mathlib.Order.Interval.Set.Basic
/-!
# Bounded and unbounded sets
We prove miscellaneous lemmas about ... | theorem bounded_gt_Ioc [Preorder α] (a b : α) : Bounded (· > ·) (Ioc a b) :=
(bounded_gt_Ioi a).mono Set.Ioc_subset_Ioi_self
| Mathlib/Order/Bounded.lean | 204 | 205 |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.CharP.Reduced
import Mathlib.RingTheory.IntegralDomain
-- TODO: remove Mathlib.Algebra.CharP.Reduced and move the last two lemmas to Lemmas
/-... | Mathlib/RingTheory/RootsOfUnity/Basic.lean | 896 | 903 | |
/-
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.Data.Set.Finite.Basic
import Mathlib.Order.Atoms
import Mathlib.Order.Grade
import Mathlib.Order.Nat
/-!
# Finsets and multisets form a graded order
This... | /-- Finsets form an order-connected suborder of multisets. -/
lemma ordConnected_range_val : Set.OrdConnected (Set.range val : Set <| Multiset α) :=
⟨by rintro _ _ _ ⟨s, rfl⟩ t ht; exact ⟨⟨t, Multiset.nodup_of_le ht.2 s.2⟩, rfl⟩⟩
| Mathlib/Data/Finset/Grade.lean | 60 | 62 |
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.Galois.Basic
/-!
# Galois Groups of Polynomials
In this file, we introduce the Galois group of a polynomial... |
theorem mul_splits_in_splittingField_of_mul {p₁ q₁ p₂ q₂ : F[X]} (hq₁ : q₁ ≠ 0) (hq₂ : q₂ ≠ 0)
(h₁ : p₁.Splits (algebraMap F q₁.SplittingField))
(h₂ : p₂.Splits (algebraMap F q₂.SplittingField)) :
(p₁ * p₂).Splits (algebraMap F (q₁ * q₂).SplittingField) := by
apply splits_mul
· rw [←
(SplittingFi... | Mathlib/FieldTheory/PolynomialGaloisGroup.lean | 288 | 318 |
/-
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... | · refine ⟨fun hn => False.elim (h.not_le hn), fun hn => ?_⟩
rw [toReal_neg_iff_sign_neg.1 h] at hn
exact False.elim (hn.not_lt (by decide))
· simp [h, sign, ← sin_toReal]
· refine ⟨fun _ => ?_, fun _ => h.le⟩
| Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | 751 | 755 |
/-
Copyright (c) 2022 Yaël Dillies, Sara Rousta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Sara Rousta
-/
import Mathlib.Logic.Equiv.Set
import Mathlib.Order.Interval.Set.OrderEmbedding
import Mathlib.Order.SetNotation
/-!
# Properties of unbundled ... | Mathlib/Order/UpperLower/Basic.lean | 1,310 | 1,312 | |
/-
Copyright (c) 2021 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, María Inés de Frutos-Fernández, Filippo A. E. Nuccio
-/
import Mathlib.Data.Int.Interval
import Mathlib.FieldTheory.RatFunc.AsPolynomial
import Mathlib.RingTheory.Binom... | by_cases h : m = n
· simp [h]
· simp [h, show m + k ≠ n + k by omega]
| Mathlib/RingTheory/LaurentSeries.lean | 143 | 146 |
/-
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... |
@[simp] theorem concat_take_get {n : ℕ} {s : Stream' α} : s.take n ++ [s.get n] = s.take (n + 1) :=
(take_succ' n).symm
theorem getElem?_take {s : Stream' α} : ∀ {k n}, k < n → (s.take n)[k]? = s.get k
| Mathlib/Data/Stream/Init.lean | 529 | 533 |
/-
Copyright (c) 2022 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.Shapes.StrongEpi
import Mathlib.CategoryTheory.LiftingProperties.Adjunction
/-!
# Preservati... | invFun s := ⟨F.preimage s.section_, by
apply F.map_injective
simp only [map_comp, map_preimage, map_id]
apply SplitEpi.id⟩
left_inv := by aesop_cat
right_inv x := by aesop_cat
@[simp]
| Mathlib/CategoryTheory/Functor/EpiMono.lean | 190 | 197 |
/-
Copyright (c) 2022 Stuart Presnell. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Stuart Presnell, Eric Wieser, Yaël Dillies, Patrick Massot, Kim Morrison
-/
import Mathlib.Algebra.GroupWithZero.InjSurj
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Algebra.... | Mathlib/Algebra/Order/Interval/Set/Instances.lean | 382 | 382 | |
/-
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.ModelTheory.Ultraproducts
import Mathlib.ModelTheory.Bundled
import Mathlib.ModelTheory.Skolem
import Mathlib.Order.Filter.AtTopBot.Basic
/-!
# First-... | exact Monotone.comp (g := Set.image ((↑) : s → α)) (f := ((↑) : Finset s → Set s))
Set.monotone_image fun _ _ => Finset.coe_subset.2
/-- Any theory with an infinite model has arbitrarily large models. -/
theorem exists_large_model_of_infinite_model (T : L.Theory) (κ : Cardinal.{w}) (M : Type w')
[L.S... | Mathlib/ModelTheory/Satisfiability.lean | 157 | 170 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Joey van Langen, Casper Putz
-/
import Mathlib.Algebra.CharP.Algebra
import Mathlib.Algebra.CharP.Reduced
import Mathlib.Algebra.Field.ZMod
import Mathlib.Data.Nat.Prime.In... | theorem sum_subgroup_pow_eq_zero [CommRing K] [NoZeroDivisors K]
{G : Subgroup Kˣ} [Fintype G] {k : ℕ} (k_pos : k ≠ 0) (k_lt_card_G : k < Fintype.card G) :
∑ x : G, ((x : Kˣ) : K) ^ k = 0 := by
rw [← Nat.card_eq_fintype_card] at k_lt_card_G
nontriviality K
have := NoZeroDivisors.to_isDomain K
rcases (ex... | Mathlib/FieldTheory/Finite/Basic.lean | 178 | 209 |
/-
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.Algebra.Category.ModuleCat.ChangeOfRings
import Mathlib.Algebra.Category.Ring.Basic
import Mathlib.RingTheory.Kaehler.Basic
/-!
# The differentials of a morphis... | @[ext]
lemma ext {M : ModuleCat B} {α β : KaehlerDifferential f ⟶ M}
(h : ∀ (b : B), α (d b) = β (d b)) : α = β := by
rw [← sub_eq_zero]
have : ⊤ ≤ LinearMap.ker (α - β).hom := by
rw [← KaehlerDifferential.span_range_derivation, Submodule.span_le]
rintro _ ⟨y, rfl⟩
rw [SetLike.mem_coe, LinearMap.mem... | Mathlib/Algebra/Category/ModuleCat/Differentials/Basic.lean | 111 | 121 |
/-
Copyright (c) 2022 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.Analysis.InnerProductSpace.Convex
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Combinatorics.Additive.AP.Three... | section NumericalBounds
theorem log_two_mul_two_le_sqrt_log_eight : log 2 * 2 ≤ √(log 8) := by
have : (8 : ℝ) = 2 ^ ((3 : ℕ) : ℝ) := by rw [rpow_natCast]; norm_num
rw [this, log_rpow zero_lt_two (3 : ℕ)]
apply le_sqrt_of_sq_le
rw [mul_pow, sq (log 2), mul_assoc, mul_comm]
refine mul_le_mul_of_nonneg_right ?_... | Mathlib/Combinatorics/Additive/AP/Three/Behrend.lean | 274 | 291 |
/-
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.NumberTheory.LSeries.RiemannZeta
import Mathlib.NumberTheory.Harmonic.GammaDeriv
/-!
# Asymptotics of `ζ s` as `s → 1`
The goal of this file is to ev... |
lemma term_of_lt {n : ℕ} (hn : 0 < n) {s : ℝ} (hs : 1 < s) :
term n s = 1 / (s - 1) * (1 / n ^ (s - 1) - 1 / (n + 1) ^ (s - 1))
- n / s * (1 / n ^ s - 1 / (n + 1) ^ s) := by
have hv : ∀ x ∈ uIcc (n : ℝ) (n + 1), 0 < x := by
intro x hx
rw [uIcc_of_le (by simp only [le_add_iff_nonneg_right, zero_le_one... | Mathlib/NumberTheory/Harmonic/ZetaAsymp.lean | 144 | 172 |
/-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying, Rémy Degenne
-/
import Mathlib.Probability.Process.Adapted
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
/-!
# Stopping times, stopped processes and stopped va... | @[deprecated (since := "2024-12-25")] alias measurableSpace_le' := measurableSpace_le
example {f : Filtration ℕ m} {τ : Ω → ℕ} (hτ : IsStoppingTime f τ) : hτ.measurableSpace ≤ m :=
hτ.measurableSpace_le
example {f : Filtration ℝ m} {τ : Ω → ℝ} (hτ : IsStoppingTime f τ) : hτ.measurableSpace ≤ m :=
hτ.measurableSpa... | Mathlib/Probability/Process/Stopping.lean | 345 | 354 |
/-
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Algebra.Ring.Int.Defs
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.Cast.Order.Basic
import Math... | | bit1 a, bit0 b => by
dsimp [cmp]
have := cmp_to_nat a b; revert this; cases cmp a b <;> dsimp <;> intro this
| Mathlib/Data/Num/Lemmas.lean | 145 | 147 |
/-
Copyright (c) 2021 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn, Antoine Labelle, Rémi Bottinelli
-/
import Mathlib.Combinatorics.Quiver.Path
import Mathlib.Combinatorics.Quiver.Push
/-!
## Symmetric quivers and arrow reversal
This file co... | Φ = Symmetrify.lift φ := by
subst_vars
fapply Prefunctor.ext
· rintro X
rfl
· rintro X Y f
cases f
| Mathlib/Combinatorics/Quiver/Symmetric.lean | 188 | 194 |
/-
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.Algebra.BigOperators.Finprod
import Mathlib.Topology.ContinuousMap.Algebra
import Mathlib.Topology.Compactness.Paracompact
import Mathlib.Topology.Sh... | rw [← Finset.sum_sdiff hI, ρ.sum_finsupport hx₀]
suffices ∑ i ∈ I \ ρ.finsupport x₀, (ρ i) x₀ = ∑ i ∈ I \ ρ.finsupport x₀, 0 by
| Mathlib/Topology/PartitionOfUnity.lean | 200 | 201 |
/-
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... | Mathlib/Geometry/Euclidean/Angle/Unoriented/RightAngle.lean | 509 | 515 | |
/-
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, Yaël Dillies
-/
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Data.Set.MulAntidiagonal
import Mathlib.Algebra.Group.Pointwise.Set.Basic
/-! # Multiplicat... | Mathlib/Data/Finset/MulAntidiagonal.lean | 114 | 126 | |
/-
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.Ordmap.Invariants
/-!
# Verification of `Ordnode`
This file uses the invariants defined in `Mathlib.Data.Ordmap.Invariants` to construct `Ordset... | · rw [splitMax_eq]
· obtain ⟨v, e⟩ := Valid'.eraseMax_aux hl
| Mathlib/Data/Ordmap/Ordset.lean | 408 | 409 |
/-
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
-/
import Mathlib.Analysis.Complex.Asymptotics
import Mathlib.Analysis.SpecificLimits.Normed
import Mathlib.Data.Complex.Trig... | Mathlib/Analysis/SpecialFunctions/Exp.lean | 502 | 505 | |
/-
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.Algebra.Category.Grp.Limits
import Mathlib.CategoryTheory.CofilteredSystem
import Mathlib.CategoryTheory.Galois.Decomposition
import Mathlib.Catego... | (fun {A B} f ↦ by
ext (φ : Aut A.obj)
dsimp
erw [evaluationEquivOfIsGalois_apply, evaluationEquivOfIsGalois_apply]
simp [-Hom.comp, ← f.comp])
lemma autIsoFibers_inv_app (A : PointedGaloisObject F) (b : F.obj A) :
(autIsoFibers F).inv.app A b = (evaluationEquivOfIsGalois F A A.pt).symm ... | Mathlib/CategoryTheory/Galois/Prorepresentability.lean | 324 | 331 |
/-
Copyright (c) 2021 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.SetFamily.Shadow
/-!
# UV-compressions
This file defines UV-compression. It is an operation on a set family that reduces its ... | theorem compress_idem (u v a : α) : compress u v (compress u v a) = compress u v a := by
unfold compress
split_ifs with h h'
· rw [le_sdiff_right.1 h'.2, sdiff_bot, sdiff_bot, sup_assoc, sup_idem]
| Mathlib/Combinatorics/SetFamily/Compression/UV.lean | 107 | 110 |
/-
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
-/
import Mathlib.Order.Filter.SmallSets
import Mathlib.Topology.UniformSpace.Defs
import Mathlib.Topology.ContinuousOn
/-!
# Basic resu... | le_of_nhds_le_nhds <| to_nhds_mono h
theorem toTopologicalSpace_comap {f : α → β} {u : UniformSpace β} :
| Mathlib/Topology/UniformSpace/Basic.lean | 448 | 450 |
/-
Copyright (c) 2024 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.Order.Hom.Ring
import Mathlib.Data.ENat.Basic
import Mathlib.SetTheory.Cardinal.Basic
/-!
# Conversion between `Cardinal` and `ℕ∞`
In this ... | ofENat m ≤ ofNat(n) ↔ m ≤ OfNat.ofNat n := ofENat_le_nat
| Mathlib/SetTheory/Cardinal/ENat.lean | 93 | 93 |
/-
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.SetTheory.Cardinal.ENat
/-!
# Projection from cardinal numbers to natural numbers
In this file we define `Cardinal.toNat` to be the natural projectio... | Mathlib/SetTheory/Cardinal/ToNat.lean | 164 | 165 | |
/-
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.MeasureTheory.Measure.Content
import Mathlib.MeasureTheory.Group.Prod
import Mathlib.Topology.Algebra.Group.Compact
/-!
# Haar measure
In this fi... | @[to_additive]
theorem prehaar_sup_le {K₀ : PositiveCompacts G} {U : Set G} (K₁ K₂ : Compacts G)
(hU : (interior U).Nonempty) :
prehaar (K₀ : Set G) U (K₁ ⊔ K₂) ≤ prehaar (K₀ : Set G) U K₁ + prehaar (K₀ : Set G) U K₂ := by
simp only [prehaar]; rw [div_add_div_same, div_le_div_iff_of_pos_right]
| Mathlib/MeasureTheory/Measure/Haar/Basic.lean | 288 | 292 |
/-
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, Yaël Dillies
-/
import Mathlib.Data.Nat.Basic
import Mathlib.Data.Int.Order.Basic
import Mathlib.Logic.Function.Iterate
import Mathlib.Order.Compare
impor... | Mathlib/Order/Monotone/Basic.lean | 1,105 | 1,113 | |
/-
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 Mathlib.Data.Set.Operations
import Mathlib.Order.Basic
import Mathlib.Order.BooleanAlgebra
import Mathlib.Tactic.Tauto
import Mathlib.Tactic.B... | Mathlib/Data/Set/Basic.lean | 2,456 | 2,458 | |
/-
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... | simp
· have Z := (h.changeOrigin hy h'y).hasFPowerSeriesWithinAt.hasFDerivWithinAt
apply (Z.mono (subset_insert _ _)).mono_of_mem_nhdsWithin
rw [nhdsWithin_insert_of_ne]
· exact self_mem_nhdsWithin
| Mathlib/Analysis/Calculus/FDeriv/Analytic.lean | 172 | 176 |
/-
Copyright (c) 2018 Michael Jendrusch. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Jendrusch, Kim Morrison, Bhavik Mehta, Jakob von Raumer
-/
import Mathlib.CategoryTheory.EqToHom
import Mathlib.CategoryTheory.Functor.Trifunctor
import Mathlib.CategoryTheo... | Mathlib/CategoryTheory/Monoidal/Category.lean | 1,005 | 1,008 | |
/-
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 | 389 | 392 | |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Kappelmann
-/
import Mathlib.Algebra.Order.Floor.Defs
import Mathlib.Algebra.Order.Floor.Ring
import Mathlib.Algebra.Order.Floor.Semiring
deprecated_module (sinc... | Mathlib/Algebra/Order/Floor.lean | 899 | 901 | |
/-
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.Algebra.Group.PUnit
import Mathlib.Algebra.Group.Subgroup.Ker
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.GroupTheory.Congruence... |
@[to_additive]
theorem comp_lift {P' : Type*} [Monoid P'] (f : P →* P') (g₁ : M →* P) (g₂ : N →* P) :
f.comp (lift g₁ g₂) = lift (f.comp g₁) (f.comp g₂) :=
| Mathlib/GroupTheory/Coprod/Basic.lean | 442 | 445 |
/-
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.Order.Group.Indicator
import Mathlib.MeasureTheory.OuterMeasure.Basic
/-!
# Operations on outer measures
In this file we defi... | map_smul' _ _ := coe_fn_injective rfl
| Mathlib/MeasureTheory/OuterMeasure/Operations.lean | 206 | 207 |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Heather Macbeth
-/
import Mathlib.Analysis.SpecialFunctions.Complex.Circle
import Mathlib.Geometry.Euclidean.Angle.Oriented.Basic
/-!
# Rotations by oriented angles.
This... | have hp := div_pos (norm_pos_iff.2 hy) (norm_pos_iff.2 hx)
constructor
· rintro rfl
rw [← LinearIsometryEquiv.map_smul, ← o.oangle_smul_left_of_pos x y hp, eq_comm,
rotation_oangle_eq_iff_norm_eq, norm_smul, Real.norm_of_nonneg hp.le,
div_mul_cancel₀ _ (norm_ne_zero_iff.2 hx)]
· intro hye
rw... | Mathlib/Geometry/Euclidean/Angle/Oriented/Rotation.lean | 285 | 292 |
/-
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... | | nil => simp
| cons a l IH =>
| Mathlib/Data/List/Basic.lean | 1,155 | 1,156 |
/-
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.Computability.Tape
import Mathlib.Data.Fintype.Option
import Mathlib.Data.Fintype.Prod
import Mathlib.Data.Fintype.Pi
import Mathlib.Data.PFun
import M... | | peek f => TM2.Stmt.peek k f
| pop f => TM2.Stmt.pop k f
/-- The effect of a stack action on the local variables, given the value of the stack. -/
| Mathlib/Computability/TuringMachine.lean | 405 | 408 |
/-
Copyright (c) 2015 Nathaniel Thomas. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nathaniel Thomas, Jeremy Avigad, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.GroupWithZero.Action.Defs
import Mathlib.Algebra.Ring.Defs
/-!
# Modules over a ring
In th... | Mathlib/Algebra/Module/Defs.lean | 579 | 579 | |
/-
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.GradedObject.Associator
import Mathlib.CategoryTheory.GradedObject.Single
/-!
# The left and right unitors
Given a bifunctor `F : C ⥤ D ⥤ D`, an ... |
@[simp, reassoc]
lemma mapBifunctorRightUnitorCofan_inj (j : J) :
(mapBifunctorRightUnitorCofan F Y e p hp X j).inj ⟨⟨j, 0⟩, hp j⟩ =
(F.obj (X j)).map (singleObjApplyIso (0 : I) Y).hom ≫ e.hom.app (X j) := by
| Mathlib/CategoryTheory/GradedObject/Unitor.lean | 175 | 179 |
/-
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.Algebra.GroupWithZero.Pointwise.Set.Basic
import Mathlib.Algebra.Ring.Pointwise.Set
import Mathlib.Topology.MetricSpace.Isometry
import Mathlib.Topol... | Mathlib/Topology/MetricSpace/IsometricSMul.lean | 458 | 459 | |
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jalex Stark
-/
import Mathlib.Algebra.Polynomial.Monic
/-!
# Lemmas for the interaction between polynomials and `∑` and `∏`.
Recall that `∑` and `∏` are notation for ... | simpa using natDegree_multiset_sum_le (s.val.map f)
lemma natDegree_sum_le_of_forall_le {n : ℕ} (f : ι → S[X]) (h : ∀ i ∈ s, natDegree (f i) ≤ n) :
| Mathlib/Algebra/Polynomial/BigOperators.lean | 57 | 59 |
/-
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... | (⋃ i ∈ I, Es i)ᶜ = ⋃ i ∈ Iᶜ, Es i := by
ext x
| Mathlib/Data/Set/Lattice.lean | 1,230 | 1,231 |
/-
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, Mario Carneiro, Johannes Hölzl
-/
import Mathlib.Algebra.Order.Group.Unbundled.Basic
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Or... | Mathlib/Algebra/Order/Group/Defs.lean | 267 | 268 | |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Joël Riou
-/
import Mathlib.CategoryTheory.ConcreteCategory.Basic
import Mathlib.CategoryTheory.Shift.Basic
import Mathlib.Data.Set.Subsingleton
import Mathlib.Algebra.Grou... | /-- The equivalence between β-graded objects and γ-graded objects,
given an equivalence between β and γ.
| Mathlib/CategoryTheory/GradedObject.lean | 181 | 182 |
/-
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.BigOperators.Group.Finset.Defs
/-! # Lemmas about coprimality with big products.
These lemmas ... |
theorem coprime_fintype_prod_right_iff [Fintype ι] {x : ℕ} {s : ι → ℕ} :
Coprime x (∏ i, s i) ↔ ∀ i, Coprime x (s i) := by
| Mathlib/Data/Nat/GCD/BigOperators.lean | 52 | 54 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yaël Dillies
-/
import Mathlib.Logic.Equiv.Set
import Mathlib.Order.CompleteLattice.Lemmas
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection.Basic
/-... |
instance OrderDual.instFrame : Frame αᵒᵈ where
__ := instCompleteLattice
__ := instHeytingAlgebra
| Mathlib/Order/CompleteBooleanAlgebra.lean | 454 | 457 |
/-
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.MeasureTheory.Measure.AEMeasurable
/-!
# Typeclasses for measurability of operations
In this file we define classes `MeasurableMul` etc and prove d... | Mathlib/MeasureTheory/Group/Arithmetic.lean | 954 | 957 |
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